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Electronic structure and spectroscopy of excited and open-shell species
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Electronic structure and spectroscopy of excited and open-shell species
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ELECTRONIC STRUCTURE AND SPECTROSCOPY OF EXCITED AND OPEN-SHELL SPECIES by Lucas Peter Koziol A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CHEMISTRY) August 2009 Copyright 2009 Lucas Peter Koziol Acknowledgements Several people made immeasurable contributions to the work presented in this thesis. The primary thanks goes to my scientific advisor, Prof. Anna Krylov, whose enthusiasm for science is always contagious. During the five years I’ve spent at USC, the Krylov group has always consisted of only the most singularly bright and engaging individuals conceivable. From the get-go, Sergey Levchenko introduced me to Q-Chem calcula- tions and electronic structure. Piotr Pieniazek and I have discussed a lot of science together, sometimes in the middle of a 15 mile run. Vadim Mozhayskiy and Vitalii Vanovschi were great office-mates. Kadir Diri is a reliable expert on countless chem- istry tools and methods, as well as an excellent lunch buddy. Melania Oana, Evgeny Epifanovsky, Anna Golubeva, Arik Landau, Prashant Manohar and Eugene Kamarchik have also become good friends. Arun Sharma, Chris Nemirow, Laura Edwards and I went through all the stages of grad school together. Igor Fedorov was my collabora- tor on the diazomethane/diazirine projects, and I learned a lot about experimental work from him. Zhou Lu and I also became good friends. I would like to thank Prof. Hanna Reisler for her support on our joint theory- experiment projects, and Profs. Curt Wittig and Stephen Bradforth for their excellent courses. At Emory, I’d like to thank Dr. Bastiaan Braams for teaching me about potential energy surface fitting, and Prof. Joel Bowman for making it possible, on two occasions, ii for me to spend some pleasurable weeks in Atlanta. Prof. Stuart Carter at Reading answered all of my questions about Coriolis terms. Outside of the office, life in Los Angeles was greatly enhanced by friendship with several extraordinary persons. From Masters swimming—Krista Zabor, Erik Tolma- choff, Jackie Corcoran, Ginny DeFrank and Michelle Arbeitman are a few of the close friends I swam and happy-houred with. From running—Pankaj Mishra, Iain Bailey and I trained together for the LA Marathon in 2006. Leslie Kharma and I also ran miles together. Gergana Kodjebacheva has been my friend from UCLA, and led me to believe that they’re really not that bad over there. From bicycle touring—Robert Yula and I have biked up several of the steepest mountains in southern California, and Peter Rankenburg and I have biked around Davis and Napa County. Finally, I would like to thank my two brothers, Conrad and Adam, and my parents, for making it all possible. iii Table of Contents Acknowledgements ii List of Tables viii List of Figures xi Abstract xviii Chapter 1 Introduction 1 1.1 Chapter 1 references 6 Chapter 2 Beyond vinyl: electronic structure of unsaturated propen-yl and buten-yl hydrocarbon radicals 7 2.1 Introduction 7 2.2 Computational details 10 2.3 Ground state equilibrium structures 12 2.4 Electronically excited states: Vertical spectra 15 2.5 Valence excited states derived from then← π , π ∗ ← n, and n← σ CC transitions 26 2.6 Equilibrium geometries and adiabatic excitation energies for then← π excited states 31 2.7 Excitedπ ∗ ← π states 33 2.8 Rydbergnl m ← n states 36 2.9 Rydbergnl m ← π states 43 2.10 Conclusions 46 2.11 Chapter 2 references 48 Chapter 3 Diazomethane I: investigations of the electronic Rydberg states: assignments and interactions 53 3.1 Introduction 53 3.2 Computational details 55 iv 3.3 Computational results 56 3.3.1 Equilibrium geometries and ionization energies 56 3.3.2 Excited electronic states 57 3.4 Discussion 62 3.4.1 The 2 1 A 2 (3p y ← π ) and 2 1 B 1 (3p z ← π ) Rydberg states and their interaction 62 3.4.2 The 3 1 A 1 (3p x ← π ) state 71 3.5 Conclusions 73 3.6 Chapter 3 references 75 Chapter 4 Diazomethane II: theoretical and experimental investigations: vibronic structure and ion core interactions in the electronic Rydberg states 78 4.1 Introduction 78 4.2 Theoretical methods and results 80 4.3 Discussion 86 4.3.1 Vibrational assignments for the 2 1 A 2 (3p y ) Rydberg state 86 4.3.2 Vibrational assignments for the 1 2 B 1 ground-state cation 89 4.3.3 Structural and vibrational motions in neutral and ionic diazomethane 90 4.4 Summary 94 4.5 Chapter 4 references 96 Chapter 5 Diazirine: multiphoton ionization and dissociation 98 5.1 Introduction 98 5.2 Computational studies of the electronically excited and ionized states of diazirine 100 5.3 Discussion 106 5.3.1 Excited states and photoionization of diazirine 106 5.3.2 Detection of ionization and dissociation products 107 5.3.3 Pathways leading to CH(X) fragments 108 5.4 Summary 116 5.5 Chapter 5 references 118 Chapter 6 The 1,2,3-tridehydrobenzene triradical: 2 B or not 2 B? The answer is 2 A! 123 6.1 Introduction 123 6.2 Theoretical methods and computational details 127 v 6.3 Results and discussion 131 6.3.1 Electronic structure and equilibrium geometries of the 2 A 1 and 2 B 2 states 131 6.3.2 Vertical and adiabatic energy differences between the 2 A 1 and 2 B 2 states 135 6.3.3 Vibrational spectrum 139 6.4 Conclusions: electronic structure and infrared spectra 149 6.5 Abstract: variational calculations on surfaces 149 6.6 Background 150 6.7 Computational details 152 6.8 Anharmonicities 153 6.9 Chapter 6 references 160 Chapter 7 Hydroxycarbene diradicals: electronic structure and infrared spectra 164 7.1 Introduction 164 7.2 Vibrational self-consistent field theory 166 7.3 Computational details 169 7.4 Results and discussion 173 7.4.1 The IR spectrum of HCOH 173 7.4.2 Deuterated hydroxycarbene: HCOD 179 7.5 Conclusions 183 7.6 Chapter 7 references 185 Chapter 8 Hydroxycarbene diradicals: ionization and photoelectron spectra 188 8.1 Introduction 188 8.2 Theory and computational details 191 8.3 Molecular orbital framework and structural effects of ionization 196 8.4 Photoelectron spectra of HCOH 198 8.5 Photoelectron spectra of HCOD 205 8.6 Comparison with the parallel-mode harmonic approximation 205 8.7 Conclusions 209 8.8 Chapter 8 references 211 Chapter 9 Future work 214 9.1 Background 214 9.2 Methodology 217 9.3 Chapter 9 references 219 Bibliography 220 vi Appendices Appendix A The ezVibe program: input and examples 234 A.1 Introduction 234 A.2 Input parameters 235 A.3 User-supplied potential energy surface 239 Appendix B The ezVibe program: code structure 241 vii List of Tables 2.1 Vertical excitation energies, oscillator strengths, and properties of the excited states of the vinyl radical 20 2.2 Vertical excitation energies, oscillator strengths, and properties of the excited states of the propen-1-yl radical 21 2.3 Vertical excitation energies, oscillator strengths, and properties of the excited states of the propen-2-yl radical 22 2.4 Vertical excitation energies, oscillator strengths, and properties of the excited states of the buten-2-yl radical 24 2.5 Vertical excitation energies, oscillator strengths, and properties of the excited states of the the 1-buten-2-yl radical 25 3.1 Vertical excitation energies (Δ E vert , eV), oscillator strengths (f L ), dipole strengths (μ 2 tr , a.u.), and changes in second dipole moment of charge dis- tributions relative to the ground stateΔ hX 2 , (a.u.) 2 for the excited states of CH 2 N 2 at EOM-CCSD/6-311(3+,+)G* 58 3.2 Calculated vertical and adiabatic excitation energies (eV) and quantum defects (δ and corresponding experimental values. 60 3.3 CCSD(T)/cc-pVTZ harmonic vibrational frequencies for cation ground state (1 2 B 1 ), (cm − 1 ) 68 4.1 Calculated equilibrium structures for the ground state of the neutral and cation and for the Rydberg states of CH 2 N 2 82 4.2 Transition energies and vibrational frequencies of neutral ground state, 3p Rydberg states, and cation of CH 2 N 2 83 4.3 Transition energies and vibrational frequencies of neutral ground state, 3p Rydberg states, and cation of CD 2 N 2 84 viii 4.4 Transition energies and vibrational frequencies of neutral ground state, 3p Rydberg states, and cation of CHDN 2 85 5.1 Calculated equilibrium structures for the ground, 1 1 B 2 , and 1 1 A 2 valence states of the neutral and for the ground state of the cation 103 5.2 Vertical excitation energies (eV), oscillator strengths, dipole strengths (a.u.), and changes in second dipole moment of charge distributions (a.u. 2 ) for the excited states of c-CH 2 N 2 at EOM-CCSD/6-311(3+,+)G* a 104 5.3 Calculated values ofΔ H 0 f of diazomethane and diazirine 109 6.1 Selected structural parameters for the 2 A 1 and 2 B 2 equilibrium struc- tures of 1. R C1− C2 is the bond length between adjacent radical carbons. A C1− C2− C3 is the angle formed by the three radical carbons. 133 6.2 Vertical and adiabatic energy differences between the 2 A 1 and 2 B 2 states of 1,2,3-tridehydrobenzene, as defined in Figure 6.4, using at EOM- CCSD/6-31G(d) equilibrium geometries unless specified otherwise. 136 6.3 Adiabatic energy differenceΔ E 3 , eV , at R-CCSD(T)/cc-pVTZ and B3LYP/cc- pVTZ equilibrium geometries. All multi-reference calculations employ a CASSCF(9,9) reference. 137 6.4 CCSD(T)/cc-pVTZ frequencies and IR intensities. 143 6.5 SF-DFT/6-311G* using 50/50 functional frequencies and IR intensities. 144 6.6 BLYP/cc-pVTZ harmonic and fundamental vibrational frequencies for the two doublet states of 1. 145 6.7 B3LYP/cc-pVTZ frequencies, IR intensities, and anharmonic correc- tions calculated to second order of perturbation theory. Anharmonic force constants up to quartic order were obtained by finite differences of Cartesian Hessians, for terms with up to three distinct indices. 58 146 6.8 2 A 1 B3LYP/6-311(+)G** harmonic frequencies and anharmonicities 155 6.9 2 B 2 B3LYP/6-311(+)G** harmonic frequencies and anharmonicities 156 6.10 2 A 1 PES harmonic frequencies and anharmonicities 157 6.11 2 B 2 PES harmonic frequencies and anharmonicities 158 7.1 Comparison of harmonic frequencies (cm − 1 ) and IR intensities (km/mol, in parentheses) for trans-hydroxycarbene. 170 ix 7.2 Comparison of harmonic frequencies (cm − 1 ) and IR intensities (km/mol, in parentheses) for cis-hydroxycarbene. 171 7.3 HCOH VCI energy levels (cm − 1 ) and IR intensities (km/mol). 174 7.4 VSCF, VCI and VMP2 results for trans-HCOH. 178 7.5 Normal coordinates, harmonic frequencies (cm − 1 ) and IR intensities (km/mol) of HCOD at the CCSD(T)/cc-pVTZ level with all electrons being correlated. 179 7.6 HCOD VCI energy levels (cm − 1 ) and IR intensities (km/mol). 182 8.1 Comparison of harmonic frequencies (cm − 1 ) and IR intensities (km/mol, in parentheses) for cis-HCOH + . 193 8.2 Comparison of harmonic frequencies (cm − 1 ) and IR intensities (km/mol, in parentheses) for trans-HCOH + . 194 8.3 Comparison of harmonic frequencies (cm − 1 ) between neutral and cation PESs. 194 8.4 HCOH + VCI vibrational levels below 3,600, and corresponding levels for HCOD + (cm − 1 ). 199 8.5 Active vibrational levels of cis-HCOH + / HCOD + in the photoelectron spectrum of cis-HCOH / HCOD. 202 8.6 Active vibrational levels of trans-HCOH + / HCOD + in the photoelec- tron spectrum of trans-HCOH / HCOD. 204 x List of Figures 1.1 Air pollution over downtown Los Angeles 2 2.1 Hydrocarbon radicals on your daily commute. 8 2.2 The CCSD(T)/6-311(2+,2+)G(d,p) ground state optimized geometries of the vinyl, propenyl, and butenyl radicals, and the molecular orienta- tion in Cartesian coordinate system. For 1-buten-2-yl, the OXY plane is the C 1 C 2 C 3 plane. 10 2.3 The σ CC , π , n, π ∗ , and 3s ROHF orbitals of vinyl. These orbitals are very similar in all the radicals. 11 2.4 Calculated vertical electronic excitation energies and oscillator strengths of the vinyl radical. The reported experimental excitation energies (see Introduction for references) are shown by hollow bars. The intensity of the experimental transitions is arbitrary. 16 2.5 Calculated vertical electronic excitation energies and oscillator strengths of propen-1-yl. 17 2.6 Calculated vertical electronic excitation energies and oscillator strengths of propen-2-yl. 17 2.7 Calculated vertical electronic excitation energies and oscillator strengths of 1-buten-2-yl. 18 2.8 Calculated vertical electronic excitation energies and oscillator strengths of trans-2-buten-2-yl. 18 2.9 Changes in the n ← π , π ∗ ← n, and n ← σ CC vertical excitation energies. 27 xi 2.10 Changes in the oscillator strengths for the n← π , π ∗ ← n, and n← σ CC transitions. The ground and vertical excited state permanent dipole moments are shown by arrows. The plane of the figure is considered as theOXY plane, and theOZ axis is perpendicular to this plane (see Fig. 2.2). For 1-buten-2-yl, the Z-component of the permanent dipole moment is shown in parentheses. 27 2.11 Changes in theπ ∗ ← π vertical excitation energies. 29 2.12 Changes in the oscillator strengths for the π ∗ ← π transitions. The ground and vertical excited state permanent dipole moments are shown by arrows. The plane of the figure is considered as the OXY plane, and the OZ axis is perpendicular to this plane (see Fig. 2.2). For 1- buten-2-yl, theZ-component of the permanent dipole moment is shown in parentheses. 29 2.13 Changes in thenl m ← n vertical excitation energies. 37 2.14 Changes in the oscillator strengths for the nl m ← n transitions. The ground and vertical excited state permanent dipole moments are shown by arrows. The plane of the figure is considered as the OXY plane, and the OZ axis is perpendicular to this plane (see Fig. 2.2). For 1- buten-2-yl, theZ-component of the permanent dipole moment is shown in parentheses. 38 2.15 Quantum defects δ for the nl m ← n Rydberg states, see Eq. (2.2). Arrows connect species that are different by a single substitution of a hydrogen by a methyl or ethyl group. Next to the arrows, the differ- ences in IPs, as well as in the excitation energies for the 3s, 3p x , 3p y , and3p z states (from top to bottom) for the connected species are shown. 39 2.16 Changes in thenl m ← π vertical excitation energies. 44 2.17 The EOM-SF-CCSD lowest excited state (n← π ) optimized geometries of the vinyl, propenyl, and butenyl radicals, and the molecular orienta- tion in Cartesian coordinate system. For 1-buten-2-yl, the OXY plane is the C 1 C 2 C 3 plane. 45 3.1 Molecular orbitals relevant to ground and excited electronic states of CH 2 N 2 . 53 xii 3.2 Left panel: ground state equilibrium structures ( ˚ A and deg) of CH 2 N 2 for: the neutral (1 1 A 1 ) at CCSD(T)/cc-pVTZ (regular print) and B3LYP/ 6-311G(2df,p) (italics) and for the cation 1 2 B 1 (∞← π ) at CCSD/6- 311G** (underlined). The corresponding nuclear repulsion energies are: 61.280112, 61.514227, and 61.118198 hartrees, respectively. Right panel: excited state equilibrium structures for the 2 1 A 2 ← 1 1 A 1 (3p y ← π ), 2 1 B 1 ← 1 1 A 1 (3p z ← π ), and 3 1 A 1 ← 1 1 A 1 (3p x ← π ) states showed in normal , italics, and underlined print, respectively. CNN and HCNN refer to respective angle and dihedral angle for the C s 3 1 A 1 ← 1 1 A 1 (3p x ← π ) state, which is the only one with other than C 2v symmetry. The corresponding nuclear repulsion energies are: 60.705257, 59.502297, and 60.715012 hartrees. Experimental parameters of ground state: CN length: 1.300 ˚ A; NN length: 1.139 ˚ A; CH length: 1.075 ˚ A; HCH angle: 126.0 ◦ . 56 3.3 Bars showing calculated vertical excitation energies of CH 2 N 2 . Allowed and forbidded transitions are indicated by filled and hollow bars, respec- tively. 58 3.4 Survey 2 + 1 REMPI spectrum in the 6.32− 7.30 eV (51,000-58,900 cm − 1 ) region. The empty bars indicate the calculated values of vertical excitation energies. 59 3.5 2+1 REMPI spectrum of diazomethane in the region of excitations to the 2 1 A 2 (3p y ← π ) and 2 1 B 1 (3p z ← π ) states obtained by measuring m/e = 42 as a function of excitation energy. The laser wavelength increment was 0.005 nm. See the text for details of the C-F bands. 63 3.6 Photoelectron image and the corresponding eKE distributions obtained at excitation wavelengthλ = 382.69 nm (2hν = 52,262 cm − 1 ; middle peak of band C). 64 3.7 Photoelectron image and the corresponding eKE distributions obtained at excitation wavelength 380.75 nm (52,528 cm − 1 ; middle peak of band D). 66 3.8 Photoelectron image and the corresponding eKE distributions obtained at excitation wavelength 379.50 nm (52,700 cm − 1 ; middle peak of band F). 67 3.9 Photoelectron image and the corresponding eKE distributions obtained at excitation wavelength 380.39 nm (52,577 cm − 1 ; band E). 69 xiii 3.10 Photoelectron image and the corresponding eKE distributions obtained at excitation wavelength 351.09 nm (56,965 cm − 1 ). 72 4.1 Two Lewis structures for diazomethane. The z-axis lies along CNN, the y-axis lies in the plane perpendicular to CNN, and the x-axis is out-of- plane. 80 4.2 2+1 REMPI spectra for (a) CH 2 N 2 , (b) CD 2 N 2 , and (c) CHDN 2 fol- lowing two-photon laser excitation at51,750− 54,900 cm − 1 . An inset in (a) for CH 2 N 2 shows a 54,900− 56,700 cm − 1 spectrum magnified 10 times, whereas an inset in (b) displays the 54,500− 55,000 cm − 1 range in x10 magnification. 87 4.3 Harmonic frequencies of the neutral and cation ground state of CH 2 N 2 compared to those of the 2 1 A 2 (3p y ) and 2 1 B 1 (3p z ) Rydberg excited states. 92 5.1 Molecular orbitals relevant to ground and excited electronic states of c- CH 2 N 2 . The three-membered ring lies in the yz plane, with the z-axis coinciding with the C 2 symmetry axis. 101 5.2 Left panel: ground-state equilibrium structures ( ˚ A and deg) of diazirine for the neutral (1 1 A 1 ) at CCSD(T)/cc-pVTZ (normal print) and B3LYP/6- 311G(2df,p) (italics) and for the cation, 1 2 B 1 (∞← n), at B3LYP/6- 311G(2df,p) (underlined). The corresponding nuclear repulsion ener- gies are 64.158275, 64.295975, and 62.797366 hartree, respectively. Right panel: excited state equilibrium structures for the 1 1 B 2 (π ∗ ← n) and 1 1 A 2 (π ∗ ← σ NN ) excited states at EOM-CCSD/6-311G** shown in normal print and italics, respectively. The corresponding nuclear repul- sion energies are 62.313876 and 61.730487 hartree. 102 5.3 Top panel: the image obtained in dissociation at 312.10 nm by moni- toring CH (X,v 0 = 0,N 00 = 9). Bottom panel: the c.m. translational energy distribution, P(E T ), of the CH(X) fragments (right axis) and the recoil anisotropy parametersβ i (E T ) (left axis). 112 5.4 Top: initial isomerization of diazirine to isodiazirine. Bottom: the cyclic intermediate oxirane (c-HCOCH) in the hydrogen scrambling of ketene. 113 6.1 The three isomers of tridehydrobenzene. 123 xiv 6.2 Matrix isolation of 1,2,3-tridehydrobenzene. Compound 4 was pho- tolyzed with 308 nm light using an XeCl excimer laser. Subsequent irradiation with 248 nm KrF or 193 nm ArF excimer lasers produced 4, as well as other side-products. 124 6.3 Frontier molecular orbitals and the leading electronic configurations of the 4 B 2 , 2 A 1 , and 2 B 2 states of 1,2,3-tridehydrobenzene. Orbital ener- gies (hartrees) for the UHF quartet reference are given at the optimized geometries of the two doublet states. 125 6.4 Vertical and adiabatic energy differences between the 2 A 1 and 2 B 2 states. 126 6.5 Equilibrium structure of the 2 A 1 (left) and 2 B 2 (right) states of 1,2,3- tridehydrobenzene. Bond lengths are in ˚ A and angles are in degrees. Geometrical parameters are listed as follows: R-CCSD(T)/cc-pVTZ; ROHF-CCSD(T)/cc-pVTZ; SF-CCSD/6-31G(d); SF-DFT/6-311G(d); BLYP/cc-pVTZ; B3LYP/cc-pVTZ 132 6.6 Energy as a function of the distance between the radical centers C1 - C3. 134 6.7 Experimental IR spectrum of 1 compared to B3LYP/cc-pVTZ calcu- lated spectra for 2 A 1 and 2 B 2 states, including anharmonic corrections. 140 6.8 Calculated frequencies for the 2 A 1 state (solid lines) compared to the three experimental absorptions (dashed lines). The intensities of the lines are scaled such that the intensities of the most intense experimental and calculated peaks are equal. Top: CCSD(T)/cc-pVTZ; middle: SF- DFT/6-311G(d) with 50/50 functional; bottom: BLYP/cc-pVTZ. 141 6.9 Calculated frequencies for the 2 B 2 state (solid lines) compared to the three experimental absorptions (dashed lines). The intensities of the lines are scaled such that the intensities of the most intense experimental and calculated peaks are equal. Top: CCSD(T)/cc-pVTZ; middle: SF- DFT/6-311G(d) with 50/50 functional; bottom: BLYP/cc-pVTZ. 142 7.1 Stationary points on the HCOH PES. The subspace connecting formalde- hyde with trans-hydroxycarbene (at 12,085 cm − 1 ) is not defined on our surface fits: at this point, CCSD(T)/cc-pVTZ energy at B3LYP/cc- pVTZ optimized transition state is shown for comparison. 165 xv 7.2 CCSD(T)/cc-pVTZ (regular print), CCSD(T)/aug-cc-pVTZ (italics), CCSD(T)/cc-pVQZ (underlined), and PES (bold) equilibrium structures of trans-(left) and cis-(right) hydroxycarbene. E nuc =30.63185899 a.u. and E nuc = 30.55794113 a.u. for trans- and cis- isomers, respectively, at the CCSD(T)/cc-pVTZ (frozen core) equilibrium structure. 172 7.3 VCI (top) and harmonic (bottom) IR spectrum for trans-HCOH. 176 7.4 VCI (top) and harmonic (bottom) IR spectrum for cis-HCOH. 176 7.5 VCI infrared spectra (black) compared to experimental data of Schreiner et al. (red), 5 for trans-HCOH (top) and trans-HCOD (bottom). 177 7.6 VCI (top) and harmonic (bottom) IR spectrum for trans-HCOD. 180 7.7 VCI (top) and harmonic (bottom) IR spectrum for cis-HCOD. 181 8.1 Stationary points on the HCOH (lower, CCSD(T)/cc-pVTZ) and HCOH + (upper) PES. Vertical arrows represent ionization to the Franck-Condon regions and vertical (regular print) and adiabatic (underline) IEs are given. Energies of stationary points are listed on each surface relative to their global minimum (trans- structure). The formaldehyde isomer was not included in our PES, and associated barrier (marked with *) was calculated with CCSD(T)/cc-pVTZ at B3LYP/cc-pVTZ optimized transition state. 189 8.2 Equilibrium structures on cation PES. CCSD(T)/cc-pVTZ (regular print), CCSD(T)/aug-cc-pVTZ (underline), CCSD(T)/cc-pVQZ (italic), and PES (bold) for cis- (left) and trans-HCOH + (right). E nuc = 31.858717 a.u. and 31.825806 a.u. at the CCSD(T)/cc-pVTZ (frozen core) geometries. 192 8.3 Equilibrium structures calculated on the PES for HCOH (regular print) and HCOH + (underline) for cis- (left) and trans- (right) isomers. 193 8.4 Highest occupied molecular orbital of cis- (left) and trans-HCOH (right). 196 8.5 Franck-Condon factors for HCOH ionization producing electronic ground state of HCOH + in the range from the ZPE (0 cm − 1 ) to 7,000 cm − 1 . Left: cis- isomer; right: trans- isomer. 198 8.6 Franck-Condon factors for HCOD ionization producing electronic ground state of HCOD + in the range from the ZPE (0 cm − 1 ) to 7,000 cm − 1 . Left: cis- isomer; right: trans- isomer. 205 xvi 8.7 Comparison between VCI (black lines) and parallel-mode harmonic oscil- lator approximation (red lines) using normal coordinates of the neutral (left) and cation (right) for the Franck-Condon factors of cis-HCOH. Harmonic intensities are not scaled to match VCI. 206 8.8 Comparison between VCI (black lines) and parallel-mode harmonic oscil- lator approximation (red lines) using normal coordinates of the neutral (left) and cation (right) for the Franck-Condon factors of trans-HCOH. Harmonic intensities are not scaled to match VCI. 207 8.9 The effect of rotations of normal coordinates on Franck-Condon fac- tors within the parallel-mode approximation. (a) The correct overlap between wave-functions on lower (q”) and upper (q’) surfaces. (b) The overlap when lower normal coordinates are rotated to coincide with upper coordinates. (c) The overlap when upper normal coordinates are rotated to coincide with lower coordinates. 208 B.1 Inheritance diagram for major classes in ezVibe. 240 xvii Abstract Radical systems, or systems with unpaired electrons, play fundamental roles in chem- istry, biology, and physics. Relevant examples include photochemical breakdown of air pollution in the lower atmosphere, carbon-carbon bond formation in synthetic and pharmaceutical chemistry, and formation of fundamental organic molecules in the inter- stellar medium. The work presented in this thesis focuses on the electronic structure and spectroscopy of open-shell species; the organization is as follows. Chapter 1 pro- vides a short background and overview. Chapter 2 presents a study of the excited states of several vinylic hydrocarbon radicals, with a focus on 3p Rydberg states and the effects of increased alkyl length on excitation energies and quantum defects. Chapters 3 and 4 extend some of these conclusions to the diazomethane molecule, work which was done in close collaboration with the Reisler group at USC. Chapter 5 focuses on diazirine, a three-membered cyclic isomer of diazomethane. The agreement between theory and experiment afforded a detailed insight into the photochemical dissociation of diazirine via a two-photon process. Chapter 6 explores the electronic structure of the 1,2,3-tridehydrobenzene triradical. Despite a nominally antibonding character in one of the electronic states, signifcant distortion of nuclear framework was observed that maximized overlap between meta carbons. Experimental isolation of this system xviii by our coworkers Winkler et al. allowed comparison to harmonic frequencies calcu- lated for two nearly-degenerate electronic states, leading to conclusive assignment of the ground state. Chapters 7 and 8 present our results on the hydroxycarbene dirad- icals. Full-dimensional potential energy surfaces were obtained. Anharmonic vibra- tional energies and wavefunctions were calculated on the surfaces, as well as intensi- ties for two types of spectroscopies. With the full-dimensional dipole moment surface, infrared spectra were obtained. Franck-Condon factors (photoelectron spectra) were calculated between neutral and cation surfaces; the photoelectron spectrum shows qual- itative differences between the two isomers, which will facilitate a future experimental identification. Lastly, Chapter 9 presents future work ideas, which concern the vari- ational approach and block diagonalization techniques for the molecular Hamiltonian. The two Appendices provide user information and code structure for ezVibe, a program developed in our group for solving the vibrational problem on potential energy surfaces. xix Chapter 1 Introduction Radicals are ubiquitous in chemistry. They occur whenever chemical bonds are bro- ken or stretched, and upon electronic excitation and ionization. Atmospheric and com- bustion chemistries are two examples which involve complex cascades of free radical reactions. In the lower troposphere (the first few miles above Earth’s surface), high-energy photons and low pressures afford relatively long lifetimes to radical species. In urban areas, hydrocarbons formed by incomplete combustion are released into the atmosphere through car exhaust. These either exit the tailpipe as radicals, or undergo a) oxidation by hydroxyl radical, or b) direct CH-bond cleavage by UV light. From there, the radi- cals enter a series of pathways that disrupts the balance of the two major natural cycles: the oxygen cycle and the nitrogen cycle. The radicals immediately react with oxygen to form peroxides. From there, several things can happen: a) they react with them- selves to form complex volatile organic compounds (VOCs), b) if sufficient oxidation occurs, they become water soluble and are removed from the atmosphere by wet depo- sition, or c) they are sequentially broken down into smaller building blocks, ending up as formaldehyde or CO 2 . In each of these steps, repeat oxidation by oxygen and reac- tion with NO X species leads to increased ozone concentration in the troposphere. In the upper atmosphere (stratosphere), ozone blocks UV radiation. In the lower troposphere, however, it is highly corrosive to human tissue. As an “international traveler,” ozone will continue having major human health and political implications in the ensuing decades. 1 These reactions themselves do not proceed only on the ground state. Photoionization studies of brominated hydrocarbons by the Butler group 1 found that a primary dissoci- ation pathway produced radicals in a low-lying excited state. Chapter 2 describes our work characterizing the electronic excitation spectrum for several prototypical vinylic systems. Figure 1.1: Air pollution over downtown Los Angeles The study of radical and open-shell systems presents unique challenges for both the- ory and experiment. Experimental studies often yield a spectrum; the positions of the lines give energy levels, while intensities correspond to some property of the associated wavefunctions. For example, photoelectron spectrum intensities can be well approxi- mated by Franck-Condon factors, overlap integrals between vibrational wavefunctions. Experimental data is often incomplete, in that it both introduces error bars (resolution, temperature effects, environment perturbations such as matrix shifts) and often provides a partial spectrum or select peaks. Even when it is complete, the spectrum can be com- plicated enough that assignment and interpretation is difficult without theoretical data. Spectroscopy is incredibly useful because it provides a unique fingerprint of a molec- ular system, and casts insight into both electronic structure and the nuclear motion. In photoelectron spectroscopy, the position of the band heads yield ionization energies, 2 information about high-lying orbitals and their energies (per Koopmans theorem), and an electronic spectrum of the ionized system. Vibrational progressions on the bound ionized states provide frequencies and anharmonicity constants, as well as information about structures. Theoretical modeling of vibrational spectra utilizes the full molecular Hamilto- nian, and is necessarily performed under several simplifying approximations. Born- Oppenheimer separation, approximate electronic structure, finite basis sets, and the har- monic approximation are some of the usual truncations required by the intractable scal- ing with system size. In several cases, this effectively precludes quantitative comparison of theory and experiment. This thesis addresses theoretical work on several challenging open shell systems, their excited states, and their vibrational spectra. Following is an expanded overview of each of the chapters, with an effort to comment on the bigger picture for each one. Chapter 2 presents a study on the excited states of a series of vinylic hydrocarbon radicals. Trends in the 3p Rydberg states, as well as quantum defects as a function of anisotropy and alkyl chain size, are discussed. Chapters 3 and 4 extend some of these ideas to diazomethane, a molecule important for the synthesis of carbenes and in interstellar chemistry. The excited states of diazomethane are characterized using state-of-the-art electronic structure methods, and harmonic frequencies on the Rydberg states are explained in terms of core-valence interactions. The topic of Chapter 5 is a cyclic isomer of diazomethane, the 3-membered ring diazirine. A detailed picture of its photodissociation dynamics is obtained through a collaboration with Reisler and coworkers at USC. Strong CH + signals were observed in the experiments, and from the velocity map images, it was deduced that CH fragments were produced with large translational energy arising from a two-photon process. Calculations showed that in the 3 relevant energy range, two-photon absorption via the 1 1 A 2 ← 1 1 B 2 ← 1 1 A 1 states pro- ceeds very efficiently. The most likely route to CH formation involves isomerization on the 1 1 A 2 state to isodiazirine, followed by dissociation to CH + N 2 . Although gener- ally inefficient, such atom shifts are unusually facile in other members of the so-called 16-electron molecules, particularly in isotopic scrambling and dissociation of ketene via oxirane intermediate. Chapter 6 explores the complicated electronic structure of the 1,2,3- tridehydrobenzene triradical, a fundamental step in the systematic decomposition of benzene. Three radical sp 2 atomic orbitals lead to three singly-occupied molecular orbitals. Despite a nominally antibonding character in one of the states, the carbon framework undergoes significant distortion to maximize overlap between meta carbons. This 2 A 1 state is the electronic ground state; however, it is nearly degenerate, adia- batically, with the 2 B 2 state and triples corrections are required to obtain the correct ordering. This was further supported by close agreement between the CCSD(T) and EOM-SF-CCSD(2,3) methods. Harmonic frequencies and anharmonic corrections were calculated and compared to experimental infrared lines obtained in Ne matrix by Michael Winkler, Wolfram Sander, and coworkers. The comparison supported assignment of the 2 A 1 ground state for this system. This research project highlighted the loss of intuition that can occur whenever the electronic structure is complicated by near-degenerate molecular orbitals that are partially occupied. Chapters 7 and 8 present our studies on the hydroxycarbene, HCOH, diradicals. In addition to electronic structure, this work also focused on accurate vibrational spec- troscopy, specifically toward the larger goal of approaching quantitative agreement between theory and experiment. To this end, full-dimensional potential energy surfaces were obtained by fitting CCSD(T) ab initio data. Specially constructed symmetrized 4 polynomial basis sets, developed by Braams and coworkers, was used to ensure permu- tational symmetry of the surfaces. Methods for solving the vibrational problem beyond the harmonic approximation—including vibrational self-consistent field and vibrational configuration interaction—were used to obtain numerically converged vibrational ener- gies and wavefunctions. These methods were implemented in a code called ezVibe, which is documented in the Appendices at the end of this work. Dipole moment sur- faces and full-dimensional Franck-Condon integrals, including Duschinsky rotations, were used to calculate the infrared and photoelectron spectra, respectively. For these systems it was shown that anharmonicities are necessary not only for capturing the intri- cate details of a spectrum, but also for a qualitative description above the lowest-energy region (0− 2000 cm − 1 ). Anharmonicities mix several vibrational states and influence both IR intensities and Franck-Condon factors. For instance, overtones and combination bands obtain strong intensity in the IR spectrum of cis-HCOH and complicate the spec- trum in the region of the fundamental stretching modes. Recently, HCOH has been iso- lated experimentally, and observed infrared bands matched our calculations extremely well. In the photoelectron spectra of HCOH, large geometrical changes upon ionization mix the normal modes, introducing large deficiencies in the parallel-mode approxima- tion as well as anharmonicity. The VCI spectra show long progressions in modes that connect the minima on the two states, in agreement with the Condon reflection principle. 5 1.1 Chapter 1 references [1] J.L. Miller, M.J. Krisch, and L.J. Butler. J. Phys. Chem. A, 109:4038–4048, 2005. 6 Chapter 2 Beyond vinyl: electronic structure of unsaturated propen-yl and buten-yl hydrocarbon radicals 2.1 Introduction Unsaturated hydrocarbon radicals have attracted attention as reactive intermediates in hydrocarbon combustion since the late 1960s 1–3 . Similar species containing the vinyl moiety have become a versatile tool in the radical synthetic chemistry 4 . The smallest unsaturated hydrocarbon radical, vinyl, has been studied extensively both experimen- tally and theoretically. The ground state structure was derived by Kanamori et al. from the infrared diode laser spectra 5 . The first absorption band, with the origin at about 2.49 eV and Franck-Condon maximum near 3.08 eV , was measured in the region 360- 530 nm by visible absorption spectroscopy 6 , cavity ring-down laser absorption spec- troscopy 7, 8 , and action spectroscopy of the jet-cooled radicals 9 , and was assigned to the ˜ A 2 A 00 ← ˜ X 2 A 0 (n← π ) transition. Two absorption features at 164.71 nm (7.53 eV) and 168.33 nm (7.37 eV) were detected from vacuum ultraviolet flash photolysis, and were assigned to a Rydberg transition 10 . A broad and featureless absorption was observed in the region 225-238 nm (5.21-5.51 eV) using room-temperature gas-phase ultraviolet spectroscopy, with a maximum cross-section at 225 nm (5.51 eV) 11 . The absorption was 7 attributed mainly to the ˜ C 2 A 0 ← ˜ X 2 A 0 (π ← π ∗ ) excitation, with a small contribution from the ˜ B 2 A 0 ← ˜ X 2 A 0 (π ∗ ← n) excitation, assuming a larger intensity for the former transition 11 . VOLUME 110 MARCH 23, 2006 NUMBER 11 http://pubs.acs.org/JPCA THE JOURNAL OF PHYSICAL CHEMISTRY JPCAFH Unsaturated Hydrocarbon Radicals That Appear as Reaction Intermediates in Combustion of Hydrocarbon Fuels May Be Produced in Electronically Excited States (see page xxxx) A PUBLISHED WEEKLY BY THE AMERICAN CHEMICAL SOCIETY MOLECULES, SPECTROSCOPY, KINETICS, ENVIRONMENT, & GENERAL THEORY Figure 2.1: Hydrocarbon radicals on your daily commute. No direct spectroscopic measurements of propenyl or butenyl radicals have been reported so far. Recently, Butler and coworkers found evidence that the C-Br fission, the primary channel of 2-bromo-1-butene photodissociation, produces about 10-201- buten-2-yl radicals. Based on our preliminary calculations, this state was assigned to then← π transition 12 . Similar behavior was observed for 2-chloro-2-butene 13 . A fair number of theoretical studies on the structures and energetics of the ground and several electronically excited states of vinyl have been reported 11, 14–19 . High level ab initio calculations of the vinyl ground state equilibrium structure were also reported by Peterson and Dunning 20 . Most of the vinyl excited states calculations were on the 8 valence n← π (the lowest electronic state of vinyl), π ∗ ← n, and π ∗ ← π states. Vertical excitation energies for several Rydberg transitions were calculated by Mebel et al. 17 , although the Rydberg states were not characterized in terms of the quantum numbersnl m . Unlike vinyl, propenyl and butenyl radicals have not been characterized theoreti- cally. These radicals are derived from vinyl by substituting one or two of its hydrogen atoms with methyl or ethyl groups, and, consequently, inherit some of its properties with slight modifications. However, the substituents can also bring around some unique prop- erties. In this work, we present accurate ab initio calculations of the ground and first excited state equilibrium structures of vinyl, propen-1-yl, propen-2-yl, 1-buten-2-yl and trans-2-buten-2-yl radicals. We also present vertical excitation energies and oscillator strengths for both valence and Rydberg states, as well as permanent dipole moments for the valence states. The changes in geometries, excitation energies and properties of the ground and excited states in the above sequence of radicals are discussed and analyzed. Moreover, a qualitative picture of the effect of methyl or ethyl group substitutions on the electronic properties is derived to provide a basis for understanding the effects of molecular size and structure on the Rydberg states’ quantum defect. In addition, we present new interesting examples of hyperconjugation in hydrocarbons. The paper is organized as follows. Section 2.2 summarizes the technical details of the calculations. Section 2.3 presents the analysis of the ground state equilibrium geometries of the radicals. The calculated vertical electronic spectra are discussed in Section 2.4. In sections 2.5, 2.7, 2.8, and 2.9, the changes in the excitation energies and properties for different groups of excited states are discussed for the vinyl→ propen-1- yl→ propen-2-yl→ trans-2-buten-2-yl→ 1-buten-2-yl sequence. Section 2.6 presents adiabatic excitation energies and optimized geometries, as well as their changes in the 9 above sequence for the lowest excited 2 A 00 ← 2 A 0 (n← π ) state. Finally, Section 2.10 summarizes our conclusions. 2.2 Computational details The equilibrium ground state geometries were optimized by CCSD(T) 21, 22 using the ACES II electronic structure program 23 . The restricted open-shell Hartree-Fock (ROHF) doublet reference was used in all the optimizations. We employed the 6- 311(2+,2+)G(d,p) basis to calculate the equilibrium structures of vinyl, propen-1-yl, and propen-2-yl, and the 6-311(+,+)G(d,p) basis for 1-buten-2-yl and trans-2-buten-2- yl. The bases were derived from the polarized split-valence 6-311G(d,p) basis 24, 25 by augmenting it with additional sets of diffuse functions. Pure angular momentum spheri- cal harmonics (5 d-functions) were used throughout this study. As shown in Fig. 2.2, all of the radicals except 1-buten-2-yl have C s symmetry. Figure 2.2: The CCSD(T)/6-311(2+,2+)G(d,p) ground state optimized geometries of the vinyl, propenyl, and butenyl radicals, and the molecular orientation in Cartesian coordinate system. For 1-buten-2-yl, theOXY plane is the C 1 C 2 C 3 plane. 10 Figure 2.3: Theσ CC , π , n, π ∗ , and 3s ROHF orbitals of vinyl. These orbitals are very similar in all the radicals. Relevant molecular orbitals are shown in Fig. 2.3. In this notation, the ground state electronic configuration is(π ) 2 (n) 1 . The vertical excitation energies are calculated using the EOM-EE-CCSD method 26, 27 with the ROHF (π ) 2 (n) 1 doublet reference, except for the π ∗ ← π states (one quartet and two doublets), for which the EOM-SF-CCSD method 28 with the ROHF (π ) 1 (n) 1 (π ∗ ) 1 quartet reference was used. For the 3s← π states, additional calculations were performed using the EOM-SF-CCSD method with the ROHF (π ) 1 (n) 1 (3s) 1 quartet reference and Hartree-Fock orbitals optimized for the ROHF doublet reference. The ionization potentials (IPs) were calculated by EOM-IP- CCSD 29–31 . The 6-311(2+,2+)G(d,p) basis was employed for all single-point excited state calcu- lations, as well as for the IPs for the ionization from the half-filled orbital n. The IPs for the ionization from theπ bonding orbital were calculated using the 6-311++G(d,p) basis set. In the excited state calculations, four lowest and four highest molecular 11 orbitals were frozen for the butenyl radicals, whereas all the orbitals were active for vinyl and propenyls. The permanent dipole moment for the ground and excited states was calculated using the non-relaxed EOM-CCSD one-particle density matrix 27, 28 . The ground state density matrix was calculated by the EOM-SF-CCSD method with the ROHF(π ) 1 (n) 1 (π ∗ ) 1 quartet reference. The geometries of the lowest excited state,n← π , were optimized using the EOM- SF-CCSD method with the unrestricted Hartree-Fock (UHF) quartet reference. The 6- 311(2+,2+)G(d,p) and 6-311(+,+)G(d,p) basis sets were used for the vinyl and propenyl radicals, and for the butenyl radicals, respectively. All orbitals were active in the excited state geometry optimizations. The assignment of the valence and Rydberg character to the excited states was based on three criteria: (i) the symmetry of the transitions, (ii) the character of the molecular orbitals in the leading EOM-CCSD amplitudes, and (iii) the second moments,hX 2 i, hY 2 i, andhZ 2 i, of the EOM-CCSD electron density. The character of the Hartree-Fock orbitals was determined using the Spartan 32 interface. 2.3 Ground state equilibrium structures The equilibrium geometries of the vinyl, propenyl, and butenyl radicals are presented in Fig. 2.2 (we calculated only the lowest-energy, that is trans-conformers, of the propen- 1-yl and 2-buten-2-yl radicals. The common feature of all the species is the C-C double bond and the unpaired electron on one of the unsaturatedsp 2 -like carbon orbitals. In the subsequent discussion, we shall refer to the carbon hosting the unpaired electron asα -C, and the other unsaturated carbon atom asβ -C. Atoms (groups) attached toα -C orβ -C will be referred to asα -atom (α -group) orβ -atom (β -group), respectively. The distance 12 between theα andβ carbons will be referred to asr CC . The notationαCC will be used for the angle between the twoσ bonds connectingα -C to other atoms. The simplest member of this family, the vinyl radical, is derived from ethylene by removing one hydrogen atom. This results in a slight contraction of the C-C bond (r CC = 1.325 ˚ A in vinyl vs. 1.330 ˚ A in ethylene) and a largerαCC angle ( 6 αCC = 136.6 o in vinyl vs. 121.7 o in ethylene). Both effects can be explained by the delocalization of the unpaired electron leading to a larger weight of the p orbital in the singly occupied sp 2 hybrid orbital onα -C, which, consequently, changes the hybridization of the other two sp 2 orbitals towardssp, with the angle closer to 180 o . The H-β -C-H angle is not affected by the hydrogen removal, while theβ -group is rotated as a whole towards theα -H atom in the plane of the molecule. This observation suggests that the steric repulsion between the neighboring H atoms is not the dominant factor in theαCC angle increase. This is also supported by a very weak dependence of the αCC angle on the distance between the neighboring hydrogens, and by the fact that this angle is, in most cases, larger for species where the distance between the neighboring H atoms is larger. Derived by the substitution of a β -hydrogen by a methyl group, the propen-1-yl radical exhibits only slight changes in the αCC angle and the r CC distance relative to vinyl. Interestingly, whereas the angle between the β -hydrogen and the methyl group remains almost the same as in vinyl and ethylene, the rotation of the β -group towards α -H is more pronounced in propen-1-yl than in vinyl. This implies that the rotation is due to the repulsion between the diffuse unpaired electron and the electron density localized between β -C and the attached atom or group, while the angle between the two β -hydrogens in vinyl, or the β -hydrogen and the methyl group in propen-1-yl is determined by the sp 2 hybridization on β -C. The repulsion is also responsible for the small increase inr CC in propen-1-yl relative to vinyl. 13 The propen-2-yl radical is derived by substituting the α -hydrogen of vinyl with a methyl group. A small increase (∼ 1 o ) in theαCC angle indicates an enhanced delocal- ization of the unpaired electron due to its repulsion from electrons localized along the C-C bonds. The delocalization leads to a slightly largerβ -group rotation than in vinyl, although not as large as in the propen-1-yl radical, where the unpaired electron interacts with more dense electronic cloud betweenβ -C and the methyl group. All the effects described above are found to be additive. If bothα -H andβ -H of vinyl are replaced with methyl groups to produce the trans-2-buten-2-yl radical, the value of the αCC angle approaches that of propen-2-yl, and the β -group rotates by the same number of degrees as in propen-1-yl. The structural consequences of the substitution of the α -hydrogen from vinyl with an ethyl group, which leads to the 1-buten-2-yl radical, are very similar to those due to a methyl group in propen-2-yl, except that in 1-buten-2-yl the vinyl moiety is slightly non-planar ( 6 6125 = 179.79 according to atom numbering from Fig. 2.2). The equilibrium orientation of methyl groups in hydrocarbons is an interesting prob- lem (see 33 for a comprehensive review). As shown by Prophristic and Goodman 34 , the major factor responsible for the staggered geometry of ethane (i.e., structure with dihe- dral angle HCCH = 60 o ) is not the steric repulsion of the C-H bonds, but rather the transfer of electrons from one methyl group to the other, leading to their participation in the C-H bonding of the other methyl group. This effect, termed hyperconjugation, stabilizes the relative orientation of the methyl groups, which maximizes the overlap of σ CH bonding orbitals on one methyl group with the antibonding orbitals on the other methyl group. The mechanism of the hyperconjugative charge transfer was first sug- gested by Weinhold 35 , and varies from one molecule to another. For example, in the 14 propene molecule, which can be derived from both propen-1-yl and propen-2-yl radi- cals by adding hydrogen to the radical center, the orientation of the methyl group is the same as in the radicals (see Fig. 2.2). This orientation (called eclipsed in 33 ), is stabi- lized by participation of the σ CH electrons of the two out-of-plane C-H bonds in the π bonding of the double bond (in other words, it is stabilized by the hyperconjugative charge transfer from theσ CH bond of the methyl group to theπ antibonding orbital of the double bond 33 ). Certainly, this type of hyperconjugation plays a role in stabiliz- ing the structures of the propenyl and trans-2-buten-2-yl radicals. However, there is a stronger hyperconjugative effect in the propen-2-yl and both butenyl radicals; namely, the transfer of electron density from the in-plane σ CH bond to the radical center. This follows from the observation that one of the hydrogens in the ethyl group of 1-buten- 2-yl is almost co-planar with the vinyl moiety, indicating that it is the overlap between the in-planeσ CH bonding orbital and the partially filled lone pair ofα -C that stabilizes the orientation of the ethyl group in 1-buten-2-yl. This is also confirmed by a small but systematic elongation of the in-planeσ CH bond in all the propenyl and butenyl rad- icals. The hyperconjugation with unpaired electrons has been suggested to stabilize radical products of bond dissociation for several other molecular systems 36 . Note that the geometry of the ethyl group in 1-buten-2-yl is staggered and is very close to that of ethane. 2.4 Electronically excited states: Vertical spectra The vertical excitation energies, oscillator strengths, and dipole strengths (squared tran- sition dipole moments) for vinyl, propen-1-yl, propen-2-yl, 1-buten-2-yl, and trans-2- buten-2-yl are summarized in Tables 2.1, 2.2, 2.3, 2.5, and 2.4, respectively. The<S 2 > values are also given as a measure of reliability of the calculated energies and, especially, 15 properties: large spin contamination indicates possible errors in the excitation energies, as well as large errors in transition strengths and permanent dipole moments. The results are also visualized in Figs. 2.4-2.8 as stick spectra. Note that only singly excited states are shown. Also, since the highestπ ∗ ← π state was calculated separately from the rest, there are more states in the energy region between this state and the second-highest one shown on the pictures. The electronic spectra of the radicals at hand are very dense and are dominated by the Rydberg excitations. Figure 2.4: Calculated vertical electronic excitation energies and oscillator strengths of the vinyl radical. The reported experimental excitation energies (see Introduction for references) are shown by hollow bars. The intensity of the experimental transitions is arbitrary. 16 456 78 0.0 0.1 0.2 3p x n n CC 3p z n 3s 3sn 3s 3p y n * Oscillatorstrength ExcitationEnergy, eV n * *n n CC Figure 2.5: Calculated vertical electronic excitation energies and oscillator strengths of propen-1-yl. 456 78 0.0 0.1 0.2 n CC 4sn 3d x 2 -y 2n 3d xy n 3p x n 3p z n 3s 3sn 3s 3p y n * Oscillatorstrength ExcitationEnergy, eV n * *n Figure 2.6: Calculated vertical electronic excitation energies and oscillator strengths of propen-2-yl. 17 456 78 0.00 0.08 0.16 * 4p x n 4p y n 3p z 3s * n CC 4sn 3dn 3p x n 3p z n 3sn 3p y n Oscillatorstrength ExcitationEnergy,eV n *n Figure 2.7: Calculated vertical electronic excitation energies and oscillator strengths of 1-buten-2-yl. 456 78 0.00 0.08 0.16 3s * * 3d x 2n n CC 3d xz n 3d y 2 -z 2n 4sn 3d xy n 3p x n 3p z n 3sn 3s 3p y n Oscillatorstrength ExcitationEnergy,eV n *n Figure 2.8: Calculated vertical electronic excitation energies and oscillator strengths of trans-2-buten-2-yl. In the following sections, we discuss the systematic changes in excitation energies and intensities of the calculated transitions. The permanent dipole moments of the radi- cals in the ground and valence excited states (at the ground state equilibrium geometries) are shown in Figs. 2.10 and 2.12. The ground state permanent dipole moments (in a.u.) are (0.106, -0.243, 0) for vinyl, (0.216, -0.164, 0) for propen-1-yl, (-0.110, -0.318, 0) 18 for propen-2-yl, (-0.113, -0.309, -0.014) for 1-buten-2-yl, and (-0.001, -0.246, 0) for trans-2-buten-2-yl. It was first pointed out by Mebel et al. 17 that the two absorption features of vinyl, at 7.37 eV and 7.53 eV , observed by Fahr and Laufer 10 , could be due to different elec- tronic states rather than vibrational spacing. Based on their calculations, Mebel et al. suggestedn← σ (excitation energy 7.31 eV) andRyd← π (excitation energy 7.48 eV) as candidates for these transitions. Their excitation energy for the Rydberg state is very close to our excitation energy for the3s← π state (7.47 eV). Our excitation energy for then← σ state of vinyl is larger (7.67 eV); however, we found another state,3p z ← n, at 7.38 eV , which is very close to one of the observed peaks. Moreover, this state, as well as the3s← π one, are much more intense than then← σ state. Thus, we suggest that the3p z ← n and3s← π states are responsible for the observed transitions. 19 Table 2.1: Vertical excitation energies, oscillator strengths, and properties of the excited states of the vinyl radical State Transition Δ E, eV f L μ 2 tr <S 2 > 2 A 00 n← π 3.31 0.0012 0.0144 0.75 4 A 0b π ∗ ← π 4.35 0 0 3.75 2 A 00 π ∗ ← n 4.93 0.0030 0.0250 0.76 2 A 0b π ∗ ← π c 5.60 0.0002 0.0011 0.75 2 A 0 3s← n 6.31 0.0051 0.0327 0.76 2 A 0 3p x ← n 6.88 0.0126 0.0748 0.76 2 A 0 3p y ← n 7.09 0.0581 0.3346 0.76 4 A 00 b 3s← π 7.31 (7.33) 0 0 3.75 2 A 00 3p z ← n 7.38 0.0096 0.0534 0.81 2 A 00 b 3s← π d 7.47 (7.63) 0.0594 0.3247 0.76 2 A 00 b 3s← π e 8.11 0.0249 0.1255 0.82 2 A 0 n← σ CC 7.67 0.0009 0.0046 0.83 4 A 00 3p x ← π 7.95 0 0 2.20 2 A 0b π ∗ ← π f 8.34 0.2204 1.0782 1.10 EOM-CCSD/6-311(2+,2+)G(d,p) level of theory using ROHF doublet reference at the geometry from Fig. 2.2 (spherical d-functions, E HF = -77.403554 hartree). a Calculated by EOM-SF-CCSD with the ROHF (π ) 1 (π ∗ ) 1 (n) 1 quartet reference (spherical d- functions, E HF = -77.287220 hartree). b Calculated by EOM-SF-CCSD with the ROHF(π ) 1 (n) 1 (3s) 1 reference (orbitals are optimized for the doublet reference, the 3s orbital is the 16thα -spin-orbital by energy in the Hartree-Fock ROHF doublet reference). The EOM-EE-CCSD excitation energy is shown in parentheses. c 2(απ )(βn )(απ ∗ )− (βπ )(αn )(απ ∗ )− (απ )(αn )(βπ ∗ ) d (βπ )(αn )(α 3s)− (απ )(αn )(β 3s) e 2(απ )(βn )(α 3s)− (βπ )(αn )(α 3s)− (απ )(αn )(β 3s) f (βπ )(αn )(απ ∗ )− (απ )(αn )(βπ ∗ ) 20 Table 2.2: Vertical excitation energies, oscillator strengths, and properties of the excited states of the propen-1-yl radical State Transition Δ E, eV f L μ 2 tr , a.u. <S 2 > 2 A 00 n← π 3.04 0.0011 0.0151 0.75 4 A 0b π ∗ ← π 4.37 0 0 3.75 2 A 00 π ∗ ← n 5.13 0.0030 0.0241 0.76 2 A 0b π ∗ ← π c 5.60 0.0002 0.0014 0.75 2 A 0 3s← n 6.14 0.0090 0.0598 0.76 2 A 0 3p y ← n 6.77 0.0611 0.3679 0.77 2 A 0 3p x ← n 6.92 0.0018 0.0103 0.77 4 A 00c 3s← π 6.79 (6.82) 0 0 3.75 2 A 00 3p z ← n 7.09 0.0166 0.0954 0.96 2 A 00c 3s← π d 6.94 (7.12) 0.0187 0.1099 0.74 2 A 00c 3s← π e 7.65 0.0005 0.0025 1.10 2 A 0 n← σ − CC 7.31 0.0029 0.0165 0.76 2 A 0 n← σ + CC 7.47 0.0093 0.0510 0.86 4 A 00 3p x ← π 7.52 0 0 1.99 4 A 00 3p y ← π 7.56 0 0 2.01 4 A 0 3p z ← π 7.57 0 0 1.31 2 A 0b π ∗ ← π f 8.15 0.1999 1.0010 1.16 EOM-CCSD/6-311(2+,2+)G(d,p) level of theory using ROHF doublet reference at the geometry from Fig. 2.2 (spherical d-functions, E HF = -116.451791 hartree). a Calculated by EOM-SF-CCSD with the ROHF (π ) 1 (π ∗ ) 1 (n) 1 quartet reference (spherical d- functions, E HF = -116.332735 hartree). b Calculated by EOM-SF-CCSD with the ROHF(π ) 1 (n) 1 (3s) 1 reference (orbitals are optimized for the doublet reference, the 3s orbital is the 16thα -spin-orbital by energy in the Hartree-Fock ROHF doublet reference). The EOM-EE-CCSD excitation energy is shown in parentheses. c 2(απ )(βn )(απ ∗ )− (βπ )(αn )(απ ∗ )− (απ )(αn )(βπ ∗ ) d (βπ )(αn )(α 3s)− (απ )(αn )(β 3s) e 2(απ )(βn )(α 3s)− (βπ )(αn )(α 3s)− (απ )(αn )(β 3s) f (βπ )(αn )(απ ∗ )− (απ )(αn )(βπ ∗ ) 21 Table 2.3: Vertical excitation energies, oscillator strengths, and properties of the excited states of the propen-2-yl radical State Transition Δ E, eV f L μ 2 tr , a.u. <S 2 > 2 A 00 n← π 3.29 0.0017 0.0205 0.75 4 A 0b π ∗ ← π 4.43 0 0 3.75 2 A 00 π ∗ ← n 4.87 0.0022 0.0186 0.76 2 A 0b π ∗ ← π c 5.61 0.0008 0.0062 0.75 2 A 0 3s← n 5.70 0.0024 0.0174 0.76 2 A 0 3p y ← n 6.39 0.0597 0.3817 0.76 2 A 0 3p x ← n 6.45 0.0036 0.0231 0.76 2 A 00 3p z ← n 6.66 0.0070 0.0429 0.78 4 A 00c 3s← π 6.75 (6.80) 0 0 3.75 2 A 00c 3s← π d 6.92 (7.01) 0.0225 0.1326 0.73 2 A 00c 3s← π e 7.34 0.0026 0.0143 0.76 2 A 0 3d xy ← n 7.10 0.0034 0.0195 0.76 2 A 0 3d x 2← n 7.22 0.0094 0.0532 0.76 2 A 0 4s← n 7.29 0.0021 0.0115 0.76 2 A 00 3d xz ← n 7.36 0.0003 0.0018 0.77 2 A 00 3d yz ← n 7.42 0.0000 0.0000 0.77 2 A 0 n← σ − CC 7.43 0.0250 0.1371 0.79 2 A 0 d y 2 − z 2← n 7.47 0.0013 0.0071 0.77 4 A 00 3p x ← π 7.52 0 0 2.01 4 A 00 3p y ← π 7.58 0 0 1.94 4 A 0 3p z ← π 7.58 0 0 1.13 2 A 0 4p y ← n 7.72 0.0167 0.0884 0.77 22 Table 2.3, continued: State Transition Δ E, eV f L μ 2 tr , a.u. <S 2 > 2 A 00 3p x ← π 7.73 0.0043 0.0229 1.66 2 A 0 4p x ← n 7.74 0.0012 0.0062 1.05 2 A 0b π ∗ ← π f 8.23 0.1886 0.9350 1.29 EOM-CCSD/6-311(2+,2+)G(d,p) level of theory using ROHF doublet reference at the geometry from Fig. 2.2 (spherical d-functions, E HF = -116.457027 hartree). a Calculated by EOM-SF-CCSD with the ROHF (π ) 1 (π ∗ ) 1 (n) 1 quartet reference (spherical d- functions, E HF = -116.335488 hartree). b Calculated by EOM-SF-CCSD with the ROHF(π ) 1 (n) 1 (3s) 1 reference (orbitals are optimized for the doublet reference, the 3s orbital is the 16thα -spin-orbital by energy in the Hartree-Fock ROHF doublet reference). The EOM-EE-CCSD excitation energy is shown in parentheses. c 2(απ )(βn )(απ ∗ )− (βπ )(αn )(απ ∗ )− (απ )(αn )(βπ ∗ ) d (βπ )(αn )(α 3s)− (απ )(αn )(β 3s) e 2(απ )(βn )(α 3s)− (βπ )(αn )(α 3s)− (απ )(αn )(β 3s) f (βπ )(αn )(απ ∗ )− (απ )(αn )(βπ ∗ ) 23 Table 2.4: Vertical excitation energies, oscillator strengths, and properties of the excited states of the buten-2-yl radical State Transition Δ E, eV f L μ 2 tr , a.u. <S 2 > 2 A 00 n← π 3.02 0.0014 0.0196 0.75 4 A 0b π ∗ ← π 4.44 0 0 3.75 2 A 00 π ∗ ← n 5.10 0.0021 0.0170 0.76 2 A 0b π ∗ ← π c 5.62 0.0007 0.0050 0.75 2 A 0 3s← n 5.64 0.0015 0.0109 0.77 2 A 0 3p y ← n 6.14 0.0650 0.4321 0.76 2 A 0 3p x ← n 6.26 0.0254 0.1654 0.77 2 A 00 3p z ← n 6.38 0.0085 0.0546 0.86 4 A 00c 3s← π 6.38 (6.44) 0 0 3.75 2 A 00c 3s← π d 6.58 (6.63) 0.0068 0.0422 0.76 2 A 00c 3s← π e 7.05 0.0004 0.0025 1.05 2 A 0 4s← n 6.89 0.0015 0.0088 0.77 4 A 00 3p x ← π 6.94 0 0 1.97 4 A 00 3p y ← π 6.99 0 0 1.94 2 A 0 3d xy ← n 6.99 0.0010 0.0056 0.76 2 A 00 3d yz ← n 7.01 0.0002 0.0014 0.77 2 A 0 3d x 2← n 7.03 0.0036 0.0212 0.77 4 A 0 3p z ← π 7.06 0 0 1.87 2 A 00 3d xz ← n 7.15 0.0001 0.0006 0.78 2 A 00 3p x ← π 7.19 0.0000 0.0001 1.74 2 A 0 n← σ CC 7.20 0.0012 0.0065 0.86 2 A 0 3d y 2 − z 2← n 7.29 0.0012 0.0068 0.85 2 A 0b π ∗ ← π f 8.38 0.4053 1.9747 1.04 EOM-CCSD/6-311(2+,2+)G(d,p) level of theory using ROHF doublet reference at the geometry from Fig. 2.2 (spherical d-functions, E HF = -155.504207 hartree). a Calculated by EOM-SF-CCSD with the ROHF (π ) 1 (π ∗ ) 1 (n) 1 quartet reference (spherical d- functions, E HF = -155.380248 hartree). b Calculated by EOM-SF-CCSD with the ROHF(π ) 1 (n) 1 (3s) 1 reference (orbitals are optimized for the doublet reference, the 3s orbital is the 26thα -spin-orbital by energy in the Hartree-Fock ROHF doublet reference). The EOM-EE-CCSD excitation energy is shown in parentheses. c 2(απ )(βn )(απ ∗ )− (βπ )(αn )(απ ∗ )− (απ )(αn )(βπ ∗ ) d (βπ )(αn )(α 3s)− (απ )(αn )(β 3s) e 2(απ )(βn )(α 3s)− (βπ )(αn )(α 3s)− (απ )(αn )(β 3s) f (βπ )(αn )(απ ∗ )− (απ )(αn )(βπ ∗ ) 24 Table 2.5: Vertical excitation energies, oscillator strengths, and properties of the excited states of the the 1-buten-2-yl radical State Transition Δ E, eV f L μ 2 tr , a.u. 2 A n← π 3.28 0.0014 0.0216 4 A a π ∗ ← π 4.42 0 0 2 A π ∗ ← n 4.81 0.0026 0.0216 2 A a π ∗ ← π c 5.58 0.0008 0.0055 2 A 3s← n 5.73 0.0024 0.0173 2 A 3p y ← n 6.27 0.0616 0.4014 2 A 3p x ← n 6.40 0.0054 0.0346 2 A 3p z ← n 6.52 0.0081 0.0505 4 A b 3s← π 6.78 (6.83) 0 0 2 A b 3s← π d 6.94 0.0300 0.1764 2 A b 3s← π e 7.35 0.0056 0.0309 2 A n← σ CC ,3s← π 7.03 0.0199 0.1157 2 A n← σ CC 7.04 0.0111 0.0645 2 A 4s← n 7.05 0.0038 0.0218 2 A 3d← n 7.10 0.0025 0.0147 2 A 3d← n 7.18 0.0021 0.0121 2 A 3d← n 7.23 0.0003 0.0015 2 A 3d← n 7.26 0.0004 0.0023 2 A 3d← n 7.42 0.0068 0.0372 4 A 3p z ← π 7.45 0.0777 0.4257 4 A 3p y ← π 7.48 0.0013 0.0073 4 A 3p x ← π 7.57 0 0 2 A 4p y ← n 7.58 0.0109 0.0585 2 A 4p x ← n 7.61 0.0033 0.0176 2 A 3p x ← π 7.63 0.0502 0.2684 2 A 3p y ← π 7.67 0.0032 0.0170 25 Table 2.5, continued State Transition Δ E, eV f L μ 2 tr , a.u. 2 A 4p z ← n 7.71 0.0011 0.0057 2 A 3p x ← π 7.77 0.0062 0.0328 2 A a π ∗ ← π f 8.04 0.2395 1.2163 EOM-CCSD/6-311(2+,2+)G(d,p) level of theory using ROHF doublet reference at the geometry from Fig. 2.2 (spherical d-functions, E HF = -155.500917 hartree). a Calculated by EOM-SF-CCSD with the ROHF (π ) 1 (π ∗ ) 1 (n) 1 quartet reference (spherical d-functions, E HF = -155.379309 hartree). b Calculated by EOM-SF-CCSD with the ROHF(π ) 1 (n) 1 (3s) 1 reference (orbitals are optimized for the doublet reference, the 3s orbital is the 17thα -spin-orbital by energy in the Hartree-Fock ROHF doublet reference). The EOM-EE-CCSD excitation energy is shown in parentheses. c 2(απ )(βn )(απ ∗ )− (βπ )(αn )(απ ∗ )− (απ )(αn )(βπ ∗ ) d (βπ )(αn )(α 3s)− (απ )(αn )(β 3s) e 2(απ )(βn )(α 3s)− (βπ )(αn )(α 3s)− (απ )(αn )(β 3s) f (βπ )(αn )(απ ∗ )− (απ )(αn )(βπ ∗ ) 2.5 Valence excited states derived from the n ← π , π ∗ ← n, andn← σ CC transitions The valence excited states derived from then← π ,π ∗ ← n, andn← σ CC transitions do not have analogues in ethylene. These states correspond to electron promotion to or from the half-filled orbital. Thus, single excitations from the doublet reference determinant provide a spin-complete zeroth-order description of these states. Therefore, EOM-EE- CCSD describes these states accurately 28, 37 . Fig. 2.9 shows changes in these vertical excitation energies in the vinyl→ propen- 1-yl→ propen-2-yl→ 1-buten-2-yl→ trans-2-buten-2-yl series. The correspond- ing changes in the oscillator strengths and permanent dipole moments are shown in Fig. 2.10. The n← π state, which corresponds to the excitation from the α -C-β -C π 26 vinyl propen-1-yl propen-2-yl 1-buten-2-yl 2-buten-2-yl 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Excitationenergy,eV n n n CC Figure 2.9: Changes in then← π ,π ∗ ← n, andn← σ CC vertical excitation energies. vinyl propen-1-yl propen-2-yl 1-buten-2-yl 2-buten-2-yl 0.000 0.005 0.010 0.015 0.020 0.025 Oscillatorstrength n n n CC (-.01) (-.01) (-.07) (-.31) Figure 2.10: Changes in the oscillator strengths for then← π ,π ∗ ← n, andn← σ CC transitions. The ground and vertical excited state permanent dipole moments are shown by arrows. The plane of the figure is considered as the OXY plane, and the OZ axis is perpendicular to this plane (see Fig. 2.2). For 1-buten-2-yl, the Z-component of the permanent dipole moment is shown in parentheses. 27 bond to the partially filledα -Csp 2 orbital, is the lowest in all the radicals. The vertical excitation energy for this state is about 3.3 eV in vinyl, propen-2-yl, and 1-buten-2-yl, whereas it is lower by 0.3 eV in propen-1-yl and trans-2-buten-2-yl. This change is con- sistent with the structural differences (see Fig. 2.2): the radicals with the sameβ -group have similar excitation energies, which means that it is not the change in the half-filled orbitals, rather the change in theπ bonding orbitals that is responsible for the change of the excitation energy. The same holds for theπ ∗ ← n state (see Fig. 2.9), although the β -group substitution has an opposite effect: the vertical excitation energy is about 4.9 eV for vinyl, propen-2-yl, and 1-buten-2-yl, but it is higher by about 0.2 eV for propen- 1-yl and trans-2-buten-2-yl. This can be rationalized as follows: the increase of electron density on the β -group upon the substitution of hydrogen with a methyl group causes the energies of the π and π ∗ orbitals to increase, while at the same time having minor relative effect on the energy of the half-filled orbital. Consequently, it becomes easier to move an electron from the π orbital to n, but harder to excite an electron from n to π ∗ . Moreover, the energy gap between theπ andπ ∗ orbitals is not affected by the sub- stitution, as follows from the structural insensitivity of the firstπ ∗ ← π doublet vertical excitation energy (see Fig. 2.11 in Sec. 2.7). The changes in then← σ CC excitation energies are more complicated. There are as many differentn← σ CC states as there are differentσ CC bonds. The molecular orbital and the EOM-CCSD amplitude analyses reveal that the different σ CC bonds form a delocalizedσ CC -system, from which the excitation occurs. Finally, then← σ CC states are fairly high in energy (above 7 eV), where the density of states is high, and they mix with the Rydberg states of the same symmetry. This explains why some of the n← σ CC states are missing, e. g.,n← σ + CC for the propen-2-yl radical. There are two n← σ CC transitions in propen-1-yl: at 7.31 eV (n← σ − CC ) and 7.47 eV (n← σ + CC ), 28 vinyl propen-1-yl propen-2-yl 1-buten-2-yl 2-buten-2-yl 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 Excitationenergy,eV 4 () 2 () 2 () Figure 2.11: Changes in theπ ∗ ← π vertical excitation energies. vinyl propen-1-yl propen-2-yl 1-buten-2-yl 2-buten-2-yl 0.0000 0.0002 0.0004 0.0006 0.0008 0.2 0.4 Oscillatorstrength (-.01) 4 () 2 () 2 () (-.03) (.20) (-.01) Figure 2.12: Changes in the oscillator strengths for theπ ∗ ← π transitions. The ground and vertical excited state permanent dipole moments are shown by arrows. The plane of the figure is considered as the OXY plane, and the OZ axis is perpendicular to this plane (see Fig. 2.2). For 1-buten-2-yl, the Z-component of the permanent dipole moment is shown in parentheses. 29 with both of them being lower than the n← σ CC transition in vinyl (7.67 eV). We interpret this in terms of reduced repulsion between the σ CC bonds, due to removal of an electron from one of the bonds. Alternatively, one can picture the resulting positive σ CC -system as being stabilized by the higher number of bonds in the system, analogous to the decreasing basicity of amines in the order tertiary, secondary, and primary. The two methods are equivalent in explaining the observed trend of decreasing n← σ CC excitation energies with increasing number ofσ CC bonds (see Fig. 2.9). Despite the delocalization of then← σ CC excitations in propen-1-yl, we can assign, based on the values of the dipole strengths, the n ← σ − CC transition at 7.31 eV as an excitation from the σ CC bond between α -C and β -C (lower transition strength, the electron is transferred to n from the closer bond), whereas the n← σ + CC transition at 7.47 eV is from the σ CC bond between β -C and the carbon atom of the methyl group (higher transition strength, the charge is transferred over larger distance). This is also consistent with changes in the permanent dipole moment upon excitation (see Fig. 2.10). The higher excitation energy for the n← σ − CC transition in propen-2-yl (7.43 vs. 7.31 in propen-1-yl) is due to increased repulsion between the additional electron on the half-filled orbital and the electrons participating in the carbonσ -bonding. The exci- tation occurs mainly from the longer C-C bond; the electrons from the C-H bonds of the methyl group compensate for the decrease in electron density between the carbons due to high polarizability of the C-H bonds. This electron transfer explains the high oscillator strength for then← σ − CC transition in propen-2-yl (see Fig. 2.10). In trans-2-buten-2-yl and 1-buten-2-yl, the increased number of σ CC bonds rela- tive to propen-1-yl and propen-2-yl results in increased repulsion between the electrons forming these bonds; and, consequently, in the decrease of then← σ CC excitation ener- gies. The lowest n← σ − CC excitation in trans-2-buten-2-yl corresponds mainly to the 30 excitation from the double bond. The positive charge created is farther from the methyl hydrogens, which results in a decreased electron donation from the methyl groups and consequent lower oscillator strength relative to propen-2-yl. Due to the lower symme- try, the n← σ CC and 3s← π states are allowed to mix in the 1-buten-2-yl radical. The EOM-CCSD amplitude analysis supports this interpretation. The driving force for this mixing is the increase of the repulsion between the lone pair on α -C and the π - bond upon the n← σ CC excitation. The possibility to remove an electron from the π -bond provides an additional degree of freedom for relaxation, which explains the lower excitation energy for then← σ CC transition in 1-buten-2-yl (7.03 eV) relative to trans-2-buten-2-yl (7.20 eV). 2.6 Equilibrium geometries and adiabatic excitation energies for then← π excited states. Equilibrium geometries of the n← π states are summarized in Fig. 2.17. The most prominent change in the geometric parameters upon the n← π excitation is the elon- gation of theα -C-β -C bond and the decrease of theα CC angle. The elongation of the C-C bond is, of course, due to the decrease in the bond order from 2 (oneσ and oneπ bond) to 1.5 (oneσ and half of aπ bond). The decrease of theα CC angle is due to the decreased delocalization of the lone pair on α -C, plus the repulsion between the lone pair and the bonds connecting α -C with neighboring atoms. Because of the decrease of the α CC angle, the steric repulsion between the α and β hydrogens becomes com- petitive with the repulsion between the lone pair on α -C and the closest bonds in the β -group, resulting in the opposite rotation of the β -group with respect to the ground 31 state (see Sec. 2.3). The α CC angle in propen-2-yl, 1-buten-2-yl, and trans-2-buten-2- yl is by about 5 o larger than in vinyl and propen-1-yl due to the repulsion of the C-C bonds. These changes in the equilibrium geometries and bond angles are much larger than the error bars for the CCSD and CCSD(T) methods 38 . Changes discussed below are comparable to the systematic and non-systematic errors of CCSD vs CCSD(T), and therefore should be taken with a grain of caution. An interesting consequence of the partial breaking of the π -bond upon the n← π excitation is the increase of electron donation from the out-of-plane C-H bonds to theα - C-β -Cπ bond, leading to the shortening of theα -C-β -C bond in the vinyl→ propen-2-yl → 1-buten-2-yl→ propen-1-yl→ trans-2-buten-2-yl sequence. This is confirmed by the increase of the out-of-plane C-H bond lengths relative to the ground state. Although the elongation of the C-H bonds is very small (∼ 0.002 ˚ A), it is likely not due to the differ- ence between the methods used to optimize excited and ground state structures, because the bonds are elongated, while CCSD systematically underestimates bond lengths rela- tive to CCSD(T) 38 . the maximum of error distribution of CCSDis shifted towards shorter bond lengths relative to CCSD(T). Also, since the excited electron is partially transferred fromβ -C toα -C, the C-C bond in theβ -group of propen-1-yl and trans-2-buten-2-yl is shortened by about 0.02 ˚ A by the excitation. The α -C-H bond in propen-1-yl and the C-C bond in theα -group of propen-2-yl, trans-2-buten-2-yl, and 1-buten-2-yl are elon- gated by∼ 0.01-0.02 ˚ A relative to the ground state because the orbitals ofα -C become moresp 3 -like hybridized, and the contribution of the2s orbital to theα -C-H orα -C-C bond decreases. The equilibrium geometry of the 1-buten-2-yl radical in the n← π excited state is slightly non-planar (dihedral angle C 4 -C 3 -C 2 -C 1 is 161 o ). The calculated energy differ- ence between the optimized planar and non-planar structures is very small (82 cm − 1 ). 32 We explain the small energy increase for the planar structure by the increased repulsion between the lone pair onα -C and theσ C-C bond of the ethyl group. The n← π adiabatic excitation energies for the vinyl, propen-1-yl, propen-2-yl, trans-2-buten-2-yl, and 1-buten-2-yl radicals are 2.47 eV , 2.24 eV , 2.39 eV , 2.14 eV , and 2.33 eV , respectively. The calculated adiabatic excitation energy of vinyl, 2.47 eV , is very close to the observed origin of the lowest electronic transition, 2.49 eV 6–8 . Note that, while the vertical excitation energies for then← π transition decrease very slowly within the vinyl→ propen-2-yl→ 1-buten-2-yl; and propen-1-yl→ trans-2-buten-2- yl groups (see Sec. 2.5 and Fig. 2.9), the adiabatic excitation energies decrease faster within these two groups (0.06-0.1 eV adiabatic vs. 0.01-0.02 eV vertical excitation energy difference between the neighboring radicals in the groups). The steeper decrease of the adiabatic excitation energies is due to the increase of the number of degrees of freedom with the system size, which facilitates the relaxation of the excited state. 2.7 Excitedπ ∗ ← π states Unlike the transitions described in Sec. 2.5, theπ ∗ ← π excitation is present in ethylene, where it yields two states: the triplet 3 B 3u state at 4.49 eV , and the singlet 1 B 3u state at 8.18 eV vertically. Ethylene vertical excitation energies were calculated using EOM- EE-CCSD/6-311(2+,2+)G(d,p) at the geometry from Saxe 39 : r CC = 1.330 ˚ A; r CH = 1.076 ˚ A; and 6 HCC = 121.7 o . Symmetry is D 4h . Due to the unpaired electron, there are threeπ ∗ ← π states in the radicals: one (low- spin) quartet and two doublets. The spin-pure zeroth-order wave functions of these states are 38 : 4 Ψ= ( απ )(αn )(βπ ∗ )+(απ )(βn )(απ ∗ )+(βπ )(αn )(απ ∗ ) 33 2 Ψ 1 =2(απ )(βn )(απ ∗ )− (βπ )(αn )(απ ∗ )− (απ )(αn )(βπ ∗ ) 2 Ψ 2 =(βπ )(αn )(απ ∗ )− (απ )(αn )(βπ ∗ ) (2.1) It is easy to see that some of the above configurations are formally double excitations from the doublet (π ) 2 (n) reference, which results in a large spin contamination of the EOM-EE-CCSD target states. The large spin contamination can cause large errors in excitation energies 40 and, especially, in oscillator strengths and permanent dipole moments. On the other hand, all of the above configurations are single spin-flipping excitations from the quartet (απ )(αn )(απ ∗ ) reference and can be described by EOM- SF-CCSD. Consequently, the spin contamination of the EOM-SF-CCSD target states is much smaller. The systematic changes in vertical excitation energies and properties of theπ ∗ ← π states are summarized in Figs. 2.11 and 2.12. The excitation energies for the quartet state of vinyl and propen-1-yl are very close (4.35 eV and 4.37 eV , respectively). The same is true for the propen-2-yl, 1-buten-2-yl, and trans-2-buten-2-yl radicals (4.43 eV , 4.42 eV , and 4.44 eV , respectively). However, the differences in excitation energy for the radicals from these two groups are larger; i. e., it is at least 0.05 eV higher for propen-2-yl and both buten-2-yl isomers. We attribute this difference to the more diffuse character of the unpaired electron in the propen-2-yl and buten-2-yl isomers (confirmed by the geometry change mentioned in Sec. 2.3), which results in a stronger Pauli repulsion for the quartet state. Indeed, this effect is observed only for the quartet π ∗ ← π state, for which the Pauli repulsion is stronger, while the transition energy for the first doublet is almost the same for all the radicals. The reason for the excitation energies to the quartet state for the radicals being lower than the corresponding excitation energy in ethylene is the better overlap of the carbon out-of-plane p-orbitals in ethylene, due to higher symmetry. 34 As shown by Mulliken 41 , the transitions from a bonding orbital to the corresponding antibonding orbital are usually very strong. However, the oscillator strength of theπ ∗ ← π transition to the first doublet in the radicals is weak (< 0.001). The origin of this was first explained by Zhang and Morokuma for the vinyl radical 18 . In the lower energy doublet of vinyl, the unpaired electron is coupled to the spin-forbidden “triplet”π ∗ ← π configuration. However, in the second doublet, the unpaired electron is coupled to the “singlet” π ∗ ← π configuration, and this transition is indeed the strongest among the calculated transitions of vinyl (see Table 2.1). This applies to the other species as well. The structural dependence of the second doublet is determined by the interplay between electron repulsion and attraction of electrons to partially positive hydrogens on the methyl/ethyl groups. Indeed, due to the Coulomb repulsion between the unpaired electron and an electron on aπ ∗ orbital in vinyl, the latter tends to be as far away from the former as possible, but it cannot go too far because of nuclear attraction. As a result, the excitation energy for the second doublet in vinyl (8.34 eV) is higher than the sin- gletπ ∗ ← π excitation in ethylene (8.18 eV). In propen-1-yl, the energy of the excited electron on the π ∗ orbital is lowered by the attraction to the out-of-plane hydrogens (see Fig. 2.2), and the excitation energy becomes 8.15 eV , even lower than in ethylene. In propen-2-yl, in which the unpaired electron and the out-of-plane hydrogens of the methyl group are on the same side of theπ bond, two trends are in opposition. Never- theless, the energy is lowered by the attraction to the hydrogens, and the second doublet in propen-2-yl (8.23 eV) is lower than in vinyl, although it is higher than in ethylene. In 1-buten-2-yl, we have an ethyl group instead of a methyl group, and the energy of an electron on the π ∗ orbital is lowered even more than in propen-2-yl, resulting in a low excitation energy of 8.04 eV . Finally, in trans-2-buten-2-yl, the stabilizing effects of the two methyl groups lead to a less diffuse distribution of an electron on the π ∗ 35 orbital, resulting in a high excitation energy of 8.38 eV , which is close to vinyl. Note that this behavior is completely different from the structural dependence of the quartet and the first doublet states, because only for the second doublet are the energetics of the transition determined by Coulomb interactions rather than exchange interactions. 2.8 Rydbergnl m ← n states The Rydberg nl m ← n states, which are well described by EOM-EE-CCSD, are pre- dominantly single excitations of the unpaired electron to a diffuse Rydberg orbital. Since there are no unpaired electrons in ethylene, these excitations are unique to the radicals. The excitation energy for the Rydberg states of small polyatomic molecules can be approximated by the Rydberg formula 42 : E ex =IP− 13.61 (n− δ ) 2 , (2.2) whereE ex is the excitation energy (in eV), IP is the ionization potential of the molecule (in eV), n is the principal quantum number, and δ is the quantum defect parameter accounting for the penetration of the excited Rydberg electron to the cation core (δ = 0.9-1.2 for s-states, 0.3-0.6 for p-states, and smaller or equal to 0.1 for d-states 42 ). The ionization potential of a molecule is determined by two factors: the energy of the orbital from which the ionization occurs, and the redistribution of the electron density in the ion core upon ionization. The quantum defect δ depends on the size and the shape of a molecule. Finally, the Rydberg states can mix with valence states and other Rydberg states of the same symmetry. The calculated IPs of vinyl, propen-1-yl, propen-2-yl, 1-buten-2-yl, and trans-2- buten-2-yl radicals are 9.63 eV , 9.28 eV , 8.79 eV , 8.66 eV , and 8.51 eV , respectively. 36 The decrease of IPs in this sequence is due to increased repulsion between an unpaired electron and other electrons as the number of electrons increases. This explains the monotonous decrease (except the 3p x ← n state in propen-1-yl, which is discussed below) of the excitation energies for thenl m ← n Rydberg states in the above sequence, as well as a larger decrease upon the vinyl α -hydrogen substitution compared to a β -hydrogen substitution (i.e., propen-2-yl versus propen-1-yl, and propen-1-yl versus trans-2-buten-2-yl), as shown in Fig. 2.13. vinyl propen-1-yl propen-2-yl 1-buten-2-yl 2-buten-2-yl 6.0 6.5 7.0 7.5 Excitationenergy,eV 3dxyn 3pzn 3pxn 3pyn 3sn Figure 2.13: Changes in thenl m ← n vertical excitation energies. However, the excitation energies for Rydberg states of many-electron systems are also influenced by the interaction of the excited electron with the cation core. The strength of this interaction is characterized by the quantum defect,δ . Despite the repul- sion of the valence electrons, there is a non-vanishing probability to find a Rydberg electron in such proximity to a heavy nucleus that it “feels” its higher positive charge, resulting in a positive δ in Eq.(2.2). Fig. 2.15 presents the summary of the quantum 37 vinyl propen-1-yl propen-2-yl 1-buten-2-yl 2-buten-2-yl 0.00 0.01 0.02 0.05 0.06 Oscillatorstrength 3pyn 3pxn 3pzn 3sn 3dxyn Figure 2.14: Changes in the oscillator strengths for the nl m ← n transitions. The ground and vertical excited state permanent dipole moments are shown by arrows. The plane of the figure is considered as the OXY plane, and the OZ axis is perpendicular to this plane (see Fig. 2.2). For 1-buten-2-yl, theZ-component of the permanent dipole moment is shown in parentheses. defects for the3s and the three components of the3p Rydberg states of vinyl, propenyl, and butenyl radicals, along with the changes in IPs and the excitation energies upon sub- stitution of hydrogens with methyl (ethyl in case of 1-buten-2-yl) groups. The general trend is that the quantum defect decreases as the system size increases, which leads to a slower excitation energy drop relative to the drop in IP. This is due to the increase in the number of electrons, which screen the nuclei in the cation core. The most intriguing result is the very large (0.76 vs. 0.69 in CH 3 , calculated at the same level) quantum defect for the 3p x (directed along the C-C bond) Rydberg state of vinyl, and its sharp drop in propen-1-yl and propen-2-yl (by about 0.16), which, as mentioned above, leads to the increased excitation energy for the propen-1-yl despite the decrease in IP. The NBO analysis 43 of the electron density for the 3p x state reveals 38 IP 0.97 3s 0.58 3p x 0.48 3p y 0.82 3p z 0.86 IP 0.35 3s 0.17 3p x -0.04 3p y 0.32 3p z 0.29 IP 0.77 3s 0.50 3p x 0.66 3p y 0.63 3p z 0.71 IP 0.84 3s 0.61 3p x 0.43 3p y 0.70 3p z 0.72 IP 0.28 3s 0.06 3p x 0.19 3p y 0.25 3p z 0.28 IP 0.13 3s -0.03 3p x 0.05 3p y 0.12 3p z 0.14 0.90 0.59 0.62 0.47 0.92 0.60 0.67 0.51 0.84 0.55 0.61 0.48 0.82 0.54 0.60 0.47 0.98 0.76 0.69 0.54 Figure 2.15: Quantum defectsδ for thenl m ← n Rydberg states, see Eq. (2.2). Arrows connect species that are different by a single substitution of a hydrogen by a methyl or ethyl group. Next to the arrows, the differences in IPs, as well as in the excitation energies for the 3s, 3p x , 3p y , and 3p z states (from top to bottom) for the connected species are shown. a large weight of carbon 2s orbitals in the Rydberg orbital (occupied by the excited electron) in vinyl, but not in propen-1-yl and propen-2-yl. We did not observe any substantial mixing of the valence and the3s or3p Rydberg states, which would manifest itself in excitations shared by two or more EOM-CCSD target states. This does not exclude, of course, the Rydberg-valence interactions due to the mixed Rydberg-valence character of the HF orbitals that reflects the diffuseness of the valence orbitals due to electron repulsion. Thus, we distinguish between the mixing of many-electron states and mixing of one-electron states. The large quantum defect 39 for the 3p x state of vinyl cannot be explained by mixing of the Rydberg and valence electronic states. To understand this quantum defect, we performed NBO 43 analysis of the 3p Ryd- berg state electron density. We also analyzed the relevant MOs by using the Spartan interface 32 . The analysis clearly demonstrates that about a half of the positive charge in the cation core is distributed among the hydrogens. Due to the high polarizability of the C-H bonds, the carbons effectively strip the hydrogens off the electrons, thus acquiring a negative charge. The MO analysis shows that the orientation of the in-plane 3p x,y components in the radicals is determined by the anisotropy of the potential created by the positively charged hydrogens. This is why one of the 3p components (3p x in our notations) is directed along the line connecting the two far-most carbons in the radicals of C s symmetry (vinyl, propenyl, and trans-2-buten-2-yl). Contrary to atoms, the quantum defect in molecules depends also on the charge dis- tribution in the cation core. To explain the role of this charge distribution, we consider two model systems: the methyl and methylene radicals. It is known 44 that the quantum defectδ for the in-plane components of the3p Rydberg states in methyl (∼ 0.7) is larger than δ for the out-of-plane component (∼ 0.6), which is close to a free carbon atom. Large δ s for the in-plane components can be explained by the finite dimension of the molecule in the plane. Since the positive charge of the cation is distributed over a finite area, the maxima of the in-plane Rydberg electron wave function tend to approach the center of the charge distribution, which is the carbon atom, resulting in a higher prob- ability of finding a Rydberg electron near the carbon nucleus. Note that both in-plane and out-of-plane Rydberg 3p electrons must have a node on the carbon nucleus in D 3h symmetry. 40 In methylene,δ s are 0.77 for the3p x state (in-plane, perpendicular to the symmetry axis), 0.64 for the 3p y state (in-plane, parallel to the symmetry axis), and 0.60 for the 3p z state (perpendicular to the molecular plane). The excitation energies and IPs for the 3p Rydberg states of methylene were cal- culated using the EOM-EE-CCSD and EOM-IP-CCSD methods, respectively, with the 6-311(2+,2+)G** basis set (spherical d-functions) and the triplet ROHF reference. We considered only the3p Rydberg states corresponding to excitation from the out-of-plane half-filled orbital. We used the following geometry: r CH = 1.077 ˚ A, 6 HCH = 134 o (C 2v symmetry). We also calculated the quantum defects at the geometries with one or both r CH = 1.096 ˚ A, and found that the quantum defects are insensitive to the C-H bond lengths. At 6 HCH = 90 o and bothr CH = 1.077 ˚ A, the quantum defects were found to be 0.72 for the 3p x state, 0.69 for the 3p y state, and 0.62 for the 3p z state, and the sum of δ s for the in-plane Rydberg states is 2.41. At 6 HCH = 179 o , the quantum defects were found to be 0.78 for the3p x state, and 0.64 for the3p y,z states. Once again, the sum ofδ s for the3p x and3p y components, 2.42, is very close to the sum at the other geometries. The largerδ for the3p x state of methylene relative to methyl is due to the fact that all the positive charge of the methylene cation core is distributed mainly in thex direction, whereas in methyl it is spread in the two dimensions of the molecular plane. Note that δ for the3p x Rydberg state of methylene is almost the same as in vinyl. The important role of the charge distribution in determining δ s for the in-plane 3p Rydberg states of methylene and methyl is confirmed by the fact that the sum of theseδ s is almost the same in both radicals (2.41 in methylene and 2.4 in methyl). Also, this sum for methylene depends only slightly on the H-C-H angle, (see above). Thus, similar to methylene, we explain the large quantum defect for the3p x Rydberg state of vinyl, which is directed mainly along the C-C bond, by the distribution of the 41 cation core’s positive charge along the molecule. In the y direction, the charge distri- bution in vinyl is similar to that of methyl, and it makesδ for the 3p y Rydberg state of vinyl (0.69) close toδ for the in-plane3p Rydberg state components of methyl (∼ 0.7). There is another effect which may contribute to the largeδ for the3p x state of vinyl. Namely, the center of the charge distribution and, consequently, the nodes of the Ryd- berg electron wave function no longer coincide with the carbon nuclei, which further increases the probability of finding the Rydberg electron on the carbon nuclei. This is confirmed by the increase of δ for larger αCC angles (up to 0.86 for αCC = 180.), contrary to methylene, in which it changes only slightly upon the increase of the H- C-H angle (see above). This is also confirmed by the increase of the α -C 2s orbital contribution to the Rydberg NBO orbital as theαCC angle increases. In propen-1-yl or propen-2-yl, as we substitute α or β -hydrogens with a methyl group, the positive charge distributed along the cation core is more effectively screened by the electrons on the carbonσ bonds. This results in a much smaller penetration of the Rydberg electron into the carbon nuclei, reflected by the large decrease in the quantum defect. Another interesting observation is a larger decrease inδ relative to vinyl for the3p y Rydberg state of propen-2-yl compared to propen-1-yl (δ = 0.62 vs. 0.67 relative to 0.69 in vinyl). We attribute this difference to the presence of two σ bonds connecting α -C to the other carbons in propen-2-yl versus one in propen-1-yl, which results in a more effective screening of α -C with electrons. Moreover, due to the partial positive charge on the hydrogens, the 3p y Rydberg electron in propen-2-yl tends to move around α -C from which it was removed, which is not the case in propen-1-yl (see Fig. 2.2). The correlation of the oscillator strengths is shown in Fig. 2.14. Note that thenl m ← n transitions are relatively strong. In fact,3p y ← n is one of the most intense transition 42 after very strongπ ∗ ← π . The large transition dipole moment for this excitation is due to the large overlap of the singly occupied orbital with the3p y Rydberg orbital. The spectral density of the nl m ← n Rydberg states increases rapidly with the decrease in IP, as seen in Figs. 2.4-2.8. For example, the 3d ← n states appear at about 7.1 eV in propen-2-yl, but are above 7.5 eV in propen-1-yl. However, the density of Rydberg states in the radicals is even larger due to Rydberg excitations from the π bonding orbital, which appear at relatively low energies. 2.9 Rydbergnl m ← π states Since thenl m ← π transitions do not involve the half-occupied orbital, they are present in ethylene as well. Similar to the π ∗ ← π transitions (see Eq. 2.1 in Sec. 2.7), the coupling of an unpaired electron with the nl m ← π excitation results in one low-spin quartet and two doublet states: 4 Ψ= ( απ )(αn )(βnl m )+(απ )(βn )(αnl m )+(βπ )(αn )(αnl m ) 2 Ψ 1 =2(απ )(βn )(αnl m )− (βπ )(αn )(αnl m )− (απ )(αn )(βnl m ) 2 Ψ 2 =(βπ )(αn )(αnl m )− (απ )(αn )(βnl m ) (2.3) and EOM-SF-CCSD should be used instead of EOM-EE-CCSD to reduce spin contami- nation. In the case ofnl m ← π excitations, however, application of the EOM-SF-CCSD method is problematic, because usually Rydberg states can only be described as an excitation to a linear combination of several diffuse Hartree-Fock molecular orbitals. Nevertheless, we attempted EOM-SF-CCSD calculations for the 3s← π states. The “3s” orbital in the high-spin quartet reference(π )(π ∗ )(3s) was taken to be the molecu- lar orbital with the largest contribution to the3s state calculated by the EOM-EE-CCSD 43 method (see Sec. 2.2). As can be seen from Tables 2.1-2.5, the difference between the EOM-SF-CCSD and EOM-EE-CCSD excitation energies for the quartet state is less than 0.1 eV , and for the two-configurational doublet it is less than 0.2 eV . Contrary to the π ∗ ← π states, of the two doublets the two-configurational one is lower in energy. The high excitation energy for the π ∗ ← π two-configurational doublet is due to the Coulomb repulsion between an electron on aπ ∗ orbital and the unpaired electron. In the case of3s← π transitions, the determining factor is the repulsion between the unpaired electron and an electron on the π orbital rather than the diffuse 3s electron. Since the three-configurational doublet has larger contribution of determinants in which the elec- trons on theπ andn orbitals have opposite spin, the Coulomb electron repulsion in this doublet is higher than in the two-configurational one. For similar reasons, the excitation energy for the quartet3s← π state is smaller than that for the corresponding doublets. Thus, one may expect the same behavior for othernl m ← π Rydberg states. vinyl propen-1-yl propen-2-yl 1-buten-2-yl 2-buten-2-yl 6.5 7.0 7.5 8.0 2 (3px) 2 (3s) 2 (3s) Excitationenergy, eV 4 (3pz) 4 (3px) 4 (3py) 4 (3s) Figure 2.16: Changes in thenl m ← π vertical excitation energies. 44 Figure 2.17: The EOM-SF-CCSD lowest excited state (n← π ) optimized geometries of the vinyl, propenyl, and butenyl radicals, and the molecular orientation in Cartesian coordinate system. For 1-buten-2-yl, theOXY plane is the C 1 C 2 C 3 plane. The changes in the nl m ← π excitation energies are shown in Fig. 2.16. The IPs for ionization from the π orbital resulting in the triplet cation state of vinyl, propen-1- yl, and propen-2-yl are 10.53 eV , 9.78 eV , and 9.76 eV , respectively. Note that the IPs for ionization resulting in the triplet cation state must be used in Eq. (2.2) to estimate excitation energies for the quartet nl m ← π states. Interestingly, the quantum defect δ for the 3p x ← π quartet state of vinyl is large (0.70), and decreases sharply in propen- 1-yl (δ = 0.55) and in propen-2-yl (δ =0.54), very similar to the 3p x ← n states (see Sec. 2.8). The vertical excitation energies for the triplet 3p x ← π , 3p y ← π , and 3p z ← π transitions in ethylene are 7.95 eV , 7.94 eV , and 8.21 eV , respectively, and the ionization potential (IP) is 10.56 eV . The vertical excitation energies for the 3p← π transitions in ethylene were calculated at the EOM-EE-CCSD/6-311(2+,2+)G** level of theory (spherical functions were used). The orientation of the molecule was as follows: the 45 carbon atoms are on thex axis,xy is the plane of the molecule, and thez axis is perpen- dicular to the plane. The ionization potential was calculated at the EOM-IP-CCSD/6- 311(2+,2+)G** level of theory. This gives quantum defectδ = 0.72 for the in-plane3p components, which is almost the same as δ for the in-plane 3p components of the methyl radical. This indicates similar cation core positive charge distributions in ethylene and methyl. The calculated quartet3p x ← π excitation energy for vinyl is the same as the triplet3p x ← π excitation energy for ethylene (7.95 eV). Also, the IP for the ionization of vinyl resulting in the triplet cation state, 10.53 eV , is very close to the IP of ethylene, 10.56 eV . However, the IP for the ionization resulting in the singlet cation state of vinyl is higher, 11.30 eV . The reason for the higher-multiplicity cation and the 3p x ← π excited state of vinyl being similar to those of ethylene is also the reduced Coulomb repulsion between an electron on the half-filled orbital and the remaining electron on theπ bonding orbital. Note that although this reduction in the Coulomb repulsion lowers the quartet 3p x ← π state of vinyl relative to the corresponding doublets, the penetration of the Rydberg electron to the cation core (δ = 0.70) is less than in the case of the3p x ← n excitation (δ = 0.76). 2.10 Conclusions We presented the results of high-level ab initio calculations of the ground and first excited state equilibrium geometries, vertical excitation energies, oscillator strengths, and valence states’ permanent dipole moments for the vinyl, propen-1-yl, propen-2-yl, 1-buten-2-yl, and trans-2-buten-2-yl radicals. The electronic spectrum of these species is very dense, and is dominated by Rydberg transitions. The electronic structure and energetics of the valence low-lying excited states are similar in all the radicals, except for then← σ states, which change substantially with the number of adjacentσ -bonds. 46 The excitation energies for the Rydberg states, however, depend strongly on the size and the structure of the radicals. The major factor responsible for the changes in the Ryd- berg states’ energies is the strong dependence of the ionization potentials on the size and geometric structure. Our results suggest that the quantum defectδ for the3p x ,3p y , and 3p z Rydberg states is determined by the geometry dependent charge distribution within the cation core. We interpret the elongation of the in-plane C-H bonds in the ground state propen- 1-yl, propen-2-yl, and trans-2-buten-2-yl radicals as indication of the hyperconjugative charge transfer from the in-plane C-H bonds to the half-filled orbital. 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[42] G. Herzberg. volume III. van Nostrand Reinhold: New York, 1966. [43] E.D. Glendening, J.K. Badenhoop, A.E. Reed, J.E. Carpenter, J.A. Bohmann, C.M. Morales, and F. Weinhold. Theoretical Chemistry Institute, University of Wiscon- sin, Madison, WI, 2001. [44] I. Mart´ ın, A.M. Velasco, and C. Lav´ ın. 86:59–66, 2002. 50 Chapter 3 Diazomethane I: investigations of the electronic Rydberg states: assignments and interactions 3.1 Introduction Diazo compounds serve as sources of carbenes in organic synthesis and participate in alkylation reactions. 1 Diazomethane is an important source of methylene and its pho- tochemistry is relevant to understanding the chemistry in atmospheres rich in N2 and methane, such as in Titan, Triton, and Pluto. 2 Not surprising, elucidating its excitation and fragmentation mechanisms has attracted interest since 1933. 3 In spite of this inter- est, its excited electronic states have not been fully assigned. The reason is, of course, the instability of diazomethane and its explosive nature. The CN bond is weak: the dis- sociation energy to triplet methylene is < 38 kcal/mol and to singlet methylene it is < 47 kcal/mole 1, 4 and the molecule is unstable. The goal of the present paper is to char- acterize the electronic structure and photophysics of diazomethane in the region of its lowest Rydberg states by using a combination of high-level electronic structure calcu- lations and photoelectron and photofragment ion imaging experiments. This would aid, among others, in developing sensitive spectroscopic diagnostics for diazomethane via its 51 Rydberg states. Therefore, our focus is also on identifying electronic state interactions and estimating dissociation rates. The electronic configuration of ground state diazomethane (1 1 A 1 ) is: [core + low-lying] 16 (7a 1 ) 2 (2b 2 ) 2 (2b 1 ) 2 (3b 2 ) 0 (8a 1 ) 0 (3b 1 ) 0 = [core + low-lying] 16 (σ ) 2 (π σ ) 2 (π ) 2 (π ∗ σ ) 0 (σ ∗ ) 0 (π ∗ ) 0 where “low-lying” refers to the 1s core orbitals, two σ CN orbitals, as well as one extended sigma-bonding and two π -type molecular orbitals (which are not involved in excitations). Relevant molecular orbitals are shown in Fig. 3.1. CH 2 N 2 lies in the yz plane with the z axis coinciding with the C 2 symmetry axis (CNN bond). The lowest valence excitations promote an electron from the π orbital to the π * or π ∗ σ orbital. 5 Because the ionization energy (IE) of diazomethane is low (experimentally determined at 9.00 eV), 6 the 3s, 3p and 3d Rydberg states are located in the same region as the valence states. 5 The ion’s excited states are at 14.13 and 15.13 eV , and the parent ion is quite stable. 5 The most intense electronic transition of diazomethane is 2 1 A 1 ← 1 1 A 1 (π ∗ ← π ). The antibonding character of the target molecular orbital makes this state unbound, dis- sociating adiabatically to N 2 ( 1 Σ + g ) + CH 2 (˜ c 1 A 1 ), in agreement with Herzberg’s observa- tions. 7 As of yet, the Rydberg states have not been well characterized. An earlier theoretical study estimated them to lie vertically at 5.89 (3s), 6.65 - 6.87 (3p) and 7.48 – 7.68 (3d) eV . 5 However, the proximity of valence states can lead to interactions with core electrons, giving rise to valence-Rydberg character and affecting quantum defects. A more recent study 8 suggested that all states at 5.31 - 7.29 eV contain some Rydberg character, and that the 2 1 A 1 (π ∗ ← π ) state, found at 5.53 eV vertically is mostly valence and has the largest oscillator strength. 8 52 z y 3p x 3p y 3p z π π σ * π* N C H H (2b ) 1 (3b ) 2 (3b ) 1 1 N2 Figure 3.1: Molecular orbitals relevant to ground and excited electronic states of CH 2 N 2 . Absorption spectra identify the strongest absorption at 260 - 190 nm (4.77 - 6.53 eV), a region that includes the intense2 1 A 1 ← 1 1 A 1 (π ∗ ← π ) transition. 9, 10 The absorption exhibits diffuse structures at 230, 218, and 214 nm, which may belong to several mixed valence/Rydberg states. At yet shorter wavelengths (154 - 193 nm), Merer observed in absorption a series of perpendicular bands, which he characterized as having 1 B 1 and 1 B 2 symmetry. 11 Some had a resolved K structure, whereas others were more diffuse. Specifically, in the region near 190 nm he assigned the bands to transitions to three Ryd- berg states, most probably 3p, which he denoted D( 1 B 1 ), E( 1 B 2 ), and F( 1 B 1 ). He also identified perturbations among some of these states, and attributed them to Coriolis cou- pling. In addition, two intense and diffuse band systems, which do not fit any Rydberg series, are centered at 175 and 140 nm and may indicate mixed states containing large contribution of valenceπ ∗ ← π excitations. 8, 11 53 To date, no molecular beam studies of diazomethane have been reported because of difficulties in preparing and delivering it intact into the interaction chamber. In an attempt to develop efficient sources of carbenes, we have recently adapted the tradi- tional MNNG (N-methyl-N’nitro-N-nitrosoguanidine) + KOH preparation method of diazomethane 12, 13 for work in molecular beams. We also modified our inlet and pulsed- nozzle systems for stable and safe delivery. We have used 2 + 1 resonance enhanced multiphoton ionization (REMPI) comple- mented by photoelectron imaging of the excited states to characterize Rydberg states in the region around 190 nm previously studied by Merer. 8, 11 We have obtained REMPI and photoelectron spectra and identified intense, K-resolved transitions not seen in the one-photon absorption experiments of Merer. We assign them to the 2 1 A 2 ← 1 1 A 1 (3p y ← π ) Rydberg transition allowed only in two-photon absorption. Moreover, we obtain for the first time photoelectron spectra from excited states of diazomethane. These spectra reveal interactions between electronic states. The accompanying high- level electronic structure calculations reported here determine ground, Rydberg, and ionic state geometries and vertical transition energies. Combined with experiments, they enable us to characterize the nature of the excited electronic states and their interactions. Our results show that the observed transitions in the region51,000− 58,900 cm − 1 (6.32-7.30 eV) can be assigned to three electronic Rydberg states: the 2 1 A 2 (3p y ← π ) state, which can be excited by a two-photon transition, and the 2 1 B 1 (3p z ← π ) and 3 1 A 1 (3p x ← π ) states, which can be accessed by both one- and two-photon transitions. We show that specific vibronic levels of the Rydberg 2 1 A 2 (3p y ) and2 1 B 1 (3p z ← π ) states interact due to accidental resonances. In addition, we show that the Rydberg 3 1 A 1 (3p x ) wavefunction is strongly mixed with a dissociative valence 2 1 A 1 (π ∗ ← π ) state. We propose that the one-photon absorptions seen by Merer in this wavelength region are due 54 primarily to vibrational progressions in the 2 1 B 1 (3p z ← π ) state, and that some levels are mixed with the 2 1 A 2 (3p y ← π ) state, which is dark in one-photon excitation. In future work, we will report assignments of other vibronic levels of the excited Rydberg states of diazomethane and its isotopologs and compare them to vibrational states of the ground state ion. The complete vibronic analysis confirms the assignments proposed in this paper. 3.2 Computational details The equilibrium geometry and vibrational frequencies of neutral CH 2 N 2 were calculated by CCSD(T), 14, 15 using the cc-pVTZ basis, 16 and by B3LYP, 17 using the 6-311G(2df,p) basis. Geometry and frequencies of the ground state of the cation were calculated by CCSD(T)/cc-pVTZ, using the unrestricted (UHF) reference. Vertical excitation energies from the ground state were calculated using EOM-CCSD27,28/6-311(3+,+)G* at the B3LYP optimized geometry. The basis sets were derived from the polarized split – valence 6-311G(d,p) basis by adding additional polarization and diffuse functions. 18, 19 The error bars 20 for equilibrium geometries for CCSD(T)/cc-pVTZ are much lower than the geometrical changes observed in this study. Nonsystematic errors for bond lengths are on the order of about 0.002 ˚ A. CCSD(T) underestimates bond angles by about 0.5 ◦ , and has a nonsystematic error of about 1 ◦ . EOM-CCSD excitation energies are accurate to within 0.1− 0.3 eV . 21 A basis set with adequate diffuse functions is necessary for states with Rydberg character. As in our previous work, 22 the assignment of valence and Rydberg character to the excited states was based on: (i) the symmetry of the transition, (ii) leading EOM-CCSD amplitudes and the character of the corresponding HF orbitals, and (iii) the second momentshX 2 i,hY 2 i, andhZ 2 i of the EOM-CCSD electron density. The character of 55 the HF orbitals was determined using the Molden interface. 23 All optimizations, fre- quencies, and excited state calculations were performed using the Q-Chem 24 and ACES II 25 electronic structure programs. Franck-Condon factors were modeled using the PES4 program. 26 3.3 Computational results N N C H H 1.292 1.298 1.362 1.132 1.139 1.115 1.077 1.071 1.086 124.07 125.00 127.80 N N C H H 1.398 1.363 1.354 1.130 1.112 1.114 1.084 1.083 1.089 128.64 129.91 124.74 CNN : 160.55 HCNN: 92.7 3p ( ): x C s Figure 3.2: Left panel: ground state equilibrium structures ( ˚ A and deg) of CH 2 N 2 for: the neutral (1 1 A 1 ) at CCSD(T)/cc-pVTZ (regular print) and B3LYP/ 6-311G(2df,p) (ital- ics) and for the cation 1 2 B 1 (∞← π ) at CCSD/6-311G** (underlined). The correspond- ing nuclear repulsion energies are: 61.280112, 61.514227, and 61.118198 hartrees, respectively. Right panel: excited state equilibrium structures for the 2 1 A 2 ← 1 1 A 1 (3p y ← π ), 2 1 B 1 ← 1 1 A 1 (3p z ← π ), and 3 1 A 1 ← 1 1 A 1 (3p x ← π ) states showed in normal , italics, and underlined print, respectively. CNN and HCNN refer to respective angle and dihedral angle for the C s 3 1 A 1 ← 1 1 A 1 (3p x ← π ) state, which is the only one with other than C 2v symmetry. The corresponding nuclear repulsion energies are: 60.705257, 59.502297, and 60.715012 hartrees. Experimental parameters of ground state: CN length: 1.300 ˚ A; NN length: 1.139 ˚ A; CH length: 1.075 ˚ A; HCH angle: 126.0 ◦ . 3.3.1 Equilibrium geometries and ionization energies The equilibrium geometries of neutral diazomethane in its ground (1 1 A 1 ) and Rydberg 3p excited states, as well as of the cation in its ground (1 2 B 2 ) state, are shown in Fig. 3.2. Both the CCSD(T) and B3LYP optimized geometries agree with experimentally deter- mined values, 27 which are also summarized in this figure. The first ionization removes 56 an electron from the highest-occupied, out-of-plane π orbital (Fig. 3.1), with vertical IE of 8.95 eV , as calculated by EOM-IP-CCSD 28–30 with 6-311G(2df,p) basis set. The experimental vertical IE is 9.00 eV , 6 and the adiabatic IE calculated at CCSD(T)/cc- pVTZ geometries and corrected by zero point energy is 8.84 eV . The second ionization energy, which corresponds to removing an electron from theπ ∗ sigma orbital (Fig. 3.1) in the plane of the molecule, is much higher and equals 14.00 eV vertically, at the EOM- IP-CCSD level. Calculated structural changes resulting from the first ionization are consistent with removing an electron from a bonding π orbital. The CN bond length increases substantially (by 0.064 ) due to a decrease in bond order. At the same time, the NN bond length slightly contracts (by 0.024 ) due to the antibonding nature of theπ orbital along the NN bond. The increase in HCH angle relative to the neutral (by 2.8 ◦ ) is due to depleted electron density along the CN bond, which allows the two CH bonds to move away from each other to minimize electrostatic repulsion. The molecule has C 2v symmetry in both the neutral and the cation ground states. 3.3.2 Excited electronic states The vertical excitation energies, oscillator strengths, transition dipole moments, and second moments of the excited states of CH 2 N 2 are listed in Table 3.3.2. The excitation energies and oscillator strengths are also plotted as a stick spectrum in Fig. 3.3. In the C 2v point group, states of A 2 symmetry have zero oscillator strength in one- photon excitation; these are depicted by hollow bars. The electronic spectrum is fairly dense in the 5.00 – 8.00 eV region and dominated by Rydberg transitions. All the calculated singlet states derive from excitations from the highest occupiedπ orbital. The lowest singlet excited state, which is well separated from the others and lies at 3.21 eV , corresponds to the valence 1 1 A 2 ← 1 1 A 1 (π ∗ ← π ) transition. In one-photon 57 3 4 5 7 6 8 Excitation energy (eV) 0.00 0.05 0.10 0.15 0.20 0.25 π σ * π π * π 3s π 3p x π 3p y π 3p z π 2 3d z π 3d yz π 4p x π 3d xz π 4s π Oscillator strength Figure 3.3: Bars showing calculated vertical excitation energies of CH 2 N 2 . Allowed and forbidded transitions are indicated by filled and hollow bars, respectively. Table 3.1: Vertical excitation energies (Δ E vert , eV), oscillator strengths (f L ), dipole strengths (μ 2 tr , a.u.), and changes in second dipole moment of charge distributions rela- tive to the ground stateΔ hX 2 , (a.u.) 2 for the excited states of CH 2 N 2 at EOM-CCSD/6- 311(3+,+)G* State Δ E vert f L μ 2 tr Δ hX 2 i Δ hY 2 i Δ hZ 2 i 1 1 A 2 (π →π ∗ σ ) 3.21 0 0 1 -2 1 1 1 B 1 (π → 3s) 5.33 0.0200 0.1186 15 10 6 2 1 A 1 (π →π ∗ ) 5.85 0.2100 1.4904 5 16 14 2 1 A 2 (π → 3p y ) 6.35 0 0 39 10 7 2 1 B 1 (π → 3p z ) 6.39 0.0100 0.0644 13 11 41 3 1 B 1 (π → 3d 2 z ) 7.08 0.0100 0.0431 24 22 74 3 1 A 1 (π → 3p x ) 7.15 0.0700 0.3821 23 69 16 3 1 A 2 (π → 3d yz ) 7.23 0 0 53 15 48 4 1 B 1 (π → 4s) 7.34 0.0010 0.0071 75 76 33 4 1 A 1 (π → 3d xz ) 7.36 0.0001 0.0005 22 68 63 5 1 A 1 (π → 4p x ) 7.77 0.0100 0.0582 69 210 82 At the B3LYP/6-311G(2df,p) optimized geometry; E CCSD = -148.426979 hartree. absorption,this state was observed as a weak, broad band centered at 3.14 eV . This opti- cally forbidden transition becomes weakly allowed by vibronic couplings with other electronic states. 58 In the5.00− 6.00 eV region, we found two transitions: the Rydberg 1 1 B 1 ← 1 1 A 1 (3s ← π ) transition at 5.33 eV , and the valence 2 1 A 1 ← 1 1 A 1 (π ∗ ← π ) at 5.85 eV . In agree- ment with the antibonding valence character of the upper state, the latter transition has oscillator strength an order of magnitude higher than the Rydberg’s (oscillator strength 0.21 compared to 0.02) due to high spatial overlap between the π and π ∗ orbitals. In this region (5.50 - 7.00 eV) Rabalais assigned a single intense, broad band centered at 5.70 eV based on known data, 31 but diffuse structures have also been reported. 9, 10 The neighboring Rydberg state is likely hidden under the intense valence band. Wavenumber (cm ) -1 51000 52000 53000 54000 55000 56000 57000 58000 Excitation energy (eV) 6.50 6.75 7.00 7.25 Ion signal 3p x π 3p y π 3p z π 3d z 2 π 3d yz π Figure 3.4: Survey2+1 REMPI spectrum in the6.32− 7.30 eV (51,000-58,900 cm − 1 ) region. The empty bars indicate the calculated values of vertical excitation energies. Between 6.00 and 7.00 eV our calculations predict only two electronic transitions: the 2 1 A 2 ← 1 1 A 1 (3p y ← π ) and 2 1 B 1 ← 1 1 A 1 (3p z ← π ) at vertical excitation energies 6.34 and 6.38 eV , respectively. In the region7.00− 7.25 eV , three electronic transitions have been calculated (Table 3.3.2): 3 1 B 1 ← 1 1 A 1 (3d 2 z ← π ) (vertical excitation energy 7.08 eV), 3 1 A 1 ← 1 1 A 1 (3p x ← π ) (7.15 eV), and 3 1 A 2 ← 1 1 A 1 (3d yz ← π ) (7.23 eV). 59 The latter has zero oscillator strength due to symmetry and could only borrow intensity. The 3 1 B 1 ← 1 1 A 1 (3d 2 z ← π ) transition has very low oscillator strength; also, its pure Rydberg character does not imply any broadening. The calculated vertical excitation energies, superimposed on the2+1 REMPI spectrum, are shown in Fig. 3.4. The 3 1 A 1 ← 1 1 A 1 (3p x ← π ) Rydberg transition has oscillator strength about one order of magnitude greater than transitions to the other 3p states, due to mixing with the valence 1 1 A 2 ← 1 1 A 1 (π ∗ ← π ) transition of the same symmetry. This is confirmed by the composition of its EOM-CCSD wavefunction, which contains a large contribution from the π ∗ ← π configuration. Both the intensity and diffuse nature of the two peaks observed in this region (see below) suggest assignment of this band as the 3 1 A 1 ← 1 1 A 1 (3p x ← π ) Rydberg transition. Table 3.2: Calculated vertical and adiabatic excitation energies (eV) and quantum defects (δ and corresponding experimental values. State Δ E vert Δ E ad Δ E vert− ad δ calc δ exp 3 1 A 1 (π → 3p x ) 7.15 6.91 0.24 0.25 0.36 2 1 A 2 (π → 3p y ) 6.35 6.24 0.11 0.71 0.68 2 1 B 1 (π → 3p z ) 6.39 6.30 0.09 0.69 0.65 1 2 B 1 cation 8.95 8.85 0.10 a Measured at 52,227 cm − 1 (K’ = 0← K” = 0 and K’ = 1← K” = 1). b Measured at 52,679 cm − 1 (K’ = 0← K” = 1). c Measured at 56,898 cm − 1 . d Ref. 6. We also report vertical and adiabatic excitation energies for the Rydberg 3p states and the cation ground state (1 2 B 1 ) and compare them with the experimental results (Table 3.3.2). The 3 1 A 1 ← 1 1 A 1 (3p x ← π ) transition shows the largest difference between its vertical and adiabatic excitation energies (0.24 eV), whereas the 2 1 A 2 ← 60 1 1 A 1 (3p y ← π ) and 2 1 B 1 ← 1 1 A 1 (3p z ← π ) transitions exhibit much smaller differ- ences (∼ 0.1 eV) and are within 0.02 eV of each other. The magnitude of the vertical- adiabatic relaxation in these states is close to that in the cation, supporting their descrip- tion as pure, cation-like Rydberg states. The 3 1 A 1 ← 1 1 A 1 (3p x ← π ) transition’s large difference indicates strong Rydberg/valence mixing. Referring to the equilibrium geometries of the states shown in Fig. 3.2, we find that the 2 1 A 2 (3p y ← π ) and 2 1 B 1 (3p z ← π ) states are more similar in structure to the 1 2 B 1 cation than to the neutral; the 3 1 A 1 (3p x ) state, in contrast, is different than both the cation and the other two 3p states. Whereas the 1 2 B 1 cation and the 2 1 A 2 , and 2 1 B 1 neutral excited states retain the C 2v symmetry of the neutral ground state, the 3 1 A 1 (3p x ← π ) state has C s equilibrium structure. The CN bond length increases and the CNN bond is no longer linear (161 ◦ , Fig. 3.2). The plane of the CH 2 sp2 center is also broken by about 3 ◦ . This twisted geometry reflects the partial population of the strongly antibondingπ ∗ orbital. The 3p y and 3p z orbitals in diazomethane are oriented to maximize overlap with the positive centers of the nuclear charge distribution. According to natural bond order (NBO) analysis, 32 about half of the +1 charge of the nuclear core is accommodated by the hydrogens. The lobes of the 3p y orbital have highest density on top of the hydrogen atoms (Fig. 3.1). Donation into the electron-deficient CH bonds leads to a contracted CH bond length relative to the cation, and a larger HCH bond angle (Fig. 3.2). A similar effect along the CNN bonds is prevented by symmetry. In contrast, the 3p z orbital has significant density along the CNN axis. The resulting electronic donation leads to contracted CN and NN bond lengths relative to the cation, and a smaller HCH angle to minimize electrostatic repulsion. Similar effects have been seen in the electronic structure of several vinyl radicals. 22 61 3.4 Discussion Promotion of an electron from the highest occupiedπ molecular orbital of CH 2 N 2 (Fig. 3.1) into 3p atomic-like orbitals generates the 3 1 A 1 (3p x ← π ), 2 1 A 2 (3p y ← π ), and 2 1 B 1 (3p z ← π ) Rydberg states. These states correlate with the ground state of the ion, 1 2 B 1 (∞← π ). For small molecules, the excitation energies of the Rydberg states can often be approximated by the Rydberg formula (in cm − 1 ): 7 E Ryd =IE− 109,737.3 (n− δ ) 2 (3.1) where E Ryd is the excitation energy of the Rydberg state (in cm − 1 ), IE is the adi- abatic ionization energy (in cm − 1 ), n is the principal quantum number, and δ is the quantum defect. The calculated and experimental quantum defects are listed in Table 3.3.2. Typical values for n = 3 Rydberg states are∼ 0.9− 1.2, 0.4− 0.7, and∼ 0.1 for s, p, and d states, respectively. The calculated and experimental quantum defects for the 3 1 A 1 (3p x ← π ) state fall outside this region, and differ from the values for the other two Rydberg states, in agreement with its partial valence composition. Below we discuss first the 2 1 A 2 (3p y ← π ) and 2 1 B 1 (3p z ← π ) Rydberg states, followed by a discussion of the 3 1 A 1 (3p x ← π ) state. 3.4.1 The 2 1 A 2 (3p y ← π ) and 2 1 B 1 (3p z ← π ) Rydberg states and their interaction The internal energy (E int ) of the ion corresponding to each measured photoelectron peak is obtained from the peak position of the eKE distribution, eKE: E int =3hν − eKE− IE (3.2) 62 53000 Wavenumber (cm ) -1 Ion signal 54000 55000 52000 D F E C Figure 3.5: 2+1 REMPI spectrum of diazomethane in the region of excitations to the 2 1 A 2 (3p y ← π ) and 2 1 B 1 (3p z ← π ) states obtained by measuring m/e = 42 as a function of excitation energy. The laser wavelength increment was 0.005 nm. See the text for details of the C-F bands. In the 2 + 1 REMPI spectrum of CH 2 N 2 shown in Fig. 3.5, the first group of three peaks (group C at52,227− 52,295 cm − 1 ) was not observed in the one-photon absorption spectrum reported by Merer. We assign this group to the origin band of the 2 1 A 2 ← 1 1 A 1 transition for the following reasons. According to electric-dipole transition selection rules in C 2v symmetry, the A 2 ← A 1 transition is optically forbidden in one-photon but allowed in two-photon excitation. The onset of this transition is close to the calculated one and its quantum defect (δ = 0.68) is typical of a Rydberg p state. The Rydberg nature of the state is also confirmed by the photoelectron image (Fig. 3.6) obtained at λ = 382.69 nm (52,262 cm − 1 ). From the correpsonding eKE distribution shown in Fig. 3.6, we conclude that the CH 2 N + 2 ions are generated in the vibrational ground state. This suggests a similarity in geometries of the neutral excited state and ion ground state, which results in a propensity for ionization via the diagonal 1 2 B 1 ← 2 1 A 2 (∞← 3p y ) 0 0 0 transition. The internal energy of the ion obtained by excitation through the 0 0 0 band is∼ 60 cm − 1 , indicating little rotational excitation in the neutral parent. Similar 63 0.2 0.4 0.6 0.8 1.0 P(E) 2000 0 4000 6000 8000 10000 eKE (cm ) -1 Figure 3.6: Photoelectron image and the corresponding eKE distributions obtained at excitation wavelengthλ =382.69 nm (2hν =52,262 cm − 1 ; middle peak of band C). photoelectron kinetic energy distributions are obtained for the other two peaks in this triad. Additional support for this assignment is obtained from the ab initio calculations, which show the presence of only two electronic states in this region (Table 3.3.2): the 2 1 A 2 (3p y ← π ) state at 6.34 eV (51,213 cm − 1 ) and the 2 1 B 1 (3p z ← π ) state at 6.38 eV (51,535 cm − 1 ). The 1 B 1 ← 1 A 1 transition is optically allowed in one- photon excitation but the the 1 A 2 ← 1 A 1 is not, and therefore the observed band located at 52,227-52,295 cm − 1 , which has not been seen in one-photon absorption, is assigned as the band origin of 2 1 A 2 ← 1 1 A 1 (3p y ← π ) transition. 64 The three components of the band are attributed to resolved K structure. CH 2 N 2 in its neutral ground state (1 1 A 1 ) is a near-symmetric prolate top (A” = 9.112 cm − 1 , B”=0.377 cm − 1 , and C”=0.362 cm − 1 ), 7 and according to calculations its 3p Rydberg states also have C 2v symmetry (except3 1 A 1 (3p x ← π )).In C 2v transitions to the vibronic bands of A 1 and A 2 symmetry are expected to be governed by Δ K = 0,± 2 selection rules with a 4A” spacing between K bands. 33 Transitions to B 1 and B 2 vibronic bands are governed by Δ K =± 1 selection rules, with a 2A” spacing. The separation of∼ 35 cm − 1 between the three peaks in this groups, which is approximately equal to 4A”, indicates that the rotational transition is governed by Δ K= 0,± 2 selection rules and that the excited state is of A symmetry. All the observed transitions originate in K = 0 and 1, the only ones significantly populated in the supersonic expansion. The first peak, in order of increasing energy, arises from the K’ = 0← K” = 0 and K’ = 1← K” = 1 transitions. The next two components of this triad correspond to the K’ = 2← K” = 0, and K’ = 3← K” = 1 transitions, respectively. In this triad, the outermost bands are more intense because they include transitions from K” = 1, which according to nuclear-spin statistics are three times more intense than those from K” = 0. The next three groups of bands at 52,513− 52,541 cm − 1 , 52,574 cm − 1 , and 52,679− 52,722 cm − 1 were observed in one-photon absorption and assigned by Merer as transitions to three different electronic states denoted as D( 1 B 1 ), E( 1 B 2 ), and F( 1 B 1 ), respectively. According to the ab initio calculations, there exist only two electronic states (2 1 B 1 (3p z ← π ) and 2 1 A 2 (3p y ← π )) in this energy region. The absence of other states within the error bars of the method make this assignment conclusive. The photoelectron spectra can aid in assigning the spectra. The bands with partially resolved rotational structure at 52,513-52,541 cm − 1 (D band) and 52,679-52,722 cm − 1 (F band) are assigned as mixed bands composed of the 9 1 0 transition to the2 1 A 2 (3p y ← 65 2000 0 4000 6000 8000 10000 0.2 0.4 0.6 0.8 1.0 eKE (cm ) -1 P(E) Figure 3.7: Photoelectron image and the corresponding eKE distributions obtained at excitation wavelength 380.75 nm (52,528 cm − 1 ; middle peak of band D). π ) state and the band origin of the2 1 B 1 (3p z ← π )← 1 1 A 1 transition. This assignment is based on the two-peak structure in the eKE distributions shown in Figs. 3.7 and 3.8. Comparing the two eKE distributions, which have equal energy separations but different peak heights, suggests that two electronic states are coupled. The difference in anisotropy of the strong inner and outer rings also suggests that the photoelectrons corresponding to these rings may be ejected from different electronic states. In Merer’s absorption experiments, the optically forbidden 9 1 0 band of the2 1 A 2 (3p y ← π )← 1 1 A 1 transition becomes allowed by vibronic coupling to the2 1 B 1 (3p z ← π ) state mediated by the non-totally symmetricν (b2) vibration. B 1 symmetry in the 2 1 A 2 state is obtained for A 2 (electronic symmetry)× b 2 (vibrational symmetry) vibronic bands. 66 2000 0 4000 6000 8000 10000 0.2 0.4 0.6 0.8 1.0 eKE (cm ) -1 P(E) Figure 3.8: Photoelectron image and the corresponding eKE distributions obtained at excitation wavelength 379.50 nm (52,700 cm − 1 ; middle peak of band F). Because the eKE distributions in Figs. 3.7 and 3.8 display only two major peaks, we use a two state approximation to obtain the energy of the deperturbed states and the coupling matrix elements. The two molecular eigenstates|ψ + i and|ψ − i are expressed as linear combinations: |ψ + i =α φ 1 n 0 0 0 (a 1 ), 1 B 1 (3p z ) oE +β φ 2 n 9 1 0 (b 2 ), 1 A 2 (3p y ) oE , (3.3) |ψ − i =− β φ 1 n 0 0 0 (a 1 ), 1 B 1 (3p z ) oE +α φ 2 n 9 1 0 (b 2 ), 1 A 2 (3p y ) oE . (3.4) 67 We obtain|α 2 | and|β 2 |from the relative peak heights in the two photoelectron dis- tributions, giving an average ratio of |β | 2 |α | 2 = 1.82. Using|α 2 | +|β 2 | = 1, we obtain α = 0.6 and β = 0.8, and putting these values into the two-state coupled equations 34 , we finally obtain: E 1 = 52,638 cm − 1 ; E 2 = 52,590 cm − 1 ; and V12∼ 83 cm − 1 , where E 1 ,2 are the deperturbed energies of the two states, and V12 is the coupling matrix ele- ment. The eKE distributions show that the D band (52,679− 52,722 cm − 1 ) has a larger contribution from the2 1 B 1 (3p z ← π ) state, and therefore E 1 = 52,638 cm − 1 is assigned as the adiabatic origin of this state. Its associated quantum defect,δ = 0.65, is typical of a Rydberg p state. Table 3.3: CCSD(T)/cc-pVTZ harmonic vibrational frequencies for cation ground state (1 2 B 1 ), (cm − 1 ) mode assignment symmetry frequency ν + 1 CH 2 symmetric stretching a 1 3164 ν + 2 NN stretching a 1 2199 ν + 3 CH 2 symmetric bending a 1 1432 ν + 4 CN stretching a 1 1001 ν + 5 CNN bending (out-of-plane) b 1 440 ν + 6 CH 2 wagging b 1 712 ν + 7 CH 2 asymmetric stretching b 2 3311 ν + 8 CH 2 rocking b 2 1133 ν + 9 CNN bending (in-plane) b 2 377 The peaks in the eKE distributions shown in Figs. 3.7 and 3.8 correspond to vibra- tional levelsν + 0 ,ν + 9 , andν + 4 +nν + 9 (n = 0− 2) in the CH 2 N ion (whereν + 9 is the CNN in-plane bend andν + 4 is the CN stretch). The frequency ofν + 9 is∼ 420± 10 cm − 1 , and the frequencies ofν + 4 +nν + 9 combination modes are 1,002± 20cm − 1 , 1,404± 30 cm − 1 , and 1,808± 40 cm − 1 for n = 0, 1 and 2, respectively. The existence of a vibrational progression suggests that there is a small geometry change between the Rydberg and the 68 cation states. Calculated vibrational frequencies for the ground state of the 1 2 B 2 cation are given in Table 3.4.1 for comparison. Theν + 4 fundamental band was observed before in the He I PES with a frequency of 970± 80 cm − 1 . 6 The three K bands of each of the mixed transitions of B 1 symmetry are separated by ∼ 14 cm − 1 and∼ 21 cm − 1 (approximately 2A”). This indicates that these transitions are governed by Δ K =± 1 selection rules and confirms that the excited states have B symmetry. The contributions to the three components, in order of increasing energy, are from the K’ = 0← K” = 1, K’ = 1← K” = 0, and K’ = 2← K” = 1 transitions. 2000 0 4000 6000 8000 10000 0.2 0.4 0.6 0.8 1.0 eKE (cm ) -1 P(E) P(E) P(E) P(E) P(E) P(E) P(E) P(E) P(E) P(E) P(E) P(E) Figure 3.9: Photoelectron image and the corresponding eKE distributions obtained at excitation wavelength 380.39 nm (52,577 cm − 1 ; band E). 69 The band at 52,574 cm − 1 [E( 1 B 2 ), Fig. 3.9] is assigned as the 5 transition to 2 1 A 2 (3p y ). The single peak in the corresponding eKE is assigned as ν + 5 (b 1 ; CNN out-of- plane bend) of CH 2 N whose frequency is 318± 10 cm − 1 . The slight difference between the value reported here and that given by Merer 11 (52,598.7 cm − 1 and 52,668.6 cm − 1 for the K’ = 2← K” = 3, and K’ = 2← K” = 1 transitions, respectively) reflects the lower rotational temperature in our experiment, where only K” = 0, 1 are significantly populated. All the other bands in the region52,000− 55,000 cm − 1 are fairly sharp and have a typical triad K structure. Their photoelectron spectra display single peaks and they are assigned to unperturbed transitions to the 2 1 A 2 (3p y ← π ) state. A detailed analysis of the vibronic spectrum will be given in a separate publication. Here we note only that from the widths of the bands, we conclude that the 2 1 A 2 (3p y ← π ) state is bound and only weakly predissociative. Finally we compare briefly our interpretation of the coupled states to the one offered by Merer. As discussed above, we see evidence only for coupling of two bands, D and F, of B 1 vibronic symmetry, and no evidence for coupling in the E( 1 B 2 ) band. In contrast, Merer concluded that all three bands interact via Coriolis coupling. This conclusion was reached in analogy with the interaction of the ν “ 5 , ν “ 6 , and ν “ 9 vibrational modes of the neutral ground state of diazomethane (1 1 A 1 ), 35 which couple the b 2 level via Coriolis interaction with two b 1 levels. The difference may derive from the rotational excitation of the parent. Merer carried out his measurements at 300 and 196 K, where K levels up to K’ = 10 are populated, whereas in our molecular beam measurements (Trot ∼ 10 K) only K’≤ 2 are significantly populated.Using the Coriolis constants given by Merer, we obtain that the values of the|VFE| and|VED| Coriolis coupling matrix elements for K’ = 10 are∼ 82 and∼ 142 cm − 1 , respectively, whereas for K’ = 2 the 70 corresponding terms are 5 times smaller (16 and 28 cm − 1 , respectively). The direct coupling matrix element between the F and D state obtained in our study is|VFD|∼ 83 cm − 1 , is much higher than the|VFE| and|VED| Coriolis coupling matrix elements for K’ = 2. These considerations and the photoelectron images support our interpretation that at low rotational temperatures there is significant interaction only between the two states of 1 B 1 symmetry. 3.4.2 The 3 1 A 1 (3p x ← π ) state In the 57,000 cm − 1 region of the one-photon absorption spectrum two strong diffuse bands spaced by∼ 430± 20 cm − 1 were observed by Merer 11 . These bands did not fit the previous progressions and it was suggested that they represent more than one electronic state. As discussed above, our EOM-CCSD study of the excited states of CH 2 N 2 using the 6-311(3+,+)G* basis set reveals several states in this region (Fig. 3.4): 3 1 B 1 (3d← π ), 3 1 A 1 (3p x ← π ), and 3 1 A 2 (3d yz ← π ). The latter cannot be observed in one-photon absorption due to symmetry considerations. The 3 1 B 1 (3d← π ) state has very low oscillator strength, and its pure Rydberg character should not result in the significant broadening observed in the measurements. The transition to the 1 A 1 (3p x ) state has the largest oscillator strength among the Rydberg states in this region due to its mixing with the valence 2 1 A 1 (π ∗ ← π ) state. Therefore the 3 1 A 1 ← 1 1 A 1 (3p x ← π ) transition is the best candidate for these bands. Calculations of the geometries of the excited states show that the geometry of the 3 1 A 1 (3p x ) state differs considerably that of 2 1 A 2 (3p y ← π ), 2 1 B 1 (3p z ← π ), and the cation 1 2 B 1 (∞← π ). The 3 1 A 1 (3p x ← π ) state has an in-plane bent geometry corresponding to C s symmetry, as discussed in Section 4(b). 71 0.2 0.4 0.6 0.8 1.0 P(E) 0 eKE (cm ) -1 0 3000 6000 9000 eKE (cm ) -1 12000 15000 Figure 3.10: Photoelectron image and the corresponding eKE distributions obtained at excitation wavelength 351.09 nm (56,965 cm − 1 ). The different geometries of the neutral 3 1 A 1 (3p x ← π ) and the cation 1 2 B 1 states should give rise to a propensity for ionization via off-diagonal transitions, resulting in multiple peaks in the photoelectron spectrum as indeed in the experiment. The eKE distribution obtained at λ = 351.09 nm (56,965 cm − 1 ) has an intense band at 8,098 cm − 1 , and a progression of three peaks at 11,020, 11,992, and 12,800 cm − 1 (Figure 3.10). This progression can be assigned to excitation of the v6 (CH 2 wag) mode in the cation. The experimental frequencies for v6 and 2v6 are 808± 30 and 1,780± 70 cm − 1 , respectively, suggesting that these modes are rather anharmonic. The theoretical harmonic frequency for the v6 mode is 712± 20 cm − 1 . The strong peak at 8,089 cm − 1 can be assigned to excitation of eitherν + 1 +ν + 3 orν + 7 +ν + 3 vibrations (4,765± 190 cm − 1 ) 72 in the cation (where ν + 1 and ν + 3 are the CH 2 symmetric stretch and bend, respectively andν + 7 is the CH 2 asymmetric stretch). The eKE distributions atλ = 351.51 nm (56,898 cm − 1 ) andλ = 349.28 nm (57,261 cm − 1 ) have similar character. The diffuse nature of the bands is attributed to their strong coupling to the 2 1 A 1 (π ∗ ← π ) valence state, which lends them oscillator strength and is repulsive. The observed dissociation products, N 2 ( 1 Σ + g ) + CH 2 (˜ c 1 A 1 ), indeed correlate with a molecular state of A 1 symmetry. 3.5 Conclusions The electronic states of diazomethane in the region3.00− 8.00 eV have been obtained by ab initio calculations, and transitions in the region 6.32− 7.30 eV have been char- acterized experimentally using a combination of 2+1 REMPI spectroscopy and pho- toelectron imaging in a molecular beam. Specifically, in the region where experiments have been carried out we find that only three Rydberg 3p states are excited, the 2 1 A 2 (3p y ← π ), 2 1 B 1 (3p z ← π ), and 3 1 A 1 (3p x ← π ). The former two states are of mostly pure Rydberg character, whereas the 3 1 A 1 (3p x ← π ) state is mixed with the valence 2 1 A 1 (π ∗ ← π ) state and is strongly predissociative. We find that the ground vibrational level of the2 1 B 1 (3p z ← π ) state is mixed with the2 1 A 2 (3p y ← π )ν 9 level, which is of B 1 vibronic symmetry. The other2 1 A 2 (3p y ← π ) vibronic states exhibit pure Rydberg character and generate ions in single vibrational states. They also predissociate rather slowly. Thus, this state can serve both as a sensitive and state-specific 2-photon diag- nostic for diazomethane and as a gateway state for preparation of diazomethane ions in specific vibrational levels. The photoelectron spectra of the 3 1 A 1 (3p x ← π ) state, on the other hand, gives rise to many states of the ion. The two-photon excitation scheme used here excited efficiently vibronic levels in the 2 1 A 2 (3p y ← π ) state, whereas vibronic 73 levels of the 2 1 B 1 (3p z ← π ) state are best reached by one-photon absorption, as done by Merer. 11 The experimental and theoretical results agree very well, both in regard to excitation energies and to vibrational modes of the ion. Analysis of the photoelectron spectra of the excited states elucidates the interactions between the Rydberg electron with the ion core and reveals mixings with valence states as well. 74 3.6 Chapter 3 references [1] W. Kirmse. McGraw-Hill Book Company, 1955. [2] F. Raulin, M. Khlifi, M. Dang-Nhu, and D. Gautier. Adv. Space. Res., 12:181, 1992. [3] F.W. Kirkbride and R.G. Norrish. J. Chem. Soc., page 119, 1933. [4] H. Okabe. Wiley, New York, 1978. [5] S.P. Walch and W.A. Goddard III. J. Am. Chem. 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By comparing the normal mode frequencies of the 3p Rydberg states to those of the ground state of the neutral (1 1 A 1 ) and the cation (1 2 B 1 ), we analyze the influence of the unpaired electron in each of the 3p orbitals on the structure and vibrational motions in the Rydberg states. The strategy adopted in this work is the following. Using high level theory we cal- culate the normal mode harmonic frequencies of the target states for CH 2 N 2 , CD 2 N 2 , and CHDN 2 , and compare them to available experimental results. Some experimental frequencies for the neutral and ion ground states and the 2 1 B 1 (3p z ) Rydberg state are available in the literature, 2–5 and we complement those with new experimental data on the normal modes of the 2 1 A 2 (3p y ) Rydberg state and the ground-state cation. In our 78 previous work, 1 2+1 REMPI spectra and photoelectron VMI of the excited states of CH 2 N 2 have been used for the first time to characterize the spectrum of diazomethane in a molecular beam, and these studies are extended here to the isotopologs of dia- zomethane and to higher excitation energies (51,750− 58,500 cm − 1 ). The excellent agreement between theory and experiment allows us to present a full discussion of the influence of the 3p Rydberg electron on the vibrational frequencies of the corresponding excited states as compared to those of the ground states of the neutral and the cation. In the 2 + 1 REMPI spectrum of the 2 1 A 2 (3p y ← π ) transition of CH 2 N 2 obtained before, 1 strong K-resolved transitions not seen in one-photon absorption 4 were observed. Using a combination of experiment and theory, the upper states of the observed transitions were assigned, in order of increasing energy, to the 2 1 A 2 (3p y ← π ), 2 1 B 1 (3p z ← π ), and 3 1 A 1 (3p x ← π ) Rydberg states. Although the out-of-plane 3p x Rydberg orbital is usually the least perturbed by the molecular core, the spectrum associated with this state is found to be more perturbed than those associated with the 3p y and 3p z states whose unpaired electrons occupy in-plane orbitals. The2+1 REMPI signal for the 3 1 A 1 (3p x ← π ) state is broader and several times lower in peak intensity than that for transitions to the 2 1 A 2 (3p y ) or 2 1 B 1 (3p z ) states. This broadening is shown by ab initio calculations to result from mixing of the 3 1 A 1 (3p x ) Rydberg state with the dissociative valence 2 1 A 1 (π ∗ ← π ) state, 1 which shortens the lifetime of this state and reduced its ionization efficiency. Also, the geometry of the 3 1 A 1 (3p x ) state is quite different from the cation (C 2v ) having C s symmetry, 1 in contrast to the 2 1 A 2 (3p y ) and 2 1 B 1 (3p z ) states, which like the cation have C 2v symmetry. In addition, analyses of photoelectron kinetic energy (eKE) distributions of CH 2 N 2 indicate that the band origin of the 2 1 B 1 (3p z ) state is mixed with the 2 1 A 2 (3p y ) ν 9 79 level, which is of B 1 vibronic symmetry. 1 However, most of the other bands in its2+1 REMPI spectrum can be assigned as pure transitions to the 2 1 A 2 (3p y ) state. Figure 4.1: Two Lewis structures for diazomethane. The z-axis lies along CNN, the y-axis lies in the plane perpendicular to CNN, and the x-axis is out-of-plane. The paper is organized as follows. Section 4.2 describes the electronic structure models, and the results of calculations of geometries and vibrational frequencies of the neutral and cation ground states and the three 3p Rydberg states for the isotopologs of diazomethane. In Section 4.3, after discussing the proposed assignments, we present a detailed analysis of the structure and normal mode frequencies of Rydberg states of diazomethane and their dependence on the Rydberg electron. The main results and conclusions are summarized in Section 4.4. 4.2 Theoretical methods and results The equilibrium geometry and vibrational frequencies of neutral diazomethane were calculated by CCSD(T) 6, 7 /cc-pVTZ 8 and B3LYP 9 /6-311G(2df,p). Geometry and fre- quencies of the ground state of the cation were calculated by CCSD(T)/cc-pVTZ using the unrestricted (UHF) reference. The basis sets were derived from the polarized split- valence 6-311G(d,p) basis by adding additional polarization and diffuse functions. 10, 11 Isotope shifts for the ground-state neutral and the cation were calculated at the B3LYP/6-311G(2df,p) level. The equilibrium structures, frequencies, and isotope shifts of the 2 1 A 2 (3p y ) and 2 1 B 1 (3p z ) Rydberg states were calculated by EOM-EE-CCSD 12, 13 80 with 6-311(3+,+)G(2df). While equilibrium structures, excitation energies, and most of the skeletal frequencies are reasonably converged with just a single polarization set, additional polarization functions are crucial for out-of-plane vibrations, probably due to re-hybridization induced by those motions. For instance, the CH 2 wagging frequency in the 2 1 A 2 (3p y ) and 2 1 B 1 (3p z ) states increases by 93 and 74 cm − 1 , respectively, upon increasing polarization from 6-311(3+,+)G(d) to 6-311(3+,+)G(2df). A smaller 6-31(2+)G(d) basis was employed for (lower-symmetry) 3 1 A 1 (3p x ) state to reduce com- putational costs. Vertical excitation energies and adiabatic IE’s were calculated using EOM-CCSD/6- 311(3+,+)G* and EOM-IP-CCSD/6-311G(2df,p), 14–16 respectively, at the B3LYP/6- 311G(2df,p) optimized geometries. All optimizations, frequencies, and excited state calculations were performed using the Q-Chem 17 and ACES II 18 electronic structure programs. The Natural Bond Orbital (NBO) program 19 was employed to analyze bonding in neutral, electronically excited and ionized diazomethane. All equilibrium geometries are summarized in Table 4.2. Note that within the Born- Oppenheimer approximation, the equilibrium geometries of all isotopologs are identical. Moreover, since the symmetry of the nuclear Coulomb potential is also the same, the C 2v point group may be used for the electronic wave functions in all cases. Tables 4.2-4.2 present calculated (harmonic) and experimental (fundamental) vibra- tional frequencies for CH 2 N 2 , CD 2 N 2 and CHDN 2 , respectively. The B3LYP and CCSD(T) results for the neutral are in good agreement, which validates the B3LYP results for isotope shifts. The comparison between the theoretical and experimental results, as well as the changes in structures and frequencies induced by ionization and electronic excitation are discussed in Section 4.3. 81 Table 4.1: Calculated equilibrium structures for the ground state of the neutral and cation and for the Rydberg states of CH 2 N 2 1 1 A 1 a 3 1 A 1 (3p x ) b 2 1 A 2 (3p y ) c 2 1 B 1 (3p z ) c 1 2 B 1 d r CN ,( ˚ A) 1.298, 1.292 1.398 1.352 1.343 1.362 r NN ,( ˚ A) 1.139, 1.132 1.130 1.105 1.107 1.115 r CH ,( ˚ A) 1.071, 1.077 1.084 1.079 1.084 1.086 6 HCH ( ◦ ) 125.00, 124.07 128.64 129.96 124.76 127.80 6 NNC ( ◦ ) 180, 180 160.55 180 180 180 6 NNHC ( ◦ ) 0, 0 92.7 0 0 0 All values remain unchanged for CD 2 N 2 and CHDN 2 . E nuc =61.514227 (61.280112), 59.502297, 61.132745, 61.137542, 61.118198 a.u., respectively, at CCSD(T)/cc-pVTZ (B3LYP/6-311G(2df,p)) level. a Left values calculated using CCSD(T)/cc-pVTZ; right values using B3LYP/6-311G(2df,p). Experimental geometrical values: r CN : 1.300 ˚ A;r NN : 1.139 ˚ A;r CH : 1.077 ˚ A; 6 HCH: 126.1 ◦ ; 6 NNC: 180 ◦ ; 6 NNHC: 0 ◦ ; Ref. 20 b EOM-CCSD/6-31(2+)G(d). c EOM-CCSD/6-311(3+,+)G(2df). d CCSD(T)/cc-pVTZ. 82 Table 4.2: Transition energies and vibrational frequencies of neutral ground state, 3p Rydberg states, and cation of CH 2 N 2 1 1 A 1 3 1 A 1 2 1 A 2 2 1 B 1 1 2 B 1 exp a exp c exp d exp e exp f calc b calc calc calc calc Band origin, IE . . . 56898 52227 52628 72620, 72585 g 77665 51213 51535 71375 ν 1 (a 1 ) CH 2 sym. stretch 3077 . . . 3059 2980 3015 3185, 3230 3202 3182 3104 3164 ν 2 (a 1 ) NN stretch 2102 . . . 2062 2142 2110, 2180 g 2203, 2173 2093 2225 2208 2199 ν 3 (a 1 ) CH 2 sym. bend 1414 . . . 1463 1246 1420 1448, 1462 1367 1424 1370 1432 ν 4 (a 1 ) CN stretch 1170 . . . 969 864 985, 970 g 1214, 1196 768 1007 1007 1001 ν 5 (b 1 ) CNN o.p. bend 564 . . . 347 . . . 320 586, 578 571 448 460 440 ν 6 (b 1 ) CH 2 wag 406 . . . (623) 819 810 413, 420 526 594 767 712 ν 7 (b 2 ) CH 2 asym. stretch 3185 . . . 3370 . . . . . . 3305, 3347 3368 3311 3290 3311 ν 8 (b 2 ) CH 2 rock 1109 . . . . . . . . . . . . 1125, 1129 1134 1136 1088 1133 ν 9 (b 2 ) CNN i.p. bend 421 . . . 356 . . . 420 432, 419 431 352 446 377 All values in cm − 1 . Tentatively assigned frequencies are enclosed in parentheses. a Data from refs. ( 2, 3 ) (frequencies rounded to integer cm − 1 ; accuracy=± 2 cm − 1 ). b Harmonic frequencies, see text. For the 1 1 A 1 state, the left values calculated by using B3LYP/6- 311G(2df,p), and the right values by CCSD(T)/cc-pVTZ. For the 3 1 A 1 (3p x ) state, symmetry is lowered to C s , b 1 modes are of a” symmetry, while all others are a’. c Uncertainty± 15 cm − 1 . d Uncertainty± 5 cm − 1 . e Fundamental frequencies are from Ref. 4 relative to the determined unperturbed value of the 2 1 B 1 (3p z ) band origin. The perturbed origin for the band origin (52,690 cm − 1 ) was defined as the average value for the1← 0 and0← 1 transitions. f Frequency accuracy defined as the spacing between the rotational lines of cation is± 50 cm − 1 . g Data from Ref. 5. Frequency accuracy is± 80 cm − 1 . 83 Table 4.3: Transition energies and vibrational frequencies of neutral ground state, 3p Rydberg states, and cation of CD 2 N 2 1 1 A 1 3 1 A 1 2 1 A 2 2 1 B 1 1 2 B 1 exp a exp c exp d exp e exp f calc b calc calc calc calc Band origin, IE . . . 56871 52214 52664 72620, 72585 g ν 1 (a 1 ) CD 2 sym. stretch 2245 . . . 2189 2183 . . . 2305 2313 2302 2244 2246 ν 2 (a 1 ) NN stretch 2096 . . . 2051 2081 2180 g 2198 2060 2213 2197 2145 ν 3 (a 1 ) CD 2 sym. bend 1213 . . . 1044 907 . . . 1267 1034 1076 1054 1108 ν 4 (a 1 ) CN stretch 970 . . . 919 766 915, 970 g 984 734 949 932 957 ν 5 (b 1 ) CNN o.p. bend . . . . . . (256) . . . (225, 275) 571 564 446 444 425 ν 6 (b 1 ) CD 2 wag 318 . . . (590) 606 . . . 327 417 457 611 592 ν 7 (b 2 ) CD 2 asym. stretch 2414 . . . 2344 . . . . . . 2470 2527 2484 2471 2440 ν 8 (b 2 ) CD 2 rock 903 . . . . . . . . . . . . 919 914 899 891 903 ν 9 (b 2 ) CNN i.p. bend . . . . . . (318) . . . 340 392 376 328 401 357 All values in cm − 1 . Tentatively assigned frequencies are enclosed in parentheses. a Data from refs. ( 2, 3 ) (frequencies rounded to integer cm − 1 ; accuracy=± 2 cm − 1 ). b Harmonic frequencies, see text. For the 1 1 A 1 state, the values were calculated using B3LYP/6- 311G(2df,p). For the 3 1 A 1 (3p x ) state, symmetry is lowered to C s , b 1 modes are of a” symmetry, while all others are a’. c Uncertainty± 15 cm − 1 . d Uncertainty± 5 cm − 1 . e Fundamental frequencies are from Ref. 4 relative to the determined unperturbed value of the 2 1 B 1 (3p z ) band origin. The perturbed origin for the band origin (52,695 cm − 1 ) was defined as the average value for the1← 0 and0← 1 transitions. f Frequency accuracy defined as the spacing between the rotational lines of cation is± 50 cm − 1 . g Data from Ref. 5. Frequency accuracy is± 80 cm − 1 . 84 Table 4.4: Transition energies and vibrational frequencies of neutral ground state, 3p Rydberg states, and cation of CHDN 2 1 1 A 1 3 1 A 1 2 1 A 2 2 1 B 1 1 2 B 1 exp a exp c exp d exp e exp f calc b calc calc calc calc Band origin, IE . . . 56936 52221 52648 . . . ν 1 (a 1 ) CD stretch 2331 . . . 2237 . . . . . . 2382 2410 2385 2343 2335 ν 2 (a 1 ) NN stretch 2097 . . . 2060 . . . . . . 2201 2076 2221 2205 2149 ν 3 (a 1 ) CHD bend 1310 . . . 1305 . . . . . . 1351 1273 1309 1256 1297 ν 4 (a 1 ) CN stretch 1157 . . . 983 . . . (960) 1196 751 1017 1012 1063 ν 5 (b 1 ) CNN o.p. bend 549 . . . . . . . . . . . . 578 565 448 455 434 ν 6 (b 1 ) CHD wag 368 . . . (624) . . . . . . 375 482 529 691 673 ν 7 (b 2 ) CH stretch 3133 . . . 2976 . . . . . . 3262 3307 3253 3209 3203 ν 8 (b 2 ) CHD rock . . . . . . . . . . . . . . . 942 963 923 908 925 ν 9 (b 2 ) CNN i.p. bend . . . . . . 326 . . . (375) 409 395 339 420 371 All values in cm − 1 . Tentatively assigned frequencies are enclosed in parentheses. a Data from refs. ( 2, 3 ) (frequencies rounded to integer cm − 1 ; accuracy=± 2 cm − 1 ). b Harmonic frequencies, see text. For the 1 1 A 1 state, the values were calculated using B3LYP/6- 311G(2df,p). For the 3 1 A 1 (3p x ) state, symmetry is lowered to C1. c Uncertainty± 15 cm − 1 . d Uncertainty± 5 cm − 1 . e Fundamental frequencies are from Ref. 4 relative to the determined unperturbed value of the 2 1 B 1 (3p z ) band origin. The perturbed origin for the band origin (52,691 cm − 1 ) was defined as the average value for the1← 0 and0← 1 transitions. f Frequency accuracy defined as the spacing between the rotational lines of cation is± 50 cm − 1 . 85 4.3 Discussion 4.3.1 Vibrational assignments for the 2 1 A 2 (3p y ) Rydberg state As stated above, one of the goals of the present work is to determine experimentally the fundamental vibrational modes of the 2 1 A 2 (3p y ) state of diazomethane and its cation in order to compare them with theoretical calculations. In the REMPI spectra, the strongest transitions to the Rydberg 2 1 A 2 (3p y ) state are those of a 1 vibrational symmetry and their assignment are robust. Bands of b 1 and b 2 symmetry are much weaker and often do not show a discernible K-structure. Their assignments, which rely mainly on calculations, are tentative. The proposed assignments for CH 2 N 2 , CD 2 N 2 , and CHDN 2 are shown in Fig. 4.2 (a)-(c) and the fundamental frequencies are listed in Tables 4.2-4.2. In assigning fundamental frequencies in the 2 1 A 2 (3p y ) state we relied on: (i) the measured positions of the REMPI vibronic bands; (ii) the K-structure of the vibronic bands; (iii) the energy positions of the diagonal peaks in the eKE distributions; (iv) changes observed for H/D isotopologs; and (v) results of ab initio calculations. As discussed previously, 1 2+1 REMPI excites mostly vibronic levels in the 2 1 A 2 (3p y ) state, and transitions of a 1 symmetry to ν 0 1 -ν 0 4 exhibit the highest intensity. For example, in the 2 + 1 REMPI spectrum of CH 2 N 2 (Fig. 4.2 (a)), the strong bands at 53,196− 53,265,53,690− 53,756,54,289− 54,358, and55,286− 55,353 cm − 1 are assigned, respectively, as the 4 1 0 , 3 1 0 , 2 1 0 , and 1 1 0 transitions to the 2 1 A 2 (3p y ← π ) state. 86 Figure 4.2: 2+1 REMPI spectra for (a) CH 2 N 2 , (b) CD 2 N 2 , and (c) CHDN 2 following two-photon laser excitation at51,750− 54,900 cm − 1 . An inset in (a) for CH 2 N 2 shows a54,900− 56,700 cm − 1 spectrum magnified 10 times, whereas an inset in (b) displays the54,500− 55,000 cm − 1 range in x10 magnification. 87 The corresponding frequencies of the totally symmetric (a 1 ) ν 0 4 , ν 0 3 , ν 0 2 , and ν 0 1 funda- mentals are 969, 1,463, 2,062, and 3,059 cm − 1 , respectively, in good agreement with the calculated values (Table 4.2). These assignments are confirmed by the positions of combination bands at54,627− 54,679 and55,783− 55,846 cm − 1 , which are assigned as 3 1 0 4 1 0 and 2 1 0 3 1 0 , respectively. CD 2 N 2 and CHDN 2 transitions involving the totally sym- metric modes are also intense, and it is easy to identify the 4 1 0 , 3 1 0 , 2 1 0 , and 1 1 0 transitions. For CD 2 N 2 (Fig. 4.2 (b)), the frequencies ofν 0 4 ,ν 0 3 ,ν 0 2 , andν 0 1 are 919, 1,044, 2,051, and 2,189 cm − 1 , respectively, and for CHDN 2 (Fig. 4.2 (b)), they are 983, 1,305, 2,060, and 2,237 cm − 1 . A further test of the reliability of the assignments is that bands involving CH or CD motions change their frequency as expected for isotopic substitution. For example, the CH and CD stretch fundamentals in CHDN 2 have a frequency ratio CH:CD ˜ 1.4. All the experimentally determined values of the fundamental vibrational frequencies of a 1 symmetry are in good agreement with the calculated harmonic frequencies (Tables 4.2-4.2). Whereas it is fairly easy to assign the totally symmetric fundamentals, this is not the case for the weak bands of b 1 and b 2 symmetry. The transitions to theν 9 (b 2 ) fundamen- tals are mixed with the 2 1 B 1 origin bands, and this mixing lends them intensity. These transitions can be deperturbed by using a two-state approximation, as described before, 1 and the deperturbed values are listed in the tables. In our previous paper we assigned the strong band of CH 2 N 2 located at 52,574 cm − 1 as the 5 1 0 transition to 2 1 A 2 (3p y ). The separation between the triad of bands in CD 2 N 2 at 52,461− 52,495 cm − 1 is∼ 17 cm − 1 , which is typical of the transitions of A 1 or A 2 rovibronic levels. However, these bands could not be assigned to any of the a 1 modes or their combinations. They are closest to the calculated frequency of the 5 1 0 transition, 88 resulting in aν 0 5 (b 1 ; CNN out-of-plane bend) frequency of 264 cm − 1 . The single peak in the corresponding eKE distribution, whose frequency is 275± 10 cm − 1 is assigned asν + 5 . A similar eKE distribution was observed in ionization through the 52,461 cm − 1 transition for which the internal energy of the cation was calculated to be 225 cm − 1 . We therefore tentatively assign the upper state of this REMPI transition asν 0 5 . The ν 0 6 and ν 0 7 normal modes of CH 2 N 2 , CD 2 N 2 , and CHDN 2 , are assigned based primarily on the closeness of observed (weak) REMPI band to the calculated vibrational frequencies and their isotopic variations. Tentative assignments are shown in parenthe- ses in Tables 4.2-4.2. All the assigned fundamental frequencies are summarized and compared with calculations in Tables 4.2-4.2. 4.3.2 Vibrational assignments for the 1 2 B 1 ground-state cation The He(II) photoelectron spectrum of diazomethane was reported before, 5 and the adi- abatic IE of the ion and the frequencies of several of its vibrational levels were deter- mined (Table 4.2). We obtained these and additional vibrational frequencies from the peak position in the eKE distribution. In assigning the ion’s vibrational modes and fre- quencies we used mainly those eKE distributions that had a single peak resulting from the diagonal Franck-Condon transition; i.e. the vibrational frequencies obtained for the excited Rydberg state and the cation were rather similar. As discussed before, in the case of the mixed levels described above, we obtained ionic vibrational frequencies from the peak separations in the eKE distributions. 1 We note that only strong transitions, whose signal was high above background, could be used reliably because we detect all pho- toelectrons produced by ionization disregarding of their origin. The eKE for the origin band places the adiabatic IE of the cation at IE=72,620± 100 cm − 1 , in excellent agree- ment with the published value of 72,585± 160 cm − 1 . 5 The uncertainty in our values 89 reflects mainly uncertainty in K of about one unit in the ionization step. As with the val- ues for neutral diazomethane, the experimental and theoretical vibrational frequencies for the ions agree very well. 4.3.3 Structural and vibrational motions in neutral and ionic diazomethane The observed changes in structure and frequencies induced by ionization and electronic excitation (Tables 4.2-4.2) can be explained by simple molecular orbital considerations in combination with NBO analysis. As expected from the wave function analysis, 1 the structures and vibrational frequencies of the 2 1 A 2 (3p y ) and 2 1 B 1 (3p z ) Rydberg states are similar to those of the cation and they both retain C 2v structure. The 3 1 A 1 (3p x ) state, however, differs considerably from both the cation and the other two 3p states due to its mixing with the valence 2 1 A 1 (π ∗ ← π ) state, and it has C s equilibrium structure. 1 Below we first compare the calculated and experimental values to validate the the- oretical results and the assignments, and then proceed to analyze differences in struc- tures and frequencies between the Rydberg states and the cation in order to understand the structural and spectroscopic signatures of Rydberg-valence, Rydberg-Rydberg and Rydberg-ion core interactions. As far as structures are concerned, the calculated bond lengths of the neutral are within 0.002 of the experimental values, 20 as expected for CCSD(T)/cc-pVTZ level of theory. The B3LYP values are also very close. The maximum discrepancy between the calculated and experimental frequencies for all three isotopologes is about 5%, which is a typical value for anharmonicities. For the cation, the three lowest frequencies exhibit larger deviations, i.e., 10− 12% for the CH 2 wag and CNN in-plane bend, and 37− 80% for the CNN out-of-plane bend. The out-of-plane vibrations involving the carbon 90 atom hosting the unpaired electron are similar to the out-of-plane mode in substituted methyl radicals, which has been found to be extremely anharmonic. 21 A similar trend is observed for the two Rydberg states as well: most of the calculated frequencies are within 12% from the experimental ones, except for the same out-of-plane modes, CNN out-of-plane bend and CH 2 wag. Overall, the observed changes in structure and vibrational frequencies are consistent with removing an electron from the bondingπ CN -orbital, which also has an antibonding character with respect to NN. To explain the differences in structures and frequencies between the cation and the Rydberg states, we analyze the interactions of the Rydberg electron with the ion core. For example, the 3p y Rydberg orbital is localized in the plane of the molecule per- pendicular to the principal rotation axis (see Fig. 1 in Ref. 1). Its electron density is greatest on top of the hydrogen atoms and the C and middle N (directly bonded to C) atoms. The A 2 symmetry imposes a nodal plane along this axis. The 2 1 A 2 (3p y ) state differs from the cation mostly in the HCH angle (129.91 ◦ relative to 127.80 ◦ in the cation). The NBO analysis of the electron density of both states reveals that about half of the +1 charge of the nuclear core is accommodated by the hydrogens. The lobes of the 3p y orbital, located directly on the hydrogen atoms in space, can interact with the positively-charged hydrogen atoms. The larger HCH angle in the 2 1 A 2 (3p y ) state is thus attributed to increased electron density along the CH bonds. The 3p y orbital does not affect the CN bond in a similar way due to symmetry restrictions, so the net effect is to increase repulsion between the hydrogens. A similar argument explains the decrease in the HCH angle in the 2 1 B 1 (3p z ) state relative to the cation (124.74 ◦ compared to 127.80 ◦ ). The occupied 3p z orbital has elec- tron density centered along the CNN axis, with one lobe centered directly in the space 91 between the two hydrogens, while the other is located on the terminal nitrogen, which appropriates almost all of the remaining total nuclear positive charge. Thus, the orien- tation of the Rydberg orbital allows its electron density to overlap with the centers of positive charge in the nuclear core. Similar examples of Rydberg orbital orientation and the anisotropy of the cation core have been observed in a series of unsaturated hydro- carbon radicals. 22 In the 2 1 B 1 (3p z ) state of diazomethane, the HCH angle decreases to maximize this interaction. The 3p z orbital, which has a node on the central nitrogen, can donate density along both the CN and NN bonds; hence the observed contraction of these bonds with respect to the cation. Figure 4.3: Harmonic frequencies of the neutral and cation ground state of CH 2 N 2 com- pared to those of the 2 1 A 2 (3p y ) and 2 1 B 1 (3p z ) Rydberg excited states. The variations in the calculated vibrational frequencies for the ground state neutral and cation, and 3p y and 3p z Rydberg states of CH 2 N 2 are depicted in Fig. 4.3. Only modes below 3,000 cm − 1 are shown: the frequencies of the symmetric and asymmet- ric CH stretches do not vary significantly with electronic excitation/ionization and are therefore omitted. To explain the observed trends in vibrational frequencies, we divide the vibrational modes into three groups: (i) those that involve displacements mainly along the CNN 92 framework (CN and NN stretches, and CNN bends); (ii) those with displacements primarily in the CH 2 moiety (CH 2 wag, rock, and bend); and (iii) the CH stretching vibrations, which are not affected by the excitation/ionization. For the different elec- tronic states, trends in the first group are due mostly to the effect of lower CN and NN bond orders, while those in the second are due to the interaction between the positively- charged hydrogens and the Rydberg electron density, and the hybridization of the car- bon. Within each group, we also observe marked differences between the in-plane and out-of-plane modes. The four modes that comprise the first group are the CN and NN stretches (both a 1 ), and the b 1 and b 2 CNN bends. As shown in Fig. 4.3, the CN stretch is strongly affected by the removal of an electron from the HOMO p orbital; whether this electron is ionized or placed in a Rydberg orbital has almost no effect on the frequency. Thus, ioniza- tion/electronic excitation results in elongation of the CN bond, and a slight contraction of the NN bond. The changes in vibrational frequencies involving CNN motions are consistent with these changes in bond order. Referring to the out-of-plane b 1 CNN bend (ν 0 5 ) mode, reducing the order of the p bond leads to a strong decrease in frequency in the Rydberg states as well as the cation; i.e., the in-plane Rydberg orbitals provide no additional contribution relative to the cation. In contrast, the in-plane b 2 CNN bend (ν 0 6 ) shows a strong frequency change between the two Rydberg states (Fig. 4.3). The trend in this mode is complementary to that in the analogous mode in the second group: the b 2 CH 2 bend (rock). For the 3p y state, the CNN bending frequency drops significantly with respect to the neutral (by 67 cm − 1 ) whereas the CH 2 bend mode increases slightly (by 7 cm − 1 ). For the 3p z state, the CNN bend frequency increases relative to its value in the 2 1 A 2 (3p y ) state by 94 cm − 1 , to 93 above the frequency of the neutral, whereas the CH 2 bend decreases by 48 cm − 1 to below that of the neutral. For the cation, the CNN mode drops by 69 cm − 1 relative to the 2 1 B 1 (3p z ) state, falling again below the neutral value, while the CH 2 bend increases by 45 cm − 1 and is, within error, the same as in the neutral. The largest difference within both modes occurs between the 2 1 A 2 (3p y ) and 2 1 B 1 (3p z ) states. For the CNN bend, the displacement moves the CNN framework off the nodal plane and into the electron density of the 3p y orbital in the yz-plane. However, this displacement moves the atoms out of the density of the 3p z orbital, which is hindered by the donation of electron density into the CN and NN bonds. Consequently, the frequency of this vibration is significantly higher in the 2 1 B 1 (3p z ) state than in the 2 1 A 2 (3p y ) state. Finally and quite surprisingly, the frequency of the CH 2 out-of-plane wag (ν 0 6 ) increases significantly upon excitation/ionization. The reason for this is the competi- tion between the two resonance forms in the ground-state wave function (Fig. 4.1) and the change in hybridization of the carbon induced by ionization/electronic excitation. The NBO analysis confirms the competition between the two resonance structures in the ground-state wave function, which gives rise to sp 2 and sp 3 hybridized carbon for the left and right structures of Fig.4.1, respectively. Removing an electron from either of these structures results in sp 2 hybridized carbon and, therefore, a reduction in the sp 3 contribution, as confirmed also by NBO analysis. The increased sp 2 character leads to a stiffer out-of-plane vibration, which is seen in the calculations. 4.4 Summary The joint experimental and theoretical investigation discusses the structure and normal mode frequencies of the ground and excited Rydberg states of diazomethane and its isotopologs and of the corresponding cations. The experimental measurements exploit 94 REMPI spectroscopy and velocity map imaging of photoelectrons from excited vibronic levels of the 2 1 A 2 (3p y ) state to obtain vibronic assignments in the 2 1 A 2 (3p y ) and 2 1 B 1 (3p z ) Rydberg states, and vibrational states of the cation. The accompanying high-level ab initio calculations determine structures and vibrational states in the ground states of the neutral and cations as well as the three Rydberg 3p states. The good agreement between the electronic structure results and the current experimental results on the 2 1 A 2 (3p y ) state and the cation, as well as previous studies on other states, allows a full anal- ysis of Rydberg-ion core interactions and trends in vibrational frequencies. Although the 2 1 A 2 (3p y ) and 2 1 B 1 (3p z ) Rydberg states have planar C 2v symme- try like the ion, they exhibit differences in geometry due to specific interactions of the electron in the 3p y or 3p z orbital with the nuclei charge distributions of the ion core. Trends in vibrational frequencies in the ground states of the neutral and ion and the 2 1 A 2 (3p y ) and 2 1 B 1 (3p z ) states are consistent with removing an electron from the bond- ing π CN -orbital, which nevertheless has an antibonding character with respect to NN. In explaining the observed trends, the vibrational modes are divided into two groups, which involve displacements mainly (i) along the CNN framework, and (ii) in the CH 2 moiety. Trends in the first group are due mostly to effects of the lower CN and NN bond orders, whereas those in the second group are due to the interaction between the positively-charged hydrogens and the Rydberg electron density, and the hybridization of the carbon. Within each group, marked differences in behavior between the in-plane and out-of-plane modes are observed. The largest changes in frequencies upon ionization are observed in the CN stretch, CH 2 wag and the two CNN bending modes. Differences in vibrational frequencies between the 2 1 A 2 (3p y ) and 2 1 B 1 (3p z ) Rydberg states reflect state-specific interactions of the charge density of the electron in the Rydberg 3p orbital with the nuclei charge density in the ion core. 95 4.5 Chapter 4 references [1] L. Koziol, I. Fedorov, G. Li, J.A. Parr, H. 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A, 110:2746–2758, 2006. 97 Chapter 5 Diazirine: multiphoton ionization and dissociation 5.1 Introduction Diazirine (c-CH 2 N 2 ) belongs to the family of isoelectronic molecules known as ”16- electron molecules”, which have attracted considerable attention for decades because of the inherent complexity of their photodissociation dynamics and certain similarities in their properties. 1, 2 In addition, HNCO, H 2 CNN and H 2 CCO, all members of this family, are known to have structural isomers. 1–16 Of these molecules, the least studied is the H 2 CN 2 group, mainly due to the instabil- ity of its prototype member, diazomethane, and its cyclic counterpart diazirine. Recently it has become possible to produce stable molecular beams of these species, and studies of their photophysics and photochemistry in highly excited states have begun, aided by high-level electronic structure calculations. In previous papers we described the two- photon excitation of diazomethane to 3p Rydberg states, and discussed its electronic structure and couplings among its excited states. 17, 18 In the present article, we report first results on the two-photon dissociation of diazirine, specifically the formation of CH products, as well as electronic structure calculations on its excited and ionized states. Diazirines contain a three-membered ring composed of one carbon atom and two double-bonded nitrogen atoms. Although diazomethane had been known since the 98 1920’s, diazirines were first synthesized only in the 1960’s. 19–22 However, due to their structural uniqueness and their roles as precursors of carbenes, much has been learned since then about their spectroscopy, photochemistry and thermal decomposition. 22 For the prototype diazirine (c-CH 2 N 2 ), the simplest member of the group, the lowest lying UV absorption bands were assigned to the structured 1 1 B 2 ← 1 1 A 1 (π ∗ ← n) system with a band origin at 31,187 cm − 1 (320.65 nm). 21, 23 Its VUV absorption spectrum shows an intense, structureless band centered at 145− 185 nm as well as diffuse structures at ˜ 120− 143 nm. 24 The vertical (adiabatic) ionization energy (IE) of diazirine was deter- mined experimentally at 10.75 eV (10.3 eV), and the ion’s low-lying excited states are at 13.25 eV (12.8 eV) and 14.15 eV (14.15 eV). 25 Paulett and Ettinger reported the 298 K heat of formation of diazirineΔ H 0 f,298 =79.3 kcal/mol, 26, 27 whereas Laufer and Okabe obtained60.6<Δ H 0 f,298 ≤ 66 kcal/mol. 24 Although significant experimental work has been carried out since the 1960’s, the level of theoretical studies during this early period was too low to provide reliable infor- mation on excited states. With the availability of high-level electronic structure com- puter codes, a renewed effort to elucidate the electronic structure of diazirines/diazo compounds and their decomposition mechanisms was initiated in the 1990’s. 10–13, 28–30 The molecular structure of c-CH 2 N 2 , its related cation, and its ionization energy were the subject of several investigations. 31, 32 Electronically excited states and energy differ- ences among isomers were also calculated. 13, 28 Whereas several aspects of diazirine’s photochemistry on the lowest excited state 1 1 B 2 have been discussed, 10, 13, 33 little is known from experiment and theory about pho- todissociation on higher electronic states, and in particular about multiphoton dissocia- tion pathways. By combining high-level electronic structure calculations and photofrag- ment laser spectroscopic experiments in molecular beams, we were able to investigate 99 the photodissociation of c-CH 2 N 2 following two-photon absorption. In the only previ- ous molecular beam study, diazirine was excited to the 1 1 B 2 ← 1 1 A 1 origin band and fluorescence, tentatively assigned to emission from high vibronic levels of the excited 1 1 B 2 state of the singlet methylene product, was observed. 34 We have adapted the traditional preparation method of diazirine 20, 35 to molecular beam studies, and used resonance enhanced multiphoton ionization (REMPI) comple- mented by dc slice velocity map imaging (VMI) 36 to detect CH (X 2 Π ) photodissocia- tion fragments. Specifically, CH products from two-photon dissociation were detected by 2+1 REMPI at the dissociation wavelength (one-color experiment) using the D 2 Π (ν 0 = 2)←← X 2 Π (ν “ = 0) intermediate transition (where the double arrow indi- cates two-photon transition). The dc sliced images of CH+ ions were anisotropic and typical of those of a perpendicular transition and fast dissociation. Several channels are discussed as possible sources of CH(X 2 Π ) fragments and the results suggest that the predominant pathway is two-photon absorption by the parent molecule followed by isomerization to isodiazirine and subsequent dissociation to CH + HN 2 . 5.2 Computational studies of the electronically excited and ionized states of diazirine The equilibrium geometry and vibrational frequencies of neutral diazirine ground state were calculated by CCSD(T) 37, 38 using the cc-pVTZ basis 39 and by B3LYP 40 using the 6-311G(2df,p) basis. The equilibrium geometry of the cation was calculated using B3LYP/6-311G(2df,p). Vertical excitation energies were calculated using EOM- CCSD 41, 42 /6-311(3+,+)G* at the B3LYP optimized geometry. This basis set was derived from 6-311G* by adding three diffuse sp functions to heavy atoms and one diffuse s 100 function to hydrogen. The assignment of valence and Rydberg character to the excited states was based on: (i) the symmetry of the transition, (ii) leading EOM-CCSD ampli- tudes and character of corresponding HF orbitals, and (iii) the second momentshX 2 i, hY 2 i, andhZ 2 i of the EOM-CCSD electron density. The character of the HF orbitals was determined using the Molden interface. 43 All EOM-CCSD excited states were dom- inated by single excitations. Figure 5.1: Molecular orbitals relevant to ground and excited electronic states of c- CH 2 N 2 . The three-membered ring lies in the yz plane, with the z-axis coinciding with the C 2 symmetry axis. The excited valence states 1 1 B 2 (π ∗ ← n) and 1 1 A 2 (π ∗ ← σ NN ) were optimized by EOM-CCSD/6-311G** under C 2v constraint using analytic gradients. 44 For the low- est state, 1 1 B 2 (π ∗ ← n), the optimized structure is a true minimum as confirmed by vibrational frequency calculations. For the 1 1 A 2 (π ∗ ← σ NN ) state, we were not able to perform frequency calculations in this basis because of the limitations of the finite differences code and the large density of states in this energy range. To validate the structure of this state, we optimized the geometry starting from the C 1 distorted structure 101 and computed frequencies using the smaller 6-31G* basis. This calculation produced a similar C 2v structure and no negative frequencies. Figure 5.2: Left panel: ground-state equilibrium structures ( ˚ A and deg) of diazirine for the neutral (1 1 A 1 ) at CCSD(T)/cc-pVTZ (normal print) and B3LYP/6-311G(2df,p) (ital- ics) and for the cation, 1 2 B 1 (∞← n), at B3LYP/6-311G(2df,p) (underlined). The corre- sponding nuclear repulsion energies are 64.158275, 64.295975, and 62.797366 hartree, respectively. Right panel: excited state equilibrium structures for the 1 1 B 2 (π ∗ ← n) and 1 1 A 2 (π ∗ ← σ NN ) excited states at EOM-CCSD/6-311G** shown in normal print and italics, respectively. The corresponding nuclear repulsion energies are 62.313876 and 61.730487 hartree. All optimizations, frequencies, and excited state calculations were performed using the Q-Chem 45 and ACES II 46 electronic structure programs. The electronic configuration of the ground state (1 1 A 1 ) of diazirine is: [core + low-lying] 16 (6a 1 ) 2 (2b 2 ) 2 (3b 1 ) 2 (1a 2 ) 0 (4b 1 ) 0 = [core + low-lying] 16 (σ NN ) 2 (π NN ) 2 (n) 2 (π ∗ NN ) 0 (σ ∗ NN ) 0 where “core + low-lying” refers to the 1s core orbitals and combinations of σ CN and π NN orbitals that are not involved in excitations. Relevant molecular orbitals are shown in Fig. 5.1. The CNN ring lies in the yz plane with the z axis coinciding with the C 2 symmetry axis. The equilibrium geometries of diazirine in its ground neutral and cation states are shown in Fig. 5.2 and the geometries of the excited states are given in Table 5.2. The neutral ground state geometry agrees well with experimental values. 47 102 Table 5.1: Calculated equilibrium structures for the ground, 1 1 B 2 , and 1 1 A 2 valence states of the neutral and for the ground state of the cation 1 1 A 1 a 1 1 B 2 (π ∗ ← n) c 1 1 A 2 (π ∗ ← σ NN ) b 1 2 B 1 c r CN ,( ˚ A) 1.477, 1.479 1.534 1.453 1.617 r NN ,( ˚ A) 1.229, 1.216 1.260 1.422 1.144 r CH ,( ˚ A) 1.075, 1.081 1.083 1.093 1.082 6 HCH ( ◦ ) 119.6, 119.5 120.3 117.1 129.5 6 NNC ( ◦ ) 65.4, 65.7 65.8 60.7 69.3 6 NCN ( ◦ ) 49.2, 48.5 48.5 58.6 41.4 a Left values calculated by CCSD(T)/cc-pVTZ, right values by B3LYP/6-311G(2df,p). Experi- mental values: 47 r CN = 1.482± 0.003 ˚ A; r NN = 1.228± 0.003 ˚ A; r CH = 1.09± 0.02 ˚ A; 6 HCH= 117± 2 ◦ . b Values calculated by EOM-CCSD/6-311G** c Values calculated by B3LYP/6-311G(2df,p). The first ionization removes an electron from the highest-occupied molecular orbital (HOMO), which is the 3b 1 orbital denoted as n in Fig. 5.1. This orbital features sigma- type bonding between an atomic p orbital on carbon and out-of-phase s orbitals on the nitrogens (Fig. 5.1). The geometry change upon ionization from this orbital is consistent with a weakening bonding interaction in the CN bonds; consequently, the cation resembles a weakly bound N 2 ··· CH + 2 as described before. 31 Indeed, at the CCSD/6-311G(2df,p) level of theory, the cation lies only 0.73 eV (16.9 kcal/mol) below the N 2 + CH + 2 asymptote. The vertical IE calculated at the B3LYP neutral geometry with the EOM-IP-CCSD/6-311G(2df,p) method is 10.71 eV . This agrees well with the experimental value, 47 indicative that triples corrections are unnecessary for this system. The Rydberg states are expected to have similar structures to the cation, thus should be weakly-bound. The vertical excitation energies, their leading configuration state functions, and one- photon oscillator strengths are summarized in Table 5.2, along with oscillator strengths 103 Table 5.2: Vertical excitation energies (eV), oscillator strengths, dipole strengths (a.u.), and changes in second dipole moment of charge distributions (a.u. 2 ) for the excited states of c-CH 2 N 2 at EOM-CCSD/6-311(3+,+)G* a state Δ E vert f L μ 2 tr Δ hX 2 i Δ hY 2 i Δ hZ 2 i 1 1 B 2 (π ∗ ← n) 4.27 0.004/(-) 0.037 0 1 -2 1 1 B 1 (3s← n) 7.32 0.0006/(0) 0.003 9 17 12 1 1 A 2 (π ∗ ← σ NN ) 7.61 0/(0.059) 0 -1 1 -1 2 1 A 1 (3p x ← n) 7.86 0.099/(0.013) 0.516 25 8 7 2 1 B 1 (3p z ← n) 7.88 0.0210/(0) 0.021 6 11 36 2 1 A 2 (3p y ← n) 7.97 0/(0.015) 0 8 33 9 a At the B3LYP/6-311G(2df,p) optimized geometry; E CCSD =− 148.417293 a.u. for transitions from the 1 1 B 2 state to higher electronic states. One-photon vertical exci- tations from the ground state show that the strongest excitation is to the 3p x Rydberg state. Transition dipole moments were also calculated between excited states in a single EOM-CCSD calculation. This allowed us to obtain oscillator strengths specifically from the 1 1 B 2 (π ∗ ← n) state to the other EOM states. The lowest excited state is a valence 1 1 B 2 (π ∗ ← n) excitation at 4.27 eV (293 nm). Well separated from this state, there is a cluster of states between 7.2 and 8.0 eV (174− 156 nm); these are the n = 3 Rydberg states and the valence 1 1 A 2 (π ∗ ← σ NN ) state. Absorption to these states agree reasonably well with the measured absorption spectrum of diazirine at165− 145 nm. 24 The oscillator strengths in excitation from the 1 B 2 state show that the dominant tran- sition is to the second valence state, 1 1 A 2 , rather than to the Rydberg states. The oscilla- tor strength for the 1 1 A 2 ← 1 1 B 2 transition is about an order of magnitude greater than that for 1 1 B 2 ← 1 1 A 1 excitation, implying high probability for a two-photon transition 104 via the intermediate 1 1 B 2 state. Excitation to 1 A 2 is forbidden in one-photon transition from the ground state, but is allowed in two-photon excitation via the 1 B 2 state. Excitation to the 1 1 A 2 (π ∗ ← σ NN ) state removes an electron from a bonding orbital along NN (Fig. 5.1), and places it into the π ∗ orbital, which is antibonding along NN. Both orbitals have electron density localized along the NN bond, and thus the excitation doubly weakens this bond. Comparing geometries of the 1 A 2 and ground state reveals elongation of the NN bond by 0.22 ˚ A, which is very close to a single NN bond length (1.45 ˚ A), and opening of the NCN angle by approximately 9 ◦ . The CN bond length decreases only slightly, by 0.024 ˚ A. Even though the geometry optimization produces a stable structure, the considerable reduction of the formal bonding character of this state suggests a small barrier to dissociation and nearly-dissociative character. Moreover, the elongated NN bond may facilitate isomerization to isodiazirine in which the NN bond is a single bond (see below). The limitations of the employed single-reference methods do not allow us to characterize the potential energy surface (PES) too far from the Franck-Condon region. This state was also found to be bound by the CASSCF calculations of Arenas et al, 13 who performed surface-hopping calculations to model photoinduced dynamics on this state, referred to by them as S2. They reported fast (∼ 40 fs) radiationless relaxation to the 1 1 B 2 state (S1 in their notation) via a conical intersection followed by dissociation to methylene and N 2 . However, the low level of theory employed in their work (i.e., small basis, and considering only valence states without dynamical correlation) can adversely affect their results. For example, the authors reported unbound equilibrium structure for the 1 1 B 2 (π ∗ ← n) state, in disagreement with our calculations and the experimental Franck-Condon progressions. 21, 23 105 5.3 Discussion 5.3.1 Excited states and photoionization of diazirine The electronic structure calculations of diazirine’s excited states show that in addition to the low-lying 1 1 B 2 (π ∗ ← n) state common to all diazirines, 21–23, 28, 48 there is a cluster of states in diazirine at7.6− 8.3 eV , which consists of then = 3 Rydberg states (s and p) and the 1 1 A 2 (π ∗ ← σ NN ) valence state. The weakly-bound Rydberg states appear to be pure and do not interact appreciably with nearby valence states. The most efficient one-photon absorption is expected to be to the 2 1 A 1 (3p x ) Rydberg state. The situation is quite different in two-photon excitation via the 1 1 B 2 intermediate state. The oscillator strength calculations from the 1 1 B 2 state suggest that the most efficient excitation is to the 1 1 A 2 (π ∗ ← σ NN ) valence state, and this oscillator strength is greater by a factor of > 4 than to the 2 1 A 1 (3p x ) and 2 1 A 2 (3p y ) Rydberg states. Thus, we believe that the main excitation process responsible for ionization and photodissociation of diazirine in our experiments is a sequential two-photon process 1 1 A 2 ← 1 1 B 2 ← 1 1 A 1 , which accesses the bound but dissociative 1 1 A 2 valence state (see Section 5.2). Accepting this as the main excitation process explains the high propensity for disso- ciative photoionization observed in the REMPI spectrum of diazirine. Both the 1 1 B 2 and 1 1 A 2 states have geometries that differ significantly from the ground-state ion and this would lead to internally excited and predissociative diazirine cations. Three photons are required for ionization of diazirine, giving rise to ions that can have> 1 eV of internal energy above the 10.3 eV adiabatic ionization energy. 25 The dissociation energy of the ion is calculated at 0.73 eV above its ground state, and its geometry resembles that of the loosely bound CH + 2 ··· N 2 complex (see Table 5.2 and Fig. 5.2); thus, the observation that CH + 2 fragment ions are the main photoionization products is not surprising. 106 5.3.2 Detection of ionization and dissociation products As discussed above, following 324− 305 nm laser irradiation multiphoton processes lead to ionization and photodissociation. The wavelength range at which CH + 2 ions are detected coincides with the 1 1 B 2 ← 1 1 A 1 structured absorption system of diazirine, but the vibronic features observed in the CH + 2 action spectrum are much broader than those in the corresponding one-photon absorption spectrum. 21, 22 This difference is rational- ized by realizing that the CH + 2 photoionization spectrum is a result of multiple photon excitation via the dissociative 1 1 A 2 (π ∗ ← σ NN ) intermediate state. Thus, there is com- petition between fast dissociation and ionization and the effective lifetime for ionization via 1 + 1 + 1 REMPI is determined by dissociation in the second intermediate state. This explains both the significant lifetime broadening and the low yield of CH + 2 ions, which are detected only at higher laser fluences. We also note that no REMPI signal from neutral N 2 and CH 2 photofragments is detected. The CH(X)2+1 REMPI spectrum observed via the D 2 Π (v 0 =2)←← X 2 Π (v “ =0) transition is well known, but its analysis is complicated by severe predissociation in the upper D state. 49, 50 As discussed above, similar REMPI spectra of CH(X) have been observed in multiphoton dissociation of several precursors. 49–55 In studies of photoion and photoelectron spectroscopy, it has been established that the predissociation rate in the D 2 Π i (v 0 =2) state increases greatly for rotational levelsN 0 ≥ 11 as a result of curve crossing with repulsive states, 49–51 and rotational transitions from CH(X) that terminate in N 0 ≥ 11 in the D state cannot be observed. As a result of this predissociation, it is impossible to determine the effective rotational line strengths and infer populations. However, it is clear from our spectra that the decrease in the observed line intensities starts atN “ <11, before fast predissociation sets in, indicating that the maximum in the rotational distribution is lower than the predissociation limit. It is also evident that the 107 lowest rotational levels have small populations, indicating that the rotational distribution is shifted to higher rotational states and is probably nonstatistical. Particularly striking is the similarity between the CH(X) REMPI spectra obtained in this work and the corresponding REMPI and LIF spectra of CH(X) obtained in two- photon dissociation of the isoelectronic ketene at comparable levels of parent excita- tion. 49–51, 56 In the next section we expand on this similarity and discuss possible disso- ciation mechanisms. 5.3.3 Pathways leading to CH(X) fragments The most intriguing experimental finding of this work is the intense REMPI spectrum assigned to CH(X) fragments. Its anisotropic angular distribution indicates that fast dissociation via a perpendicular transition is responsible for its formation. From the observation that all the rotational branches in the CH(X) spectrum in the region310− 316 nm can be detected, we infer that the absorption by the CH(X) precursor is broad and not state-specific. Although we cannot offer a definitive mechanism for the production of CH(X), we describe below several possible pathways and discuss our strong preference for one of them. From the high CH(X) translational energies determined from the photofragment images, we conclude that this fragment must be generated by absorption of at least two photons. One-photon production of CH(X) from ground state diazirine requires wave- lengths< 207 nm (48,300 cm − 1 ; 138 kcal/mol), whereas CH(X) production is observed with 314 nm excitation (31,800 cm − 1 ) and the fragments are born with substantial trans- lational energies. Before discussing possible reaction pathways, an assessment of the dissociation energy of diazirine to produce CH fragments is needed. The largest uncertainty derives 108 Table 5.3: Calculated values ofΔ H 0 f of diazomethane and diazirine diazomethane diazirine 0 K 298 K 0 K 298 K references 65.42, 65.68 63.18, 64.15 77.9, 74.10 76.11, 74.10 ( 57 ) 66.7 65.3 ... ... ( 58 ) 68.0 ... 77.7 ... ( 59 ) 64.3 63.1 74.5 73.0 ( 11, 12 ) from the value of the heat of formation of diazirine. The two experimental values (obtained over 35 years ago) differ greatly from each other,60.6− 66 kcal/mol 24 and 79.3 kcal/mol, 26, 27 and each determination is associated with experimental difficulties. The NIST Chemistry Webbook 60 gives both values and the issue has remained the subject of debate. 22, 24, 58, 59, 61, 62 In the past 15 years, with theoretical methods achieving chemi- cal accuracy, the heats of formation of diazirine and its structural isomer diazomethane (whose heat of formation is just as controversial) were calculated using high-level elec- tronic structure methods, and a re-evaluation of the accepted values was called for. 58, 59 In Table 5.3 we summarize the calculated heats of formation, Δ H 0 f,0 , which are much more convergent than the experimental ones. Several authors have calculated heats of formation of diazomethane and diazirine through various pathways. Gordon and Kass, 11, 12 employed the atomization reaction and two isodesmic reactions using the G2 model chemistry, giving average values of 64.3 and 74.5 kcal/mol forΔ H 0 f,0 of diazomethane and diazirine at 0 K, respectively. Catoire 57 reported CBS-Q and G2 heats of formation for several species produced by the decomposition reactions of monomethylhydrazine. Ab initio atomization reactions and atomic heats of formation at 0 K (gas phase) were calculated, giving values of 65.4 109 and 65.7 kcal/mol for CBS-Q and G2 methods for diazomethane, respectively, and 78.0 and 76.0 for diazirine. Walch 59 reported heats of formation using CASSCF methods with double-zeta Dun- ning basis sets for geometries, and internally contracted configuration interaction (ICCI) for energetics with double-, triple-, and quadruple-zeta Dunning bases. He recom- mended the value ofΔ H 0 f,0 =77.7 kcal/mole for diazirine. Dixon et al. 58 reported 65.3 and 66.7 kcal/mol for the heat of formation of diazomethane at 298 and 0 K, respec- tively, computed using the CCSD(T) method and CBS extrapolation, which is the most accurate theoretical estimate. All calculations included zero point corrections via har- monic frequency calculations. For diazomethane, both G2 calculations 11, 57 are within 1 kcal/mol of the result of Dixon et al., 58 suggesting similar accuracy for the respective diazirine values. Thus, for diazomethane the preferred theoretical value isΔ H 0 f,0 =67± 3 kcal/mole, much closer to the experimental value of 67 kcal/mol recommended by Setser and Rabinovitch, 63 than to values given in the NIST Chemistry Webbook. 60 Moreover, in all the calcu- lations diazirine is to found to lie 10± 1 kcal/mole above the ground state of dia- zomethane, allowing us to adapt the thermochemistry of diazomethane reactions to the case of diazirine (see below). 11–13, 57, 59 We conclude that theoretical heats of formation for diazirine are 75± 2 and 77± 2 kcal/mol at 298 and 0 K, respectively. These values are much closer to those recommended by Paulett and Ettinger, 26 than to those of Laufer and Okabe. 24 In agreement with the calculations, in what follows we have adopted the value of 77 ± 3 kcal/mol as the heat of formation of diazirine at 0 K. We use the calculated value Δ H 0 f,0 = 60.8 kcal/mole for the heat of formation of HN 2 58 and the accepted values 110 of Δ H 0 f,0 for H, CH, and CH 2 (1 1 A 1 ) of 51.63, 64 141.61± 0.14, 65 and 102.37± 0.38 kcal/mol, 65 respectively. The most direct dissociation process leading to CH(X) formation is: c-CH 2 N 2 (1 1 A 1 )+2hν → CH(X 2 Π )+ HN 2 (1 2 A”);Δ H 0 r =125± 4 kcal/mol (I) The HN 2 product is metastable and is calculated to lie 9 kcal/mole above the thermo- chemical threshold for H + N 2 ; 66 thus the overall dissociation process is: c-CH 2 N 2 (1 1 A 1 )+2hν → CH(X 2 Π )+H( 2 S)+N 2 (X 1 Σ + g ) (Ia) which likely evolves in sequential steps. In order to assess the feasibility of reaction (I), the maximum allowed translational energy release in the CH(X) product needs to be determined. The best estimate is obtained from Fig. 5.3, since this image was obtained for state-selected CH(X,N “ =9). For this image, 2hν = 64,082 cm − 1 , and forN “ = 9 of CH(X),E rot = 13,000 cm − 1 (B(CH)= 14.457 cm − 1 ). 67 Thus, the energy required for reaction (I) terminating in CH(X,N “ = 9) is 43,900± 1,400 cm − 1 (125± 4 kcal/mol) plus 1,300 cm − 1 . Sub- tracting this energy from the photon energy, we obtain that the maximum allowed c.m. translational energy isE T =18,900± 1400 cm − 1 , in good agreement with the observed value of20,500± 1,000. A different route to assess the maximum allowed c.m. translational energy asso- ciated with CH(X, N” = 9) in reaction (I) is to start with the calculated value for the dissociation of diazomethane: 68 CH 2 N 2 (1 1 A 1 )→ CH 2 (1 1 A 1 )+ N 2 (X 1 Σ + g );Δ H 0 r =32.6 kcal/mol add to it the energy required to dissociate CH 2 (1 1 A 1 ) to CH(X,N 00 =9) + H and subtract 9 kcal/mole for the formation of the HN 2 product. Taking into account the 10 kcal/mol difference between the heats of formation of diazirine and diazomethane, we obtain that reaction (I) should require 126 kcal/mole, which corresponds to a c.m. translational 111 Figure 5.3: Top panel: the image obtained in dissociation at 312.10 nm by monitoring CH (X,v 0 = 0,N 00 = 9). Bottom panel: the c.m. translational energy distribution, P(E T ), of the CH(X) fragments (right axis) and the recoil anisotropy parametersβ i (E T ) (left axis). energy of≤ 19,900 cm − 1 , again in good agreement with the experimental result. We conclude, therefore, that our results are well explained by reaction (I). Moreover, on the basis of our results and the recent theoretical calculations, we argue that the heat of formation of diazirine at 0 K should be revised upward to77± 3.0 kcal/mol and that the diazomethane value should be lower by 10 kcal/mol. We note that using the calculated value for the heat of formation of the diazirinyl radical of117± 1.0 kcal/mol 57, 69, 70 and the calculated D 0 = 93 kcal/mol for c-CH 2 N 2 (1 1 A 1 )→ c-CHN 2 (1 2 A 2 ) + H( 2 S), 31 we obtain the heat of formation of diazirine as 76 kcal/mol. 112 Reaction I is the only one that can generate CH(X) via absorption of two photons, and thus should be the favored dissociation pathway. Inspection of the recoil anisotropy parameters shows that in the region where the intensity of CH(X) is substantial, theβ 2 parameter is fairly constant and is typical of one-photon excitation via a perpendicular transition. This suggests that the anisotropy is determined largely in the second step, i.e. excitation from the long-lived intermediate 1 1 B 2 state to the dissociative 1 1 A 2 state. The fluence dependence of CH(X) production in these one-color experiments is not very revealing. At the fluence levels where reasonable signals are obtained (> 0.15 mJ, 40 cm f.l. lens) the intensity dependence is slightly higher than quadratic; however, this dependence reflects mainly the two-photon nature of the CH D← X excitation rather than the1+1 photon excitation process in diazirine. The most likely mechanism of reaction (I) involves initial isomerization to isodi- azirine, with transfer of a hydrogen atom from carbon to nitrogen. (Fig. 5.4, top) Figure 5.4: Top: initial isomerization of diazirine to isodiazirine. Bottom: the cyclic intermediate oxirane (c-HCOCH) in the hydrogen scrambling of ketene. Such isomerization is in general inefficient; however, it is known that in the family of the so-called 16-electron molecules, atom shifts are unusually facile. 1, 3, 4, 6, 16, 71 In par- ticular, in the photodissociation of isotopically labelled ketene (H 12 2 C 13 CO), scrambling between the two carbon isotopes in the CO product has been ascribed to the formation 113 of the cyclic intermediate oxirane, c-HCOCH (Fig. 5.4, bottom) 16, 71 accompanied by a hydrogen shift. Moreover, in the two-photon excitation of ketene at comparable wavelengths (279.3 and 308 nm), 56 Ball et al. observed efficient production of CH(X) fragments and obtained their rotational distributions by LIF. The rotational distributions are bell- shaped, with widths of 10− 15 N and maxima that shift to higher N values at higher excitation energies. The authors favor a mechanism in which isomerization of ketene to the formylmethylene isomer (HCCHO) via the cyclic oxirane precedes dissociation to CH(X). In contrast, in the one-photon dissociation of ketene at 157.6 nm, the predominant product channel is CH 2 +CO, while the H + HCCO and CH + HCO channels account for less than 5% of the products. 72 One-photon dissociation, in this case, is assumed to proceed via excitation to the 3d Rydberg state, and the mechanism is different than in 1+1 photon excitation at about the same level of energy. It appears, therefore, that one- and two-photon dissociation processes in ketene at comparable excitation energies proceed via different mechanisms. In conclusion, a pathway that is initiated by two-photon absorption in diazirine and evolves via isomerization and H-shift to dissociation (either simultaneously or sequen- tially) agrees well with our experimental observations and can also explain the high internal energies deposited in the co-fragment. We point out that in the 1 1 A 2 state of diazirine, the calculated NN bond length is 1.422 ˚ A, close to the single bond length of isodiazirine, calculated at1.56− 1.63 ˚ A. 73–75 This should facilitate isomerization either directly on the excited state, or via a conical intersection with the ground state. 114 Below we discuss briefly two other possible routes to CH(X) and assess their feasi- bility. The first is a sequential pathway initiated by breaking one CH bond by two-photon absorption followed by one-photon dissociation of the diazirinyl radical, c-HCNN: c-CH 2 N 2 (1 1 A 1 )+2hν → c-HCNN (1 2 A “ )+ H( 2 S);Δ H 0 r =93 kcal/mol (IIa) c-HCNN (1 2 A “ )+hν → CH (X 2 Π )+ N 2 (X 1 Σ + g );Δ H 0 r =26 kcal/mol (IIb) Assuming that the maximum CH(X) translational energy allowed by the thermo- chemisty in each step is achieved, translational energies higher than 21,000 cm − 1 may be observed. In step (IIa), both the cyclic and the open-chain HCNN may be generated. To date, there is no experimental information on the diazirinyl radical, but calcu- lations show that it is a stable species, close in its heat of formation to the open-chain HCNN, HNCN, and CNNH structural isomers. 69, 70, 76–79 The HCNN system has recently attracted attention because of its relevance to the CH + N 2 reaction mechanism, and the facile isomerization among different structural isomers has been discussed. 69, 70, 76–79 Experimentally, while no studies of diazirinyl photodissociation are available, Neumark and coworkers have studied the photodissociation of jet-cooled open-chain HCNN radi- cals via a parallel transition at wavelengths that coincide with the photon energies used in the current experiments. 80 HCNN exhibits broad absorption features at26,000− 40,000 cm − 1 , with an onset of dissociation at about 25,400 cm − 1 , resulting in a broad transla- tional energy distribution in the CH(X) radical. While this mechanism can explain the observed high translational energies, it does not explain the observed anisotropy param- eter. Also, the requirement for three-photon absorption plus two photon for detection makes this pathway less likely than mechanism (I), and the importance of reaction (IIa) needs to be established independently. Another reaction sequence that may, in principle, lead to formation of CH(X) with high translational energies is initial one-photon dissociation on the 1 1 B 2 state of 115 diazirine to generate CH 2 fragments in one of the three lowest singlet states, followed by two-photon absorption in CH 2 to generate CH(X). We feel that such a sequence is unlikely, because (i) the one-photon 1 B 2 ← 1 A 1 absorption in diazirine is highly struc- tured; and (ii) the subsequent photon absorption by the small radical CH 2 should also be structured and not necessarily coincide with the CH D← X absorption at all wave- lengths. Thus, the broad and even nature of the absorption spectrum in our studies leads us to believe that the first step involves 1 + 1 excitation to the 1 A 2 state followed by dissociation to CH(X). 5.4 Summary Multiphoton ionization and dissociation processes in diazirine have been studied both experimentally (via304− 325 nm two-photon absorption) and theoretically by using the EOM-CCSD and B3LYP methods. The electronic structure calculations identified two valence states and four Rydberg states in the region 4.0− 8.5 eV . In one-photon exci- tation, the calculated strongest absorption is to the 2 1 A 1 (3p x ) Rydberg state, whereas in two-photon absorption at comparable energies via the low-lying 1 1 B 2 valence state, the strongest absorption is predicted to reach the dissociative valence 1 1 A 2 state. The diazirine ion should be rather unstable, with a binding energy of only 0.73 eV and a geometry that resembles a weakly bound CH + 2 ··· N 2 complex. On the basis of the electronic structure calculations, we conclude that two-photon absorption in diazirine is very efficient. Weak absorption to the 1 1 B 2 state is immedi- ately followed by more efficient absorption of another photon to reach the 1 1 A 2 state from which competition between ionization and fast dissociation takes place. Absorp- tion of a third photon leads to dissociative photoionization with the formation of CH + 2 fragment ions. No parent diazirine ions are detected. 116 Two-photon dissociation on the 1 1 A 2 state leads to efficient detection of CH(X) frag- ments. We propose that the most likely route to CH(X) formation is isomerization to isodiazirine followed by dissociation. 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Phys., 124, 2006. 122 Chapter 6 The 1,2,3-tridehydrobenzene triradical: 2 B or not 2 B? The answer is 2 A! 6.1 Introduction Triradicals 1 are of fundamental interest due to their complex electronic structure and unusual bonding patterns arising from the interaction of three formally unpaired elec- trons. The electronic structures of the phenyl radical and the three benzyne isomers, which are formally derived from benzene by removal of two hydrogen atoms, have been the subject of many investigations 2–11 . However, tridehydrobenzenes 1 - 3 (Fig. 6.1), the next step in the systematic decomposition of benzene (as discussed by Wenthold) 12 , are characterized less extensively. Their increased multiconfigurational character chal- lenges both theory and experiment. . . . . . . . . . 1 2 3 Figure 6.1: The three isomers of tridehydrobenzene. 123 Previous theoretical studies by Krylov and coworkers 1, 13, 14 demonstrated that the unpaired electrons form partial bonds between the radical centers, with stabilizing inter- actions ranging from 0.5 to 32 kcal/mol 14 . All three isomers possess low-spin doublet ground states. The doublet-quartet splitting is largest in 1,2,3-tridehydrobenzene (1 in Fig. 6.1), which indicates strongest interaction between adjacent radical sites. Isomers 2 and 3 are less stable than 1 by approximately 2.5 and 17.3 kcal/mol, respectively. These values are in agreement with heats of formation experimentally determined in gas phase by Wenthold and coworkers 15 . O I O O I O O O − CO 2 , − CO 1 4 308 nm 5 6 308 nm − I − CO 2 , − CO 248 nm − I ∆ = 8 K ∆ = 8 K Figure 6.2: Matrix isolation of 1,2,3-tridehydrobenzene. Compound 4 was photolyzed with 308 nm light using an XeCl excimer laser. Subsequent irradiation with 248 nm KrF or 193 nm ArF excimer lasers produced 4, as well as other side-products. Recently, Sander and coworkers reported the preparation and infrared (IR) spectro- scopic characterization of 1 isolated in cryogenic neon matrices at 3 K 16 , summarized in Fig. 6.2. 1,2,3-tridehydrobenzene was prepared by photolysis of 3-iodophthalic anhy- dride 4 resulting in formation of monoradical 5 and benzyne 6 that, upon subsequent short-wavelength irradiation, produced a species with three characteristic absorptions at 524, 753, and 1698 cm − 1 (among other side-products). Upon annealing the matrix at 124 8 K, recombination with iodine atoms was observed, leading to the disappearance of the three absorptions and back-formation of 6 (at the same time, the concentration of precursor 4 increased at the expense of 5). 10 a 1 11 a 1 7 b 2 2 B 2 2 A 1 4 B 2 - 0.587 - 0.434 - 0.182 - 0.548 - 0.355 - 0.432 Figure 6.3: Frontier molecular orbitals and the leading electronic configurations of the 4 B 2 , 2 A 1 , and 2 B 2 states of 1,2,3-tridehydrobenzene. Orbital energies (hartrees) for the UHF quartet reference are given at the optimized geometries of the two doublet states. Although the two lowest doublet states of 1,2,3-tridehydrobenzene, 2 A 1 and 2 B 2 (Fig. 6.3), are well separated in energy vertically 14 , they are nearly degenerate adia- batically, according to calculations by Sander and coworkers 16 . This large difference between the vertical and adiabatic gaps (see Fig. 6.4) originates in different bonding patterns in the two states leading to significant geometric relaxation. The relative adia- batic ordering of the two states depends strongly on the methods employed and a weak preference (1− 2 kcal/mol) for the 2 A 1 state was obtained at higher levels of theory 16 . The two isomers are separated by a barrier of only 4− 5 kcal/mol. By comparison of 125 calculated and measured vibrational spectra, the matrix-isolated species was identified as 2 A 1 − 1. ΔE 3 2 A 1 2 B 2 R C1-C3 H H H ΔE 1 ΔE 2 Figure 6.4: Vertical and adiabatic energy differences between the 2 A 1 and 2 B 2 states. This contribution presents a series of high-level calculations aiming to quantify the 2 A 1 − 2 B 2 splitting and to elucidate the electronic structure of the ground state of 1,2,3- tridehydrobenzene. Also, the measured IR data is compared to calculated vibrational spectra for both states at various levels of theory, including a discussion of anharmonic corrections at the BLYP/cc-pVTZ and B3LYP/cc-pVTZ levels. The structure of the paper is as follows. The next section outlines theoretical methods and computational details. Sec. 6.3 discusses equilibrium structures, vertical and adiabatic energy sepa- rations, and vibrational frequencies of the two states. Our final remarks are given in Sec. 6.4. 126 6.2 Theoretical methods and computational details Triradicals – species with three unpaired electrons distributed over three nearly degenerate orbitals – feature extensive electronic degeneracies that result in multi- configurational wave functions 1 . Similarly to diradicals 17–20 , the wave functions of low-spin states (M S = 1 2 ) are multi-determinantal, whereas the high-spin quartet state (M S =+ 3 2 ) can be well described by a single-reference wave function. In general, these low spin states are not accessible by the traditional ground state single-reference meth- ods, unless energy gaps between the frontier MO are large enough to stabilize one of the determinants. The M S = 1 2 states can be described by multi-reference methods employing a com- plete active space self-consistent field (CASSCF) reference wave function 21 . Dynam- ical electron correlation can be subsequently included by perturbation theory to sec- ond (RS2) or third (RS3) order 22 , or by truncated configuration interaction including single and double excitations (CISD) from the reference configurations. 23, 24 In all CI calculations discussed in this work, internal contraction is applied to keep the number of configurations in the CI expansion manageable, whereas only the doubly external excitations are contracted in perturbation theory. 22 Due to lack of size-consistency in truncated CI, the latter approach requires additional corrections, e.g., the Davidson cor- rection, denoted as CISD+Q. 25 We also apply corrections inspired by the cluster expan- sion, i.e., the multi-reference average-quadratic coupled cluster (AQCC) 26 and multi- reference averaged coupled pair functional (ACPF) methods 27 . Alternatively, all these states can be described accurately by the spin-flip (SF) mod- els 28–34 . In the SF approach, low-spin states are described as spin-flipping excitations 127 from a well-behaved high-spin reference state. In the case of triradicals, the SF method describes target states as: Ψ d,q M S =1/2 = ˆ R M S =− 1 ˜ Ψ q M S =3/2 , (6.1) where ˜ Ψ q M S =3/2 is the ααα high-spin reference determinant, e.g., 4 B 2 from Fig. 6.3, ˆ R M S =− 1 is an excitation operator that flips the spin of an electron (α →β ), andΨ d,q M S =1/2 stands for the wave functions of the doublet. e.g., 2 B 2 and 2 A 1 , and quartet target states. Since all the configurations (withM S = 1/2) present in the low-lying triradical states are formally obtained from the M S = 3/2 reference state by single excitations including a spin-flip, the SF method provides a balanced description of all the triradical states within a single-reference formalism. An attractive feature of the SF approach is that non-dynamical and dynamical cor- relation are included in a single computational step. The description of the latter can be systematically improved by employing theoretical models of increasing complexity for the reference wave function 28, 30, 32, 33, 35 . In this work, we use a density-functional based method, SF-DFT, which was shown to yield accurate equilibrium structures and frequencies 31, 36 , and two coupled-cluster based models 33, 35 : equation-of-motion SF coupled-cluster model with single and double substitutions [EOM-SF-CCSD or EOM- SF(2,2)], and the EOM-SF-CC method in which triple excitations are included in the EOM part, EOM-SF(2,3). When the energy separations between the frontier orbitals are sufficiently large, the closed-shell doublet wave functions become single-determinantal, and can also be well described by single-reference methods. The analysis of the 2 B 2 and 2 A 1 wave functions reveals that they are dominated by a single electronic configuration at the respective equilibrium geometries, i.e., the leading EOM-SF-CCSD amplitudes are 0.95 and 0.89 128 for the 2 A 1 and 2 B 2 states, respectively. This allows us to employ single-reference CCSD with perturbative inclusion of triple excitations, CCSD(T), 37, 38 to calculate equilibrium geometries and vibrational frequencies of these states. CCSD(T) calculations were performed using spin-unrestricted (UHF) and spin- restricted (ROHF) references. In the latter case, two different methods were employed: (i) ROHF-CCSD(T), which uses non-spin-adapted cluster excitation operators ˆ e ai ; and (ii) spin-adapted CCSD(T) [termed R-CCSD(T)], which employs the spin-coupled oper- ators ˆ E ai = ˆ e ˜ a ˜ i + ˆ e ai , where tilde and overline refer to α and β spin orbitals, respec- tively (see ref. 39 ). Although total energies are slightly different, the resulting structures, frequencies, and relative energies are very close. For instance, the adiabatic energy gap between the two lowest states is by 0.026 eV (0.6 kcal/mol) lower in the ROHF- CCSD(T) calculation. Equilibrium geometries and vibrational frequencies of these two low-lying states were calculated using the following methods: CCSD(T) 37, 38 with cc-pVTZ basis, 40 SF- DFT 31 with 6-311G(d) 41 basis and a 50/50 functional (50% Hartree-Fock + 8% Slater + 42% Becke for exchange and 19% VWN + 81% LYP for correlation) 31 , and B3LYP 42 with cc-pVTZ basis. The unrestricted (10a 1 ) 1 (7b 2 ) 1 (11a 1 ) 1 quartet reference was used for all spin-flip calculations. In CCSD(T) calculations, the ROHF reference was used in the optimization of the 2 A 1 and 2 B 2 states and UHF reference was employed in frequen- cies calculations. UHF-CCSD(T) geometries were also obtained, and are very similar to those obtained with ROHF reference (bond lengths within 0.005 ˚ A, angles within 0.01 o for 2 B 2 and 0.5 o for 2 A 1 ). Pure angular momentum spherical harmonics (5 d-functions) were used throughout. Single point CCSD(T)/cc-pVTZ calculations employed frozen core for the lowest six orbitals. 129 CCSD, CCSD(T) and SF-CC single point energies were obtained at either EOM-SF- CCSD/6-31G(d) or ROHF-CCSD(T)/cc-pVTZ optimized geometries. Effects of triple excitations were characterized by the EOM-SF(2,3) methods. To approximate the energies of more expensive methods with large basis sets, an extrapolation scheme based on energy separability was used: E large EOM− SF(2,3) =E large EOM− SF(2, ˜ 3) +(E EOM− SF(2,3) − E EOM− SF(2, ˜ 3) ) small (6.2) E large EOM− SF(2,3) =E large EOM− SF(2,2) +(E EOM− SF(2,3) − E EOM− SF(2,2) ) small (6.3) where E large and E small are total energies calculated in relatively large and small bases, respectively. This scheme assumes that changes in total energy due to basis set effects are similar for the different methods. In the EOM-SF(2, ˜ 3) method, triple excitations are included only within a specified active space. This active space was chosen to include the three orbitals at the radical centers and the six benzene valence orbitals. R-CCSD, R-CCSD(T), and multi-reference single point energies were also calcu- lated by CAS-RS2, CAS-RS3, MR-CISD, AQCC, and ACPF using systematically larger basis sets (cc-pVDZ, cc-pVTZ, and cc-pVQZ) for both R-CCSD(T) and B3LYP/cc- pVTZ geometries. CASSCF calculations employed the same active space as in the EOM-SF(2, ˜ 3) calculations, i.e., nine electrons in nine orbitals, CASSCF(9,9). We also calculated anharmonic corrections to harmonic frequencies at the BLYP/cc- pVTZ and B3LYP/cc-pVTZ levels. Three methods were used: Transition-Optimized Shifted Hermite Theory (TOSH), vibrational perturbation theory (VPT2) 43 , and vibra- tional configuration interaction (VCI) 44 . In VPT2, the third and fourth power terms in the expansion of the potential are treated as perturbations, and the matrix elements of harmonic wave functions are computed. However, this method breaks down for near degenerate modes. In TOSH, the harmonic wave functions are shifted along the normal 130 modes, by a distance σ . Comparing the first-order perturbation energy for this shifted wave function, with the second-order energy for the ordinary wave function, allows an approximation toσ that gives the second-order energy at only first-order cost. In VCI, the nuclear wave functions are expanded in the basis of a product of harmonic oscilla- tors. VCI(n) refers to the inclusion of all basis functions in which the sum of excitations in all the modes is equal to n. Based on benchmark studies, VCI(1) and VCI(3) usually overestimate the true energy, while VCI(4) is close to the converged result. At USC, calculations were performed using the Q-Chem 45 and ACES II 46 electronic structure packages. At UCLA and RUB, the Gaussian03and MOLPRO 47 packages were used. 6.3 Results and discussion 6.3.1 Electronic structure and equilibrium geometries of the 2 A 1 and 2 B 2 states 1,2,3-tridehydrobenzene, which is derived from benzene by removing three hydrogen atoms, has three adjacent sp 2 -hybridized radical centers. Frontier molecular orbitals (MOs) and electronic configurations of the 4 B 2 , 2 A 1 and 2 B 2 states are shown in Fig. 6.3. The leading electronic configurations are (10a 1 ) 1 (7b 2 ) 1 (11a 1 ) 1 , (10a 1 ) 2 (11a 1 ) 1 , and (10a 1 ) 2 (7b 2 ) 1 , respectively. The two lowest electronic states of 1,2,3-tridehydrobenzene are closed-shell doublets with the bonding 10a 1 orbital being doubly occupied (Fig. 6.3). The states differ in the single occupation of either the 7b 2 ( 2 B 2 state) or 11a 1 ( 2 A 1 state) orbitals. 131 The 10a 1 , 7b 2 , and 11a 1 orbitals (Fig. 6.3) are of overall bonding, nonbonding, and antibonding character, respectively. Thus, the 2 B 2 state, in which the bonding and non- bonding orbitals are occupied, appears to be a reasonable candidate for the ground state. However, the 11a 1 orbital also has bonding character between the C1 and C3 radical centers, and this character increases upon benzene ring deformations that bring the two centers closer together. The competition between C1-C2-C3 and C1-C3 interactions results in the two states being nearly degenerate adiabatically. 1.352 1.332 1.348 1.342 1.354 1.352 1.352 1.332 1.348 1.342 1.354 1.352 1.376 1.368 1.374 1.371 1.380 1.377 1.376 1.368 1.374 1.371 1.380 1.377 1.083 1.072 1.073 1.078 1.079 1.085 1.083 1.072 1.073 1.078 1.079 1.085 1.404 1.395 1.404 1.404 1.410 1.413 1.404 1.395 1.404 1.404 1.410 1.413 1.089 1.078 1.079 1.084 1.085 1.091 1.089 1.078 1.079 1.084 1.085 1.091 1.677 1.642 1.629 1.692 1.692 1.769 1.677 1.642 1.629 1.692 1.692 1.769 108.7 109.0 109.6 109.8 109.6 110.9 108.7 109.0 109.6 109.8 109.6 110.9 157.3 157.0 155.5 154.6 155.2 151.6 157.3 157.0 155.5 154.6 155.2 151.6 78.2 76.9 74.8 75.4 77.3 87.7 78.2 76.9 74.8 75.4 77.3 87.7 113.0 112.7 112.9 113.1 113.1 113.3 113.0 112.7 112.9 113.1 113.1 113.3 H H H H H H R C1-C3 : 1.295 1.287 1.307 1.281 1.300 1.295 1.295 1.287 1.307 1.281 1.300 1.295 1.395 1.385 1.396 1.394 1.402 1.404 1.395 1.385 1.396 1.394 1.402 1.404 1.076 1.086 1.075 1.081 1.082 1.087 1.076 1.086 1.075 1.081 1.082 1.087 1.406 1.408 1.398 1.406 1.412 1.416 1.406 1.408 1.398 1.406 1.412 1.416 2.355 2.360 2.326 2.342 2.367 2.359 2.355 2.360 2.326 2.342 2.367 2.359 130.9 130.3 129.1 130.9 131.1 131.2 130.9 130.3 129.1 130.9 131.1 131.2 122.9 122.7 123.0 123.1 123.1 122.9 122.9 122.7 123.0 123.1 123.1 122.9 117.4 117.8 117.0 117.2 116.8 117.0 117.4 117.8 117.0 117.2 116.8 117.0 115.9 116.1 116.1 115.9 116.0 116.0 115.9 116.1 116.1 115.9 116.0 116.0 H H H H H H 1.078 1.089 1.077 1.083 1.084 1.090 1.078 1.089 1.077 1.083 1.084 1.090 R C1-C3 : Figure 6.5: Equilibrium structure of the 2 A 1 (left) and 2 B 2 (right) states of 1,2,3- tridehydrobenzene. Bond lengths are in ˚ A and angles are in degrees. Geometrical parameters are listed as follows: R-CCSD(T)/cc-pVTZ; ROHF-CCSD(T)/cc-pVTZ; SF- CCSD/6-31G(d); SF-DFT/6-311G(d); BLYP/cc-pVTZ; B3LYP/cc-pVTZ The optimized geometries of the two doublet states are shown in Fig. 6.5. They reveal large structural differences, in agreement with the characters of the corresponding MOs. With respect to the meta radical centers (C1 and C3), the 11a 1 orbital is bonding, while the 7b 2 orbital is antibonding (see Fig. 6.3). This leads to a contracted C1-C3 bond length (1.68-1.69 ˚ A) in the 2 A 1 state, as compared to2.36− 2.37 ˚ A in the 2 B 2 state. This bonding interaction in the 2 A 1 state is much stronger than in singlet m-benzyne, where the corresponding distance is slightly above 2 ˚ A. 14, 48–50 The C1-C2 distance is longer by about 0.1 ˚ A relative to o-benzyne (1.24 ˚ A), reflecting the antibonding character of 132 the occupied 11a 1 orbital along these bonds. In the 2 B 2 state, the C1-C2 distance is 1.30 ˚ A, slightly longer than in o-benzyne, but shorter than in the 2 A 1 state of 1, showing that in this state the ortho- as well as meta- interaction is weaker than in the corresponding benzynes. The interaction of the three unpaired electrons in this state might therefore be called sigma allylic, in analogy with the nodal properties of the allyl system. Table 6.1: Selected structural parameters for the 2 A 1 and 2 B 2 equilibrium structures of 1. R C1− C2 is the bond length between adjacent radical carbons. A C1− C2− C3 is the angle formed by the three radical carbons. 2 B 2 state 2 A 1 state Method R 1− 3 R 1− 2 A 1− 2− 3 R 1− 3 R 1− 2 A 1− 2− 3 R-CCSD(T)/cc-pVTZ 2.367 1.300 131.1 1.692 1.354 77.3 ROHF-CCSD(T)/cc-pVTZ 2.355 1.295 130.9 1.677 1.348 76.9 EOM-SF-CCSD/6-31G(d) 2.361 1.307 129.1 1.642 1.353 74.77 SF-DFT/6-311G(d0 50/50 fnl 2.326 1.281 130.3 1.629 1.332 75.39 BLYP/cc-pVTZ 2.359 1.295 131.2 1.769 1.352 87.7 B3LYP/cc-pVTZ 2.342 1.287 130.9 1.692 1.342 78.2 For well-behaved methods, e.g. CCSD(T) and SF, the structures of the two states are not very sensitive to the method used. Table 6.1 lists selected structural param- eters obtained at various levels of theory. Despite a modest basis set, 6-31G(d), the EOM-SF-CCSD structures are in close agreement with the CCSD(T)/cc-pVTZ ones. Surprisingly, there are non-negligible differences between the two CCSD(T) methods. For bondlengths in well-behaved closed-shell molecules, the errors of the CCSD(T)/cc- pVTZ are about 0.008 ˚ A(see Ref. 51 . In the challenging case of the triradical, however, the differences in bondlengths calculated by the two CCSD(T) methods are of the same order, i.e., 0.006 ˚ A and 0.015 for the C1-C2 and C1-C3 distances in the 2 A 1 state. Relative to CCSD(T)/cc-pVTZ, SF-DFT/6-311G(d) (with 50/50 functional) slightly underestimates bond lengths and angles (by about 0.02 ˚ A and 1 o ). BLYP/cc-pVTZ 133 slightly overestimates them (by as much as 0.09 ˚ A and 4 o in the 2 A 1 state), while B3LYP/cc-pVTZ yields slightly tighter structures, consistent with the presence of Hartree-Fock exchange. Overall, for the 2 A 1 state, the B3LYP value for the C1-C3 bondlengths agrees with CCSD(T) better that the BLYP one, while for the C1-C2 bond in the 2 B 2 state the opposite is true. Relatively strong dependence of the DFT results on the fraction of Hartree-Fock exchange is common in open-shell species and systems with vibronic interactions, and has been rationalized in terms of the self-interaction error 52, 53 . Figure 6.6: Energy as a function of the distance between the radical centers C1 - C3. Following our earlier studies on m-benzynes 6, 50 , we also calculated PES scans along the reaction coordinate connecting the two minima, as sketched in Fig. 6.4. This is done by constraining the C1-C3 distance (at intervals of± 0.01 ˚ A around the minimum energy structures), and optimizing all other degrees of freedom at the DFT level. Subsequently, higher-level calculations were carried out at these geometries (Fig. 6.6). Regarding the C1-C3 equilibrium distance, the R-CCSD(T) structure (1.69± 0.01 ˚ A compared to 134 1.692 ˚ A in full optimization) is nicely flanked by the CAS-RS2 (1.71± 0.01 ˚ A) and CAS-RS3 (1.67± 0.01 ˚ A) distances. Our most reliable ROHF-CCSD(T)/cc-pVTZ and R-CCSD(T) estimates of the dis- tance between the meta radical centers in 2 B 2 and 2 A 1 -1 are 2.355− 2.367 ˚ A and 1.677− 1.692 ˚ A, respectively. 6.3.2 Vertical and adiabatic energy differences between the 2 A 1 and 2 B 2 states Fig. 6.4 defines the vertical and adiabatic energy separations of the two states. The corresponding values for the vertical electronic energiesΔ E 1 andΔ E 2 , and for the adi- abatic energy difference Δ E 3 , are listed in Tables 6.2 and 6.3. The ZPE correction is given separately and is not included inΔ Es. The vertical energy gaps between the two states are fairly consistent for all levels of theory, and rather large: at the 2 A 1 geometry (Δ E 1 ), the states are separated by 4.8 eV , and at the 2 B 2 (Δ E 2 ) geometry by 1.4 eV . However, the adiabatic difference (Δ E 3 ) is much smaller due to geometrical relaxation, and the two states are nearly degenerate. In contrast toΔ E 1 andΔ E 2 ,Δ E 3 depends strongly on the basis set employed in the calculation. Regardless of method, 6-31G yields an adiabatic energy difference of about 0.9 eV . Inclusion of polarization functions to carbon atoms is necessary, which lowers Δ E 3 to 0.1 eV or less. Increasing from double-zeta to triple-zeta basis, e.g. 6-31G(d)→ 6-311G(d), has a comparatively small effect (0.01 eV). The effect of additional polar- ization, e.g. 6-311G(d)→ 6-311G(2df), is also minor. Overall, for this system the EOM-SF method is converged at 6-31G(d). All EOM-SF(2,2) calculations place the 2 B 2 below the 2 A 1 state, but the latter is less stable by only 0.1 eV or less with polarized basis sets. Inclusion of triples further brings 135 Table 6.2: Vertical and adiabatic energy differences between the 2 A 1 and 2 B 2 states of 1,2,3-tridehydrobenzene, as defined in Figure 6.4, using at EOM-CCSD/6-31G(d) equilibrium geometries unless specified otherwise. Method Δ E a 1 , eV Δ E a 2 , eV Δ E b 3 , eV 1 EOM-SF-CCSD/6-31G 4.74 1.60 0.96 2 EOM-SF-CCSD/6-31G(d) 4.90 1.43 0.09 3 EOM-SF-CCSD/6-311G(d) (on C) 4.86 1.43 0.08 4 EOM-SF-CCSD/6-311G(2df) 4.86 1.43 0.08 5 EOM-SF(2, ˜ 3)/6-31G 4.70 1.55 0.90 6 EOM-SF(2,3)/6-31G 4.71 1.55 0.89 7 EOM-SF(2, ˜ 3)/6-31G(d) 4.84 1.43 0.003 8 EOM-SF(2,3)/6-31G(d) 4.85 1.43 -0.02 9 EOM-SF(2,3)/6-31G(d), extrapolated c 4.87 1.38 0.02 10 EOM-SF(2,3)/6-31G(d), extrapolated d 4.85 1.43 -0.01 11 EOM-SF(2,3)/6-311G(2df), extrapolated e 4.83 1.38 0.01 12 EOM-SF(2,3)/6-311G(2df), extrapolated f 4.81 1.43 -0.03 13 UHF-CCSD/cc-pVTZ ··· ··· -0.17 14 ROHF-CCSD/cc-pVTZ ··· ··· -0.17 15 UHF-CCSD(T)/cc-pVTZ ··· ··· -0.26 16 ROHF-CCSD(T)/cc-pVTZ ··· ··· -0.07 17 Δ ZPE g ··· ··· 0.006 a absolute value b E( 2 B 2 ) - E( 2 A 1 ). Negative values correspond to 2 A 1 ground state. c EOM(2,3)/6-31G(d)≈ EOM(2,2)/6-31G(d)+[EOM(2,3)-EOM(2,2)] 6− 31G d EOM(2,3)/6-31G(d)≈ EOM(2, ˜ 3)/6-31G(d)+[EOM(2,3)-EOM(2, ˜ 3)] 6− 31G(d) e EOM(2,3)/6-311G(2df)≈ EOM(2,2)/6-31G(2df)+[EOM(2,3)- EOM(2,2)] 6− 31G. f EOM(2,3)/6-311G(2df)≈ EOM(2,2)/6-31G(2df)+[EOM(2,3)- EOM(2,2)] 6− 31G(d). g at UHF-CCSD(T)/cc-pVTZ level. the 2 A 1 energy down relative to 2 B 2 . Restricting triple excitations to an active space (the three singly occupied and six totalπ andπ ∗ orbitals) at 6-31G(d) results in adiabatic difference of about 0.003 eV . Full triple excitations at the 6-31G(d) level predict 2 A 1 - 1 to be 0.02 eV more stable than the 2 B 2 state. 136 Table 6.3: Adiabatic energy difference Δ E 3 , eV , at R-CCSD(T)/cc-pVTZ and B3LYP/cc-pVTZ equilibrium geometries. All multi-reference calculations employ a CASSCF(9,9) reference. Method cc-pVDZ cc-pVTZ cc-pVQZ R-CCSD a -0.22 -0.23 -0.22 R-CCSD b -0.22 -0.23 -0.23 R-CCSD(T) a -0.10 -0.10 -0.09 R-CCSD(T) b -0.09 -0.10 -0.09 CASSCF a 0.66 0.68 0.68 CASSCF b 0.69 0.71 0.71 CAS-RS2 a -0.02 -0.03 -0.04 CAS-RS2 b -0.01 -0.03 -0.04 CAS-RS3 a -0.03 -0.04 -0.05 CAS-RS3 b -0.02 -0.04 -0.05 MR-CISD a 0.18 0.20 0.20 MR-CISD b 0.20 0.21 0.22 MR-CISD+Q a 0.00 0.01 0.01 MR-CISD+Q b 0.02 0.02 0.02 AQCC a 0.00 0.00 0.00 AQCC b 0.01 0.00 0.00 ACPF a -0.03 -0.03 -0.03 ACPF b -0.02 -0.03 -0.03 a B3LYP/cc-pVTZ equilibrium geometry. b R-CCSD(T)/cc-pVTZ equilibrium geometry. The systematic effect of partial and full triples onΔ E 3 suggests taking advantage of extrapolation schemes, Eqs. (6.2) and (6.3), to approximate the effect of full triples in the large basis set limit. Within a given basis set, the inclusion of active space triples affectsΔ E 3 significantly, while full triples provide only a smaller additional correction. For example, with the 6-31G basis set, the active space triples lower Δ E 3 by 0.06 eV relative to EOM-SF(2,2), while full triples contribute an additional 0.01 eV . With the 6-31G(d) basis set, EOM-(2, ˜ 3) lowersΔ E 3 by 0.09 eV , while EOM-(2,3) contributes an 137 additional 0.02 eV . Although absolute values ofΔ E 3 are very different (0.9 vs. 0.1 eV), this similarity indicates that (i) active space triples sufficiently account for most of the correlation provided by full triples, and (ii) this effect is fairly consistent across different basis sets. In terms of energy separability, Eq. (6.2) provides a better extrapolation scheme than Eq. (6.3), yielding a very accurate result forΔ E 3 in the 6-31G→ 6-31G(d) example (within 0.01 eV , see entries 8 and 10 in Table 6.2). The EOM-SF(2,3)/6-311G(2df) energy was extrapolated using the calculated EOM- SF(2,2)/6-311G(2df) value and the difference between the EOM(2,3) and EOM(2,2) val- ues with the 6-31G(d) basis set [Eq. (6.3)]. This gives− 0.03 eV , or− 0.69 kcal/mol, for Δ E 3 . Lowering by 0.03 eV to account for the difference between active and full space triples schemes yields a final approximated EOM-SF(2,3)/6-311G(2df) Δ E 3 of− 0.06 eV , or− 1.38 kcal/mol. The ROHF-CCSD(T)/cc-pVTZ and R-CCSD(T) energy differ- ences are− 0.07 and− 0.10 eV , or -1.61 and -2.3 kcal/mol, respectively. Both CCSD(T) methods agree very well with each other. Moreover, the excellent agreement between these two very different approaches, CCSD(T) and EOM-SF(2,3), is very encouraging. Both methods include triple excitations. CCSD(T) is very accurate for systems with single configurational wave functions, while EOM-SF-CCSD gives a balanced descrip- tion of a general triradical wave function, both in the limit of small and large energy separations between the frontier MOs. In addition to R-CCSD and R-CCSD(T) data, Table 6.3 summarizes results of multi- reference calculations. While Hartree-Fock, which systematically overestimates bond- ing interactions, places 2 A 1 significantly below 2 B 2 (see Ref. 16 ), CASSCF overestimates the contributions of antibonding configurations and reverses the state ordering. In both 138 cases, inclusion of dynamical correlation brings the two states closer in energy. Multi- reference perturbation theory to second or third order givesΔ E 3 =− 0.05 eV , in agree- ment with the EOM(2,3) and CCSD(T) results. Multi-reference CI approaches yield larger discrepancies. MR-CISD still favors the 2 B 2 state by as much as 0.20− 0.22 eV , while the Davidson correction, leads to a nearly vanishing energy gap, indicating that size-extensivity effects are significant for a proper description of the relative energy of the two doublet states. The Davidson-corrected gaps are in agreement with almost perfect degeneracy predicted by AQCC. ACPF, which is considered to be a particularly suitable compromise between multi-reference and size-extensivity effects, returns to a moderate preference of 0.03 eV for the 2 A 1 state. Using different equilibrium geometries [R-CCSD(T) vs B3LYP], as well as basis sets beyond cc-pVTZ level have negligible effect on adiabatic energy gaps. The inclu- sion of zero point energies provides a minor correction of 0.006 eV . To summarize, the best estimate of the adiabatic energy difference between the two states (including ZPE) is0.03− 0.09 eV , or0.69− 2.07 kcal/mol. Based on numerous benchmark studies 51, 54, 55 , the conservative estimate of the error bar for ourΔ E 3 is± 1 kcal/mol, or about 0.02 eV . 6.3.3 Vibrational spectrum The three experimentally observed absorptions at 1698, 753, and 524 cm − 1 have been assigned to the 2 A 1 ground state of the title triradical, based on a comparison with spec- tra calculated at the BLYP/cc-pVTZ level for both states 16 . Although two lower frequen- cies are reproduced reasonably well for both states, the third intense line, 1698 cm − 1 , is absent in the 2 B 2 state, which was a decisive argument in favor of 2 A 1 . According to these calculations, the fourth and fifth most intense absorptions of textbf1 in the 2 A 1 139 Figure 6.7: Experimental IR spectrum of 1 compared to B3LYP/cc-pVTZ calculated spectra for 2 A 1 and 2 B 2 states, including anharmonic corrections. state are around 399 and 1467 cm − 1 with rather low relative intensities of 17 and 18%, respectively. To further complete the IR spectrum of 1, refined measurements in the mid-IR region, as well as FIR measurements down to 200 cm − 1 , have been carried out. No additional signals that could unequivocally be assigned to the tridehydrobenzene were identified in these spectral ranges, however. On the computational side, we conducted series of calculations aimed to refine the- oretical frequencies and, most importantly, to establish error bars for the theoretical predictions. Due to the open-shell character of the triradical, the benchmark results obtained for the closed-shell molecules 51 are not directly transferable. Moreover, anhar- monicities and possible matrix-induced shifts further complicate the comparison with the experiment. The rest of this section presents our analysis of harmonic frequencies calculated by DFT, SF-DFT, and CCSD(T), as well anharmonic corrections evaluated by the VPT2, VCI, and TOSH approaches using DFT potential energy surfaces. 140 0 500 1000 1500 2000 2500 3000 3500 0 20 40 60 80 100 Intensity frequency, cm -1 calc. exp. 0 500 1000 1500 2000 2500 3000 3500 0 20 40 60 80 100 120 140 160 Intensity frequency, cm -1 calc. exp. 0 500 1000 1500 2000 2500 3000 3500 0 20 40 60 80 100 Intensity frequency,cm -1 calc. exp. Figure 6.8: Calculated frequencies for the 2 A 1 state (solid lines) compared to the three experimental absorptions (dashed lines). The intensities of the lines are scaled such that the intensities of the most intense experimental and calculated peaks are equal. Top: CCSD(T)/cc-pVTZ; middle: SF-DFT/6-311G(d) with 50/50 functional; bottom: BLYP/cc-pVTZ. Calculated harmonic frequencies and anharmonic corrections for the 2 A 1 and 2 B 2 states are summarized in Tables 6.4, 6.5, tab:anharm and 6.7, and shown as stick spectra with intensities (alongside experimental data) in Figs. 6.8 and 6.9. Results are given for: CCSD(T)/cc-pVTZ, SF-DFT/6-311G(d) with 50/50 functional, BLYP with VPT2, VCI2 141 0 500 1000 1500 2000 2500 3000 3500 0 20 40 60 80 100 Intensity frequency, cm -1 calc. exp. 0 500 1000 1500 2000 2500 3000 3500 0 20 40 60 80 100 Intensity frequency,cm -1 calc. exp. 0 500 1000 1500 2000 2500 3000 3500 0 20 40 60 80 100 Intensity frequency,cm -1 calc. exp. Figure 6.9: Calculated frequencies for the 2 B 2 state (solid lines) compared to the three experimental absorptions (dashed lines). The intensities of the lines are scaled such that the intensities of the most intense experimental and calculated peaks are equal. Top: CCSD(T)/cc-pVTZ; middle: SF-DFT/6-311G(d) with 50/50 functional; bottom: BLYP/cc-pVTZ. and TOSH anharmonicities, and B3LYP/cc-pVTZ with VPT2 anharmonic corrections. Among the electronic structure methods, CCSD(T) frequencies are expected to be the most accurate. For well-behaved sys- tems, CCSD(T)/cc-pVTZ harmonic frequencies are within 2% (based on benchmark studies of diatomics) of the experimental values, which translates into 11, 15, and 34 142 Table 6.4: CCSD(T)/cc-pVTZ frequencies and IR intensities. 2 A 1 state 2 B 2 state Symmetry Frequency I I rel Frequency I I rel 1a 1 474 31 36 579 1 1 2a 1 823 1 1 841 3 4 3a 1 1079 3 3 1038 0 0 4a 1 1103 1 1 1130 13 18 5a 1 1436 4 5 1431 1 1 6a 1 1803 45 52 1577 0 0 7a 1 3201 9 10 3195 1 1 8a 1 3273 1 1 3237 7 10 1a 2 603 0 0 460 0 0 2a 2 827 0 0 921 0 0 1b 1 391 1 1 448 0 0 2b 1 598 1 1 590 2 3 3b 1 805 87 100 800 72 100 4b 1 980 1 1 993 0 0 1b 2 550 64 74 433 21 29 2b 2 923 11 13 1120 1 1 3b 2 1101 8 9 1143 10 14 4b 2 1301 1 1 1283 0 0 5b 2 1323 12 14 1370 7 10 6b 2 1545 12 14 1492 1 1 7b 2 3269 1 1 3229 3 4 cm − 1 error bars for the three strongest transitions. However, comparison with exper- iment is not straightforward due to anharmonicities. Typical anharmonic corrections are about 2% of the harmonic frequencies (e.g., see a benchmark study of diatomics 56 ), although anharmonicities of3− 5% are rather common, yielding a correction of about 15− 30 cm − 1 for a 500 cm − 1 harmonic mode. Moreover, anharmonicities may mix normal modes and significantly alter intensities. 143 Table 6.5: SF-DFT/6-311G* using 50/50 functional frequencies and IR intensities. 2 A 1 state 2 B 2 state Symmetry Frequency I I rel Frequency I I rel 1a 1 539 42 27 625 1 1 2a 1 857 2 1 893 3 3 3a 1 1107 2 1 1069 0 0 4a 1 1133 0 0 1165 16 19 5a 1 1491 6 4 1496 1 1 6a 1 1886 32 20 1649 0 0 7a 1 3287 11 7 3278 2 2 8a 1 3339 2 1 3305 13 15 1a 2 555 0 0 468 0 0 2a 2 791 0 0 939 0 0 1b 1 432 6 4 445 1 1 2b 1 645 2 1 601 3 3 3b 1 875 157 100 798 86 100 4b 1 999 0 0 1006 1 1 1b 2 611 49 31 435 23 27 2b 2 951 8 5 1158 5 6 3b 2 1130 7 4 1203 5 6 4b 2 1344 1 1 1350 1 1 5b 2 1386 9 6 1402 4 5 6b 2 1600 9 6 1544 4 5 7b 2 3335 1 1 3298 10 12 Vibrational frequencies for the 2 A 1 state are shown in Fig. 6.8. The strongest (most intense) calculated peak lies within 6.5 % of the strongest experimental peak, for both states and all methods except SF-DFT/6-311G(d) (which shows a 14 % difference for the 2 A 1 state). For all methods the strongest frequency corresponds to an out-of-plane (b 1 ) wagging mode of the three hydrogens. All methods also reproduce the experimentally observed frequency at 1698 cm − 1 , which is an a 1 breathing mode of the three radical 144 carbons, in the 2 A 1 state only. The differences are 5.8, 9.9, and 0.6% for CCSD(T), SF-DFT, and anharmonically-corrected B3LYP frequencies, respectively. Table 6.6: BLYP/cc-pVTZ harmonic and fundamental vibrational frequencies for the two doublet states of 1. 2 A 1 state 2 B 2 state Symmetry ν harm TOSH VPT2 VCI(2) ν harm TOSH VPT2 VCI(2) 1a 1 412 403 402 410 569 550 545 555 2a 1 810 802 803 808 817 804 804 809 3a 1 1042 1021 1011 1028 994 983 982 988 4a 1 1050 1037 1038 1043 1094 1082 1080 1091 5a 1 1375 1347 1335 1354 1369 1344 1330 1352 6a 1 1715 1683 1693 1692 1519 1503 1510 1516 7a 1 3081 2901 2904 2944 3092 2826 2877 2929 8a 1 3147 3000 3001 3065 3116 2954 2966 2995 1a 2 550 545 546 550 431 438 435 444 2a 2 772 785 593 810 833 853 733 871 1b 1 359 325 268 334 384 380 362 388 2b 1 557 549 548 554 535 548 534 556 3b 1 735 710 543 737 729 726 568 748 4b 1 898 826 684 850 910 897 813 913 1b 2 519 512 499 520 437 432 377 449 2b 2 894 867 872 875 1092 1056 1041 1067 3b 2 1081 1044 1027 1053 1143 1124 1125 1134 4b 2 1258 1229 1212 1237 1260 1230 1212 1242 5b 2 1323 1288 1283 1300 1390 1337 1317 1350 6b 2 1474 1435 1428 1444 1475 1397 1433 1439 7b 2 3144 2913 2984 2967 3109 2865 2932 2921 The lowest frequency peak, experimentally observed at 524 cm − 1 , cannot be defi- nitely assigned by the calculations. At the CCSD(T) and SF-DFT levels especially, two modes of sizeable intensity are calculated within the close energy region: the 1a 1 and 1b 2 modes, both in-plane deformations of the carbon skeleton with large amplitudes on 145 the meta radical centers. At the CCSD(T) level, the 1b 2 mode (at 550 cm − 1 ) is within 4.7 % of the experimental frequency and comparable in relative intensity. However, the 1a 1 mode (at 474 cm − 1 ) is within 9.5 % and also has considerable (though less than the 1b 2 ) intensity. Table 6.7: B3LYP/cc-pVTZ frequencies, IR intensities, and anharmonic corrections cal- culated to second order of perturbation theory. Anharmonic force constants up to quar- tic order were obtained by finite differences of Cartesian Hessians, for terms with up to three distinct indices. 58 2 A 1 state 2 B 2 state Symmetry ν harm I I rel ν anharm δν ν harm I I rel ν anharm δν 1a 1 471 27 33 452 -19 448 0 0 441 -6 2a 1 829 1 1 818 -10 883 0 853 0 -30 3a 1 1070 4 5 1047 -24 1121 9 13 1107 -15 4a 1 1086 1 1 1064 -22 1122 14 20 1101 -20 5a 1 1423 4 5 1390 -34 1424 5 8 1400 -25 6a 1 1784 37 45 1708 -77 1562 0.2 0 1510 -52 1a 2 581 0 0 570 -11 584 2 2 562 -22 2a 2 802 0 0 787 -15 847 3 4 825 -21 1b 1 367 0.3 0 362 -5 416 1 1 408 -8 2b 1 581 2 2 571 -10 563 3 5 551 -11 3b 1 772 82 100 756 -16 766 70 100 748 -18 4b 1 950 0.2 0 930 -19 1027 0 0 1007 -19 1b 2 548 65 79 535 -13 469 31 45 456 -14 2b 2 916 9 11 892 -23 964 0.1 0 944 -21 3b 2 1098 4 5 1081 -17 1167 4 6 1142 -26 4b 2 1294 0.5 1 1265 -29 1300 1 1 1277 -23 5b 2 1346 6 8 1302 -45 1418 0.3 0 1385 -33 6b 2 1522 14 17 1482 -40 1512 1 1 1475 -38 At the SF-DFT level, both low-frequency modes shift to higher frequencies: the 1a 1 mode, at 539 cm − 1 , is slightly higher than experiment, and the 1b 2 is at 611 cm − 1 . This contradicts the CCSD(T) results. 146 Frequencies for the 2 B 2 state are shown in Fig. 6.9. The two low-frequency observed modes are both unequivocally assigned for all methods. However, the third peak at 1698 cm − 1 is absent in all calculated spectra. There is a region around 1100 cm − 1 where one or two states consistently appear with some intensity; however, no experimental peak is recorded within 350 cm − 1 , or about 30 %, of this frequency region. Overall, the frequencies of this electronic state are much less sensitive to the method, compared to 2 A 1 . Finally, in order to compare different treatments of anharmonicities, anharmonic corrections by several methods were calculated at the BLYP/cc-pVTZ level (see Table 6.6). Enormous corrections at the VPT2 level suggest the failure of this approach: for example, the frequencies at 735 and 772 cm − 1 are both shifted by over 150 cm − 1 . The TOSH and VCI(2) corrections are in relatively agreement for all modes. For the 2 A 1 state, TOSH gives an average correction of 3.1%, while VCI(2) gives a 1.8% correction. For the 2 B 2 state, TOSH gives 2.2% correction, while VCI(2) gives only 0.6%. With regard to anharmonically corrected BLYP frequencies, the VCI(2) results pro- vide the best agreement with experiment. The two lower modes (at 519 and 735 cm − 1 , harmonically) remain essentially unchanged. This is notable since these modes already closely reproduce experiment at the harmonic level. Moreover, the third mode (at 1715 cm − 1 , harmonically) is corrected to 1692 cm − 1 , very close to the experimental frequency (1698 cm − 1 ). We also conducted B3LYP calculations of the harmonic frequencies and anharmonic corrections. Similarly to equilibrium geometries, B3LYP frequencies agree better than BLYP with the CCSD(T) values for the 3 most intense transitions (6a 1 , 3b 1 , and 1b 2 ). The BLYP frequencies for these modes are lower than the CCSD(T) ones by 88, 70, and 31 cm − 1 , respectively, whereas the differences between B3LYP and CCSD(T) are only 147 19, 33, and 2 cm − 1 . Relatively strong dependence of the DFT results on the fraction of Hartree-Fock exchange is common in open-shell species and systems with vibronic interactions, and is due to self-interaction error. Although the agreement between the calculated B3LYP anharmonic frequencies and the experiment is remarkable – they are within 10, 3, and 11 cm − 1 of each other, the discrepancies between the CCSD(T) and DFT harmonic frequencies, as well as functional dependence, suggest more conservative estimate of the error bars of10− 30 cm − 1 . The fourth most intense absorption is calculated at 470 cm − 1 (B3LYP and CCSD(T)). The anharmonic correction for this line is estimated to be 2− 20 cm − 1 . However, despite refined measurements in the IR and FIR regions, no signal in this spectral range could unequivocally be assigned to the triradical. Whether the calculated (harmonic) intensities for this mode are too high (at CCSD(T), SF-DFT, and DFT lev- els of theory), or whether the signal escapes detection for experimental reasons (e.g. line-broadening due to side-splitting, or a matrix-induced shift), cannot be answered conclusively on the basis of the available data. Nevertheless, the overall agreement of the three most intense experimentally observed absorptions with the high-level calculations described in this work, allows the matrix-isolated species to be identified as the 2 A 1 ground state of the 1,2,3- tridehydrobenzene triradical. Moreover, this is also supported by calculated adiabatic energy differences between the two electronic states. 148 6.4 Conclusions: electronic structure and infrared spec- tra A variety of high-level quantum chemical methods were employed to characterize the two lowest electronic states of the 1,2,3-tridehydrobenzene triradical and to determine their relative energy. According to the nodal characteristics of the singly occupied molecular orbitals, the 2 A 1 state of 1 shows a C1 - C3 bonding interaction, with a dis- tance between the formal radical centers around1.68− 1.69 ˚ A, whereas this distance is much larger in the 2 B 2 state (2.36− 2.37 ˚ A). As estimated by several correlated methods, the 2 A 1 state is adiabatically 0.03− 0.09 eV lower in energy. Assignment of the 2 A 1 ground state is also supported by comparison of calculated vibrational frequencies with the measured matrix-IR spectrum of the molecule. The three absorptions assigned to the triradical are reproduced by ab initio and anharmonically-corrected DFT calculations. The extremely small energy gap between the 2 A 1 and 2 B 2 states suggests that the character of the ground state can be easily manipulated by introduction of appropriate substituents. In view of the different bonding patterns, the two electronic states of 1 are expected to differ considerably in their properties and reactivity. Investigations of substituted tridehydrobenzenes and attempts to understand substituent effects in these systems are currently in progress in our laboratories. 6.5 Abstract: variational calculations on surfaces Anharmonic vibrational levels for the two lowest electronic states of 1,2,3- tridehydrobenzene are investigated using full-dimensional potential energy surfaces, fit- ted to energy data at the B3LYP/6-311(+)G** level. The two states are the 2 B 2 and 149 2 A 1 doublet states, which feature different interactions between the unpaired electrons. Tridehydrobenzene has 21 normal modes, with several modes displaying anharmonic- ities of 4− 6 %. This provides a good benchmark system for comparison of different methods including vibrational self-consistent field, vibrational perturbation theory, and vibrational configuration interaction. The large dimensionality for this system leads to large basis sets in the variational approach and high density of states. 6.6 Background In a previous article we characterized the electronic structure of 2 B 2 and 2 A 1 states of 1,2,3-tridehydrobenzene, and calculated the adiabatic energy gap using high-level methods and incorporating triples corrections. Harmonic frequencies were calculated and compared to experimental IR bands, allowing assignment of 2 A 1 as the ground state, with an adiabatic energy gap of 0.7− 2.1 kcal/mol. In this chapter, we fit full- dimensional potential energy surfaces (PESs) around the minima of these states and focus on describing the anharmonicity in the vibrational structure. For medium systems (greater than∼ 6 atoms), the dimensionality of the vibrational problem makes the variational approach unwieldy. In general, VCI is necessary to achieve a quantitative description of overtone and combination bands (compared to scal- ing methods or perturbation theory), and even for fundamental bands (0→ 1) when degeneracies exist or mode-mode coupling is strong. However, for single-molecule sys- tems with modest anharmonicity, it can be hoped that perturbation theory, or even VSCF, can recover most of the correlation. This is the case for the fundamental levels of 1,2,3- tridehydrobenzene. The fundamentals often the most important for spectroscopy. In infrared spectroscopy, fundamental bands are usually an order of magnitude larger than combinations and overtones due to selection rules of the harmonic oscillator. 150 Another issue is the size of the variational basis required for convergence of the energy levels. Calculating intramolecular vibrational redistribution (IVR) and rate con- stants utilizes statistical models that require vibrational energies for the total density of states ρ (E); these are usually binned harmonically even at high energies (>10,000 cm − 1 ). The study of IVR using variational approaches for large molecules has a long history. Stuchebrukhov and Marcus developed a tier model based on perturbative cou- pling orders and used it to study IVR from excited overtones of the acetylinic H-stretch in (CH 3 ) 3 XCC-H, where X = C or Si. It was observed that for X = C, IVR proceeded via many intermediate off-resonant transitions facilitated by multimode couplings (in 3rd and 4th order), and that this was more important than ρ (E). In the case where X = Si, an accidental absence of strong resonances in the region of the excited overtones leads to a bottleneck of the IVR, despite the fact thatρ (E) is 30 times larger than for X = C! In general, an excited “light” state excited by overtone spectroscopy is not an eigen- state, and it couples to the manifold of “dark” states with a decay rate constant given by the golden rule. This coupling is proportional to both the density of vibrational states and the coupling element between wavefunctions. Several studies have investigated the IVR of benzene. Sibert et al. used the 21 planar modes of benzene in classical and quan- tum studies for high-energy overtones. 57, 58 These indicated that 7 ring normal modes acquire significant energy from the v = 6 CH overtone. Their force field included up to cubic terms and neglected intra-ring couplings. In another study, a 30 normal mode model was employed. 59 It was found that energy flow into normal modes under 1200 cm − 1 was not significant; however, significant energy redistribution was seen into the in-plane CH wags in the1300− 1800 cm − 1 range. 151 6.7 Computational details The potential energy surfaces were fitted to ab initio data at the B3LYP/6-311(+)G** level of theory. The PESs are 7th degree polynomials in Morse variables of the set of interatomic distances, represented in a specially constructed basis invariant to permuta- tions of like nuclei. The Morse variables are defined as: y(i,j) = e − r(i,j) λ . r(i,j) is the internuclear distance between atoms i and j, and the value ofλ is 2 bohr. The polynomi- als each contain 1,353 terms. The 2 B 2 PES was fitted to 10,432 B3LYP energies, with 7,596 points in the range [0,0.1) a.u. above the global minimum, 1,169 points in the range [0.1,0.2) a.u., and 1,667 points above 0.2 a.u. The 2 A 1 PES was fitted to 11,275 energies, with 4,567 points in the range [0,0.1) a.u. above the global minimum, 2,425 points in the range [0.1,0.2) a.u., and 4,283 points above 0.2 a.u. For both surface, points above 0.2 a.u. were included to enforce asymptotes for highly twisted configurations and small internuclear distances. During the fitting process, extensive classical trajectories were run at a total energy of 0.22 hartree (∼ 48,000 cm − 1 ) to capture unphysical holes in the 21-dimensional surface. The harmonic frequencies on the PESs reproduce the ab initio frequencies very closely. The average absolute difference is 10.7 cm − 1 for the 2 A 1 state, and 8.2 cm − 1 if the three high-frequency hydrogen stretches are omitted, which are less well reproduced by the PES (within 37, 20, and 20 cm − 1 difference from ab initio, in order of frequency). On the 2 B 2 PES, the average absolute difference is 11.1 cm − 1 , with a maximum differ- ence between ab initio and PES of 37 cm − 1 . Vibrational energies and wavefunctions were calculated on the PESs using four methods: individual diagonalization of the 1D potentials, vibrational self-consistent field (VSCF), vibrational Moller-Plesset to second order (VMP2), and vibrational con- figuration interaction (VCI). The basis for the VSCF optimized modals was the set of 152 harmonic oscillators, with v = 0− 15 and corresponding frequency in each normal coordinate. For calculation of the fundamental transitions, the post-VSCF basis set for VMP2 and VCI consisted of up to 4 excitations from the ground VSCF reference. Nor- mal mode coupling was restricted to 2-mode coupling terms, and ro-vibrational coupling terms were neglected. The numerical quadrature was performed using Gauss-Hermite quadrature using 20 points per dimension. To compare with results on the PES, anharmonic corrections were also calculated by several methods using Q-Chem, 45 by first fitting a quartic force-field and then calcu- lating analytic matrix elements over the harmonic oscillator basis. Three methods were employed: vibrational perturbation theory (VPT2) 43 , transition-optimized shifted Her- mite theory (TOSH), 60 and VCI up to the fourth order of excitation. The parameters for the force field were calculated by finite differences, and 3rd and 4th order terms with all unique indexes were omitted. All electronic structure calculations were performed using the Q-Chem code. 45 All vibrational calculations were performed using ezVibe, 61 a new code available on the web. Correctness was benchmarked against Multimode, 62 with agreement in VSCF and VCI energies to within1− 2 cm − 1 . 6.8 Anharmonicities The molecular orbitals have been discussed in detail in the first part of this chapter. The equilibrium structures at the B3LYP/6-311(+)G** level are similar to the DFT and CCSD(T) structures described earlier. Harmonic frequencies and anharmonic calculations utilizing the quartic force field at the B3LYP/6-311(+)G** level are shown in Tables 6.8 and 6.8. Comparison between harmonic, perturbative (VPT2 and TOSH), and variational (VCI[4]) shows that approx- imate methods to include anharmonicity account for the largest part of the correlation. 153 For the 2 A 1 state, the average absolute percent difference between harmonic frequencies and VCI frequencies is 2.22%. The average differences between VPT2/TOSH frequen- cies and VCI are 0.62 and 0.49%, respectively. Very similar numbers are seen for the 2 B 2 state. In all tables, the percent difference of a given energy for each method is defined as the difference between the energy and corresponding VCI energy, divided by the VCI energy. The average of the absolute values over all 21 states are listed as the bottom entry in the tables. It can be noted that in both cases, TOSH is an improvement over VPT2. The two sets of perturbative methods both tend to overcompensate for cor- relation, giving lower energies than VCI, as opposed to the harmonic frequencies which are generally higher than VCI. 154 Table 6.8: 2 A 1 B3LYP/6-311(+)G** harmonic frequencies and anharmonicities No. E Harm. Δ E Harm. E VPT2 Δ E VPT2 E TOSH Δ E TOSH E VCI[4] v1 367 -2 370 1 370 1 369 v2 470 9 454 -7 456 -5 461 v3 540 0 536 -4 536 -4 540 v4 576 6 567 -3 567 -3 570 v5 576 0 574 -2 574 -2 576 v6 763 -54 830 13 830 13 817 v7 793 -50 852 9 852 9 843 v8 827 4 817 -6 817 -6 823 v9 913 6 902 -5 902 -5 907 v10 941 -29 977 7 977 7 970 v11 1066 10 1048 -8 1055 -1 1056 v12 1083 11 1064 -8 1072 0 1072 v13 1097 14 1080 -3 1080 -3 1083 v14 1286 11 1272 -3 1272 -3 1275 v15 1348 20 1322 -6 1322 -6 1328 v16 1415 17 1393 -5 1395 -3 1398 v17 1520 19 1496 -5 1496 -5 1501 v18 1786 32 1741 -13 1750 -4 1754 v19 3167 185 2986 4 2967 -15 2982 v20 3222 183 3032 -7 3032 -7 3039 v21 3226 132 3074 -20 3088 -6 3094 %diff a ··· 2.22 ··· 0.62 ··· 0.49 0 a The average of the absolute percent difference, defined for each method, frequency as E method− VCI /E VCI 155 Table 6.9: 2 B 2 B3LYP/6-311(+)G** harmonic frequencies and anharmonicities No. E Harm. Δ E Harm. E VPT2 Δ E VPT2 E TOSH Δ E TOSH E VCI[4] v1 410 0 407 -3 407 -3 410 v2 441 -5 444 -2 444 -2 446 v3 469 -1 467 -3 467 -3 470 v4 552 -10 560 -2 560 -2 562 v5 582 10 568 -4 571 -1 572 v6 754 -50 822 18 822 18 804 v7 840 11 824 -5 828 -1 829 v8 871 -39 915 5 915 5 910 v9 950 -43 995 2 995 2 993 v10 1024 7 1009 -8 1015 -2 1017 v11 1120 8 1106 -6 1108 -4 1112 v12 1125 24 1099 -2 1099 -2 1101 v13 1169 13 1152 -4 1152 -4 1156 v14 1298 15 1279 -4 1279 -4 1283 v15 1413 20 1390 -3 1391 -2 1393 v16 1421 24 1396 -1 1396 -1 1397 v17 1506 15 1482 -9 1482 -9 1491 v18 1557 20 1531 -6 1537 0 1537 v19 3170 181 2990 1 2981 -8 2989 v20 3190 195 2996 1 2996 1 2995 v21 3196 122 3043 -31 3069 -5 3074 %diff a ··· 2.33 ··· 0.52 ··· 0.39 0 a The average of the absolute percent difference, defined for each method, frequency as E method− VCI /E VCI 156 Table 6.10: 2 A 1 PES harmonic frequencies and anharmonicities No. E Harm. Δ E Harm E 1D Δ E 1D E VSCF Δ E VSCF E VMP2 Δ E VMP2 E VCI v1 361 -1 371 9 362 0 362 0 362 v2 480 22 490 32 456 -2 458 0 458 v3 553 7 555 9 542 -4 546 0 546 v4 568 5 569 6 559 -4 563 0 563 v5 569 2 574 7 566 -1 567 0 567 v6 771 -29 840 40 803 3 798 -2 800 v7 784 -14 838 40 801 3 795 -3 798 v8 830 11 829 10 816 -3 819 0 819 v9 916 -1 917 0 916 -1 917 0 917 v10 925 -4 973 44 932 3 927 -2 929 v11 1082 12 1089 19 1075 5 1066 -4 1070 v12 1092 11 1091 10 1081 0 1069 -12 1081 v13 1099 -32 1140 9 1129 -2 1131 0 1131 v14 1307 13 1313 19 1292 -2 1293 -1 1294 v15 1338 6 1346 14 1329 -3 1332 0 1332 v16 1415 18 1419 22 1397 0 1397 0 1397 v17 1522 16 1525 19 1502 -4 1505 -1 1506 v18 1782 37 1785 40 1745 0 1744 -1 1745 v19 3204 222 3156 174 2987 5 2983 1 2982 v20 3242 236 3307 301 3028 22 3000 -6 3006 v21 3246 154 3190 98 3109 17 3084 -8 3092 %diff ··· 2.19 ··· 2.84 ··· 0.30 ··· 0.14 0 157 Table 6.11: 2 B 2 PES harmonic frequencies and anharmonicities No. E Harm. Δ E Harm. E 1D Δ E 1D E VSCF Δ E VSCF E VMP2 Δ E VMP2 E VCI v1 415 0 423 8 413 -2 415 0 415 v2 442 -4 445 -1 444 -2 447 1 446 v3 469 -5 472 -2 471 -3 474 0 474 v4 535 -9 542 -2 542 -2 544 0 544 v5 545 -7 549 -3 549 -3 552 0 552 v6 748 -20 799 31 775 7 766 -2 768 v7 829 12 828 11 815 -2 818 1 817 v8 864 -3 901 34 870 3 865 -2 867 v9 959 3 983 27 954 -2 955 -1 956 v10 1016 -7 1018 -5 1026 3 1023 0 1023 v11 1129 26 1129 26 1108 5 1118 15 1103 v12 1131 16 1137 22 1114 -1 1115 0 1115 v13 1186 26 1188 28 1158 -2 1160 0 1160 v14 1290 21 1297 28 1268 -1 1268 -1 1269 v15 1421 23 1427 29 1398 0 1397 -1 1398 v16 1445 41 1448 44 1404 0 1404 0 1404 v17 1524 21 1537 34 1501 -2 1503 0 1503 v18 1539 26 1540 27 1516 3 1513 0 1513 v19 3156 147 3048 39 2992 -17 3017 8 3009 v20 3187 197 3262 272 3013 23 2988 -2 2990 v21 3189 108 3141 60 3086 5 3065 -16 3081 %diff ··· 1.94 ··· 2.22 ··· 0.34 ··· 0.16 0 158 Harmonic frequencies and anharmonic calculations utilizing the symmetrized PESs are shown in Tables 6.10 and 6.11, for the 2 A 1 and 2 B 2 states, respectively. Comparison between harmonic, 1D diagonalized, VSCF, VMP2, and VCI is listed. For both states, the 1D diagonalization provides a worse description relative to VCI than the harmonic approximation. This supports the opinion that multimode effects are equal in magnitude to diagonal anharmonicity. The diagonal energies also exclusively overestimate the fre- quency, in contrast with perturbation theory. The average (absolute) percent difference in frequency relative to VCI is 2.19 and 2.84 cm − 1 for 2 A 1 , and 1.94 and 2.22 cm − 1 for 2 B 2 , for harmonic and diagonal approximations, respectively. The VSCF, VMP2, and VCI energies are all extremely close. At the VSCF level, the average % difference from VCI is 0.30 and 0.34% for 2 A 1 and 2 B 2 states, respectively. On the 2 A 1 surface, the maximum differences are 22 and 17 cm − 1 for two of the high- frequency stretches, and 5 cm − 1 for one mode below 3,000 cm − 1 . 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It is believed to play a role in the pho- tochemistry of formaldehyde and in the interstellar media 1–3 . Recently, possible role of hydroxycarbene in the reaction of carbon atom with water has been considered 7, 8 . Due to its relatively mild diradical character (the ground state is a singlet, and the ST gap is about 1 eV), this is a well behaved system from the electronic structure point of view, and it has been characterized computationally 2–5, 7, 8 . The H 2 CO→HCOH iso- merization was discussed in classical and quantum MD simulations of the lowest disso- ciation channel, H 2 CO→H 2 + CO 9–11 . These isomers are also contained in the ab initio global potential energy surface of Zhang et al. 12 and they play a role in the interesting ”roaming” dynamics in the H 2 CO photodissociation 13 . HCOH can be derived from formaldehyde following protonation of the oxygen 5, 6 . Hydroxycarbene has been produced in the photodissociation of CH 2 OH 14 , which enabled the determination of its heat of formation. As we learned during the final stages of the manuscript preparation, Schreiner and coworkers have successfully synthe- sized trans-HCOH and HCOD and recordered their IR spectra in argon matrix 15 . They 164 reported that HCOH is relatively short-lived even in cryogenic environment and rear- ranges to formaldehyde with the half-life of about two hours. The deuterated species, HCOD, however, has been found to be very stable at these conditions. The assignment of the observed IR band origins was supported by variational calculations of vibrational energy levels using CCSD(T)/cc-pCVQZ quartic force field 15 . The differences between the computed and experimental values were less than 1% for all the observed transitions except for the two fundamental transitions,ν 1 in HCOH andν 2 in HCOD, for which the deviations were 1.7% and 1.4%, respectively. 12,085 H C O H H C H O H C O H H C O H H C O H H C O H … 0.0 -18,060 1,616 9,363 12,868 E (cm -1 ) Figure 7.1: Stationary points on the HCOH PES. The subspace connecting formalde- hyde with trans-hydroxycarbene (at 12,085 cm − 1 ) is not defined on our surface fits: at this point, CCSD(T)/cc-pVTZ energy at B3LYP/cc-pVTZ optimized transition state is shown for comparison. The lowest energy isomer, trans-hydroxycarbene, is 0.20 eV below the cis-isomer, and is separated by a relatively low barrier of 0.96 eV , as shown in Fig. 7.1. The observed fast rearrangement of HCOH to formaldehyde via tunneling through a higher barrier of 1.50 eV suggest even shorter life-times of cis-HCOH, which, however, might be increased by deuteration. 165 To be of a benchmark value for experimentalists, anharmonic corrections should be taken into account even for the fundamental transitions in rigid molecules, e.g., in formaldehyde they are of 3− 5% of the harmonic frequencies 16, 17 . Their magnitude increases for overtone transitions. For the higher-energy isomers, cis- and trans- HCOH, the PES exhibits stronger anharmonicities, and the harmonic IR spectrum is quite dif- ferent from the exact one. Lower symmetry of HCOH can also contribute to larger anharmonicities. We report computed infrared (IR) spectra of the two hydroxycarbene isomers. We combine high-level ab initio calculations of potential energy and dipole moment sur- faces (PES and DMS, respectively) with numerically accurate calculations of vibrational levels and wave-functions. The latter are used to evaluate IR intensities. Excellent agree- ment between the experimental spectra and our results for the trans-isomer suggests that the predicted spectra for cis-HCOH/HCOD are accurate as well, and can aid in a future experimental characterization of the cis-isomer. Moreover, our full-dimenional PES covering both isomers and the transition barriers between them can be used to investi- gate isomerization dynamics. 7.2 Vibrational self-consistent field theory A brief description of some of the methods used to obtain anharmonic energies and wavefunctions on potential energy surfaces is described below. The Vibrational self- consistent field (VSCF) method provides an optimized basis set, akin to Hartree-Fock in electronic structure, for so-called post-VSCF treatments such as vibrational Moller- Plesset (VMP2) and vibrational configuration interaction (VCI), which are utilized 166 extensively throughout this chapter and the following. These methods were imple- mented in a code called ezVibe. This code is freely available for download; the Appen- dices provide user information and code structure. Early corrections for anharmonicity consisted of various scale factors based on com- parison with experimental data. This is possible because most methods systematically overestimate frequencies. Calculations for different groups of molecules have yielded scale factors for several methods including Hartree-Fock, MP2, QCISD, and several DFT functionals in a wide variety of basis sets. 18 However, these fail for sets of molecules for which sufficient experimental data does not exist, or for unusual single cases. For the explicit treatment of anharmonicity, perturbation theory has provided the most popular approach. 19, 20 Generally, coupled-cluster CCSD(T) treatment with triple- zeta quality basis sets are required for maximum errors to be within 10 cm − 1 of experi- mental data. 21 However, this method requires calculation of quartic force constants with respect to displacements along nuclear coordinates. Although analytic energy gradi- ents and second derviatives are available for many quantum chemistry packages, higher derivatives must be calculated by a finite differences approach. The numerical calcula- tion of derivatives is subject to a host of problems. Perturbation theory also breaks down for strongly anharmonic systems, for molecular clusters, and for large molecules due to increasing resonance effects. 21 Another method to explicitly treat anharmonicity is the variational approach. It avoids the calculation of higher derivatives, and instead requires an accurate represen- tation of the potential energy surface. In this paper, the vibrational self-consistent field method will be discussed. 167 In VSCF, the target vibrational wavefunction is written as a product of one- dimensional “modal” wavefunctions: Ψ (n) (Q 1 ,...,Q N ) = N Y j=1 ψ (n j ) j (Q j ) (7.1) The superscript(n) refers to a vector of quantum numbers(n 1 ,...,n j ,...n N ) and labels a given vibrational state. The goal is to find the “best” modal wavefunctions, ψ (n j ) j , such that they minimize the total energyhΨ (n) |H|Ψ (n) i. The Hamiltonian in normal coordinates for a nonrotating molecule, neglecting vibration-rotation interaction terms, is H =− 1 2 N X j=1 ∂ ∂Q 2 j +V(Q 1 ,...,Q N ) (7.2) This leads to the set of single-mode equations, h (n) j ψ (n j ) j (Q j ) = (n) j ψ (n j ) j (Q j ) (7.3) h (n) j = − 1 2 ∂ ∂Q 2 j +V (n) j (Q j ) j =1,...,N These are one-dimensional Fock-like equations in Q j . They look like independent eigen- value equations; however, they are coupled through the effective potential terms, which depend on all other modal functions: 168 V (n) j (Q j ) = * N Y l6=j ψ (n l ) l (Q l )|V(Q 1 ,...,Q N )| N Y l6=j ψ (n l ) l (Q l ) + . (7.4) Eqn. (7.4) can be interpreted as averaging out the effect of all other modes along mode j, by integrating the potential energy over the other coordinates. V(Q 1 ,...,Q N ) is the eigenenergy of the electronic Hamiltonian, calculated by a quantum chemistry program. After a suitable representation for the potential energy is obtained, Eqs. (7.3) are solved self-consistently. The modal wavefuntions are usually expressed in the basis of primitive harmonic oscillator or Gaussian functions, and coupled algebraic equations are solved in analogy with the Roothan implementation for Hartree-Fock. 22, 23 It has been shown that the basis choice has only a small effect compared to the quality of the potential. 21 Two independent factors limit the accuracy of the VSCF approach. First is the valid- ity of a separable vibrational wavefunction. Second is the accuracy of the potential energy over the entirety of configuration space sampled by quadrature over the one- particle (harmonic) basis functions. 7.3 Computational details The vibrational levels and wave-functions of HCOH were calculated by diagonalizing the full Watson Hamiltonian 24 forJ = 0 (pure vibration) in the basis constructed using vibrational self-consistent field (VSCF) functions. The VSCF calculations employed harmonic oscillator wave-functions along the normal coordinates as the 1D basis set for the optimized VSCF modals. Basis functions with quantum numbers from 0 to 15 were included for each modal, with corresponding harmonic frequency. These were contracted to 6 numerical functions. Multimode interactions in the PES were included 169 up to the 4-mode level and ro-vibrational (Watson correction and Coriolis) interactions up to 3-mode level. The basis for VCI calculations consisted of all VSCF product wave- functions with maximum of 5-tuple excitations from the reference in each mode, with maximum of 4 modes excited, with the added restriction of a maximum sum of quanta of 10. The matrix elements of the Hamiltonian were calculated numerically by Gauss- Hermite quadrature with 10 integration points. Table 7.1: Comparison of harmonic frequencies (cm − 1 ) and IR intensities (km/mol, in parentheses) for trans-hydroxycarbene. Mode cc-pVTZ a cc-pVTZ b PES c aug-cc-pVTZ b cc-pVQZ b ν 6 o.p. wag (a”) 1101 (117) 1091 1098 1086 1093 ν 5 i.p. bend (a’) 1218 (139) 1215 1214 1209 1218 ν 4 CO stretch (a’) 1339 (64) 1327 1326 1312 1318 ν 3 i.p. bend (a’) 1520 (19) 1514 1508 1506 1513 ν 2 CH stretch (a’) 2882 (139) 2855 2853 2863 2863 ν 1 OH stretch (a’) 3773 (81) 3754 3754 3740 3754 a ACES II using analytic gradients, all electrons are correlated. b MOLPRO using total energies, core electrons are frozen. c Finite-differences calculations using PES fitted to the cc-PVTZ (frozen core) results. This procedure was employed to calculate vibrational energies and intensities that were used to compute the IR spectra presented below. Most of the CI energies are converged to within 1 cm − 1 . Only for seven of the states above 3000 cm − 1 for the cis-HCOH and cis-HCOD isomers, the convergence is about2− 3 cm − 1 . For benchmark purposes, we also conducted vibrational Rayleigh-Schr¨ odinger per- turbation theory calculations for trans-HCOH. We employed second-order Mø ller- Plesset scheme (VMP2) as described in Refs. 25, 26 . These calculations included three- mode couplings in the potential and neglected rovibration terms. They are compared with VSCF and VCI results obtained at the same level of theory. All these calculations 170 employed 10 basis functions in each mode. These VCI/VMP2 calculations included up to 8-tuple excitations. Table 7.2: Comparison of harmonic frequencies (cm − 1 ) and IR intensities (km/mol, in parentheses) for cis-hydroxycarbene. Mode cc-pVTZ a cc-pVTZ b PES c aug-cc-pVTZ b cc-pVQZ b ν 6 o.p. wag (a”) 1028 (27) 1016 1014 1011 1020 ν 5 i.p. bend (a’) 1238 (44) 1235 1238 1228 1235 ν 4 CO stretch (a’) 1345 (69) 1331 1335 1315 1331 ν 3 i.p. bend (a’) 1491 (42) 1483 1476 1471 1479 ν 2 CH stretch (a’) 2794 (195) 2764 2768 2772 2797 ν 1 OH stretch (a’) 3662 (17) 3650 3655 3628 3651 a ACES II using analytic gradients, all electrons are correlated. b MOLPRO using total energies, core electrons are frozen. c Finite-differences calculations using PES fitted to the cc-PVTZ (frozen core) results. Vibrational calculations employed a full-dimensional PES. The PES covers the trans- and cis- isomers and the two barriers connecting them, see Fig. 7.1. It is an 8 th degree polynomial in the set of internuclear bond lengths, represented in a specially constructed basis invariant to permutations of like nuclei. The polynomial has 1613 terms, which are fitted by least squares to 17262 ab initio single points, calculated by the CCSD(T) method 27, 28 with the cc-pVTZ basis set 29 . There are 12953 points in the energy range [0,0.1) hartrees above the lower minimum (the trans- equilibrium struc- ture), 2337 points in the range [0.1,0.2) hartrees, and 1542 points above 0.2 hartrees (these latter points are mostly fragment data and highly twisted configurations to enforce asymptotes). The r.m.s. fitting errors in these energy regions are 44, 108, and 81 cm − 1 , respectively. It reproduces harmonic frequencies at the two minima within 7 cm − 1 of the ab initio values (see Tables 7.3 and 7.3) values and correctly describes asymptotic behavior of the CO, CH, and OH stretches. Numerical frequencies on the PES are calcu- lated by five-fold central finite differences with step size of 0.01 ˚ A. Similarly constructed 171 PESs have been used in several dynamics and spectroscopy studies, and details of con- structing the symmetrized polynomial basis are given elsewhere 30 . The DMS was fitted to the same set of geometries as the PES, at the CCSD/6-311G** level of theory. H C O H H C O H 1.319 1.320 1.317 1.318 102.0° 102.1 102.1 102.0 107.2° 107.6 107.3 107.2 1.317 1.319 1.312 1.316 106.7° 106.6 106.7 106.6 113.9° 114.1 114.0 113.8 1.114 1.114 1.113 1.107 0.967 0.968 0.965 0.967 1.121 1.121 1.117 1.113 0.971 0.972 0.969 0.971 Figure 7.2: CCSD(T)/cc-pVTZ (regular print), CCSD(T)/aug-cc-pVTZ (italics), CCSD(T)/cc-pVQZ (underlined), and PES (bold) equilibrium structures of trans-(left) and cis-(right) hydroxycarbene. E nuc = 30.63185899 a.u. and E nuc = 30.55794113 a.u. for trans- and cis- isomers, respectively, at the CCSD(T)/cc-pVTZ (frozen core) equilibrium structure. Basis set effects were investigated by calculating equilibrium geometries and har- monic frequencies with the aug-cc-pVTZ and cc-pVQZ bases 29 (see Tables 7.3 and 7.3 and Fig. 7.2 for comparison). In both isomers, changes in equilibrium angles are less than 0.1 o . In the trans- isomer, bond lengths are converged to 0.002 ˚ A. In the cis-isomer, bond lengths slightly decrease with increasing basis set size. The largest change is in the CO bond, with 0.005 ˚ A difference between the cc-pVTZ and cc-pVQZ bases. For the trans-isomer, changes in harmonic frequencies are less than 10 cm − 1 between the cc-pVTZ and cc-pVQZ bases, and slightly larger between cc-pVTZ and aug-cc-pVTZ. For the cis-isomer, the differences are within 4 cm − 1 for all the modes except the CH stretch that changes by 32 cm − 1 between cc-pVTZ and cc-pVQZ. This mode is also very sensitive to core electron treatment. 172 Single point energies and dipole moments for the fitting were calculated using MOL- PRO 31 and Q-Chem 32 , respectively. Harmonic frequencies and infrared intensities were calculated using MOLPRO and ACES II 33 , respectively. The core orbitals were frozen in all MOLPRO calculations, and correlated in ACES II and Q-Chem calculations. MOL- PRO harmonic vibrational frequencies were computed by finite-differences using total energies, whereas ACES II calculations employed first analytic derivatives 34 . CCSD dipole moments were computed using analytic gradients and properties code 35 . The MULTIMODE program was used for all vibrational energy and intensity calcu- lations 36 . Additional VSCF, VCI and VMP2 calculations were performed using a new code 37 developed at USC. 7.4 Results and discussion 7.4.1 The IR spectrum of HCOH Harmonic frequencies and intensities for the HCOH isomers are given in Tables 7.3 and 7.3. VCI energies below 4000 cm − 1 and corresponding anharmonic intensities are given in Table 7.3. Intensities are reported within a threshold of 0.01 km/mol. The two sets are also depicted as stick spectra in Figs. 7.3 and 7.5. VMP2 results are presented in Table 7.4.1, which also contains VCI and VSCF results obtained at the same level of theory (see Section 7.3 for the details). 173 Table 7.3: HCOH VCI energy levels (cm − 1 ) and IR intensities (km/mol). trans- cis- No. State label Energy Intensity Energy Intensity 1 ν 6 1060 119.66 978 34.90 2 ν 5 1177 131.74 1189 47.00 3 ν 4 1295 75.53 1299 76.00 4 ν 3 1475 22.32 1442 41.55 5 2ν 6 2097 1.01 1942 1.30 6 ν 5 +ν 6 2240 0.10 2166 0.45 7 2ν 5 2347 2.37 2368 3.43 8 ν 4 +ν 6 2352 0.03 2268 0.03 9 ν 4 +ν 5 2461 1.96 2471 2.10 10 ν 3 +ν 6 2521 0.33 2421 0.74 11 2ν 4 2569 0.76 2579 0.04 12 ν 3 +ν 5 2622 23.25 2650 69.19 13 ν 2 2691 116.08 2552 161.07 14 ν 3 +ν 4 2776 24.00 2736 1.42 15 2ν 3 2952 1.92 2885 7.88 16 3ν 6 3113 0.00 2895 0.05 17 ν 5 +2ν 6 3280 0.00 3124 0.17 18 ν 4 +2ν 6 3384 0.01 3221 0.00 19 2ν 5 +ν 6 3416 0.00 3333 0.04 20 3ν 5 3510 0.34 3545 0.44 21 ν 4 +ν 5 +ν 6 3525 0.02 3441 0.02 22 ν 1 3553 51.59 3397 10.90 23 ν 3 +2ν 6 3545 2.65 3385 0.07 24 2ν 4 +ν 6 3622 0.00 3536 0.01 25 ν 4 +2ν 5 3620 0.00 3631 0.13 174 Table 7.3, continued: trans- cis- No. State label Energy Intensity Energy Intensity 26 ν 3 +ν 5 +ν 6 3676 0.08 3622 0.57 27 2ν 4 +ν 5 3724 0.00 3730 0.05 28 ν 2 +ν 6 3743 0.35 3519 2.13 29 ν 3 +2ν 5 3770 0.01 3830 0.47 30 3ν 4 3824 0.00 3841 0.01 31 ν 3 +ν 4 +ν 6 3825 0.08 3712 0.01 32 ν 2 +ν 5 3853 0.71 3713 1.69 33 ν 3 +ν 4 +ν 5 3923 0.11 3935 0.10 34 ν 2 +ν 4 3973 0.35 3849 0.35 35 2ν 3 +ν 6 3984 0.00 3861 0.02 36 ν 2 +ν 3 ··· ··· 3952 0.33 37 4ν 6 ··· ··· 3943 0.02 The four lowest transitions lying below 1600 cm − 1 are 0-1 transitions for out-of- plane wag, two in-plane bends, and the CO stretch. Anharmonic corrections for the frequencies and intensities of these modes are small and result in slightly lower fre- quencies. VMP2 and VCI are within 5 cm − 1 from each other. In the 1600− 3000 cm − 1 region, there is only one harmonic frequency, theν 2 CH stretch. However, when anharmonicities are taken into account, the spectra become more complex. In the trans- isomer, ν 2 has intensity of 116.1 km/mol, and two combination bands also have non- negligible intensity: ν 3 +ν 5 and ν 3 +ν 4 , at 23.3 and 24.0 km/mol, respectively. VMP2 behavior becomes more erratic, especially for higher overtones, e.g., the VMP2-VCI differences of60− 75 cm − 1 were observed for 2 overtones in this energy range. For the seven experimental IR bands reported, VCI calculations closely match the experiment. Differences are typically less than 12 cm − 1 (less than 1%), with the excep- tion of the OH stretch,ν 1 , where VCI frequency is 52 cm − 1 higher than the experimental 175 1000 1500 2000 2500 3000 3500 1E-3 0.01 0.1 1 10 100 Frequency, cm -1 A(v), km/mol 1000 1500 2000 2500 3000 3500 1E-3 0.01 0.1 1 10 100 Figure 7.3: VCI (top) and harmonic (bottom) IR spectrum for trans-HCOH. 1000 1500 2000 2500 3000 3500 1E-3 0.01 0.1 1 10 100 Frequency, cm -1 A(v), km/mol 1000 1500 2000 2500 3000 3500 1E-3 0.01 0.1 1 10 100 Figure 7.4: VCI (top) and harmonic (bottom) IR spectrum for cis-HCOH. value. This frequency was similarly overestimated in the CCSD(T)/cc-pCVQZ quartic force field calculations 15 . The close agreement between the two sets of theoretical data 176 suggests that the discrepancies between the theoretical and experimental values are larg- erly due to the matrix-induced shifts. 1000 1500 2000 2500 3000 3500 1E-3 0.01 0.1 1 10 100 Frequency, cm -1 A(v), km/mol 1000 1500 2000 2500 3000 3500 1E-3 0.01 0.1 1 10 100 Frequency, cm -1 A(v), km/mol Figure 7.5: VCI infrared spectra (black) compared to experimental data of Schreiner et al. (red), 5 for trans-HCOH (top) and trans-HCOD (bottom). Intensities of the transitions calculated using VCI wave-functions are also in excel- lent agreement with the experiment, i.e., the computed values are within 2− 7% of experiment except for theν 2 stretch, for which the observed intensity is lower than the theoretical value by a factor of 2.4. The wave function of this state is heavily mixed and include significant contributions from three VSCF configurations. We expect that the degree of mixing, which is crucial for determining the intensity, might be strongly affected by the matrix environment. In the cis-isomer, only the ν 3 +ν 5 transition has significant intensity (69.2 km/mol), compared to the cis-ν 2 stretch of 161.1 km/mol. Several transitions in this region also acquire intensities of1− 3 km/mol, well within experimental reach. 177 Table 7.4: VSCF, VCI and VMP2 results for trans-HCOH. No. State label VSCF VMP2 a VCI VMP2-VCI 1 ν 6 1054 1042 1042 0 2 ν 5 1186.72 1166 1171 -5 3 ν 4 1297.37 1294 1293 1 4 ν 3 1482.75 1471 1473 -2 5 2ν 6 2107.29 2116 2058 58 6 ν 5 +ν 6 2239.31 2196 2208 -12 7 2ν 5 2385.78 2383 2333 50 8 ν 4 +ν 6 2345.81 2346 2330 16 9 ν 4 +ν 5 2479.69 2458 2453 5 10 ν 3 +ν 6 2521.08 2497 2499 -2 11 2ν 4 2574.87 2570 2567 3 12 ν 3 +ν 5 2681.75 2681 2606 75 13 ν 2 2674.41 2678 2688 -10 14 ν 3 +ν 4 2775.25 2773 2773 0 15 2ν 3 2968.24 2947 2947 0 16 3ν 6 3170.07 3175 3056 119 17 ν 5 +2ν 6 3289.24 3289 3221 68 18 ν 4 +2ν 6 3395.17 3380 3342 38 19 2ν 5 +ν 6 3433.34 3346 3370 -24 20 3ν 5 3594.08 3532 3490 42 21 ν 4 +ν 5 +ν 6 3527.62 3525 3485 40 22 ν 1 3507.17 3517 3543 -26 a ezVibe calculations, see text. In the harmonic approximation, there is only one line,ν 1 (OH stretch), with nonzero intensity above 3000 cm − 1 . In trans-HCOH, the VCI ν 1 ffundamental transition has the highest intensity (51.6 km/mol), followed by a neighboring combination/overtone, ν 3 +2ν 6 (2.7 km/mol). Transitions to all other states have intensity below 1 km/mol. In cis-HCOH, ν 1 has intensity of 10.9 km/mol, and two states acquire 1− 2 km/mol intensity. 178 The anharmonic corrections for the two hydrogen stretches are about 200− 250 cm − 1 for both isomers; their intensities are also affected more strongly than those of the four lowest modes. VMP2 reproduces this value very accurately. Most notable is the OH stretch for the trans- isomer, which is 81 km/mol in the double-harmonic approximation (DHA) and 52 km/mol when computed with VCI wave-functions and DMS. Consistently, anharmonicities slightly lowered intensities for all fundamentals. In the high energy region, the harmonic approximation breaks down qualitatively. 7.4.2 Deuterated hydroxycarbene: HCOD Harmonic frequencies and intensities for the HCOD isomers are given in Table 7.5. Lower vibrational frequencies cause a higher density of states; thus, VCI states up to 3600 cm − 1 are reported (Table 7.6). The spectra are visualized in Figs. 7.6 and 7.7 Table 7.5: Normal coordinates, harmonic frequencies (cm − 1 ) and IR intensities (km/mol) of HCOD at the CCSD(T)/cc-pVTZ level with all electrons being correlated. Mode trans-HCOD cis-HCOD ν 6 o.p. wag (a”) 941 (81) 889 (0) ν 5 i.p. bend (a’) 951 (67) 952 (12) ν 4 CO stretch (a’) 1331 (104) 1333 (88) ν 3 i.p. bend (a’) 1459 (9) 1456 (32) ν 2 OD stretch (a’) 2744 (66) 2660 (29) ν 1 CH stretch (a’) 2885 (116) 2800 (172) Harmonic frequencies for HCOD are smaller than HCOH due to the higher nuclear mass of deuterium. Within the adiabatic approximation, potential energy and dipole surfaces (including equilibrium structures) are the same for the two species; only the normal coordinate vectors differ, as the nuclear masses affect the Hessian. Intensities are related to mass indirectly through the normal coordinates and their derivatives. 179 1000 1500 2000 2500 3000 3500 1E-3 0.01 0.1 1 10 100 Frequency, cm -1 A(v), km/mol 1000 1500 2000 2500 3000 3500 1E-3 0.01 0.1 1 10 100 Figure 7.6: VCI (top) and harmonic (bottom) IR spectrum for trans-HCOD. Behavior of the lowest four modes is similar to HCOH. However, the energy region of the stretches (2500− 3000 cm − 1 ) exhibits different patterns. Anharmonicities bring the two stretches closer in frequency. Also, for both isomers, they lower the intensity of ν 2 and increase that of ν 1 , which reverses the relative ratio of intensities in trans- HCOD, i.e., the ratio of the intensities of the ν 2 and ν 1 fundamental transitions is 0.2 and 1.8 at the DHA and VCI levels, respectively. The latter value agrees well with the experimental ratio of 2.9. Combination bands also play a larger role in the stretch region, possibly due to an increased density of states facilitating the couplings between the modes. In trans- HCOD, the transitions to the ν 3 +ν 4 and 2ν 3 states acquire intensities of 30.8 and 16.2 km/mol, respectively, compared to 94.4 and 51.5 km/mol for ν 2 and ν 1 states. Both these wave-functions have large (0.2− 0.5) contributions from the ν 1 VSCF state. In cis-HCOD, the spectrum is more harmonic relative to the trans- isomer, similarly to HCOH. The same combination bands as for trans-HCOD have the only non-negligible 180 1000 1500 2000 2500 3000 3500 1E-3 0.01 0.1 1 10 100 1000 1500 2000 2500 3000 3500 1E-3 0.01 0.1 1 10 100 Frequency, cm -1 A(v), km/mol Figure 7.7: VCI (top) and harmonic (bottom) IR spectrum for cis-HCOD. intensities, although their magnitude (12.5 and 14.2 km/mol) are smaller than for theν 2 andν 1 fundamentals (54.3 and 136.2 km/mol, respectively). 181 Table 7.6: HCOD VCI energy levels (cm − 1 ) and IR intensities (km/mol). trans- cis- No. State label Energy Intensity Energy Intensity 1 ν 6 908 78.80 847 1.54 2 ν 5 925 60.61 921 11.77 3 ν 4 1287 119.56 1288 102.37 4 ν 3 1419 10.75 1414 32.37 5 2ν 6 1796 0.13 1682 0.70 6 ν 5 +ν 6 1831 0.06 1767 0.22 7 2ν 5 1842 0.62 1826 2.30 8 ν 4 +ν 6 2190 0.06 2126 0.02 9 ν 4 +ν 5 2205 0.02 2202 0.09 10 ν 3 +ν 6 2321 0.28 2261 0.68 11 ν 3 +ν 5 2339 0.31 2326 2.91 12 2ν 4 2552 0.60 2552 0.39 13 ν 2 2622 94.38 2516 54.29 14 ν 1 2669 51.54 2584 136.20 15 3ν 6 2667 0.00 2505 0.01 16 ν 3 +ν 4 2713 30.82 2704 12.53 17 ν 5 +2ν 6 2710 7.01 2598 20.12 18 2ν 5 +ν 6 2749 0.00 2665 0.00 19 3ν 5 2754 0.03 2720 0.21 20 2ν 3 2851 16.15 2834 14.20 21 ν 4 +2ν 6 3072 0.03 2953 0.02 22 ν 4 +ν 5 +ν 6 3108 0.0 3043 0.00 23 ν 4 +2ν 5 3116 0.00 3099 0.08 24 ν 3 +2ν 6 3204 0.03 3094 0.05 25 ν 3 +ν 5 +ν 6 3242 0.00 3173 0.08 182 Table 7.3, continued: trans- cis- No. State label Energy Intensity Energy Intensity 26 ν 3 +2ν 5 3251 0.00 3230 0.02 27 2ν 4 +ν 6 3450 0.02 3382 0.01 28 2ν 4 +ν 5 3463 0.00 3461 0.00 29 ν 2 +ν 6 3526 0.05 3358 1.23 30 ν 2 +ν 5 3536 0.94 3428 2.53 31 ν 1 +ν 6 3573 0.76 3421 0.85 32 ν 1 +ν 5 3584 0.13 3500 0.29 33 4ν 6 3578 0.00 3379 0.01 Eight experimental IR bands for trans-HCOD were reported 15 ; our VCI calculations show similar agreement with frequencies and intensities as for HCOH. The largest dif- ference is a 34 cm − 1 higher calculated frequency forν 2 (OD stretch). Similar result was obtained in the CCSD(T)/cc-pCVQZ quartic force field calculations 15 . 7.5 Conclusions We report accurate configuration interaction vibrational levels of the cis- and trans- isomers of HCOH and HCOD. The IR spectra in the region below 4000 cm − 1 are strongly affected by anharmonicities. Strong effects were observed for IR intensities, as anharmonic mode couplings mix bright and dark states. For trans-HCOH/nnHCOD, frequencies and intensities are in excellent agreement with the recently reported IR spec- tra 15 . This agreement validates accuracy of our PES and gives predictive power to the calculated spectra of the cis-isomers, as well as isomerization barriers. For the lowest four fundamentals, the DHA reproduces both frequencies and inten- sities rather accurately. However, for combinations and overtones, and even stretch 183 fundamentals, inclusion of anharmonicity is necessary. Combination/overtone bands acquired intensity even in the low energy (<2500 cm − 1 ) region, which could compli- cate assignment of fundamental bands. For fundamental transitions, VMP2 reproduces the VCI results very accurately (less than 10 cm − 1 differences forν 2 -ν 6 , and 24 cm − 1 forν 1 ); however, the errors as large as 60-70 cm − 1 were observed for some overtones, and one high-overtone state (3ν 6 ) was overestimated by 120 cm − 1 . Overall, HCOH isomers exhibit higher anharmonicities than formaldehyde, a more rigid molecule of higher symmetry. For the fundamental transitions, average anhar- monicity in frequencies is 3.0% in formaldehyde 16, 17 , whereas in trans- and cis-HCOH it is 3.7% and 4.6%, respectively. The overtones deviate more from the harmonic values for the HCOH isomers also; the average difference between the ν = 2 overtones and twice theν = 1 values is 8 cm − 1 for formaldehyde, 13 cm − 1 for trans-HCOH, and 10 cm − 1 for cis-HCOH. In HCOD, the higher nuclear mass led to a denser spectrum, where the effects of multimode contributions to the PES, as well as state mixing, increased significantly. Anharmonicities generally lowered the frequencies relative to the harmonic approx- imation, and they affected different modes very differently, so that the spacing between vibrational levels changed. This leads to Fermi resonances not described by the DHA. Finally, the effects of anharmonicities on the intensities were found to be important. Relative to the DHA, anharmonicities tend to lower the intensity of strong peaks and increase the intensity of weak peaks owing to the state mixing. 184 7.6 Chapter 7 references [1] P.L. Houston and C.B. Moore. J. Chem. Phys., 65:757–770, 1976. 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Krylov. ezVibe, http://iopenshell.usc.edu/downloads/. 187 Chapter 8 Hydroxycarbene diradicals: ionization and photoelectron spectra 8.1 Introduction Hydroxycarbene, HCOH, is a high-energy diradicaloid isomer of formaldehyde. It is believed to play a role in formaldehyde photochemistry and its “roaming hydro- gen” dynamics, the interstellar medium, and reactions of carbon atom with water 1–5 . HCOH production is a major channel in the photodissociation of hydroxymethyl radi- cal, CH 2 OH, in the 3p Rydberg state 6 . Reisler and coworkers determined the heat of formation of the deuterated isotope HCOD to be24± 2 kcal/mol 7 . Recently its synthe- sis and spectroscopic characterization were reported by Schreiner et al 8 , who isolated the trans-HCOH and HCOD in argon matrix at 11K and identified several infrared (IR) band origins. The experiment was supported by variational calculations of the anhar- monic energies using the CCSD(T)/cc-pVQZ quartic force field. In an independent study, the vibrational levels and IR intensities for the ground states of neutral cis- and trans-HCOH were reported 9 . The calculated lines and intensities matched the experi- mental data of Schreiner et al. closely. It was found that anharmonicities were crucial for correctly describing IR intensities as well as energies. The harmonic approximation described accurately only the lowest fundamental frequencies; however, it overestimated 188 the stretching modes by approximately 200 cm − 1 in both isomers. Several combina- tion/overtone bands acquired intensity in the low-energy region (0− 3,000 cm − 1 ) and complicated the spectrum. E (c m -1 ) 0 1318 6190 7085 14159* H C O H H C O H H C O H H C O H H C O H -1811 … H C H O -18060 0 1616 10663 12907 12085* H C O H H C O H 9.45 (eV ) 8.76 (eV ) … 9.44 (eV ) 8.79 (eV ) … Figure 8.1: Stationary points on the HCOH (lower, CCSD(T)/cc-pVTZ) and HCOH + (upper) PES. Vertical arrows represent ionization to the Franck-Condon regions and vertical (regular print) and adiabatic (underline) IEs are given. Energies of stationary points are listed on each surface relative to their global minimum (trans- structure). The formaldehyde isomer was not included in our PES, and associated barrier (marked with *) was calculated with CCSD(T)/cc-pVTZ at B3LYP/cc-pVTZ optimized transition state. The cation HCOH + has also been studied. Berkowitz 10 and also Burgers 11 observed the species by mass spectroscopy in the dissociative photoionization of methanol. Near the dissociation threshold of hydrogen elimination, HCOH + , rather than H 2 CO + , was the dominant product 10 . HCOH tunnels effectively through the barrier to formaldehyde 8 . The calculated rate constant for the forward reaction is almost an order of magnitude higher than for the reverse reaction 12 , as would be expected from energetics. Whereas on 189 the neutral surface, HCOH is much higher than formaldehyde, the energy gap between HCOH + and H 2 CO + is much smaller (1,811 cm − 1 , see Fig. 8.1). Thus, it might be easier to observe HCOH isomer in the ionized rather than neutral state. From the electronic structure point of view, HCOH is an example of substituted carbenes, diradical species playing important role in organic chemistry 13 . Spectroscop- ically, prototypical substituted carbenes have been studied by Reid and co-workers 14–17 . Using high-resolution spectroscopy, they characterized the singlet-triplet gaps, spin- orbit couplings, and mode-specific dynamics of several triatomic carbenes 14–17 . Halo- gen substitution reduces diradical character resulting in the singlet ground state. The OH group has similar effect — the ground state of hydroxycarbene is a singlet, and the singlet-triplet gap is about 1 eV 2–5, 18, 19 . In this report we calculate the vibrational levels of ground-state HCOH + , and the associated photoelectron spectra from ground vibrational states of cis- and trans-HCOH. Photoelectron spectroscopy is a sensitive tool that provides information about electronic structure and nuclear motion. Positions of band heads yield ionization energies (IEs), information about the orbitals via Koopmans theorem, and an electronic spectrum of the ionized system. Vibrational progressions give frequencies and anharmonicities on the upper state, and information about structural differences between the two states. The structure of the paper is as follows. Sec. 8.2 discusses theory and computational details, including basis set convergence, details of the vibrational configuration inter- action (VCI) basis, and calculation of the Franck-Condon factors. Sec. 8.3 discusses the molecular orbitals and structural changes upon ionization, as well as barriers on the PES. Sec. 8.4 discusses the vibrational levels of HCOH + and presents the photoelectron spectra, and Sec. 8.5 does the same for the deuterated isomers HCOD. Sec. 8.6 com- pares the VCI photoelectron spectra with the harmonic parallel-mode approximation, 190 and shows that better accuracy is obtained by calculating displacements along the cation normal coordinates. Finally, Sec. 8.7 presents our conclusions. 8.2 Theory and computational details The calculations employed potential energy surfaces (PESs) for the neutral and cation ground states. The neutral PES is described in our early publication 9 . The new cation surface covers the cis- and trans- wells and the space connecting them. The PES is a 9 th degree polynomial in Morse variables of the set of interatomic distances, represented in a specially constructed basis invariant to permutations of like nuclei. The Morse vari- ables are defined as: y(i,j) =e − r(i,j) λ . r(i,j) is the internuclear distance between atoms i and j, and the value of λ is set to 2 bohr. The polynomial contains 2,649 terms fitted by weighted data to 26,221 ab initio single point energies, calculated by the CCSD(T) method 20, 21 with the cc-pVTZ basis set 22 . The restricted orbital Hartree-Fock (ROHF) was used as a reference. The PES was fitted to 22,263 points in the range [0,0.1) a.u. above the global minimum (trans-HCOH + equilibrium structure), 1,903 points in the range [0.1,0.2) a.u., and 1,894 points in the range [0.2,0.5) a.u. The least squares was weighted to ensure low-energy points were well fitted and harmonic frequencies repro- duced ab initio values. The rms fitting errors are 24 cm − 1 below 3,000 cm − 1 , 44 cm − 1 below 5,000 cm − 1 , and 62 cm − 1 below the highest barrier at 7,085 cm − 1 . Points above 0.1 a.u. were included to enforce asymptotes for fragmentation and small internuclear distances. Similarly constructed PESs have been used in several dynamics and spec- troscopy studies 23–29 ; details of constructing the symmetrized polynomial basis are given elsewhere 30 . Basis set effects were considered by examining equilibrium structures and frequen- cies with aug-cc-pVTZ and cc-pVQZ bases (Fig. 8.2). Bond lengths and angles are well 191 H C O H H C O H 0.989 0.990 0.987 0.989 1.225 1.226 1.221 1.225 1.099 1.099 1.098 1.099 117.0° 117.2 117.3 117.1 124.4° 124.4 124.5 124.6 0.991 0.993 0.990 0.991 1.220 1.220 1.215 1.219 1.100 1.101 1.099 1.100 119.5° 119.7 119.9 119.9 131.6° 131.6 131.5 132.1 Figure 8.2: Equilibrium structures on cation PES. CCSD(T)/cc-pVTZ (regular print), CCSD(T)/aug-cc-pVTZ (underline), CCSD(T)/cc-pVQZ (italic), and PES (bold) for cis- (left) and trans-HCOH + (right). E nuc = 31.858717 a.u. and 31.825806 a.u. at the CCSD(T)/cc-pVTZ (frozen core) geometries. converged at the cc-pVTZ level. The largest differences are a 0.005 ˚ A decrease in CO bond length and a 0.4 ◦ increase in the HOC angle. Harmonic frequencies are also well converged. Average absolute differences are 7.9 cm − 1 between cc-pVTZ and aug-cc- pVTZ and 5.0 cm − 1 between cc-pVTZ and cc-pVQZ. Basis set convergence is better with respect to polarization than diffuse functions, implying some diffuse character of the electron density. The OH stretch is most sensitive to this: its frequency decreases by 16 cm − 1 upon adding diffuse functions, but remains unchanged with added polarization. The PES replicates equilibrium CCSD(T)/cc-pVTZ bond lengths to 0.001 ˚ A, and bond angles to 0.2 ◦ in trans- and 0.5 ◦ in cis-HCOH + (Fig. 8.2). Frequencies on the PES were calculated numerically using 5-point central difference formulas. They reproduce CCSD(T) finite difference frequencies with an average (absolute) difference of 5.8 and 4.3 cm − 1 for cis- and trans-HCOH + , respectively, and maximum differences of 14 and 12 cm − 1 (Tables 8.2, 8.2). Vibrational energies and wave-functions were calculated by diagonalizing the Wat- son Hamiltonian 31 for J = 0 (pure vibration) in a basis of vibrational self-consistent 192 H C O H H C O H 0.967 0.989 1.318 1.225 1.107 1.099 107.2 117.1 102.0 124.6 0.971 0.991 1.316 1.219 1.113 1.100 113.8 119.9 106.6 132.1 Figure 8.3: Equilibrium structures calculated on the PES for HCOH (regular print) and HCOH + (underline) for cis- (left) and trans- (right) isomers. Table 8.1: Comparison of harmonic frequencies (cm − 1 ) and IR intensities (km/mol, in parentheses) for cis-HCOH + . Mode cc-pVTZ a cc-pVTZ b PES c aug-cc-pVTZ b cc-pVQZ b ν 6 op. wag (a”) 935 (32) 931 931 921 924 ν 5 ip bend (a’) 988 (65) 987 996 984 984 ν 4 ip bend (a’) 1171 (176) 1173 1159 1166 1169 ν 3 CO stretch (a’) 1733 (73) 1716 1711 1710 1724 ν 2 CH stretch (a’) 3085 (45) 3055 3054 3047 3054 ν 1 OH stretch (a’) 3464 (339) 3448 3442 3428 3446 a ACES II using analytic gradients, all electrons are correlated. b MOLPRO using total energies, core electrons are frozen. c Finite-differences calculations using PES fitted to the cc-pVTZ (frozen core) results. field 32 (VSCF) functions. The basis for VSCF optimized modals was the set of har- monic oscillator wave-functions along the normal coordinates, with quantum numbers from 0 to 15. Multimode interactions in the PES were included up to the 4-mode level. The rovibrational corrections were treated in an approximate manner. The Watson cor- rection was calculated up to the 4-mode level. Coriolis coupling terms that coupled two modes were integrated over a 2-mode representation of the inverse moment of inertia tensor. The basis for VCI calculations consisted of all VSCF product wave-functions with maximum of 10 total quanta excited from the VSCF ground state reference, with a 193 Table 8.2: Comparison of harmonic frequencies (cm − 1 ) and IR intensities (km/mol, in parentheses) for trans-HCOH + . Mode cc-pVTZ a cc-pVTZ b PES c aug-cc-pVTZ b cc-pVQZ b ν 6 op wag (a”) 966 (157) 965 970 965 958 ν 5 ip bend (a’) 998 (213) 999 997 994 987 ν 4 ip bend (a’) 1257 (33) 1255 1254 1249 1246 ν 3 CO stretch (a’) 1706 (87) 1692 1694 1685 1699 ν 2 CH stretch (a’) 3097 (58) 3073 3069 3066 3073 ν 1 OH stretch (a’) 3529 (413) 3511 3499 3495 3511 a ACES II using analytic gradients, all electrons are correlated. b MOLPRO using total energies, core electrons are frozen. c Finite-differences calculations using PES fitted to the cc-pVTZ (frozen core) results. maximum of 5 modes simultaneously excited. Matrix elements of the Hamiltonian were calculated numerically using Gauss-Hermite quadrature with 20 integration points for 1D and 2D integrals, 15 points for 3D integrals, and 10 points for 4D integrals. Table 8.3: Comparison of harmonic frequencies (cm − 1 ) between neutral and cation PESs. op wag ip bend ip bend CO stretch CH stretch OH stretch cis-HCOH 1014 1238 1476 1335 2768 3655 cis-HCOH + 931 996 1159 1711 3054 3442 trans-HCOH 1098 1214 1508 1326 2853 3754 trans-HCOH + 970 997 1254 1694 3069 3499 Franck-Condon factors were calculated as full-dimensional (i.e., 6-dimensional) integrals over the normal coordinates of the cation PES. The neutral ground-state wave- function at each point was obtained by aligning the molecules according to center of mass and the principal axis system, transforming between the normal coordinates, and evaluating the VCI wave-function. Thus, no approximations were made in evaluating 194 Franck-Condon factors via full-dimensional integration conducted using exact transfor- mation between the two sets of normal coordinates. Only transitions from the ground vibrational states of the neutral are considered in photoelectron spectrum calculations as these are most likely to be of relevance to future experiments. Non-zero Franck-Condon factors were calculated for levels up to 7,000 cm − 1 above the zero-point energy. With the present VCI basis, convergence in the VCI energies was converged to 1 cm − 1 for most states below 4,000 cm − 1 , with the exception of four combination/overtones of ν 6 , which are converged to about 2 cm − 1 . This mode leads towards the out-of-plane transition state connecting cis- and trans-; large VCI bases lead to inefficient convergence probably because they sample this flat region. Above 4,000 cm − 1 , convergence in these states is about5− 10 cm − 1 . ν 6 is the only out-of-plane mode and is not active in the photoelectron spectrum. The active states are converged to about 5 cm − 1 up to 7,000 cm − 1 . Single point energies for the PES fitting were calculated using MOLPRO 33 and Q- Chem 34 , respectively. Harmonic frequencies and infrared intensities were calculated using MOLPRO and ACES II 35 , respectively. The core orbitals were frozen in all MOL- PRO calculations, and correlated in ACES II and Q-Chem calculations. MOLPRO har- monic vibrational frequencies were computed by finite-differences using total energies, whereas ACES II calculations employed first analytic derivatives 36 . Vibrational wave functions and energy level were computed using ezVibe code 37 . For benchmark purposes, we compared VCI levels from ezVibe with the MULTIMODE program 38 . Agreement in the energies was within 1 cm − 1 for states below 6,000 cm − 1 (approximately 160 states), and within 2 cm − 1 below about 7,300 cm − 1 (300 states). 195 8.3 Molecular orbital framework and structural effects of ionization The smallest carbene, methylene (CH 2 ), has a triplet ground state, with two unpaired electrons on the divalent carbon atom. The singlet-triplet gap is 0.39 eV 39, 40 . Substi- tuted carbenes have diverse properties, for example in the stereospecifity of their reac- tions 13, 41–43 . The differences in reactivity can often be explained in terms of the singlet versus triplet character of the ground state. Figure 8.4: Highest occupied molecular orbital of cis- (left) and trans-HCOH (right). The triplet state in carbenes has two electrons in nonbonding orbitals on carbon, one σ and oneπ . The singlet state has the electrons paired in theσ orbital, with theπ orbital unoccupied. The effect of substituents can be explained using simple molecular orbital considerations 44, 45 : substituent groups withπ type lone pairs (N,O atoms) lead to singlet ground states because these lone pairs can mix with carbon’sπ orbital. This can raise it enough so that pairing the electrons inσ becomes energetically favorable. For example, in HCOH the singlet state is about 1 eV below the triplet. The vertical (adiabatic) IEs of HCOH are 9.45 (8.76) and 9.44 (8.79) eV for the cis- and trans- isomers, respectively, as computed at the CCSD(T)/cc-pVTZ level (ZPE excluded). The highest occupied molecular orbital (HOMO) on HCOH is a lone pair on carbon with a minor contribution on OH which provides antibonding character along 196 the CO bond. (Fig. 8.4). The first ionization removes an electron from the HOMO, with large geometrical changes in equilibrium structure. The CO bond is shortened by 0.097 and 0.093 ˚ A in the cis- and trans- isomers, respectively. Ionization from an sp 2 orbital on carbon increases the carbon’s overall s character; the HCO angle increases by 25.5 and 22.6 ◦ . The HOC angle also increases, by 6.1 and 9.9 ◦ . The displaced equilibrium structures strongly affect the shape of the PES, and har- monic frequencies show strong differences upon ionization (Table 8.2). The largest change is in the CO stretch, which increases by 370 cm − 1 in both isomers upon ion- ization. This is due to the shortening of the CO bond. The CH stretching frequency increases upon ionization, by 286 and 216 cm − 1 . The remaining four frequencies decrease. The OH stretch decreases by 213 and 255 cm − 1 ; this follows from the longer OH bond in the cation, due to increased donation into the electron-depleted carbon. The remaining three are bending modes involving the OH group; the oxygen lone pairs encounter less steric hindrance with a single electron on C in these motions. Two barriers on the HCOH + PES, which separate the cis- and trans- wells, are 6,190 and 7,085 cm − 1 above the trans- minimum (Fig. 8.1). The respective transition states represent in-plane and out-of-plane rotation of H around the oxygen, respectively. These transition states are lower in energy relative to the neutral (by 6,717 cm − 1 for the linear, and by 3,578 cm − 1 for the out-of-plane). This also is due to decreased repulsion between the electrons on O and C: in out-of-plane rotation, the HOC angle remains essentially constant. In in-plane-rotation, this angle changes and the oxygen’s electron density is brought closer to the carbon center. The ionized carbon atom presents a much smaller barrier for this interaction, hence the disproportionate effect of ionization on the two barriers. 197 It should be noted that Fig. 1 in our previous paper 9 had a typographical error: energies of the two barriers connecting cis- and trans-HCOH were incorrectly labeled relative to the cis- minimum, rather than to the trans- as indicated. In addition, the neutral PES was optimized to replicate harmonic frequencies; we have since created a similar PES that replicates barrier heights accurately with only a moderate decline in the accuracy of the harmonic frequencies. 8.4 Photoelectron spectra of HCOH Vibrational levels of HCOH + up to 3,600 cm − 1 are listed in Table 8.4. Considering the fundamental excitations, the first four levels (up to 1,700 cm − 1 ) are accurately described by the harmonic approximation, with an average deviation between harmonic and VCI excitation energies of 35 cm − 1 . The higher stretches show large deviations from the har- monic approximation; VCI decreases the CH and OH stretch fundamental frequencies by approximately 150 and 190 cm − 1 , respectively. 0 1000 2000 3000 4000 5000 6000 7000 0.01 0.02 0.03 0.04 0 1 2 3 4 5 6 7 8 9 10 11 12 13 -- -- 14 ---- -- -- -- -- -- -- 17 -- -- 19 20 -- -- -- -- -- -- -- 21 -- -- -- -- -- 22 -- -- -- -- 23 -- -- -- -- 24 25 -- -- -- -- -- 26 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 27 -- -- -- -- -- -- -- -- -- -- -- -- -- 28 -- -- -- -- 29 -- -- -- -- -- -- -- -- -- -- -- -- 30 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 31 -- -- -- -- -- -- -- -- -- -- -- -- -- -- 32 -- -- -- -- -- -- -- -- 33 -- -- -- -- -- -- 34 -- 18 16 15 ΔE (cm -1 ) 0 1000 2000 3000 4000 5000 6000 7000 0.01 0.02 0.03 0.04 0.05 0 -- 1 2 3 -- -- -- 4 -- 5 -- 6 -- -- 7 -- -- 8 -- -- -- 9 10 -- -- -- -- 11 -- -- -- -- 12 -- -- -- -- -- -- -- 13 -- -- 14 15 -- -- -- -- -- ---- 16 -- 17 -- -- -- ---- -- -- -- 18 -- -- -- -- -- ---- -- -- -- 19 -- -- -- -- -- -- -- -- -- -- -- -- -- -- 20 -- -- ---- 21 -- -- -- -- -- -- -- -- -- -- -- 22 -- -- -- -- -- 23 -- -- -- -- -- -- -- -- -- -- -- 24 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 25 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 26 -- -- -- -- -- -- -- -- -- -- -- -- -- -- 27 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 28 -- ΔE (cm -1 ) Figure 8.5: Franck-Condon factors for HCOH ionization producing electronic ground state of HCOH + in the range from the ZPE (0 cm − 1 ) to 7,000 cm − 1 . Left: cis- isomer; right: trans- isomer. 198 Table 8.4: HCOH + VCI vibrational levels below 3,600, and corresponding levels for HCOD + (cm − 1 ). No. State label cis-HCOH + cis-HCOD + trans-HCOH + trans-HCOD + 1 ν 6 905 746 935 781 2 ν 5 949 822 967 822 3 ν 4 1126 1099 1211 1143 4 ν 3 1684 1671 1664 1655 5 2ν 6 1811 1475 1858 1547 6 ν 5 +ν 6 1874 1575 1915 1604 7 2ν 5 1885 1641 1933 1644 8 ν 4 +ν 6 2045 1835 2149 1912 9 ν 4 +ν 5 2052 1947 2141 1982 10 2ν 4 2224 2167 2405 2259 11 ν 3 +ν 6 2582 2436 2597 2417 12 ν 3 +ν 5 2624 2488 2626 2477 13 3ν 6 2711 2193 2779 2301 14 ν 5 +2ν 6 2782 2306 2826 2365 15 ν 3 +ν 4 2801 2753 2863 2785 16 3ν 5 2823 2456 2908 2463 17 2ν 5 +ν 6 2825 2395 2903 2430 18 ν 2 2896 2384 2933 2478 19 ν 4 +2ν 5 2954 2784 3054 2818 20 ν 4 +2ν 6 2962 2557 3100 2671 21 ν 4 +ν 5 +ν 6 2990 2691 3090 2751 22 2ν 4 +ν 5 3122 3038 3298 3109 23 2ν 4 +ν 6 3156 2877 3340 3016 24 ν 1 3248 2900 3328 2927 25 3ν 4 3300 3191 3578 3347 26 2ν 3 3347 3321 3306 3287 27 ν 3 +2ν 6 3481 3176 3519 3177 28 ν 3 +ν 5 +ν 6 3542 3253 3574 3285 29 ν 3 +2ν 5 3560 3304 3589 3301 199 The photoelectron spectra for the two isomers are shown in Fig. 8.5, and positions and intensities are tabulated in Tables 8.4 and 8.4. The intensities are unitless; intensity of 1 corresponds to full overlap between the neutral and cation wave-functions. The cis-HCOH photoelectron spectrum is given in Fig. 8.5 and Table 8.4. In the low energy region (0− 2,000 cm − 1 ), the lowest-frequency mode ν 6 has no intensity. This is the only mode which is not fully symmetric; in the absence of normal mode coupling in the PES, transitions to odd levels of this mode are forbidden by symmetry. The other four fundamentals in this range have appreciable intensity, withν 5 and 2ν 5 the strongest. In cis-HCOH + , ν 5 is a scissoring of OH and CH which moves the molecule toward linearity. From Fig. 8.3, displacement along this mode brings the cation into the Franck-Condon region. ν 4 increases one angle and decreases the other one. It is active because the difference in HCO angles in neutral and cation structures is much larger than the difference in HOC angles. The third active mode isν 3 , which is the CO stretch. In the high energy range (2,000− 7,000 cm − 1 ), six peaks have significant intensity (greater than approximately 0.025). Peaks labeled 5, 7, and 17 are composed of primar- ily one VSCF state, and correspond to combination bands of two modes, ν 5 and either ν 4 (5 and 17, more intense) or ν 3 (7, less intense). Peaks 23, 26, and 29 are combina- tions in all three active modes and mix several (2− 8) VSCF states. All of the intense peaks in the cis-HCOH spectrum represent vibrational states with multiple quanta in combinations ofν 5 andν 4 , and to a smaller extent,ν 3 . The trans-HCOH spectrum is given in Fig. 8.5 and Table 8.4. It is qualitatively different than for cis-HCOH: there are fewer intense progressions, and they occur at much lower energies. ν 4 at 1,211 cm − 1 dominates the low-energy part of the spectrum, with ν 3 having about half the intensity. ν 4 is the bend which brings the molecule to linearity, and ν 3 is the CO stretch. Above 2,000 cm − 1 , the five strongest peaks occur 200 below 4,050 cm − 1 . The dominant progression throughout the spectrum is an overtone ofν 4 . Other peaks with lesser intensity are combination bands involving excited quanta in ν 3 and ν 4 . Compared to the cis- isomer, trans-HCOH shows a less dense spectrum with most of the intensity inν 4 . 201 Table 8.5: Active vibrational levels of cis-HCOH + / HCOD + in the photoelectron spec- trum of cis-HCOH / HCOD. cis-HCOH + cis-HCOD + No. State label Energy Intensity State label Energy Intensity 0 0 0 0.0104 0 0 0.0098 1 ν 5 949 0.0202 ν 5 746 0.0091 2 ν 4 1126 0.0121 ν 4 1099 0.0222 3 ν 3 1684 0.0144 ν 3 1671 0.0148 4 2ν 5 1885 0.0163 ν 4 +ν 5 1835 0.0222 5 ν 4 +ν 5 2052 0.0312 2ν 4 2167 0.0279 6 2ν 4 2224 0.0075 ν 2 2384 0.0068 7 ν 3 +ν 5 2624 0.0265 ν 4 +2ν 5 2557 0.0076 8 ν 3 +ν 4 2801 0.0062 ν 3 +ν 4 2753 0.0267 9 3ν 5 2823 0.0140 2ν 4 +ν 5 2877 0.0342 10 ν 4 +2ν 5 2954 0.0218 3ν 4 3191 0.0226 11 ν 4 +2ν 6 2962 0.0186 2ν 3 3321 0.0107 12 2ν 4 +ν 5 3122 0.0173 ν 2 +ν 4 3461 0.0156 13 2ν 3 3347 0.0102 ν 2 +ν 4 3511 0.0084 14 ν 3 +2ν 5 3560 0.0208 2ν 4 +2ν 5 3589 0.0151 15 ν 3 +ν 4 +ν 5 3714 0.0185 ν 3 +2ν 4 3809 0.0205 16 4ν 5 3776 0.0184 3ν 4 +ν 5 3859 0.0298 17 ν 4 +3ν 5 3865 0.0295 ν 2 +ν 4 +ν 5 4180 0.0101 18 ν 3 +2ν 4 3891 0.0176 2ν 3 +ν 4 4389 0.0149 19 2ν 4 +2ν 5 4008 0.0069 ν 3 +2ν 4 +ν 5 4501 0.0236 20 ν 2 +ν 4 4022 0.0100 3ν 4 +2ν 5 4565 0.0233 21 2ν 3 +ν 5 4279 0.0165 ν 3 +3ν 4 4844 0.0386 22 ν 3 +3ν 5 4502 0.0171 2ν 3 +ν 4 +ν 5 5096 0.0112 23 ν 3 +ν 4 +2ν 5 4619 0.0244 ν 3 +2ν 4 +2ν 5 5217 0.0131 24 5ν 5 4730 0.0196 2ν 3 +2ν 4 5438 0.0099 25 ν 2 +ν 5 +ν 6 4740 0.0083 ν 3 +3ν 4 +ν 5 5482 0.0160 202 Table 8.4, continued: cis-HCOH + cis-HCOD + No. State label Energy Intensity State label Energy Intensity 26 ν 4 +5ν 5 4814 0.0319 2ν 3 +3ν 5 5486 0.0122 27 2ν 3 +2ν 5 5215 0.0116 4ν 4 +2ν 5 5549 0.0187 28 ν 3 +4ν 5 5464 0.0238 ν 3 +4ν 4 5839 0.0174 29 ν 3 +ν 4 +3ν 5 5553 0.0351 2ν 3 +2ν 4 +ν 5 6139 0.0120 30 ν 3 +2ν 4 +2ν 5 5723 0.0240 ν 3 +3ν 4 +2ν 5 6236 0.0112 31 2ν 3 +3ν 5 6166 0.0074 ··· ··· ··· 32 ν 2 +ν 3 +2ν 5 6389 0.0083 ··· ··· ··· 33 ν 3 +5ν 5 6466 0.0209 ··· ··· ··· 34 ν 3 +ν 4 +4ν 5 6529 0.0177 ··· ··· ··· The qualitative differences in the two photoelectron spectra can be rationalized by differences in equilibrium structures upon ionization (Fig. 8.3). As discussed in Sec. 8.3, the opening up of the HOC angle is the most prominent effect, followed by a smaller opening of the HCO angle. The third difference is a shortening of the CO bond in both isomers. The intensity in the photoelectron spectra of HCOH is dominated by combination bands of the two hydrogen bending modes, whose primary displacement is changing these angles. A smaller amount of intensity is seen in the CO stretch. Thus to a great extent the progressions in the photoelectron spectrum are due to the Condon reflection principle 46 . 203 Table 8.6: Active vibrational levels of trans-HCOH + / HCOD + in the photoelectron spectrum of trans-HCOH / HCOD. trans-HCOH + trans-HCOD + No. State label Energy Intensity State label Energy Intensity 0 0 0 0.0155 0 0 0.0144 1 ν 5 967 0.0062 ν 4 1143 0.0419 2 ν 4 1211 0.0390 ν 3 1655 0.0193 3 ν 3 1664 0.0200 2ν 4 2259 0.0644 4 ν 4 +ν 5 2141 0.0162 ν 3 +ν 4 2785 0.0483 5 2ν 4 2405 0.0515 2ν 3 3287 0.0090 6 ν 3 +ν 5 2626 0.0071 3ν 4 3347 0.0703 7 ν 3 +ν 4 2863 0.0371 ν 3 +2ν 4 3894 0.0676 8 ν 2 2933 0.0102 2ν 3 +ν 4 4411 0.0786 9 2ν 4 +ν 5 3298 0.0124 ν 2 +2ν 4 4714 0.0075 10 2ν 3 3306 0.0218 ν 3 +3ν 4 4979 0.0642 11 3ν 4 3578 0.0450 5ν 4 5459 0.0336 12 ν 3 +ν 4 +ν 5 3794 0.0150 2ν 3 +2ν 4 5512 0.0361 13 ν 3 +2ν 4 4049 0.0427 ν 2 +3ν 4 5791 0.0084 14 ν 2 +ν 4 4144 0.0118 ν 2 +ν 3 +ν 4 +ν 5 6046 0.0220 15 2ν 4 +2ν 5 4207 0.0126 ν 3 +4ν 4 6047 0.0357 16 3ν 4 +ν 5 4449 0.0179 6ν 4 6520 0.0174 17 2ν 3 +ν 4 4498 0.0225 ··· ··· ··· 18 4ν 4 4736 0.0312 ··· ··· ··· 19 ν 3 +2ν 4 +ν 5 4950 0.0216 ··· ··· ··· 20 ν 3 +3ν 4 5221 0.0277 ··· ··· ··· 21 3ν 4 +2ν 5 5372 0.0131 ··· ··· ··· 22 4ν 4 +ν 5 5606 0.0109 ··· ··· ··· 23 2ν 3 +2ν 4 5674 0.0195 ··· ··· ··· 24 5ν 4 5883 0.0091 ··· ··· ··· 25 3ν 3 +ν 4 6114 0.0166 ··· ··· ··· 26 ν 3 +4ν 4 6378 0.0218 ··· ··· ··· 27 2ν 3 +2ν 4 +ν 5 6587 0.0113 ··· ··· ··· 28 2ν 3 +3ν 4 6845 0.0161 ··· ··· ··· 204 8.5 Photoelectron spectra of HCOD The photoelectron spectra for cis- and trans-HCOD are depicted in Fig. 8.6 and tab- ulated in Tables 8.4 and 8.4. Compared to HCOH, the energies of all the states are decreased due to the larger mass of D; the relative intensities also change because the different mass affects the normal modes. The normal modes tend to localize vibration into H and suppress it in D. This effect is largest in trans-HCOD. 0 1000 2000 3000 4000 5000 6000 7000 0.01 0.02 0.03 0.04 0 1 -- 2 -- -- -- 3 4 -- 5 -- -- 6 -- -- -- 7 -- 8 -- 9 -- -- -- -- -- -- -- -- 10 -- -- -- -- -- 11 -- 12 13 -- -- 14 -- -- -- -- -- -- 15 -- 16 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 17 -- -- -- -- -- -- -- 18 -- -- -- -- -- -- 19 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 21 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 22 -- -- -- -- -- -- -- -- -- 23 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 24 -- -- -- -- 25 26 -- -- -- -- -- 27 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 28 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 29 -- -- -- -- -- -- -- -- -- -- -- -- -- 30 20 ΔE (cm -1 ) 0 1000 2000 3000 4000 5000 6000 7000 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0 -- -- 1 -- -- -- 2 ---- 3 -- -- -- -- -- -- -- ---- 4 -- -- -- -- -- -- -- -- -- -- -- -- -- 5 -- 6 ---- -- -- -- -- -- -- -- -- -- -- -- 7 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 8 -- -- -- -- -- -- -- -- -- -- -- -- -- -- 9 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 10 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 12 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 13 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 14 15 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 16 -- 11 ΔE (cm -1 ) Figure 8.6: Franck-Condon factors for HCOD ionization producing electronic ground state of HCOD + in the range from the ZPE (0 cm − 1 ) to 7,000 cm − 1 . Left: cis- isomer; right: trans- isomer. 8.6 Comparison with the parallel-mode harmonic approximation Franck-Condon factors between two electronic states are often approximated by assum- ing that (a) the vibrational wave-functions are harmonic and (b) all normal coordinates on the two surfaces are parallel, i.e. completely neglecting the Duschinsky rotations 47 . 205 In this case the Franck-Condon factors are products of 1D integrals over harmonic oscil- lator wave-functions, which are shifted by displacementΔ Q between equilibrium struc- tures along that normal coordinate. Because of the neglect of rotations, Δ Q depend on the choice of normal modes used for calculation. The spectra of HCOH calculated using the two sets of normal coordinates are compared to the VCI spectrum for both isomers. All spectra were generated using CCSD(T)/cc-pVTZ frequency calculations and the ezSpectrum program 48 . 0 1000 2000 3000 4000 0.00 0.01 0.02 0.03 0.04 0.05 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 17 18 19 20 21 1 2 3 4 5 6 7 8 9 10 12 14 15 1618 17 20 19 0 1516 21 ΔE (cm -1 ) CO stretch 0 1000 2000 3000 4000 0.00 0.01 0.02 0.03 0.04 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 17 19 20 21 1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 18 17 20 19 0 1516 21 ΔE (cm -1 ) Figure 8.7: Comparison between VCI (black lines) and parallel-mode harmonic oscil- lator approximation (red lines) using normal coordinates of the neutral (left) and cation (right) for the Franck-Condon factors of cis-HCOH. Harmonic intensities are not scaled to match VCI. Fig. 8.7 compares parallel-mode spectra with the VCI spectrum for the cis- isomer. The parallel-mode spectra are calculated using normal coordinates of the neutral (left column in Fig. 8.7) and cation (right column) normal coordinates. The displacements differ significantly only along one coordinate, the CO stretch;Δ Q equals 0.08 and 0.21 ˚ A √ amu for neutral and cation normal coordinates, respectively. The differences are due to rotations (mixing); the CO bond is longer in the neutral and other modes, espe- cially the stretches, have relative displacements along CO in their motion. Since the 206 0 1000 2000 3000 4000 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 2 3 4 5 6 7 8 10 9 11 12 13 14 17 0 ΔE (cm -1 ) i.p. bend ΔE (cm -1 ) 0 1000 2000 3000 4000 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0 1 2 3 4 5 6 7 8 10 9 11 12 13 14 15 16 17 0 Figure 8.8: Comparison between VCI (black lines) and parallel-mode harmonic oscil- lator approximation (red lines) using normal coordinates of the neutral (left) and cation (right) for the Franck-Condon factors of trans-HCOH. Harmonic intensities are not scaled to match VCI. bends have to be displaced significantly to account for the change in HOC and HCO angles (by 0.35 and 0.33 ˚ A √ amu in neutral and cation, respectively), Δ Q along the CO stretch is smaller in the neutral coordinates. Consequently the photoelectron spec- trum using the neutral normal coordinates shows negligible intensity in the CO stretch fundamental (peak 3 in Fig. 8.7) and underestimates the intensity of all states with quanta in this mode. Fig. 8.8 compares this approximation with VCI for trans-HCOH. The same effect is seen, except that theΔ Qs differ in one of the bending modes rather than the CO stretch (peak 1 in Fig. 8.8). The displacements are 0.05 and 0.15 ˚ A √ amu in the neutral and cation normal coordinates, respectively. The effect of normal coordinate rotation on the wave-function overlap between states is shown in Fig. 8.9. On the lower state, only the ground vibrational wave-function is considered (in the absence of hot bands). The errors in FCFs due to rotation of the ground vibrational wave-function depend on two factors: the displacementΔ Q and the 207 difference in frequencies of the active normal modes: if these frequencies are very sim- ilar, errors are small [column (b) in Fig. 8.9]. In HCOH, the three active frequencies are within 238 and 294 cm − 1 of each other for cis- and trans-, respectively. On the upper state, all of the wave-functions are considered. For excited vibrational wave-functions, even small rotations can significantly affect the overlap due to the nodal structure [col- umn (c) in Fig. 8.9]. Therefore, for large relative rotations of normal coordinates, it can be more accurate to use the normal coordinates of the cation within the parallel-mode approximation, especially if the active modes have similar frequencies on the neutral state. Figure 8.9: The effect of rotations of normal coordinates on Franck-Condon fac- tors within the parallel-mode approximation. (a) The correct overlap between wave- functions on lower (q”) and upper (q’) surfaces. (b) The overlap when lower normal coordinates are rotated to coincide with upper coordinates. (c) The overlap when upper normal coordinates are rotated to coincide with lower coordinates. 208 8.7 Conclusions We report accurate configuration interaction calculations of vibrational levels of the cis- and trans- isomers of HCOH + and HCOD + . The photoelectron spectra from the ground vibrational wave-functions of the two isomers are also presented. HCOH + is derived by removing an electron from a doubly-occupied lone pair orbital on the carbon atom (Fig. 8.4), with antibonding contribution along CO. This leads to large structural changes upon ionization, including shortening of the CO bond and increase in HCO angle due to increased s hybridization on C. Changes in harmonic frequency are due to structural changes and in the reduced repulsion between electrons on O and the C center in the cation. VCI fundamental excitations are harmonic for the lowest four normal modes, while the CH and OH stretches show anharmonicities over 150 cm − 1 . Due to the large dif- ference in equilibrium structures on the neutral and cation surfaces, non-zero Franck- Condon factors are calculated for energies up to 7,000 cm − 1 . The progressions are localized into select frequencies, namely two in-plane bends and the CO stretch. This is rationalized in terms of the geometrical differences. Photoelectron spectra for the HCOD isotopes are significantly different than for HCOH; this is due to the suppression of D motion in the normal mode vibrations. The photoelectron spectra in the parallel-mode harmonic approximation were also calculated, and compared with the VCI spectra. This approximation was fairly accurate for the low-energy part of the spectrum, especially in duplicating intensities of the three active fundamental excitations in both isomers. For combinations and overtones, the harmonic intensities for the strong peaks are only accurate to within a factor of 2 for cis- HCOH. However, the parallel-mode harmonic approximation is slightly more accurate for trans-HCOH than for cis-. 209 The calculated photoelectron spectra for cis- and trans-HCOH are qualitatively dif- ferent, which should make an experimental identification possible. Moreover, these differences are present even in the low-energy part of the spectrum (below 2,000 cm − 1 ) where the VCI method is expected to have the highest accuracy. 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The package also contains modified versions of the MOLECULE Gaussian integral program of J. Alml¨ of and P.R. Taylor, the ABACUS integral derivative program written by T.U. Helgaker, H.J.Aa. Jensen, P. Jørgensen and P.R. Taylor, and the PROPS property evaluation integral code of P.R. Taylor. [36] J. Gauss, J.F. Stanton, and R.J. Bartlett. J. Chem. Phys., 95:2623–2638, 1991. [37] L. Koziol and A.I. Krylov. ezVibe, http://iopenshell.usc.edu/downloads/. [38] J.M. Bowman, S. Carter, and X. Huang. 22(3):533–549, 2003. [39] P. Jensen and P.R. Bunker. J. Chem. Phys., 89:1327–1332, 1988. [40] W.H. Green Jr., N.C. Handy, P.J. Knowles, and S. Carter. J. Chem. Phys., 94:118, 1991. [41] L. Salem and C. Rowland. Angew. Chem. Int. Ed. Engl., 11(2):92–111, 1972. [42] W.T. Borden, editor. Wiley, New York, 1982. [43] V . Bonaˇ ci´ c-Kouteck´ y, J. Kouteck´ y, and J. Michl. Angew. Chem. Int. Ed. Engl., 26:170–189, 1987. [44] A.D. Walsh. J. Chem. Soc., pages 2260–2266, 1953. [45] R. Hoffmann, G.D. 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Vibrational calculations often utilize a potential energy surface, which is either global or fit to some force field expression. In this case the electronic structure part of the problem has already been solved. The nuclear wavefunctions are expanded in a basis using the normal coordinates, and matrix elements of the Watson Hamiltonian are evaluated. As the final step, this Hamiltonian is diagonalized to yield the vibrational energies and wavefunctions. It is this last step we will focus on. As the system size increases, diagonalization creates a steep computational bottleneck. This is the standard problem for variational approaches. For example, consider a system with N M normal 214 modes. The number of states that are N E -tuply excited from the N M - dimensional reference state equals N E +N M N M − 1 ! ≈ 3× N A +N E 3× N A ! (9.1) Without introducing reduced dimensionality schemes, it is currently possible to calcu- late accurate vibrational spectra (states up to several thousand cm − 1 above the ZPE) for systems containing5− 6 atoms. However, in recent years highly accurate global PESs have been developed for systems of up to 9 atoms. For example, we briefly consider two systems: 4-atom hydroxycarbene molecule and 9-atom water trimer system. For the hydroxycarbene radical cation, HCOH + , up to about 10 excitations from an optimized VSCF reference were needed to converge vibrational eigenvalues up to∼ 7000 cm − 1 above the ZPE. For this 4-atom system, the dimension of the Hamiltonian matrix was ∼ 11,000× 11,000, which took approximately 12 hours to diagonalize directly using BLAS routines and 6 hours using a block-Lanczos algorithm for the lowest several hun- dred eigenstates. Water trimer has 9 atoms and 21 normal modes. A modest sized basis set consisting of up to 6 excitations, with additional limits of 4 quanta per mode and 4 modes simultaneously excited, leads to a matrix of dimension 24,564. For systems with ∼ 6 atoms, converged vibrational calculations are not possible for these systems with- out introducing reduced-dimensionality schemes which neglect mode coupling between several coordinates. This should be a valid approach for many molecular systems, but can pose problems for dimers and molecular clusters. Much previous work utilized approximations in variational calculations for vibra- tions. 1–6 In 1993, Stuchebrukhov et al. 1 introduced a “tier model” for coupling based on anharmonic couplings to third and fourth order in quartic force fields. Each tier 215 consists of the set of states which are coupled by a given order in perturbation the- ory. For instance, two states differing in six vibrational quantum numbers can be cou- pled by sixth-order anharmonicities in first order, or by cubic anharmonicities in second order. Spectroscopically, line broadening due to intramolecular vibrational redistribu- tion (IVR) should be strongly dependent on the density of states in the first case, while this dependence is less clear in the second case. For computational purposes, the sec- ond case is strongly preferrable, and the question was whether this model was valid in real life. Application of the tier model to high overtones of acetylinic H-stretch in (CX 3 ) 3 YCCH, where X = H, D and Y = C, Si, with diagonalization of states within tiers and spectral linewidths obtained by the Golden Rule expression, achieved impressive agreement with experiment. More recently, Benidar et al. 6 calculated vibrational spectra of cyclopentadi- enylphosphine, a 13-atom system, using DFT, and achieved remarkable agreement with experiment. However, the vibrational states were partitioned beforehand according to so-called “spectral windows,” which are defined by iterative applications of an excitation operator to an initial set of states. Twenty-one spectral windows between 200− 3500 cm − 1 were chosen, and agreement with experiment was within 30 cm − 1 for all the states. Several works address prescreening of the VCI basis based on perturbation theory applied to the harmonic or VSCF functions. 3, 5 This allows iterative buildups of active space Hamiltonians which significantly reduce dimensionality or achive block diagonal- ization. The method described below calculates the entire Hamiltonian matrix, and then block diagonalizes it using the natural sparsity. Several of the previous works avoid explicit evaluation of the entire Hamiltonian, which we do not. In the n-mode repre- sentation of the PES, 2-mode, etc. integrals are calculated up to a maximum 1-mode quantum numbern max and stored in tensors; the Hamiltonian is then built up by simply 216 summing over indexes of the tensors. For increasingly large systems, perhaps 15− 20 atoms, this method becomes unfeasible and prescreening of the basis before building up the Hamiltonian would be necessary. In the following section, we suggest below that the nature of the normal coordinate basis and harmonic oscillator basis naturally facilitates a block structure. Even division into2− 3 blocks introduces large savings, and a pilot implementation suggests that this is the case. 9.2 Methodology Matrix diagonalization scales asN 3 , whereN is the dimension of the matrix. For block diagonalization into M equally-sized blocks, the scaling reduces to M − 2 × N 3 . The usefulness equals the extent to which the vibrational matrix is block diagonalizable. This depends on coupling terms between matrix elements. The following section provides rationalization for why nuclear Hamiltonians in normal coordinates lend themselves extremely well to this separability. As a consequence, a simple algorithm for composing the blocks is proposed, which should provide significant savings in computational cost with small loss in accuracy. The normal coordinates used in the Watson Hamiltonian provide a separable approx- imation to the wavefunction. Multimode effects provide a correction which can be fairly substantial. However, these are local in the sense that their Taylor series converges with the number of dimensions: 3-mode effects are smaller than 2-mode effects, and 4-mode effects are even smaller. Consider a harmonic PES with a nonzero coupling term only between modes 1 and 2,V 12 . Then, a matrix element must have left and right quantum numbersn q i =m q i for allq i =3,4,...,N M in order to be non-zero, and the VCI matrix 217 will reduce to one large diagonal block and one block of sizeN 1 × N 2 , whereN i equals the number of quanta allowed inq i . In general several of the coupling terms are nonzero, however partitioning the Hamil- tonian into sets of coupled normal modes is facilitated. Furthermore, within a set of terms coupled by the potential, one can observe that each quantum number mixes only with a small subset of the total quanta. For instance, the integralhn 1 =0|V 1 |m 1 =10i is likely to vanish. This second-level partitioning further separates the basis. Following is a simple algorithm to block diagonalize the vibrational Hamiltonian. First, choose a threshold for setting a matrix element to 0. A value between 1− 10 cm − 1 is suggested. Second, for each basis indexu, consider the setS u of basis indexes v such thatH uv 6=0. 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The code calcu- lates numerical integrals of the kinetic energy, potential energy,and the inverse moment- of-inertia tensor. The PES is treated within the so-called n-mode representation (see Chapter 7) which provides a systematic expansion of the full-dimensional surface into 1-, 2-, 3-, etc. dimensional terms, with increasing difficulty to integrate but decreasing contribution to energy. The program is freely available as an executable which can be used with any of the Bastiaan Braams symmetrized potential energy surfaces, about 20 of which can also be downloaded. It can also be used with general user-supplied PESs, with modification of the general “PES” class and recompilation. The well-commented source code (via Doxygen program) is also available freely upon request. Although not standard in the download, this code has also been used for calcula- tion of infrared intensities (requiring a dipole moment surface) as well as calculation of Franck-Condon factors between multiple surfaces. These are fairly easy to implement and further information is available. 233 Download and information are available on the iOpenShell website, http://iopenshell.usc.edu/ A.2 Input parameters This appendix provides user details for ezVibe, a program to calculate vibrational self- consistent field (VSCF), Moller-Plesset (VMP2) and configuration interaction (VCI) anharmonic energies and vibrational wavefunctions on potential energy surfaces. User must supply an input file in XML format. Please see samples provided with the distribution. Following is a list and short description of all input fields. User must provide (i) geometrical parameters: optimized geometry and set of normal coordinates, or starting geometry as an initial guess in optimization and normal mode analysis, (ii) quadrature details: degree of mode coupling and size of quadrature grids, and (iii) parameters defining the VSCF and post-VSCF calculations: inclusion of ro-vibrational (Watson and Coriolis) corrections, size of harmonic basis sets for VSCF, and number of excitations to include in post-VSCF calculations. Running the executable requires one command line argument, the address of the XML input file. For example, ./ezVibe input1.xml prints to standard output, and ./ezVibe input1.xml ¿ output1.txt prints into plain text file. In addition, two XML tags in the input file specify the interface with the PES: ¡pes moltype¿ molecular type (for default ezPES surfaces). Atom types in decreas- ing multiplicity, in x, y, z order (see ezPES documentation or following examples). Depends only on the nuclei, therefore isomers and charged species receive the same 234 tag. For example, H 2 CO and HCOH + are both specified as x2y1z1. All isomers of tridehydrobenzene, C 6 H 3 are specified as x6y3. ¡pes directory¿ pathway towards PES coefficients from the current directory (for default ezPES surfaces; for user-supplied surface, see below). For example, pes dirs/H20/ is an appropriate tag. For example, two sample commands, for HCOH and C 6 H 3 surfaces: ./ezVibe x2y1z1 HCOH-PES/ hcoh1.xml ./ezVibe x6y3 C6H3-PES/ input/c6h3.xml For tags that start with ”if”, either a 0 (false) or positive integer (true) is required. <ifOptimize> If true, search for critical point on PES. Optimized structure is translated to center-of-mass and rotated to principle-rotation axes. If false, structure provided in input is used for calculations. Note that center-of-mass coordinates are required; alignment to principle-rotation axes is optional, but diagonalizes the moment-of-inertia tensor. Geometry must be provided in bohr. <ifCalc Normal Modes> If true, Hessian is calculated by central finite differences (5- point formula). Normal coordinates and frequencies are calculated and used as basis set for calculations. If false, normal coordinates and frequencies must be provided in input. Normal coordinates should be un-weighted (reduced mass = 1 for each mode; divide eigenvecs of Hessian by square-root mass). Frequencies should be provided in wavenumbers. <geom opt stepsize> Optimization is performed by Newton minimization using the Hessian calculated at each step. The step size can be too large in early iterations lead- ing to a geometry outside the well or in an unphysical region (i.e. atoms very close together) on the surface. The displacement vector is contracted by this stepsize at each 235 step. Default value is 0.33; increase for faster convergence, decrease if the optimization diverges. 0.2− 1.0 is a standard range. <ifPrint opt step> If true, prints structure at every step. <geom opt convergence> Structure is optimized when PES between subsequent itera- tions is below threshold (default 0.0001 cm − 1 ). <findiff stepsize> Step size for finite difference formula for numerical second deriva- tives in Hessian calculation (default 0.0025 bohr). <ifPrint molstate> If true, prints out the normal coordinates. <number quadrature points 1d> <number quadrature points 2d> <number quadrature points 3d> <number quadrature points 4d> Number of Gauss-Hermite quadrature points for quadrature over the PES and inverse- moment-of-inertia. For an integrand which is a polynomial of degree 2N, N or greater points ensures exact integration. For an integral over the modals, we have contributions from the Hermite polynomial of the highest contributing harmonic oscillator (v1 and v2 from left and right) and from the potential (v3 = 2 in harmonic potential, etc.) For example, using harmonic oscillator basis up to v = 10, and assuming the potential is 5th degree, this gives 13 points required for exact integration. In principle, 15 points for 1D and 2D grids is usually sufficient for VCI states up to doubly excited, and 20 is sufficient up to 300 states for the HCOH radicals. 3D and 4D grids require less quadrature points because the n-mode representation of the PES subtracts out contributions from lower n. For a 3D calculation, set quadrature points 4d to 0. To diagonalize the 1D potentials and neglect multimode couplings entirely, set quadrature points 1d greater than 0 and all others to 0. 236 <ifCoriolis> If true, Watson correction term is added to the potential up to the level of mode-coupling (from number quadrature points nd tags, and Coriolis correction is included at the 2-mode level. If false, ro-vibrational terms are neglected. <vscf harmonic basis> The VSCF modals are expanded in harmonic oscillators. v= 10− 15 is recommeded for higher than fundamental excitations. Note that extremely high numbers can cause problems if PES has unphysical holes far from the minimum or if there are barriers nearby. <vscf occupation> Label for VSCF state to be calculated. For instance, for 3-mode sys- tem, ground-state calculation specify 000 in this field. To calculate state with 1 quanta in first mode, specify100. <vscf convergence energy> Iterations are terminated when subsequent energy is below this threshold. Units are cm − 1 and default is 0.1 <ifVMP2> If true, VMP2 energy is calculated for the VSCF state which was optimized. <ifVCI> If true, VCI calculation is performed. VSCF basis wavefunctions are created on a grid, matrix elements over the PES and optionally Coriolis terms are stored, and Watson Hamiltonian matrix in VSCF basis is diagonalized. <max post excitation> This sets the maximum total number of excitations (quanta) allowed in the VSCF basis. For instance, for three-mode system, setting this tag to 5 would include states500 and221. <max post per mode> This sets the total number of excitations allowed in each normal coordinate. For instance, for three-mode system, if max post excitation = 5 and setting this tag to 3,300 would be included as well as311. 500 would not be included. <max post modes excited> This sets the total number of modes excited. For three- mode system, max post excitation = 5 and max post per mode = 3 and setting this tag to 2 will include320 but not311. 237 <vci number states> Number of VCI states to print. Default is all of them (size of VCI matrix). <ifVCI harm basis> ezVibe can calculate VCI using harmonic oscillators instead of optimized VCI modals. If true, the VSCF calculation is skipped. This option gives less converged VCI energies for a given basis. Also, VMP2 method is not applicable if this option is true. <ifVCI block> ezVibe can block diagonalize the basis set. The Hamiltonian matrix contains 0 elements due to symmetry, orthogonality of the basis sets, and couplings. If true, the ordering of the basis indexes is rearranged and block diagonal matrices are formed and diagonalized. Since this option is most relevant for large matrices, the Hamiltonian matrix is not calculated. <VCI block thresh> If the ifVCI block option is true, matrix elements below this threshold energy (in cm − 1 ) will be set to 0, which potentially increases the number of blocks and decreases computational time. The default is 0.0 cm − 1 ; which introduces no approximation and gives quantitatively equal results to ifVCI block = false. A low threshold, such as10 − 5 , will ensure block diagonalization by point group symmetry by accounting for tiny errors due to numerical integrals. From a few test cases, a threshold of 1 cm − 1 produces agreement within 0.1 cm − 1 for all states, and 10 cm − 1 provides significant savings with agreement within 2− 3 cm − 1 . For particular systems, this threshold should be tested for small VCI bases to establish rough error bars before using this method for extremely large calculations. A.3 User-supplied potential energy surface Handlers for the PES are supplied in the GenPES class (GenPES.h and GenPES.C), which contains one function for the PES calls and one for initialization. All that the 238 VCI code requires is the function pes sp, which returns the value of the potential in hartrees for a supplied geometry: given as an array of coordinates in x1, y1, z1, x2, y2, z2 ... order (in bohr). In the simplest case, substituting a stand-alone expression (such as a function of bond-lengths and angles)into pes sp and recompiling would suffice. In the case of Fortran code, please see structure of Fortran 90 subroutines and associated wrappers in accessory code/fortran 90 routines/ directory. Note that providing a gen- eral user-supplied PES (except all of the Bas Braams symmetrized PES library) require recompiling the ezVibe program. 239 Appendix B The ezVibe program: code structure This appendix details code structure for ezVibe. The main classes utilized in ezVibe and their purpose are detailed below. A simple inheritance diagram for the major classes is shown in Fig. B.1; each subsequent step in the calculation uses previous objects (i.e. the sets of VSCF solutions) both to provide a basis set and to re-use relevant integrals (i.e. VMP2 requires VSCF effective Hamiltonian). Figure B.1: Inheritance diagram for major classes in ezVibe. Basic class: this structure holds information from the input file including molecular, quadrature, and basis set data. It also contains handler functions for accessing Atom, NormMode, GaussHermite, HarmOsc data types. Holds pointers to wavefunction grids for: harmonic oscillators and VSCF bases. 240 Harmonic oscillator class: Holds wavefunctions, grids, 1st and 2nd derivatives. Also holds expressions for calculating kinetic energy matrix elements analytically. Gauss Hermite: the integration throughout the PES is done as summation over dis- crete points in 1 dimension with associated weights. Weights and points differ per normal coordinate. The points and weights are assigned in functions held here. Dif- ferent quadrature schemes can be implemented here; currently basic Gauss-Hermite quadrature is used; this requires an e − ax 2 term in the integrand. This fits naturally due to the choice of harmonic oscillators as one-particle basis for the wavefunctions. See Appendix A for details about accuracy of this method. Optimized “HEG” quadrature of Carter et al. was also used but found to offer same results with comparable efficiency for the model systems tested (H 2 O and HCOH). NM Setup: Calculates geometry optimization and normal mode analysis. Hessians are calculated with 5-point central difference formula. Also translates structure into center-of-mass coordinates and aligns along principal axes. Potential: This holds the quadrature points for the potential energy surface in the n- mode representation. A large complication is that different number of quadrature points are allowed for different n, so that subtracting contributions from lower n requires sepa- rate grids. To this end there are A PP (point potential) and A MP (mean potential) grids; the former are temporary and contain different numbers of points in each dimension, and the latter are the final, square n-mode representations. The Watson correction term is calculated along with the potential and treated as an “addition to the PES”, while for the Coriolis correction, the inverse moment of inertia (MOI) tensor is calculated up to the 2-mode representation and stored in A Cor utens. The diagonal inertia tensor is not assumed (which corresponds to alignment along principal rotation axes). A note about the Coriolis terms: they are calculated in a sum-over-states expression. This leads to 241 two relevant 2-dimensional integrals: firstly the modals over theQ 1 ∗ P 2 operator, and secondly the modals over the inverse-MOI operator held in A Cor utens. VCI Main: this is the main program for VSCF, VMP2, and VCI calculations. Fol- lowing steps are performed: (i) XML input file is parsed and assigned to MolState object (molecular data) and Basic object (other data). (ii) Array of harmonic oscillator objects is assigned and grids are created. (iii) Arrays of Gauss-Hermite quadrature points and weights are created based on the frequencies of each mode. (iv) Array of VSCF modals is created for the VSCF state to be optimized. (v) One-dimensional kinetic energy and potential matrixes created for each normal coordinate. (vi) VSCF effective potentials are calculated and stored. For each normal coordinate Qi, partial integration over the modals Q j , etc. is done. For instance, for a 2-mode potential V ij , integration along Q j is done over φ j ; these are summed over all j 6= i and kept in a 1D array along Q i . Finally to create a VSCF matrix element for Q i , this 1D-array is integrated over harmonic oscillators u and v. For u = v terms, terms in the VSCF expansion not containing modei must also be included as a summation. (vii) The VSCF matrixes are diagonalized and energy expression is evaluated. If convergence is not reached, the modal objects are reset and step (vi) is repeated. (viii) if either VMP2 or VCI is to be calculated (<ifVMP2> and/or<ifVCI> tags in xml input file) then VMP2 object is constructed. The size of the VMP2 and VCI expan- sions are determined by the three tags<max post excitation>,<max post per mode>, and <max post modes excited> in the input file. Please see Appendix A for detailed explanation. The 1D modal grids are calculated using the optimized modals. Note that 242 these must be calculated also for the excited eigenvectors of the VSCF eigenstates (even though VSCF calculation optimizes only a single state), with quantum numbers in each normal coordinate ranging from 0 to min(max post excitation,max post per mode). This is unique from where each VSCF basis state is optimized through a separate VSCF calculation. The latter becomes time-consuming for large post-VSCF bases and also encounters optimization problems. Also the improvement gained from reducing the basis size due to improved VSCF states is modest. (ix) VMP2 energy is calculated. Note that VMP2 does not support Coriolis terms at this time. (x) If VCI calculation is specified, VCI Tensor object is created. This holds tensors for the multi-dimensional matrix elements over the potential. For a 2D tensor element, the indexes are: q1, q2, n1, n2, n3, n4 corresponding to <φ n1 q1 φ n2 q2 |V q1,q2 |φ n3 q1 φ n4 q2 > Tensors are stored so that elements q1> q2, n1> n3, and n2> n4. The functions get Tens return the correct element from any ordering of the coordinates and quanta. (xi) The Watson Hamiltonian is formed by summing over analytic kinetic energies (in VCI Tens class) and collecting matrix elements from the tensors. It is Hermitian, diagonalization is done by Herm Diag in KMatrix class. A block Lanczos routine for diagonalization of lowest several states (up to a few hundred) would be extremely useful. 243
Abstract (if available)
Abstract
Radical systems, or systems with unpaired electrons, play fundamental roles in chemistry, biology, and physics. Relevant examples include photochemical breakdown of air pollution in the lower atmosphere, carbon-carbon bond formation in synthetic and pharmaceutical chemistry, and formation of fundamental organic molecules in the interstellar medium. The work presented in this thesis focuses on the electronic structure and spectroscopy of open-shell species
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Creator
Koziol, Lucas Peter
(author)
Core Title
Electronic structure and spectroscopy of excited and open-shell species
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Chemistry
Publication Date
07/14/2009
Defense Date
05/15/2009
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University of Southern California
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chemistry,electronic structure,OAI-PMH Harvest,radicals,spectroscopy
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English
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Advisor
Krylov, Anna I. (
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), Haas, Stephan (
committee member
), Reisler, Hannah (
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koziol@usc.edu,lucas.koziol@gmail.com
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Tags
chemistry
electronic structure
radicals
spectroscopy