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Interfacial vibrational and electronic excited states of photovoltaic materials probed using ultrafast nonlinear spectroscopy

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Interfacial vibrational and electronic excited states of photovoltaic materials probed using ultrafast nonlinear spectroscopy

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Content Interfacial Vibrational and Electronic Excited States of Photovoltaic Materials Probed Using Ultrafast Nonlinear Spectroscopy by Dhritiman Bhattacharyya A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CHEMISTRY) December 2019 ii Dedicated to my loving parents, Atanu Bhattacharyya and Chitra Bhattacharyya iii Acknowledgements Surviving through the graduate school at USC has not only helped me shape my scientific acumen, but also left an ever-lasting imprint on every aspect of my life. It presented a learning podium in my front where life lessons were also given along with conventional education. The graduate curriculum completely transformed me as a person, it made me more agnostic over years and taught me to question everything before accepting it. The two people who mostly deserve the credit for helping me through this journey are Prof Stephen Bradforth and Prof Alexander Benderskii. Working with them as a joint student is undoubtedly one of the best decisions of my life so far. They both are great scientists, and yet very humble, simple and down to earth. I would like to thank both of them for giving me this wonderful opportunity and for guiding my way over the years. There are a lot to learn from Steve. On one side, he is very passionate about his research, and on the other side, he is a wonderful administrator. One should really learn some time management skill from him. The best thing that I learnt while interacting with him about science is that devil always lies in the details. He motivated me to extract the simple physical interpretation out of a very detailed and messy work so that it can be more accessible to the scientific community, particularly to those who are not very familiar with our field of study. Apart from being a perfect mentor, he was also a friend with a wonderful sense of humor and always enjoyed listening to the embarrassing stories of our Friday night-outs. I am really blessed that I got someone like him as my supervisor in pursuit of my graduate studies. Working with Alex has truly been a fantastic experience. He always gave me utmost freedom to work in the lab and try different projects. He was extremely supportive with everything and I have never heard him say ‘NO’ to anyone. Alex is also a wonderful teacher. Things that I learnt in his Nonlinear Spectroscopy course have helped me throughout my PhD career and will be an asset forever. iv I also take this opportunity to thank Prof Jahan Dawlaty. He is the person who has always motivated me throughout this journey. Working with him as a Burg Foundation teaching fellow had been an awesome experience to remember. The way he answers a question starting from the basics is really a treat to watch. He taught us the Introductory Spectroscopy course and built the foundation for my being an independent thinker. I will always be grateful to him for helping me find the postdoctoral position. When I first joined the group, there were seniors who helped me overcome the first hurdle of getting accustomed to the new environment. Anirban Roy, Saptaparna Das and Purnim Dhar really emerged as the examples of ideal seniors. Starting from teaching me how to align the laser, giving inputs on the qualifying exam proposal and attending my horrible practice talks; their contributions were enormous. Not only being there during the academic hours, but also travelling with them to different National parks and joining them for the home-cooked lunch/dinner, really made me feel at home. I cherish my friendship with Gaurav Kumar and Angelo Montenegro. The friendship started with solving the assignments together and continued to be stronger over the years. Angelo’s wedding was the first and so far, the only American wedding I have been a part of and being his ‘Best Man’ during the ceremony had really been a tremendous experience. I would also like to thank all of my group members: Chayan Dutta, Konstantin Kudinov, Muhammet Mammetkuliyev, Jimmy Joy, Laura Estergreen, Ariel Nessel, Ryan Mcmullen and Michael Kellogg. I will surely miss their support and companionship. I express my sincere gratitude to Dr Kaushik Nanda, Tirthendu Sen and Sahil Gulania from Prof Anna Krylov’s group for helping me with QChem calculations and giving me a lot of feedback on different theory-related problems. The acknowledgment would have been incomplete without thanking the Department of Chemistry at University of Southern California. No amount of good words is enough to express the help and support provided by the administrative staff of the department, Michele Dea, Magnolia Benitez and all the other persons who made our day-to-day life a lot easier. I would also like to thank Prof Surya Prakash, Prof Hanna Reisler and Prof Curt Wittig for being there from the very beginning and constantly guiding me through v the journey. I also want to acknowledge National Science Foundation (CHE-1665532) for providing me with research assistantship during my graduate studies. Now, I would like to acknowledge that one person who was always there with me during my entire graduate studies: Subhasish Sutradhar; my senior, friend and roommate. We used to discuss everything starting from science to politics and educate each other constantly. He is that friend who was always there through thick and thin, and guided me as my elder brother thought-out. I hope this bonding between us will stay forever. Finally, it is the time to acknowledge those two human beings for whom I am what I am today: my loving parents, Atanu Bhattacharyya and Chitra Bhattacharyya. Being a single child has never been easy because of the amount of expectations my parents had from me. But what is more difficult is to let that only child move to a different continent for studies. To my parents, my career has always been a top priority. I have gotten ample encouragement and support from them to pursue my PhD at USC which most of the single children in India do not get. The way they brought me up since childhood amidst all odds and the way they sacrificed their own happiness for my career know no bounds. Today, on the verge of finishing my studies at the graduate school I feel that I have finally done justice to their lifelong dreams and I thank them for bringing me up as a good human being. Last, but not the least, I would like to thank my beautiful wife, Chandrama Mukherjee, for being a constant source of inspiration. Dhritiman Bhattacharyya July 2019 Los Angeles, CA, USA vi TABLE OF CONTENTS List of Figures ……………………………………………………………………………… viii List of Tables ………………………………………………………………………………. xv Chapter 1. Electronic processes in Organic Photovoltaics I. Introduction …………………………………………………………………… 1 II. Band-Bending at the Interface ………………………………………………… 3 III. Sum Frequency Generation Spectroscopy: VSFG vs ESFG …………………. 4 IV. Conventional way of doing ESFG experiments ………………………………. 7 V. Our proposed scheme of ESFG experiment …………………………………... 8 VI. Experimental Section: White Light Generation ………………………………. 10 VII. Outline of the thesis ………………………………………………………. 14 Chapter 1 Bibliography …………………………………………………………… 15 Chapter 2. Molecular Orientation of Poly-3-hexylthiophene at the Buried Interface with Fullerene I. Introduction …………………………………………………………………… 20 II. Results and Discussion ………………………………………………………… 22 III. Conclusions ……………………………………………………………………. 40 Chapter 2 Bibliography ……………………………………………………………. 41 Chapter 3. Vibrational Sum Frequency Generation (VSFG) Spectroscopy Measurement of the Rotational Barrier of Methyl Groups on Methyl- Terminated Silicon (111) Surfaces I. Introduction …………………………………………………………………… 45 II. Results and Discussion ………………………………………………………… 47 III. Conclusions ……………………………………………………………………. 56 Chapter 3 Bibliography ……………………………………………………………. 57 Chapter 4. A Vibrational Sum Frequency Generation Study of the Interference Effect in a Thin Film of 4,4′-Bis(N-Carbazolyl)-1,10-Biphenyl (CBP) and the Interfacial Orientation I. Introduction …………………………………………………………………… 62 II. Results and Discussion ……………………………………………………….. 64 A. Spectroscopic characterization of CBP film ……………………………….. 64 B. Modeling optical interference for CBP films ………………………………. 70 III. Conclusions ……………………………………………………………………. 80 Chapter 4 Bibliography ……………………………………………………………. 81 vii Chapter 5. Electronic Structure of Liquid Methanol and Ethanol from Polarization- Dependent Two-Photon Absorption Spectroscopy I. Introduction …………………………………………………………………….. 85 II. Background: 1PA spectrum of methanol and ethanol …………………………. 89 III. Methods ………………………………………………………………………… 92 A. Experimental ………………………………………………………………... 92 B. Computational ……………………………………………………………… 93 IV. Results and Analysis …………………………………………………………… 96 A. Calculation of 2PA cross-section …………………………………………… 96 B. Experimental 2PA spectra and polarization ratios …………………………... 101 V. Discussion ………………………………………………………………………. 105 A. Characterization of excited states: 1PA vs 2PA spectra ……………………. 105 B. Comparing 2PA spectra of water, methanol and ethanol ………………… 110 C. Simulation of 2PA spectrum ……………………………………………… 112 VI. Discussion ……………………………………………………………………. 122 Chapter 5 Bibliography …………………………………………………………… 126 Chapter 6. Electronic Structure of Liquid Hexane and Cyclohexane from Polarization- Dependent Two-Photon Absorption Spectroscopy I. Introduction …………………………………………………………………... 133 II. Background: 1PA spectrum of cyclohexane and hexane ……………………. 135 III. Results: Experimental and Computational …………………………………… 137 IV. Analysis ………………………………………………………………………. 144 V. Discussion ……………………………………………………………………. 154 VI. Conclusion ……………………………………………………………………. 157 Chapter 6 Bibliography …………………………………………………………… 159 Chapter 7. Teaching Entropy from Phase Space Perspective: Connecting the Statistical and Thermodynamic Views Using a Simple 1-D Model I. Introduction …………………………………………………………………… 162 II. Motivation …………………………………………………………………… 166 III. Introduction to Phase Space …………………………………………………… 169 IV. The Phase Space Approach to Describe Entropy ……………………………… 172 V. Examples ………………………………………………………………………. 178 A. Adiabatic Volume Change ………………………………………………… 178 B. Isothermal Volume Change ………………………………………………. 182 C. Expansion into Vacuum …………………………………………………… 183 VI. Conclusion ……………………………………………………………………. 184 Chapter 7 Bibliography …………………………………………………………… 185 viii List of Figures 1.1 Schematic diagram of (a) bilayer and (b) bulk heterojunction solar cells. The active layer is composed of electron donor and electron acceptor molecules. The donor-acceptor moiety of the bilayer solar cell is shown in the inset. ……………………………………………. 2 1.2 Representative band diagram for metal/n-type semiconductor before and after equilibrium. Evac, Ev, EC, EF,m, EF,s, stand for vacuum energy, energy of valence band, energy of conduction band, Fermi energy of metal and Fermi energy of the semiconductor respectively. The work functions of the metal and the semiconductor are represented by φm and φs respectively. This figure is adapted from Chem. Rev. 2012, 112 (10), 5520–5551. ……………………… 4 1.3 Schematic energy diagram for (a) VSFG and (b) ESFG spectroscopy. ……………….. 6 1.4 This figure is adapted from a publication by Yamaguchi et al. (J. Chem. Phys. 2006, 125 (19), 194711. https://doi.org/10.1063/1.2375093). ESFG spectra of Oxazine 750 at the air/water interface measured with three different wavelengths of the up-converting pulse; 800 nm, 740 nm and 692 nm. ………………………………………………………………………… 7 1.5 Our proposed scheme to perform ESFG spectroscopy. The left side and the right side show the state diagram of excitation and the corresponding Feynman diagram respectively; ‘v’ and ‘e’ stand for the virtual state and the excited electronic state. ……………………………. 9 1.6 NIR spectrum of the white light continuum generated by focusing the 800 nm pulse in water, sapphire and vanadate crystals. In case of water and sapphire, OD=1 filter is used before the recording the white light spectrum. ……………………………………………………. 11 2.1 FTIR (red) and Raman (blue) spectra of P3HT/C 60 bilayer sample. The vibrational modes are shown using colored sticks: C-C inter ring stretch (faded blue line, 1380 cm -1 ), C=C symmetric stretch modes (gray line, 1410-1490 cm -1 ) and C=C asymmetric stretch (yellow line, 1510 cm - 1 ) of the thiophen ring. …………………………………………………………………. 23 2.2 Calculated SSP and PPP intensity as a function of orientation angle for both P3HT/air and P3HT/C60 interface. ……………………………………………………………………. 25 2.3 (a) SFG spectrum of P3HT spin-coated on CaF2 substrate. Both PPP (red) and SSP (blue) spectra are shown. All IR active modes of P3HT appear in SFG spectrum as well: C-C inter ring stretch (faded blue line, 1380 cm -1 ), C=C symmetric stretch (gray line, between 1430- 1490 cm -1 ) and C=C asymmetric stretch modes (yellow line, 1510 cm -1 ) of the thiophene ring. (b) SFG spectrum of P3HT/C60 bilayer sample. In addition to the peaks of P3HT, the sharp feature at 1425 cm -1 (green line) appears due to the IR active mode of C60. (c) Calculated PPP (red), SSP (blue) SFG intensity and PPP/SSP intensity ratio (green) for P3HT/CaF 2 interface as a function of the tilt angle between the plane of the thiophene ring and the surface normal (d) Calculated plot for P3HT/C60 interface. …………………………………………… 26 2.4 SSP spectra of C60 vapor-deposited on CaF2. …………………………………………. 27 ix 2.5 Calculated PPP/SSP ratio for C=C symmetric stretch of the thiophene ring at P3HT/C 60 and P3HT/CaF2 interface. Here we the vary the interfacial refractive index for the IR wavelength. For P3HT/C60, n' IR is varied in between 1.97 and 2.12; whereas for P3HT/CaF2 it is varied in between 1.35 and 2.12. ………………………………………………………………. 29 2.6 Calculated PPP/SSP ratio for C=C symmetric stretch of the thiophene ring at P3HT/CaF2 interface. Here we the vary the interfacial refractive index for both the visible and SFG wavelengths. n'VIS is varied in between 1.43 and 2.27; and n'SFG is varied in between 1.43 and 2.37. ………………………………………………………………………………. 30 2.7 Calculated PPP/SSP ratio for C=C symmetric stretch of the thiophene ring at P3HT/C60 interface. Here we the vary the interfacial refractive index for both the visible and SFG wavelengths. n'VIS is varied in between 2.02 and 2.27; and n'SFG is varied in between 2.09 and 2.37. ……………………………………………………………………………….. 31 2.8 Calculated SSP and PPP intensity of C=C symmetric stretch mode of the thiophene ring of P3HT as a function of the tilt angle of the thiophene backbone with respect to the laboratory Z-axis. Calculations are done using three different βaac values.; from which βaac = 70 is used for our reported orientational angle. …………………………………………………… 32 2.9 SFG spectra of P3HT-C60 bilayer sample where the spin-coating solution of P3HT is (a) 11 mg/mL and (b) 5.5 mg/mL. ……………………………………………………………. 37 2.10 10(a) and 10(b): Comparison of SFG spectrum of P3HT-C60 bilayer sample before and after thermal annealing at 155 0 C for one hour for SSP and PPP polarization respectively. (c) The normalized SSP spectra in the range 1430-1500 cm -1 before (red) and after (blue) thermal annealing. (d) Cartoon figure of the orientation of P3HT at fullerene surface before and after annealing. ………………………………………………………………………………. 39 3.1 VSFG spectra of methyl-terminated Si(111) for PPP and SPS polarizations measured at five different temperatures. Spectra at different temperatures (blue (21 o C), purple (38 o C), green (56 o C), yellow (85 o C) and red (118 o C)) are stacked vertically, after adding a constant offset to each spectrum. In going from 21 o C to 118 o C; the offset values for PPP were -804, -400, 500, 1000, 1500 and for SPS were: -50, -167, -50, 50, 100. Black lines show spectral fitting as described in the text. ………………………………………………………………… 48 3.2 Plot of PPP and SPS linewidths of the asymmetric −CH3 stretch vs. temperature. ……. 52 3.3 ln(krot) vs 1/T plot; where krot is the rotational rate constant and T is the temperature in Kelvin. The plot is fitted with a straight line (red). The slope and the intercept of the fitted line are used to calculate the rotational barrier (Erot) and the attempt frequency (k0). ……………….. 54 4.1 FTIR, Raman and Polarization-selective VSFG spectra of a 100 nm thin film of CBP. For our work, we are interested in the mode that is highlighted in the figure (blue band). The mode is x identified as the C=C symmetric stretching mode localized mainly on the biphenyl backbone of CBP. The structure of CBP is shown in the inset. …………………………………. 63 4.2 Azimuthal anisotropy study of the CBP film. The CBP film was kept on a rotational stage and VSFG spectra were measured changing the azimuthal angle of the film by 20 0 . The black crosses correspond to the normalized VSFG intensity. The blue line connecting the black crosses traces a circle, suggesting that the VSFG intensity doesn’t change with the azimuthal angle. ………………………………………………………………………………….. 65 4.3 VSFG spectra at PPP and SSP polarization combinations as a function of film thickness. Spectra of 20 nm, 50 nm, 100 nm, 150 nm and 200 nm films are stacked vertically from bottom to top. The film thicknesses are indicated to the right side of each spectrum. The black lines correspond to the fits of the spectra. …………………………………………………… 66 4.4 This figure is adapted from a publication by Tong et al. (J. Chem. Phys. 2010, 133 (3), 034704.). Here, schematic diagram of the optical interference is shown assuming a three-layer model where medium I, II and III represent air, CBP film and CaF2 respectively. 'd' is the thickness of the film. SFI and SFII represent the VSFG signals generated from air/CBP and CBP/CaF2 interfaces respectively. ……………………………………………………… 69 4.5 Plot of phase (φ2) of the VSFG signal from the buried interface as a function of film thickness, for both PPP and SSP polarization combinations. ……………………………………… 69 4.6 Simulated Fresnel factors as a function of film thickness: (left panel) CBP/air interface and (right panel) CaF2/CBP interface. Fresnel factors are also plotted as a function of interfacial refractive index; n′ = 1.7 (green), n′ = 1.6 (black), n′ = 1.5 (blue), n′ = 1.4 (red), n′ = 1.3 (grey), n′ = 1.2 (pink), n′ = 1.1 (nude), n′ = 1.0 (brown). For CBP/air interface; 1.0 ≤ 𝑛 𝐶𝐵𝑃 /𝑎𝑖𝑟 ′ ≤ 1.7 and for CaF2/CBP interface; 1.4 ≤ 𝑛 𝐶𝐵𝑃 /CaF 2 ′ ≤ 1.7 …………………………………… 71 4.7 Relation between molecular (a, b, c) and laboratory (X, Y, Z) coordinate frames for CBP. 𝜃 is the tilt angle of the transition dipole with respect to the laboratory Z axis. 𝜓 is the twist angle with respect to the long molecular axis and 𝜑 is the azimuthal angle in the X-Y plane. … 73 4.8 Plots of polarizability (𝛼 ) and IR transition dipole moment (𝜇 ) with respect to nuclear displacement for the 140 0 dihedral angle conformer of CBP. Six polarizability elements are shown, as 𝛼 𝑐𝑎 = 𝛼 𝑎𝑐 ; 𝛼 𝑐𝑏 = 𝛼 𝑏𝑐 ; 𝛼 𝑎𝑏 = 𝛼 𝑏𝑎 .The blue markers and the red lines represent the calculated numbers and the polynomial fits respectively. Derivatives are calculated from the fitted equation at the zero nuclear displacement. ………………………………………… 74 4.9 Plots of (top panel) 𝜒 𝑃𝑃𝑃 𝐼 , 𝜒 𝑆𝑆𝑃 𝐼 , 𝜒 𝑃𝑃𝑃 𝐼 /𝜒 𝑆𝑆𝑃 𝐼 and (bottom panel) 𝜒 𝑃𝑃𝑃 𝐼𝐼 , 𝜒 𝑆𝑆𝑃 𝐼𝐼 , 𝜒 𝑃𝑃𝑃 𝐼𝐼 /𝜒 𝑆𝑆𝑃 𝐼𝐼 with respect to the orientation angle of the transition dipole from the laboratory Z axis as shown in Figure 4.7. The horizontal dotted black lines on the top and the bottom panels represent the PPP/SSP amplitude ratio of the Lorentzians for air/CBP and CBP/CaF2 interfaces; as obtained from the global fitting procedure. For the air/CBP interface (top panel), the interfacial refractive index is varied from 1.0 ≤ 𝑛 𝐶𝐵𝑃 /𝑎𝑖𝑟 ′ ≤ 1.7 ; whereas for CBP/ CaF2 interface xi (bottom panel), it is varied in between 1.4 ≤ 𝑛 𝐶𝐵𝑃 /CaF 2 ′ ≤ 1.7; with n′ = 1.7 (green), n′ = 1.6 (black), n′ = 1.5 (blue), n′ = 1.4 (red), n′ = 1.3 (grey), n′ = 1.2 (pink), n′ = 1.1 (nude), n′ = 1.0 (brown). All calculations are shown for 𝜎 = 20. ………………………………………. 77 4.10 Plots of (left panel) 𝜒 𝑃𝑃𝑃 𝐼 /𝜒 𝑆𝑆𝑃 𝐼 and (right panel) 𝜒 𝑃𝑃𝑃 𝐼𝐼 /𝜒 𝑆𝑆𝑃 𝐼𝐼 as a function of orientation angle. The top and the bottom panels represent the distribution width of the tilt angle to be 𝜎 = 5 0 and 𝜎 = 20 0 respectively. …………………………………………………………… 79 4.11 Plots of (left panel) 𝜒 𝑃𝑃𝑃 𝐼 /𝜒 𝑆𝑆𝑃 𝐼 and (right panel) 𝜒 𝑃𝑃𝑃 𝐼𝐼 /𝜒 𝑆𝑆𝑃 𝐼𝐼 as a function of orientation angle. The top and the bottom panels represent the distribution width of the tilt angle to be 𝜎 = 5 0 and 𝜎 = 20 0 respectively. …………………………………………………………… 79 5.1 Gas phase 1PA and liquid phase 1PA and 2PA spectra of water, methanol and ethanol. The water gas phase spectrum is digitized from Ref: Chem. Phys. Lett. 2005, 416, 152–159 and is converted to ϵ (L mol -1 cm -1 ) using the equation ϵ = (NA/ln(10))σ1PA, where σ1PA and NA are 1PA cross section and the Avogadro constant respectively. The alcohol gas phase spectra are digitized from Ref 58 and converted to ϵ using A= ϵcl equation, where the path length l is 1μm. The water liquid phase 1PA spectrum is from Ref: J. Chem. Phys. 1974, 60, 3483–3486 and Ref: Nat. Commun. 2017, 8, 15435, and the 1PA spectra of liquid alcohols are from Ref 61 . In (c) the 2PA spectra for water, methanol and ethanol are shown. Water 2PA spectrum is from Ref: J. Chem. Phys. 2009, 130, 084501; the data for the alcohols are from this work. Spectra recorded with both a 4.6 eV pump and a 6.2 eV pump are shown in the same figure; parallel (blue) and perpendicular polarization (red). The purple and the green arrows show the position of the vertical ionization energies in gas phase and in liquid phase, respectively. The VIE of water vapor is 12.62 eV and is outside the spectral range shown in the above plot. (Insets) For liquid MeOH, the 1PA and 2PA showing detail from 6.5-8.5 eV. …………………… 88 5.2 Liquid phase 2PA spectra of propanol and butanol. Spectra recorded with both a 4.6 eV pump and a 6.2 eV pump are shown in the same figure; parallel (blue) and perpendicular polarization (red). …………………………………………………………………………………. 89 5.3 Calculated 1PA and 2PA transitions for methanol and ethanol at the EOM-CCSD/d-aug-cc- PVDZ level of theory. The 2PA cross sections are calculated using 6.2 eV pump in both cases. The calculated gas phase vertical ionization energy (VIE) for methanol and ethanol are also shown. ……………………………………………………………………………….. 101 5.4 The experimental polarization ratio (ρ) for water, methanol and ethanol plotted against the total excitation energy with (blue) 4.6 eV and (red) 6.2 eV pump. ……………………….. 103 5.5 The experimental polarization ratio (ρ) for propanol and butanol plotted against the total excitation energy with (blue) 4.6 eV and (red) 6.2 eV pump. ………………………... 105 5.6 NTOs corresponding to the strong 1PA and 2PA transitions in methanol. An isovalue of 0.015 is used for rendering orbital surfaces. The yellow arrow points to the position of the oxygen atom………………………………………………………………………………… …. 109 xii 5.7 NTOs corresponding to the strong 1PA and 2PA transitions for ethanol. An isovalue of 0.015 is used for rendering orbital surfaces. The yellow arrow points to the position of the oxygen atom. …………………………………………………………………………………… 110 5.8 Simulation of 4.6 eV pump 2PA spectrum using EOM-CCSD transition energies and strengths; the transition widths are 2.0 eV FWHM; (b) result obtained by applying 0.9 eV blue shift to the lowest 1A" ←1A' transition; the transition widths are 1.1 eV FWHM (c) simulation of the 6.2 eV pump 2PA spectrum using the EOM-CCSD results; the transition widths are 1.4 eV FWHM; (d) simulated spectrum obtained by blue-shifting the first transition by 0.9 eV; 4A"←1A' and 12A'←1A' transitions are blue-shifted by 1.2 eV and all other transitions by 0.33 eV; peaks modeled with Gaussian FWHM of 1.0 and 1.75 eV. Green dotted lines correspond to an identical simulation except 2A'←1A' and 12A"←1A' transitions are now blue shifted by 1.2 eV. ……………………………………………………………………….. 114 5.9 Simulation of the1PA spectrum of liquid MeOH (a) by using the same parameters used to simulate Fig 6(d); (b) now with selected transitions (2A'←1A' and 12A"←1A') also given a different blue-shift compared to the remaining transitions. …………………………….. 117 5.10 Simulation of polarization ratio (ρ) spectrum of methanol for (a) 4.6 eV and (b) 6.2 eV pump. The excitation energies have been shifted and use two different Gaussian widths, as described in the text. ……………………………………………………………………... 118 5.11 Simulation of parallel 2PA spectra of ethanol; for (a) 4.6 and (c) 6.2 eV pump. Simulation of ethanol polarization ratio (ρ); for (b) 4.6 and (d) 6.2 eV pump. ………………………. 119 6.1 Gas phase 1PA spectra of (a) cyclohexane and (b) n-hexane reproduced from Ref 1 . The gas phase ionization energies are indicated by the green arrow pointing towards the X-axis…. 137 6.2 Calculated 1PA and 2PA transitions for cyclohexane, hexane-all-trans (TTT) and hexane- trans-gauche-trans (TGT) at EOM-CCSD/d-aug-cc-PVDZ level of theory. 2PA cross sections are calculated using 4.6 eV pump energy. Character of strong 1PA and 2PA transitions are written right beside the peaks. …………………………………………………………. 141 6.3 Liquid 1PA spectra of (a) cyclohexane and (b) n-hexane. Liquid 2PA spectra of (c) cyclohexane and (d) n-hexane reported in this work. The 2PA cross sections are measured with both parallel and perpendicular polarizations of the pump and probe beams. The pump photon energy is 4.6 eV for our experiment. ………………………………………………….. 142 6.4 2PA polarization ratio plot of (a) cyclohexane and (b) n-hexane reported in this work. The pump photon energy used in our experiment is 4.6 eV. ………………………………. 143 6.5 NTO plots corresponding to the strong transitions of cyclohexane in 1PA and 2PA. …… 145 6.6 Simulation of the 2PA spectra of cyclohexane for (a) parallel and (b) perpendicular polarization. The simulated polarization ratio is shown in panel (c). The black dots and the red xiii line stand for the raw data and the simulated plot respectively. The pump photon energy used in our experiment is 4.6 eV. ………………………………………………………….. 147 6.7 Simulation of the liquid 1PA spectrum of cyclohexane. The raw data (black dots) is digitized from Ref 23 and the red line stands for the simulated plot. …………………………….. 147 6.8 Simulation of the liquid 1PA spectrum of n-hexane using the theoretically calculated results for the TTT and TGT conformers. …………………………………………………….. 149 6.9 NTO plots corresponding to the strong transitions of hexane_TTT in 1PA and 2PA. ….. 151 6.10 NTO plots corresponding to the strong transitions of hexane_TGT in 1PA and 2PA…. 151 6.11 Simulation of the parallel 2PA spectra of n-hexane using theoretical calculated results for (a) TTT and (b) TGT conformers. The simulation of the polarization ratio is obtained using theoretical calculated results for (c) TTT and (d) TGT conformers. The black dots and the red line stand for the raw data and the simulated plot respectively. The parameters of these simulations are mentioned in the text. Revised simulation (dashed green line) is performed by giving ~0.18 eV blueshift to the 2Ag←1Ag transition in the TTT conformer and 2A←1A transition in the TGT conformer. ………………………………………………………. 153 7.1 Demonstrating the importance of accounting for increase in momenta, rather than apparent spatial randomness, in describing entropy increase. (a) A snapshot of a gas, showing apparent random spatial distribution. Another snapshot after addition of heat (increase in entropy according to Clausius's definition), qualitatively has the same random spatial distribution and does not convey the idea that entropy has increased. (b) Considering the momenta of particles (indicated by arrows) resolves this conflict. Addition of heat allows the particles to explore a larger range of momenta. In this instance, it is the increase in momentum configurations that is related to entropy change. …………………………………………………………. 167 7.2 Demonstrating that accounting for spatial configurations alone creates a conflict between the Clausius' definition and the statistical definition of entropy. When a gas is adiabatically compressed, it does not exchange heat with the surrounding and therefore, according to Clausius, its entropy must not change. However, the gas manifestly occupies less volume after compression and therefore it has fewer spatial configurations, implying that its entropy must have reduced. This conflict is resolved after one accounts for the increase in the momentum configurations of the gas particles after compression which exactly balances the decrease in the available spatial configurations…………………………………………………... 168 7.3 (left panel) A single particle moving in one-dimension confined to a string of length a. (right panel) The phase space corresponding to this particle with three representative points….. 172 7.4 (left panel) Two non-interacting particles riding on two separate strings, both are constrained to move in one-dimension. (middle and right panel) The phase space corresponding to the red and the blue particle respectively, each phase space is two dimensional. ……………… 173 xiv 7.5 (left panel) N non-interacting particles are confined to move along strings in one dimension. (right panel) The phase space corresponding to each of these N particles. The total phase space area is the multiplication of the individual phase space areas. …………………………. 176 7.6 Adiabatic reversible compression of an ideal gas in a cubic container. (a) The compression is shown in real space where the length of the cube is decreased from a to a/2. (b) The effect of compression on momentum space. Due to increase in temperature from T1 to T2, the volume of the momentum space increases. (c) The phase space volume, which is the multiplication of the real space and momentum space volumes, does not change due to the balance between the spatial compression and momentum expansion. Hence there is no entropy change consistent with the Clausius' definition. …………………………………………………………… 180 7.7 Isothermal reversible expansion of an ideal gas kept in a cubic container. (a) The expansion is shown in real space where the length of the cube increases from a to 2a. (b) There is no e_ect of expansion in momentum space since temperature doesn't change during isothermal process. (c) The _nal volume of phase space is greater than that of the initial volume, hence entropy increases during isothermal expansion process. ………………………………………… 182 xv List of Tables 1.1 Combination of photons (NIR + up-conversion pulse) and the sum frequency wavelength. …9 1.2 Input power of 800 nm, threshold power for the generation of white light and power of the NIR portion of the white light are tabulated for three different materials. ……………………. 10 1.3 Power of input 800 nm, white light (~400-1500 nm), NIR portion of white light (~850-1500 nm) are tabulated. ………………………………………………………………………… 12 2.1 Fitting parameters of the VSFG spectra of P3HT/C60/CaF2 and P3HT/CaF2 samples for both PPP and SSP polarizations. A 60 nm P3HT layer is spin-coated on top of a 50 nm vapor- deposited C60 film. The concentration of the spin-coating solution of P3HT is 11 mg/mL in THF. The PPP/SSP ratio is written in the parenthesis. ……………………………………. 34 2.2 Fitting parameters of the VSFG spectra of P3HT/C60/CaF2 before and after annealing (155 0 C) for both PPP and SSP polarizations. A 25 nm P3HT layer is spin-coated on top of a 50 nm vapor-deposited C60 film. The concentration of the spin-coating solution of P3HT is 5.5 mg/mL in THF. The PPP/SSP ratio is written in the parenthesis. …………………………………. 35 2.3 Optical parameters used in SFG orientation analysis ……………………………………… 36 2.4 Upon annealing, the change in angle of different conformers of P3HT (with different vibrational frequencies) are different. For this bilayer sample, the spin-coating solution of P3HT is 5.5 mg/mL. ………………………………………………………………………... 38 3.1 Fitting parameters of the VSFG spectra for SPS polarization at five different temperatures…51 3.2 Fitting parameters of the VSFG spectra for PPP polarization at five different temperatures…52 3.3 Torsional barrier of methyl groups in different organic compounds. ……………………….. 54 4.1 Fitting parameters obtained from the global fitting procedure as described in the text. VSFG spectra are recorded at PPP and SSP polarization combinations for five different thicknesses. 𝜑 2 is reported in degree. …………………………………………………………………….. 68 4.2 𝜇 ′ and 𝛼 ′ values are calculated for different conformers of CBP. The molecule with dihedral angle of ~142 0 has a D2 symmetry, all other conformers are of C1 symmetry. …………….. 75 5.1 Calculated electronic transitions of methanol (EOM-CCSD/d-aug-cc-PVDZ) with 6.2 eV pump energy. ………………………………………………………………………………. 97 xvi 5.2 Calculated electronic transitions of ethanol (EOM-CCSD/d-aug-cc-PVDZ) with 6.2 eV pump energy. …………………………………………………………………………………….. 98 5.3 Calculated electronic transitions of methanol (EOM-CCSD/d-aug-cc-PVDZ) with 4.6 eV pump energy. ……………………………………………………………………………… 99 5.4 Calculated electronic transitions of ethanol (EOM-CCSD/d-aug-cc-PVDZ) with 4.6 eV pump energy. …………………………………………………………………………………….. 100 5.5 Absolute 2PA cross sections (in GM) for parallel polarization at 7.14 eV and 9.4 eV for methanol and ethanol. ………………………………………………………………………102 5.6 Linear polarization ratio at 7.92 eV (266nm+380nm) for methanol and ethanol. ………….104 5.7 Key parameters from NTO calculations for methanol and ethanol. Excitations to the states labeled in red and black are bright in 2PA and 1PA respectively. …………………………. 106 5.8 1PA and 2PA absorption thresholds of liquid water, methanol and ethanol (in eV). ……... 111 6.1 Calculated electronic transitions of cyclohexane (EOM-CCSD/d-aug-cc-PVDZ) with 4.6 eV pump energy. ……………………………………………………………………………….138 6.2 Calculated electronic transitions of hexane-TTT (EOM-CCSD/d-aug-cc-PVDZ) with 4.6 eV pump energy. ……………………………………………………………………………….139 6.3 Calculated electronic transitions of hexane-TGT (EOM-CCSD/d-aug-cc-PVDZ) with 4.6 eV pump energy. ……………………………………………………………………………….140 1 Chapter 1 Electronic processes in organic photovoltaic materials I. Introduction Efficiency always comes with a price and this fact is very well known in solar cell community. Inorganic solar cells based on crystalline silicon assembly are abundantly used in the semiconductor industry because of their enhanced device efficiency (~27%). 1 Organic solar cells, on the other hand, have a maximum power conversion efficiency of ~11%, 2 but they are way cheaper as compared to inorganic solar cells. 3 Such trade-off between cost and efficiency has always motivated researchers to delve into the intricacies of power conversion process, with the aim to make it more efficient while cutting down the cost of production. Such continuous effort has gradually paid off over a period of last 60 years, as the researchers have managed to increase the power conversion efficiency of silicon based inorganic solar cells from ~6% 4 to ~25%. 1 Similar improvement has become possible for organic semiconductor based solar cells as well. 5–9 Till the first reported power conversion efficiency of ~1%, 10 organic semiconductor based solar cells now have an efficiency of ~11%. 2 There are still a lot that can be done to boost up this efficiency, which gives us the motivation to study the mechanistic details of the power conversion process in organic photovoltaic materials. Before going any further, let us first review how do we get power (electric current) out of a solar cell, organic solar cell in particular. A typical organic solar cell is composed of an electron donor and an electron acceptor material that are sandwiched between two electrodes, a cathode 2 and an anode. When an organic photovoltaic material absorbs sunlight, electrons are promoted to higher excited states leaving a hole in the ground electronic state. These electron-hole pairs are bound to each other by electrostatic forces and are commonly known as excitons. The typical binding energy of an exciton in organic semiconducting materials is ~0.5 eV, 11,12 which is higher than the thermal energy (𝑘 𝐵 𝑇 ) at room temperature, ~0.03 eV. As a result, excitons do not spontaneously dissociate at room temperature giving rise to “free” electrons and “free” holes, which are needed for the generation of electricity. Rather, after generation, the exciton diffuses to the donor-acceptor interface where it dissociates leaving an electron in the acceptor and a hole in the donor. 13–15 The driving force for exciton dissociation at the interface comes from the difference in dielectric constant between the donor and the acceptor materials. 16 The free electrons and holes subsequently migrate to the respective electrodes to generate electricity. Schematic diagram of two different types of solar cells are shown in Figure 1.1. 17–19 Figure 1.1. Schematic diagram of (a) bilayer and (b) bulk heterojunction solar cells. The active layer is composed of electron donor and electron acceptor molecules. The donor-acceptor moiety of the bilayer solar cell is shown in the inset. 3 II. Band-Bending at the Interface Now, let us focus on the exciton dissociation process which is one of the key steps for the generation of electricity from an organic photovoltaic material. As discussed before, this process takes place at the donor-acceptor interface. The efficiency of this process depends on several factors like: 1. the relative energies of the valence and conduction bands of the donor and the acceptor molecules. 20 2. Orientation of the donor-acceptor molecules at the interface. 21 The information about the bandgap of a material can be obtained from linear absorption spectroscopy. Unfortunately, the bandgap reported by UV-VIS measurement is not the same as the bandgap at the surface/interface, because of the well-known band-bending phenomenon. 22–28 The concept of band-bending was first developed by Schottky and Mott 29–31 and can be understood by taking a simple example of metal/n-type semiconductor interface, as shown in Figure 1.2. 20 Due to energetic difference between the Fermi levels of a metal and a semiconductor, there is a flow of electron from one side to the other when these two materials are kept in contact. The direction of the flow depends on the magnitude of the work function (φm for metal and φs for semiconductor). If φm > φs, the electrons will flow from the semiconductor to the metal. 20 This flow will continue till the Fermi levels of the metal and the semiconductor are aligned. As a result of such electron flow to equilibrate the Fermi levels, the metal side of the interface accrues some negative charge, while the electron density in the semiconductor side gets depleted. This gives rise to an interfacial electric field which shifts the energy levels of the band edges. Depending on whether φm > φs or φm < φs, the band edges bend upward or downward towards the interface. 20 Band bending thus changes the interfacial band gap from that observed in the bulk, which directly influences processes like exciton dissociation and charge carrier recombination. 4 Figure 1.2. Representative band diagram for metal/n-type semiconductor before and after equilibrium. Evac, Ev, EC, EF,m, EF,s, stand for vacuum energy, energy of valence band, energy of conduction band, Fermi energy of metal and Fermi energy of the semiconductor respectively. The work functions of the metal and the semiconductor are represented by φm and φs respectively. This figure is adapted from Ref 20 III. Sum Frequency Generation Spectroscopy: VSFG vs ESFG To extract information about the interfacial properties of organic photovoltaic materials, we have adopted Sum frequency generation (SFG) spectroscopy in our lab. 21,32–35 It is a second order nonlinear optical technique which is electric dipole forbidden in a centrosymmetric environment and hence does not carry any contribution from the bulk phase of the isotropic material. 21,36 In SFG experiments, a second order polarization 𝑃 (2) is induced in the sample using 5 two input electric fields (𝐸 1 ,𝐸 2 ) of frequency 𝜔 1 and 𝜔 2 . The sum frequency field is then emitted in the phase matching direction and recorded by a liquid nitrogen cooled CCD detector. 𝑃 (2) = 𝜒 (2) : 𝐸 1 𝐸 2 𝑎𝑛𝑑 𝜔 𝑆𝐹𝐺 = 𝜔 1 + 𝜔 2 𝜒 (2) is the proportionality constant and is known as second order nonlinear susceptibility of the material. Details about 𝜒 (2) have been discussed in Chapter 4. SFG can be used to get information about the surface vibrational states or surface electronic states and depending on that it is named Vibrational SFG (VSFG) or Electronic SFG (ESFG) spectroscopy. In VSFG, a femtosecond IR pulse is first used to excite a vibrational coherence in the molecule. A picosecond visible pulse upconverts that coherence in a non-resonant fashion and the sum frequency light is detected in the phase matching direction. 37,38 The narrowband up- conversion pulse determines the frequency resolution in the VSFG experiment. ESFG spectroscopy is carried out exactly the same way except the first pulse, in this case, excites an electronic coherence instead of a vibrational coherence. The difference between VSFG and ESFG scheme is shown in Figure 1.3. The VSFG selection rule dictates that for a vibrational mode to be VSFG active, it has to be both IR and Raman active. 34,39,40 As a result, the second order nonlinear response of the molecule, known as molecular hyperpolarizability (β), can be written as a multiplication of IR transition dipole moment (μ) and Raman polarizability tensor elements (α); as shown in equation (1). 34,39 𝛽 𝑖𝑗𝑘 = − 1 2ħ ∑ 𝜇 𝑛 0 𝑘 𝜇 0𝑛 𝑖𝑗 𝜔 𝑛 − 𝜔 𝐼𝑅 − 𝑖 Г 𝑛 𝑛 (1) 6 Here, 𝑛 is an index used to designate vibrational normal modes of energy 𝜔 𝑛 . 𝜔 𝐼𝑅 is the frequency of the input IR light, Г 𝑛 is the linewidth of the 𝑛 𝑡 ℎ transition and i, j, k are molecular coordinates. Similar selection rule is applicable for the ESFG spectroscopy. The only difference is, for ESFG, 𝜇 𝑛 0 𝑘 term stands for the transition dipole of the one photon absorption process and 𝜔 𝐼𝑅 term is replaced by 𝜔 𝑉𝐼𝑆 1 in the denominator of equation (1). ESFG can give us information about the bandgap of a semiconductor material at the interface, which is different from the bulk bandgap as discussed earlier. Figure 1.3. Schematic energy diagram for (a) VSFG and (b) ESFG spectroscopy. 7 IV. Conventional way of doing ESFG experiments Several groups are working on ESFG spectroscopy using the scheme shown in Figure 1.3. 41– 43 In a typical broadband ESFG experiment, a white light supercontinuum is used to create an electronic coherence between the ground and excited states, which then gets upconverted by a narrowband picosecond visible pulse and the ESFG signal is collected in the phase matching direction. Yamaguchi et al. 41 reported the ESFG spectra of a dye molecule, oxazine 750, adsorbed at the air/water interface. They observed that ESFG spectral feature is strongly dependent on the frequency of the upconverting pulse (𝜔 1 ), as shown in Figure 1.4. Two prominent bands are observed in the ESFG spectrum at 341 nm and 366 nm, using an 800 nm up-conversion pulse. The 366 nm band appears to shift to the blue side of the spectra and eventually disappears, as the wavelength of the up-conversion pulse is changed from 800 nm to 740 nm to 692 nm. 41 Figure 1.4. This figure is adapted from a publication by Yamaguchi et al. 41 ESFG spectra of Oxazine 750 at the air/water interface measured with three different wavelengths of the up- converting pulse; 800 nm, 740 nm and 692 nm. 8 To interpret such excitation wavelength dependence of the ESFG spectra, a complicated global fitting procedure was adapted to separate contributions from one and two photon resonances, which often make the spectral analysis difficult. Moreover, since the first excitation by the white light continuum is resonant with an electronic state, any excited state dynamics that takes place within the cross-correlation timescale of the two input pulses (which can vary from 100 fs to ~ps) can potentially contribute to the ESFG spectra and thereby makes the spectral analysis even harder. Presence of conical intersection in the first excited electronic state (for example, in essential amino acid like tryptophan) can further complicate the excited state dynamics, as process like internal conversion can take place in even <50 fs timescale. 44 To simplify the analysis of the ESFG spectra, we decided to follow a slightly different scheme 45 of optical excitation as shown in Figure 1.5. V. Our proposed scheme of ESFG experiment For organic photovoltaic materials, the lowest electronic states usually lie in the visible region, between 400-800 nm. 46 In our proposed ESFG scheme, the electronic coherence between the ground state and the excited state is achieved following a two-photon excitation of the material, where one of the photons come from the supercontinuum and the other photon comes from the up- conversion pulse. The NIR portion of the supercontinuum (900-1500 nm) is used for this purpose. The frequency of the second pulse is chosen based on which part of the visible spectrum we want to excite. The frequency combination of the two pulses is shown in Table 1.1. 9 Table 1.1. Combination of photons (NIR + up-conversion pulse) and the sum frequency wavelength NIR portion of white light (nm) Up-conversion pulse (nm) SFG wavelength (nm) 900-1500 800 420-522 900-1500 1000 470-600 900-1500 1200 515-667 900-1500 1400 548-724 900-1500 1600 576-775 At this point, it is very important to investigate which material is most suitable for the generation of the NIR portion of the white light. The best material would be the one which produces maximum bandwidth of white light in the NIR region along with considerable intensity. The scientific details of the generation of white light continuum will not be discussed here, and readers are encouraged to go through the pioneering book by Robert Alfano. 47 Figure 1.5. Our proposed scheme to perform ESFG spectroscopy. The left side and the right side show the state diagram of excitation and the corresponding Feynman diagram respectively; ‘v’ and ‘e’ stand for the virtual state and the excited electronic state. 10 VI. Experimental Section: White Light Generation We have used three different materials (water, sapphire and vanadate crystal) to study the generation of white light in the NIR region. The 800 nm pulses with a repetition rate of 1 KHz are focused to the sample and the generated white light is collimated by a 90 0 off-axis parabolic mirror. After that, the white light is passed through an 835 nm LP (Long Pass) filter with 70% transmittance and two consecutive 800 HR’s (high reflectors). It helps to cut down the visible portion of the white light spectrum (wavelengths below 835 nm) and the residual 800 nm driving field. The light is then focused to a monochromator by a 10 cm focusing lens and recorded by InGaAs photodiode array. Our results showed that the thresholds of the white light generation are different in different materials, as tabulated in Table 1.2. The optimal pump power is chosen such that it generates the most stable white light continuum. The spectrum of the NIR portion of the white light is shown in Figure 1.6. Before recoding the white light spectrum from water and sapphire, an OD=1 filter is used; otherwise the high intensity of the generated white light continuum saturates the detector. Table 1.2. Input power of 800 nm, threshold power for the generation of white light and power of the NIR portion of the white light are tabulated for three different materials. These measurements are done with 1 KHz laser system. Material Threshold (mW) Pump power (mW) NIR continuum power (mW) Water 0.5 2.95 0.12 Sapphire 1.3 2.75 0.11 Vanadate 0.9 2.17 0.06 11 Figure 1.6. NIR spectrum of the white light continuum generated by focusing the 800 nm pulse in water, sapphire and vanadate crystals. In case of water and sapphire, OD=1 filter is used before the recording the white light spectrum. These measurements are done with 1 KHz laser system. Our results indicate that although the white light generated from the vanadate crystal has the maximum span in the NIR region, up to 1650 nm, its intensity is ~10% as compared to that of water and sapphire. White light generated from sapphire and water are of similar intensity and they span up to 1550 nm and 1400 nm respectively. Comparing both spectral width in the NIR region and intensity, sapphire seems to be the best candidate for the white light generation. Despite that, we decided to use water to generate white light for our ESFG experiment, and the reasons are as follows: 1. We can control the temperature of water which has significant effect on the white light generation process, 2. For crystals, the input power of 800 nm that we can use, is limited by the damage threshold of the crystal. For water, this problem can be easily avoided by continuously circulating it through a flow cell. 3. With increasing input power of 800 nm, the spectral range in 12 the NIR can extend up to 1500-1550 nm in case of water. Using a 5 KHz laser system, we have performed an input power dependence study of white light generation from water and the results are tabulated in Table 1.3. Table 1.3. Power of input 800 nm, white light (~400-1500 nm), NIR portion of white light (~850- 1500 nm) are tabulated. A 5 KHz repetition rate amplifier system is used for these measurements. Power of 800 nm (mW) White light power (~400- 1500 nm) (mW) Power of NIR portion of white light (~900-1500 nm) (mW) 13.4 0.6 0.13 32 2.4 0.8 59.4 5.6 2 80.3 8 2.9 120 14 5 152 19.9 7.7 To generate white light continuum from water, 800 nm pulse trains (pulse width ~60 fs) from a 5KHz amplifier system were focused onto a 1 cm water flow cell using a 10 cm focusing lens. An external chillier was used to flow water into the cuvette and the temperature of the chiller water was kept at ~13 0 C, as reported by Pandey et al. 42 The white light was first collimated using a lens of focal length 3 cm and passed through two 800 HR’s to get rid of the residual 800 nm. The power of the white light was measured using a thermal power meter. To measure the power of the NIR region of the white light, an 850 nm LP filter was placed just before the power meter. We conclude from our power dependence study that the input power of 800 nm can be set in between 13 120-150 mW (for a 5KHz repetition rate system), which will produce NIR white light pulse energy ~1-1.5 μJ. Such pulse energy should be enough for conducting an ESFG experiment using our proposed excitation scheme. There are some key points I want to mention for our current and future lab members who want to pursue this project further. 1. I faced a lot of trouble to properly focus the white light to the sample stage, even after using the 20 cm spherical concave mirror. I noticed that it is extremely important to align the white light to the principal axis of the concave mirror, in order to achieve the perfect focusing condition. It is better to align the concave mirror using the 800 nm beam before switching it to the white light beam. 2. Visual inspection of the focal spot size of the white light is difficult as it is extremely bright and over-saturates our eyes. It is better to decrease the intensity of the white light before looking at the focal spot and it can be done by passing it through a neutral density filter. 3. White light collimation has always been a challenge for this experiment. It is even harder working with the 900-1500 nm region of the white light spectrum because of the visibility issue. Similarly, it is also difficult to collimate the ESFG signal right after the sample stage. A collimating lens of smaller focal length (~ 3cm) is recommended for this purpose, as it is to be placed very close to the sample stage allowing less room for the generated ESFG signal to diverge. Also, to minimize the issue of divergence, it is better to place the CCD detector as close to the sample stage as possible. VI. Outline of the thesis The main goal during my graduate studies was to investigate the interfacial band gap which is not the same as the bulk band gap and it directly influences the exciton dissociation process. To get this information, we proposed a slightly different ESFG excitation scheme which 14 is guided by a different selection rule. According to our proposed experiment, for an electronic state to be ESFG active, it must be bright both in one photon and two photon absorption processes. 48 This allows us to speculate the interfacial band gap of a material from the bulk band gaps obtained using 1PA and 2PA spectroscopy. On the other hand, efficiency of the processes like exciton dissociation, charge carrier recombination and charge extraction, also depend on the orientation of donor-acceptor molecules in the interface of an organic photovoltaic material and the orientation of these molecules in the electrode surfaces as well. 18,49–52 A small change in molecular orientation in the donor-acceptor interface can have a large effect on the efficiency of charge transfer process, as shown by nonadiabatic quantum MD simulations. 52 As a result, information about molecular orientation at the interface is very crucial in building the structure- function relationship of organic photovoltaic materials. Surface selective vibrational sum frequency generation (VSFG) spectroscopy is popularly used to obtain such orientational information. 21,53–55 Using this technique, in Chapter 2, we have determined how an electron donor polymer material, Poly-3-hexylthiophene (P3HT), is oriented at the interface with an electron acceptor, fullerene (C60). 21 In Chapter 4, using the same technique, we have calculated the orientation of 4,4′-Bis(N-Carbazolyl)-1,10-Biphenyl (CBP), popularly used in OLED materials, at the surface of CaF2 which serves as a coating for the basic electrodes. In Chapter 3, we have demonstrated a unique application of VSFG spectral line-shape analysis to visualize the mechanistic details of a hindered methyl rotation on a methyl-terminated silicon (111) surface. 56 Chapter 5 and Chapter 6 are dedicated to the two photon absorption spectroscopy measurements of common solvents; alcohols 57 and alkanes (hexane and cyclohexane). 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Introduction The efficiency of an organic photovoltaic (OPV) device depends not only on the favorable alignment of the energy levels between the donor and the acceptor molecules, but also on the structure, morphology and the efficient packing of molecules in the active layer of a bulk heterojunction solar cell which control charge carrier generation and mobility. 1-6 Optimizing a solar cell material requires the optimization of every step that leads to the generation of electricity, including exciton dissociation, charge recombination and charge carrier mobility. The two competing processes, the exciton dissociation and the charge recombination, are guided by the orientation of the donor and the acceptor molecules at the interface and similarly the charge extraction processes and the contact resistance also depend on how molecules are oriented in contact with the electrode surfaces. 7-11 Nonadiabatic quantum molecular dynamics simulation by Mou et.al. 11 showed that a small change in molecular orientation at the interface can have a large effect on the charge transfer and the charge recombination rates in OPV materials. Several groups 12-15 have separately studied electron donor and electron acceptor molecules at various dielectric surfaces by vibrational sum frequency generation (VSFG) spectroscopy. To the best of our knowledge, the orientation of electron donor molecules at the interface with an electron acceptor has not been reported. Knowledge of the orientation of the donor and acceptor moieties at the interface can help gain insight into the mechanisms of the exciton dissociation and charge recombination dynamics, and potentially make better devices by molecular design and material 20 processing. 4 Here, we focus on measuring the orientation of the backbone of poly-3- hexylthiophene (P3HT) in contact with the fullerene surface. High carrier mobility and high extinction coefficient in the visible range make P3HT an excellent donor material to be used in the active layer of organic photovoltaic devices and in organic field-effect transistors. 16-18 The thiophene rings of adjacent polymer chains are stacked together through π- π interactions. 19-21 On the other side, fullerene and fullerene derivatives are well-known electron acceptor materials used in solar cells. The higher dielectric constant of fullerene (~ 4) compared to that of popular conjugated polymers (~ 3) makes it an ideal choice to facilitate exciton dissociation. 22 Also, high molecular symmetry of fullerene, which leads to triply degenerate LUMO levels 23 and strong polarizability, also result in efficient charge mobility in this conjugated system. Thus, a P3HT-C60 bilayer sample is a good model of the materials used in OPV devices. To extract information about the molecular organization of the P3HT/C60 buried interface in a bilayer sample, we have used a surface selective non-linear spectroscopic technique, vibrational sum frequency generation (VSFG). 24-27 VSFG is a second order nonlinear optical process which is electric dipole forbidden in a centrosymmetric environment, 28 and hence does not have any contribution from the bulk phase of an isotropic material. In VSFG, a broadband tunable IR pulse is used to create vibrational coherence 29 of all the frequencies at the interface; a narrowband visible pulse is used to upconvert the coherences; and the sum frequency light is detected in the phase matching direction. 29-31 The VSFG spectrum is recorded for different polarizations of the input and the output beams and the change in VSFG signal intensity with polarization is exploited to obtain the molecular orientation of different functional groups at the interface. 26 For our experiment, we have used two different polarization combinations : PPP and 21 SSP; where each combination denotes the polarization of the input IR light, input visible light and the output sum frequency light, from right to left. The broadband mid-infrared pulse around 1450 cm -1 is generated by routing the output of the TOPAS optical parametric amplifier (OPA) to the different frequency generator (NDFG). The bandwidth of the output IR frequency varies in between 300-350 cm -1 . The OPA was pumped with 800 nm pulse with the pulse energy of 1.2 mJ, generated from a 5 KHz Ti-Sapphire amplifier system. Another portion of the amplifier output is passed through a 4f stretcher to generate picosecond 800nm pulse which gives the frequency resolution in our experiment (~8-10 cm -1 ). At the sample stage, the power of the IR light is 3 μJ/pulse and the visible light is 6 μJ/pulse, with a 5 KHz repetition rate. The spot-size of the IR and the visible beam are 200 μm and 220 μm at the sample stage. The angle of incidence of the visible and the IR beam is 67 0 and 62 0 with respect to the surface normal. The SFG spectra is recorded using a liquid nitrogen cooled CCD detector (2048 X 512 pixels). The details of the entire experimental set-up can be found in one of the previous publications by our group. 32 II. Results and Discussion The VSFG selection rule dictates that for a vibrational mode to be SFG active, it must be both IR and Raman active. IR and Raman spectra (Fig. 2.1) were collected for a P3HT/C60 bilayer sample prepared by vapor-depositing 50 nm of C60 on CaF2 substrate and a P3HT film spin-coated from tetrahydrofuran solvent on top of it. We have made two different thicknesses of P3HT (60 and 25 nm) by changing the concentration of P3HT in the spin-coating solution, 11 mg/mL and 5.5 mg/mL. In the IR spectrum, the C-C inter-ring stretch, C=C symmetric and asymmetric stretch of the thiophene rings appear at 1380 cm -1 , 1410-1490 cm -1 and 1510 cm -1 respectively. In P3HT, there exists π- π conjugation between the thiophene monomers that extends up to 10-15 monomers or even more. 33-35 Different P3HT chains have different extents of such π conjugation, which leads 22 to slightly different vibrational frequencies and hence inhomogeneous broadening in the IR and Raman spectra. The sharp feature at 1425 cm -1 is due to the tangential mode of C60 where the opposite pentagons contract and expand out of phase. 15,36 This feature is absent in the Raman spectrum. As previously reported by several groups, 37-39 the Raman spectrum of C60 shows a peak at 1469 cm -1 assigned to the Ag pentagonal pinch mode. 14 This peak is not distinguishable in the Raman spectrum we recorded, as it is probably buried under the intense Raman peak originated due to C=C symmetric stretch of the thiophene ring. Figure 2.1. FTIR (red) and Raman (blue) spectra of P3HT/C60 bilayer sample. The vibrational modes are shown using colored sticks: C-C inter ring stretch (faded blue line, 1380 cm -1 ), C=C symmetric stretch modes (gray line, 1410-1490 cm -1 ) and C=C asymmetric stretch (yellow line, 1510 cm -1 ) of the thiophen ring. Vibrational mode of C60 appears at 1425 cm -1 , shown by the green line. 23 After identifying the IR and Raman active modes of P3HT and C60, polarization dependent VSFG experiments were performed to figure out the orientation of the P3HT backbone in contact with the fullerene surface. A previous publication by our group 12 showed that at P3HT/SiO2 and P3HT/AlOx, the majority of the SFG signal comes from the buried interface. Our Frensel factor calculation has also shown that the contribution of the C=C symmetric stretch of P3HT at P3HT/C60 buried interface to the overall SFG intensity is ~100 times stronger than that of P3HT/air interface, as shown in Figure 2.2. As the P3HT/air interface contributes very little, in our current study we can safely assume that most of the collected SFG signal is coming from the P3HT/C 60 interface. The SFG spectra for P3HT/CaF2 and P3HT/C60/CaF2 samples are shown in Figure 2.3. In both cases, signals from PPP and SSP polarizations are shown. In the case of P3HT/CaF2 samples, the SFG spectrum contains all three IR allowed vibrational modes as shown in Figure 2.3(a). In the region between 1430 to 1500 cm -1 , several SFG peaks are observed and all of them can be assigned to C=C symmetric stretch of the thiophene ring. The peak that appears at 1510 cm -1 is due to the C=C asymmetric stretch. The SFG intensity for the PPP polarization shows higher intensity than that of SSP, in the C=C symmetric stretch region. 24 Figure 2.2. Calculated SSP and PPP intensity as a function of orientation angle for both P3HT/air and P3HT/C60 interface. 25 Figure 2.3. (a) SFG spectrum of P3HT spin-coated on CaF2 substrate. Both PPP (red) and SSP (blue) spectra are shown. All IR active modes of P3HT appear in SFG spectrum as well: C-C inter ring stretch (faded blue line, 1380 cm -1 ), C=C symmetric stretch (gray line, between 1430-1490 cm -1 ) and C=C asymmetric stretch modes (yellow line, 1510 cm -1 ) of the thiophene ring. (b) SFG spectrum of P3HT/C60 bilayer sample. In addition to the peaks of P3HT, the sharp feature at 1425 cm -1 (green line) appears due to the IR active mode of C60. (c) Calculated PPP (red), SSP (blue) SFG intensity and PPP/SSP intensity ratio (green) for P3HT/CaF2 interface as a function of the tilt angle between the plane of the thiophene ring and the surface normal (d) Calculated plot for P3HT/C60 interface. SFG spectra of 10 nm C60 vapor-deposited on CaF2 and SiO2 substrates have been reported by Massari and co-workers. 14 Both IR active F1u mode (1425 cm -1 ) and Raman active Ag mode (1469 cm -1 ) show SFG activity for C60/CaF2 sample. The relative intensity of these two peaks is different in our experiment as we have used a 50 nm film of C60. Changing the thickness changes the interference between the signals coming from the top and the bottom interface of the C60 film, as shown in Figure 2.4. 26 Figure 2.4. SSP spectra of C60 vapor-deposited on CaF2 for two different thicknesses; 10 nm and 50 nm. After identifying the peaks in the SFG spectrum of C60, we move on to assign the spectra of the P3HT/C 60 bilayer sample (spin-coating solution of P3HT: 11 mg/mL in THF) which shows pronounced differences compared to P3HT/CaF2. The SSP polarization shows higher intensity between 1430 to 1460 cm -1 ; above that frequency, intensity for PPP polarization is higher. Also, the SSP spectrum shows a sharp peak at 1425 cm -1 which is not present in P3HT/CaF2 spectrum. This feature is coming from the SFG activity of the F1u mode of fullerene. The SFG active peak of C60 at 1469 cm -1 is weak in intensity and probably buried under the SFG signal coming from the C=C symmetric stretch of P3HT. It is worth mentioning that in going from P3HT/CaF2 sample to P3HT/C60/CaF2 sample, the signal intensity for SSP polarization increases by a factor of 3-4, whereas in PPP there is only a slight increase in intensity (Fig. 2.3). Qualitatively, this points to the different orientation of P3HT in contact with C 60. Quantitative orientational analysis is presented below. 10000 8000 6000 4000 2000 0 SFG intensity 1550 1500 1450 1400 1350 Frequency / cm -1 2000 1500 1000 500 0 SFG intensity 50 nm 10 nm SSP of C 60 27 According to the model proposed by Anglin et. al., 13 the adjacent thiophene rings of P3HT are connected with an average dihedral angle ∠S-C-C-S = 165 0 . As a result, for C=C symmetric stretch, the lateral component of the transition dipole of the neighboring thiophene rings gets canceled, while the out-of-plane components add up to give the net transition dipole orthogonal to the average plane of the ring pair, as shown in Figure 2.10(d). Hence, defining tilt angle θ between the net transition dipole with respect to the surface normal will help us extract orientation information about the P3HT backbone. θ=0 0 corresponds to “face-on” orientation of the thiophene rings, whereas θ=90 0 corresponds to “edge-on” orientation. For C=C symmetric stretch, only three hyperpolarizability tensor elements contribute towards the VSFG spectra if C2v symmetry is assumed for the thiophene ring. 26 We also assume a Gaussian distribution of the tilt angle θ. 40,41 The calculated SFG intensity for PPP and SSP polarization along with the PPP/SSP intensity ratio are plotted as a function of the tilt angle for both P3HT/CaF2 and P3HT/C60 interface, as shown in Figure 2.3(c) and 2.3(d) respectively. The main uncertainty in this simulation comes from the unknown refractive index of the interfacial layer (nꞌ). 26,31 As a result, while calculating the SFG intensity from an interface, nꞌ is varied for IR, visible and sum frequency wavelengths; for P3HT/CaF2 interface: 1.35<nꞌ IR<2.12; 1.43<nꞌVIS<2.27; 1.43<nꞌSFG<2.37; and for P3HT/C60 interface: 1.97<nꞌ IR<2.12; 2.02<nꞌVIS<2.27; 2.09<nꞌSFG<2.37. We found out that PPP/SSP intensity ratio changes drastically with changing nꞌVIS and nꞌSFG; but it doesn’t depend on nꞌ IR. The confidence interval of ‘θ’ reported in this paper is estimated based on the assumed range of nꞌ. 28 Figure 2.5. Calculated PPP/SSP ratio for C=C symmetric stretch of the thiophene ring at P3HT/C60 and P3HT/CaF2 interface. Here we the vary the interfacial refractive index for the IR wavelength. For P3HT/C60, n' IR is varied in between 1.97 and 2.12; whereas for P3HT/CaF2 it is varied in between 1.35 and 2.12. 29 Figure 2.6. Calculated PPP/SSP ratio for C=C symmetric stretch of the thiophene ring at P3HT/CaF2 interface. Here we the vary the interfacial refractive index for both the visible and SFG wavelengths. n'VIS is varied in between 1.43 and 2.27; and n'SFG is varied in between 1.43 and 2.37. 30 Figure 2.7. Calculated PPP/SSP ratio for C=C symmetric stretch of the thiophene ring at P3HT/C60 interface. Here we the vary the interfacial refractive index for both the visible and SFG wavelengths. n'VIS is varied in between 2.02 and 2.27; and n'SFG is varied in between 2.09 and 2.37. Second order hyperpolarizability value (β) is an important parameter for theoretically calculating the SFG intensity. For our purpose, we have used the previously reported β values by Anglin et al.; 13 βaac/ βccc = 70, βbbc/ βccc = -1. Sensitivity analysis has been performed by drastically 31 changing the βaac value. We observed that although the change in βaac changes the calculated SFG intensity in SSP and PPP polarization combinations, the PPP/SSP intensity ratio stays the same. So, any error in calculating the β value has least affect in the reported orientational angle of the thiophene backbone. Figure 2.8. Calculated SSP and PPP intensity of C=C symmetric stretch mode of the thiophene ring of P3HT as a function of the tilt angle of the thiophene backbone with respect to the laboratory Z-axis. Calculations are done using three different βaac values.; from which βaac = 70 is used for our reported orientational angle. VSFG intensity is proportional to the square of the second order non-linear susceptibility (χ (2) (ω)) of the material. χ (2) (ω) has both resonant and non-resonant part. The resonant part can be 32 approximated with Lorentzians and non-resonant part has some amplitude and a phase phase- factor associated with it. Incorporating all these, the overall fitting equation looks like the following: |χ (2) (ω)| 𝟐 = |𝐴 𝑁𝑅 𝑒 −𝑖 ∅ + ∑ 𝑏 𝑛 𝛤 𝑛 ω − ω 𝑛 + 𝑖 𝛤 𝑛 𝑛 | 𝟐 where, 𝐴 𝑁𝑅 is non-resonant-amplitude, ∅ is the phase-factor. 𝑏 𝑛 is the resonant amplitude at the vibrational frequency ω 𝑛 , with a linewidth 𝛤 𝑛 . We have used eight Lorentzians to fit the SFG spectra for P3HT/CaF2 and P3HT/C60 samples. Two Lorentzians are used to fit the C-C inter-ring stretch region (1350-1400 cm -1 ), five Lorentzians are used to fit the C=C symmetric stretch region (1400-1500 cm -1 ) and one Lorentzian is used to fit the C=C asymmetric stretch which appears above 1500 cm -1 . As pointed out by other groups, 34,42 the C=C symmetric stretching mode of P3HT is very sensitive to the degree of molecular ordering which gives rise to the inhomogeneous broadening in the SFG spectra. Using five Lorentzians to fit the C=C symmetric stretch region will help us figure out how the orientation of the P3HT backbone changes depending on the extent of their molecular ordering. The fitting parameters of the VSFG spectra and the optical parameters used in SFG orientational analysis are tabulated in Table 2.1-2.3 respectively. 33 Table 2.1. Fitting parameters of the VSFG spectra of P3HT/C 60/CaF2 and P3HT/CaF2 samples for both PPP and SSP polarizations. A 60 nm P3HT layer is spin-coated on top of a 50 nm vapor- deposited C60 film. The concentration of the spin-coating solution of P3HT is 11 mg/mL in THF. The PPP/SSP ratio is written in the parenthesis. P3HT/CaF2 P3HT/C60/CaF2 PPP SSP PPP SSP b1 5.0 -11.0 -30.5 -8.8 b2 38.5 18.5 23.0 10.5 b3 1.5 -0.9 3.7 35.2 b4 18.0 ± 1.0 28.5±0.5(0.63) 13.4±1.1 37.7±1.1(0.36) b5 18.5 ± 1.1 17.3±0.6(1.07) 26.3±0.9 55.0±0.9(0.48) b6 37.4 ± 0.7 18.6±0.5(2.01) 34.3±0.8 39.4±1.0(0.87) b7 29.1 ± 0.8 9.0 ± 0.7(3.23) 26.1±0.7 21.8±1.4(1.20) b8 -49.9 -29.5 -57.4 -21.8 Γ1, cm -1 13.6 13.6 13.7 13.7 Γ2, cm -1 6.1 6.1 4.8 7.0 Γ3, cm -1 8.2 8.2 3.8 3.8 Γ4, cm -1 16.0 16.0 6.2 6.2 Γ5, cm -1 5.7 5.7 7.5 7.5 Γ6, cm -1 6.9 6.9 7.5 7.5 Γ7, cm -1 10.8 10.8 12.4 12.4 Γ8, cm -1 7.2 8.7 7.0 7.0 ω1, cm -1 1359.7 1359.7 1344.9 1345.1 ω2, cm -1 1379.8 1379.8 1376.7 1380 ω3, cm -1 1423.0 1423.0 1425.8 1425.8 ω4, cm -1 1444.0 1444.0 1444.0 1444.0 ω5, cm -1 1455.1 1455.1 1454.2 1454.2 ω6, cm -1 1465.9 1465.9 1464.6 1464.6 ω7, cm -1 1480.7 1480.7 1477.4 1477.4 ω8, cm -1 1510.3 1508.4 1511.0 1509.6 34 ANR -6.7 -4.3 8.5 14.6 φ, rad 31.8 31.2 -20.1 -23.1 Table 2.2. Fitting parameters of the VSFG spectra of P3HT/C60/CaF2 before and after annealing (155 0 C) for both PPP and SSP polarizations. A 25 nm P3HT layer is spin-coated on top of a 50 nm vapor-deposited C60 film. The concentration of the spin-coating solution of P3HT is 5.5 mg/mL in THF. The PPP/SSP ratio is written in the parenthesis. Before Annealing After Annealing PPP SSP PPP SSP b1 -6.3 12.2 12.3 14.0 b2 5.2 4.2 4.4 8.2 b3 34.2 54.9 34.5 78.2 b4 8.0 ± 0.9 16.3±0.8(0.49) 12.8 ± 1.0 26.7±1.7(0.48) b5 27.9 ± 0.6 51.5±0.5(0.54) 31.7 ± 0.8 23.2±1.9(1.37) b6 14.0 ± 0.6 13.9±0.7(1.0) 16.1 ± 0.9 -6.2±1.4(2.60) b7 5.2 ± 0.5 6.1±0.8 (0.85) 4.9 ± 1.0 -2.8±1.1(1.75) b8 -23.0 -15.7 -25.0 -10.2 Γ1, cm -1 8.0 11.0 5.2 5.2 Γ2, cm -1 6.0 3.3 4.2 4.2 Γ3, cm -1 5.5 4.7 4.3 4.3 Γ4, cm -1 4.1 4.1 6.4 6.4 Γ5, cm -1 10.2 10.2 10.3 10.3 Γ6, cm -1 5.5 5.5 5.6 5.6 Γ7, cm -1 5.5 5.5 8.0 8.0 Γ8, cm -1 7.7 7.7 11.6 9.3 ω1, cm -1 1379.1 1378.0 1380.6 1380 ω2, cm -1 1412.8 1412.8 1416.2 1413.4 ω3, cm -1 1428.1 1428.1 1427.9 1428 35 ω4, cm -1 1444.4 1444.4 1441.0 1442.7 ω5, cm -1 1451.9 1451.9 1450.6 1449.4 ω6, cm -1 1463.1 1463.1 1462 1462.2 ω7, cm -1 1476.4 1476.3 1480 1480 ω8, cm -1 1511.0 1508.3 1508.8 1508 ANR 10.5 8.1 -1.9 13.4 φ, rad 0.6 1.8 24.1 1.8 Table 2.3. Optical parameters used in SFG orientation analysis Wavelength Refractive index P3HT 12,43,44 CaF2 45 C60 14,46 air IR (6850 nm) 2.12 1.37 1.97 1.0 VIS (805 nm) 2.27 1.43 2.02 1.0 SFG (720 nm) 2.37 1.43 2.09 1.0 Comparing the experimental intensity ratio (PPP/SSP) to that of the theoretically calculated one, we find that there is a broad distribution of the orientation of the thiophene rings in contact with the CaF2 surface. While most of the thiophene rings (in between 1450-1500 cm -1 ) are stacked vertically (edge-on) on CaF2 substrate, thiophene rings with C=C symmetric stretch frequency below 1440 cm -1 are significantly tilted towards the CaF2 substrate making an average angle θ=42 0 (33 0 - 52 0 ) between its transition dipole with respect to the surface normal. In contrast, at the P3HT/C60 interface, all the thiophene rings, considering both long and short conjugation length polymer (the frequency of C=C symmetric stretch of thiophene ring red-shifts with increasing conjugation in the polymer), are tilted towards the C60 moiety so that the net transition dipole of C=C symmetric stretch makes an average angle θ=49 0 from the surface normal with an 36 orientational distribution between 38 0 and 64 0 . The orientation of the thiophene rings of P3HT at the fullerene surface, although slightly, depends on the processing condition of the P3HT film. We have prepared two different samples by varying the spin-coating concentration of P3HT: 11mg/mL (Fig. 2.9 (a)) and 5.5 mg/mL (Fig. 2.9 (b)). Performing orientation analysis suggests that thiophene rings in the P3HT sample prepared from 11 mg/mL spin-coating solution are 2 0 -5 0 more tilted towards the fullerene surface as compared to the other P3HT sample spin-coated from a 5.5 mg/mL solution. Figure 2.9. SFG spectra of P3HT-C60 bilayer sample where the spin-coating solution of P3HT is (a) 11 mg/mL and (b) 5.5 mg/mL 37 To investigate annealing induced structural change in P3HT/C60 interface, the bilayer sample (P3HT spin-coating concentration 5.5 mg/mL) was annealed at 155 0 C for one hour. As shown in Figure 2.10(a) and 2.10(b); upon annealing, the SSP intensity decreases by ~45% between 1430–1500 cm -1 , whereas the PPP intensity increases by ~15% between 1430–1460 cm - 1 (gray shade) and decreases by ~45% between 1460-1500 cm -1 (green shade), as compared to the un-annealed sample. Fitting the annealed spectra and performing orientational analysis confirmed that the thiophene rings of P3HT, irrespective of long and short conjugation length, tilt back from the fullerene surface by 12 0 -19 0 making an average angle of θ=61 0 between the net transition dipole and the surface normal. Table 2.4. Upon annealing, the change in angle of different conformers of P3HT (with different vibrational frequencies) are different. For this bilayer sample, the spin-coating solution of P3HT is 5.5 mg/mL. Frequency, cm -1 Orientation Angle Change in Angle Before Annealing After Annealing 1444.4 45 0 (41 0 ,50 0 ) 45 0 (41 0 ,49 0 ) 0 0 1451.9 47 0 (42 0 ,51 0 ) 60 0 (54 0 ,67 0 ) 13 0 1463.1 55 0 (50 0 ,61 0 ) 74 0 (65 0 ,90 0 ) 19 0 1476.4 53 0 (48 0 ,58 0 ) 65 0 (58 0 ,73 0 ) 12 0 The SSP spectra before and after thermal annealing, normalized to intensity at 1450 cm -1 , are shown in Fig. 2.10(c). Annealing results in a red shift of ~6 cm -1 , and narrower bandwidth. Different groups 34,42,47-49 have used UV-VIS and Raman spectroscopy to show that thermal annealing leads to a higher degree of crystallinity (increasing molecular ordering) in P3HT. All those measurements were bulk-sensitive. There was no clear evidence so far that proves whether 38 such behavior occurs at the interface. The narrower spectral line and the red shift in the VSFG spectra of the annealed sample (Fig. 2.10(c)) suggest that the effect of increased crystallinity in P3HT upon annealing propagates all the way to the interface. The effect of crystallinity is further corroborated by the increase in signal intensity of the characteristic C60 peak at 1425 cm -1 upon thermal annealing. This information may facilitate the use of annealing as a tool to tune the degree of molecular ordering and orientation of donor molecules at the interface with an electron acceptor. Figure 2.10. 10(a) and 10(b): Comparison of SFG spectrum of P3HT-C60 bilayer sample before and after thermal annealing at 155 0 C for one hour for SSP and PPP polarization respectively. (c) The normalized SSP spectra in the range 1430-1500 cm -1 before (red) and after (blue) thermal annealing. (d) Cartoon figure of the orientation of P3HT at fullerene surface before and after annealing. 39 III. Conclusion We have demonstrated that interfacial orientation of the P3HT backbone depends on the nature of the underlying substrate. In contact with CaF2 substrate, the thiophene rings tend to align themselves almost perpendicularly (‘edge-on’) to the plane of the substrate. At an interface with fullerene film, there is a significant tilt of the thiophene rings towards C60 (‘face-on’). Such tilt can be explained by the possibility of the π-π stacking between the thiophene rings and the conjugated C60 molecules. This arrangement would be favorable for efficient charge transfer observed between electron donor (P3HT) and electron acceptor (C60) molecule. 50-52 Upon thermal annealing, the thiophene backbone of P3HT tilts away from the surface of C60 by about Δθ=12±6 0 on average, adopting more edge-on orientation. This is accompanied by the higher degree of crystallinity of P3HT achieved upon thermal annealing, and the associated extending of the π conjugation over greater number of thiophene monomers in the P3HT chain, 34,42 as manifested by the red-shift and narrowing of the annealed spectra, indicating reduced inhomogeneous broadening. Annealing allows more efficient packing of the P3HT chains via π-π stacking interactions of neighboring P3HT molecules, which may overcome weaker interactions between P3HT and fullerene. Thus, upon annealing, the P3HT orientation at C60 changes towards that at the weakly interacting CaF2 interface (more ‘edge-on’). Our results show that the molecular orientation at the donor-acceptor interface is the result of the interplay between donor-donor and donor-acceptor interactions, and preparation conditions. 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Introduction Functionalized Si(111) surfaces have important applications ranging from inorganic photovoltaics to nanoscale lithography. 1-14 A monolayer of organic adsorbate covalently bonded to the surface Si atoms not only enhances the resistivity of the surface towards air oxidation, 15,16 but also allows tuning of the chemical reactivity at the surface. 5 The methyl-terminated Si(111) surface is a singular example of such chemical functionalization, with all of the Si atoms at the topmost layer of the crystalline Si surface being covalently bonded to methyl groups through Si-C bonds. Previous theoretical 17-19 and experimental studies 20-24 have shown that these terminal methyl groups are oriented perpendicular to the Si substrate. Use of a two-step halogenation/methylation process via Grignard chemistry 5,15,16 enables nearly 100% coverage of methyl groups on the atop Si atoms, resulting in a well-ordered, densely-packed, chemically well- defined organic monolayer. Steric interactions between the adjacent methyl groups may hinder the rotation of –CH3 groups along the Si-C bond, and therefore the orientation of the methyl groups might be interlocked at the interface. 5 Several groups have studied the three-fold azimuthal anisotropy of Si(111) surfaces by second harmonic generation (SHG) spectroscopy. 25-27 SHG probes the electronic resonances of the 45 Si substrate and therefore is not sensitive to the attached chemical groups. In contrast, vibrational sum frequency generation (VSFG) spectroscopy has provided information about the covalently attached functional groups on methyl-terminated 28 and propynyl-terminated Si(111) substrates. 29 Along with the three-fold anisotropy of the Si(111) surface probed by SHG spectroscopy, a three- fold rotational anisotropy of covalently attached methyl and propynyl groups has been observed by VSFG spectroscopy, corroborating the proposition by Yamada et.al. 5 that the orientation of the methyl groups is interlocked at the interface due to steric interactions with the neighboring ligands. Line-shape analysis of VSFG spectra has also revealed that methyl groups on the methyl- terminated Si(111) surface undergo hindered rotation between the three isoenergetic equilibrium orientations at the interface. 28 In this work, we used surface-selective VSFG spectroscopy to investigate the energy barrier of the hindered rotation of methyl groups of the methyl-terminated Si(111) surface. Methyl- terminated Si(111) surfaces were prepared and characterized as described previously. 30-32 To remove impurities, the samples were rinsed with water, acetone, methanol and again with water, and were heated in vacuo overnight at 450 0 C. VSFG experiments before and after annealing the sample indicated that additional peaks associated with -CH2 vibrational frequencies disappeared upon annealing, and are thus consistently ascribed to impurities. A broadband femtosecond IR pulse centered around 2900 cm -1 with a FWHM of ~300 cm -1 was used to excite the symmetric and asymmetric stretching modes of the methyl groups. A narrowband picosecond 800 nm pulse upconverted the coherences, and the sum-frequency light was collected in the phase-matching direction. VSFG spectra were collected for two different polarization combinations of input and output beams: PPP and SPS, where the polarizations of the output SFG field and the input visible and IR fields are written from left to right. The sample was kept on top of a Peltier cooling/heating 46 element that was glued over a water-cooled Al block, to allow measurement of VSFG spectra at five different temperatures between 20 0 C and 120 0 C. The sample chamber was continuously purged with dry air during the experiment. A thermocouple as well as an IR heat sensor were used to measure the temperature of the sample. II. Results and Discussion Infrared spectroscopy measurements of the methyl-terminated Si(111) samples showed characteristic peaks for the symmetric and asymmetric stretches of the methyl groups at 2910 cm - 1 and 2975 cm -1 , respectively. 20,21 Figure 3.1 shows the VSFG spectra for PPP and SPS polarizations. In the case of PPP polarization, two peaks were present, corresponding to the two IR active modes of methyl groups; for SPS polarization, the peak corresponding to the symmetric stretching vibration was absent. The methyl groups are oriented perpendicular to the Si(111) substrate, so the transition dipole of the symmetric stretch (along its C3v symmetry axis) is also perpendicular to the surface, whereas the transition dipole of the asymmetric stretch lies in the plane of the silicon substrate. As a result, S-polarized IR light, with the polarization in the plane of the substrate, cannot couple with the transition dipole of the symmetric stretch, but can couple with the transition dipole of the asymmetric vibrational stretch. The effect of rotational dynamics on the lineshapes of the VSFG spectra has been described in detail by Vinaykin et. al. 33 The basic assumption is that the vibrational dephasing is independent of the orientation of the molecule, hence the second-order time-domain response function ( 𝜒 (2) (𝑡 )) can be written as a multiplication of the vibrational dephasing term 𝑉 (𝑡 ) and the rotational relaxation dynamics term 𝑅 (𝑡 ) ( 𝜒 (2) (𝑡 ) = 𝑉 (𝑡 ) ∗ 𝑅 (𝑡 )). 34 The vibrational dephasing was moreover assumed to be exponential; i.e. 𝑉 (𝑡 ) ∝ 𝑒 −𝛤 𝑣 𝑡 cos (𝜔 0 𝑡 ) , where 𝛤 𝑣 is the 47 damping constant. A general theoretical framework was presented to calculate the orientational part of the response function 𝑅 (𝑡 ) within the small-angle rotational diffusion approximation. 35-37 In the weak-confinement model, the orienting potential is assumed to be on the order of 𝑘 𝐵 𝑇 (where kB is Boltzmann’s constant) and the rotational diffusion in the surface is approximated by free anisotropic diffusion with different relaxation rates in-plane (𝐷 ∥ ) and out-of-plane (𝐷 ⏊ ). 38 For the methyl-terminated Si surface, the methyl groups rotate with respect to the 𝐶 3𝑣 symmetry axis perpendicular to the surface plane; hence, there is zero out-of-plane diffusion, i.e. 𝐷 ⏊ = 0. In this case, the SFG response function calculated by Vinaykin et. al. 33 can be re-written as: Figure 3.1: VSFG spectra of methyl-terminated Si(111) for PPP and SPS polarizations measured at five different temperatures. Spectra at different temperatures (blue (21 o C), purple (38 o C), green (56 o C), yellow (85 o C) and red (118 o C)) are stacked vertically, after adding a constant offset to each spectrum. In going from 21 0 C to 118 0 C; the offset values for PPP were -804, -400, 500, 1000, 1500 and for SPS were: -50, -167, -50, 50, 100. Black lines show spectral fitting as described in the text. 48 χ 𝑃𝑃𝑃 (2) (𝑡 ) = (𝐶 1 𝑃𝑃𝑃 + 𝐶 3 𝑃𝑃𝑃 𝑒 −𝐷 ∥ 𝑡 ) ∗ 𝑒 −𝛤 𝑣 𝑡 cos(𝜔 0 𝑡 ) (1) χ 𝑆𝑃𝑆 (2) (𝑡 ) = 𝐶 3 𝑆𝑃𝑆 𝑒 −𝐷 ∥ 𝑡 ∗ 𝑒 −𝛤 𝑣 𝑡 cos(𝜔 0 𝑡 ) (2) where the term 𝐶 1 𝑃𝑃𝑃 represents pure vibrational dephasing, and the other two terms (𝐶 3 𝑃𝑃𝑃 𝑎𝑛𝑑 𝐶 3 𝑆𝑃𝑆 ) have contributions from both rotational and vibrational relaxations. For a molecule oriented vertically to an azimuthally isotropic surface (tilt angle ≈ 0 0 from the surface normal) and undergoing rotational diffusion about an axis perpendicular to the surface plane, the PPP line-shape is dominated by the vibrational dephasing term (𝐶 1 𝑃𝑃𝑃 ) and is insensitive to the reorientation dynamics on the same timescale (𝐶 3 𝑃𝑃𝑃 = 0). 33 The modified response function for PPP polarization is thus: χ 𝑃𝑃𝑃 (2) (𝑡 ) = 𝐶 1 𝑃 𝑃𝑃 ∗ 𝑒 −𝛤 𝑣 𝑡 cos(𝜔 0 𝑡 ) (3) In contrast with the small-angle rotational diffusion approximation, the rotational relaxation on the CH3−Si(111) surface presumably occurs via 120 0 jumps of the methyl groups between three isoenergetic conformations, and cannot be considered per se as a small-step diffusion. Hence, assuming first-order decay kinetics, the response functions (1-3) are the same, with the in-plane diffusion constant (𝐷 ∥ ) replaced by the rate constant (𝑘 𝑟𝑜𝑡 ) of such hindered rotation. The frequency domain response functions are obtained by Fourier transforming equations (2) and (3), and the line-shapes are Lorentzian: χ 𝑆𝑃𝑆 (2) (𝜔 ) = 𝐴 𝑆𝑃𝑆 (𝜔 − 𝜔 0 ) + 𝑖 𝛤 𝑆𝑃𝑆 𝑤 ℎ𝑒𝑟𝑒 𝛤 𝑆𝑃𝑆 = 𝛤 𝑣 + 𝑘 𝑟𝑜𝑡 (4) 49 χ 𝑃𝑃𝑃 (2) (𝜔 ) = 𝐴 𝑃𝑃𝑃 (𝜔 − 𝜔 0 ) + 𝑖 𝛤 𝑃𝑃𝑃 𝑤 ℎ𝑒𝑟𝑒 𝛤 𝑃𝑃𝑃 = 𝛤 𝑣 (5) where 𝐴 is the amplitude and 𝛤 is the HWHM of the Lorentzian. The linewidth of the SPS spectra has contributions from both the vibrational dephasing and in-plane orientation dynamics, whereas the PPP linewidth only has a vibrational dephasing contribution. For methyl-terminated Si(111), the −CH3 groups are oriented perpendicular to the substrate, and the transition dipole of the asymmetric stretch lies parallel to the surface plane. Rotation of the methyl groups along the 𝐶 3𝑣 symmetry axis results in the in-plane rotation of the asymmetric stretch transition dipole, which manifests itself as a line broadening in the SPS spectra. The linewidth of the PPP spectra, however, is not affected by such rotation. The difference between SPS and PPP linewidth for the asymmetric stretch vibration contains information about the reorientation dynamics of the molecule: 33, 39 Γ 𝑆𝑃𝑆 − Γ 𝑃𝑃𝑃 = 𝑘 𝑟𝑜𝑡 (6) Figure 3.1 also shows the SFG spectra of the H3C−Si(111) surface measured at five different temperatures. The SFG spectra were fitted using the following equation: 40 𝐼 𝑆𝐹𝐺 (𝜔 ) = |𝐴 𝑁𝑅 𝑒 𝑖𝜑 + ∑ 𝐵 𝑗 𝜔 − 𝜔 𝑗 + 𝑖 Γ 𝑗 𝑛 𝑗 =1 | 2 (7) where 𝐴 𝑁𝑅 and φ are the amplitude and phase, respectively, of the non-resonant background relative to the resonant contribution. The resonant part for the 𝑗 th vibrational mode is expressed as a Lorentzian with an amplitude of 𝐵 𝑗 /Γ 𝑗 , a center frequency 𝜔 𝑗 , and a line-width Γ 𝑗 . Each PPP spectrum was fitted with two Lorentzians (for the symmetric and asymmetric stretch modes), and one Lorentzian (asymmetric stretch) was used to fit the SPS spectra. The fitting parameters are tabulated in Table 3.1 and Table 3.2. For the asymmetric stretch mode, the change in ΓPPP and ΓSPS 50 with temperature is shown in Figure 3.2. ΓSPS exhibited a monotonic increase with increasing temperature, whereas ΓPPP did not show a substantial change between 20 0 C and 90 0 C, with however a noticeable decrease in linewidth above 90 0 C. Because the methyl groups in the methyl- terminated Si(111) sample have a rotational barrier, 28 an increase in temperature populates higher rotational levels of the molecule, and leads to faster rotation. As a result, rotational relaxation becomes more rapid with increase in temperature, giving rise to the increase in SPS linewidth. The PPP spectrum, however, is not affected by the reorientation dynamics, hence Γ PPP is almost constant within the temperature range explored in our work. The slight decrease observed in the PPP linewidth at higher temperature could be due to motional narrowing. 41,42 Table 3.1. Fitting parameters of the VSFG spectra for SPS polarization at five different temperatures. Temperature ( 0 C) 21 0 C 38 0 C 56 0 C 85 0 C 118 0 C 𝐴 𝑁𝑅 16.3±0.2 20.6±0.2 19.9±0.2 19.7±0.3 19.9±0.3 𝜑 1.3±0.1 1.5±0.1 1.5±0.1 1.7±0.2 1.9±0.2 𝐵 -39.5±4.0 -52.2±6.9 -56.8±6.6 -63.2±8.5 -75.1±12.5 𝜔 2980.5±1.3 2976.8±1.7 2975.5±1.7 2972.3±1.9 2968.8±2.5 Γ 13.5±1.1 16.2±1.5 17.0±1.3 17.6±1.4 18.4±1.6 51 Table 3.2. Fitting parameters of the VSFG spectra for PPP polarization at five different temperatures. Temperature ( 0 C) 21 0 C 38 0 C 56 0 C 85 0 C 118 0 C 𝐴 𝑁𝑅 41.8±0.7 46.8±1.2 44.3±1.4 44.0±1.3 48.4±1.2 𝜑 19.7±0.1 20.0±0.1 20.0±0.1 20.1±0.1 20.2±0.1 𝐵 1 -251.3±14.6 -295.8±26.4 -370.9±31.5 -465.4±28.4 -488.0±28.1 𝜔 1 2915.4±0.4 2911.4±0.5 2911.5±0.5 2909.4±0.4 2908.5±0.4 Γ 1 12.5±0.5 13.1±0.8 14.1±0.7 13.7±0.5 13.9±0.5 𝐵 2 -315.3±21.6 -267.8±29.1 -306.9±33.1 -341.2±27.2 -291.8±21.7 𝜔 2 2974.1±0.4 2968.9±0.5 2968.9±0.5 2966.8±0.4 2965.3±0.4 Γ 2 11.6±0.5 12.1±0.8 12.3±0.8 12.0±0.6 10.7±0.5 Figure 3.2: Plot of PPP and SPS linewidths of the asymmetric −CH3 stretch frequency vs. temperature. 52 Following equation 6, 𝑘 𝑟𝑜𝑡 can be calculated at every temperature, which upon Fourier transform gives the timescale of rotational relaxation (τ rot ). At 21 0 C, the rotational relaxation timescale is 3±2 ps, whereas, at 118 0 C, τ rot is 730±160 fs. The measurements are consistent with the room-temperature rotational dephasing of methyl groups for methyl-terminated Si(111) reported by Malyk et al., 28 in which 𝑘 𝑟𝑜𝑡 = 3.5±1.5 cm -1 , indicating a rotational timescale of τ rot = 1~2 ps. Use of a classical correlation function 43 for the free-rotors predicts that the timescale of rotational dephasing for a free methyl rotor is ~100 fs at room temperature. The rotational barrier in methyl-terminated Si(111) thus decreases the methyl rotation by more than an order of magnitude relative to a free rotor. Figure 3.3 shows the natural logarithm of the rate constant, ln(𝑘 𝑟𝑜𝑡 ) plotted against 1/T according to Arrhenius equation: ln(𝑘 𝑟𝑜𝑡 ) = ln(𝑘 0 ) − 𝐸 𝑟𝑜𝑡 𝑅 1 𝑇 (8) where T is the temperature in Kelvin; 𝐸 𝑟𝑜𝑡 is the activation energy, which in our case, is the barrier of methyl rotation; and 𝑘 0 is the attempt frequency. The activation barrier of the methyl rotation was calculated from the slope of the fitted straight line, − 𝐸 𝑟𝑜𝑡 𝑅 , yielding a value of 𝐸 𝑟𝑜𝑡 = 2.4±1.0 kcal/mol, which is equivalent to 9.9±4.3 kJ/mol, or 830±360 cm -1 . Using density functional perturbation theory, Brown et al. 44 calculated the upper bound of the rotational barrier of methyl groups on a methyl terminated Si(111) surface. On a fully methylated surface, a single methyl group was rotated while holding the neighboring methyl groups fixed, not allowing them to undergo any kind of correlated motion to minimize the steric interactions. The theoretically calculated upper bound, i.e. 112 meV ~ 900 cm -1 , is in excellent agreement with our experimentally determined rotational barrier. The methyl internal rotational barrier in some common organic 53 compounds ranges from 10 cm -1 (toluene) 45,46 to 514 cm -1 ( 35 Cl-o-cholorotoluene) 47 to ~1000 cm - 1 (ethane) 48 (Table 3.3). The attempt frequency (𝑘 0 ) can be calculated from the intercept, ln (𝑘 0 ), and is estimated to be 2.9  10 13 s -1 for CH3−Si(111), which corresponds to a frequency of 150 cm - 1 for the restricted rotation. Figure 3.3: ln (𝑘 𝑟𝑜𝑡 ) vs 1/T plot; where 𝑘 𝑟𝑜𝑡 is the rotational rate constant and T is the temperature in Kelvin. The plot is fitted with a straight line (red). The slope and the intercept of the fitted line are used to calculate the rotational barrier (𝐸 𝑟𝑜𝑡 ) and the attempt frequency (𝑘 0 ). 54 Table 3.3: Tortional barrier of methyl groups in different organic compounds tabulated from the literature. Compound Methyl rotational barrier (cm -1 ) Toluene 10 45,46 o-fluorotoluene 227 49 35 Cl-o-chlorotoluene 514 47 37 Cl-o-chlorotoluene 507 47 o-methylanisole 444 50 o-xylene 518 51 Acetaldehyde 408 52-55 anti-o-cresol 370 56 syn-o-cresol methylsilane 661 56 500 44 ethane 1100 48 As mentioned above, three preferential orientations of the methyl groups are present on the methyl-terminated Si(111) surface, and the methyl groups switch between these three isoenergetic conformations by hindered rotation along the torsional vibrational coordinate with respect to the C-Si bond. A simple mathematical function that gives the correct threefold symmetric potential can be written in the form: 𝑉 (𝜑 ) = 1 2 𝐸 𝑟𝑜𝑡 (1− cos(3𝜑 )); 43 where φ is the torsional angle and 𝐸 𝑟𝑜𝑡 is the barrier of rotation. Solving the Schrodinger equation for this potential and assuming that the thermal energy is smaller compared to the rotational barrier, i.e. 𝑘 𝐵 𝑇 ≪ 𝐸 𝑟𝑜𝑡 , yields an expression for the harmonic frequency (𝜈 ) of the torsional motion as: 43 𝜈 = 3 2𝜋 ( 𝐸 𝑟𝑜𝑡 2𝐼 𝑟 ) 1 2 (9) 55 where 𝐼 𝑟 is the reduced moment of inertia of a single methyl group with respect to the axis along the C-Si bond. Using 𝐼 𝑟 = 3.260 amuÅ 2 and 𝐸 𝑟𝑜𝑡 = 9.9 kJ/mole, the frequency of torsional motion is calculated to be 190 cm -1 . This calculated torsional frequency for a single methyl group matches well with the experimentally determined attempt frequency (150 cm -1 ) of the methyl-terminated Si(111) sample. This agreement suggests that the rotation of the methyl groups is independent from each other, as opposed to a gear-like concerted rotation of multiple methyl groups on the surface. The rotational relaxation of a single methyl group thus seems to occur without any correlation with the adjacent methyl moieties. III. Conclusions We have used VSFG spectroscopy to determine the rotational barrier of methyl groups of the methyl-terminated silicon(111) surface. The methyl groups are oriented perpendicular to the Si substrate and cannot rotate freely due to steric interactions with the adjacent methyl moieties. As a result, the VSFG response from the C-H stretches of the methyl groups showed a three-fold rotational anisotropy. The room temperature PPP vs SPS line-shape analysis predicts a possibility 56 of hindered rotation of the methyl groups between three isoenergetic equilibrium configurations. For a “rod-like molecule” standing perpendicular to the substrate and rotating with respect to the surface normal, the PPP spectral line-shape is mainly dominated by the vibrational relaxation, whereas the SPS spectrum carries information about both the vibrational dephasing and rotational dynamics. Increasing temperature results in faster rotational relaxation, which manifests itself in the increase in SPS linewidth. Therefore, the difference between SPS and PPP linewidth increases with temperature and measures the contribution of the reorientation dynamics in the VSFG spectra for CH3−Si(111) surface. Fitting the rate constant of rotational relaxation (𝑘 𝑟𝑜𝑡 ) to the Arrhenius equation yields the energy barrier of rotation, 𝐸 𝑟𝑜𝑡 = 830± 360 cm -1 , as well as the attempt frequency, 𝑘 0 = 2.9  10 13 s -1 . The experimentally determined attempt frequency matches well with the calculated harmonic torsional frequency of a single methyl group. This accord suggests that the methyl rotation on the CH3−Si(111) surface is not a concerted gear-like motion involving multiple methyl groups but rather represents the rotation of an individual methyl group uncorrelated from the rest of the surface functionality. 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J.; Godefroid, M., The ground and first torsional states of acetaldehyde. J. Mol. Spectrosc. 1992, 153 (1-2), 578-586. 55. Kundu, T.; Pradhan, B.; Singh, B. P., Origin of methyl torsional potential barrier—an overview. J. Chem. Sci. 2002, 114 (6), 623-638. 56. Welzel, A., Hellweg, A., Merke, I. and Stahl, W.,, Structural and torsional properties of o- cresol and o-cresol-OD as obtained from microwave spectroscopy and ab initio calculations. J. Mol. Spectrosc. 2002, 215(1), 58-65. 60 Chapter 4 A Vibrational Sum Frequency Generation Study of the Interference Effect in a Thin Film of 4,4′-Bis(N-Carbazolyl)- 1,10-Biphenyl (CBP) and the Interfacial Orientation Abstract Molecular organization of vapor-deposited organic molecules in the active layer of organic light emitting diodes (OLEDs) have been a matter of great interest as they directly influence various optoelectronic properties and the overall performance of the devices. Contrary to the general assumption of isotropic molecular orientation in vacuum deposited thin-film OLEDs, it is possible to achieve an anisotropic molecular distribution at or near the surface under controlled experimental conditions. In this study, we have used interface-specific vibrational sum frequency generation (VSFG) spectroscopy to determine the orientation of a low molecular weight OLED material, 4,4′-Bis(N-carbazolyl)-1,1′-biphenyl (CBP) at the free (air) and the buried (CaF2) interfaces. VSFG spectra were measured at four different polarization combinations for five different thicknesses of the CBP film. The spectral shift and the VSFG intensity change with the film thickness can be accurately modeled by considering the optical interference effect of the signals coming from the CBP/air and CBP/CaF2 interfaces. A global fitting of the experimental spectra for all thicknesses along with the theoretical simulation reveal that the long molecular axis of CBP is oriented at an angle of ~58 0 (47 0 - 70 0 ) from the surface normal at the air/CBP interface, whereas at CBP/CaF2 interface, the angle is ~48 0 (43 0 - 52 0 ). Such a change in angle (~10 0 ) suggests that the CBP molecule tends to orient more vertically (edge-on) at the buried CaF2 interface, which may be attributed to. the intermolecular 𝜋 -𝜋 stacking interaction between adjacent CBP molecules. 61 I. Introduction In the vapor deposited thin OLED films, it is generally assumed that the molecular orientation is isotropic. Later, several groups reported anisotropic molecular orientation of the emitters, hole transport and electron transport materials in the amorphous films. 1–8 Although, large planar-shaped molecules are known to have a preference for horizontal orientation, 3,8 it cannot be extrapolated to relatively low molecular weight OLED materials, such as 4,4′-Bis(N-carbazolyl)-1,1′-biphenyl (CBP) and N,N′-Di(1-naphthyl)-N,N′-diphenyl-(1,1′-biphenyl)-4,4′-diamine (NPD). 9 The vapor deposited neat films of CBP and NPD which are excellent hole transport materials, were considered to have a completely random molecular orientation. 3,10 But under controlled experimental conditions, an anisotropic molecular distribution is possible to achieve for these molecules. 11–15 Controlling the substrate temperature has been identified as one of the key factors behind such stable molecular arrangements. 11,14–16 Variable angle spectroscopic ellipsometry (VASE) is popularly used to understand the orientation and conformation of the molecules in the vapor deposited OLED films. But VASE technique is not interface specific and hence, reports the bulk-averaged orientation. To obtain information about the molecular orientation at or near the surface, we resort to vibrational sum frequency generation (VSFG) spectroscopy, which is forbidden in centrosymmetric media (bulk of a material) under electric dipole approximation, but is allowed at interfaces where local symmetry is broken. 17–20 In this work, films of different thicknesses were prepared by vapor-depositing CBP material on IR grade CaF2 windows. FTIR and Raman spectra are collected as shown in Figure 4.1. VSFG was performed by tuning the IR pulse to be centered at ~1550 cm -1 and upconverting the vibrational coherence with picosecond (narrowband) visible laser pulse fixed at ~795 nm. The VSFG spectra were measured at four different polarization combinations : PPP, SSP, SPS and PSS; and the detailed experimental set 62 up is described elesewhere. 21 Figure 4.1. FTIR, Raman and Polarization-selective VSFG spectra of a 100 nm thin film of CBP. For our work, we are interested in the mode that is highlighted in the figure (blue band). The mode is identified as the C=C symmetric stretching mode localized mainly on the biphenyl backbone of CBP. The structure of CBP is shown in the inset. II. Results and Discussion A. Spectroscopic characterization of CBP film As dictated by the selection rule of VSFG spectroscopy, for a vibrational mode to be VSFG active, it must be both IR and Raman active. The vibrational mode at ~1602 cm -1 is both IR and 63 Raman active and will be the focus of our SFG analysis. It is assigned to the C=C symmetric stretch mode localized mainly on the biphenyl backbone of CBP and coupled to the in-plane C-H deformation modes of the molecule. 22 The transition dipole associated with this mode lies along the long molecular axis of CBP and thus, the orientation of CBP at the interface can be described by the orientation of this transition dipole with respect to the laboratory fixed Z-axis (normal to the surface of the film). 9 Figure 4.1 shows the VSFG spectra of a 100 nm CBP film recorded with all four polarization combinations mentioned above. The band at ~1602 cm -1 only appears for PPP and SSP polarization combinations (where the IR field is P-polarized); but not in case of SPS and PSS (where the IR field is S-polarized). It suggests that the transition dipole corresponding to ~1602 cm -1 vibrational mode has a net component along the laboratory Z axis and the vapor deposited films of CBP has substantial anisotropy at or near the surface. However, there is no in- plane azimuthal anisotropy observed, as shown in Figure 4.2. No VSFG activity of the 1602 cm -1 mode for SPS and PSS polarization combinations (IR field is S-polarized, i.e. polarized in the plane of the substrate) results from this in-plane symmetry of the film. Figure 4.2: Azimuthal anisotropy study of the CBP film. The CBP film was kept on a rotational stage and VSFG spectra were measured changing the azimuthal angle of the film by 20 0 . The black crosses correspond to the normalized VSFG intensity. The blue line connecting the black crosses traces a circle, suggesting that the VSFG intensity doesn’t change with the azimuthal angle. 64 To further investigate the origin of VSFG response from the vapor deposited CBP films, VSFG spectra are recorded for PPP and SSP polarization combinations as a function of film thickness, as shown in Figure 4.3. 9 The VSFG intensity increases with increase in film thickness from 20 nm to 100 nm, stays almost same for the 150 nm film and then decreases as we increase the film thickness to 200 nm. Such a change in intensity as a function of the sample thickness is also associated with a noticeable spectral shift which suggests that the measured VSFG spectra result from the optical interference of the signals coming from the free CBP/air and the buried CaF2/CBP interfaces. 9 Change in thickness of the sample changes the interference pattern between these two signals giving rise to a spectral shift and a non-monotonic change in VSFG intensity. Figure 4.3. VSFG spectra at PPP and SSP polarization combinations as a function of film thickness. Spectra of 20 nm, 50 nm, 100 nm, 150 nm and 200 nm films are stacked vertically from bottom to top. The film thicknesses are indicated to the right side of each spectrum. The black lines correspond to the fits of the spectra. 65 A global fitting procedure was used to simultaneously fit all the spectra for both PPP and SSP polarization combinations. Here we assumed that the orientation of the CBP molecule at either interface does not change with the thickness of the film. As a result, for a surface/interface, although the PPP and SSP signal intensities can vary independently with film thickness, the PPP/SSP intensity ratio should stay the same. The ratio, however, can be different for different interfaces which points to the difference in orientation. Two Lorentizians were used to fit the 1600 cm -1 region of the spectra. These two Lorentzians correspond to the same IR transition, but for two different interfaces, air/CBP and CBP/CaF2. The Lorentzian used for the buried interface has a 𝑒 𝑖 𝜑 2 phase factor associated with it. The phase 𝜑 2 accounts for the thickness of the sample. A third Lorentzian at ~1620 cm -1 was used to accurately reproduce the shoulder in the high frequency side of the spectrum. The appearance of this shoulder in the VSFG spectra corresponds to the weak IR transition in the same region, as shown in Figure 4.1. The fitting equation is: 𝐼 𝑆𝐹 𝐺 = 𝑦 0 + |𝐴 𝑒 𝑖𝜑 + 𝑏 1 𝜔 − 𝜔 1 + 𝑖 Г 1 + 𝑏 2 𝑒 𝑖 𝜑 2 𝜔 − 𝜔 2 + 𝑖 Г 2 + 𝑏 3 𝜔 − 𝜔 3 + 𝑖 Г 3 | 2 (1) where, 𝑦 0 is a constant. 𝐴 and 𝜑 are the amplitude and the phase of the non-resonant background with respect to the resonant contribution. As described above, three Lorentzians were used to fit the resonant part of the VSFG response, where 𝑏 𝑖 , 𝜔 𝑖 and Г 𝑖 are the amplitude, frequency and linewidth of each Lorentzian. The first and second Lorentzians were for the air/CBP and CBP/CaF2 interfaces respectively. The fitting parameters are tabulated in Table 4.1 and the fitted spectra are shown in Figure 4.3. The amplitude ratio of the Lorentzians for PPP and SSP polarization combination turns out to be 2.3 for air/CBP interface ( 𝑏 1 𝑃𝑃𝑃 𝑏 1 𝑆𝑆𝑃 ) and 0.6 for CBP/CaF2 interface ( 𝑏 2 𝑃𝑃𝑃 𝑏 2 𝑆𝑆𝑃 ). Such difference in ratio points to the different orientations of the CBP molecule at the free surface 66 vs the buried interface. We have performed theoretical analysis to calculate these orientation angles, presented in the next section. Table 4.1. Fitting parameters obtained from the global fitting procedure as described in the text. VSFG spectra are recorded at PPP and SSP polarization combinations for five different thicknesses. 𝜑 2 is reported in degree. A schematic diagram of optical interference is shown in Figure 4.4, where medium I, II and III represent air, CBP and CaF2 respectively. The VSFG signal generated from Interface II (SFII) is associated with a phase (𝜑 2 ) with respect to that generated from Interface I (SFI). 9 This phase factor accounts for the geometric path difference between the VSFG signals originating from the free air/CBP and buried CBP/CaF2 interfaces. As can be seen from Figure 4.5, 𝜑 2 changes linearly with the film thickness for both PPP and SSP polarization combinations, and the slopes are also similar in both cases. 67 Figure 4.4. This figure is adapted from a publication by Tong et al. 23 Here, schematic diagram of the optical interference is shown assuming a three-layer model where medium I, II and III represent air, CBP film and CaF2 respectively. 'd' is the thickness of the film. SFI and SFII represent the VSFG signals generated from air/CBP and CBP/CaF2 interfaces respectively. Figure 4.5. Plot of phase (𝜑 2 ) of the VSFG signal from the buried interface as a function of film thickness, for both PPP and SSP polarization combinations. 68 B. Modeling optical interference for CBP films We now use a theoretical model to describe thickness dependent interference effects in VSFG spectroscopy. 23–31 Amongst several established approaches, we adopted the interference model proposed by Tong et al, 23,27 which is based on a three layer model consisting of two interfaces, CBP/air and CaF2/CBP. 32 Fresnel factors were calculated considering multiple reflections within the CBP film. 9 The second order hyperpolarizability tensors for both SSP and PPP polarization combinations can then be expressed as: 23 𝜒 𝑆𝑆𝑃 ∝ [𝐿 𝑌𝑌𝑍 𝐼 𝑠𝑖𝑛 𝜃 𝐼𝑅 𝜒 𝑌𝑌𝑍 (2),𝐼 + 𝐿 𝑌𝑌𝑍 𝐼𝐼 𝑠𝑖𝑛 𝜃 𝐼𝑅 𝜒 𝑌𝑌𝑍 (2),𝐼𝐼 ] (2) 𝜒 𝑃𝑃𝑃 ∝ [ −𝐿 𝑋𝑋𝑍 𝐼 𝑐𝑜𝑠 𝜃 𝑆𝐹𝐺 𝑐𝑜𝑠 𝜃 𝑉𝐼𝑆 𝑠𝑖𝑛 𝜃 𝐼𝑅 𝜒 𝑋𝑋𝑍 (2),𝐼 − 𝐿 𝑋𝑍𝑋 𝐼 𝑐𝑜𝑠 𝜃 𝑆𝐹𝐺 𝑠𝑖𝑛 𝜃 𝑉𝐼𝑆 𝑐𝑜𝑠 𝜃 𝐼𝑅 𝜒 𝑋𝑍𝑋 (2),𝐼 +𝐿 𝑍𝑋𝑋 𝐼 𝑠𝑖𝑛 𝜃 𝑆𝐹𝐺 𝑐𝑜𝑠 𝜃 𝑉𝐼𝑆 𝑐𝑜𝑠 𝜃 𝐼𝑅 𝜒 𝑍𝑋𝑋 (2),𝐼 + 𝐿 𝑍𝑍𝑍 𝐼 𝑠𝑖𝑛 𝜃 𝑆𝐹𝐺 𝑠𝑖𝑛 𝜃 𝑉𝐼𝑆 𝑠𝑖𝑛 𝜃 𝐼𝑅 𝜒 𝑍𝑍𝑍 (2),𝐼 −𝐿 𝑋𝑋𝑍 𝐼𝐼 𝑐𝑜𝑠 𝜃 𝑆 𝐹𝐺 𝑐𝑜𝑠 𝜃 𝑉𝐼𝑆 𝑠𝑖𝑛 𝜃 𝐼𝑅 𝜒 𝑋𝑋𝑍 (2),𝐼𝐼 − 𝐿 𝑋𝑍𝑋 𝐼𝐼 𝑐𝑜𝑠 𝜃 𝑆𝐹𝐺 𝑠𝑖𝑛 𝜃 𝑉𝐼𝑆 𝑐𝑜𝑠 𝜃 𝐼𝑅 𝜒 𝑋𝑍𝑋 (2),𝐼𝐼 +𝐿 𝑍𝑋𝑋 𝐼𝐼 𝑠𝑖𝑛 𝜃 𝑆𝐹𝐺 𝑐 𝑜𝑠 𝜃 𝑉𝐼𝑆 𝑐𝑜𝑠 𝜃 𝐼𝑅 𝜒 𝑍𝑋𝑋 (2),𝐼𝐼 + 𝐿 𝑍𝑍𝑍 𝐼𝐼 𝑠𝑖𝑛 𝜃 𝑆𝐹𝐺 𝑠𝑖𝑛 𝜃 𝑉𝐼𝑆 𝑠𝑖𝑛 𝜃 𝐼𝑅 𝜒 𝑍𝑍𝑍 (2),𝐼𝐼 ] (3) where, 𝜃 𝐼𝑅 ,𝜃 𝑉𝐼𝑆 𝑎𝑛𝑑 𝜃 𝑆𝐹𝐺 are the angles of incidence of the IR, visible and SFG beams respectively. 𝐿 𝐼 and 𝐿 𝐼𝐼 are the total Fresnel factors for CBP/air and CaF2/CBP interfaces, and they can be written as a multiplication of individual Fresnel coefficients corresponding to the IR, VIS and SFG frequencies; 𝜔 𝐼𝑅 , 𝜔 𝑉𝐼𝑆 and 𝜔 𝑆𝐹𝐺 respectively. 𝐿 𝑖𝑗𝑘 = 𝐿 𝑖𝑖 (𝜔 𝑆𝐹𝐺 )𝐿 𝑗𝑗 (𝜔 𝑉𝐼𝑆 )𝐿 𝑘𝑘 (𝜔 𝐼𝑅 ) The simulated Fresnel factors for the CBP/air (𝐿 𝐼 ) and CaF2/CBP interfaces (𝐿 𝐼𝐼 ) are shown in Figure 4.6; as a function of film thickness and refractive index of the interfacial layer (n′). The oscillatory behavior of the Fresnel coefficients results from the change in interference pattern with the film thickness. 9 The interfacial refractive index n′ is varied in between the bulk refractive indices of the materials: 1.0 ≤ 𝑛 𝐶𝐵𝑃 /𝑎𝑖𝑟 ′ ≤ 1.7 and 1.4 ≤ 𝑛 𝐶𝐵𝑃 /CaF 2 ′ ≤ 1.7. A comparison of the Fresnel coefficients at both interfaces reveals that the VSFG spectra are dominated by the CBP/air 69 interface, as 𝐿 𝐼 is at least 10 times stronger or more than 𝐿 𝐼𝐼 . Nevertheless, contributions from both the interfaces were considered in our calculation. Figure 4.6. Simulated Fresnel factors as a function of film thickness: (left panel) CBP/air interface and (right panel) CaF2/CBP interface. Fresnel factors are also plotted as a function of interfacial refractive index; n′ = 1.7 (green), n′ = 1.6 (black), n′ = 1.5 (blue), n′ = 1.4 (red), n′ = 1.3 (grey), n′ = 1.2 (pink), n′ = 1.1 (nude), n′ = 1.0 (brown). For CBP/air interface; 1.0 ≤ 𝑛 𝐶𝐵𝑃 /𝑎𝑖𝑟 ′ ≤ 1.7 and for CaF2/CBP interface; 1.4 ≤ 𝑛 𝐶𝐵𝑃 /CaF 2 ′ ≤ 1.7 70 𝜒 𝐼 and 𝜒 𝐼𝐼 in equation (2) and (3), are the second order nonlinear susceptibility tensors for the C=C symmetric stretch mode (~1602 cm -1 ) of CBP at CBP/air and CaF2/CBP interfaces respectively. It is to point out that 𝜒 is the macroscopic susceptibility with respect to the laboratory axis, and it is related to the molecular hyperpolarizability tensor (𝛽 ) by the following equation: 𝜒 𝑋𝑌𝑍 (2) = ∑𝑅 𝑋𝑎 ,𝑌𝑏 ,𝑍 𝑐 (𝜃 ,𝜑 ,𝜓 ) 𝑎𝑏𝑐 𝛽 𝑎𝑏𝑐 (2) (4) where 𝑅 (𝜃 ,𝜑 ,𝜓 ) is the Euler rotation matrix used for the transformation of molecular coordinate system (a, b, c) to the laboratory frame (X, Y, Z), as shown in Figure 4.7. The elements of 𝛽 for CBP were calculated by the following equation: 𝛽 𝑎𝑏𝑐 𝑞 ∝ 𝜕𝛼 𝑎𝑏 𝜕𝑄 𝑞 𝜕𝜇 𝑐 𝑞 𝜕𝑄 𝑞 (5) Here 𝜇 𝑐 𝑞 is the IR transition dipole moment for the 𝑞 𝑡 ℎ vibrational mode and 𝛼 𝑎𝑏 is the Raman polarizability. To calculate 𝛽 , the geometry of the CBP molecule was first optimized at B3LYP/6- 311G** level of theory. The optimized geometry of CBP has a D2 point group symmetry, with a dihedral angle of ~142 0 with respect to the C-C single bond connecting two benzene rings of the biphenyl chromophore. The frequency and the polarizability of the optimized molecule were calculated at the same level of theory. The calculated frequencies were scaled by a factor of 0.97 before comparing with the experimental spectrum. The IR transition dipole corresponding to ~1602 cm -1 mode lies along the long molecular axis of CBP. To calculate 𝜕𝛼 𝑎𝑏 𝜕𝑄 𝑞 and 𝜕𝜇 𝑐 𝑞 𝜕𝑄 𝑞 , four different geometries were generated upon stretching and compressing the optimized molecule along the ~1602 cm -1 normal mode coordinate. For each geometry, 𝜇 𝑐 𝑞 and 𝛼 𝑎𝑏 were calculated and their derivatives with respect to 𝑞 𝑡 ℎ normal mode coordinate were obtained numerically. The 71 𝛽 𝑎𝑏𝑐 𝑞 values were then calculated by multiplying these derivatives, as shown in equation (5). All calculations were performed using Q-Chem Electronic Structure Package codes, version 5.1.2. Figure 4.7. Relation between molecular (a, b, c) and laboratory (X, Y, Z) coordinate frames for CBP. 𝜃 is the tilt angle of the transition dipole with respect to the laboratory Z axis. 𝜓 is the twist angle with respect to the long molecular axis and 𝜑 is the azimuthal angle in the X-Y plane. It is well-known that amorphous materials like CBP tend to have several molecular conformations in their vapor-deposited film as they lack long-range periodic structures mostly seen in crystalline films. 3 To generate different conformations of CBP, geometries were optimized by changing the dihedral angle of the benzene rings of the biphenyl chromophore between 130 0 and 150 0 , with an increment of 5 0 . It is worth pointing out that any deviation from the optimized dihedral angle (~142 0 ) makes the molecule adopt a C1 symmetry. Following the same procedure mentioned above, β values were calculated for all these conformations. Our calculations reveal that there are nine non-zero β tensor elements that contribute to the SFG spectra of CBP. It is because, IR transition dipole has only one non-zero component (𝜇 𝑐 , along the ‘c’ molecular axis, as shown in Figure 4.6); resulting to a single non-zero derivative, 𝜇 𝑐 ′ (𝜇 𝑎 ′ , 𝜇 𝑏 ′ = 0). On the contrary, the polarizability (𝛼 ) is represented by a 3x3 matrix which has nine non-zero elements, out of which 𝛼 𝑎𝑏 = 𝛼 𝑏𝑎 , 𝛼 𝑏𝑐 = 𝛼 𝑐𝑏 , 𝛼 𝑎𝑐 = 𝛼 𝑐𝑎 that result into six unique β values. The calculated polarizability 72 and the IR transition dipole moment for the 140 0 dihedral angle conformer were plotted as a function of the displacement from the equilibrium geometry, and are shown in Figure 4.8. The plots shown in Figure 4.8 were fitted with polynomials and the derivatives were calculated from the fitted equations at zero nuclear displacement. The 𝜇 ′ and 𝛼 ′ values for different conformers of CBP are tabulated in Table 4.2. It is clear that 𝛼 𝑎𝑏 ′ and 𝛼 𝑏𝑎 ′ values are ~20 to 100 times larger than the rest, making 𝛽 𝑎𝑏𝑐 and 𝛽 𝑏𝑎𝑐 the only dominant second-order susceptibility tensor elements in this case. Figure 4.8. Plots of polarizability (𝛼 ) and IR transition dipole moment (𝜇 ) with respect to nuclear displacement for the 140 0 dihedral angle conformer of CBP. Six polarizability elements are shown, as 𝛼 𝑐𝑎 = 𝛼 𝑎𝑐 ; 𝛼 𝑐𝑏 = 𝛼 𝑏𝑐 ; 𝛼 𝑎𝑏 = 𝛼 𝑏𝑎 .The blue markers and the red lines represent the calculated numbers and the polynomial fits respectively. Derivatives are calculated from the fitted equation at the zero nuclear displacement. 73 Table 4.2. 𝜇 ′ and 𝛼 ′ values are calculated for different conformers of CBP. The molecule with dihedral angle of ~142 0 has a D2 symmetry, all other conformers are of C1 symmetry. Dihedral angle 𝛼 𝑐𝑐 ′ 𝛼 𝑎𝑎 ′ 𝛼 𝑏𝑏 ′ 𝛼 𝑐𝑎 ′ or 𝛼 𝑎𝑐 ′ 𝛼 𝑐𝑏 ′ or 𝛼 𝑏𝑐 ′ 𝛼 𝑎𝑏 ′ or 𝛼 𝑏𝑎 ′ 𝜇 𝑐 ′ 142 0 4.3255 -0.1555 -0.0528 0.0439 0.3571 23.681 0.0667 130 0 0.034 -0.0034 -0.0028 -0.0027 1.4748 22.695 -1E-11 135 0 1.0788 -0.0966 -0.0535 -0.0046 0.9143 23.487 2E-11 140 0 0.7609 -0.0431 -0.0249 0.0136 0.3318 24.272 5E-11 145 0 1.113 -0.1668 -0.1223 0.0215 -0.1962 25.111 -2E-11 150 0 0.8973 -0.1069 -0.0771 0.0188 -0.7259 26.12 2E-11 The calculated β values (molecular hyperpolarizability) were used to compute the macroscopic second-order susceptibilities (𝜒 2 ), as shown in equation (4). There are four 𝜒 2 elements that contribute to the VSFG spectra for PPP polarization combination (𝜒 𝑋𝑋𝑍 (2) , 𝜒 𝑋𝑍𝑋 (2) , 𝜒 𝑍𝑋𝑋 (2) and 𝜒 𝑍𝑍𝑍 (2) ); but only one contributes to that of SSP (𝜒 𝑌𝑌𝑍 (2) ). For C1 point group symmetry, the 𝜒 2 values are obtained by the following equations: 𝜒 𝑌𝑌𝑍 (2) = 𝜒 𝑋𝑋𝑍 (2) = 1 2 𝑁 𝑠 [〈𝐴 〉𝛽 𝑐𝑐𝑐 + 〈𝐵 〉(𝛽 𝑎𝑎𝑐 + 𝛽 𝑏𝑏𝑐 ) − 〈𝐶 〉𝛽 𝑏𝑏𝑐 − 〈𝐷 〉𝛽 𝑎𝑎𝑐 + 〈𝐸 〉(𝛽 𝑎𝑏𝑐 + 𝛽 𝑏𝑎𝑐 ) − 〈𝐹 〉(𝛽 𝑏𝑐𝑐 + 𝛽 𝑐𝑏𝑐 ) + 〈𝐺 〉(𝛽 𝑎𝑐𝑐 + 𝛽 𝑐𝑎𝑐 ) + 〈𝐻 〉(𝛽 𝑏𝑐𝑐 + 𝛽 𝑐𝑏𝑐 ) − 〈𝐼 〉(𝛽 𝑎𝑐𝑐 + 𝛽 𝑐𝑎𝑐 )] (6a) 𝜒 𝑋𝑍𝑋 (2) = 1 2 𝑁 𝑠 [〈𝐴 〉𝛽 𝑐𝑐𝑐 − 〈𝐶 〉𝛽 𝑏𝑏𝑐 − 〈𝐷 〉𝛽 𝑎𝑎𝑐 + 〈𝐸 〉(𝛽 𝑎𝑏𝑐 + 𝛽 𝑏𝑎𝑐 ) − 〈𝐹 〉𝛽 𝑏𝑐𝑐 + 〈𝐺 〉𝛽 𝑎𝑐𝑐 + 〈𝐻 〉(𝛽 𝑏𝑐𝑐 + 𝛽 𝑐𝑏𝑐 ) − 〈𝐼 〉(𝛽 𝑎𝑐𝑐 + 𝛽 𝑐𝑎𝑐 )] (6b) 𝜒 𝑍𝑋𝑋 (2) = 1 2 𝑁 𝑠 [〈𝐴 〉𝛽 𝑐𝑐𝑐 − 〈𝐶 〉𝛽 𝑏𝑏𝑐 − 〈𝐷 〉𝛽 𝑎𝑎𝑐 + 〈𝐸 〉(𝛽 𝑎𝑏𝑐 + 𝛽 𝑏𝑎𝑐 ) − 〈𝐹 〉𝛽 𝑐𝑏𝑐 + 〈𝐺 〉𝛽 𝑐𝑎𝑐 + 〈𝐻 〉(𝛽 𝑏𝑐𝑐 + 𝛽 𝑐 𝑏𝑐 ) − 〈𝐼 〉(𝛽 𝑎𝑐𝑐 + 𝛽 𝑐𝑎𝑐 )] (6c) 𝜒 𝑍𝑍𝑍 (2) = 𝑁 𝑠 [〈𝐽 〉𝛽 𝑐𝑐𝑐 + 〈𝐹 〉(𝛽 𝑏𝑐𝑐 + 𝛽 𝑐𝑏𝑐 ) − 〈𝐺 〉(𝛽 𝑎𝑐𝑐 + 𝛽 𝑐𝑎𝑐 ) + 〈𝐶 〉𝛽 𝑏𝑏𝑐 + 〈𝐷 〉𝛽 𝑎𝑎𝑐 − 〈𝐸 〉(𝛽 𝑎𝑏𝑐 + 𝛽 𝑏𝑎𝑐 ) − 〈𝐻 〉(𝛽 𝑏𝑐𝑐 + 𝛽 𝑐 𝑏 𝑐 ) + 〈𝐼 〉(𝛽 𝑎𝑐𝑐 + 𝛽 𝑐𝑎𝑐 )] (6d) where 〈𝐴 〉 = 〈𝑠𝑖𝑛 2 𝜃 𝑐𝑜𝑠𝜃 〉; 〈𝐵 〉 = 〈𝑐𝑜𝑠𝜃 〉; 〈𝐶 〉 = 〈𝑠𝑖𝑛 2 𝜃 𝑐𝑜𝑠𝜃 𝑠𝑖𝑛 2 𝜓 〉; 74 〈𝐷 〉 = 〈𝑠𝑖 𝑛 2 𝜃 𝑐𝑜𝑠𝜃 𝑐𝑜𝑠 2 𝜓 〉; 〈𝐸 〉 = 〈𝑠𝑖𝑛 2 𝜃 𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜓 𝑐𝑜𝑠𝜓 〉; 〈𝐹 〉 = 〈𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜓 〉; 〈𝐺 〉 = 〈𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜓 〉; 〈𝐻 〉 = 〈𝑠𝑖𝑛 3 𝜃𝑠𝑖𝑛𝜓 〉; 〈𝐼 〉 = 〈𝑠𝑖𝑛 3 𝜃𝑐𝑜𝑠𝜓 〉; 〈𝐽 〉 = 〈𝑐𝑜𝑠 3 𝜃 〉; 𝑁 𝑠 is the number density of the molecules at the interface. 〈𝐴 〉 through 〈𝐽 〉 are the expectation values and can be calculated by the following equation: 〈𝐴 〉 = ∭ 𝐴 ∗ 𝑓 (𝜃 ,𝜓 ,φ)𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜓𝑑 φ ∭𝑓 (𝜃 ,𝜓 ,φ)𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜓𝑑 φ (7) Here, 𝜃 goes from 0 to 𝜋 2 , 𝜓 and φ are integrated from 0 to 2𝜋 . We have aligned the CBP molecule such that the transition dipole for 1602 cm -1 mode lies along the 'c' molecular axis, which is tilted by angle 𝜃 with respect to laboratory Z axis (see Figure 4.7). A uniform distribution over 0 to 2π is assumed for azimuthal angle (φ) and twist angle (ψ). To describe the molecular orientation at the interface, a Gaussian distribution of the said transition dipole is considered as a function of tilt angle 𝜃 . 𝑓 (𝜃 ,𝜓 ,φ) = 1 √2𝜋 𝜎 2 𝑒 − (𝜃 −𝜃 0 ) 2 2𝜎 2 (8) where 𝜃 0 and 𝜎 are the average tilt angle from the surface normal and the width of the distribution of the tilt angle, respectively. Armed with all the equations mentioned above, for both CBP/air and CBP/CaF2 interfaces, 𝜒 𝑃𝑃𝑃 (2) , 𝜒 𝑆𝑆𝑃 (2) and 𝜒 𝑃𝑃𝑃 (2) /𝜒 𝑆𝑆𝑃 (2) ratio are calculated as a function of average tilt angle (𝜃 0 ), interfacial refractive index (𝑛 ′) and width of tilt angle (𝜎 ). Comparison of theoretically calculated ratio with the PPP/SSP amplitude ratio of the Lorentzians obtained from the global fitting exercise of the experimental spectra suggests that the long molecular axis of CBP is oriented at an angle of ~58 0 75 (47 0 ,70 0 ) from the surface normal at the air/CBP interface, whereas at CBP/CaF 2 interface, the angle is ~48 0 (43 0 ,52 0 ). So, in going from the buried interface to the free interface, the CBP molecule tends to orient more horizontally (face-on), as evident from the ~10 0 change in the orientational angle (𝜃 ). The confidence interval of 𝜃 for both the interfaces are determined from the assumed range of 𝑛 ′ and 𝜎 . The results for 140 0 dihedral angle conformer are shown in Figure 4.9. Figure 4.9. Plots of (top panel) 𝜒 𝑃𝑃𝑃 𝐼 , 𝜒 𝑆𝑆𝑃 𝐼 , 𝜒 𝑃𝑃𝑃 𝐼 /𝜒 𝑆𝑆𝑃 𝐼 and (bottom panel) 𝜒 𝑃𝑃𝑃 𝐼𝐼 , 𝜒 𝑆𝑆𝑃 𝐼𝐼 , 𝜒 𝑃𝑃𝑃 𝐼𝐼 /𝜒 𝑆𝑆𝑃 𝐼𝐼 with respect to the orientation angle of the transition dipole from the laboratory Z axis as shown in Figure 4.6. The horizontal dotted black lines on the top and the bottom panels represent the PPP/SSP amplitude ratio of the Lorentzians for air/CBP and CBP/CaF2 interfaces; as obtained from the global fitting procedure. For the air/CBP interface (top panel), the interfacial refractive index is varied from 1.0 ≤ 𝑛 𝐶𝐵𝑃 /𝑎𝑖𝑟 ′ ≤ 1.7 ; whereas for CBP/ CaF2 interface (bottom panel), it is varied in between 1.4 ≤ 𝑛 𝐶𝐵𝑃 /CaF 2 ′ ≤ 1.7; with n′ = 1.7 (green), n′ = 1.6 (black), n′ = 1.5 (blue), n′ = 1.4 (red), n′ = 1.3 (grey), n′ = 1.2 (pink), n′ = 1.1 (nude), n′ = 1.0 (brown). All calculations are shown for 𝜎 = 20. 76 The orientation plots for all other conformers of CBP molecule are shown in the following. 𝜒 𝑃𝑃𝑃 /𝜒 𝑆𝑆𝑃 is plotted with respect to the orientation angle of the transition dipole of the 1600 cm -1 vibrational mode of CBP from the laboratory Z axis; for both air/CBP (𝜒 𝑃𝑃𝑃 𝐼 /𝜒 𝑆𝑆𝑃 𝐼 ) and CBP/CaF2 (𝜒 𝑃𝑃𝑃 𝐼𝐼 /𝜒 𝑆𝑆𝑃 𝐼𝐼 ) interfaces. The horizontal dotted black lines represent the PPP/SSP amplitude ratio of the Lorentzians for air/CBP and CBP/CaF2 interface; as obtained from the global fitting procedure. For the air/CBP interface, the interfacial refractive index is varied between 1.0 ≤ 𝑛 𝐶𝐵𝑃 /𝑎𝑖𝑟 ′ ≤ 1.7 ; whereas for CBP/ CaF2 interface, it is varied in between 1.4 ≤ 𝑛 𝐶𝐵𝑃 /CaF2 ′ ≤ 1.7; with n′ = 1.7 (green), n′ = 1.6 (black), n′ = 1.5 (blue), n′ = 1.4 (red), n′ = 1.3 (grey), n′ = 1.2 (pink), n′ = 1.1 (nude), n′ = 1.0 (brown). Orientation plot for 150 0 dihedral angle conformer of CBP Figure 4.10: Plots of (left panel) 𝜒 𝑃𝑃𝑃 𝐼 /𝜒 𝑆𝑆𝑃 𝐼 and (right panel) 𝜒 𝑃𝑃𝑃 𝐼𝐼 /𝜒 𝑆𝑆𝑃 𝐼𝐼 as a function of orientation angle. The top and the bottom panels represent the distribution width of the tilt angle to be 𝜎 = 5 0 and 𝜎 = 20 0 respectively. 77 Orientation plot for 130 0 dihedral angle conformer of CBP Figure 4.11: Plots of (left panel) 𝜒 𝑃 𝑃 𝑃 𝐼 /𝜒 𝑆𝑆𝑃 𝐼 and (right panel) 𝜒 𝑃𝑃𝑃 𝐼𝐼 /𝜒 𝑆𝑆𝑃 𝐼𝐼 as a function of orientation angle. The top and the bottom panels represent the distribution width of the tilt angle to be 𝜎 = 5 0 and 𝜎 = 20 0 respectively. III. Conclusion In this work, surface selective vibrational sum frequency generation spectroscopy (VSFG) was used to probe the structural order of vapor deposited thin films of CBP at the CaF 2 interface. As mentioned earlier, CBP molecule is centrosymmetric in its optimized geometry (D2 symmetry) and hence, its IR active transitions should be Raman inactive and vice versa. Despite that, the appearance of ~1602 cm -1 mode in both FTIR and Raman spectra (Figure 4.1) indicate the possibility of symmetry breaking of the CBP molecules in the film and the presence of multiple conformational structures as well. Such symmetry breaking leads to the mixing of the degenerate/quasi-degenerate normal modes via anisotropic perturbation at the interface. 33–35 Nonlinear response from both the air/CBP and CBP/CaF2 interfaces contribute to the overall VSFG activity of the molecule. Fresnel factor calculation suggests that the VSFG response is dominated by the air/CBP interface, ~10 times stronger than that of the buried CBP/CaF2 interface. Thickness 78 dependent VSFG intensity and spectral change can be accurately modeled by taking the optical interference effect into account and the orientation of the CBP molecule at both the interfaces were extracted. 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Sci. 2007, 601 (12), 2420–2425. (48) Dürr, A. C.; Nickel, B.; Sharma, V.; Täffner, U.; Dosch, H. Observation of Competing Modes in the Growth of Diindenoperylene on SiO2. Thin Solid Films 2006, 503 (1–2), 127– 132. 82 Chapter 5 Electronic Structure of Liquid Methanol and Ethanol from Polarization-Dependent Two-Photon Absorption Spectroscopy I. Introduction One photon (1PA) and two photon absorption (2PA) spectroscopies are two complementary techniques as they are governed by different selection rules. Information from both techniques together can help provide a complete picture of electronic excited states of a molecule or material. Moreover, the polarization dependence of two-photon absorption spectroscopy gives information about the symmetry of the electronic excited states. 2PA has been widely used in many condensed phase applications. 1–13 However, the 2PA cross sections (σ2PA) of common solvents, which are typically obtained at a single wavelength via the traditional z-scan technique, are sparse in the literature. 14–21 In comparison, broadband spectra analogous (and complementary) to conventional linear UV-visible 1PA spectra are only available for a handful of much larger systems such as retinal, 22 C60, 23 porphyrins, 24–27 ZnS nanocrystals, 28,29 stilbene and phenanthrene, 30–32 and hydroxyphenacyl phototriggers. 33 Previously, we reported the broadband 2PA spectrum of water in the 7-10 eV range. 34 In this paper, we have extended our study to the prototypical alcohols: methanol and ethanol. The main focus will be to analyze the nature of the electronic excited states and how the 2PA spectrum changes due to substitution of one of the hydrogens in water by an alkyl group. The 2PA cross sections of common solvents have been reported by several groups at discrete energies. 16–18 However, a comprehensive understanding of the excited states or even the 83 determination of 2PA band gap is next to impossible with such a few data points. Liquid jet photoelectron spectroscopy experiments on methanol show that the vertical ionization energy (VIE) decreases from 10.94 eV to 9.99 eV on going from gas to liquid phase. 35 In spite of this, liquid methanol can be adiabatically ionized down to its optical absorption edge at ~ 4.7 eV which is ~5.3 eV less than the VIE. 36 A similar phenomenon is true for liquid water. 37–39 Explaining such phenomena demands a better visualization of electronic structure and understanding the fate of the optically bright Rydberg states in the condensed phase. There is abundant literature 40–44 concentrating on the interaction of a Rydberg state with other Rydberg and/or valence states in isolated molecular systems. Describing such interactions has emerged as a challenging problem in electronic structure theory. Failing to accurately quantify the coupling between these states can dramatically affect the calculated electronic properties, such as the transition dipole and the shape of the potential energy surfaces. 45,46 Regarding Rydberg states in condensed environments, pioneering work has been carried out by Chergui and coworkers, who investigate the fate of Rydberg orbitals in condensed media by trapping the NO molecule in both molecular matrices 47,48 and rare gas matrices. 49,50 Optical excitation to a Rydberg state leads to an increase in the spatial distribution of the electrons giving rise to a large change in the molecule-matrix interaction. The gas to matrix shift of absorption energies and the broadening of absorption bands are a function of the size of the matrix cage and the nature of the excited Rydberg orbital (s, p or d type). 51 In the case of NO, lengthening of the fluorescence lifetime of a molecular Rydberg state when going from isolation to condensed media 52 points to the fact that the local structure of the solvent environment has considerable influence on the transition dipole. Furthermore, due to mixing of Rydberg orbitals with the solvent continuum, the characterization of these orbitals below the VIE provides invaluable information on the competing ionization and dissociation channels. 36,53,54 84 Albrecht and coworkers considered liquid benzene and benzene diluted in hexane via two-photon excitation spectra. 55,56 This study pointed that the dominance of a Rydberg excitation between 6 – 7 eV which was well known in the gas phase, also played an important role in liquid solution. The favorable selection rules for 2-photon excitation were important in uncovering this state. To get a better picture of the Rydberg contribution to the electronic structure of other important bulk molecular liquid solvents, the simple sigma bonded liquids methanol and ethanol are amenable to analysis because of the lack of low-lying valence excitations. Here we report on the continuous 2PA spectra of these liquid alcohols in the energy range 6.8-10.2 eV. Two different pump energies (4.6 and 6.2 eV) have been used to cover the region of the spectra. By changing the relative polarization of pump and probe photons, both parallel and perpendicularly-polarized spectra have been reported, along with the polarization ratio (ρ) which is defined as (σpara/σperp), where σpara and σperp are parallel and perpendicular 2PA cross sections respectively. Theoretical 2PA cross sections for transitions from ground to several excited states are calculated for isolated methanol and ethanol molecules and they provide a useful starting point in assigning the nature of the transitions that have significant contribution towards the 2PA spectra. 85 Fig. 5.1. Gas phase 1PA and liquid phase 1PA and 2PA spectra of water, methanol and ethanol. The water gas phase spectrum is digitized from Ref 57 and is converted to ϵ (L mol -1 cm -1 ) using the equation ϵ = (NA/ln(10))σ1PA, where σ1PA and NA are 1PA cross section and the Avogadro constant respectively. The alcohol gas phase spectra are digitized from Ref 58 and converted to ϵ using A= ϵcl equation, where the path length l is 1μm. The water liquid phase 1PA spectrum is from Ref 59 and Ref 60 , and the 1PA spectra of liquid alcohols are from Ref 61 . In (c) the 2PA spectra for water, methanol and ethanol are shown. Water 2PA spectrum is from Ref 34 ; the data for the alcohols are from this work. Spectra recorded with both a 4.6 eV pump and a 6.2 eV pump are shown in the same figure; parallel (blue) and perpendicular polarization (red). The purple and the green arrows show the position of the vertical ionization energies in gas phase 35 and in liquid phase, 62 respectively. The VIE of water vapor is 12.62 eV 35 and is outside the spectral range shown in the above plot. (Insets) For liquid MeOH, the 1PA and 2PA showing detail from 6.5-8.5 eV. 86 Fig. 5.2. Liquid phase 2PA spectra of propanol and butanol. Spectra recorded with both a 4.6 eV pump and a 6.2 eV pump are shown in the same figure; parallel (blue) and perpendicular polarization (red). II. Background: 1PA spectrum of methanol and ethanol The gas phase spectrum of alcohols is a very good starting point for identifying excited electronic states before we delve into the more complicated liquid system. The gas phase 1PA spectra of methanol and ethanol, 58 shown in Fig. 5.1(a), reveal the lowest energy band at ~6.7 eV. This band is very weak in intensity and is due to the 3s←2pz orbital promotion on oxygen (z axis 87 is defined as out-of-plane), as pointed out by Cheng et al. 53 The broadness of this band comes from the dissociative nature of the potential energy surface of the 1A" upper state; although the 1A" state is predominantly Rydberg in nature, it acquires anti-bonding character along the O-H vibrational coordinate due to interaction with states of the same symmetry. 53,63 The oscillator strength of the similar character transition is almost 10 times stronger in the case of water. A similar result is obtained in our ab initio calculation (vide infra), where the oscillator strength of this transition is seen to be reduced from 0.05 in water to 0.004 in methanol. This reduction is the consequence of the delocalization of the 3s Rydberg orbital of oxygen over the alkyl groups of the alcohols which makes the transition moment integral ⟨𝑛 0 |𝑒 .𝑟 |3𝑠 ⟩ decrease with the increasing size of the alkyl group. 63 The first gas-phase 1PA band (3s←2pz) is red-shifted by 0.7 eV as compared to water. The alkyl group, being electron donating in nature, can be considered to push electron density towards the oxygen atom which results in increasing electron-electron repulsion and thereby destabilizes the lone pairs on oxygen (HOMO and HOMO-1). This is consistent with the drop of first vertical ionization potential from water to methanol to ethanol. 64,65 As it can be seen in Fig. 5.1(a), in MeOH, the two clusters of sharp bands in the region of 7.7 – 9.0 eV are transitions from HOMO (2pz) to the 3p Rydberg orbitals. 53,66,67 These lie much lower in energy as compared to those for water (sharp features at > 9.8 eV). 57 In the case of ethanol, the energy splitting of the Rydberg 3p orbitals are very small and the overlapping transitions in this region give rise to a broad feature between 7.5-8.5 eV. 58,68–70 A similar broadening is observed with the addition of alkyl groups in the electronic spectra of aldehydes. 71 A completely different set of spectral features are observed in the 1PA spectrum of liquid methanol, as shown in Fig. 5.1(b). In the literature, the location of the first absorption band (1A" ←1A'), a transition that is clearly observed in the methanol gas phase spectrum at ~ 6.7 eV, is 88 unclear in the liquid or solid phase spectra. Jung et al. 61 proposed that isolated-molecule 1A" ←1A' (3s←2pz) transition is blue-shifted by 1.7 eV and 1.6 eV in methanol and ethanol respectively, which gives rise to a broad band around 8.3 eV in the absorption spectra of alcohols in the liquid phase. In comparison, Kuo et al. 72 assigned the same 8.3 eV band observed in the 1PA spectra of solid methanol recorded at 10 K to the 2A" ←1A' (3px←2pz) transition. Kuo et al. 72 also pointed out that given the small oscillator strength of the 1A" ←1A' transition observed in the gas phase, it is possible that with a large blue-shift, the lowest excitation may overlap with stronger features arising from 2A" ←1A' transition (~8.3 eV) and thus is not clearly discernable. Interestingly, by carefully examining their solid phase MeOH spectra, we noticed a subtle shoulder at ~7.4 eV, but the authors did not comment on it at all in their article. We speculate that the shoulder may originate from 1A" ←1A' transition. However, if our speculation is correct, the blue-shift of the first electronic transition in methanol (~0.7 eV) is significantly less than that reported by Jung et al. (~1.7 eV). 61 In support of our assertion, we note a similar blue shift (0.9 eV) for the first electronic band of water when going from gas to condensed phase (7.3 to 8.2 eV) which has been reported by several groups 34,57,59,60 and is shown in Fig. 5.1(b). We can expect a similar line of argument will be true for liquid ethanol as well. Beyond 9 eV, there is a monotonic increase of the 1PA cross section with the appearance of a shoulder ~9.5 eV. Rydberg transitions with n ≥ 3 contribute to the higher energy region (> ~9 eV), but because of their overlapping nature, it is difficult to assign each transition separately. 53,72 As can be seen from the low lying electronic states of ROH system described above, there is still much to learn about the evolution of the electronic structure in the liquid phase. In this work, 2PA spectroscopy allows us to obtain a better picture of the electronic structure of liquid alcohols and helps us identify the important transitions in the energy range 6.5-10 eV. The 89 key concepts we will be addressing in this paper are as follows: (1) How does the electronic structure of methanol and ethanol deviate from that of water and what is the effect of an alkyl substitution on the intense transitions in 1PA and 2PA? (2) To what extent do allowed electronic transitions shift and broaden upon solvation? In particular, what happens to the lowest electronic transition? (3) How important are the Rydberg transitions in liquid phase and do the transition properties for Rydberg excitations change dramatically upon solvation? III Methods A. Experimental In our experiment, the 7.0 – 10.0 eV energy region is accessed by simultaneous absorption of one deep-UV photon and one UV-visible photon. Both pump and probe are overlapped spatially and temporally on a wire guided gravity jet. 73 Two different pump energies are used: 4.6 eV (266 nm) and 6.2 eV (200 nm). A white light continuum from 1.8 eV (690 nm) to 3.9 eV (315 nm) is used as the probe. The 4.6 eV pump spans a total two-photon energy of 6.7 - 8.5 eV, whereas 6.2 eV pump extends the spectra to 10.1 eV. The generation of pump and probe beams has been discussed elsewhere. 34 At the sample, the 6.2 eV and 4.6 eV pumps were attenuated to 1.5 μJ and 6 μJ respectively, with spatial full width at half maximum (FWHM) of 400-500 μm as determined by the knife-edge technique. The sample was placed at the focal point of the probe continuum (typically 315 < λprobe < 690 nm), where the spatial FWHMs were 60-150 μm across the continuum spectrum. The spot size and the pulse energy of the pump beam are adjusted to avoid transient absorption signal from two-photon ionization of the material. 74 The wavelength dependence of the spot size of the probe beam is compensated mathematically. The thickness of the liquid jet varies from 40-50 μm, with an estimated 20% error, as measured by group velocity delay method. 73 The 90 polarization purity is better than 200:1 across the range of probe wavelengths. The polarization purity of 4.6 eV pump is better than 70:1 as measured by extinction of the light through a calcite polarizer, whereas for 6.2 eV pump, the purity is at least 40:1 as measured using a calibrated stack of 9 silica plates at Brewster angle. After passing through the sample, the broadband continuum was dispersed and projected onto a 256-channel silicon photodiode array. Differential absorbance (Apump-on – Apump-off) is recorded for all the wavelengths in the continuum by measuring the intensity of the probe with and without the pump pulse. The 2PA coefficient β (in cm/W) is calculated as following: 𝛽 = ln (10) 𝑙𝑓 𝐸 𝑝𝑢𝑚𝑝 ∫𝛥 𝐴 2𝑃𝐴 (𝜏 )𝑑𝜏 (1) where 𝜏 is the time delay between pump and probe pulses, 𝛥 𝐴 2𝑃𝐴 (𝜏 ) is the differential absorption with time delay, 𝑙 is the path length of the sample (in our experiment, it is the thickness of the liquid jet), Epump is the energy of the pump pulse in μJ and f is the overlap factor accounting for the different spot size of pump and probe beams. The signal is integrated from negative to positive time delay in order to eliminate the contribution from the cross-phase modulation signal. 75 The absolute 2PA cross section is calculated by 23 𝜎 2𝑃𝐴 = ℏ𝜔 𝑝𝑢𝑚𝑝 𝑁 𝛽 (2) where 𝜔 𝑝𝑢𝑚𝑝 is the frequency of the pump pulse and N is the number density of the liquid sample. 𝜎 2𝑃𝐴 is expressed in Goeppert-Mayer (1GM=10 -50 cm 4 s molecule -1 photon -1 ). 91 B. Computational The two-photon transition moment is given by the following equations: 76 𝑀 𝑏𝑐 𝑘 ←0 = −∑( ⟨𝑘 |𝜇 𝑐 |𝑛 ⟩⟨𝑛 |𝜇 𝑏 |0⟩ Ω 𝑛 0 − 𝜔 1 + ⟨𝑘 |𝜇 𝑏 |𝑛 ⟩⟨𝑛 |𝜇 𝑐 |0⟩ Ω 𝑛 0 − 𝜔 2 ) 𝑛 (3) 𝑀 𝑏𝑐 0←𝑘 = −∑( ⟨0|𝜇 𝑏 |𝑛 ⟩⟨𝑛 |𝜇 𝑐 |𝑘 ⟩ Ω 𝑛 0 − 𝜔 1 + ⟨0|𝜇 𝑐 |𝑛 ⟩⟨𝑛 |𝜇 𝑏 |𝑘 ⟩ Ω 𝑛 0 − 𝜔 2 ) 𝑛 (4) where 𝜇 is the dipole moment operator, Ω 𝑛 0 is the transition energy between ground state ‘0’ and intermediate state ‘n’, 𝜔 1 and 𝜔 2 are the frequencies used to make two photon transitions. The rotationally averaged 2PA coefficient is calculated as: 77,78 𝛽 = 𝐹 30 ∑𝑆 𝑎𝑎 ,𝑏𝑏 𝑎 ,𝑏 + 𝐺 30 ∑𝑆 𝑎𝑏 ,𝑎𝑏 𝑎 ,𝑏 + 𝐻 30 ∑𝑆 𝑎𝑏 ,𝑏𝑎 𝑎 ,𝑏 (5) where S is transition strength matrix; 𝑆 𝑎𝑏 ,𝑐𝑑 = 0.5 ∗ (𝑀 𝑎𝑏 0←𝑘 𝑀 𝑐𝑑 𝑘 ←0 + 𝑀 𝑐𝑑 0←𝑘 𝑀 𝑎𝑏 𝑘 ←0 ) (6) And F, G, H are integer constants that depend on the relative polarization of the two photons. For parallel linearly polarized light, F=G=H=2; for perpendicular linearly polarized light, F= -1, G= 4, H= -1. 78 92 In the case of degenerate photons, the macroscopic 2PA absolute cross section is calculated as: 79 𝜎 2𝑃𝐴 = 𝜋 3 𝛼 𝑎 0 5 (2𝜔 ) 2 𝑐 𝛽 𝑆 (2𝜔 ,𝜔 0 ,𝛤 ) (7) Where α is the fine structure constant, a0 is Bohr radius, ω is the photon energy, c is the speed of light and S (2ω, ω0, Γ) is the line shape function to account for the spectral broadening. In case of non-degenerate photon energies, equation (7) gets slightly modified: 𝜎 2𝑃𝐴 = 2𝜋 3 𝛼 𝑎 0 5 𝜔 𝜎 2 𝑐 𝛽 𝑆 (𝜔 𝜎 ,𝜔 0 ,𝛤 ) (8) where ωσ is the sum of the photon energies. Calculated excitation energies based on a coupled-cluster singles and doubles (CCSD) ground state are generally accurate for both valence and Rydberg states that are primarily single electron promotions with an error typically around 0.1 − 0.3 eV. 80 Electronically excited states can be calculated at a similar level of theory as the CCSD ground state using equation-of-motion (EOM) methods. 81,82 For our purpose, we have used the excitation energy (EOM-EE-CCSD) method with d-aug-cc-PVDZ basis set for calculating the excitation energies for transitions from ground to excited states and the 2PA cross sections associated with each transition in methanol and ethanol. The details of this method have been described elsewhere. 76 The calculated wave functions and the transition energies are used in equations that are formally equivalent to (3) and (4) for calculating the 2PA transition moments. The orientationally averaged 2PA strength β2PA and macroscopic 2PA cross section σ2PA (in GM unit) are calculated for both parallel and perpendicular 93 polarization. All calculations are performed using Q-Chem electronic structure program 83 at the equilibrium geometry of an isolated methanol or ethanol molecule optimized at the CCSD/cc- PVTZ level of theory. IV. Results and Analysis A. Calculation of 2PA cross-section The point group symmetry of isolated methanol and ethanol molecules in their lowest energy conformer is Cs, which has two irreducible representations, A' and A". The 2PA cross sections have been calculated for 42 excited states (up to 11.5 eV) for methanol and 26 states (up to 10.17 eV) for ethanol. The smaller energy range for ethanol is due to increasing computational cost for the larger molecule. The calculated 2PA cross section with 6.2 eV pump is higher than that of 4.6 eV pump. This can be easily understood from equations (3) and (4); as the energy of one of the photons becomes closer to the excitation energy, the denominator decreases giving rise to a higher 2PA transition moment. Fig. 5.2 shows the calculated 1PA oscillator strength and the 2PA cross section for parallel polarization at 6.2 eV pump energy. The 1PA oscillator strength has a maximum at a different energy region as compared to the 2PA cross section. This is a result of the different transition moment integrals in 1PA and 2PA spectroscopy. We also calculated the 2PA polarization ratio corresponding to each transition. For an orientationally averaged sample, a totally symmetric transition has a polarization ratio ≥ 4/3, whereas in case of a non-totally symmetric transition, it is ≤ 4/3. 78,84–86 The ground states for all ROH molecules have A' symmetry. Therefore, a transition to any excited state of A' irreducible representation is totally symmetric and hence the polarization ratio is greater than 4/3. Conversely, a transition from ground to any excited state of A" irreducible representation is a non-totally 94 symmetric transition. Computed results for the total excitation energy (Eex), 1PA oscillator strength (fL), 2PA cross sections for parallel polarization; microscopic (β2PA), macroscopic (σ2PA) and the polarization ratio (ρ) corresponding to each transition are presented in Table 5.1 and Table 5.2 for methanol and ethanol respectively. The absolute 2PA cross-section (σ2PA, in GM) is calculated from the relative 2PA cross section (β2PA, in atomic units) by the equation: 76 𝜎 2𝑃𝐴 = 2𝜋 2 𝛼 𝑎 0 5 (𝜔 1 + 𝜔 2 ) 2 𝛽 2𝑃𝐴 𝑐 ∗ Γ ∗ 10 −50 ∗ 27.2107 (9) where 𝛼 is the fine structure constant, 𝑎 0 is the Bohr radius in units of cm, Γ is the lifetime broadening (taken to be 0.1 eV), 𝜔 1 and 𝜔 2 are the energies of the two photons in eV, and 𝑐 is the speed of light in cm/s. Table 5.1. Calculated electronic transitions of methanol (EOM-CCSD/d-aug-cc-PVDZ) with 6.2 eV pump energy. State Eex (eV) fL (  10 -3 ) β2PA (atomic unit) a σ2PA (GM) a 2PA Polarization ratio ( ρ) a 1A" 6.73 4.0 74 2.5 1.0 2A" 7.90 30.0 36 1.7 1.3 2A' 8.31 26.6 18 0.9 1.4 3A" 8.33 6.4 13 0.7 0.7 3A' 8.75 1.2 28 1.6 1.8 4A" 8.95 0.2 80 4.7 1.0 5A" 9.14 0.8 15 0.9 0.6 4A' 9.28 1.0 37 2.4 2.1 6A" 9.30 0.9 22 1.4 1.3 7A" 9.38 0.1 33 2.1 0.9 5A' 9.44 0.5 330 21.6 11.2 8A" 9.49 5.6 0.3 0.02 0.5 6A' 9.63 22.0 33 2.3 6.7 7A' 9.68 44.3 11 0.8 1.7 95 a 6.2 eV pump energy Table 5.2. Calculated electronic transitions of ethanol (EOM-CCSD/d-aug-cc-PVDZ) with 6.2 eV pump energy. 9A" 9.68 2.3 0.9 0.1 0.9 10A" 9.98 2.1 9 0.6 1.0 8A' 10.15 36.3 13 1.0 11.1 11A" 10.17 2.4 0.1 0.004 0.4 12A" 10.20 10.4 3 0.2 1.3 13A" 10.42 0.5 4 0.3 0.9 9A' 10.47 10.7 8 0.6 5.1 14A" 10.50 0.7 0.4 0.03 1.1 State Eex (eV) fL (  10 -3 ) β2PA (atomic unit) a σ2PA (GM) a 2PA Polarization ratio ( ρ) a 1A" 6.73 2.8 82 2.7 1.0 2A" 7.87 17.0 55 2.5 1.2 2A' 8.07 27.6 33 1.6 1.7 3A" 8.15 19.3 35 1.7 1.2 3A' 8.67 0.9 70 3.9 6.5 4A" 8.85 0.6 7 0.4 0.3 5A" 8.91 1.1 24 1.4 1.3 6A" 9.03 0.8 30 1.8 1.1 7A" 9.08 2.6 13 0.8 0.6 4A' 9.08 0.1 53 3.2 2.1 5A' 9.11 0.3 242 14.8 12.5 8A" 9.33 3.7 1 0.1 0.6 6A' 9.38 10.6 4 0.3 13.8 9A" 9.44 6.8 1 0.1 0.9 7A' 9.47 17.6 162 10.7 87.9 10A" 9.71 2.6 4 0.3 0.9 11A" 9.74 31.9 3 0.2 1.3 8A' 9.77 101.1 21 1.5 7.6 96 a 6.2 eV pump energy Table 5.3. Calculated electronic transitions of methanol (EOM-CCSD/d-aug-cc-PVDZ) with 4.6 eV pump energy. 12A" 9.79 0.2 12 0.9 1.2 9A' 9.85 0.9 58 4.1 10.3 13A" 9.85 0.2 3 0.2 0.7 14A" 9.90 1.1 3 0.2 1.1 15A" 10.01 1.4 5 0.3 1.3 10A' 10.06 3.7 40 3.0 9.8 16A" 10.15 1.7 4 0.3 1.0 11A' 10.17 10.5 5 0.4 33.4 States Eex (eV) fL (  10 -3 ) β2PA (atomic unit) σ2PA (GM) Polarization ratio 1A" 6.73 4.0 20 0.67 1.3 2A" 7.90 30.0 24 1.08 1.3 2A' 8.31 26.6 4 0.22 2.4 3A" 8.33 6.4 2 0.08 1.3 3A' 8.75 1.2 20 1.13 1.8 4A" 8.95 0.2 26 1.56 1.3 5A" 9.14 0.8 8 0.51 1.3 4A' 9.28 1.0 18 1.16 2.1 6A" 9.30 0.9 14 0.89 1.3 7A" 9.38 0.1 17 1.07 1.3 5A' 9.44 0.5 197 12.92 12.9 8A" 9.49 5.6 0.1 0.005 1.3 6A' 9.63 22.0 27 1.82 6.9 7A' 9.68 44.3 9 0.05 1.3 9A" 9.68 2.3 1 0.60 1.6 10A" 9.98 2.1 4 0.30 1.3 8A' 10.15 36.3 12 0.89 11.4 11A" 10.17 2.4 0.05 0.003 1.0 97 Table 5.4. Calculated electronic transitions of ethanol (EOM-CCSD/d-aug-cc-PVDZ) with 4.6 eV pump energy. 12A" 10.20 10.4 2 0.19 1.3 13A" 10.42 0.5 3 0.23 1.2 9A' 10.47 10.7 7 0.58 5.3 14A" 10.50 0.7 0.3 0.03 1.2 States Eex (eV) fL ( 10 -3 ) β2PA (atomic unit) σ2PA (GM) Polarization ratio 1A" 6.73 2.8 24 0.81 1.3 2A" 7.87 17.0 30 1.37 1.3 2A' 8.07 27.6 5 0.24 1.3 3A" 8.15 19.3 9 0.44 1.3 3A' 8.67 0.9 49 2.70 6.4 4A" 8.85 0.6 1 0.06 1.3 5A" 8.91 1.1 9 0.53 1.3 6A" 9.03 0.8 16 0.96 1.3 7A" 9.08 2.6 6 1.36 2.2 4A' 9.08 0.1 23 0.38 1.3 5A' 9.11 0.3 138 8.43 13.8 8A" 9.33 3.7 0.1 0.01 1.3 6A' 9.38 10.6 3 0.22 22.2 9A" 9.44 6.8 1 0.05 1.3 7A' 9.47 17.6 125 8.23 72.1 10A" 9.71 2.6 2 0.17 1.3 11A" 9.74 31.9 2 0.15 1.3 8A' 9.77 101.1 17 1.22 8.4 12A" 9.79 0.2 7 0.48 1.3 98 Fig. 5.3. Calculated 1PA and 2PA transitions for methanol and ethanol at the EOM-CCSD/d-aug- cc-PVDZ level of theory. The 2PA cross sections are calculated using 6.2 eV pump in both cases. The calculated gas phase vertical ionization energy (VIE) for methanol and ethanol are also shown. B. Experimental 2PA spectra and polarization ratios The experimental 2PA spectra of liquid methanol and ethanol are presented in Fig. 5.1(c). The absolute 2PA cross section σ2PA is plotted against the total energy of the two photons. Parallel polarization gives rise to higher cross section throughout the entire spectrum than perpendicular polarization, with a polarization ratio > 4/3 above 7.5 eV. This observation alone points to the fact that the spectrum is mainly dominated by totally symmetric transitions, i.e. transitions to A' upper states. In the overlapping region, around 8.5 eV, the 6.2 eV pump has a higher 2PA cross section than 4.6 eV pump due to pre-resonant enhancement, as expected from equations (3) and (4). 34 The 99 comparison of σ2PA values between our study and the literature is summarized in Table 5.5. The difference in absolute values at 9.4 eV likely originates from the fact that 264 nm photons are further away from resonance. At 7.1 eV, the absolute values reported here are in reasonable agreement with the literature. Table 5.5. Absolute 2PA cross sections (in GM) for parallel polarization at 7.14 eV and 9.4 eV for methanol and ethanol. 2PA energy (photon combination, nm) 7.14 eV 7.14 eV (266/500) a (347/347) b 9.4 eV 9.4 eV (200/387) c (264/264) d methanol 0.15±0.06 0.32 5.1 ± 2.0 1.7 ± 0.2 ethanol 0.30±0.12 0.45 7.6 ± 3.0 3.1 ± 0.3 a This study: 266nm pump and 500nm probe (selected from the continuum); b 2PA coefficient (β, in cm/GW) taken from Ref 16 ; converted to 2PA cross section (σ, in GM) using the following parameters: M methanol=32 g/mol; ρ methanol=0.791 g/cm 3 ; M ethanol = 46 g/mol; ρ ethanol = 0.785 g/cm 3 ; the error was not reported by the author. c This study: 200nm pump and 387nm probe (from the continuum) d 2PA coefficient (β, in cm/GW) taken from Ref 17,18 ; converted to 2PA cross section (σ, in GM) using the parameters shown in footnote b. 10% error was reported by the authors. Fig. 5.3 shows the polarization ratio as a function of total 2PA energy for both 4.6 eV and 6.2 eV pump. The polarization ratio is ~3.0 at 10.0 eV and decreases to ~2.0 at 8.3 eV. Such a trend is preserved in the lower energy region covered by 4.6 eV pump, where polarization ratio decreases from ~2.75 at 8.5 eV to ~1.0 at 7.0 eV. This observation suggests that the contribution from non-totally symmetric transitions becomes more prominent in the lower energy region: i.e., ground to 1A'' state (3s←2pz) and to the 2A'' state (3px←2pz). As can be seen in Fig. 5.3, transitions to equivalent states in water also have low polarization ratios. 34 The overall trend of methanol polarization ratio bears more resemblance to water than higher alcohols, where the polarization 100 ratio spectrum is rather flat (The 2PA spectra of propanol, butanol are shown in Fig. 5.2 and their polarization ratio are shown in Fig. 5.5). 87 For example, in ethanol, the polarization ratio shows less variation within our experimental spectral range; the ratio is ~ 3.0 in the lower energy region and maintains a value greater than 2.5 throughout. This suggests that transitions to the excited states of A" symmetry (non-totally symmetric transition) have relatively small cross section in the 2PA spectra of ethanol, even in the lower energy region. One explanation is that the ethyl group has a larger effect to reduce the molecular symmetry. Alternatively, unlike methanol, the electronic structure in higher alcohols is less dominated at lower energy by transitions involving oxygen- centered orbitals and thus increasingly deviates from water. Fig. 5.4. The experimental polarization ratio (ρ) for water, methanol and ethanol plotted against the total excitation energy with (blue) 4.6 eV and (red) 6.2 eV pump. We observe that in the overlapping region, the polarization ratio with a 4.6 eV pump is higher than that with a 6.2 eV pump for methanol, whereas in the case of ethanol, they are almost the same. For both alcohols, we identify from the electronic structure calculations that transitions to 2A' and 3A" excited states are most likely to contribute to the overlapping region of the spectra. For methanol, when going from 4.6 to 6.2 eV pump energy, the theoretically predicted polarization ratio decreases by a factor of 1.6 and 1.9 respectively for these two states. Whereas for ethanol, 101 the polarization ratio increases by a factor of 1.3 for 2A' state and decreases by a factor of 1.1 for 3A" state when a higher energy pump photon is used. These two competing factors cancel out, resulting in identical polarization ratio of EtOH in the overlapping region for both 4.6 and 6.2 eV pump excitation. We can compare the polarization ratio at 7.92 eV to that obtained by Rasmusson et al. 88 using an identical photon combination (Table 5.6). Our experimental values are close to, but consistently higher than their values. This discrepancy may be explained by the fact that the measurement by Rasmusson et al. 88 includes a considerable amount of background transient absorption signal, most likely due to 2- (pump-) photon ionization of the solvent. Since the absorption due to photoionization photoproducts (e.g., solvated electrons) is isotropic, the background signal will reduce the 2PA polarization ratio. In light of this, it is worthwhile to stress that using appropriate pump irradiance to eliminate transient absorption (pump-probe) signal is essential for accurately measuring 2PA polarization ratios. Table 5.6. Linear polarization ratio at 7.92 eV (266nm+380nm) for methanol and ethanol Methanol Ethanol This work a 1.9 3.2 Rasmusson b 1.5 3.1 a 266nm pump and 380 nm probe (selected from the continuum). b Data taken from Ref 88 . 102 Fig. 5.5. The experimental polarization ratio (ρ) for propanol and butanol plotted against the total excitation energy with (blue) 4.6 eV and (red) 6.2 eV pump. V. Discussion A. Characterization of excited states: 1PA vs 2PA spectra An initial characterization of the excited states can be gleaned from a natural transition orbital (NTO) analysis - identifying the orbitals participating in a particular transition. The NTOs for important transitions in isolated methanol and ethanol are displayed in Fig. 5.4 and Fig. 5.5 respectively. Several key parameters 89–91 related to the excited state wavefunctions are reported in Table 5.7. They include the absolute mean separation |𝑑 ℎ→𝑒 ⃗⃗⃗⃗⃗⃗⃗⃗⃗ | between the average position of electron and hole, the size of the electron (σe) and hole (σh), and the change in size of the 103 wavefunction following the excitation, Δ<R 2 >. 31 For transitions to valence states, Δ<R 2 > is small, but a large value is expected for transitions to Rydberg states. The NTO participation ratio (PRNTO), as shown in equation 10, indicates how many significant NTO pairs are needed to describe a transition. 𝑃𝑅 𝑁𝑇𝑂 = (∑ 𝜆 𝑖 𝑖 ) 2 ∑ 𝜆 𝑖 2 𝑖 (10) Where, λi is the weight of a respective configuration obtained by the singular value decomposition of the transition density matrix. 90,91 Table 5.7. Key parameters from NTO calculations for methanol and ethanol. Excitations to the states labeled in red and black are bright in 2PA and 1PA respectively. State E ex (eV) |𝒅 𝒉 →𝒆 ⃗⃗⃗⃗⃗⃗⃗⃗⃗ | (Å) σ e (Å) σ h (Å) σ e - σ h (Å) PR NTO Δ<R 2 > Methanol 1A" 6.73 1.27 2.51 1.18 1.33 1.01 6.0 4A" 8.95 2.23 4.86 1.20 3.66 1.01 27.5 5A' 9.44 0.55 5.33 1.22 4.11 1.02 25.7 2A" 7.90 2.05 3.42 1.25 2.17 1.01 12.2 2A' 8.31 0.74 4.19 1.22 2.97 1.01 15.2 7A' 9.68 1.78 4.62 1.34 3.28 1.4 20.9 8A' 10.15 0.42 4.16 1.33 2.83 1.02 14.6 Ethanol 1A" 6.73 1.44 2.58 1.22 1.36 1.01 8.0 2A" 7.87 1.92 3.69 1.34 2.35 1.03 12.4 3A' 8.67 1.66 2.90 1.42 1.48 1.04 8.8 5A' 9.11 0.71 6.06 1.29 4.77 1.03 33.0 7A' 9.47 1.63 4.02 1.49 2.53 1.16 14.3 2A' 8.07 0.71 4.20 1.29 2.91 1.01 15.1 8A' 9.77 0.5 4.52 1.49 3.03 1.05 16.5 104 The first distinct difference between the 1PA and 2PA spectra of methanol is the presence of a shoulder in the 2PA spectrum at around 7.6 eV, but not in 1PA; as shown in the inset of Figs. 1(b) and 1(c). This can be readily explained by comparing the calculated 1PA and 2PA transition strengths in Fig. 5.2. In the 6.5 - 9 eV region, the ratio of the 1PA oscillator strengths for 2A" ← 1A' and 1A" ← 1A' transitions is 8, whereas the intensity ratio of the same transitions is 0.7 in the 2PA spectrum. Thus, the relative intensities of the two transitions are predicted to be different by a factor of 11 between the 1PA and 2PA spectra. As a result, the lowest excitation band (1A" ← 1A') centered at 6.7 eV should be revealed in 2PA spectrum, whereas the existence of this first transition is buried in the tail of the strong band centered around 7.9 eV in the 1PA spectrum. Another interesting point is observed from the theoretical calculation of 1PA and 2PA for methanol. In the 2PA spectrum, the 5A' ←1A' transition has the highest cross section, whereas the 7A' ← 1A' transition accounts for the highest oscillator strength in the 1PA spectrum. Clearly, different transitions are favored in 2PA than 1PA, which is in line with the different selection rules. In 2PA, the 5A' ←1A' transition is mainly governed by the promotion of the electron from 2a" orbital (HOMO) to 5a" orbital (LUMO+10), as shown in Fig. 5.4. The HOMO in methanol mainly consists of the nonbonding (out of plane) O(2pz) mixed with C(2pz). The 5a" orbital is the O(3pz) mixed with C(3pz). Therefore, the 5a" ← 2a" promotion, in essence, is pz-pz transition on each atom, a transition with no change in orbital angular momentum that is unfavorable in 1PA, but strongly allowed in 2PA. Equivalently, one can consider these molecular orbitals centered between the carbon and oxygen and as both resemble d orbitals around this center, there is no change in angular momentum. On the other hand, the 7A' ← 1A' transition, which shows the highest 1PA oscillator strength, can be described by two participating NTO pairs. The relative contribution (λ) of each NTO pair 31 is shown in Fig. 5.4. The NTO pair which has the major contribution towards 105 this transition involves exciting an electron from orbital 7a' (HOMO-1) to 14a' (LUMO+8). The HOMO-1 has mostly the (in plane) character of O(2py) mixed with C(2py) and 1s orbitals of the methyl hydrogen atoms. The 14a' orbital has mainly ‘3s’ Rydberg character, so the 14a' ← 7a' promotion is a 3s ← 2py type transition with 1 change in angular momentum that is favorable for 1PA. For ethanol, the 5A' ← 1A' transition has the highest 2PA cross section and originates from a pz-pz transition like methanol; whereas the 8A' ← 1A' transition has the highest 1PA oscillator strength and involves promoting an electron from in-plane 2py orbital to the Rydberg 3s orbital. This observation leads us to conclude that although the transitions in methanol and ethanol are mainly governed by the molecular selection rules applied to the C s point group, transitions involving Rydberg orbitals can be realized by merely considering the atomic selection rule. 106 Fig. 5.6. NTOs corresponding to the strong 1PA and 2PA transitions in methanol. An isovalue of 0.015 is used for rendering orbital surfaces. The yellow arrow points to the position of the oxygen atom. 107 Fig. 5.7. NTOs corresponding to the strong 1PA and 2PA transitions for ethanol. An isovalue of 0.015 is used for rendering orbital surfaces. The yellow arrow points to the position of the oxygen atom. B. Comparing 2PA spectra of water, methanol and ethanol A comparison of the 2PA spectrum of H2O with those of MeOH and EtOH reveals several significant points. The first 1PA band of water is very weak in 2PA as reported by Elles et al. 34 But in case of MeOH and EtOH, this transition gains intensity. This can be reconciled by considering the contribution of multiple atoms to the relevant molecular orbitals in alcohols. It has been shown by experimental and theoretical studies 92–94 that the HOMO of methanol is a combination of 70% oxygen p character and 30% carbon p and hydrogen s characters. Our calculations also confirm these results. Further, the LUMOs of the alcohols also have contributions from both oxygen and carbon. 53,58,95 As a result, the atomic selection rules that describe the electronic transitions in water surprisingly well is now relaxed, and the molecular selection rule in the Cs framework starts to play an important role making the lowest energy 3s ← 2pz transition 108 acquire considerable 2PA cross section. This effect helps bringing the 2PA threshold energy down to ~6.9 eV in methanol and ethanol compared to that of water (~7.8 eV), as shown in Table 5.8. An important consequence of the higher 2PA threshold in water is an almost 1 eV broader window for 2PA spectroscopy of dissolved solutes compared with the alcohols. Table 5.8. 1PA and 2PA absorption thresholds of liquid water, methanol and ethanol (in eV) water Methanol Ethanol 1PA(99%T) ~6.4 a 5.6 b 5.6 b 2PA(99%T) c 7.8 6.9 6.9 a Data taken from Ref 96 . The extinction coefficient at 6.4 eV is ~2.25×10 -4 M -1 cm -1 , corresponding to a transmission slightly greater than 99% in a 0.1 cm cell. b Data taken from Ref 97 . The thresholds were the energies where the transmission of the liquid was ~99% in a 0.1 cm cell. c This study. The 2PA thresholds are the energies where the transmission of the probe is ~99% (equivalent to absorbance of ~4 mOD) in a 0.1 cm path length with at peak irradiance of the 267 nm pump pulse = ~100 GW/cm 2 . This is estimated based on the energies where the absorbance was ~0.2 mOD with a ~50 μm pathlength. It was shown in the case of water 34 that the transition from the ground state to the 3 1 A1 excited state accounts for the highest cross-section below ~11 eV in the 2PA spectrum. This transition (2b1 ← 1b1), situated at 10.23 eV, involves promoting an electron from 2pz orbital of oxygen to 3pz orbital (z axis is out of plane). For both MeOH and EtOH, the 5A' ← 1A' transition shows the maximum 2PA cross section and they too are due to 3pz ← 2pz excitation. This means that substituting one of the hydrogens in water by a methyl or an ethyl group does not significantly change the nature of the dominant transition in this region of the 2PA spectrum. As shown in Table 5.7, the Δ<R 2 > value for this transition is quite large for both methanol and ethanol (25.7 and 33.0 Å 2 , respectively), reflecting the diffuse 3pz orbitals. Thus, totally symmetric transitions to Rydberg-like states dominate the 2PA spectra of water and alcohols. 109 C. Simulation of 2PA spectrum Simulating fully condensed phase 2PA spectra for an extended liquid, just as for 1PA electronic spectra, is a state-of-the-art problem in electronic structure theory. 92–96 Although computations that accurately include the influence of the environment are possible for solutes, 98– 103 the quantum mechanical treatment of a large number of solvent molecules, without edge effects or deficiencies in TD-DFT, is beyond the scope of current electronic structure. Instead, we calculate the 2PA properties of the isolated gas-phase molecule as a guide for interpreting the spectroscopy of the liquid. Considering the single molecule as a perturbed central chromophore has been shown to be a reasonable starting point for water. 34 There are several effects that are important to consider in translating gas phase calculations to describe the liquid phase, including the shifting of transition energies due to local environment around (or specific interactions with) the central molecular chromophore, and the energy broadening of each transition. An important consideration is the fate of Rydberg orbitals in a condensed medium. 39,47,104 A potential problem is that different excited states are likely to behave differently upon solvation. We start with the excitation energies and 2PA cross sections for transitions to the set of higher excited states obtained from the EOM-CCSD calculations as described in Fig. 5.3 and Tables 5.1-5.4. A simulation of the parallel 2PA spectra of methanol with 4.6 eV pump is shown in Figs. 5.8(a) and 5.8(b). States up to 10A" (9.99 eV) have been included in this simulation and the resulting spectrum in Fig. 5.8(a) is obtained directly from the calculated results without any adjustments to intensities or positions. Normalized Gaussians are centered around the calculated excitation energies and each transition is, somewhat arbitrarily, given a fixed Gaussian width, ~2.0 eV FWHM. The simulation accurately predicts the mostly featureless rise of the 2PA cross section 110 with increasing energy up to ~7.5 eV, but fails to reproduce more subtle features of the experimental spectra. The simulation in Fig. 5.8(b) is obtained by keeping the calculated positions and cross sections of all electronic transitions unchanged except for applying a 0.9 eV blue-shift to the position of the first electronic transition, 1A" ←1A'. The justification for such a large shift comes from the previous work, 34 where there is good agreement that the first electronic transition in water blue-shifts by 0.9 eV when going from gas to liquid. 59,67 For both water and methanol, the first transition is due to the promotion of an electron from 2pz to 3s Rydberg orbital of oxygen, so it is reasonable to expect such a blue shift in methanol too. Interestingly, shifting the energy of the lowest transition alone is sufficient to reproduce most of the subtle curvature in the experimental 2PA spectra of methanol in this spectral region, as shown in Fig. 5.8(b). The simulation gives even better agreement by reducing the bandwidth to 1.1 eV FWHM. Such a blue shift of low-n Rydberg transitions is consistent with the observation by Vigliotti et al. for NO in rare gas matrices. 51 We compare this reduced bandwidth and 0.9 eV shift of the lowest transition energy with the 1PA spectrum of methanol below. It is worth mentioning that the calculated absolute 2PA cross sections (σ2PA), as in Table 5.1-5.4, are different from the experimental values, by a factor of ‘3’ for methanol and ‘1.4’ for ethanol. Such discrepancy in absolute magnitude may result from both uncertainty in the experimental parameters such as laser spot size, and the simplicity of the theoretical spectral model. Therefore, a free scaling parameter is applied to match the overall spectral intensity in the simulation with experiment. 111 Fig. 5.8. (a) Simulation of 4.6 eV pump 2PA spectrum using EOM-CCSD transition energies and strengths; the transition widths are 2.0 eV FWHM; (b) result obtained by applying 0.9 eV blue shift to the lowest 1A" ←1A' transition; the transition widths are 1.1 eV FWHM (c) simulation of the 6.2 eV pump 2PA spectrum using the EOM-CCSD results; the transition widths are 1.4 eV FWHM; (d) simulated spectrum obtained by blue-shifting the first transition by 0.9 eV; 4A"←1A' and 12A'←1A' transitions are blue-shifted by 1.2 eV and all other transitions by 0.33 eV; peaks modeled with Gaussian FWHM of 1.0 and 1.75 eV. Green dotted lines correspond to an identical simulation except 2A'←1A' and 12A"←1A' transitions are now blue shifted by 1.2 eV. Simulations of the parallel 2PA spectra with 6.2 eV pump are shown in Fig. 5.8(c) and 5.8(d). States up to 12A' (10.91 eV) have been included in this simulation. As before, the simulation shown in Fig. 5.8(c) is obtained by using the calculated EOM-CCSD results without any adjustment in position or cross section of any transition. The simulation does not accurately reproduce the experimental intensity distribution above ~9.3 eV. While the experimental spectrum shows a monotonic rise in cross section from 8.3 eV to 10.2 eV, the simulation shows a maximum at around 9.4 eV, with the cross section decreasing at higher energy. The decreasing cross section 112 at higher energy in the simulation is not simply a result of truncating the calculations, because the calculated transition energies extend 0.8 eV above the energy range shown. To understand how the electronic energy levels are shifted from gas to the condensed phase, the absorption spectra of solid methanol is a good starting point. Only three Gaussians are necessary to reproduce the experimental absorption spectrum of solid methanol at 10K, as reported by Kuo et al. 72 The first two Gaussians are centered at around 8.45 and 10.5 eV with bandwidths of 1.25 eV and 3 eV, respectively. We followed the same procedure to fit the liquid 1PA spectrum of methanol measured by Jung et al. 61 Simulation of the liquid spectrum also requires three Gaussians, centered at 8.15, 10.05 and 11.66 eV, with bandwidths of 0.97 and 2.07 eV, respectively, for the first two Gaussians. Comparing the peak positions obtained by fitting the experimental solid and liquid phase 1PA spectra of MeOH to the results of our gas phase EOM- CCSD calculations, as shown in Fig. 5.3 (top), clearly reveals that in both cases, all strong transitions in 1PA are blue-shifted. Another important result revealed by this fitting exercise is that the Gaussian width increases almost by a factor of 2-2.5 from the lower to the higher energy region of the spectra. An improved simulation of the experimental 2PA spectra can certainly be done by incorporating all of these factors. Moreover, careful inspection of the NTO plots for MeOH shown in Fig. 5.6 reveals that the upper orbitals of the 4A"←1A' and 12A'←1A' transitions are identical and have significant Rydberg 3s character centered on oxygen. The Rydberg character is similar to the upper orbital of the first transition (1A"←1A'). Therefore, it follows that 4A"←1A' and 12A'←1A' transitions will have a similar blue-shift upon solvation. We account for these effects in the simulation of the 6.2 eV 2PA spectrum in Fig. 5.8(d), where the first transition (1A"←1A') is blue-shifted by 0.9 eV, the 4A"←1A' and 12A'←1A' transitions are blue-shifted by 1.2 eV, and all other transitions are given a uniform blue-shift of 0.33 eV (see below). States up to 9 eV are 113 given similar Gaussian width (~ 1.0 eV) and beyond that a different Gaussian width (~1.75 eV) is applied for the rest of the transitions. Now, to provide some confidence as to how reasonable the shift of the electronic states and the adoption of two different bandwidths are, we simulate the liquid 1PA spectra using the gas phase 1PA oscillator strengths calculated by EOM-CCSD method and keeping the same set of contraints used for the 2PA simulation. As shown in Fig. 5.9(a), the simulation does not immediately reproduce the experimental 1PA spectrum well. But if we simply allow a different blue-shift (1.2 eV) for the 2A'←1A' transition, the experimental spectra and the simulation show remarkable resemblance. Moreover, as shown in Fig. 5.6, the upper orbital corresponding to 12A"←1A' transition is the same as that of 2A'←1A' transition. Hence, a similar blue-shift is expected for the 12A"←1A' transition as well. The justification for a larger shift for the 2A'←1A' and 12A"←1A' transitions is discussed later. Keeping all these factors in mind, the revised 1PA simulation in Fig. 5.9(b) includes blue- shifts of 0.9 eV for the lowest transition, 1.2 eV for the transitions to 2A', 4A", 12A" and 12A', and 0.33 eV for all other transitions. States up to 9 eV have a Gaussian width of 1 eV, and the rest are given a Gaussian width of 1.75 eV. Now to check the consistency, these parameters are used to re-simulate the 2PA plots. The revised 2PA simulations are shown as the green dotted lines in Figures 5.8(b) and 5.8(d), and they are little changed from the earlier simulations (red, blue). This again reflects the complementarity of the transitions having significant intensity in 1PA versus 2PA spectra. The simulation procedure clearly establishes our hypothesis that the first electronic excited state blue-shifts from gas to liquid phase; the blue-shift (0.9 eV) is almost half of that predicted by Jung et al. 61 (1.7 eV). This smaller blue-shift provides an explanation for the shoulder at ~7.4 eV that we observed in the solid-phase methanol spectrum measured by Kuo et al. 72 114 Fig. 5.9. Simulation of the1PA spectrum of liquid MeOH (a) by using the same parameters used to simulate Fig 6(d); (b) now with selected transitions (2A'←1A' and 12A"←1A') also given a different blue-shift compared to the remaining transitions. We have previously shown that the polarization ratio is a sensitive tool to analyze experimental 2PA spectra, especially when there are a number of overlapping transitions in the spectral range. 34 If there is a single transition that dominates the experimental 2PA spectrum, then the polarization ratio has a constant value. Any change in the polarization ratio in the spectral region signifies contributions from additional transitions. For methanol, as we have noted, the polarization ratio increases monotonically from 7 eV to 10 eV. Fig. 5.10(a) and 5.10(b) show the simulated polarization ratio for 4.6 eV and 6.2 eV pump photons, respectively, using polarization- dependent intensities for each transition from the EOM-CCSD calculations (equation 5). Thus, we use the simulation parameters reported above to simultaneously reproduce both the parallel and perpendicular polarized 2PA spectra, as well as the 1PA spectrum of liquid methanol. Thus, the polarization simulation helps us evaluate the degree of shifting of the electronic states and their broadening upon solvation. It should be kept in mind that the reported shift and broadening of the electronic states are only meant to capture the overall picture of the molecule upon solvation. The shift and the broadening of each individual excited state likely differs; determination of these properties from first principles is beyond the scope of this paper. 115 Fig. 5.10. Simulation of polarization ratio (ρ) spectrum of methanol for (a) 4.6 eV and (b) 6.2 eV pump. The excitation energies have been shifted and use two different Gaussian widths, as described in the text. In the case of ethanol, calculation of the 2PA cross section is limited to states up to 10.15 eV. To accurately reproduce the rising edge of the experimental spectrum beyond 10 eV, a Gaussian centered around 10.95 eV is added to the simulation (with a polarization ratio of 2). Every calculated state is given a uniform blue-shift of 0.2 eV except the first state which is given a blue-shift of 0.9 eV. States up to 9 eV are broadened by 0.7 eV and the rest have a broadening of ~2.6 eV. We suspect that this larger width for the higher transitions (as well as the arbitrary additional peak) is necessary simply because of the truncation of excitations computed for EtOH. The difference, however, in the lower energy transition linewidth is significant; methanol has ~50% broader transitions than ethanol. To rationalize this difference in the lower energy part of the spectrum, we refer readers to the partial radial distribution functions, 𝑔 𝑂𝑂 (𝑟 ), obtained by molecular-dynamics (MD) simulations of liquid methanol 105,106 and ethanol 107 at room temperature (298K). As a general trend, the first maxima corresponding to 𝑔 𝑂𝑂 (𝑟 ) function appears at the same position in both methanol and ethanol; but the height of the maximum is greater in case of ethanol, 107 even though the molecular number density of methanol at 298K (1.48 ∗ 10 −2 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 Å −3 ) is higher than that of ethanol (1.03∗ 10 −2 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 Å −3 ). 107 This leads to 116 the conclusion that at room temperature, ethanol is more structured than methanol; for identical character electronic transition this should give rise to a smaller inhomogeneous transition linewidth in the case of ethanol, exactly as required in our spectral simulation. The simulated 2PA spectrum and the polarization ratio of EtOH are shown in Fig. 5.11. Although our simulation reproduces the experimental 2PA spectra with 4.6 and 6.2 eV pump quite well, it fails to capture some subtle features in the polarization ratio. For example, the simulation does not reproduce the slight dip in experimental polarization ratio around 9.5 eV. This again emphasizes the fact that unlike the absolute one- and two-photon absorption spectra, the 2PA polarization ratio is an extremely sensitive experimental quantity; an accurate simulation of the polarization ratio requires knowledge of the shift and broadening of every transition. Nevertheless, the overall trend in polarization ratio helps us determine the transitions that contribute the most to the experimental 2PA spectra, especially when there are several overlapping transitions. Fig. 5.11. Simulation of parallel 2PA spectra of ethanol; for (a) 4.6 and (c) 6.2 eV pump. Simulation of ethanol polarization ratio (ρ); for (b) 4.6 and (d) 6.2 eV pump. 117 Finally, we return to the fate of Rydberg transitions upon solvation. Although several experimental 49–52,55,56,104,108 and theoretical works 45,46 have been carried out to address this problem, the issue of Rydberg states in the condensed phase is very much an open question. However, our simulations of the spectra for liquid methanol and ethanol provide important clues based on the magnitudes of the shifts assigned to different electronic transitions from the isolated molecule reference. For both alcohols, the first excited state originating from a 3s ← 2pz transition requires blue-shifting by a relatively large 0.9 eV – we speculate this is because the Rydberg 3s orbital is more compact than higher-lying and higher angular momentum Rydberg orbitals, experiencing particularly large Pauli repulsion. 34 This observation is consistent with the blue- shifting of low-n Rydberg transitions for NO in rare gas matrices observed by Vigliotti et al. 51 These authors comment that the n=1 exciton, i.e., the exciton obtained following an excitation to the lowest Rydberg orbital, is somewhat different from the remaining states (n ≥ 2), as it has a smaller characteristic radius than the solid-state unit cell. 51 In methanol, we applied a large blue shift to transitions to the 2A', 4A" and 12A' states, in addition to the first excited state. 1 Inspecting the NTO plots corresponding to 4A"←1A' and 12A'←1A' transitions, we notice that in both cases, the upper orbitals also have a significant amount of ‘s’ character centered on oxygen. These transitions have considerable 2PA cross section for an isolated molecule, and shifting these states are absolutely necessary to reproduce the experimental 2PA spectra and the polarization ratio in the liquid. Since the oxygen atom acts as a H-bond acceptor, it makes sense that charge density centered on oxygen will face enhanced Pauli repulsion from closer-coordinated solvent, leading to a larger blue-shift upon condensing. On the other hand, the 2A'←1A' transition also needs to be 1 The 12A"←1A' transition was also shifted by ~1eV to be consistent, as the upper orbital for this transition is the same as 2A' ←1A' transition. 118 blue-shifted by ~1 eV to simulate the liquid 1PA spectrum of methanol. NTO plots show that the upper orbitals corresponding to these transitions are of ‘3p’ type, where ‘3p’ orbitals centering on carbon and oxygen constructively interfere with each other. While we cannot currently rationalize this large shift, interestingly such a large shift is not required for the similar transitions in ethanol. To understand the latter point, we refer to Table 5.7. NTO calculations provided a number of excitation characteristics including the size of the electron (σe) and hole (σh), the absolute mean separation between the average position of electron and hole (|𝑑 ℎ→𝑒 ⃗⃗⃗⃗⃗⃗⃗⃗⃗ |), and the change in size of the wavefunction following the excitation (Δ<R 2 >). In going from methanol to ethanol, for a transition involving similar lower and upper orbitals (as can be seen from NTO plots), all these parameters (σe, σh, |𝑑 ℎ→𝑒 ⃗⃗⃗⃗⃗⃗⃗⃗⃗ |, Δ<R 2 >) increase in magnitude which is consistent with the increase in the size of the molecule. The 2A'←1A' transition, however, does not follow this trend. The size of the electron (σe) corresponding to this transition is almost the same in both methanol and ethanol; 4.19 and 4.20 Å , giving rise to the Δ<R 2 > value of 15.2 and 15.1 Å 2 ; and the mean electron-hole separation (|𝑑 ℎ→𝑒 ⃗⃗⃗⃗⃗⃗⃗⃗⃗ |) of 0.74 and 0.71 Å respectively. This leads to a relatively smaller Pauli repulsion for the Rydberg electron in ethanol than methanol, leading to a smaller blue shift for ethanol. In ethanol, the spectrum can be reproduced from the isolated molecule calculation simply by blue-shifting all the states equally by 0.2 eV, except the first transition which, as for water and methanol, has a larger blue shift (0.9 eV here). Unlike methanol, no corrective shift is needed for any other transition, even for the transitions having an apparent ‘s’ character on the oxygen atom. We suspect that is due to the increase in the size of the alkyl group, where, increasing carbon orbital character is involved in the upper state. Only the first transition preserves the majority O(3s) character. For transitions involving higher Rydbergs, the generalized blue-shift for ethanol 119 (relative to gas phase) is less than that for methanol (0.33 eV). These Rydberg states of ethanol are more delocalized over the molecular framework, and from Table 5.7 we can see that they also reach further into the solvent. Such diffuseness results in less shielding of the cationic core in ethanol leading to greater polarization stabilization. Moreover, a decrease in the number density going from methanol to ethanol contributes to smaller Pauli repulsion of the excited electrons to the Rydberg orbitals. 109 These two effects explain a smaller blue-shift in ethanol as compared to methanol. Alcohols have proven to be ideal candidates for studying the effect of solvation on Rydberg transitions, as their excited states in the gas phase are mainly of Rydberg type because of the lack of π bonding in these molecules. In case of both alcohols studied here, the Rydberg states are central in characterizing the one and two photon spectra, even at the lowest transition energies. For example, our calculations show that the highest cross-section 2PA transition (5A' ← 1A') involves dominant Rydberg character. A related transition can be identified in the liquid 2PA spectrum, based on the high polarization ratio that is required to reproduce the experimental 2PA polarization spectrum. It is fascinating that knowledge of the isolated molecule Rydberg states which contribute high cross-section transitions to both the 1PA or 2PA spectrum, can provide a reasonable description for the electronic structure of the bulk liquid. We believe this justifies our approach in using a gas phase electronic structure as a starting point to elucidate the nature of the transitions contributing to the condensed phase electronic spectra. VI. Conclusion Broadband 2PA spectra for liquid methanol and ethanol have been presented for the first time. 2PA spectra are also simulated from first principles for isolated methanol and ethanol 120 molecules at the EOM-CCSD/d-aug-cc-PVDZ level of theory. The calculations form the basis for assigning the relatively featureless liquid phase 2PA spectra but are particularly helpful in interpreting the polarization ratio. Relatively modest modifications of the computed in vacuo properties are necessary to reproduce the general features of the experimental spectra. For both the alcohols, the first excited state due to 3s ← 2pz transition is blue-shifted by 0.9 eV as the upper Rydberg 3s orbital experiences Pauli repulsion due to confinement by the solvent shell surrounding a single alcohol molecule. 34 Such shift was known for water, but was never confirmed for alcohols. As mentioned before, there was an apparent contradiction in the literature 61,72 concerning the gas to condensed phase shift of the first electronic state in methanol. Our simulation of the 2PA spectra suggests that the 0.9 eV blueshift for the first transition in alcohols indeed mirrors the similar shift in water. Among all the other transitions, those with an upper orbital having ‘s’ character centered on the oxygen atom experience a similar blueshift of ~1 eV upon solvation. This is consistent with literature observations for the diatomic NO in solid matrices. 48,51 In contrast, our result indicates that an upper ‘s’ orbital centered or spread onto carbon does not experience as large of a blue-shift. This can be rationalized as follows: as the two oxygen lone pairs take part into hydrogen bonding, promotion of an electron from the ‘out-of- plane’ O(2pz) orbital to a primarily O(3s) Rydberg orbital (with a more spherical electron distribution) increases electron density in the direction where the hydrogen bonding is present. Therefore, the upper state electron wavefunction experiences large Pauli repulsion that gives rise to such large blue shifts. Transitions involving promotion of an electron into ‘p’ or ‘d’ type diffuse Rydberg orbitals, experience a smaller blue shift with correspondingly less Pauli repulsion and greater stabilization of their cationic core by the surrounding polar molecules. The higher-lying electronic states in ethanol experience relatively less destabilization, leading to less blue-shifted 121 transitions (0.2 eV) as compared to methanol (0.33 eV). For the bigger alkyl substituent, the Rydberg state is more delocalized over the molecular framework and the cation core is less screened for polarization stabilization by the surrounding dielectric. Ab initio calculations show that the most intense peaks in the 2PA spectra correspond to the 3 1 A1 transition in water and the 5A'←1A' transition in methanol and ethanol; and all of them are due to totally-symmetric 2pz-3pz excitations. This means that substituting one of the hydrogens in water by a methyl or an ethyl group does not change the nature of the strongest transition dominating the 2PA spectra. Nevertheless, the calculated absolute 2PA cross-section for this transition drops from ~22 GM in MeOH to ~15 GM in EtOH. This is because, ethyl group has a larger effect to reduce the molecular symmetry, leading to a larger deviation of 2pz-3pz transition from the ‘totally-symmetric’ regime. However, over the entire spectral region, the absolute 2PA cross section increases from methanol to ethanol. This points to the fact that an increasing number of electronic states contribute to the overall 2PA spectra, with larger alkyl groups thus compensating for the apparent decrease in cross-section of the most intense transition. We observe that the excitation-energy variation of the polarization ratio is similar for water and methanol. The decrease in polarization ratio in the lower energy region results from an increasing contribution of the non-totally symmetric transitions (transitions to A" states; for example, 1A"←1A'). On the contrary, for ethanol (and longer alkyl substituents), the polarization ratio starts at ~3 in the lower energy region and shows less variation in the experimental window maintaining a value > 2.5 throughout. This suggests that transitions to the excited states of A" symmetry have relatively small cross sections in the 2PA spectra of higher alcohols, and the 2PA spectrum is dominated by totally symmetric transitions (transitions to A' states), even in the lower energy region. Such a difference in polarization ratio helps us draw a very important conclusion. 122 The electronic structure in higher alcohols, unlike methanol, is less dominated at lower energy by transitions involving oxygen-centered orbitals. The ethyl group blends in more carbon character to the lower and upper orbitals involved in the relevant electronic transitions and thus increasingly deviates from water. 87 This provides a good demonstration of the value of the polarization ratio in 2PA spectroscopy and how sensitive it is in capturing the character of electronic transitions in 2PA spectra. Needless to say, the electronic structure of alcohols is much more complicated in the higher energy region, especially in the region approaching the ionization continuum. We can however use the gas phase ab initio calculations and NTO analysis to speculate on the nature of the excitations across the recorded spectra. First, it is important to emphasize that the intense Rydberg transitions in the alcohol spectra fall well within our experimental spectral range, even after experiencing a general blueshift upon condensation to bulk. Removing these Rydberg transitions from the simulation, or drastically changing their cross-section from isolated molecule calculations results in a poor “fit” of the experimental spectra and the polarization ratio. This emphasizes the persistence of transitions in condensed phase that resemble the properties of the Rydberg transitions in the gas phase. Our work shows that the Rydberg-like states seem to preserve their identity and the high cross-section of their transitions from the ground state in liquid phase; this is a surprising result considering the large perturbations from increasing number density, polarization of the surrounding environment, and the near degeneracy of orbital energies across the neat liquid. In fact, the agreement between the experimental 2PA spectra and simulations, which are based entirely on isolated-molecule calculations, indicates that the transitions can still be considered within a single molecule (atom-based) picture in the liquid phase possibly due to symmetry constraints, rather than being better understood from a solid-state framework. 123 The broadband 2PA spectra presented here have provided a more complete picture of the electronically excited states for simplest solvents that are heavily exploited for laser spectroscopy of dissolved solutes specifically because they lack π excitations. We have been able to comment on the 2PA bandgap for these solvents, otherwise not possible through single color two-photon measurement. The 2PA threshold goes down to ~6.9 eV in methanol and ethanol as compared to that of water. Despite maintaining a very favorable effective bandgap, we note that water has a 1 eV broader window that can be used for 2PA or UV pump-probe spectroscopy of dissolved solutes. We stress the importance of measuring the continuous polarization ratio spectrum, as it carries a valuable imprint of the individual transitions that matter the most in the 2PA spectra. Finally, perhaps surprisingly, isolated molecule electronic structure calculations provide a very reasonable starting point to analyze the spectra. 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Introduction In saturated molecules, i.e. the molecules that lack π bonding, the valence electrons are involved in the formation of covalent bonds or can exist as non-bonding lone pairs. The vacuum ultraviolet spectra of the molecules that contain lone-pair electrons, are dominated by sharp transitions to higher Rydberg orbitals from the non-bonding electrons in ground state. Such Rydberg transitions are often associated with n→ σ* type valence transitions as well. 1 Water and alcohols are classic examples of such systems. 2,3 In contrast, relatively featureless diffuse vacuum ultraviolet spectra are obtained for the saturated molecules without any lone pair electrons. 1 The diffuseness of these transitions may come from the displacement of the excited state potential energy surface (PES) with respect to the ground state. 4 The shallow depth of the excited state PES may lead to the pre-dissociation of molecule along one of the normal mode coordinates, which contributes to the broadness of the transitions as well. 1,4 Alkanes and cycloalkanes are excellent candidates for this category. Electronic structure and spectroscopy of these molecules are less explored as compared to their conjugated or aromatic counterparts. In the alkane spectra, several discrete absorption peaks are observed which originate from the excitation to Rydberg orbitals. These Rydberg transitions are not well resolved in linear alkanes, even for small chain molecules. 5– 7 Cyclic alkanes, on the contrary, show distinct Rydberg transitions, as previously demonstrated for adamantane, cyclopropane, cyclobutane, cyclohexane, bicyclo[2.2.2]octane etc. 8–10 These 131 sharp Rydberg transitions usually rest upon broad featureless continuum, which arises due to the σ→ σ* type valence transitions. 11 The vibronic features associated with these Rydberg transitions are difficult to interpret and there are only quite a few examples in the literature that address this issue. 8,11,12 Raymonda and Simpson adopted independent-systems theory to predict the energy levels of the lower electronic states of alkanes. 7 In this method, the C-C bonds are treated separately than C-H bonds, and starting with the excitation band (σC-C→ σ*C-C) of the monomer, i.e. ethane (with only one C-C bond), the energy levels of higher alkanes can be calculated by introducing the nearest neighbor interactions and breaking the degeneracy. Although this exciton model works surprisingly well for substituted/ unsubstituted alkanes and cycloalkanes, it falls short in explaining the weak band observed in most of the alkanes and specially in branched alkanes in the region between 7-8 eV. Their assignment of this band as intramolecular charge transfer between adjacent C-C segments was later contradicted by Lombos et. al 6 and Robin 1,11 who suggested this transition due to excitation to Rydberg 3s orbital. One factor that often complicates the analysis of the alkane spectra is the presence of several equilibrium conformations at room temperature. Although the absorption frequencies of different conformers are not expected to deviate much from one another, the oscillator strengths may vary drastically due to the change in molecular point group symmetry that directly influences the transition selection rule. Spectral analysis becomes further complicated in going from gas to liquid phase, as sharp Rydberg transitions get broadened and overlap with neighboring transitions, giving rise to an apparently featureless spectrum. To get a better understanding of the electronic structures in alkanes and cycloalkanes in liquid phase, we have adopted two photon absorption (2PA) spectroscopy which is governed by a different selection rule than one photon absorption (1PA). The continuous 2PA spectra are 132 measured for liquid hexane and cyclohexane in the energy range between 7-8.5 eV, for both parallel and perpendicular polarization of pump and probe photons. Both molecules can exist in several conformations in liquid state. For cyclohexane, there are two conformers present: chair and boat conformer. Molecular dynamics (MD) simulation studies predicted that cyclohexane mainly exists in ‘chair’ conformation in liquid phase. 13 The situation is markedly different in case of n- hexane. There are twelve conformers possible in case of a n-hexane molecule. Ab-initio and density functional calculations suggest that the lowest energy conformation is all-trans (TTT), which is followed by nearly isoenergetic trans-gauche-trans (TGT) and gauche-trans-trans (gtt) conformers. 14 Further up in energy is trans-gauche-gauche (tgg) conformer. 14 At room temperature, in deuterated liquid hexane, there is almost an equal distribution of all-trans and all- gauche conformations present, as predicted by MD simulation and Neutron diffraction studies. 15 On the contrary, low frequency isotropic Raman studies predict that ~20% of the conformers are present in all-trans form, whereas ~50% conformers have at least one gauche C-C bond in it. 16 Clearly, the literature reports on the relative population of different conformers in liquid hexane are in conflict, which further complicate the analysis of its electronic spectra. II. Background: 1PA spectrum of cyclohexane and hexane The gas phase 1PA spectra of cyclohexane shows two or more transitions in the region between 6.7 to 7.7 eV. 8 The lowest transition has a maximum at 7.13 eV and is assigned to the excitation from HOMO to Rydberg 3s orbital (LUMO). 8 It is to be noted that the chair conformation of cyclohexane has a D3d point group symmetry. Our ab-initio calculation reveals that the HOMO of cyclohexane is of eg symmetry, while LUMO is a1g. According to Laporte selection rule, a1g←eg transition is one-photon forbidden, but two-photon allowed. In spite of that, excitation to 3s Rydberg orbital shows up in the 1PA spectrum via vibronic coupling with an a2u 133 mode which has a ground state vibrational frequency of 522 cm -1 . 4,10,17 The cluster of sharp peaks following the first transition are due to excitation to Rydberg 3p orbitals with a maximum at ~7.55 eV. The n=4 Rydberg transitions appear above 8.7 eV. The sharp peak ~ 8.9 eV can be assigned to the promotion of electron to 4p Rydberg orbital. This 4p transition and other higher energy transitions show a vibronic progression of 1076 cm -1 , which can be associated with a combination mode of C-C breathing vibration and C-C-C torsional mode with the frequency of 802 cm -1 and 386 cm -1 respectively. 8 The frequency of vibronic progression matches well with that observed in the photoelectron spectrum, 0.14 eV ≈ 1100 cm -1 . Other lower frequency progressions observed in the 1PA spectrum of cyclohexane show irregular spacing; and this is a consequence of Jahn Teller splitting of the excited electronic states arising from the doubly degenerate σ bonding orbitals (HOMO) and the overtone/combination splitting due to anharmonicity of the potential energy surface. 11 In contrast to the 1PA spectrum of cyclohexane gas, the hexane spectrum doesn’t show any sharp feature in the experimental range shown in Fig. 6.1(b). Starting with the 9.3 eV band in ethane and applying independent system theory (bond additivity model) as mentioned above, Raymonda et al. 7 predicted the excitation energy in higher monologs of the alkane series. The calculated transition energies at 8.28 eV, 8.86 eV and 9.79 eV correlate well with the observed transitions at 8.37 eV, 8.99 eV and 9.55 eV. The observed transition energies were extracted by fitting the gas phase 1PA spectra of n-hexane. They assigned these transitions to σC-C→ σ*C-C type, involving mainly C-C bonds of the molecule. 7 A shoulder at a lower energy ~7.5 eV was assigned to the charge transfer transition between adjacent C-C segments. 7 Later, Au et al. 18 noticed the presence of this shoulder in the 1PA spectrum of ethane as well, which has only one C-C bond and cannot have any charge transfer transition of this type. They assigned this transition to HOMO→3s 134 transition. 1,18 Looking at the similarity of the transitions in n-alkane (n=3-8) and performing term- value analysis, Au et al. 18 assigned the 8.2 eV transition to the combination of HOMO → 3p and HOMO-1 → 3s excitations and the 8.8 eV transition to a HOMO-1→ 3p excitation. Figure 6.1: Gas phase 1PA spectra of (a) cyclohexane and (b) n-hexane reproduced from Ref 1 . The gas phase ionization energies are indicated by the green arrow pointing towards the X-axis. III. Results: Experimental and Computational To accurately assign the nature of the transitions in gas phase cyclohexane and n-hexane molecules and to predict the fate of these transitions upon solvation, ab-initio calculations have been performed for an isolated molecule. The geometries are optimized at RI-MP2/cc-PVTZ level 135 of theory. The excited electronic states are calculated for the optimized geometry using equation- of-motion couple cluster (EOM-CC) method. 19,20 The transition energies, 1PA and 2PA cross sections are calculated using the excitation energy method (EOM-EE-CCSD), with d-aug-cc- PVDZ basis set. Chair conformer of cyclohexane is used for the calculation, whereas for n-hexane, two different conformers are taken: all-trans (TTT) and trans-gauche-trans (TGT). The macroscopic 2PA cross section σ2PA (in GM) is calculated from the orientationally average 2PA strength (β2PA) by the following equation: 21,22 𝜎 2𝑃𝐴 = 2𝜋 2 𝛼 𝑎 0 5 (𝜔 1 + 𝜔 2 ) 2 𝛽 2𝑃𝐴 𝑐 ∗ Г ∗ 10 −50 ∗ 27.2107 where, 𝛼 is the fine structure constant, 𝑎 0 is the Bohr radius (in cm), Г is lifetime broadening (0.1 eV), 𝜔 1 and 𝜔 2 are the energies of the two photons in eV (𝜔 1 = 4.6 eV for our purpose) and 𝑐 is the speed of light (in cm/s). The calculated transition energies and 1PA and 2PA cross sections are tabulated in Table 6.1-6.3 and are shown as the stick spectra in Figure 6.2. Table 6.1. Calculated electronic transitions of cyclohexane (EOM-CCSD/d-aug-cc-PVDZ) with 4.6 eV pump energy. State Eex (eV) fL (  10 -3 ) β2PA (atomic unit) a σ2PA (GM) a 2PA Polarization ratio (ρ) a 2Ag 7.83 0 18.6 0.8 1.3 1Bg 7.83 0 18.7 0.8 1.3 2Bu 8.25 0.7 0.0 0.0 - 2Au 8.32 28.2 0.0 0.0 - 3Bu 8.32 28.2 0.0 0.0 - 3Ag 8.54 0 221.4 11.9 3.3 4Ag 8.87 0 21.3 1.2 1.3 2Bg 8.87 0 21.3 1.2 1.3 4Au 8.88 2.9 0.0 0.0 - 4Bu 8.88 2.9 0.0 0.0 - 5Bu 8.89 234.9 0.0 0.0 - 5Ag 8.92 0 489.1 28.6 22.6 6Ag 8.97 0 8.5 0.5 3.7 7Ag 8.98 0 9.4 0.6 1.3 4Bg 8.98 0 9.4 0.6 1.3 136 8Ag 9.00 0 8.6 0.5 1.3 5Bg 9.00 0 8.6 0.5 1.3 6Bg 9.01 0 0.0 0.0 - 9Ag 9.28 0 0.2 0.0 1.3 7Bg 9.28 0 0.2 0.0 1.3 8Bg 9.35 0 16.2 1.0 1.3 10Ag 9.35 0 16.2 1.0 1.3 5Au 9.39 0.2 0.0 0.0 - 6Bu 9.39 0.2 0.0 0.0 - 7Bu 9.42 2.4 0.0 0.0 - 6Au 9.43 3.3 0.0 0.0 - 8Bu 9.43 3.3 0.0 0.0 - 7Au 9.44 0 0.0 0.0 - 9Bg 9.52 0 0.1 0.0 1.3 11Ag 9.52 0 0.1 0.0 1.3 a 4.6 eV pump energy Table 6.2. Calculated electronic transitions of hexane-TTT (EOM-CCSD/d-aug-cc-PVDZ) with 4.6 eV pump energy. State Eex (eV) fL (  10 -3 ) β2PA (atomic unit) a σ2PA (GM) a 2PA Polarization ratio (ρ) a 2Ag 8.12 0.0 23.0 1.1 2.0 1Au 8.55 0.1 0.0 0.0 - 1Bu 8.56 6.2 0.0 0.0 - 2Bu 8.57 36.9 0.0 0.0 - 1Bg 8.64 0.0 103.8 5.7 1.3 3Ag 8.78 0.0 368.8 20.9 21.3 2Au 8.80 0.5 0.0 0.0 - 4Ag 9.03 0.0 383.4 22.9 5.5 3Bu 9.05 19.6 0.0 0.0 - 3Au 9.13 194.6 0.0 0.0 - 4Bu 9.17 296.4 0.0 0.0 - 5Bu 9.20 0.2 0.0 0.0 - 2Bg 9.20 0.0 5.4 0.3 1.3 4Au 9.22 3.2 0.0 0.0 - 5Ag 9.22 0.0 30.2 1.9 2.3 3Bg 9.24 0.0 7.6 0.5 1.3 6Ag 9.26 0.0 68.4 4.3 4.3 4Bg 9.39 0.0 7.7 0.5 1.3 5Bg 9.40 0.0 6.4 0.4 1.3 7Ag 9.41 0.0 4.8 0.3 1.5 8Ag 9.48 0.0 32.5 2.1 3.6 6Bu 9.54 28.9 0.0 0.0 - 9Ag 9.60 0.0 6.4 0.4 1.6 137 5Au 9.60 7.0 0.0 0.0 - 10Ag 9.61 0.0 53.2 3.6 2.7 6Au 9.66 3.3 0.0 0.0 - 6Bg 9.66 0.0 5.0 0.3 1.3 a 4.6 eV pump energy Table 6.3. Calculated electronic transitions of hexane-TGT (EOM-CCSD/d-aug-cc-PVDZ) with 4.6 eV pump energy. State Eex (eV) fL (  10 -3 ) β2PA (atomic unit) a σ2PA (GM) a 2PA Polarization ratio (ρ) a 2A 8.35 0.0 107.7 5.5 7.3 1B 8.37 0.2 23.7 1.2 1.3 2B 8.65 36.5 1.1 0.1 1.3 3A 8.68 22.5 104.4 5.8 2.9 4A 8.72 19.6 94.2 5.3 4.6 3B 8.73 29.9 2.4 0.1 1.3 4B 8.74 6.8 0.3 0.0 1.3 5A 8.79 0.0 1.9 0.1 10.2 6A 8.84 8.5 166.5 9.6 82.6 5B 8.92 21.9 5.2 0.3 1.3 7A 9.07 2.6 3.8 0.2 1.5 6B 9.09 219.6 0.6 0.0 1.3 8A 9.29 3.7 299.3 19.0 6.2 7B 9.31 7.2 49.3 3.1 1.3 8B 9.37 2.4 34.5 2.2 1.3 9A 9.38 3.3 45.5 2.9 3.3 10A 9.41 10.3 62.8 4.1 8.5 9B 9.44 10.7 15.5 1.0 1.3 11A 9.45 2.0 40.0 2.6 1.3 12A 9.47 46.3 94.8 6.2 14.2 10B 9.48 12.9 14.5 1.0 1.3 13A 9.50 1.2 12.6 0.8 1.7 11B 9.51 1.1 21.4 1.4 1.3 14A 9.55 8.2 13.4 0.9 1.3 12B 9.58 0.5 2.2 0.2 1.3 13B 9.60 1.6 4.6 0.3 1.3 a 4.6 eV pump energy 138 Figure 6.2: Calculated 1PA and 2PA transitions for cyclohexane, hexane-all-trans (TTT) and hexane-trans-gauche-trans (TGT) at EOM-CCSD/d-aug-cc-PVDZ level of theory. 2PA cross sections are calculated using 4.6 eV pump energy. Character of strong 1PA and 2PA transitions are written right beside the peaks. The point group symmetries of the chair conformation of cyclohexane and the TTT conformer of n-hexane are D3d and C2h respectively. Because of the presence of inversion symmetry in both these conformations, the 1PA and 2PA transitions are complementary to each other, i.e. 1PA active transitions are not 2PA allowed and vice versa. The symmetry of the ground electronic state is 1Ag, hence transitions to electronic states of ‘gerade’ and ‘ungerade’ symmetries are 2PA and 1PA allowed respectively. For example, in cyclohexane, 5Ag ←1Ag transition is most intense in 2PA, whereas 5Bu ←1Ag transition is strongest in 1PA. The TGT conformer of n-hexane, on the other hand, has a C2 point group symmetry. The lack of inversion symmetry in this point group allows all the excited electronic states of A and B character to be both 1PA and 2PA active. 139 The 1PA and 2PA spectra of liquid cyclohexane and n-hexane are shown in Figure 6.3. The 2PA spectra are reported for both parallel and perpendicular polarizations of the pump and probe photons, where the energy of the pump pulse is 4.6 eV. For both the molecules, parallel polarization of the two photons give rise to a higher cross section as compared to that of perpendicular polarization. It is also apparent from the spectra that cyclohexane has a lower energy 1PA and 2PA threshold as compared to n-hexane. This is consistent with our ab-initio calculation where the lowest excited state of cyclohexane appears at 7.8 eV, whereas in hexane-TTT it is at 8.1 eV and in hexane-TGT conformer, it is placed further up at 8.35 eV. Figure 6.3: Liquid 1PA spectra of (a) cyclohexane and (b) n-hexane digitized from Ref 23 and Ref 24 respectively. Liquid 2PA spectra of (c) cyclohexane and (d) n-hexane reported in this work. The 2PA cross sections are measured with both parallel and perpendicular polarizations of the pump and probe beams. The pump photon energy is 4.6 eV for our experiment. 140 Another interesting difference is observed by examining the polarization ratio plots of these molecules, as shown in Figure 6.4. In case of cyclohexane, the polarization ratio increases almost linearly in the energy range explored here, starting from ~1.7 at 7.0 eV, it reaches ~4 at 8.5 eV. Having a polarization ratio >4/3 in the entire spectral region suggests that transition in cyclohexane is mainly dominated by totally-symmetric type excitations. The decrease in polarization ratio in the lower energy region is a consequence of the contributions from 2Ag ←1Ag and 1Bg ←1Ag transitions arising due to the promotion of electron from doubly degenerate HOMO to the 3s Rydberg orbital (3s ←eg) of cyclohexane (shown in Figure 6.5). The polarization ratio of hexane, on the other hand, quickly increases from ~1.0 to ~2.5 in the energy range 7-8 eV and then flattens down maintaining an almost constant value in the higher energy region. It is worth pointing out that trace amount of photoionization products may also contribute to the decrease in polarization ratio in the lower energy side of the spectra. 25–27 Figure 6.4: 2PA polarization ratio plot of (a) cyclohexane and (b) n-hexane reported in this work. The pump photon energy used in our experiment is 4.6 eV. 141 IV. Analysis Before we address the issue of the shifts of the electronic states in going from the gas to liquid phase and the broadening they experience upon solvation, let us first compare the gas phase 1PA spectrum of cyclohexane with the excited energy levels obtained by our ab-initio calculation of a single cyclohexane molecule. Transition from HOMO to Rydberg 3s orbital appears at ~7.13 eV in the gas phase spectrum of cyclohexane. Our ab-initio calculation of cyclohexane predicts the first two transitions at 7.83 eV due to HOMO →Rydberg 3s type transitions (2Ag ←1Ag and 1Bg ←1Ag). The degeneracy of the excited states comes from the doubly degenerate HOMO (eg symmetry) in the cyclohexane molecule. According to Raymonda et al. 8 , HOMO→3p type Rydberg transition appears at 7.77 eV in the gas phase spectrum of cyclohexane. The energy of the corresponding transition is predicted to be at 8.3 eV according to our calculation. In the similar line, energies of all of our theoretically calculated transitions are appeared to be overestimated as compared to the experimental transition energies in the gas phase. The error in calculating excited state energies by EOM-EE-CCSD method is typically 0.1-0.2 eV for excitations to states by a single electron promotion. Calculation of energies of the excited states having substantial double excitation character may lead to an error up to ~ 1 eV. The degeneracy of the bonding molecular orbitals in cyclohexane results to significant contribution from double excitations during a particular transition, leading to an overestimation of the transition energies. In order to match with gas phase transitions in cyclohexane, the calculated excitation energies are red-shifted by ~0.6 eV, except the first transition which is forced to line-up with the lowest energy excitation in cyclohexane at ~7.13 eV. The NTO plots of the strong transitions of cyclohexane in 1PA and 2PA are shown in Figure 6.5. 142 Figure 6.5: NTO plots corresponding to the strong transitions of cyclohexane in 1PA and 2PA. Jung et. al 23 reported the 1PA spectrum of liquid cyclohexane in the range between 6-11 eV. The absorption threshold was estimated to be 7.0±0.1 eV. Gaussian fitting of the liquid 1PA spectra revealed the first peak at ~7.8 eV with the FWHM of ~0.8 eV and the second peak ~8.7 eV with ~1.4 eV FWHM. 23 The first peak was assigned to the transition to 3p Rydberg states and the second peak was due to the transition to 3d Rydberg orbitals which gets blue-shifted upon solvation. They however, did not see any signature of the transition to 3s Rydberg states in their experiment. It is to be noted that transition from HOMO → 3s Rydberg orbitals is one photon parity forbidden in cyclohexane, and yet, this transition shows up in gas phase 1PA spectrum due to vibronic coupling. An interesting observation from the analysis of the liquid 1PA spectrum of 143 cyclohexane by Jung et al. 23 is the higher Gaussian width of the transitions at higher energy, which interestingly matches with our observation for liquid water 28 and alcohols. 22 Analysis of natural transition orbitals (NTO) reveal that 1PA and 2PA allowed transitions are dominated by the excitation to ‘p’ and ‘d’ type Rydberg orbitals respectively, except the first three 2PA allowed transitions: degenerate transitions 2Ag ←1Ag and 1Bg ←1Ag due to HOMO → 3s excitation and 3Ag ←1Ag transition due to HOMO-1 → 3s Rydberg excitation. Previous publications by our group 22,28 showed that in water and alcohols, the low-lying Rydberg 3s orbitals get blue-shifted more than other Rydbergs. Keeping these factors in mind, we perform the simulation of liquid cyclohexane 2PA spectra and the polarization ratio using the results obtained from our ab-initio calculation. Before performing the simulation, the calculated transition energies are red-shifted by ~0.6 eV to match them up with the gas phase transitions. The common sets of parameters that simultaneously reproduce all the spectrum are reported here: the transitions up to ~8 eV are broadened by ~0.7 eV, and above that, a FWHM of ~1.1 eV is used. Excitations involving ‘3s’ type upper Rydberg orbitals (2Ag ←1Ag, 1Bg ←1Ag and 3Ag ←1Ag) are given a blue- shift of 0.55 eV and all the other transitions are given ~0.05 eV gas to liquid blue-shift in our simulation. The simulated plots are shown in Figure 6.6. It is to point out that the general blue- shift of ~0.05 eV is not a very strict parameter in our simulation. The simulations reproduce the experimental spectra equally well with a general shift anywhere between -0.05 and 0.05 eV. The gas to liquid energy shifts and the broadening of the excited states obtained by simulating the 2PA spectra can now be used to simulate the liquid 1PA spectra of cyclohexane. As shown in Table 6.1, the 1PA transitions in cyclohexane are guided by a completely different sets of excitations that involve the excited states of ‘ungerade’ symmetry. Despite such mutually exclusive transitions in 1PA and 2PA, liquid 1PA spectrum can be reproduced by allowing similar shifts and broadening 144 to the 1PA active transitions as done for the 2PA active transitions during 2PA simulation. To accurately reproduce the 1PA spectra beyond 9 eV, a transition centering ~9.5 eV is added to the simulation with an oscillator strength of 0.36. The simulated 1PA spectrum is shown in Figure 6.7. Figure 6.6: Simulation of the 2PA spectra of cyclohexane for (a) parallel and (b) perpendicular polarization. The simulated polarization ratio is shown in panel (c). The black dots and the red line stand for the raw data and the simulated plot respectively. The pump photon energy used in our experiment is 4.6 eV. Figure 6.7: Simulation of the liquid 1PA spectrum of cyclohexane. The raw data (black dots) is digitized from Ref 23 and the red line stands for the simulated plot. 145 Simulation of liquid hexane spectrum is far more complicated as there are several conformations of the n-hexane molecule present at room temperature and the relative populations of these conformers are in conflict according to the literature reports. Moreover, the liquid n- hexane used in our experiment are ~95% pure. The impurity comes due to the presence of a mixture of branched alkanes and cyclic alkanes. Nevertheless, for the purpose of demonstration, we have performed EOM-CCSD calculations to obtain the excited state energies and 1PA and 2PA cross sections for two different conformers of n-hexane; all-trans (TTT) and trans-gauche-trans(TGT), as shown in Figure 6.2 and Table 6.2-6.3. The calculated excited state energies differ by 0.1-0.3 eV between these two conformers. The all-trans conformer (TTT) has an inversion symmetry resulting the 1PA active transitions to be 2PA inactive and vice versa. The lack of inversion symmetry in the TGT conformer allows all the excited electronic states to be both 1PA and 2PA active. Moreover, different point group symmetry is expected to directly influence the experimental transition strengths which can also be confirmed by comparing the calculated 1PA and 2PA cross sections of different conformations. The same situation is true for the rest of the twelve conformations of n-hexane molecule as well. We first start with simulating the liquid 1PA spectra using the theoretically calculated results for the TTT and TGT conformers of n-hexane. For the TTT conformer, the simulated plot is obtained by red-shifting the energies of the calculated transitions by 0.78 eV and each transition is broadened by 0.76 eV. For the TGT conformer, the excited energy levels are red-shifted by 0.65 eV with a broadening of 0.66 eV given to each transition. Such red-shift of the calculated transitions to simulate the liquid 1PA spectra arises from several factors: 1. the intrinsic limitation of the EOM-CCSD approach that doesn’t include higher order excitations (triple and above); 29–31 2. not accounting for the vibronic transitions in the polyatomic molecule. While the theoretically 146 calculated excitations are vertical in nature, the maximum absorbance of a polyatomic molecule in gas phase often corresponds to the adiabatic transition resulting to a red-shift of the calculated energy levels, 32,33 and most importantly, 3. the solvatochromic shift of the transitions in going from gas to liquid phase. The simulated plots are shown in Figure 6.8. The lesser broadening of the transitions while simulating the liquid 1PA spectra using the TGT conformer is expected as higher number of excited states participate in this simulation as compared to that of the TTT conformer. Although, both simulations predict the increase in absorbance in going from the lower to the higher energy region of the spectrum with a maximum at ~8.4 eV, qualitative comparison between the simulated plots reveal that the TTT conformer reproduces the experimental spectra a little better than that of the TGT conformer. Figure 6.8: Simulation of the liquid 1PA spectrum of n-hexane using the theoretically calculated results for the TTT and TGT conformers. 147 It is worth pointing out that the experimental 1PA spectra of n-hexane shown here is measured using an ATR-FUV spectrometer and is not identical to the corresponding transmittance spectra, 34 as the ATR technique is known to cause a shift in the peak position as well as the peak intensity. 24 As a result, to extract quantitative information like gas to liquid energy shifts of the transitions and the broadening, we resort to the simulation of our measured 2PA spectra reported in this paper. As discussed before, Au et al. 18 assigned the 8.2 eV transition of an isolated hexane molecule to the combination of HOMO → 3p and HOMO-1 → 3s excitations and the 8.8 eV transition to a HOMO-1→ 3p excitation. Comparing these transitions to the natural transition orbitals calculated for the TTT and TGT conformers (Fig. 6.9-6.10) reveal that the theoretically computed energies are typically overestimated by 0.3-0.4 eV for both these conformers, similar to the case of cyclohexane. By carefully analyzing the second derivative spectra of liquid hexane 1PA, Tachibana et al. 24 assigned the shoulder at ~7.7 eV to the HOMO → 3s transition and the intensity ratio of the main peak to the shoulder is about 10 in the liquid state, as is also the case for the gas phase. 18 For the TTT conformer of n-hexane, 2Ag←1Ag transition accounts for promoting an electron from HOMO to the 3s Rydberg orbital and is symmetry forbidden in 1PA. The equivalent transition for the TGT conformer is 2A←1A which also has zero oscillator strength according to our gas phase EOM-CCSD calculation. The closely spaced 1B←1A transition also stands for exciting an electron to the 3s Rydberg orbital with an oscillator strength of 2×10 -4 , which is ~200 times weaker than that of 2B←1A transition. This again reinforces the idea of the presence of multiple conformations of n-hexane molecule in the liquid state at room temperature and points to the fact that how difficult it is to draw a quantitative conclusion by simulating the liquid phase spectra of n-hexane using only two representative conformers. 148 Figure 6.9: NTO plots corresponding to the strong transitions of hexane_TTT in 1PA and 2PA. Figure 6.10: NTO plots corresponding to the strong transitions of hexane_TGT in 1PA and 2PA. 149 We begin with the simulation of liquid hexane 2PA spectra using the theoretically calculated transition energies and the 2PA cross sections as tabulated in Table 6.2-6.3. For both TTT and TGT conformers, all the transitions are red-shifted by 0.35 eV to compensate for the overestimation of the calculated excited state energies. An additional red-shift of ~0.13 eV is also necessary to best reproduce the experimental 2PA spectra. As we allow the bandwidth of each transition float freely during our simulation, it adopts a value of 0.9 eV and 0.6 eV for the TTT and the TGT conformers respectively. The reduction in bandwidth in the simulation using the TGT conformer mirrors the result obtained by simulating the liquid 1PA spectra. The simulated plots for the hexane 2PA spectra at parallel polarization are shown in Figure 6.11(a), 6.11(b). As evident from the hexane simulation, although both the conformers can capture the general feature of the 2PA spectra, i.e. increase in 2PA cross section with increasing energy, the TGT conformer alone can reproduce the experimental spectra quite well as compared to that of TTT conformer. The simulation of 2PA polarization ratio is also conducted using the similar parameters that are used for simulating the parallel 2PA spectra and the results are shown in Figure 6.11(c), 6.11(d). The simulation using the results for the TGT conformer shows a relatively flat polarization ratio in the experimental range shown here and fails to reproduce the decrease in polarization ratio in the lower energy part of the spectrum. The accordance between the experimental and the theoretically simulated polarization ratio can be drastically improved if we allow a blueshift of ~0.18 eV to the 2A←1A transition which accounts for promoting an electron from HOMO to the 3s Rydberg orbital. Such blueshift of the first transition is not unprecedented as we find similar blueshift in case of cyclohexane (0.55 eV) and for water 28 and alcohols. 22 In the all-trans (TTT) conformer of n-hexane, 2Ag←1Ag transition is responsible for HOMO→3s Rydberg excitation. For consistency, this transition is also shifted by similar amount as done for that of the TGT conformer and the 150 revised simulations are shown as green line in Figure 6.11. The general parameters that simultaneously reproduce the 2PA spectra for both parallel and perpendicular polarization combination, along with the polarization ration are the following : all the theoretically calculated energy levels are red-shifted by 0.35 eV (for TTT) and 0.4 eV ( for TGT) to account for the intrinsic computational error of the EOM-CCSD method; an additional solvatochromic red-shift of ~0.15- 0.2 eV is given to each energy level except the transition responsible for HOMO→3s Rydberg excitation (2Ag←1Ag in TTT and 2A←1A in TGT conformer)which is given a blueshift of ~0.18 eV. The transition bandwidths for the TTT and the TGT conformers turn out to be 0.9 eV and 0.65 eV respectively. Figure 6.11: Simulation of the parallel 2PA spectra of n-hexane using theoretical calculated results for (a) TTT and (b) TGT conformers. The simulation of the polarization ratio is obtained using theoretical calculated results for (c) TTT and (d) TGT conformers. The black dots and the red line stand for the raw data and the simulated plot respectively. The parameters of these simulations are mentioned in the text. Revised simulation (dashed green line) is performed by giving ~0.18 eV blueshift to the 2Ag←1Ag transition in the TTT conformer and 2A←1A transition in the TGT conformer. 151 V. Discussion Let us start the discussion by pointing to the fact that in going from cyclohexane to n- hexane, the gas phase ionization potential increases from ~9.9 eV to ~10.2 eV which is consistent with the blueshift of the first absorbance band. 34 Costner et al. 34 showed that the absorption spectra of linear alkanes and cycloalkanes shift to the lower energy with the increase in the number of carbon atoms in the alkyl or cycloalkyl chain. By performing term value analysis, they concluded that the addition of carbon atoms destabilizes the HOMO while the energy of Rydberg LUMO orbitals does not get affected, resulting to a redshift of the absorption spectra as well as the ionization potential. In the similar line, the smaller ionization energy of cycloalkane as compared to its linear alkane homologue is usually attributed to the destabilization of the HOMO in the former due to ring strain. This logic, however, doesn’t work for cyclohexane as it is devoid of any ring strain because of the puckering of its backbone. Nevertheless, the redshift in ionization energy and the first absorption band in going from n-hexane to cyclohexane can be qualitatively explained by considering one extra C-C interaction that is present in the cyclic homologue, and thereby causing further destabilization of the HOMO. Our gas phase ab-initio calculation also captures this result where the first absorption band of cyclohexane is predicted to be ~0.3 eV lower in energy of that of the all-trans (TTT) conformer of n-hexane. It is apparent from the liquid phase spectra of cyclohexane and hexane (Figure 6.3) that the former has a lower energy 2PA threshold (~7.3 eV) as compared to the later (~7.6 eV). These 2PA thresholds were calculated based on where the absorption intensity reaches 10% of the peak value within our spectral range. Comparison with other solvents reveal that the 2PA threshold of both alkanes are higher than that of alcohols (~7.2 eV) and thereby will allow ~0.1-0.4 eV broader 152 window for 2PA spectroscopy on dissolved solutes. The threshold, however, is smaller than that of water (~8.2 eV). For cyclohexane, the transitions that involve the promotion of electrons to the Rydberg 3s orbital are 2Ag ←1Ag and 1Bg ←1Ag (degenerate, HOMO → 3s) and 3Ag ←1Ag (HOMO-1 → 3s). These transitions experience a gas to liquid blue-shift of ~0.55 eV, which resembles ~0.9 eV blueshift of 1A′′←1A′ transition (2pz → 3s) in water and alcohols. Such blueshift is a result of the Pauli repulsion experienced by the excited orbital due to the confinement by the surrounding solvent molecules. 28 In case of water and alcohols, the participating orbitals involved in the 1A′′←1A′ transition is mainly centered around the oxygen atom where the hydrogen bonding takes place. As a result, promotion of electron from an out of plane 2pz orbital to a spherically distributed 3s orbital increases electron density in the direction of the hydrogen bonding 22 and thereby subjects to a larger blueshift. There are two factors that determine the energetic shifting of a transition in going from gas to liquid phase: 1. Pauli repulsion experienced by the excited electron due to surrounding solvent molecules, which destabilizes the excited state; 2. Stabilization of the cationic core via instantaneous polarization by the solvent molecules, that causes a red-shift to the gas phase transitions. 35 A blueshift of ~0.55 eV for the HOMO/HOMO-1 → 3s type transitions in cyclohexane clearly dictates that the Pauli repulsion plays a dominant role in this case. Similar transitions in n-hexane (2Ag ←1Ag in TTT and 2A ←1A in TGT) also experience a solvatochromic blue-shift, but of smaller magnitude, ~0.18 eV. Unlike cyclohexane, the excited orbitals for these transitions are not pure Rydberg 3s type; mixed with valence antibonding 𝜎 𝐶 −𝐻 ∗ , as shown in Figure 6.9. The density of all the hexane isomers are approximately equal, ranging between 0.645-0.657 g/cm 3 at room temperature. 13 The density of liquid cyclohexane is 0.774 g/cm 3 which is ~20% 153 higher than that of liquid hexane. 36 As a result, the excited electrons in the 3s Rydberg orbitals of cyclohexane experience higher Pauli repulsion resulting to a larger gas to liquid blueshift as compared to that of n-hexane. In the similar line, the rest of the transitions in cyclohexane experience zero solvatochromic shift, whereas for n-hexane, the transitions are red-shifted by ~0.15-0.2 eV. For transitions to higher level Rydberg orbitals, the excited electrons are more spread out in space leaving the cationic core easily accessible by the surrounding solvent molecules and thereby giving rise to a higher polarization stabilization of the excited electronic state. As a consequence, for the higher energy Rydberg transitions in cyclohexane, the Pauli repulsion and the cationic stabilization nearly cancel each other, resulting to a zero gas to liquid energy shift. In case of n-hexane, the higher Rydberg excitations experience lesser Pauli repulsion (due to smaller liquid density than that of cyclohexane) and the polarization stabilization of the cationic core is even more facile due to higher polarizability of n-hexane (11.94 Å 3 ) as compared to that of cyclohexane (11.04 Å 3 ). 37 These two factors combined result to a general redshift of the higher energy Rydberg transitions in going from the gas to liquid phase of n-hexane. In both hexane and cyclohexane, the strong transitions in 1PA and 2PA are dominated by the excitation of electrons to the Rydberg ‘p’ and ‘d’ type orbitals respectively. In cyclohexane, 5Ag ←1Ag transition (Rydberg 3d excitation) is most intense in 2PA, whereas 5Bu ←1Ag transition (Rydberg 3p excitation) is strongest in 1PA. The 5Ag ←1Ag transition also shows the highest polarization ratio. The experimental 2PA spectra is mainly dominated by this transition as evident from the steady increase in the 2PA cross section and an almost linear increase in the polarization ratio in our experimental spectral range. The decrease in polarization ratio in the lower energy region is a consequence of the contributions from 2Ag ←1Ag and 1Bg ←1Ag transitions arising due to the promotion of electron from doubly degenerate HOMO to the 3s Rydberg orbital (3s ←eg) 154 of cyclohexane (shown in Figure 6.5). The 3s ←eg type transitions are only allowed in 2PA, but are absent in 1PA due to symmetry reasons. This can be confirmed by the presence of a shoulder in the 2PA spectra at ~7.6 eV, but no such shoulder is found in the 1PA spectra of cyclohexane (Figure 6.3). Natural transition orbital analysis reveals that the strongest 2PA transitions in the TTT (4Ag ←1Ag) and the TGT (8A ←1A) conformers of n-hexane also result from the Rydberg 3d excitation. This leads to the conclusion that cyclization of n-hexane doesn’t change the nature of the highest cross section transition in 2PA. VI. Conclusion In this work, we have presented the broadband 2PA spectra of liquid cyclohexane and n- hexane for the first time. The experimental 2PA spectra are simulated using the transition energies and the cross sections calculated for an isolated cyclohexane and n-haxane molecule at EOM- CCSD/d-aug-cc-PVDZ level of theory. Simulation of liquid hexane spectra are performed for two different conformers of n-hexane (TTT and TGT) separately. To reproduce the general feature of the experimental spectra, minor modifications of the computed properties are necessary, and it gives us a reasonable starting point to assign relatively featureless 2PA spectra and most importantly, the polarization ratio. For cyclohexane, transitions involving excitation of electrons to the 3s Rydberg orbitals are blue-shifted by ~0.55 eV, and all other transitions experience zero gas to liquid blueshift. In n- hexane, the Rydberg 3s transitions are blue-shifted by ~0.18 eV and the rests are red-shifted by ~0.15-0.2 eV upon solvation. The smaller blueshift of the Rydberg 3s transitions in n-hexane is attributed to lesser Pauli repulsion due to lower density of liquid hexane as compared to that of liquid cyclohexane. The general redshift of the electronic states of n-hexane upon solvation results 155 from the greater stabilization of the cationic core due to higher polarizability of n-hexane as compared to that of cyclohexane. Cyclohexane molecule is present predominantly in the chair conformation in the liquid phase and it belongs to highly symmetric D3d point group with a center of inversion. The gas phase selection rule for this molecule dictates that the 1PA and 2PA active transitions are complementary to each other. We find that the liquid phase 1PA and 2PA spectra can be wonderfully reproduced by taking the gas phase 1PA and 2PA active transitions respectively as a basis and with relatively modest modifications of the computed in vacuo properties. This emphasizes the fact that the gas phase selection rule about the complementarity of the transitions between 1PA and 2PA, is still valid in the liquid phase and the electronic structure calculation of an isolated molecule gives us a reasonable starting point to analyze the liquid phase spectra. The simulation of the liquid n-hexane spectra is relatively tricky as there are multiple conformers present in the liquid phase. Nevertheless, we attempted simulating the hexane spectra using the theoretically computed properties for the TTT and TGT conformers. The TTT conformer has a center of inversion, but the TGT conformer doesn’t. This allows all the electronic states in the TGT conformer to be both 1PA and 2PA active. The simulation using the results of the TGT conformer is in better accord with the experimental spectra as compared to the TTT conformer; reflecting the presence of considerable amount of lower symmetry conformer in the liquid phase. None of these conformers can reproduce the experimental polarization ratio well, reinforcing the idea that the polarization ratio is an extremely sensitive quantity in capturing the character of the electronic transitions in the 2PA spectra. Our ab-initio calculations and the NTO analysis clearly reveal that transitions in hexane and cyclohexane are mainly dominated by Rydberg excitations, even in the lower energy region. 156 The most intense transitions in both alkanes are also Rydberg type, and they fall well within our experimental spectral range even after experiencing the solvatochromic shift mentioned in this work. These gas phase Rydberg transitions seem to preserve their identity in the liquid phase, as the experimental 2PA spectra and the polarization ratio cannot be well reproduced by removing these transitions from the simulation, or by changing their oscillator strength drastically. This result mimics our conclusion about the Rydberg transitions in water and alcohols and reinforces the idea that transitions in liquid phase can still be envisioned from a single-molecule framework and the gas phase transition properties don’t get drastically altered upon solvation. Chapter 6 bibliography (1) Robin, M. B. 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(37) Bosque, R.; Sales, J. Polarizabilities of Solvents from the Chemical Composition. J. Chem. Inf. Comput. Sci. 2002, 42 (5), 1154–1163. https://doi.org/10.1021/ci025528x. 159 Chapter 7 Teaching Entropy from Phase Space Perspective: Connecting the Statistical and Thermodynamic Views Using a Simple 1-D Model Abstract Connecting the thermodynamic definition of entropy 𝑑𝑆 = 𝑑𝑄 /𝑇 (Clausius’ equation) with the statistical definition 𝑆 = ln (𝛺 ) (Boltzmann’s equation) has been a persistent challenge in chemical education at the undergraduate level. Not meeting this challenge results in students taking away the meaning of entropy in a vague and subjective way as a measure of “disorder” or increase in number of configurations without any meaningful way of connecting it to heat. To address this challenge, we present a simple model that connects these two definitions. This approach relies centrally on emphasizing that the number of configurations 𝛺 includes configurations in both real space and momentum space, collectively known as the phase space. Without including momentum configurations (i.e. how fast the particles move), connecting heat to entropy change is not possible. We construct the phase space for an ensemble of simple one-dimensional systems at equilibrium and show that delivery of heat 𝑑𝑄 to the system results into increase in the number of momentum configurations and consequently an expansion of the phase space area by 𝑑𝛺 . Relating 𝑑𝑄 to 𝑑𝛺 is the linchpin between the two views, and when integrated, leads to the Boltzmann’s equation. We further show that understanding entropy in terms of volume of phase space removes common ambiguities in teaching the subject. Amongst other examples, we show that understanding adiabatic compression results into contradictions if treated using the usual approach, which are resolved if a phase space view is adopted. We propose this approach at the undergraduate physical chemistry and physics level. I. Introduction Understanding the thermodynamics and statistical foundations of entropy is important for all equilibrium thermodynamics phenomena. The thermodynamics definition of entropy was proposed by Clausius as: 160 𝑑𝑠 = 𝑑𝑄 𝑇 (1) which shows that addition or removal of heat dQ to a system at temperature T, changes its entropy by dS. A consequence of this definition is that all spontaneous heat exchanges between a cold body and a hot body always leads to increase in the total entropy of the two bodies. The second law of thermodynamics generalizes this idea and asserts that all spontaneous irreversible processes must result in increase in total entropy of the system and its surrounding. Understanding the statistical underpinning of entropy is a major achievement of 19 th century science. Twelve years after Clausius proposed the above, the statistical definition was proposed by Boltzmann: 1 𝑆 = 𝑘 𝐵 𝑙𝑛𝛺 (2) where, 𝑘 𝐵 is the Boltzmann constant and 𝛺 is the total number of microscopic options for motion. The more spatial configurations, and the more momentum values that are available to a system of particles the larger the 𝛺 and consequently the larger the entropy of the system. In mechanics, phase space is a construct used for keeping track of positions and momenta of particles in a combined way. Therefore, 𝛺 can be interpreted as the volume in phase space accessible to a system of particles. These concepts will be defined and discussed in more detail later in this paper. Boltzmann's entropy equation is the foundation of Boltzmann's distribution and consequently an important cornerstone of all equilibrium thermodynamics. Therefore, its misunderstanding can have long-term educational consequences for students. Connecting the above two definitions has always posed a major pedagogical challenge. It has been customary to qualitatively describe entropy as a measure of ‘disorder’ or ‘randomness’. 2-5 Several works in the chemical education literature have cautioned that while interpreting entropy as randomness may seem convenient, it 161 introduces conceptual challenges and paradoxes, and at times is downright incorrect. 6-12 At least two facets of these difficulties are the following. First, randomness is often introduced in a qualitative way leaving it open to subjective interpretation and difficult to quantify. Second, texts often present examples of randomness in spatial arrangements of particles in real space (i.e. where are the particles?), without considering their distribution in the momentum space (i.e. how fast are the particles moving?). This divorces entropy from heat which can influence both positions and momenta of the particles. Sometimes pedagogical aids from probability theory such as dice, playing cards, and coin tosses are used to deliver the ideas related to entropy. While they are valuable at some level, without quantitatively connecting them with heat and energy, they fall short of our goal. To coherently and consistently understand both equation 1 and 2, one must connect to heat. Just resorting to randomness in the spatial configuration cannot achieve that. Since delivery of heat to a system, and the resulting increase in entropy, is closely related to changing the momenta of particles, it is not only desirable, but unavoidable to use the concept of distributions in phase space to describe entropy in a way that is consistent between the above two definitions. 13,14 We emphasize that historical development of the statistical definition of entropy, and all modern advanced presentations of the topic 15,16 are based on the phase space perspective. However, it is almost always avoided at undergraduate level due to the usual complexity of such presentations, at the cost of introducing a significant misunderstanding. Our purpose is to make this view accessible at the undergraduate-level. A different approach in recent chemical education literature and texts 17-19 has been to distinguish between entropy associated with spatial configurations (configurational entropy) and that associated with energy (thermal entropy). Similarly, the terms matter dispersal and energy dispersal have been used to describe entropy generation. 17,20-23 While 162 there may be some merit in creating this distinction, it is not necessary since the founding equations 1 and 2 do not require a distinction between matter and energy dispersal and are valid in a general sense. The approach in this work does not need to resort to this distinction. Rather, it deals with both matter and energy dispersal on the same footing using the concept of phase space as intended by Boltzmann. 13,24,25 Addition of heat can result into expansion of the available phase space volume both in real space (as in expansion of a gas) and in momentum space (increasing the temperature of a gas). Our work, to our knowledge, is a first concise pedagogical approach to introduce the concepts of phase space for describing entropy at the undergraduate physical chemistry and physics level. In particular, the following features of this approach are important. First, it does not resort to qualitative description of entropy as randomness, and rather interprets it as related to phase space volume 𝛺 . Second, it connects and emphasizes both the dispersal of energy and dispersal matter descriptions of entropy. Third, it connects the Clausius' and Boltzmann's definitions, and relates heat to the change in phase space volume. Finally, it is built upon a simple one-dimensional model and avoids unnecessary mathematical complications arising from a multi- dimensional phase space integral. We also emphasize that the statistical interpretation of entropy is sometimes built upon quantum mechanical systems with discrete energy levels. While this approach is valuable, it may give the wrong impression to students that Boltzmann's equation is only valid for quantum systems. Our approach is classical and highlights the generality of Boltzmann's equation which predates the introduction of quantum mechanics. Finally, we point out that the material here is not a proof of a fundamental connection between statistical mechanics and thermodynamics. Rather, we show that once a relation between temperature and average kinetic energy is taken as a starting point (which is readily accepted by 163 students and well-described in many existing textbooks), a connection between entropy and phase space volume easily follows. The details of this idea will be elaborated later in this paper. This paper is arranged as follows. First, to motivate the subject, we present a couple of examples of paradoxes and inconsistencies that arise from the common approach of presenting entropy. Second, we present the definition and properties of phase space, using the example of particles moving in one dimension. Third, we present how the phase space volume changes with delivering heat and how it is connected to entropy, followed by a derivation of Boltzmann's equation. Finally, we show how the new approach resolves the paradoxes that arise from an incomplete interpretation of entropy. II. Motivation We describe two examples which showcase the need for a revised view of teaching entropy at the undergraduate level. First, consider a box filled with an ideal gas, as shown in figure 7.1.a. The spatial distribution of gas particles in the box is indeed related to its entropy. An ordered configuration of gas particles, for example in a lattice confined to a corner of a box, is a state of low entropy. A ‘random’ distribution that ensures a uniform average density everywhere in the box is a state of high entropy. A gas at a finite temperature, on average, uniformly occupies the box and is in a state of high entropy. Successive snapshots of this gas, will be very similar to each other, showing a uniform distribution. Now consider adding heat 𝑑𝑄 to this system, which will make the particles move faster. However, stationary snapshots of this system after the addition of heat will be very similar to those before, showing uniform occupation of the entire volume (figure. 7.1.a). From comparing snapshots of spatial arrangement of the particles before and after heating, one cannot tell if the 164 system's entropy has increased. However, according to Clausius' definition addition of heat 𝑑𝑄 must have increased the entropy of the system by 𝑑𝑄 /𝑇 . Merely accounting for where the particles are located is not sufficient to describe the change in entropy and is at odds with the Clausius' definition. Upon addition of heat, the average momentum of the particles increases, and larger values of momenta are opened up for exploration, even though the gas is still confined to the same real space. To fully describe the number of both spatial and momentum configurations, the concept of phase space, as will be described shortly, is needed. It is the distribution of particles in the phase space that determines 𝛺 in the Boltzmann's equation, and consequently the entropy. Figure 7.1: Demonstrating the importance of accounting for increase in momenta, rather than apparent spatial randomness, in describing entropy increase. (a) A snapshot of a gas, showing apparent random spatial distribution. Another snapshot after addition of heat (increase in entropy according to Clausius's definition), qualitatively has the same random spatial distribution and does not convey the idea that entropy has increased. (b) Considering the momenta of particles (indicated by arrows) resolves this conflict. Addition of heat allows the particles to explore a larger range of momenta. In this instance, it is the increase in momentum configurations, and not the increase in randomness of spatial arrangements, that is related to entropy change. Another example that showcases an apparent discrepancy is shown in figure 7.2. If we compress a gas but not allow it to exchange heat with the surrounding (adiabatic compression), according to Clausius' definition its entropy should not change, since 𝑑𝑄 = 0 and consequently 𝑑𝑆 = 0. However, the volume of the compressed gas manifestly decreases and the number of spatial configurations available for the gas particles is clearly reduced. Therefore, if one only counts spatial configurations, the system must lose entropy (𝑑𝑆 < 0) according to Boltzmann's 165 equation. Why are these views contradictory? Another twist to this story is that in an adiabatic compression, concurrent with the volume decrease, the temperature of the system also rises. This is because no heat can escape, and the compression work is entirely converted to an increase of the internal energy which is manifested as an increase in temperature. Given this information, three students could arrive at three different conclusions about entropy change in this process. Student ‘A’ might think that the entropy of the system should increase as the temperature of the gas particles are increasing which makes the gas particles more random. Student ‘B’ may counter this logic by saying that as the volume decreases during the compression, the gas particles have a smaller volume to roam in which means entropy decreases. Student ‘C’ may argue that because there was no heat exchanged, there should not be any entropy change based on Clausius' definition. Which student is right? This apparent paradox is resolved if we count the number of configurations in the phase space, as we will quantitatively show later in this work. We encourage instructors to use the model of three students debating the concepts described above prior to introducing the material in the main body of this paper. The students should be encouraged to exchange ideas and propose arguments for and against the cases mentioned above. After explanation of the material, the instructors are encouraged to re-engage the students, particularly those with opposing views, to examine the problem in view of the new material. Such an approach will likely produce a stronger and longer-lasting impression. 166 Figure 7.2: Demonstrating that accounting for spatial configurations alone creates a conflict between the Clausius' definition and the statistical definition of entropy. When a gas is adiabatically compressed, it does not exchange heat with the surrounding and therefore, according to Clausius, its entropy must not change. However, the gas manifestly occupies less volume after compression and therefore it has fewer spatial configurations, implying that its entropy must have reduced. This conflict is resolved after one accounts for the increase in the momentum configurations of the gas particles after compression which exactly balances the decrease in the available spatial configurations. III. Introduction to Phase Space Phase space is a construct to keep track of all possible combinations of positions and momenta. In simple terms it is a book-keeping system of “where?” and “how fast?” for each particle. For a single particle of mass m moving in three dimensional space, three position coordinates (x,y,z) and three momentum values (projection of momentum vector on three axes px; py; pz) are necessary to uniquely specify its motion. Therefore, its phase space is six dimensional. Imagining this abstract six-dimensional space is difficult and for the purpose of our work unnecessary. For simplicity we resort to a simpler model of a single particle moving in one dimension confined to a string of length a (Figure 7.3). The phase space for this system is two- dimensional comprised of the position values x, ranging from 0 to a, and the possible momenta px as shown in the right panel of Figure 7.3. Any point in the phase space carries information about the position and the momentum of the particle. Three points in the phase space have been highlighted for the purpose of demonstration. Point 1 describes a snapshot of the particle where it is at the right side of the box and has a positive momentum (moving to the right). Point 2 167 corresponds to the particle in the middle of the box and not moving. Point 3 describes the particle at the left side of the box with negative momentum (moving left). Every point in this x-px plane describes one unique configuration of the particle. The larger the area of this phase space, a larger number of positions and momenta combinations will be possible. We can increase the phase space area either by increasing the size of the string (increasing a, i.e. expanding the size of the system), or allowing the particles to have a larger range of kinetic energies (i.e. delivering heat to the system) and therefore larger possible values of momentum. The area of the phase space represents the possible configurations of the particle. It is reasonable to assume that when the particle does not move at all, it will only have one configuration in the phase space at px = 0 and a fixed position. This statement will be useful later when we integrate the number of configurations from T = 0 to a finite temperature. If the maximum attainable kinetic energy for the particle is K, then the maximum magnitude for the momentum from 𝐾 = 𝑝 2 2𝑚 is: |𝑝 | = √2𝑚𝐾 (3) The area of the phase space becomes: 𝐴 = 2𝑎 √2𝑚𝐾 (4) The factor 2 comes from the fact that the momentum of the particle is a vector quantity and can have both positive and negative values. Soon, we will need to enumerate the number of points, i.e. position-momentum combinations, in this space. If we consider that space and momentum are both continuous, obviously, we will end up with infinite number of position and momentum combinations for any finite area of phase space, as there are infinite number of points in any area. To avoid this problem, Boltzmann argued that we may imagine a very small “unit” area in the 168 phase space with area 𝐴 𝑜 = ∆𝑥 ∆𝑝 𝑥 and divide the phase space area A by that unit. This converts the area of phase space to a countable number of position-momentum combinations ω. 𝜔 = 𝐴 𝐴 𝑜 (5) The choice of the unit area was not clear in Boltzmann's days. However, it did not pose a problem since regardless of the unit of measurement, it was clear that a larger A would correspond to more options of position-momenta than a smaller A and it was the relative change that mattered for the phase space area and eventually for entropy. With the advent of quantum mechanics a few decades later, it was clear that Boltzmann's foresight in quantizing the phase space was indeed justified. There is a true minimum area of phase space and it is set by the uncertainty principle 𝐴 𝑜 = ∆𝑥 ∆𝑝 𝑥 ≥ ħ 2 . However, to keep this presentation accessible and to avoid notational complexity, we use phase space area and the number of momentum-position combinations interchangeably. With the unit of measurement implied, we can write the phase space area as: 𝜔 = 2𝑎 √2𝑚𝐾 (6) Now, let us consider two non-interacting particles riding on two separate strings, shown in Figure 7.4 (left). The position and momentum configurations attainable by the red particle is shown by the phase space shaded red. Similarly, the configurations that the blue particle can adopt is represented by the area of the phase space shaded blue. What will be the total number of configurations for the two-particle system? Since, the movement of the red and the blue particles are independent from each other, for every configuration of the red particle all of the configurations of the blue particle are possible. Hence, the total number of configurations will be the product of the configurations attainable by the red and the blue particles which is the product of their individual phase space areas. If the phase space area for each particle is ω 1 and ω2 respectively, the total area of phase space corresponding to the combined possible configurations is: 169 𝛺 = ω 1 × ω 2 (7) Armed with this knowledge, we will approach the problem of phase space of an ensemble of particles on strings in the next section. Figure 7.3: (left panel) A single particle moving in one-dimension confined to a string of length a. (right panel) The phase space corresponding to this particle with three representative points. IV. The Phase Space Approach to Describe Entropy Imagine a box of length a, where N particles are confined to move along strings in one dimension as shown in Figure 7.5. We choose this simple model of particles on strings in one dimension to allow easy construction of their combined phase space without resorting to multi- dimensional integrals. We assume that the particles can exchange energy with each other by colliding with a background gas and therefore can thermally equilibrate. We begin with a scenario 170 in which the box is at temperature T. In this scenario, the total internal energy of the system of particles on the strings can be written as, 𝑈 = 𝑁 < 𝐾 > (8) Figure 7.4: (left panel) Two non-interacting particles riding on two separate strings, both are constrained to move in one-dimension. (middle and right panel) The phase space corresponding to the red and the blue particle respectively, each phase space is two dimensional. where, U is the total internal energy and <K> is the average kinetic energy of one particle. For an ideal gas of particles moving in three dimensions, the average kinetic energy is related to temperature by <K>=3/2kBT. This result is not derived here and is assumed to be presented to the students separately in the discussion of ideal gases. The number 3 in this expression arises from considering the three independent directions in space for the 3-dimensional ideal gas. For the case of 1-D gas in our model, the average kinetic energy is: < 𝐾 >= 1 2 𝑘 𝐵 𝑇 (9) 171 and hence the internal energy of the system becomes: 𝑈 = 𝑁 . 1 2 𝑘 𝐵 𝑇 (10) We point out that equation 9 is the fundamental link between temperature, which is a thermodynamic quantity and average kinetic energy, <K>, which is a statistical quantity. In what follows, we show that starting with this link, we can connect the thermodynamic concept of entropy (equation 1) with the volume of phase space. Following equation 3, the momentum corresponding to the average kinetic energy is: |𝑝 | = √2𝑚 < 𝐾 > (11) Note that this value is not the average momentum, but rather the root mean square momentum. The range of accessible momenta is ±|p|. We may skip this paragraph and begin using these limits to calculate the phase space area. But readers may be interested in the justification for this choice. First, at thermal equilibrium, it would seem unlikely that all values of momenta between ±|p| (including zero) would be equally accessible. It would seem that at a reasonably high temperature, the particles would not venture in the low momentum region of phase space and certainly would not cease to move. That is indeed true. This issue is a relatively modest price that we pay for choosing a one-dimensional system, which can be resolved easily as follows. If the particles were to be allowed to move in two (or three) dimensions, at thermal equilibrium there would be no preferred direction of motion. Some particles would be moving in the x direction, some in the y direction, and others equally covering all angles in between. In this scenario, if we were to calculate the projection of momentum on the x axis, we would find that it indeed ranges between all values from -|p| to +|p|. Finding a very low value (or zero) for px would not imply that the particle has stopped, but rather that it is moving mostly in the y direction. Therefore, for any high-dimensional system, plotting the projection of momentum on a single axis will cover the 172 entire range of momenta. Second, using the ±|p| as sharp boundaries is not exactly appropriate for a system at equilibrium. There is a distribution of values around these limits, the particles venture outside these limits every once in a while. However, it is reasonable that at the limit of large systems at equilibrium the total energy would be equally distributed amongst all particles. For a particle to move far beyond the ±|p| limits would imply that it has taken far more than its share of energy. Such excursions away from equilibrium would occur very rarely. Therefore, the width of the distribution of momenta would remain reasonably narrow, justifying using the mid points ±|p| as limits. With these issues out of the way, we move on to calculating the phase space area. Based on the above, the area of phase space per particle which is proportional to the number of configurations attainable by one particle is: 𝜔 = 𝑎 2|𝑝 | = 𝑎 2√2𝑚 < 𝐾 > (12) We wish to relate the area of phase space per particle to the internal energy of the system. Therefore, by substituting equation 8 into the above, we obtain: 𝜔 = 2𝑎 √ 2𝑚 𝑈 𝑁 = √ ( 8𝑚 𝑎 2 𝑁 )𝑈 (13) An important step in our derivation is finding out how much the phase space area increases if the internal energy of the system is increased by an amount 𝜕𝑈 , while the length of the box a is kept constant. Therefore, we calculate the partial derivative (𝜕𝜔 /𝜕𝑈 )a, which is the rate of expansion of phase space with respect to addition of energy. For notational simplicity, in the following we will drop the subscript a: 𝜕𝜔 𝜕𝑈 = 1 2 √ ( 8𝑚 𝑎 2 𝑁 ) 1 √𝑈 (14) 173 Figure 7.5: (left panel) N non-interacting particles are confined to move along strings in one dimension. (right panel) The phase space corresponding to each of these N particles. The total phase space area is the multiplication of the individual phase space areas. By appropriate substitution, we can simplify the right-hand side. We will replace √8𝑚 𝑎 2 /𝑁 from equation 13. 𝜕𝜔 𝜕𝑈 = 1 2 𝜔 √𝑈 1 √𝑈 (15) The above interesting relation shows that the rate of expansion of the phase space with increasing internal energy is directly proportional to the area of the phase space and inversely proportional to the internal energy in the system. We can replace U from equation 10 to obtain: 𝜕𝜔 𝜕𝑈 = 1 2 𝜔 𝑁 1 2 𝑘 𝐵 𝑇 = 𝜔 𝑁 𝑘 𝐵 𝑇 (16) 174 From the first law of thermodynamics, we know that in the absence of work, the change in internal energy is equal to the amount of heat given to or taken out of the system, i.e. dU = dQ. Substituting this in the above: 𝜕𝜔 𝜕𝑄 = 𝜔 𝑁 𝑘 𝐵 𝑇 (17) This relation informs us that the sensitivity of the phase space area is inversely proportional to the temperature. The phase space of colder systems expand more rapidly with addition of heat compared to those of hotter system. Separating variables, we arrive at: 𝜕𝑄 𝑇 = 𝑁 𝑘 𝐵 𝜕𝜔 𝜔 (18) The left-hand side of this equation is nothing but the Clausius definition of entropy (equation 1), while the right hand side is related to the differential increase in phase space area. Therefore: 𝜕𝑆 = 𝑁 𝑘 𝐵 𝜕𝜔 𝜔 (19) By integrating equation 19 with proper limits where entropy goes from 0 to S while the phase space area changes from 1 to 𝜔 we obtain: ∫ 𝜕𝑆 𝑆 0 = 𝑁 𝑘 𝐵 ∫ 𝜕𝜔 𝜔 𝜔 1 (20) 𝑆 = 𝑁 𝑘 𝐵 ln (𝜔 ) (21) 𝑆 = 𝑘 𝐵 ln 𝜔 𝑁 (22) Based on equation 7, the total phase space area for N particles will be obtained by multiplying the phase space area for each particle. 𝛺 = ω 1 × ω 2 …… × ω 𝑁 = 𝜔 𝑁 (23) 175 Substituting this into equation 22 results into: 𝑆 = 𝑘 𝐵 ln 𝛺 (24) Equation 24, which can also be represented in exponential form 𝛺 = 𝑒 𝑆 /𝑘 𝐵 , is the well-known Boltzmann equation. The Boltzmann distribution, and all that relies on Boltzmann distribution, are constructed upon this equation. Therefore, it is the founding principle of equilibrium thermodynamics. We emphasize that the above is, by no means, a proof of a connection between the statistical mechanics and thermodynamics. Notice that we already started with an assumed connection between temperature and average kinetic energy in equation 9. That relation, on its own, is the fundamental postulated link between thermodynamics and statistical mechanics. The arguments above show us that if equation 9 is taken to be a starting point, one can show that the thermodynamic entropy (according to Clausius) is related to volume of phase space. The Botlzmann equation connects entropy S with the area of the phase space. We are going to discuss a couple of examples which will establish that connecting entropy to the phase space area is a far better way to describe thermodynamic phenomena. It does away with qualitative, and sometimes subjective, measures of entropy such as disorder and randomness. Often this perspective resolves apparent paradoxes in book-keeping of entropy. V. Examples A. Adiabatic Volume Change The dilemma of entropy change upon compression when no heat is allowed to escape or enter was presented in an earlier section. In brief, adiabatic compression decreases the volume, and increases the temperature, but does not allow heat exchange. The points of view of the three students discussed in that section is reviewed here with understanding entropy from the phase space volume perspective. The increase in entropy of the system due to increase in temperature (as proposed by 176 student A) is exactly balanced by the decrease in entropy of the system due to smaller volume (as proposed by student B), resulting into zero entropy change (as proposed by student C). Therefore, student C is right. In an adiabatic compression the volume of the phase space does not change, because the decrease in the possible range of position coordinates due to compression is exactly balanced by the increase in the range of possible momentum values due to higher temperature. Below, we will prove this assertion quantitatively for an ideal gas. An ideal monoatomic gas is kept in a cubic box of length a. The box is adiabatically compressed to the length a/2 in all dimensions. The initial volume of the box is a 3 , and the final volume is (2a) 3 =8a 3 . The box was initially at temperature T1, the final temperature of the particles inside the compressed box is T2. In an adiabatic expansion/compression process, the initial and final temperature and volume are related by the following equation. 𝑇 1 𝑉 1 𝛾 −1 = 𝑇 2 𝑉 2 𝛾 −1 (25) where T1, V1, T2, V2 are the initial temperature, initial volume, final temperature and final volume respectively. 𝛾 is the ratio of heat capacities at constant pressure (Cp) and constant volume (Cv); (𝛾 = Cp / Cv). For an ideal monoatomic gas, for example, 𝛾 = 5/3. This equation has not been derived here and is assumed to be presented to the students earlier in their class. As described before, for an ideal gas of particles moving in 1D the average kinetic energy is related to temperature by <K>=1/2kBT. Substituting this in equation 11: |𝑝 | = √𝑚 𝑘 𝐵 𝑇 (26) 177 Figure 7.6: Adiabatic reversible compression of an ideal gas in a cubic container. (a) The compression is shown in real space where the length of the cube is decreased from a to a/2. (b) The effect of compression on momentum space. Due to increase in temperature from T1 to T2, the volume of the momentum space increases. (c) The phase space volume, which is the multiplication of the real space and momentum space volumes, does not change due to the balance between the spatial compression and momentum expansion. Hence there is no entropy change consistent with the Clausius' definition. which sets the range of accessible momenta as -|𝑝 | to +|𝑝 |. In this example, we have a 3D box, therefore the above applies to each one of the momentum directions px, py and pz. The range of accessible momenta define a cube in the momentum space, with side 2|𝑝 |. The volume of the cube represents all possible momentum values achievable by the particles. When the box is at temperature T1, the accessible volume of the momentum space is (2|𝑝 |)3 = 8|𝑝 | 3 . The phase space volume becomes: 𝛺 = 8|𝑝 | 3 𝑎 3 = 8(√𝑚 𝑘 𝐵 𝑇 1 ) 3 𝑎 3 (27) Following equation 25 the expression of the final temperature of the system is 178 𝑇 2 = 𝑇 1 ( 𝑎 3 𝑎 3 /8 ) 𝛾 −1 (28) 𝑇 2 = 8 𝛾 −1 .𝑇 1 (29) The new volume after 𝛺 after in phase space after compressing the box adiabatically becomes 𝛺 𝑎𝑓𝑡𝑒𝑟 = 8(√𝑚 𝑘 𝐵 𝑇 2 ) 3 ( 𝑎 2 ) 3 (30) 𝛺 𝑎𝑓𝑡𝑒𝑟 = 8(√𝑚 𝑘 𝐵 (8 𝛾 −1 .𝑇 1 )) 3 ( 𝑎 2 ) 3 (31) 𝛺 𝑎𝑓𝑡𝑒𝑟 = 8(√𝑚 𝑘 𝐵 𝑇 1 ) 3 𝑎 3 ( 8 𝛾 −1 2 2 ) 3 (32) 𝛺 𝑎𝑓𝑡𝑒𝑟 = 𝛺 𝑏𝑒𝑓𝑜𝑟𝑒 ( 8 𝛾 −1 2 2 ) 3 (33) As mentioned above, for an ideal monoatomic gas, 𝛾 =5/3, and ( 8 𝛾 −1 2 2 ) 3 = 1, hence 𝛺 𝑎𝑓𝑡𝑒𝑟 = 𝛺 𝑏𝑒𝑓𝑜𝑟𝑒 Therefore, we prove that there is no change in the phase space volume of the system as it undergoes adiabatic compression. This shows the entropy should not change in adiabatic compression since the volume of phase space does not change, and that there is no contradiction between the Cluasius and Boltzmann definitions of entropy. Without this approach, and merely relying on interpreting entropy qualitatively as randomness of spatial configurations, a glaring inconsistency between the two definitions would continue to confuse students and instructors alike. 179 Figure 7.7: Isothermal reversible expansion of an ideal gas kept in a cubic container. (a) The expansion is shown in real space where the length of the cube increases from a to 2a. (b) There is no e_ect of expansion in momentum space since temperature doesn't change during isothermal process. (c) The _nal volume of phase space is greater than that of the initial volume, hence entropy increases during isothermal expansion process. B. Isothermal Volume Change Imagine a system is in contact with a bath of constant temperature and its temperature does not change. Therefore, the average kinetic energy, the average momentum, and consequently the momentum coordinates in its phase space do not change (Figure 7.7.b). If the gas is allowed to expand by a differential amount 𝑑𝑉 against an external pressure 𝑃 , it performs work −𝑃𝑑𝑉 , at the expense of its internal energy. However, since it is in contact with a bath of constant temperature, it absorbs enough heat from the bath to replenish the lost energy until it is isothermal with the bath again. In this process, the volume of the box increases in real space, therefore the phase space volume should increase, even though the temperature has remained constant. In this example, the increase in entropy arises from the increase of spatial degrees of freedom. The heat transferred to 180 the system during expansion to maintain the temperature constant is all consumed to perform work that has given the system a larger quantity of real space volume. Comparison of the adiabatic and isothermal volume change shows that, to account for entropy change one must consider both spatial and momentum degrees of freedom. Either the spatial or the momentum dimension (or both) can change during a process and can influence the phase space volume. C. Expansion into Vacuum The last example is the most paradoxical and arguably the most difficult to explain. However, it is very valuable for highlighting the generality of Boltzmann's definition of entropy as related to the volume in phase space. Imagine an ideal gas that is allowed to expand into vacuum and is not allowed to exchange heat with the surrounding. In this case, the internal energy of this system will remain constant since pushing against vacuum does not require work and the gas does not exchange heat. It may be tempting to say that based on Clausius' definition, its entropy must not change. However, it is clear that this expansion is spontaneous, irreversible, and consistent with increase in entropy. This increase of entropy arises from a larger volume in real space that has become available to the system without any heat. Since no energy is lost, the momentum dimension of the phase space does not change. Therefore, the volume of the phase space will increase in exactly the same way as in the isothermal expansion. Note that in the case of isothermal expansion, the absorbed heat was used to perform work on the surrounding to make space for the system, which translated to increase in its entropy. In this case, the extra space is gained without the need to absorb heat, and just as before corresponds to an increase in the phase space volume and entropy. 181 This example shows that thinking of entropy in terms of phase space volume is more general than the Clausius' definition. We note that explaining entropy generation upon expansion into vacuum has been sometimes explained in terms of the quantum mechanics of the particles, which predicts larger number of quantized states within a given energy range for a larger size system. While this view is correct, it is not particularly illuminating. It is just the wave mechanics way of saying that the system has a larger size. Furthermore, the essence of the definition of entropy and the second law is independent of the underlying mechanics. It was developed successfully using classical mechanics, and holds true for quantum systems as well. It is not necessary to resort to quantum mechanics to explain the entropy increase upon expansion into vacuum. This may be especially advisable for students who have not been introduced to quantum mechanics, and have a difficult time grasping the wave nature of the translational wavefunction of composite particles such as atoms and molecules. VI. Conclusion We have demonstrated a simple pedagogical tool that shows the relationship between entropy and the volume of phase space. This relation can be established quantitatively and does not invoke subjective measures such as disorder. This approach combines the dispersal of energy and dispersal of matter views and does not favor one over the other. Furthermore, a simple derivation of the Boltzmann's equation, which is the foundation of all equilibrium thermodynamics, is made possible by considering the expansion of the phase space volume 𝑑𝛺 upon introduction of heat 𝑑𝑄 . Misconceptions about entropy and its interpretation are very common at all levels, ranging from the undergraduates to professional scientists. As a step to remedy this problem, we 182 recommend the approach in this work for junior and senior level physics and chemistry students. We hope that it finds its way into standard textbooks of physics and physical chemistry. Chapter 7 bibliography (1) Boltzmann, L. Vorlesungen iiber Gastheorie, JA Barth, Leipzig (1896); English translation: Lectures on Gas theory (SG Brush, transl.); Univ. California Press, Berkeley, 1964. (2) Callen, H. Thermodynamics, and an Introduction to Thermostatics, 2nd Edition; Wiley, Hoboken, NJ, 1985. (3) Denbigh, K. Principles of Chemical Equilibrium: with applications in chemistry and chemical engineering, 3rd Edition; Cambridge University Press, New York, 1971. (4) Ben-Naim, A. Entropy demystified: The second law reduced to plain common sense; World Scientific, Singapore, 2008. (5) Baker, D. General Chemistry, 5th ed. (Ebbing, Darrell D.). J. Chem. Educ. 1997, 74, 1049. (6) Ben-Naim, A. Entropy: Order or information. J. Chem. Educ. 2011, 88, 594-596. (7) Lechner, J. H. Visualizing entropy. J. Chem. Educ. 1999, 76, 1382. (8) Lambert, F. L. Disorder-A cracked crutch for supporting entropy discussions. J. Chem. Educ. 2002, 79, 187. (9) Leff, H. S. Removing the mystery of entropy and thermodynamics Part V. The Physics Teacher 2012, 50, 274-276. (10) Styer, D. F. Insight into entropy. Am. J. Phys. 2000, 68, 1090-1096. (11) Swendsen, R. H. How physicists disagree on the meaning of entropy. Am. J. Phys. 2011, 79, 342-348. (12) Jaynes, E. T. Gibbs vs Boltzmann entropies. Am. J. Phys. 1965, 33, 391-398. (13) Sciamanda, R. J. Expansion of Available Phase Space and Approach to Equilibrium. Am. J. Phys. 1969, 37, 808-809. (14) Penrose, O. Foundations of statistical mechanics. Rep. Prog. Phys. 1979, 42, 1937. (15) Pathria, R. K.; D., B. P. Statistical Mechanics; Butterworth-Heinemann; Linacre House, Jordan Hill, Oxford OX2 8DP, 1996. (16) Van Vliet, C. M. Equilibrium and Non-Equilibrium Statistical Mechanics; World Scientific Pub Co Inc; Revised edition, Singapore, 2008. (17) Lambert, F. L. Configurational entropy revisited. J. Chem. Educ. 2007, 84, 1548. (18) Atkins Peter, K. J., De Paula Julio Atkins' Physical Chemistry, Eleventh Edition; Oxford University Press, New York, USA, 2018. (19) Ellis, D. C.; Ellis, F. B. An experimental approach to teaching and learning elementary statistical mechanics. J. Chem. Educ. 2008, 85, 78. (20) Kozliak, E. I. Consistent application of the boltzmann distribution to residual entropy in crystals. J. Chem. Educ. 2007, 84, 493. (21) Kozliak, E. I. Introduction of entropy via the Boltzmann distribution in undergraduate physical chemistry: A molecular approach. J. Chem. Educ. 2004, 81, 1595. (22) Kozliak, E. I. Overcoming misconceptions about configurational entropy in condensed phases. J. Chem. Educ. 2009, 86, 1063. 183 (23) Lambert, F. L. Entropy is simple, qualitatively. J. Chem. Educ. 2002, 79, 1241. (24) Kittel, C. Elementary statistical physics; Courier Corporation; North Chelmsford, Massachusetts, USA, 2004. (25) MacDonald, D. K. C. Introductory statistical mechanics for physicists; Courier Corporation; North Chelmsford, Massachusetts, USA, 2006.

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Abstract Molecular orientation at the donor‐acceptor interface influences the efficiency of the processes like exciton dissociation, charge carrier recombination and charge extraction

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Creator Bhattacharyya, Dhritiman (author)

Core Title Interfacial vibrational and electronic excited states of photovoltaic materials probed using ultrafast nonlinear spectroscopy

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Chemistry

Publication Date 12/02/2020

Defense Date 07/01/2019

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Tag interface,OAI-PMH Harvest,photovoltaic materials,surface,two photon absorption,ultrafast nonlinear spectroscopy,vibrational sum frequency generation

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Advisor Benderskii, Alexander V. (committee chair), Bradforth, Stephen E. (committee chair), Nakano, Aiichiro (committee member)

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