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Configuration interaction calculations on the resonance states of HF- the excited states of HF, and the Feschbach states of HF-

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Content CONFIGURATION INTERACTION CALCULATIONS ON THE RESONANCE STATES OF HF", THE EXCITED STATES OF HF, AND THE FESHBACH STATES OF HF' by Kat hieen Wol f A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In .P a rtia l Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY ( Chemi s t r y ) August 1980 UMI Number: DP21879 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI DP21879 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 - 1346 U N IV E R S IT Y O F S O U T H E R N C A L IF O R N IA TH E G RADUATE SC H O O L U N IV E R S IT Y PARK LOS A NG ELES. C A L IF O R N IA 9 0 0 0 7 f h . T ) . C 'SI w & 5 k This dissertation, written by Katheleen Wolf * ........................................................................................................... under the direction of h.S.T ... Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillm ent of requirements of the degree of D O C T O R O F P H I L O S O P H Y d Dean DISSERTATION C O M M ITTEE To a l l th e years. i i ACKNOWLEDGEMENTS I am especially indebted to Professor Gerald Segal for his help and guidance, his friendship and encouragement over the past six years. I would also lik e to express my gratitude to my research group, Ross Wetmore, Chevy Goldstein, Brian Luke, Dean Liskow, Jim Diamond, and Vijaya Saraswathy for th e ir assistance. I w ill always remember the support given to me by Morlie Grauhard, Gary M il ls , Richard Lawson, and David Dows. Particular thanks go to Michele Dea and Anne Boren. Finally , I would lik e to express my gratitude to my father who gave me my technical capability and my mother who gave me the a b i l i t y to write about i t . TABLE OF CONTENTS Page DEDICATION........................................................................... ii ACKNOWLEDGEMENTS ...................................................................................... i i i LIST OF TABLES........................................................................................... vi LIST OF FIGURES. ......................... v i i i ABSTRACT.......................................................................................................... ix INTRODUCTION ......................... . . . . . ......................................... 1 References..................................................................................... 7 CHAPTER I HISTORICAL BACKGROUND............................................... 8 References.................................................................. 23 I I METHOD.................................................................................. 25 References. . . . . . ................................... 31 I I I RESONANCES OF POLAR MOLECULES........................... 34 Single Particle Resonances......................... 36 Core-Excited Resonances ............................... 47 References.................................................................. 50 IV CONFIGURATION INTERACTION CALCULATIONS ON THE RESONANCE STATES OF HF"............................... 52 Introduction............................................................. 52 Method............................................................................ 57 Results and Discuss ion.................................... 59 General Procedure .............................................. 60 Lim its ............................................................................ 68 ______ iv CHAPTER Page Other S t a t e s ............................................................. 70 Comparison with HC1~ ....................................... 7] Proof of Resonance C h a r a c t e r ........................ 73 The Bound S ta t e ........................................................ 82 Conclusions............................................................ . 86 R e fe re n c e s .................................................................. 87 V CONFIGURATION INTERACTION CALCULATIONS ON THE RYDBERG’STATES OF HF AND THE FESHBACH STATES OF HF'.............................................................................132 I n t r o d u c t i o n ............................................. 132 Background........................................................................133 M e th o d ...........................................................................139 Results and Discussion . ................................141 Conclusions........................................................................160 R e fe re n c e s ........................................................................162 \l LIST OF TABLES Table Page IV-1 Gaussian Basis S e t ............................................................90 IV - 2 SCF Results for HF (R = 1 . 732 bohr)....................91 IV-3 Calculated Cl Energy Points. . . ............................ 93 IV-4 Configuration Bases for the Ground States of HF and HF" ...................................................................95 IV-5 Summary of Observed and Calculated Results . 97 IV.-.6 Original and Altered Basis Set Description . 98 IV-7 Z+ M.O. SCF Eigenvalues for Original and Altered Basis Sets..................................................... 99 IV - 8 SCF Energy and Eigenvalues for Original and Altered Basis Sets ( a u ) ...........................100 IV-9 Cl Energy Within the Original and Altered Basis Set (e V )......................... 101 V-l Gaussian Basis Set . . , . .........................................164 V-2 SCF Results for HF (R = 1.732 bohr)...................165 V-3 SCF Molecular Orbital Designation........................167 V-4 Calculated Cl Energy Points for H F ...................169 V-5 Generating Bases for HF Potential Curves . . 170 V- 6 Calculated Cl Energy Points for HF" -- 9 to 12 eV.........................................................................172 V-7 Generating Bases for HF" Potential Curves -- 9 to 12 eV............................................................ . . . 1 7 3 v i Table Page V- 8 Calculated Cl Energy Points for HF” -- 12 to 14 e V ............................................................ 174 V-9 Generating Bases for HF" Potential Curves -- 12 to 14 e V ........................................ 175 V-10 Energy and Configuration Weighting of the 52t t HF” State and the 3 ^ t t HF State. . . . 176 V-11 Energy and Configuration Weighting of the 4 and 5 2t t HF” States and the 5^t t HF S t a t e .................... .................................................................178 v i i LIST OF FIGURES Figure Page I I - l Partitioning of the Hamiltonian Matrix . . . 32 IV-1 Gaussian Basis S e t ..................................... 102 IV - 2 M.O. 6 - - B a s i s Sets 1, 2 and 3 .....................................104 IV-3 M.O. 9 - - B a s i s Sets 1, 2 and 3 .....................................106 IV-4 M.O. 10 - - Basis Sets 1, 2 and 3 .............................108 IV-5 M.O. 13 - - Basis Sets 1, 2 and 3 . . . . . . 110 IV - 6 M.O. 14 - - Basis Sets 1 , 2 and 3 .............................112 IV-7 New M.O. -- Basis Set 2.................................................114 IV- 8 New M.O. 1 - - Basis Set 3 .............................. . . . 116 IV — 9 New M.O. 2 - - Basis Set 3. . .......................................118 IV -1 0 M.O. 6 - - Basis Sets 1, 4 and 5 .....................................120 TV-11 M.O. 9 -- Basis Sets 1, 4 and 5.....................................122 IV-1 2 M.O. 10 -- Basis Sets 1 , 4 and 5 ....................... 124 IV-13 M.O. 13 -- Basis Sets 1, 4 and 5 ........................126 IV-14 M.O. 14 -- Basis Sets 1, 4 and 5 ....................... 128 IV-15 Alternate M.O. 6 - - Basis Sets 1, 4 and 5. . 130 V-l Calculated Potential Energy Curves for HF. . 181 V-2 Calculated Potential Energy Curves for HF” -- 9 to 12 eV...................................................................... 183 V-3 Calculated Potential Energy Curves for HF” -- 12 to 14 e V ................................... 185 v i i i ABSTRACT The behavior of certain chemical systems is examined through large-scale Configuration Interaction calculations. The u t i l i t y of ab i n i t i o calculations of this type in studying the chemistry of negative ion molecules is cle a rly demonstrated. The low energy states of the negative ion HF- are one subject of considerable study. Using SCF and Configuration I nteraction methods, potential curves for several states are calculated. They include those states which dissociate to H + F~ and H~ + F, in addition to those which dissociate to HF~ + e". The results provide an interpretation of the observed experimental features and in some cases, predict certain other features that have not yet been the subject of experimental investigation. Although the calculations indicate that the lowest HF" state is bound, there is not s u ffic ie n t evidence to support the conten tion that the real HF molecule can indeed bind an electron. Resonance character is attributed to five HF" states on the basis of detailed q u a lita tiv e examination. When the basis set is expanded by the addition of diffuse functions, all five states appear to exhibit resonance behavior. Plots of the density functional for each molecular orbital against internuclear distance demonstrate that the additional electron has a high probability of residing ____________________________________________________________________ix_ in the region of the target molecule. Each state also exhibits sinosoidal behavior at longer distances. This is the expected behavior of a resonance. The results of applying Configuration Interaction techniques to the study of the higher energy Feshbach states of HF" are also presented. This investigation included calculation of the potential curves for several excited states of the: neutral molecule HF. These states of HF are referred to as the parent states of the corresponding HF” Feshbach states. Good agreement with the experimental data for the excited HF states was found. One a t tra c tiv e HF" Feshbach state is id e n tifie d in the 12 to 14 eV energy range. The characteristies of this state agree in all major respects with the experi mentally observed features. INTRODUCTION The principles that form the basis of all quantum mechanics were f i r s t introduced in 1926 by Schroedinger.^ Since that time these principles have been extended to become the framework of the theory used today. Although the sim plicity of the theory cannot be questioned, practical implementation of the basic concepts has often proven d i f f i c u l t . For certain applications, including the hydrogen atom as well as other wee, electron systems, the Schroedinger Theory allows exact analytic solution. For larger systems, approximations are necessary. The f i r s t approximation that is generally incorporated into the Schroedinger Theory is known as the Born Oppen- 2 heimer Approximation. The use of this approximation allows the separation of electronic and nuclear motions. Further sim plification can be achieved through a method for dealing with the electron-electron interaction po tential. This approach was f i r s t suggested by 3 4 Hartree, and was subsequently extended by Fock to include exchange effects. In molecular orbital theory, an electron orbital is replaced by a one-electron wave function del ocalized over the whole molecule. This 5 concept, together with the Pauli principle and the work of Hartree and Fock allow the wavefunction to be expressed in a single S la te r 6 determinant. In minimizing 1 the energy of the determinant through the Variation Principle, a set of in te g ro d iffe re n tia l equations called the Hartree-Fock equations are obtained. These equations must be solved i t e r a t i v e l y , and the lowest energy single determinant wavefunction that can be constructed is known as the Self-Consistent Field (SCF) wavefunction. The Roothaan equations^ provide the means of translating the in te g ro d iffe re n tia l equations into algebraic form. The approach allows application of the SCF method to molecules where linear combi nations of atomic orbitals (LCAO) are assumed to represent the molecular o rb ita ls . The difference between the Hartree-Fock energy for a single determinant wave function and the true energy is known as the correlation energy. The magnitude of this energy can be quite large even for small molecules, and the technique most commonly used to calculate the correlation energy is Configuration Interaction (C l). This method involves expanding the electronic wavefunction to include a linear combination of a ll possible products of Slater determinants. Cl is best described as the c a l culation of the optimum linear combination based on the O variational approach of Ritz. Although the Cl technique can be used in principle for diverse chemical problems, in practice full solution of the Cl problem is severely lim ite d. Fortunately, i t is possible to obtain r e l a t i v e l y good accuracy by lim itin g the fu ll problem 2 to a subset of the fu ll determinental basis. Indeed a reasonable estimate of the significant corrections to the SCF basis is possible i f the space includes only 9 single and double excitations. Full solution of most problems at the double e xcita tion leyel is s t i l l p ro h ibitive, and techniques for further optimizing the choice of configurations must be u t i l i z e d . The Cl method used in this work is based on the fact that in any system, there will be a small number of configurations which dominate the final wave- function. These configurations are id e n tifie d and all other configurations are ranked according to their i n t e r action with the small group of important configurations using Raleigh Schroedinger Perturbation Theory, The configurations that interact most strongly are treated exactly while the remaining configurations are treated as a perturbation. I t is this technique of solving only a subset of the entire problem exactly that makes Cl c a l culations on medium-sized molecules possible. Chapter I of this work provides some detail on the historical development of Configuration Interactio n . The specific method of Cl used to obtain the results of the applications presented here is described in Chapter I I . In Chapter I I I , theoretical and experimental cons'i- derations central to the examination of compound states of polar molecules are discussed. Compound states are formed 3 when an incident electron interacts with a molecule. The electron is temporarily captured in the neighborhood of the molecules and the resulting complex is known as a negative ion or resonance state. In the lower energy region, polar molecules can form resonances that are largely dipole-supported. At somewhat higher energies, trad itio n a l valence-type resonances occur. At s t i l l higher energies, the incident electron occupies a Rydberg orbital of the neutral excited molecule, giving rise to a Feshbach state. In Chapter I I I , the general background information on the various types of compound states is presented. In addition, the theoretical basis for the actual binding of an electron by a polar molecule is considered. In Chapter IV, the results of Cl. calculations on the lower energy (< 9 eV) negative ion states of HF" are presented. We find that the lowest HF- state lies below the ground state of HF at all internuclear distances and is therefore capable of binding an electron. Because the calculated binding energy is less than the error inherent within the method used in the calculations, it cannot be stated with certainty that there is a bound state of HF". The minimum dipole concept indicates that a stable negative ion state w ill be formed i f the dipole moment of the neutral molecule exceeds 1.625 D. The HF molecule, with an experimental dipole moment of 1.82 D, 4 is thus expected to support a bound state. When nuclear considerations are taken into account, the minimum dipole moment for binding an electron is increased s li g h t ly . Our calculated dipole moment for HF is 2.01 D, higher than the experimental value by a few tenths of a Debye. The fact that we find HF" to be s lig h tly bound is not unexpected, but is also not s u ffic ie n t proof that the lowest HF~ state is tru ly bound. The results of the calculated potential curves for four other HF“ states are also presented in Chapter IV. The lowest three of these states are formed by occupying a Rydberg orbital with the additional electron; the extra electron occupies: a valence orbital in the fourth state. Since experimental evidence on these states of HF" is v i r t u a l l y nonexistence, i t is not clear whether or not they are resonances. Nevertheless, we present q u a lita tiv e arguments that support the assignment of resonance character to all four states. A detailed examination of the. h ig h e r energy C9-14 eV); Feshbach states of HF" is presented in Chapter V. We f i r s t discuss the results o f.C l.c a lc u la tio n s on the excited states of the neutral molecule, HF. The results of these calculations agree well with the available experimental data. We have performed CT calculations on the Feshbach states of HF" which are formed by the attachment of an additional electron to the excited states 5 of the netural molecule. We find evidence of one a ttra c tiv e state with features that agree well with the experimental observations. There is no doubt that the results of the studies presented in Chapters IV and V cle a rly i l l u s t r a t e the u t i l i t y of the Cl method for detailed examination of the compound states of polar molecules. The work is also s ig nifican t for two further reasons. F irs t , a good explanation of the observed experimental features of the HF" negative ion states is provided. Second, the results predict certain unobserved experimental features and serve as a guide for both experimentalists and theoreticians for further investigation of HF" in p a rtic u la r, and other polar molecules'in;general. References 1. E. Schroedinger, Ann. Physik. 7_9, 361 , 489, 734 (1926). 2. M. Born and J.R. Oppenheimer, Ann. Physik. 84^ 457 (1927). 3. D.R. Hartree, Proc. Camb. Phil. Soc. 2A_, 89, 111 (1928). 4. V. Fock, Z. Physik. 6J_, 1 26 (1 930) and £2 , 795 (1 930). 5. W. Pauli, Z. Physik. 43, 601 (1927). 6 . J.C. S la te r, Phys. Rev. 34, 1293 (1929). 7. C.C.J. Roothaan, Rev. Mod. Phys. 2J3, 69 (1 951 ) and 32, 179 (1960). 8 . W. R itz, J. Reine Angew. Math. 13 5, 1 (1909). 9. A. Pipano and J. Shavitt, 7 CHAPTER I HISTORICAL BACKGROUND The solution of v i r t u a l l y all chemical problems involving atomic and molecular structure is based on the Schrodinger Equation:^ Hy = E¥ ( 1 . 1 ) H, the Hamiltonian operator, represents the total energy of a system. I t is specified by summing the contributions to the Kinetic Energy (T) and the Potential Energy (V). Solutions are obtained in the form of eigenvalues describing the energy of each state of the system. For molecules consisting only of the lig h te r atoms, spin-orbit coupling and r e l a t i v i s t i c effects can be reasonably neglected. With this in mind, the Kinetic and Potential 'k Energy Operators in atomic units can be expressed as T - I - Z L + I - i - Tn + Te (J . 2) A 2MA a 2 ZAZB v ZA . r T I - + I ~ d . 3) A>B r AB Ab r Ab a>b r ab V + V + V N EN EE * In this system of units, ft, the mass of the electron, and the electronic charge are unity. The unit of length - 8 is the bohr (1 bohr = .52918 x 10“ cm), while the unit of energy is the hartree (1 hartree = 27.205 eV). 8 where the Nuclear contributions are denoted as A, B and the electronic contributions by a, b. is the mass of nucleus A, and is the charge of nucleus A. and represent the nuclear and electronic kinetic energy respective!y. describes the repulsion between the nuclei, is the repulsion between electron pairs, and is the e le c tro s tatic attra c tion of all nuclei for all electrons. The Hamiltonian operator can thus be written H = T + T + V +V +V (14) M .N E NN EN EE Specifying the operator for a given system is s t r a ig h t forward, Solution of the eigenvalue problem of Equation ( 1 . 1 ) is more d i f f i c u l t , and has been accomplished exactly in only two instances, those of the hydrogen atom and the hydrogen molecule ion. For systems with more than one electron, adoption of the Born Oppenheimer Approxima- 2 tion leads to a sim p lific a tio n of the problem. This approximation is based on the premise that the nuclei, having mass far greater than that of the electrons, move much more slowly than the electrons. Thus, at any instant, the motion of the electrons is the same as i f the nuclei were in a fixed position in space. When the nuclei are in fixed positions, T^ = 0 , = constant, and the Hamiltonian can be written HE - TE + VEN + VE + VNN 0 - 5 > where H£ now represents the electronic Hamiltonian. Equation ( 1 .5 ) , in which the kinetic energy of the nuclei has been set equal to zero, is the electronic Hamiltonian. I t can be solved for any fixed geometry of the nuclei, and when solutions a re o b ta in e d for a range of nuclear coordinates, a potential surface is defined. The energy eigenvalues for each point on the potential surface can then be used for solution of the nuclear wave functions and the total energy. For the remainder of this work, we will devote our attention to the solution of equation (1 .5 ) . Although nuclear motion has indeed been found to play an i n s i g n i f i cant part in some chemical problems, i t is more important in certain others. In the binding of electrons to polar molecules, for example, i t has been found that including consideration of vibration does not a lte r the conclusions 3 on s t a b i l i t y of negative ions. In .contrast, the condi tions for binding an electron are altered s lig h t ly when rotation is taken into account.^ The Born Oppenheimer Approximation considerably sim plifies the Schroedinger Equation. Nevertheless, i t is s t i l l too complex to allow complete solution of most practical problems, and further approximation is necessary. Over the years, Molecular Orbital Theory has emerged as the most useful approximation for understanding most molecular systems. 10 5 The Variation Principle states that i f y is any well behaved function that s a tis fies the boundary conditions of a problem, then fT*HE4'dx > Eq (1.6) where Eq is the true value of the lowest energy eigenvalue of H^. The "best" set of T's for solving equation (1.6) is that which provides the lowest energy. The true energy can only be obtained by using the exact wavefunction, ¥, which would require an i n f i n i t e set of functions. In practice, i t has been found that lin ea r combinations of atomic o rbitals (LCAO) centered on each atom form a reasonable approximation to the true wavefunction in molecules. These one-electron functions are known as Molecular Orbitals (MO's). The simplest approximate wave function for an N-electron system is the product of N one-electron basis functions f i r s t introduced by Hartree.^ T (1 . . . N) = < |> 1 (l)<J>2 (2)...<j>N(N) (1.7) The 4>(N) are products of a space component and a spin component, a or $ . 7 The additional requirement that the product wave functions must be antisymmetric to the exchange of any pair of electrons leads to the expression 8 of the wavefunction in the form of a Slater determinant. 11 ¥ / N ! <f>1 ( 1 ) < f> -| ( 2 ) < j> N ( N ) ^ ( 1 ) ^ ( 2 ) .................( M ) <j>2 ( 3 )<f>2 ( 2 ) ............. < j> 2 ( N ) V N) ( 1 . 8 ) This wave function, called the Hartree-Fock wave functions is the "best" function that can be constructed by assigning each electron to a separate o r b i t a l . In minimizing the energy of equation ( 1 . 6 ) , a set of in te g ro d iffe re n tia l equations called the Hartree-Fock equations are derived. The true Hamiltonian and wave function involve the coordinates of all N electrons, while the Hartree-Fock Hamiltonian is a one electron operator. The Hartree- Fock equations are He f f (l)<J>p(l) e < j> ( 1 ) P P (1.9) where e is the orbital energy. The Hartree-Fock P Hamiltonian can be written ,,2 , ef f H - I — - I — + I J, - I K , a 2 Aa r Aa i ( 1 . 10 ) The f i r s t term is the operator for the kinetic energy of each electron; the second term is the potential energy operator for the attra c tio n between each electron and the nuclei. The Coulomb operator, J ., is the operator representing the e le c tro s ta tic repulsion between each 1 2 electron and all other electrons. I t has the form i j J 1 r ab 1 The fourth term, K., known as the Exchange Operator, has no simple physical in te rp re ta tio n , but arises from the requirement of antisymmetry of electrons to exchange. I t can be written K.4>i = 4>-jdT (T.12) 1 3 1 ~ r ab J The Hartree-Fock equations must be solved i t e r a t i v e l y . Generally, an i n i t i a l guess provides a set of i n i t i a l wave functions, which in turn lead to an improved potentia l. This improved potential is then used to obtain improved wave functions. The process is repeated until the energy reaches a reasonable s t a b i l i t y . The Hartree-Fock energy is EHF = 2 I ei " I I J i j " Ki j ) 0 - 13) T 1 J The Hartree-Fock equations can be solved in closed form only for a limited number of systems. Numerical 9 integration provides f a i r l y accurate results for atoms, while for molecules, analytic basis functions are generally employed. The s e lf - c o n s is t e n t - f ie ld (SCF) wave function is the lowest energy single determinant wave function that can be constructed within a f i n i t e basis. The analytic approach developed by Roothaan, ^ 0 involves expanding the o rb ita ls as a complete set of basis 13 functions, f^. That is, ♦i ■ p n f k (1- 14) Substitution of this expression into equation (1.9) gives £ cik »e f f f k - ei I c ik f k M u ltip lica tio n by f . and subsequent integration leads to J I Cik {Hj k f " ei Sjk^ = 0 j = l , 2 , . . . (1.16) i\ where " j " - i f *i H e f f f kdv ( 1 - 1 7 ) and sjk = -ff j f kdv (1- 18) Equations (1.16) are a set of simultaneous lin e ar homo-: geneous equations in the unknown c o e ffic ie n ts , C ^ . In order to obtain a nontrivial solution, d0t ( H j ( f “ e i Sjk^ = 0 (U 1 9 ) must be s a tis fie d . Equations (1.16) must be solved p " P -F i t e r a t i v e l y since depends on the orbitals < |> . , which in turn depend on the coefficients Cik* The advan tage of this approach is that i t reduces the eigenvalue problem to a matrix problem. In molecular systems, the choice of a set of basis 14 functions, the of equation (1 .1 4 ), is extremely important. These basis functions are generally centered on each atom, but e f fe c tiv e ly span the appropriate molecular space. Exponential functions of the form Nr 11-1 e"a r Y™(e ,cf>) (1.20) are known as Slater-type o r b i t a l s . ^ N is a normalization constant, n is the principal quantum number, and a is the screening parameter. Slater o rb ita ls are useful for many applications, but they do not possess the proper number of radial nodes and therefore do not provide a good representation of the inner part of the o r b i t a l. In addition, for large molecular calculations, they are not economically practical because of the d i f f i c u l t y in solving the two electron integrals. In contrast, these same integrals can be evaluated r e l a t i v e l y easily i f Gaussian functions are used. These functions are of the form Mxay bzce" 3 r 2 ( 1 . 2 1 ) where the x, y, z represent the angular dependence, and a, b, and c are integers. These functions are less appropriate for describing molecular systems than Slater type functions. Frequently, however, the descrip tion may be improved by contracting several Gaussians together to act as one function. In spite of the fact that greater accuracy can almost universally be obtained from the use of Slater-type O rb itals , Gaussians have received wide use because of the re la tiv e ease of the two-electron integral evaluation. The smallest basis set that can be used for any system is a minimum basis set. in these cases, either Slater type Orbitals or contractions of three Guassians are generally employed. For the oxygen atom, for example, a minimum basis set would include Is, 2 s, and 2 p functions. Since the calculated energies obtained from the mini mum basis sets are usually rather far above the Hartree-Fock energies, larger basis sets are often u t i l i z e d . One- type of basis set, the double zeta, contains twice as many functions as the minimum basis. Any basis set larger than the double zeta is an extended basis. Extended basis sets comprised of prim itive Gaussian functions can require an often prohibitive amount of time for solution of the SCF equations, and may not lead to convergence of the energy. To ameliorate this problem, contracted 1 2 gaussians or linear combinations of gauassians with fixed coefficie nts are frequently used. A further step for improvement in molecular description can be obtained by inclusion of polarization 1 3 functions. These are functions of higher £ value than required by any atom in the molecule. For the molecule, for example, a polarization function would be 16 a p function. Many properties of chemical in te re s t, including dissociation energies and dipole moments,can be r e lia b ly calculated only i f polarization functions are included. Regardless of the size or type of basis set, the Hartree-Fock approximation neglects an important c o n t r i bution to the total energy. The correlation energy, or the energy of the instantaneous repulsions between pairs of electrons, is the difference between the Hartree-Fock and the exact energy of a system. In spite of the fact that this energy represents only a small percentage of the total energy (less than 1 percent for lig h te r atom s^), its absolute magnitude may be as high as 10 eV. The Hartree-Fock energy is p a rtic u la rly inadequate when calculating potential curves of small molecules where the correlation may vary considerably as the molecule is stretched. The most frequently used technique for dealing with the problem of electron correlation is Configuration Interaction (C l). The e a r l i e s t calculations of this type 1 5 were performed in 1927, and were necessarily limited to very small systems. With the advent of the computer, the a b i l i t y to handle larger systems, atoms and their 16 17 negative ions, as well as small molecules, was established. The advantage of the Cl method lie s in its sim plicity and universal a p p lic a b i lit y . I t can be 1 7 used to calculate ground and excited state energies for molecules, atoms, negative ions, or transition complexes. In addition to the Hartree-Fock o r b ita ls , there are an i n f i n i t e number of other o rb itals that are eigen functions of for any atom or molecule. These addi tional o rbitals can be used to construct configurations other than the Hartree-Fock configuration. With the inclusion of all configurations formed from an i n f i n i t e number of o r b ita ls , the Schroedinger Equation can be solved exactly. In practice, of course, a complete set of these configurations cannot be used and the problem is generally truncated at some reasonable lev el. The C .I. wave function has the form * = 2 Cn* n ( 1 . 2 2 ) where the (f>1 s are an orthonormal set of n-electron configu rations. Equation (1.22) is a lin ear variation function, and the co e fficients are determined to minimize the energy. Application of the variation, principle leads to the determinantal equation ( H - El ) C = 0 (1.23) where the E are the eigenvalues and the C1s are the matrix of eigenvectors. The matrix element H represents the interaction between two determinants or configurations, D and D P q 18 (1.24) In a pa rtic u lar Cl c alcu latio n, i f N configurations are included, the N energies or eigenvalues w ill be obtained in the solution of equation (1 .24 ). A set of co e ffic ie n ts , C , which define the Cl wave function is associated with each energy. The solution of equation (1.24) is greatly sim pli fied by the fact that the matrix elements between J pq two configurations p and q of d iffe r e n t symmetry are id e n t ic a lly zero. Moreover, an.additional sim plific a tio n results from the expression of these matrix elements as the sum of integrals in the orthonormal one electron basis. I t can be written in terms of the one-electron operator of the Hamiltonian, The determinants of equation (1.24) may be equivalent or they may d i f f e r in occupation by one or more spin o r b ita ls . In the case where the determinants are id e n tic a l, the matrix element between them is hi;) (- (1 .25) and the two-electron Hamiltonian, Vi jki = J^i (b) j ( b) {-p—}' < J > k (a)<|>-| (a)dxadTb (1.26) a b occ so occ i j j i 1 9 For the case where the determinants d i f f e r by the occupa tion of one spin orbital ( i > j ) , OCC SO Hmm = hi j + I (Vi j k k “ Vikk j ^ ( 1 - 28) Occupations which d i f f e r by two spin o rb ita ls can be represented as Hmn ■ <v i j k i - vm j > ' K Z 9 > I f the two determinants d i f f e r by more than two spin- o rb ita ls , equation (1.24) is id e n tic a lly zero, which s ig n ific a n tly reduces the potential size of the calcula tion. Nevertheless, within any basis set larger than a minimum basis, Cl calculations even for small molecules can be extremely large. I t is therefore important to id e n tify techniques for further simplifying the problem. One tra d itio n a l method for lim itin g the size of a Cl calculation is the frpzen-core approximation. This method involves freezing the occupation of the M.O.'s that play a neglgible part in contributing to the energy of the desired property. For example, in calculating the transitio n energy from the ground to an excited state of the CO molecule, the Is M.O.'s of carbon and oxygen might be dropped from consideration. They contribute very l i t t l e to the energy difference, since they are basically atomic in nature. In cases of this type, where the M.O.'s 20 dropped from consi deration have only a t r i v i a l e ffe c t on the results, the frozen core approximation is j u s t i f i e d . A lte rn a tiv e ly , attempts to separate the sigma and pi space of certain molecules using this approximation have been 1 8 unsuccessful. I t has been found, for example, that the correlation from the sigma space contributes unequally to the pi states of some systems. Therefore, both parts of the space must be included to obtain reasonable results. Another method of lim itin g the Cl problem is to r e s t r i c t the number of configurations. Generally, a p a rticu lar state is dominated by only a few main configura tions which together comprise more than 90 percent of the final wavefunction. The only configurations which in te rac t d ire c tly with the dominant configurations are those that d i f f e r by two or few er.occupations. Those configurations d iffe r in g by more than two occupations in teract d ire c tly with the corrections, but only in d ire c tly with the dominant configurations. In a study q of the BH3 molecule, only a few t r i p l e and quadruple excitations from the ground state were found to c o n t r i bute to the final wave function. Indeed, i t is common practice in Cl calculations to include only single and double excitations from the dominant configurations. Other techniques for reducing the size of the Cl problem focus on simplifying the construction and diagonalization of the HamiItonian matrix. One method 21 that handles the problem in d i r e c t l y is based on selection 20 of a small subspace of the f u l l matrix for diagonlization. The remaining space is treated as a perturbational sum. The threshold for selection, based on the interaction with a few dominant configurations, is varied. The total energies obtained by varying the threshold can be extrapo lated to obtain the energy of the f u ll space. A second method for lim itin g the problem involves the direct calculation: of only the diagonal matrix elements and a 21 small strip of the fu l l matrix. Another technique for reducing the problem to manageable proportions is described in the next chapter. This method was used to obtain the results of the studies presented in Chapters IV and V. I t involves a p a rtitio ning of the Hamiltonian so that direct diagonalization can be avoided and, in addition, presents an e f f i c i e n t method for calculating matrix elements. The f u l l details of this technique are presented in references 22 and 23. 22 References 1. E. Schroedi nger, Anna!. Physik, _79, 361 , 489, 734 (1926), Phys. Rev. 28, 1049 (1926). 2. M. Born and J.R. Oppenheimer, Anna!. Physik, 8^, 457 (1927). 3. O.H. Crawford, Mol. Phys. 20, 585 (1 971 ). 4. O.H. Crawford, J. Chem. Phys. 66>, 4068 (1 977). 5. I . N. Levine, Quantum Chemistry, Vol. I, Quantum Mechanics, and Molecular Electronic S truc ture , A llyl and Bacon, Boston, 1970, p. 184. 6 . For a review see D.R. Hartree, Repts. Prog. Phys. JJ_, 1 1 3 (1 946-7). 7. W. Pauli, Z. Physik, 43, 601 (1927). 8 . J.C. S la te r, Phys. Rev. 3_5, 21 0 ( 1 930). 9. D.R. Hartree, The Calculations of Atomic Structures, John Wiley and Sons, New York, 1957. 10. L.C.J. Roothaan, Rev. Mod. Phys. 13, 69 (1951). 11. J.C. S late r, Phys. Rev. _36 , 57 (1 930). 12. E. Clementi and D.R. Davis, J. Comput. Phys. 2 , 223 (1967). 13. R.K. Nesbet, Rev. Mod. Phys. 32, 272 (1960). 14. P.O. Lowdin, Adv. Chem. Phys. 1, 207 (1959). 15. E.A. Hylleras, Z. Physik, 48, 469 (1928). 16. S.F. Boys and V.E. Price, Phil. Trans. Roy. Soc. (London), A246, 451 (1954). 23 17. S.F. Boys, G.B. Cook, C.M. Reeves, I. Shavitt, Nature, 178, 1207 (1956). 18. S.D. Peyerimhoff, R.J. Buenker, W.E. Kramer, and H. Hsu, Chem. Phys. L e tt. £5, 1 29 ( 1 971 ). 19. A. Pipano and J. Shavitt, In t. J. Org. Chem. 2 , 741 (1968). 20. R.J. Buenker and S.D. Peyerimhoff, Theor. Chim. Acta, 35, 33 (1974) and 39, 217 (1975). 21. Z. Gershgorn and I. Shavitt, In t. J. Org. Chem. 2 , 75. (1968). 22. R.W. Wetmore and 6 .A. Segal, Chem. Phys. Lett. 36, 478 (1975). 23. G.A. Segal, R.W. Wetmore and K. Wolf, Chem. Phys. 30, 296 (1978). / 24 CHAPTER I I METHOD The construction and diagonalization of the matrices necessary for the solution of the Cl problem for small and medium sized molecules (> 20 M.O.'s) is p ro h ibitive. Various techniques for truncating the space so that fu ll solution can be avoided have already been mentioned. The purpose of this chapter is to describe one such technique which has been used with considerable success in numerous cases. The method described here re lie s on the fact that for most chemical systems of interest-, there exists a small set of configurations that together dominate the Cl wavefunction. This set of configurations is called the core. All remaining configurations, deemed the t a i l , have a less sig n ific a n t contribution to the final wave- function. The main configurations can be id e n t ifie d either through a preliminary calculation or by scanning the Cl matrix diagonal for the terms of lowest energy. These configurations can be gathered together to form the nucleus of the ca lculation. The interaction of all remaining configurations with those of the nucleus is then tested using Raleigh; Schroedinger Perturbation theory. The configurations which in te ra c t strongly with the nucleus are id e n tifie d and grouped with the nucleus to complete the core. Other configurations which interact less strongly are also grouped together, in this case to form the t a i l . These basic concepts suggest a pa rtitio n in g of the matrix- eigenvalue problem as f i r s t proposed by Lowdin.^ I f the core is denoted by H and the ta il by H .u, this 17 aa J bb can be expressed H H . aa a b H, H, , ba bb c C a a = to c b c b ( 2.1 ) where C and C. represent the c o e ffic ie n ts . This matrix a. D equation can be reformulated into two simultaneous equations in two unknowns. H C + H . C. ~aa ~a ~ab ~b H, C + H., C. ~ba ~a ~bb ~b to C ( 2 . 2 ) (2.3) Solving for in equation ( 2 . 3 ) , and substitution of the result into equation ( 2 . 2 ) leads to H , _ C + H , [ to I - H.u ] _1 H. C = ojC ~aa ~a ~ab ~ ~bbJ ~ba ~a ~a i-l (2.4) where the term [wl - H^^]- corresponds to a kind of optical p o ten tial. Although equation (2 .4) reduces the diagonalization problem to the size of the core block, H , evaluation a a of the inverse term is s t i l l at least as d i f f i c u l t as the original d iag o n aliza tio n . The inverse matrix w ill 26 always exist and can be made diagonally dominant provided that all terms of Hbb whose diagonal elements, hbb, l i e close to a) are placed in the core block, H . Assuming a a this to be the case, the inverse can then be expressed in the following manner. Substitution of equation (2.5) into equation (2.4) leads to Although this expansion is energy dependent through the terms in w, the dependence arises only in the diagonal terms, D. Diagonalization of the H _ block can a a provide an estimate, o j , of the true eigenvalue, w. The diagonal can again be expanded, this time in terms of the difference between the approximate and the true eigenvalue, Aw. Thus, [wl - Hb b ] -1 = D' 1 + D_1OD_1 + D“1OD"1OD" 1 + where 0 represents the off-diagonal terms ( 2 . 6 ) and is the inverse diagonal D " hbbP (2.7) ( 2 . 8 ) D"1 (w) = D"1 (o>) I ( D - ^ w l A w ) " ~ u ~ u n = 0 ( 2 . 9 ) 27 Substituting this result into equation (2.8) gives {” aa + + 5 ' 1 (“ o'??"’ <“ o> + " ^ U b a + Ha b [ D _ 2 (o)o )Aa) + D " 2 (a)o ) O D _1 ( w 0 )Ao) + D ^ ( t o _ ) O D 2 (to )Aco + • • • ] H, + . . . }coC = wC ~ O ~ ~ O - D a ~ d ~ a ( 2 . 10 ) The advantages of equation (2.10) are more readily apparent i f i t is written in another way 00 , . CHaa + I V‘ " ' Ac"] C = UC (2.11) -aa n = 0 -aa -a -a In equation (2 .1 1 ) , the dominant contributions to the fin al wavefunction, and those terms that interact with them strongly, together form HL . This subspace can be a a treated f u l l y by exact diago n aliza tio n . All other terms, which in teract less s ig n if ic a n t ly , can be treated simply as a p o te n tia l, V(to). Equation (2.11) could, in principal be solved i t e r a t i v e l y for a complete solution to the problem. Fortunately, reasonable accuracy in the solution of most problems can be attained without f u ll solution by truncation of the p o te n tia l. Close examination of equation ( 2 . 8 ) reveals that the terms have a one-to-one correspondence with successively higher B r i 11 ouin-Wigner Perturbation corrections. The term, H i D ^ H . C '„; is ab ba a related to the second order B r i 11ouin-Wigner perturbation 28 correction; the next term is related to the third order correction, and so on. Tests performed on small, medium, and large matrices indicate that for most practical problems, s u ffic ie n t accuracy can be achieved by retention 3 of corrections through second order. I t is this result that makes the approach of value for the large scale Cl calculations described in Chapters IV and V, The method is p i c t o r i a l l y described in Figure 11 -1 and can be summarized as follows. The configurations which are most important to a p a rtic u la r calculation are id e n tifie d and gathered together in the upper l e f t hand corner of the core block, . The interaction of each a a of the configurations in the nucleus with all other con figurations is evaluated and becomes part of Hab' Using Raleigh-Schroedinger Perturbation Theory, the most important of the remaining configurations are gathered together with the nucleus to complete H, . the core a a block. The interaction of these configurations in the core block with all other configurations forms the balance of H b . The remaining configurations form the t a i l , H ^ . The Hfla block is then diagonalized to provide an i n i t i a l guess for the energy. The potential function is evaluated, and f i n a l l y , the f u l l problem is solved through it e r a t io n . In addition to providing a unique solution for solving the eigenvalue problem, the method used in this work also includes techniques for simplifying calculation of the matrix elements, the other time- consuming step in a Cl ca lculation. A complete descrip tion of the approach can be found in references 2 and 3. 30 References 1. P.O. Lowden, Adv. Chem. Phys. 2 , 207 (1959). 2. G.A. Segal and R.W. Wetmore, Chem. Phys. L ett. 36, 478 (1975). 3. G.A. Segal, R.W. Wetmore and K. Wolf, Chem. Phys. 30, 296 (1978). 31 Figure 11 -1 P artitioning of the' Hamiltonian Matrix 32 Haa Core Hab = Hba 50 Hbb Tail Hba Full Space CHAPTER I I I RESONANCES OF POLAR MOLECULES In recent years, the electron scattering and binding properties of polar molecules have been the subject of considerable in te re s t. A number of interesting effects resu lt from the very long-range interaction between a molecule with a permanent dipole moment and a charged p a r t i c l e . The Configuration Interaction Method described in the la s t chapter has been used with considerable effectiveness to study the binding and scattering characteristics of one such polar molecules, hydrogen flu o rid e . The results of this study are presented in Chapter IV for the lower energy region (< 9 eV) and Chapter V for the higher energy range (9 to 14 eV). Before moving to this discussion, however, a basic under standing of the nature of compound states is useful. I t is the purpose of this chapter to provide some general background on negative ion resonances. A compound state is formed by the interaction of an incident electron with a target molecule. The incident electron is temporarily captured within the neighborhood of the molecule and a complex, called a temporary negative ion or a resonance, is formed. The term resonance implies a d e fin ite energy, and sharp structure is observed in the cross section. The f i r s t evidence on the existence 34 of compound states appeared in 1921, but i t was not until the 1960's that the resonance model was applied to mol ecu! es Molecular compound states have lifetim es in the range of 10~^° to 1 0 ~ ^ sec. The lif e tim e x can be described as T = -p ( 3 . 1 ) where r is the width. These states decay by the emission of an electron into various fin a l states where they can be detected experimentally. The decay channels include r o ta tio n a l, v ib ra tio n a l, and electronic e x c ita tio n , e la s tic scattering and dissociative attachment, to name a few. A shape or single p a rtic le resonance is formed when an electron is trapped in the potential or behind the centrifugal ba rrier of the mol ecu!arstate . These types of resonances; occur at energies below about 10 eV and have been observed in D£, 0^, HD, N£, NO, and C0.^ Core excited resonances, which occur at energies above 10 eV, consist of a "hole" in one of the normally occupied o rb ita ls of the molecule and two "particles" in normally unoccupied o r b ita ls . The excited neutral molecule is called the parent of the negative ion state, while the positive ion is referred to as the grandparent. Core excited resonances that l i e below th e ir parent are e n title d Feshbach resonances. They have life tim e s that are long compared to a vibrational period and can ___________________________________________________ 3 5 therefore give rise to band structure. The two outer electrons are held in Rydberg o rbitals which l i e fa r from the ion core and as a re s u lt, these bands exhibit v ib ra tional structure similar to the grandparent. Single P a rtic le Resonances In recent years, low energy electron impact experiments have revealed pronounced structure in the 2-4 vibrational excitation cross sections of polar molecules. Within about 0.5 eV of threshold, the observed cross sections are larger by 10 to 100 times than would be predicted by the Born approximation. That is, these very large cross sections cannot be simply a ttrib u te d to c o llisio n a l momentum tran sfe r. The sharp peaks are rather the re suit of pronounced distortion of the incident electron by the potential well of the target molecule. The resonances that arise from the interaction are domina ted by very few p a rtia l waves and the symmetry of the resonant state is re fle cte d in the angular dependence of the cross section. There has been considerable in te re s t in the question of whether or not a neutral polar molecule is capable of binding an electron to form a stable negative ion state. 5-10 Several authors have shown that the e l e c t r ic dipole f i e l d of such a molecule can bind an electron i f the dipole moment is greater than 1.625 D. To understand 36 this resu lt we can consider the scattering of an electron by a point dipole. In a polar molecule with e le c t r ic dipole moment y and an electron at distance r, the potential is given by: (3.2) where 0 represents the angle between the vectors y and r Schroedinger' s equation becomes V 2 + R2 e cos v fj ( r ) = 0 (3.3) Setting o 2m eE .2 _ o-e " F and the dimensionless dipole moment, . 2 Cm y 2me y 0 ea (3.4) (3.5) where aQ is the bohr radius, equation (3.3) takes the simple form 0 y„ cos 0 9 v 2 + + k2 ip ( r ) = 0 (3.6) Separation of equation (3.6) in terms of spherical coordinates leads to equations in the three variables R, 0, $. 37 dr' 2 d_ r dr + k' A J— — sine — R ( r ) = 0 M‘ sin 0 d e de o - - \in cos e - A sin2 e 0 and ♦ M2 d 4 > $ ( 4>) = 0 (3.7) e(e) = o (3.8) (3.9) where A is a separation constant and M, an integer*.- For the lim itin g case, as E 0, equation (3.7) becomes d + 2 d A 2 "2 dr r dr r R ( r ) = 0 (3.10) A stationary solution of equation (3.10) takes the 12 form R( r ) ^ r . Substitution of this solution leads to S(S+1) - A = 0 Solutions with A < 1/4 have the form (3.11) R ( r ) = (3.12) where B is a constant. Solutions with A > take the form R(r) r - ^ ^ cos ( A - l / 4 ) ^ ^ log r (3.13) I t should be noted that equation (3.13) produces an i n f i n i t e number of zeros and thus an i n f i n i t e number of nodes, while equation (3.12),, leads to only one node. 38 We now return to equation ( 3 .8 ) . For M = 0, the solutions are $(0) = I Cfi,P£ ( C0S 0) (3.14) the Legendre Polynomials. Substitution of these solutions into equation (3.8) leads to where the £ are integers > _ 0. When A = 1/4, equation (3.15) can be written where y (MTN) is the minimum dipole moment. For this case, and for A < 1/4, there are no negative energy levels and therefore no bound states. A lte r n a t iv e ly , when A > 1/4, there are an i n f i n i t e number of negative energy le vels, and thus an i n f i n i t e number of bound states. The implications of this result are in terestin g . The case where A 5 1/4, since i t has no zeros, corresponds to the state with E = 0, the lowest le v e l. As E -> ■ 0, the negative ion becomes degenerate with the neutral molecule, and the state for the case where A = -1 /4 has the minimum dipole moment for binding. Both analytic and numerical techniques have been employed to determine the value of + C ^ £(£+1 ) - C„A + y (3.15) ( 2 £ - l ) b£-l + ( &+1 ) (2£+3)b£+l 0 (3.16) 39 y0 (min)in equation (3 .1 6 ). The solutions lead?, to the conclusion that when the dipole moment is less than 1.625 D, there are no bound states, while for a dipole moment greater than this value, an i n f i n i t e number of bound states ex ist. Although the c r i t i c a l value of the dipole moment for binding an electron has the value 1.625 D in the conven tional Born-Oppenheimer treatment, this result is modified to some extent with the inclusion of the rotational degrees of freedom of the nuclei. To account for rotation, the Hamiltonian for a symmetric rig id rotor would contain the p j 2 ^ 2 2^2 term - ^j where h j is the operator for the square of the angular momentum and I is the moment of i n e r t i a . This term does not contribute to the ground state energy of the neutral system. However, when an extra electron is present, the interaction of this electron with the added term in the Hamiltonian acts to raise the energy of the 1 3 ground state of the ion r e la t iv e to neutral system. This follows from the fact that the rotational angular momentum of the dipole and the orbital angular momentum of the incident electron are coupled to give the total angular momentum which is conserved. The resu lt is that some or all of the bound states of the electron are moved to the continuum. The exact value of the minimum dipole for binding varies, depending on the values of I and the _______ : _________________________________________ 40 internuclear distance in the molecule. However, no binding is found to occur for dipole moments less than about 2D. The stationary dipole has a very small electron a f f i n i t y when the dipole is less than 2D. For dipole moments in the range of about 2.1 to 2.3 D, the electron a f f i n i t y is much large r. When the e f fe c t of rotation is included, only one bound state is supported for a dipole moment of 2 D. For higher values of the dipole moment, two or more bound states e x is t. The effects of molecular vibration on the binding 1 4 of polar molecules have also been investigated. As long as the dipole moment of interest is the average of the dipole moment over the ground state vib ra tio n , the minimum dipole moment for binding is not a lte re d , except in cases where exothermic dissociative attachment can occur. There is some indication that induced dipole forces are a very important contribution to the energy of the weakly bound electron, even for systems with larger per manent dipole moments. Garrett suggests that a strongly polar molecule with a dipole moment greater than about 4D w ill almost surely form a stable negative ion. For molecules with dipole moments in the range of 2 to 3.5 D, a negative ion with binding energy greater than 0.01 eV w ill be formed i f the p o la r iz a b i1ity is between 20 and 40 aQ3 (3 and 6 A3 ) , 15 41 As a test of the minimum dipole moment concept, ab i n i t i o studies of the electron a f f i n i t i e s of polar mole- 16-18 cules with various dipole moments have been performed. One source of d i f f i c u l t y in calculations of this type is the lack of accuracy in the computed dipole moments. Even with wavefunctions of near Hartree-Fock q u a l i t y , c a lc u la ted dipole moments are generally higher than the experi mental values by several tenths of an eV. Because of the strong dependence of the electron a f f i n i t y on the dipole moment when the dipole moment is close to the c r i t i c a l value, i t might be thought that calculations of this type are not p a rtic u la r ly useful. This, however, is not the case. Ab i n i t i o techniques, while they cannot provide completely accurate binding energies or predic tions of- binding, can indeed serve as at least a q u a lit a tiv e guide on the binding c a p a b ilitie s of polar molecules. One group in p a rticu lar has performed ab i n i t i o 17 18 calculations on several polar molecules including LiH“ , ’ NaH- , 1 7 , 1 8 BeO~ , 17 LiF" , 17 and LiC l" . 1 6 , 1 7 The results of these theoretical studies were used, together with the available experimental data, to assess the v a l i d i t y of the simple fixed f i n i t e dipole model for predicting binding. The technique used in these investigations is straightforward. Hartree-Fock calculations were performed for the negative ions lis te d above. Special attention was paid to choosing an adequate basis set for each 42 p a rticular anion. Diffuse functions with optimized expo nents were added to the e le c tro -p o s itiv e atom to permit the extra electron to attach to the positive end of the polar molecule. The difference between the Hartree-Fock energies of the neutral molecule and the negative ion give the electron a f f i n i t y . The orbital energy for the lowest unoccupied orbital (LUMO) for the neutral molecule in the Hartree-Fock calculation can be used to estimate the magnitude of the binding c a p a b ility . A negative orbital energy-imp!ies a stable negative anion w ill be formed, while a positive o rbital energy implies that the neutral molecule is not capable of binding an electron. This is simply Koopman's Theorem Approximation. For LiH, BeO, NaH, and L i F with respective dipole moments of 5.88D, 7.41D, 6 . 9 8 D, and 6.33D, negative orbital energies were obtained. This is not surprising, since all of these molecules have dipole moment s ig n if ic a n tly in excess of the minimum dipole moment for binding. Unfortunate!y comparison of the ab i n i t i o results with experiment is not possible since the energies for binding an electron to these molecules.have not yet been deter mined. One molecule for which the experimental binding 1 9 energy has been measured is LiCl, which has a dipole moment of 7,13 D. The calculated binding energy for this molecule of 0.54 e V ^ compares rather well with the experimental value of 0.61 eV. From this example, the 43 only case where an experimental binding energy is a v a i l able, we can conclude that ab i n i t i o studies are capable of providing f a i r l y accurate values for binding energies. Jordan et a l . discuss in detail the lim ita tion s of 17 18 th e ir method for calculating exact binding energies. ” The Hartree-Fock calculations neglect two very important contributions to the binding energy: orbital relaxation and correlation corrections. In the calculations, the electron a f f i n i t y was calculated by taking the difference between the Hartree-Fock energies of the neutral molecule and the anion. The binding energy of 0.54 eV for LiCl cited above was obtained in this manner. To derive estimates of the contributions not included in the Hartree-Fock technique, Jordan et al . u t iliz e d the Equation of Motion (EOM) Method. In this method, electron a f f i n i t i e s are calculated d i r e c t ly without the need for performing calculations on both the neutral molecule and its anion. The electron a f f i n i t y attained using this method inherently includes the second order correlation and o rbital relaxation corrections as well as the third and higher order corrections to the Koopman's Theorem 1 5 estimates. The difference between the orbital energy of the LUMO of the neutral molecule in the Hartree-Fock calculations and the electron a f f i n i t i e s from the EOM calculations is thus the orbital relaxation and correlation corrections. For the molecules LiH, LiF, BeO, and NaH, 44 the binding energies obtained through the EOM calculations indeed give a larger value for the binding energy than those obtained through the Hartree-Fock calculations. For LiCl, for example, the binding energy obtained through the EOM method is higher by 0.13 eV than that obtained from the Hartree-Fock calculations. Jordan et a l . ^ maintain that the orbital relaxation and correlation effects would be expected to be small in the case of LiCl. This follows from the fact that the additional electron is located prim arily behind the ele ctrop os itive atom in a molecular orbital that is largely nonbonding in nature. This nonbonding MO does not co rrelate strongly with the other electrons in the molecule. The good agreement between the calculated binding energy and the experimental data c e rtain ly support this in te rp re ta tio n . The contributions not included in the Hartree-Fock calculations are included in configura tion in teraction calculations. As we shall see in Chapter IV, Cl techniques have been applied with considerable success to determine the binding energy of HF. This molecule has a dipole moment of 1.82 D, only s lig h t ly larger than the minimum dipole for binding and, as such, is an excellent candidate for investigation. The calculated results for LiCl suggest that rather good agreement between ab i n i t i o studies and experiment for binding energies is possible. The investigations on 45 LiCl also provide the oportunity to evaluate the c a p a b ili ties of the fixed f i n i t e dipole model in predicting binding energies. Jordan and Luken have examined the accuracy 1 8 of the dipole model in some d e t a il . For a molecule with a dipole moment of 7.2 D {the dipole moment of L i C l ) , the model predicts a binding energy of 0.08 eV. This is well below the experimental binding energy of 0.6 eV, and poses serious questions about the a p p lic a b i lit y of the model. Jordan and Luken a t trib u t e at least part of the discrepancy to what can be called penetration e ffe c ts . In the Lithium atom, the 2s electron is "pulled in". That is , the 2s electron penetrates the Is shell and is permitted to feel the nuclear charge of +3. The fixed f i n i t e dipole model f a i l s to account for this behavior. I f the ground state of the dipole model is correlated with the negative ions of molecules, incorrect nodal behavior is predicted. One must instead correlate the negative ions with the f i r s t excited state of the dipole model. Correlation with this state, however, leads to an underestimate of the binding energy of a real molecule. From the LiCl study, i t can be concluded that although the simple dipole model may be of some q u a lit a tiv e use., its a b i l i t y to predict binding energies of real molecules appears somewhat lim ite d. In Chapter IV, where we examine the binding c a p a b ility of HF, this has important im plications. The binding energy of HF ________________________________________________________________________ 46 _ 5 as predicted by the dipole model is on the order of 10 eV, while our calculations indicate the binding energy to be higher by several orders of magnitude. Core-Excited Resonances Core-excited resonances, as mentioned e a r l i e r , are characterized by two electrons in normally unoccupied MO1s and a "hole" in a normally occupied MO. When these negative ion resonances l i e below the excited state of the parent or neutral excited molecule, they are known as Feshbach states and the parents are said to exhibit a positive electron a f f i n i t y . The binding energy of a Feshbach state is defined as the difference in energy between the positive ion state and the negative ion state formed by adding two outer electrons to the positive ion core. The primary experimental means of detecting Feshbach states is electron transmission spectroscopy-. This technique involves measurement of the unscattered trans- mitted current as a function of electron energy when mono- energetic electrons are accelerated into a c o llis io n cham ber f i l l e d with the target gas. Experimental results are generally presented with the ordinate representing the derivative of the transmitted current and the abscissa, the electron energy. In many cases, sharp structure which mimics the vibratio nal spacing of the grandparent ____________________________________________________ 47 state is observed. Core-excited resonances of atoms have been studied 20 extensively, p a r t ic u l a r ly those of the rare gases. These resonances have also been detected in many diatomic 21 molecules. In p rin c ip le , a core-excited state of a molecule may consist of an electron temporarily bound e ither to a valence or Rydberg excited state. Only the Rydberg excited states, however, lead to negative ion states that have a positive electron a f f i n i t y for a fixed internuclear separation in the Franck-Condon region. Thus for Rydberg excited states, sharp resonances which l i e somewhat below the Rydberg excited states are expected. The vibrational spacing of the resonances should be sim ilar to that of the grandparent, since the two excited electrons re side rather fa r from the positive ion core and therefore perturb i t only s li g h t l y . In many cases, vibrational progressions overlap leading to confusion in id e n t i f i c a t i o n . A further complication is that the width of a core-excited resonance can change due to the opening of a new decay channel. I f , for example, the new decay channel is a repulsive negative ion state , vibrational progressions may be observed only for a lim ited number of levels ; that is, the vibrational progression may p a r t i a l l y predissociate. This situation can arise from an avoided crossing between two states of the same symmetry. As we shall see in Chapter V, this 48 explanation may apply to an experimental observation where only one vibrational level is excited in a study of the Feshbach states of HF . 49 Reference s 1. G.J. Schulz, Rev. Mod. Phys. 4_5, 423 ( 1 973). 2. K. Rohr and F. Linder, J. Phys. B: Atom. Molec. Phys. 9, 2521 (1976). 3. K. Rohr, J. Phys. B: Atom. Molec. Phys. 1_0, 1175 (1977). 4. K. Rohr, J. Phys. B: Atom. Molec. Phys. 1_1_, 1849 (1 978) . 5. M.H. Mittleman and V.P. Myerscough, Phys. L e tt. 23, 545 (1966). 6 . J.E. Turner and K. Fox, Phys. L ett. 2J3, 547 (1 966). 7. O.H. Crawford and A. Dalgarno, Chem. Phys. Lett. 1_> 23 (1967). 8 . W.B. Brown and R.E. Roberts, J. Chem. Phys. 4J5, 2006 (1967). 9. O.H. Crawford, Proc. Phys. Soc. London, 9J_, 279 (1 967) . 10. J.M. Levy-Lebbond, Phys. Rev. 1 53 , 1 (1 967). 1 1. 1 .625 D = 0.639 ea where 1 D = 1 0 " ^ esu cm. 12. L.D. Landau and E.M. L i f s h i t z , Quantum Meehan i cs, (Pergammon Press, London), 1958, p. 119. 13. W.R. G a rrett, Chem. Phys. Lett. J5, 393 (1 970). 14. O.H. Crawford, Mol. Phys. 26, 139 (1973). 15. W.R. G arrett, J. Chem. Phys. j59 , 26251 (1 978). 16. K.M. G r i f f i n g , J. Kenney, J. Simons, and K.D. Jordan, J. Chem. Phys. 63, 4073 (1975). ________________________________________ : ________________ 50 17. K.D. Jordan, K.M. G r if f i n g , J. Kenney, E.L. Andersen and J. Simons, J. Chem. Phys. 6j4, 2760 (1 976). 18. K.D. Jordan and W. Luken, J. Chem. Phys. j[4, 2760 ( 1 976) . 19. J.L. Carlsten, J.R. Peterson and W.C. Lineberger, Chem. Phys. L e tt. 3_7, 5 (1 976). 20. L. Sanche and G.J. Schulz, Phys. Rev. A, J3, 1672 (1972). 21. L. Sanche and G.J. Schulz, Phys. Rev. A, §_, 69 (1972). CHAPTER IV CONFIGURATION INTERACTION CALCULATIONS ON THE RESONANCE STATES OF HF~ Introducti on In the last chapter, a review of the theory of resonances of polar molecules was given. In this chapter, we present the results of ab i n i t i o calculations on one such molecule in the low energy region. Before describing these results, however, i t w ill be useful to summarize a few pertinent concepts. The electron scattering and electron binding proper ties of polar molecules have recently been the topic of considerable experimental and theoretical a tte n tio n . The very long range interaction potential for a charged p a rtic le and a molecule with a permanent dipole moment leads to interesting e ffe c ts . I t has been demonstrated by 1-4 a number of researchers that the e le c tr ic dipole fie ld can support an i n f i n i t e number of bound states i f the 5 dipole moment is greater than 1.625 D. For a dipole moment less than this value, no bound states e x is t. Other work has shown that i f a molecular system is treated dynamically by calculating non-Born-Oppenheimer rotational g degrees of freedom, the number of bound states is f i n i t e . For the non-stationary dipole, the c r i t i c a l dipole for binding an electron is from 10-30% greater than 1.625 D. ^ 52 There is also an indication that induced dipole forces make very important contributions to the energies of weakly bound electrons, even for systems with larger permanent dipole moments. Garrett suggests that a strongly polar molecule with a dipole greater than about 4 D w ill surely form a stable negative ion. For a molecule with a dipole moment in the range of 2 to 3.5 D, a negative ion with binding energy greater than 0.01 eV w ill be formed, i f the p o l a r i z a b i l i t y is from 20-40 a^ (3-6 A^).^ O ' Ab i n i t i o calculations have been performed on several polar molecules. Based on the premise that the electron a f f i n i t y may be r e l i a b l y estimated by the negative of the o rb ital energy of the lowest unoccupied molecular orbital (LUMO) of the neutral molecule, a series of ionic mol ecu!es have been investigated. For LiH, LiF, LiCl, NaH, N a F» NaCl, BeO, MgO, LiCN, LiNC, L i OH, and L i CH 3 which have dipoles ranging from 4.6 to 9.5 D, the calculated 9 electron a f f i n i t i e s were between 0.2 to 0.7 e V . - Another examination of the nonionic molecules, ( HF) 3 , HCN, HN03, CH^CN, H2 Q, and HF indicates stable anions for the f i r s t four are formed, while they are not for H2 0 and HF. 10 Since the f i r s t four of these molecules have dipole moments greater than about 3.5 D, they would be expected to form stable negative ions. In spite of the fact that th.e calculated dipole moments of HF and H20 are stated to be too high jby Q. 5 D (and thus larger than the c r i t i c a l value by more than 0.8 D), the study finds no stable anion for the two m o le c u le s .^ The HF molecule has a dipole moment of 1.82 D , ^ just s li g h t ly greater than that necessary to bind an electron. Studies of the negative ion states of this molecule are of importance both because of the minimum dipole concept, and because of the recent in te re st in resonant states of polar molecules in general. We have performed extensive configuration interaction calculations on the lower energy states of HF- . Although the molecule has not been widely studied, the available experimental and theoretical data are summarized below. Experimental studies of HF have t r a d i t i o n a l l y been d i f f i c u l t because of the corrosive e f fe c t of the vapor on the surfaces of optical components. Nevertheless, a few relevant studies have been performed. (a) Dissociative Attachment e" + HF F" + H The onset for F’ formation is reported to have a threshold 1 2 of 1.88 eV and a maximum at 4 eV. The dissociation energy of HF is 6.1 e V ^ (Dq = 5.84 ± .01 eV) while the 1 4 electron a f f i n i t y of F is 3.448 ± .005 eV. The thermo dynamic l i m i t , therefore, requires an appearance potential of 2.65 eV. The observed threshold of 1.88 eV implies that F- is formed as soon as is thermodynamically possible. 54 (b) Associative Detachment F" + H -> HF (v * j ) + e" This process is fast and occurs at a rate close to the Langevin rate constant with no temperature dependence. This implies l i t t l e or not activation energy for the 1 5 process and an a t t r a c t i v e potential energy curve. (c) Vibrational Excitation e" + HF e" + HF ( v , j ) -1 5 2 In each fin a l vibrational state, a Targe (10 cm ), 1 6 sharply peaked cross section is observed at threshold. A cross section of this magnitude was also observed for HC1. The cross section of both molecules have a maximum of much larger width which decreases monatonically there a f t e r . Since the cross sections are isotropic in the regions of the peaks, they do not re su lt from electron- dipole long range interactions (d ire c t vibrational e x c ita tio n ). The spectra presented by Rohr and Linder indeed indicate a very pronounced maximum for HC1 in the region of about 2 to 3 eV. For HF, however, this second maximum is hardly discernible. I t is therefore questionable whether these data support the existence of a broad maximum in the case of HF, although they c le arly do for HC1 . Theoretical studies of HF" are rather more prevalent than experimental investigation s, but are nevertheless far from complete. Two ab i n i t i o calculations on the 55 1 + - 2 + ground state of HF( E ) and the ground state of HF ( E ) find the HF“ state repulsive for a ll values of in te r - 17 18 nuclear separation. ’ Bondebey et a l ., however, point out that the exclusion of diffuse functions from the basis 1 8 set necessarily leads to a "diabatic" potential curve. Since diffuse funetions were not employed in e ith e r c a l culation, the fact that the negative ion state was found to be repulsive is not surprising. Although the two studies do agree q u a l i t a t i v e l y , o n e ^ finds the crossing of the HF” (^E+) potential with that of the HF(^E+) at 3.9 1 8 bohr, while the other claims i t to occur at 2.7 bohr. 1 9 20 In contrast to this fin din g, two other studies ’ reach an a lte rn a tiv e conclusion. Both of the l a t t e r — 2 + calculations find the HF ( E ) ground state bound for a considerable range of R. The fundamental approach to the calculation of 21 resonances is the s t a b il iz a t io n method. Using square integrable (Gaussian) functions and Configuration I n t e r action* we have performed extensive Cl calculations on the ground state of HF and the states of HF- lying below about 9 eV. Since the procedure involves representing the l i m i t of HF plus an electron at i n f i n i t e separation within a basis set of functions of f i n i t e extent, i t is necessary to establish each stable root as a resonance, and not simply as an a r t i f a c t of the calc u la tio n. The results of this study provide an in terp reta tio n for the experimentally observed features and c le a rly resolve the disagreement arising from the conclusions of previous theoretical calculations. Certain of the results also allow prediction of some ch aracteristics not yet measured experim entally. Techniques lik e those employed here were used success- 22 f u l l y in an e a r l i e r study of HC1 . Although there are s im i l a r i t i e s between the states of HC1~ and HF- , there are also important differences, and throughout this chapter we w ill refer to the previous work on HC1 for comparison. We begin with a b r ie f description of the computational method, then present the results for all calculated states, discuss the evidence of resonance character in the calculated negative ion states, and f i n a l l y , examine the c a p a b ility of the HF molecule to bind an electron. Method The atomic orbital basis set was chosen to be s u f f i c i e n t l y f l e x i b l e to represent both HF and HF- for a range of internuclear distances. A Dunning basis the 2 3 (9S/5P) prim itive gaussian basis of Huzinaga, contracted to (3S/2P) proved adequate, while the hydrogen was repre- 2 4 sented by the Dunning (2S) basis. In addition to these functions, uncontracted Gaussian Rydbergs functions of exponent 0.036 and 0.0066, and P functions of exponent 57 0.074, 0.029 and 0.0054 were added to F. F in a lly , a P type polarization function of exponent 0.9 was added to hydrogen and a d type, of exponent 1.15, to flu o rin e . The complete atomic orbital basis set is shown in Table IV-1 . The SCF wave functions and energy for HF were c a l culated within this basis set. Of the 31 molecular orb ita ls comprising the basis, only one, that of lowest energy, was dropped from consideration in the Cl calc u la tions. Table IV-2 provides a l i s t of the molecular orbitals and th e ir eigenvalues. In the second column of this ta b le , the M.O. id e n t ific a tio n system used in the text is given. The total SCF energy of HF at its e q u i l i brium internuclear distance is -100.04905 a .U ., which can be compared with the near Hartree-Fock value of -100.0705 a . u . 25 The SCF'virtual o rb ita ls of HF were assumed to form an adequate basis for Cl calculations on HF~. This is 22 j u s t i f i e d by previous work on HC1, which confirms the accuracy of this assumption. The SCF v irtu a l orb itals of HF are eigenfunctions of the f u l l n electron HF poten t i a l including its permanent dipole moment. Since, in a case of this type, the scattering is long range, the target molecule is r e l a t i v e l y l i t t l e perturbed by the scattering event. Consequently, the HF v irtu a l orb ita ls are a good representation of the natural o rb ita ls of HF~. 58 Indeed, as we shall see l a t e r , this is generally v e rif ie d by the dominance in the fin a l Cl wave functions by a single configuration. Configuration Interaction calculations were performed for the ground states of HF and HF", as well as the excited states of HF". The HF" states were described by populating the appropriate HF virtu al orbital with a single electron. In general, one or more seed configurations were selected to represent a given state of in te re s t. All single and double hole p a rt ic le excitations r e la t iv e to these few seed configurations were then generated to form the Cl space. Solution of the Cl problem was accomplished by the p a rtitio n in g technique which was discussed in Chapter I I . Results and Discussion Full Cl calculations on HF and HF" were carried out at a number of internuclear distances. Depending upon the p a rtic u la r state, f u l l calculations were performed at more than one internuclear distance including 1.5 bohr, 1.732 bohr (e q u ilib riu m ), 2.0 bohr, 2.5 bohr, 3.0 bohr, 4.0 bohr, 5.0 bohr, and 7.0 bohr. The potential curves resulting from these calculations are displayed in Figure IV -1 , and th eir energies at the various internuclear distances are presented in Table IV-3. We w ill f i r s t describe the general procedure used in a ll calculations __________ 59 and then discuss the results for each specific state separately. General Procedure In a study of this type, two important concepts must be considered. F i r s t , at any p a rtic u la r point on the potential surface, the HF and HF” calculations must be balanced with respect to one another. This w ill allow reasonable in terp re ta tio n of results involving the r e la t iv e location of the HF and the HF” states. Second, the calculations must permit proper dissociation behavior for the HF and the HF” ground states, independent of one another. This second consideration is important for insuring confidence in the absolute results. At the equilibrium internuclear distance, the HF ground state is dominated by the configuration 2 2 2 4 lo 2a 3 a 1 tt , while the dominant configuration of the lowest HF” state (1^E+) is 1a^2a^3a^l 7t^4 a . The higher HF” E + states resu lt from promotion of the additional electron to the higher E+ molecular o rb ita ls . At 7.0 bohr, which we use to represent the dissociation l i m i t , the molecular orbital picture has altered considerably. The molecular orbital designated at 1 t t now f a l l s .lower in energy than the 3 a molecular o r b i t a l . Thus, at 7.0 bohr, 2 2 4 2 the ordering of the occupied M.O.'s is l a 2 a lir 3 a . 60 The 3a M.O., at this large distance, is composed of contributions from the H Is and the F p atomic o r b ita ls . a The 4a M.O. is simply the antibonding complement to the 3a M.O., and is only s li g h t ly higher in energy. The HF 2 2 molecule which dissociates to H( S) + F( P ) , is repre sented at 7.0 bohr by a lin e a r combination of three elec- 2 2 4 2 tron configurations. The f i r s t is la 2a In 3a , the 2 2 4 SCF base. The second, la 2a 1 i t 3a4a, contains one electron each in the bonding and antibonding H-F M.O.'s. The t h ir d , 2 2 4 2 la 2a 17T 4a , has both electrons in the anti bonding H-F M.O. In the case of HF- , one electron already 2 + occupies the antibonding H-F M.O. Thus the 1 E HF state, 9 - 1 which dissociates to H( P) + F ( S), is represented simply by a lin e a r combination of the two configurations 1c^2a^lu^3a^4a and 1a^2a^lu^3a4a^. A higher HF state, 2 + the 5 E given in Figure IV-3 is one of the states - 1 2 leading to the l i m i t H ( s) + F( p). This l a t t e r state is presented by the same two electron configurations 2 + as the 1 E HF state, this time, with opposite signs. For the dissociation behavior of HF and HF- to be adequately represented, this picture must be taken into account in the Cl calculatio ns. A proper calculation for HF must include all single and double excitations generated 2 2 4 2 from three configurational bases, To 2a 1 t t 3a , the SCF 2 2 4 2 2 4 , 2 base, as well as la 2a 1 t t 3a4a and la 2a 1 t t 4 a , the two configurations representing the H-F bond. For both the 61 2 + 2 + 1 z and the 5 E states at 7.0 bohr, generation of all single and double excitations from the two configuration bases, 1 o 2 2 a 2 l T r^3a2 4 a and 1 a 2 2 a 2 l i r ^ 3 a 4 a 2 is adequate for decribing dissociation. With the dissociation l i m i t requirements in mind, the structure of the calculations for shorter internuclear separations is defined. At the equilibrium internuclear distance, the dominant contribution to the wavefunction is the SCF base. Although, at this point, the two additional bases used to generate configurations do not contribute as heavily to the wavefunction as at 7.0 bohr, they must be included to insure balance across the potential surface. Thus, at 1.732 bohr, the calculations on HF consisted of a ll single and double excitations from 2 2 4 the SCF base in addition to two bases, l a 2 a 3alTr 10a 2 2 4 2 and l a 2 a 1 t t 1 0 a . At this shorter distance, the 1 0 a M.O. has e s s e n tia lly the same character as the 4a; M.O. at 7 . 0 bohr. For the lowest state of HF", the bases included l a 2 2 a 2 3 a 2 l T r ^ 4 a and l a 2 2 a 2 l-rr2 4 a l 0 a 2 . At in te r - mediate points on the potential surface, the calculations were performed in an analogous manner. Generally, in the HF c alcu latio ns, a ll single and double excitations were generated from three bases, the SCF base, the base repre senting the configuration contributing most strongly to the wavefunction (a double excitation from the SCF base), and one other base representing a sort of "cross term" __________ 62 (a single excitation from the SCF base). For the lowest HF“ state, two generating bases were used at.each point. The f i r s t was formed by placing the additional electron in the lowest unoccupied molecular orbital of the SCF base for HF. The second was the same doubly excited configuration used in the HF ca lculations. Table IV-3 presents the configuration bases used at various distances for HF and the lowest HF~ state. The higher states of Hf~ were treated in the same 2 + manner as the 1 E HF state, with the extra electron simply occupying successively higher E + molecular o rb ita ls . The procedure resulted in a total space of about 8,400 configurations for the HF calculations. The HF- c a lc u la tions generally included between 11,000 and 13,000 total configurations. This procedure resulted in a reasonable balance in co rrela tio n energy between the HF ground state and the HF" Rydberg states. At the equilibrium i n t e r nuclear separation, the correlation energy of the HF" states should be s li g h t ly greater than that of the HF ground state. The presence of the extra electron leads to a small po la rizatio n contribution to the energy, in addition to the correla tion energy i t s e l f . We indeed find the co rrela tio n energy of all HF- states to be s li g h t ly greater than that of HF. HF (1 1 E+ ) . This state is described by the configura- 2 2 2 4 tion la 2a 3a 1 t t which represents about 99% of the __________ 63 fin a l wavefunction at the equilibrium internuclear 1 2 2 separation. I t goes smoothly to the limits’ H( S) + F( P) with a calculated dissociation energy of 6.02 eV which can be compared to the experimental value for Dg of 6 . 1 eV. 13 _ 2 + 2 + HF (1 z ) . The 1 E state was generated through occupation of M.O. 6 , an S type Rydberg function. At the equilibrium nuclear separation, this configuration repre sents 99% of the total wavefunction. I t has already been mentioned th at, at larger internuclear distances, this HF" state leads to the l i m i t H(^S) + F~(^S). The state is c le a rly bound with respect to these l i m i t s , at least q u a l i t a t i v e l y confirming the results of two previous 19 20 17 20 ca lculations, ’ and refuting those of two others. ’ This finding also agrees with the dissociative attachment results which indicate the state to be a t t r a c t i v e into the autodetaching region. The calculated electron a f f i n i t y of HF at 7.0 bohr is 3.17 eV which agrees rather well with 1 4 the experimental electron a f f i n i t y for F of 3.45 eV. I t should be noted from Figure IV-1 and the values of Table IV-3 that the 1 ^E + state of HF" appears to be bound at a ll distances r e la t iv e to the HF ground state, implying a positive electron a f f i n i t y of 0.010 eV for HF. This is in contrast to the results of e a r l i e r study on 2 + HCl, where the 1 E state of HC1 was found to l i e about 2 2 0.12 eV above the HCl ground state. The unbound nature of the HCl” state is not unexpected, however, since the dipole moment of HCl (1.1 D) is well below the c r i t i c a l dipole moment for binding an electron. We return la t e r to a complete discussion of the bound or unbound nature 2 + of the 1 E HF a fte r presenting the results of the other calculated HF- states. 2 + HF (2 Z ) . This state occurs at the equilibrium internuclear distance through occupation of M.O. 9, e s s e n tia lly a Pa Rydberg function. This configuration has a weight of 99% in the fin al Cl wavefunction at 1.732 bohr. At this same distance, the state lie s only 0.32 2 + eV above the 1 E HF ground state and corresponds to the 22E + mimic state of HCl. 2^ HF~ (3 2E+) . The 3 2Z+ state results from occupation of M.O. 10, also a Pa type Rydberg function at the equilibrium internuclear distance. I t , lik e the lower HF“ states, comprises 99% of the fin a l Cl wavefunction. This state lie s about 1.7 eV above HF at equilibrium , and 2 + corresponds to a state in HCl , the A E , that was not considered to be a resonance. 2 + HF (4 E ) . This state occurs at equilibrium through occupation of M.O. 13, p rin c ip a lly an S type Rydberg function. I t lie s about 2.3 eV above the HF ground state. 2 + This state, lik e the 3 E state ju st discussed, is analogous in occupation to a second state of HCl", the 2 + 22 B E , that was not considered to have resonance character. _______________________________________________________________________________ _ M - 2 + 2 + HF~ (5 E ) . The 5 E state results from the occupa tion of M.O. 14 at equilibrium , largely an S type function. I t is more r a d ia lly contracted than the lower energy 2 + M.O.'s and corresponds in character to the 3 E state of HCl- which is responsible for the broad resonance 1 6 observed in the HCl vibratio nal excitation spectrum. At the equilibrium internuclear distance, this state lies well above the ground state of HF, at about 6.7 eV. At 2 + longer distances, i t crosses the repulsive 7 E HF state which w ill be discussed shortly. - 2 + 2 + HF (6 E ) . The 6 E state occurs at the equilibrium internuclear distance through occupation of M.O. 17 a pa type M.O. which is valence in character. At short 2 + distances, this state also crosses the repulsive 7 E state. We have indicated only a portion of this curve in Figure IV -1 - 2 + HF (7 E ) . This state is one of the Feshbach states of HF- and is considered in more detail in the next 2 + chapter. I t crosses the 6 E at about 9.7 eV and the 5^E+ at approximately 7.3 eV. At 7.0 bohr, the 7^E+ 2 + state (which has now become the 5 E state) leads, 2 together with the 2 tt state discussed below, to the l i m i t - 1 2 H ( S) + F( P). We have indicated the adiabatic curves in Figure IV -1 for the crossings of this state with the 2 + 2 + 5 E ; for its crossing with the 6 E state we show the 2 + 2 + diabats. From the crossing of the 7 E and the 5 E states 66 we would predict H production to have a v e rtica l onset with a maximum about 7.3 eV above the HF ground state. In HCl” , the situation is somewhat d i f fe r e n t . In this case, H” production, which occurs at about 6.9 2 + eV, results from a crossing of two Z states, only one of which corresponds in electron configuration to the 22 - two HF states. The ascending state of HCl is a E+ state lying above the broad resonance state in energy. 2 + The ascending state of HF , the 5 £ is the state that would lead to experimental observation of a broad resonance. The broad resonance in HF” lie s much higher in energy than the broad resonance in HCl” by some 3.7 eV. I t is 2 + therefore reasonable to assume that the 5 £ HF state is responsible for H~ production. 2 HF (3 tt) . This state occurs at the equilibrium internuclear distance through occupation of the tt M .O .' s 15 and 16. I t is shown in Figure IV-2 simply because i t 2 crosses the 4 it state. There are two it states that l i e 2 lower in energy than the 3 tt formed by occupation of M.O.'s 7 and 8 (1 ^ t t) and M.O.'s 11 and 12 ( 22tt). Although we have not performed f u l l Cl calculations on these states, i n i t i a l single configuration calculations indicate that 2 2 + 2 + 2 the 1 it lie s between the 1 Z and the 2 E and the 2 tt 2 + 2 + lie s between the 3 E and the 4 E . - 2 2 + HF (4 tt) . This state , together with the 7 E state, leads to the l i m i t H” (^S) + F ( 2P) at 7.0 'bohr. I t , lik e the ________67 2 + - 7 Z HF state, is a Feshbach state of HF . I t has an 2 avoided crossing with the 3 t t state at shorter internuclear distances, indicated a d ia b a tic a lly in Figure IV-1. 2 The 4 t t HF state corresponds in electron configura- 2 tion to the t t state of HCl shown in Figure I I of reference 22. In the dissociative attachment of HCl", 2 the t t state produces a second overlapping, gaussian shaped peak at 9.2 eV. 22 In HF", the 4 2 tt state could lead to a sim ilar peak that would l i e at about 10.2 eV. Table IV-5 summarizes the results of the calculations and also presents the existing experimental data for comparison. We also show our estimates of certain experimental parameters and suggest the type of experi mental investigation that might be used to determine them. Limits In this study, we have performed f u l l Cl calculations 2 - 1 on the states leading to three lim its : H( S) + F ( S ) , H( 2S) + F ( 2 P), and H"(^S) + F ( 2P). With the aid of a simple MO picture, i t is clear that the lim its are pro duced by the states so indicated. 2 + One state, the 1 Z state, leads to the l i m i t H( 2S) + F~(^S). At 7.0 bohr, the fin a l wavefunction of this state is dominated by two electron configurations, 1 a 2 2a2l iT^3a24a and 1 ct2 2 o2 1 Tr^3a4c2 with signs that are in-phase with one another. At the true l i m i t , these 68 configurations would have exactly equal weighting. At large distances, the 3c r M.O. is the bonding combination of the H 1s AO and the F pa AO; the 4a is es sen tia lly its antibonding complement. The lin e a r combination of the two configurations in terms of the F pa AO and the H Is AO leads to equal weighting of (H-js + ^ ^ (H, - F„ ) and (H, + F ) (Hn - F„ ) . Upon expansion Is pa Is pa ' Is pa7 K ^ and summation of the two terms, we obtain H, F" , which I s p a simply represents an H atom and an F" atom. 2 + 2 Two states, the 5 £ and the 3 i t , lead to the l i m i t H~(^S) + F(^P) at 7.0 bohr. The 5^£ + at long internuclear distances is largely composed of two configurations, 2 2 4 ? 2 2 4 ? la 2a 1 t t 3a4a and la 2a 1 t t 3a 4a. These are the same 2 + two configurations as those representing the 1 £ state. In this case, however, the configurations have signs that are out-of-plase with one another. Expansion of the configurations in terms of the A.O.'s leads to the occupa tion H js Fpa which represents an H" atom and an F atom. 2 The 3 t t state, at large internuclear separations is 2 2 3 2 2 dominated by the configuration la 2a 1 i t 3a 4a . Expansion of this configuration in terms of the A.O.'s also leads to the occupation H^s Fpa . From this simple M.O. picture, 2 2 + - i t is reasonable to assume that the 4 7t and the 7 £ HF states indeed lead to the l i m i t H"(^S) + F(^P). The H(^S) + F( 2P) l i m i t is produced by two states. One of these states is the HF(^£+) ground state discussed 69 e a r l i e r . The other, a t t state, is an excited state of HF considered in detail in the next chapter. Other States In i n i t i a l single excitation calculations we observe three states, A, B and C, which f a l l between the lim its H( 2S) + F " ( 1 S) and H(2 S) + F ( 2 P) at 7.0 bohr. Two other 2 + 2 states, the 5 z and the 3 tt, as already indicated, also * fall within this energy region and lead to the l i m i t 1 2 H ( S) + F( P). The simple M.O. picture presented e a r l i e r supports the fact that these two states indeed lead to the H“ (^S) + F( 2 P) l i m i t . Of the three other states f a l l i n g within the same energy region two, A and B, are E+ states, while the th ir d , C, is a tt state. The electron configurations contributing to the 2 2 4 A and B states are la 2a 1 tt 3 a4 an6 , where n 5. When n > 5, the na M.O.'s are dominated by A.O. contributions e ither from F„ or F . Formulation of these electron pa s configurations in terms of the A.O.'s leads to (H, + F„ ) (H, - F„ ) F*, where F* represents an excited Is pa ls pa r Fpa or Fg AO. Expansion of the configurations produces terms in F~~. To our knowledge, F- - is not known to exist and we can only conclude that the states leading to the F lim its also do not e x is t. 2 A sim ilar situation arises in the case of the c t t state. A f u l l Cl calculation at 7.0 bohr places this __________________ 70 state between the H” (^S) + F ( 2 P) and the H( 2S) + F ( 2P) lim its in energy. The state is dominated by the electron 2 2 4 configuration la 2a 1 t t 3a4a2Tr. Expansion of this configuration in terms of the AOs again yields terms in F~“ . We therefore feel that all three states are not re a l. All other states l i e above the H( 2S ) + F ( 2 P) lim i t in energy and can probably be assumed to be re a l. Comparison with HCl” In l ig h t of the current HF- study, we have reviewed - 2 2 the results of the e a r l i e r study on HCl for comparative purposes. Several points are of p a rtic u la r in te re s t. 2 + F i r s t , the 1 E state of HF is found to be bound by some 0.01 eV, while the 12E + state of HCl" lie s above the HCl ground state. The unbound nature of the lowest HCl* state is reasonable, given that the dipole moment of HCl is much less than that required for binding an electron. The second point of in te re s t involves the higher 2E + states of HCl” and HF” . In the HCl study, only those states that could be experimentally reported were calculated. Two states of HCl” , the A ( 2 E+) and B( 2 E+ ), 2 + 2 + that l i e in energy between the reported 2 E and 3 E states were therefore not investigated. The states of _ 2 + HF corresponding to these two HCl states are the 3 E 2 + and the 4 E . On the basis of the appearance of plots 7] of the density functional against distance from the HCl * _ molecule, the two HCl states were not considered to be 22 resonances. We now question this conclusion. The plots 2 + of these states, when compared with those of the 3 E and 2 + 4 E HF states, do not seem to d i f f e r s ig n if ic a n tly . In the next section we return to this general point with a discussion of the d i f f i c u l t y of distinguishing those HF- states that have resonance character from those which are simply continuum funetions. The fina-1 issue of importance in the comparison of HF“ and HCl- involves the 2^E+ and 3^E+ HCl- states and 2 + 2 + th e ir counterparts, the 2 E and the 5 E states of HF . In the work on HCl- , these states were stated to merge with the continuum at longer internuclear distances. From the results of a series of single configuration Cl c a lc u la tions, i t now appears that these states can s t i l l be followed at very large separations a f te r an intermediate 2 + region of confusion. This is g ra t ify in g since the 2 E , 2 " j* 2 *j* 2 -j- 3 E , 4 E , and 5 E HF- states also remain r e l a t i v e l y pure in calculations of this type to 7.0 bohr. This 2 + suggests that the behavior of the E states of HCl and those of HF- is s im ila r. 2 + One fu rther difference should be noted. The 3 E HCl- valence state responsible for the broad resonance observed in the vibrational ex citation spectrum has a minimum approximately 2.3 eV above the ground state of 72 30 HCl. According to Domcke and Cederbaum, the h a lf width of this state is 2.3 eV. In HF", the state corresponding to the 3^z+ HCl" state is the 5^E+ which lie s about 6 eV above the HF ground state. This is far higher in energy than the HCl" state. The HF" state was not observed / in the vibrational excitation experiment of Rohr and};" L i n d e r . ^ This is not surprising, however, since the reported energy range extended only to about 3 eV. The results of Domcke and Cederbaum in the case of HF” would require the state to have a half width of about 6 eV, u u • 30 which seems excessive. Proof of Resonance Character A resonance or a temporary negative ion state is formed by the interaction of a target molecule with an incident electron. The electron is temporarily captured within the neighborhood of the molecule. The attachment of the electron can occur at a d e fin it e energy in which case, sharp structure is observed in the cross section. These states have a l i f e t i m e , x, of between 1 0 " ^ and -1 5 1 0 sec where x = h / r with r representing the width of the state. Experimental study of these states is possible when they decay into in e la s tic channels ( r o t a tional and vib ratio nal e x c ita tio n , electronic e x c ita tio n , dissociative attachment, and so on). When the cross section is dominated by in e la s tic processes, then the ________________________ 73 resonance contribution can be observed without i n t e r ference from the d irec t scattering mechanism. In addition to examining the energy dependence of the cross se ctio n, experimentalists can also observe the angular dependence of the cross section for the purpose of studying resonances. In studies of this type, the angular d is trib u tio n can be uniquely determined through comparison of symmetries of the i n i t i a l , resonant, and final state. When the resonant state is expanded in terms of spherical harmonics, the contribution from the lowest allowed value of I predominates. In heteronuclear diatomic molecules, mixtures of these waves, called p a rtia l 31 waves, are possible. I t has been demonstrated that pure pa rtia l waves of pa or p-rr symmetry e x h ib it c h a ra c te ris tic p wave shapes leading to a minimum in the cross section at 90°. Pure waves of da, dur, or d6 symmetry a lt e r n a t iv e ly produce a maximum at 90°. Mixtures of p a rtia l waves are also possible in heteronuclear molecules. Theoretical methods for determining whether or not a state is a resonance have also been developed. In the 21 original s ta b il iz a t io n method, frequently used for this purpose, heavy weights of a single or few configurations were used as a c r it e r io n to id e n tify stable roots. These ca lculations, however, were generally performed on mole cules without permanent dipole moments, so that no dominant f ie ld s were present. I t is our experience, in the case _________________________________________________________________________74_ 2 2 of molecules with permanent dipole moments that purity of the vector is not p a r t ic u l a r ly useful for establishing s t a b i l i t y . In Configuration Interaction calculations on the negative ion states of these molecules, a ll states seem to give r e l a t i v e l y pure Cl vectors which represent successive Rydberg states in the f i e l d of a dipole. One study on the states of LiF" u t i l i z e s orbital amplitude plots for q u a lit a t iv e speculation on the 32 character and width of states thought to be resonances. In our e a r l i e r work on HCl", resonance character was a ttrib u te d by the appearance of " s t a b ilit y " with the addition of very diffuse functions to the original basis 22 set. This approach was based on the assumption that the r e la t iv e purity of the Cl vectors indicates that the MO occupied by the scattered electron is a f a i r approxima tion to the natural o rb ita l of the electron. A graph of the density functional of this MO plotted against distance for each state with and without addition of the diffuse functions provided q u a lit a tiv e "proof" of resonance character. States that are attempting to place the scattered electron at i n f i n i t y appear increasingly sinusoidal with increasing f l e x i b i l i t y of the basis set. A lt e r n a tiv e ly , those states possessing resonance character seem to show a high p ro b a b ility of the electron in the area of the target with sinusoidal behaviour at longer di stance. __________75 33 Another recent study, which is also based on the assumption that calculated energy values for stable resonances w ill show only small changes with basis set v a ria tio n , has adopted a somewhat d i f fe r e n t approach. Rather than enlarging the basis set, the study rather recommends varying the basis set continuously. This involves, for example, going from an original basis, a i = ao5^ t 0 3 shl*f ‘tGd basis, a: = aQ6 1+^. For a stable eigenvalue, avoided crossings w ill appear. Based on the previous work, we employed basically two general techniques for investigating the resonance character of each p a rtic u la r HF" state. I t was assumed throughout that the M.O. produced in an SCF calculation is a good representation of the natural orbital of the scattered electron. Thus MOs 6 , 9, 10, 13 and 14 are representative of the 1^E+ , 2^Z+ , 3^£+ , 4^E+ and 5^Z+ HF" states respectively. The f i r s t technique simply involves comparison of the SCF energy and the M.O. eigenvalues in an altered bases set to those in the original basis set. I f an M.O. indeed represents a resonant state, then the energy (eigenvalue) would be expected to remain r e l a t i v e l y constant. The second technique consists of adding basis functions to the original basis set and v is u a lly observing the density of the M.O. This was accomplished by p lo ttin g , in confocal e l l i p t i c a l coordinates, the variatio n of the density functional with 76 distance from the two-center target molecule. We plot 2 2 2 3 ip (p -v ) D / 8 , where p and v are functions of two coordinates x and y against the coordinate along the internuclear axis, x. The F atom lie s at x=0 and the H atom is located a distance D(1.732 bohr) from the F atom in the positive x d irec tio n . The five variations of the original basis set which were considered are described in Table IV- 6 . Table IV-7 presents the SCF eigenvalues for the M.O.'s of E+ symmetry resulting from the original basis set (basis set # 1 ) and the varied basis sets (basis set # 2 through basis set #5). The f i r s t varia tion we w ill con sider is the addition of very diffuse functions. In basis set # 2 , an s function with exponent 0 . 0 0 1 was added to flu o rin e . In basis set 3#, this function and an additional p function with the same exponent were added to flu o r in e . Since the added functions are much more diffuse than any contained within the original basis set, i t would be expected that new M.O.'s would appear, with eigenvalues lower than those of the M.O.'s in the original basis set. The results of the SCF calculations indeed v e rify th is . In basis set #2, the new M.O. f a l l s much lower in energy and is dominated by the added function. Adding both S and P functions, as in basis set #3, creates two new M.O.'s of E+ symmetry which also f a l l below the energy of the M.O.'s in the original basis set. ___________________________________________ 77 From the data for basis sets #2 and #3 in Table IV - 6 and IV-7, we cle a rly observe the s t a b i l i t y of the SCF energy and of the energy of the individual M.O.'s in the original basis set. For example, the eigenvalue of M.O. 6 in the original basis set changes by only about 0.002 A.U. (0.06 eV) in basis set #2, and by only s li g h t ly more, 0.003 (0.08 eV), in basis set #3. The change in eigenvalue for M.O. 6 on moving to the augmented basis sets is, in fa c t, the largest change in eigenvalue for any of the M.O.'s. This is to be expected since the added diffuse functions in te ra c t most strongly with the lowest lying M.O. These data indicate strongly th at, since the energy of any given M.O. varies by at most, 0.08 eV, energy s t a b i l i t y of all states considered here is preserved upon variation of the basis set. In addition to the energy s t a b i l i t y c r i t e r i a , we have 22 also made use of a technique used previously for examining the change in appearance of the M.O.'s on moving to the augmented basis set. A graph of the density functional of the M.O. with distance along the internuclear axis provides an adequate two dimensional picture of each state. Figures IV-2 through IV - 6 present these plots for M.O.'s 6 , 9, 10, 13 and 14 respectively. Each state is shown for the original and the two augmented basis sets. The change in scale on the ordinate for the various M.O.'s should be noted. Figure IV-7 presents the _____________78 one additional £+ M.O. created in basis set #2 and Figures IV - 8 and IV-9 show the two additional M.O.'s formed in basis set #3. In each of these plots, the flu orine atom is at the zero point of the abcissa with the hydrogen 1.73 bohrs in the positive d irec tio n. The character of the M.O.'s of the original basis set changes very l i t t l e upon addition of the very diffuse functions, as i l l u s t r a t e d by the plots. For an M.O. which is attempting to place the scattered electron at i n f i n i t y , the density plots should show the electron density moving to further distances from the molecule. In none of the plots presented here, does this seem to occur. Rather, the addition of the more f l e x i b l e functions appears to tighten the main peaks in each case. Comparison of the density plots of the states considered to be resonances with those for the M.O.'s formed with the addition of the more ftexibTeifunctions points to an important difference. The newly formed M.O.'s appear to be a clear attempt to place the electron at i n f i n i t y , since they show the bulk of th e ir electron density at very large distance from the target. The second variation we consider is the addition of diffuse functions to the original basis set. In basis set #4, two s and two p functions with exponents 0.003 and 0.006 were added, this time to the hydrogen atom. This addition resulted in three new E+ M.O.'s between M.O. 6 and M.O. 14, and :one new E* M.O. below M.O. 6 in 79 M.O. 14, and one new Z+ M.O. below M.O. 6 In energy. Upon addition of these functions, the SCF energy was lowered by only 0.00005 a.u. (.001 eV). The largest va ria tio n in eigenvalue is observed for M.O. 13 for which the energy increased by about 0.03 a.u. (0.85 eV). Basis set #5 represents the addition of both very diffuse and diffuse fu netions to the original basis sets. The SCF energy, in this case, was lower by 0.00008 a.u. (.002 eV) than the SCF energy of the original basis set. The M.O. that was again most affected by the changes was M.O. #13. The eigenvalue of this M.O. changed by some 0.04 a.u. (1.12 eV ). The density plots for the M.O.'s of basis sets #4 and #5 are presented in Figures IV-10 through IV-14 together with those of the original basis set for compari son. In both basis set #4 and basis set #5, the new M.O.'s f a l l below M.O. 10 in energy. We should therefore expect that the M.O.'s most perturbed by the augmentation would be M.O. 5 and M.O. 9. Figures IV-10 through IV-14 indeed show this to be the case. The curves of Figure IV-10 i l l u s t r a t e that M.O. 6 may not be a resonance. Addition of diffuse functions causes the main peak to move much fa rth er from the molecule, behavior that is not expected in a resonance. We have considered the possi1 ' ’ b i l i t y that these M.O.'s of basis sets #4 and #5 thought 80 to be the same as M.O. 6 in the original basis set have not been correctly assigned. I f , instead ttie M.O.'s with eigenvalue 0.01225 a.u. of basis set #4 and eigen value of 0.00967 a.u. of basis set #5 in Table IV-7 are assigned, we obtain the density functionals of Figure I V-15 . This a lt e r n a tiv e assignment supports the a t t r i b u tion of resonance character to M.0. 6 . In basis sets #4 and #5, the main peak is in approximately the same location as i t was in basis set #1. Although in the two altered basis sets, density is building at longer range, this is not unexpected for a resonance when diffuse functions are introduced. M.O.'s 9, 10, 13, and 14 as shown in Figure IV-11 through IV-14 also changed minimally upon introduction of the additional functions. We have examined numerous other basis set additions that are not presented here. New functions of both diffuse and valence character were added to both H and F. In a ll cases, the M.O.'s of the original basis set with which the added functions interacted strongly were the most affected by the additions. This suggests that a lte rin g the basis set and observing l i t t l e change in the appearance of the M.O. is not a s u f f ic ie n t c rite rio n for judging resonance character. Our experience indicates- that the appearance of any M.O. can be changed considerably simply by judicious choice of additional functions. Further studies of this type on molecules 81 with known resonances are necessary to establish d e f i n i tiv e c r i t e r i a for assignments of resonance character. The Bound State _ Because the bound or unbound nature of the lowest HF state is a subject of much debate, we sought to better understand the results of our calculations by further investigation. The resonances of HF“ that result from the addition of one electron to a Rydberg orbital of HF arise prim arily from the permanent and induced dipolar 2 + forces. The 1 2 HF state is one such resonance, as mentioned e a r l i e r . In the HF molecule, for which the permanent dipole moment is only s l i g h t l y greater than that necessary for binding an electron, the binding energy is undoubtedly extremely small. In C .I. calculations of the type described here, the error in energy may be as high as 0.1 eV. This is a factor of 10 larger than the binding energy we calculate (0.01 eV). Because this is so, our results do not show unequivocally that HF is capable of binding an electron. One method of investigating the r e l i a b i 1ity of the calculated results is to consider the forces contributing to the bound nature of a state. One of these forces is the permanent dipole moment. A calculation performed for HF at the equilibrium internuclear distance at the Cl", level yielded a value of 2.01 D for the dipole moment which 82 is well above the experimental value of 1.82 D. The other factor .contributing to the forces generated in the mole cule is the po 1 a r i za bi 1 i ty . The experimental value of o oo oq this variable for HF is about 16.6 aQ (2.5 A"). According to reference 8 , a molecule with a permanent dipole moment as low as 2D and an average molecular 3 po1 a r i z a b i 1 i t y of between 2 0 and 40 aQ should have the c a p a b ility to bind an electron by 0.01 eV. This result also allows rotation of the molecule which contributes p o sitiv ely to the energy of the negative ion. The Cl calculations used in this work do not include the effects of ro ta tion . Our calculated dipole moment of 2.01 D, 3 together with the experimental p o la r iz a b i 1 it y of 16.6 aQ implies that our calculations should find HF bound by something less than 0.01 eV. Our calculated electron a f f i n i t y (0.01 eV), given the inherent inaccuracy of the method, seems to be at least of the proper order of magnitude. U nfortunately, because the value obtained for the electron a f f i n i t y is so close to zero, i t cannot be stated with ce rta in ty from the results of the calculations that the HF~ state is bound. Another method of testing the bound or unbound nature of the state was employed. This technique involved augmenting the original basis set with a diffuse s function of exponent .001 on the H atom. In the SCF ca lcu la tio n , the new E+ molecular orb ita l fe l l energetically 83 below the molecular o rbital occupied by the single electron 2 + in the original 1 2 HF state. Table IV - 8 presents the eigenvalues and total SCF energy for the original and altered basis set at 1.732 bohr.- Cl calculations were then performed for HF- at the equilibrium internuclear separa tion within the new basis set. In the f i r s t c a lc u la tio n , the additional electron occupied the original M.O. (the LUMO + 1). In the second ca lc u latio n , the additional electron occupied the new E+ M.O. (the LUMO). In the third ca lcu latio n, the HF ground state was calculated within the new basis set. Table IV- 9 gives the results of these calculations and, for com pari son, the results of the Cl calculations in the original basis set. In the' altered basis, the HF ground state fa lls below the original HF ground state by 0.005 eV. The HF~ state with the extra electron occupying the new £+ M.O. lie s below the new HF ground state by 0.168 eV. The HF” state with [the extra electron occupying the original 2 + M.O. now lies above the new HF ground state by 0.046 eV. Since the original and new HF ground state energies d i f f e r 2 + by only 0.005 eV, the HF 1 E state remained constant in energy to within 0.051 eV. The implications of this exercise are important. I t may be that the functions necessary for properly describing the HF" ground state were not included in the original basis set. The additional f l e x i b i l i t y of the _______ 84 bacis set achieved through the addition of a diffuse function may in fact provide the true picture of an unbound HF” ground state. The results are simply not 2 + d e f i n i t i v e regarding the bound nature of the 1 E HF state. I t could be th a t, since the energy of the original state rose above the HF ground state a f te r augmentation of the basis set, the state is not bound. Indeed, this is probably so. Whether or not the HF mole cule can, in r e a l i t y , bind an electron is not in question. Theory has shown that i t is c e rta in ly possible. Our calculated dipole moment exceeds the experimental value by some 0.2 D. This fact makes i t more l i k e l y that the results of our calculations w ill lead to a bound HF" state. That we find a bound state within our original basis set is therefore not surprising. Although our results do not allow an unequivocal conclusion on the bound or unbound nature of HF, they are s ig n ific a n t for at least two other reasons. F i r s t , we have i l l u s t r a t e d through extensive Cl calculations th a t, within a basis set that includes diffuse functions, the HF mole cule can probably bind an electron i f the calculated ✓ dipole moment is as high as 2.0 D. We cannot speculate on what the results would be i f the dipole moment were lower. The second reason for the importance of these calculations lie s in the fact that they provide guidance for other ab i n i t i o studies. Polar molecules with 85 permanent dipole moments s l i g h t l y in excess of HF should c e rta in ly be the subject of further in vestigation . Conclus ions The calculated potential curves presented in this work are in good agreement with the existing .experimental data. Moreover, some of the results of the study are useful for guidance in future experimental investigations. I t appears that the^computational techniques employed in this study are e ffe c tiv e for theoretical study of the negative ion states of polar molecules. The presence of a permanent dipole moment in HF gives rise to presumed resonances that cannot be intuited from the separated atom l i m i t s . That all the states described here are in fact resonances cannot be proven d e f i n i t i v e l y . Nevertheless, the technique we have u t i l i z e d for probing the problem provides at least a strong indication that the states of HF” indeed possess resonance character. 2 + Whether or not the 1 2 HF state is bound also cannot be unequivocally determined from this study. We find' this state bound by only 0.010 eV, a value which is small compared with the accuracy of the calculational procedure. We can only conclude.w ith'certainty that the 2 + - 1 2 state of HF lie s extremely close in energy to the HF ground state and is therefore e ither s l i g h t l y bound or s l i g h t l y unbound. 8 6 References 1. M.H. Mittleman and V.P. Myerscough, Phys. Lett. 2 3 , 545 ( 1 966 ) . 2. J.E. Turner and K. Fox, Phys. Lett. 23_, 547 ( 1 966 ). 3. W.B. Brown and R. E. Ro be rt s , J . Chem. Phys. 46_, 2006 ( 1 967) . 4. O.H. Crawford, Proc. Phys. Soc. jH.', 279 ( 1 967 ). _ 1 O 5. y = 1.625 D = 1.625 debye, where 1 D = 10" esu cm. 6 . W.R. G arrett, Mol. Phys. 2j0 , 751 ( 1 971 ). 7. W.R. G arrett, Phys. Rev. A3^, 961 ( 1 971 ). 8 . W.R. G arrett, J. Chem. Phys. 6£ , 2621 (1978). 9. Jordan and Wendolowski, paper given, r e f. 13. 10. K.D. Jordan and J.J. WendoToski, Chem. Phys. 2J_, 145 ( 1 977) . 11. R. Weiss, Phys. Rev. 113, 659 (1963). 12. O.C. Frost and C.A. McDowell, J. Chem. Phys. 2_9, 503 ( 1 958) . 13. J. Berkowitz, W.A. Chupka, P.M. Guyon, J.H. Holloway, and R. Spoler, J. Chem. Phys. 5j4 , 51 65 (1 971 ). 14. R.S. Berry and C.W. Reimann, J. Chem. Phys. _38 , 1 440 ( 1 963) . 15. F.C. Fehsenfeld et al . , J. Chem. Phys. 5_8, 5841 ( 1 973) . 16. K. Rohr,and F. Linder, J. Phys. B, Atom. MOlec. Phys. 9, 2521 (1976). 87 17. H.H. Michels, F.E. Harris and J.C. Browne, J. Chem. Phys. 48 , 2821 (1 968). 18. V. Bondybey, P.K. Pearson and H.F. Shaefer I I I , J. Chem. Phys. 57 , 1 1 23 ( 1 972) . 19. A.W. Weiss and M. Krauss, J. Chem. Phys. 52^, 4363 ( 1 969 ) . 20. W.M. Hartmann, T.L. G ilb e r t, K.A. Kaiser and A.C. Wahl, Phys. Rev. B, ? 1140 (1970). 21. H.S. Taylor and A. Hazi , Phys. Rev. A, 1_4, 2071 (1976). 22. E. Goldstein, G.A. Segal and R.W. Wetmore, J . Chem. Phys. 68_, 271 ( 1978) . 23. S. Hazinaga, J. Chem. Phys. 4_2 , 1293 (1969). 24. T.H. Dunning and P.J. Hays, "Modern Theoretical Chemistry", Volume 3, H.F. Shaefer Ed., Plenum Press* New York , N . Y. ( 1 977) . 25. A.D. McLean and M. Yoshtmine, J. Chem. Phys. 4 7, 3256 ( 1 967) . 26. G.A. Segal and R.W. Wetmore, Chem. Phys. L e tt. 32, 556 (1975). 27. G.A. Segal and R.W. Wetmore, Chem. Phys. Lett. 36, 478 (1975). 28. G.A. Segal, R.W. Wetmore and K.A. Wolf, Chem. Phys. 30, 269 (1977). 88 29. J.O. Hi. rschf el der , C.F. Curtiss and R.B. Bird, Molecular Theory of Gases and Liquids (John Wiley and Sons, New York, 19 54). 30. W. Domcke and L.S. Cederbaum,to be published. 31. F.H. Read, J. Phys. B1, 893 (1968). 32. W.J. Stevens, J. Chem. Phys., to be published. 33. A. Macias and A. Riera, J. Sociela Chemica I t a l i a n a , 1 08 , 329 ( 1 978). Table IV-1 Gaussian Basis Set Fluorine Hydrogen (/) C5s] I s [ 2s] 9995.0 0.001166 13.36 0.032828 1506.0 0.008870 2.013 0.231204 350.3 0.042380 0.4538 0.817226 104.1 0.142929 34.84 0.355372 0.1233 1 .0 0 0 0 0 0 1 2 .2 2 0.462085 4.369 0.140848 [lp ] 1 2 .2 2 -0.148452 1 .0 0 0 0 0 0 1 .0 0 0 0 0 0 1.208 1.05527 0.3634 1 .0 0 0 0 0 0 0.036 1 .0 0 0 0 0 0 0.0066 1 .0 0 0 0 0 0 i-p [5p] 44.36 0.020876 10.08 0.130107 2.996 0.396166 0.9383 0.620404 0.2733 1 .0 0 0 0 0 0 0.074 1 .0 0 0 0 0 0 0.029 1 .0 0 0 0 0 0 0.0054 1 .0 0 0 0 0 0 £ d [Id ] 1.15 1 .0 0 0 0 0 0 90 Table IV-2 SCF Results for HF (R = 1.732 bohr) M.O. Text Notation Symmetry Eigenvalue (a.u.) 1 1 a Z+ -26.29879 2 2 a E + — 1.60169 3 3a E + -0.76886 4 1 TT IT -0.65063 5 77 -0.65063 6 4a z + 0.00696 7 2 tt TT 0.01335 8 7r 0.01335 9 5a E + 0.01511 10 6 a I + 0.06850 11 3 tt TT 0.07965 12 77 0.07965 13 7 a z + 0.09187 14 8 a E + 0.26703 15 4 tt 77 0.30504 16 TT 0.30504 17 9a z + 0.32973 18 10a ' E+ 0.92540 19 11a Z + 1 .33003 20 ■ :5 tt TT 1.34772 21 TT 1.34772 22 12 a I + 1 .71036 23 6 tt TT 1.84189 24 TT 1 .84189 25 13a z + 2.75878 26 1A A 2.91399 27 A 2.91399 91 Table IV-2 (continued) M.0 . Text Notation Symmetry Eigenvalue (a.u.) 28 7 TT TT 3.35095 29 IT 3.35095 30 14a £+ 4.17831 31 15a £+ 5.65648 Total Energy: -100.04905 a.u. Nuclear Repulsion Energy = 5.19630 a.u. Total Electronic Energy = -105.24535 a.u. Table IV-3 Calculated Cl Energy Points a 1 .5 bohr 1.732 bohr 2.000 bohr 2.5 bohr 3.0 bohr 4.0 bohr 5.0 bohr 7.0 bohr HF i'z+ 0.771 0.01 0 0.346 2.056 3.644 — 5.932 6.025 HF~ i V 0.739 0 0.246 1.661 2.311 — 2.782 2.852 2V 1.088 0.324 0.510 2.114 4.559 — — 32Z+ 2.526 1.698 1.776 2.626 4.841 — — 42Z+ 3.038 2.298 2.662 4.423 6.277 - - — 52Z+ 8.105 6.736 5.869 6.078 7.693 5.819 — 62Z+ — 8.811 10.324 - - - - - - 72Z+ — 13.611 11.338 8.726 — — — 3 2 tt 8.808 8.117 — 9.952 — 5.824 5.626 CO 4 2 tt _ _ 10.209 8.701 7.230 — — — Table IV-3 (continued) a Relative to HF~ at equilibrium, E = 2727.440 eV. VO 4 ^ Table IV-4 Configuration Bases for the Ground States of HF and HF" R (bohr) HF HF" 2 2 2 4 lcT2a 30 1t t 2 2 2 4 la 2a 3a I tt 4a 1.5 la 22a^3al t t 4 1 0a • la 22a2lu410a2 2 2 4 2 la 2a I t t 4al0a la 22a23a2lTr4 2 2 2 4 la 2a 3a I t t 4a 1.732 2 2 4 la 2a 3al7T 10a la 22a2lfT410a2 2 2 4 2 la 2a I t t 4al0a la 22a23a2lTT4 , 20 20 2, 4. la 2a 3a In 4a 2.0 2 2 4 la 2a 3al7T 8a la 22a2l-rr48a2 la 22a2l7T44a8a2 la 22a23a2l7r4 la 22a23a2l7r44a 2.5 2 2 4 la 2a 3alTrT6a 2 2 4 2 la 2a 1 i t 6 a la 22a2lTr44a6a2 la 22a2lTt43a2 la 22a2lTr43a24a 3.0 la 22a2l7T43a4a 1 22 2 1 44 2 'a co i t t 4a la 22a2lTr43a4a2 , 2 2 4„ 2 «a co I t t 3 a l a 2 2 a 2 l T T 4 3 a 2 4 a 5.0 1 a22a2lir4 3a4a 1 2 2 2 1 4 2 la co I t t jo l a 2 2 a 2 l 7 T 4 3 a 4 a 2 _ 95_ Table IV-4 (continued) R (bohr) la 22a2lTr4 3a2 , 2 0 2, 4- 2 , la 2a I tt 3a 4a 7.0 2 2 4 la 2a I tt 3a4a 2 2 4 2 ;a 2a I tt 3a4a 7 2 4 2 la 2a I tt 4a 9 _ 6 Table IV-5 Summary of Observed and Calculated Results Observed Calculated Suggested Experiment HF" 12Z+ > 1 . 8 8 - 0 .01 HF" 52E+ 5.87 Vibrational Excitation HF" 52E+ 7.3 Electron Impact HF" 42t t 1 0 .2 F" + H 2.9 F + H" 7.3 Dissociative Attachment Electron A ffin ity F 3.45 3.17 Dissociation Energy HF 6 .1 6 . 0 2 97 Original Table IV- 6 and Altered Basis Set Description Basis SCF Total Set # Variation Energy (au) 1 Original -100.04905 Very Diffuse 2 Original + ^ = .001 F(s) -100.04905 3 Original o o II + F(s,p) -100.04906 Diffuse 4 Original + f, = .006, .003 H(s,p) -100.04911 Diffuse + Very Diffuse 5 Original + i f =.006, .003, .002, .001 H(s) + < . = , .006 ^ . 003 H .( p) -100.04913 98 Table IV-7 £+ M.O. SCF Eigenvalues for Original and Altered Basis Sets M.O. # 1 2 Basis Set 3 4 5 1 -26.29879 -26.29880 -26.29881 -26.29709 -26.29788 2 - - 1 .60169 -1.60170 -1.60171 -1.60113 -1.60143 3 -0.76886 -0.76887 -0.76887 -0.76834 -0.76858 0.00093 0.00077 0.00 12 1 0.00274 0.00217 0.00337 6 0.00696 0.00924 0.00977 0.00615 0.00647 0.01225 0.00967 0.01978 9 0.01511 0.01592 0.01735 0.02001 0.02196 0.02161 0.02242 0.03603 0.04617 10 0.06850 0.06967 0.07051 0.08117 0.08168 13 0.09187 0.09396 0.09437:: 0.12321 0.13316 14 0.26703 0.26782 0.26847 0.27758 0.27779 99 Table IV-8 SCF Energy and Eigenvalues fo r O riginal and Altered Basis Sets (au) Original Basis Set Altered Basis Set SCF Energy rlOO.04905 -100.04905 Eigenvalues New M.O. — +0.00114 Original M.O. +0.00696 +0.00936 100 Table IV-9 Cl Energy within the Original and Altered Basis Set (eV) Original Basis Set Altered Basis Set HF Ground State Energy -2727.430 -2727.435 HF“ State Energy (New M.O.) — -2727.603 HF- State Energy (Original M.O.) -2727.440 -2727.389 101 Figure I V-1 Gaussian Basis Set 102 12 HF ( 6 T ) 10 8 HF“(5 V ) O ' H+F H'+F 6 4 H + F 2 1.0 2.0 3.0 4.0 5.0 6.0 O C O R (bohr) Figure IV-2 M.O. 6 . - - Basis Sets 1, 2 and 3 104 .020 .018 .016 J ? .014 Q «T -012 S 0 1 0 . «T -008 ■s- .006 .004 .002 X (bohr) o cn Figure IV-3 M.O. 9 - - Basis Sets 1, 2 and 3 1 06 107 .028 .026 .024 .022 .020 .018 .016 .014 .012 .010 .008 .006 .004 X ( bohr) M.O. 10 Figure IV-4 - Basis Sets 1, 2 and 3 1 08 .05 .04 .01 2,3 X (bohr) o ID Figure I V- 5 M.O. 13 -- Basis Sets 1, 2 and 3 1:10 .05 .04 .02 .01 \ \ I I j . / 1..- J N 28 -24 -20 -16 -12 -8 -4 0 4 8 1 2 X (bohr F l g ur e IV - 6 M.O. 14 - - Basis Sets 1, 2 and 3 11 2 . 10 .09 .08 k <S ? .07 o < < r .06 cm' .05 .04 .03 .02 2,3 .01 28 -24 -20 -16 -12 -8 - 4 0 4 8 12 1 6 20 24 28 X(bohr) Figure IV- 7 New M.O. - - Basis Set 2 114 . 020- .018 go .016 — Z . -0 1 4 - < V J * .012 - C M O O • r i ) — ' CM ^ .008 - .006 - .004 — . 0 0 2 = - - ^ i L-— -j. -28 -24 -20 -16 -12 -8 _]__I___ <k , \___ I___ I___ I I L -4 0 4 8 1 2 1 6 20 24 28 X( bohr) Fi gure IV -8 New M.O. 1 -- Basis Set 3 116 .020 .018 o® .016 Q cr -on c J .012 4 . C M . 0 1 0 .008 .006 .004 .002 X(bohr) Fi gure IV-9 New M.O. 2 - - Basis Set 3 118 .020 .018 -016 ~ .014 .012 CJ .010 .008 .006 .004 .002 X(bohr) Figure IV- 1 0 M.O. 6 - - Basis Sets 1, 4 and 5 1 20 .020 .018 .016 .014 .012 ~ .010 .008 .006 .004 .002 -28 -24 -20 -16 -12 -8 -4 0 4 8 X( bohr) Figure IV-11 M. 0. 9 - - Basi s Sets 1 , 4 and 5 1 22 I/2) . 0 4 0 . 0 3 8 . 0 3 6 . 0 3 4 . 0 3 2 . 0 3 0 . 0 2 8 . 0 2 6 co ° Q . 0 2 4 .022 .020 . 0 1 8 . 0 1 6 . 0 1 4 .012 .010 . 0 0 8 . 0 0 6 . 0 0 4 .002 X(bohr) 1 23 Figure IV- 1 2 1 0 Basis Sets 1 1 24 PO tn .06 . 0 5 0 4 .02 .01 X(bohr) M.O. 13 Figure IV -1 3 - Basis Sets 1, 4 and 5 1 26 .05 .04 C s J .02 .01 X(bohr) M.O. 14 Figure IV-14 Basis Sets 1, 4 and 5 1 28 .10 .09 .08 00 "a .07 ^ .06 1 04 5 -05 C M .04 .03 .02 ,4,5 .01 l / ' N i 28 -24 -20 -16 -12 -8 -4 0 4 8 1 2 1 6 20 24 28 X(bohr) Figure IV-15 Alternate M.O. 6 - - Basis Sets 1, 4 and 5 130 .020 .018 o o .016 .014 .006 .004 .002 Vi '■'Z X(bohr) 00 CHAPTER V CONFIGURATION INTERACTION CALCULATIONS ON THE RYDBERG STATES OF HF AND THE FESHBACH STATES OF HF~ Introducti on In Chapter I I I , a background discussion of the nature of core-excited resonances was presented. Before de sc ri bing our work in this area, we b r i e f l y review the relevant concepts. Resonances in electron scattering can be c la s s ifie d into two general categories. The f i r s t type of resonance, called e ith e r a temporary negative ion or simply a resonance, is formed when an incident electron is temporarily captured in the region of a target molecule. The second type of resonance, referred to as a core-excited resonance, is characterized by a "hole" in a normally occupied orbital and two "particles" in normally unoccupied o rb ita ls .^ The neutral Rydberg electronic state associated with a p a rtic u la r resonance is called the parent state, while the positive ion core is called the grandparent state. The second type of resonance, commonly known as a Feshbach state is the subject of this chapter. Core-excited resonances can l i e e ith e r above or below the parent. Feshbach Type I resonances, with which we are concerned here l i e below the parent state and thus e x hib it a positive electron a f f i n i t y . , These resonances have life tim e s that are long compared to a vibrational _______________________________________________________________________132 period. This c h a ra c te ris tic is manifested through bands with vibrational structure sim ilar to the grandparent. Since the two electrons trapped by the ion core reside in Rydberg o rb ita ls located fa r from the core, i t is not unreason able to expect the negative ion and the positive i on to show sim ilar vi brati onal progressi ons. Resonance spectra in the rare gases have been studied 2 in great d e t a i l . Feshbach states have also been detected and examined for various diatomic molecules , including 2 H2 , CO, N2 , NO, and 0 2 - Because of the general in tere st in electron scattering from polar diatomic molecules, experimental investigations on the Feshbach states of HF 3 4 have recently been performed. * The purpose of this chapter is to present the results of configuration interaction calculations on the Rydberg states of HF and the Feshbach resonances of HF"". Before discussing the results of the study, we b r i e f l y review the pertinent experimental data on this subject and give a description of the methods employed in our calculational procedu r e . Background HF. In spite of the corrosive nature of the vapor, the excited states of the HF molecule have been the subject of some experimental investigations. The lowest energy 133 j ------------------------------------------------------------------------------------------------------------------------------------------ excited state of HF, the so-called B1 Z+ state, has been studied the most widely. Johns and Barrow f i r s t found evidence of the state in 1959 through a vacuum u l t r a - 5 v io le t absorption experiment. Through observation of the f i r s t six vibrational le v e ls , values for of 0.145 o eV, and r fi of 2.09 A (3.95 bohr) were determined. DiLonardo and Douglas la t e r reexamined the B^Z+ state r c * 7 both through absorption and emission, * and detected vibrational structure through aboutj v=73. The v=0 level was found to l i e 83,305 cm~^ (10.33 eV) above the HF ^z + ground state. According to DiLonardo and Douglas, the vibrational levels are well behaved up to v=26, which lie s at an energy of 103,880 cm-1 (12.9 e V ). Above v=26, severe perturbations are observed as the bands of the B state become mixed with other Rydberg bands. DiLonardo 3 1 + and Douglas have detected a tt-X Z band in the region of the v=27 level of the B^Z+ state which they hypothesize as being responsible for the perturbations. They also suggest the p o s s ib ilit y of perturbations beginning as low as v=24, and speculate that lower unobserved vibrational 3 levels of the ir state may be the cause. The B^Z+ state has also been observed through electron O energy loss. In this study, Salama and Hasted place the v=5 vibrational level at 11.3 eV, the v = l1 level at 12.2 eV, and report an o > e of 0.15 eV. This is in d is agreement with the data of DiLonardo and Douglas which __________ : ________________ : _____________________________________________134 shows v=5 at about TT.0 eV. Other excited states of HF have also been observed below about 14 eV. Salama and Hasted id e n t if ie d two Ryd berg series superimposed upon a dissociation continuum 8 which commences at 11.15 eV. The f i r s t series,assigned as an s Rydberg series, is reported to originate at 11.72 eV with the 3s member. An g o of 0.35 eV and an e O r g of 1.207 A (2.28 bohr) for this member were determined. The second series, te n ta tiv e ly c la s s ifie d as a p series, is reported to origina.te at 12.82 eV with the 3p member. The quantum defect of this state is rather high for a p series (0 .9 5 ) , and Salama and Hasted do not rule out an a lte r n a tiv e assignment as an s series. The o)g for this series was determined to be 0.30 eV. DiLonardo and Douglas have also observed three singlet Rydberg states which l i e above a continuum in the 1400 A (8.9 eV) region.^ Between about 12.9 and 14.4 eV, they find evidence of two strong ^Z+-X^E+ bands as well 1 1 + 6 as a strong well-behaved tt-X E system. Although these states have not been analyzed in d e t a i l , the " * t t state is reported to originate at 13.03 eV with v ib ra tional spacing of 2656" cm”" * (0.33 eV).^ The B^E + state of HF appears to be well characterized, and there is good agreement in a ll respects among the reported experimental results. This does not hold true for the Rydberg excited states of HF. There is clear 135 disagreement between Salama and Hasted, who report two series commencing at 11.72 eV and 12.82 eV, and DiLonardo and Douglas who observe no Rydberg states below 12.9 eV. Ihe results of our calculations, which are discussed sh ortly, c le a rly support the findings of DiLonardo and Douglas and put into question those of Salamar and Hasted. HF+ . The ch aracteristics of the positive ion states of HF are important in the study of the negative ion Feshbach resonances. The nature of the core excited HF" states can be better understood by id e n t if ic a tio n with th e ir grandparent HF states. Two ionization potentials of HF have been i d e n t i f i e d , both through photoelectron spectro scopy. + 2 The lowest energy ionization potential is the HF t t 9 10 which is observed at about 16 eV. ’ The m and h of e e this state are reported by Berkowitz to be 3016 cm~^ ° ID (0.37 eV) and 1.026 A (1.94 bo h r) respectively. There appears to be some disagreement regarding the loca- 2 + + tion of the second ionization p o te n tia l, a £ HF state. Lempka et a l. place its origin at 18.6 eV,^° 9 while Berkowitz reports i t to originate at 19.1 eV. 2 + -1 The vibrational spacing o f t h e £ state is 1550 cm (0.19 eV), about half that of the 2 HF+ s t a t e . 9 ° 10 Berkowitz reports a value of 1.20 A (2.3 bohr) for r g . 136 HF~ . The Feshbach states of HF” have been the subject of two experimental studies, both of which u t i l i z e d the technique of electron transmission spectroscopy. Spence and Noguchi find evidence of a Feshbach state originating 3 at 12.825 eV. Four vib ra tio nal levels with a spacing of 0.355 eV are analyzed. Because of the excellent agree ment of the vib rational spaeings with those of the HF+ 2 2 t t ionization p o te n tia l, the state is assigned as a t t . Spence and Noguchi also report a severe perturbation on the f i r s t vibrational level of this state and provide two a lt e r n a t iv e explanations for the cause. F i r s t , depending upon the p r o b a b ilitie s for tr a n s it io n , they speculate that p e rfere n tia l decay to the B^Z+ state of HF may occur from one or more of the vibrational levels of the observed HF" state. Indeed, this explanation appears to be reasonable in l i g h t of the perturbations detected in the HF B^£+ state spectrum discussed e a r l i e r . The second explanation put forth is that the state may be crossed by a second negative ion state. The interaction of the two states would persumably cause a perturbation in the transmission spectrum. Although Spence and Noguchi analyzed the spectrum from about 11.5 to 15.5 eV, they found no evidence of any other HF" Feshbach states. In a second experimental study, Mathur and Hasted observe a sharp dip in the transmission spectrum at 137 4 10.05 eV. Although no vibrational structure is apparent, 2 + i t is t e n t a t iv e ly c la s s ifie d as a Z state. According to Mathur and Hasted, the lack of vibrational structure can possibly be explained by ah avoided crossing between two states of the same symmetry. The vibrational progres sion would end a f t e r one level with the opening of a new channel, that of a repulsive negative ion state. Mathur and Hasted also report a d i s t in c t Feshbach resonance at higher energy. This state originates at 12.388 eV, and based on an analysis of fiv e vibrational levels is stated to have an co e of 0.132 eV. Mathur and Hasted discuss the fact that the vibrational spacing does not agree well with that of the f i r s t ionization p o te n tia l, 2 + the it HF state. Although i t is not mentioned, neither does the spacing agree with the spacing of the f i r s t 2 + + excited ionization p o te n tia l, the Z HF state. Neverthe- 2 + less, the state is t e n ta t iv e ly c la s s ifie d as a Z HF Feshbach state. Mathur and Hasted discuss the disagreement of th eir results with those of Spence and Noguchi. Mathur and Hasted, who performed th e ir measurements at a la t e r date, analyzed the spectrum in the region between 12.8 and 13.5 eV. They found i t only marginally possible to detect structure in spite of the fact that Spence and Noguchi find clear evidence of a resonance in that energy range. A lt e r n a tiv e ly , in Spence and Noguchi's in v e s tiga tion , ______________________________________________________________ 138 performed earlier? no structure is reported between 12.3 and 12.8 eV, the region in which Mathur and Hasted detect a Feshbach state. In lig h t of the experimental data, we have attempted to provide some insights on the in te rp re ta tio n of the experimental observations between 9 and 14 eV through configuration interaction calculations on the excited states of HF and the Feshbach states of HF~. Before describing our results and comparing them with the experi mental observations, we provide a b r ie f summary of the method used in our calculations. Method Although some aspects of the method used in the c a l culations described here are the same as those given in Chapter IV, they w ill be repeated for convenience. The atomic o rbital basis set was chosen to be s u f f i c i e n t l y f le x ib le to represent both HF and HF“ for a range of internuclear distances. A Dunning basis, the (9s/5p) prim itive gauss tan. .basis of Huzinaga, ^ 1 contracted to 1 2 (3s/2p) was adopted, while the hydrogen was repre- 1 2 sented by the Dunning (2s) basis. In addition to these functions, uncontracted Gaussian Rydberg s functions of exponent 0.036 and 0.0066, and p functions of exponent 0.074, 0.02 9 and 0.0054 were added to F. F in a lly , a p type polarization function of exponent 0.9 was added to 139 hydrogen and a d type, of exponent 1.15, to F. The com plete atomic orbital basis set is shown in Table V -l. The SCF wavefunction and energy for HF were calculated within this basis set. Of the 30 molecular o r b ita ls com prising the basis, only one, that of lowest energy, was dropped from consideration in the Cl ca lcu latio ns . Table V-2 provides a l i s t of the molecular o rb ita ls and th e ir eigenvalues as well as the energy of HF at its equilibrium internuclear distance. The SCF total energy of -100.04905 a.u. can be compared with the near Hartree Fock energy of -1 00.0705 a . u . ^ The SCF v irtu a l o rb ita ls of HF were assumed to form an adequate basis for performing Cl calculations on HF- . 1 3 Previous work on HC1 confirms that this assumption is j u s t i f i e d . The SCF v irtu a l o rb ita ls of HF are eigen functions of the f u l l n electron HF potential including its permanent dipole moment. Since, in a case of this type, the scattering is long range, the target molecule is r e l a t i v e l y l i t t l e perturbed by the scattering event. Consequently, the HF v irtu a l o rb ita ls are assumed to be a good representation of the natural o rb ita ls of HF". Configuration Interaction calculations were performed on the ground state of HF, the excited states of HF, the 2 i t positive ion state, and the core-excited states of HF . The l a t t e r states were derived by populating the appro priate HF v irtu a l o rbital with two electrons. In general, _______________ 140 a few seed configuration were selected to represent the given state of in te re s t. Then all single and double hole p a r t ic le excitations r e l a t i v e to those few seed configurations were generated to form the total Cl space. Solution of the Cl problem was accomplished by the p a r t i tioning technique which was discussed in Chapter I I 15-17 and elsewhere. Results and Discussion Full Cl calculations on the states of HF and HF” lying between about 9 and 14 eV were performed at a number of internuclear distances. The equi1ibrium internuclear separation of the HF molecule is 1.732 bohr, and we con sider 7.0 bohr as representative of the dissociation l i m i t . Before providing the results of our calculations, we present a b r ie f description of our M.O. notation system, which should considerably simplify the discussion that f o l 1 ows. The values of the second column in Table V-2 number the M.O.'s of the same symmetry sequentially, and each degenerate pair of tt and A M.O.'s is assigned one number. We w ill r e f e r , in the following discussion, for example, to M.O. number 3 as the 3a and f M.O.'s number 4 and 5 as the 2ir. Although the M.O.'s have the order shown in Table V-2 at the equilibrium internuclear separation, at distances larger than 2.5 bohr the order has a ltered. _______________________________________________________________________ 141 Table V-3 presents the M.O. notation system for two separa tions, 1.732 bohr (e q u ilib riu m ), and 4.0 bohr, for reference purposes. In the Cl c a le u la tio n s , all single- and double-hole p a rtic le excitations were generated from more than one base. In each case, these generating bases were id e n tifie d by th e ir strong contribution to the final wavefunction. For each state, we w ill present the occupa tions of the bases in the notation described above. H_F. Table V-4 provides the de tails of the calculated potential curves for the HF ground and excited states. Figure V-l displays these data p i c t o r i a l l y . The generating bases for each HF state considered here, together with the weighting of the configuration in the final wavefunction for a few appropriate internuclear separations are pre sented in Table V-5. We w ill discuss the ch aracteristics of each HF state in turn. The potential curve for the 1^£+ HF ground state is identical to that presented in Chapter IV. We w ill therefore not-repeat the analysis except to r e it e r a t e that the calculated dissociation energy of 6.02 eV compares 1 8 well with the experimental value for Dg of 6.1 eV. The 2^E+ state indicated in Figure V-l is the so- called B state of HF. Its minimum, at 3.96 bohr is calculated to l i e approximately 10.62.eV above the ground state of HF. This is only s l i g h t l y higher than the 142 5 experimental value for Te of 10.51 eV. Through a parabola f i t , we obtain an we of 0.15 eV which also agrees well with the experimental value of 0.14 eV. 7 At long internuclear distances, near the equilibrium distance for this state, the ? } E + B state is c le a rly valence in character. In this region, i t is apparently quite ionic in nature and lie s close in energy to the H + F l i m i t of 16.07 eV. 7 At shorter distances, however, the state is Rydberg in character and represents the f i r s t member of an s Rydberg series leading to the f i r s t excited ioniza- + 2 + \ tion l i m i t HF ( E ) at 19.1 eV. This change in nature can be understood more easily by re fe rrin g to the data of Table V-5. The configurations contributing most strongly at 2.3 bohr are those repre senting a positive ion core with one electron populating successively higher, E+ M.O.'s. At 4.0 bohr, the dominant configuration 9 9 4 is la 2a I t t 3a4a. At this longer distance, the 4a M.O. has become valence in character. The fact that the 2^E+ B state is the f i r s t member of a Rydberg series leading 2 + to the E ionization l i m i t at shorter internuclear dis tances raises the question of the higher members. In exploratory Cl c a lcu latio ns , we do id e n t ify higher members. Since all l i e well above 14 eV in energy in the Franck Condon region, we have not investigated them fu rth e r. The 3^E+ state is a Rydberg state leading to the + 2 ' + HF ( t t ) ionization l i m i t . As Table V-5 indicates, i t is ________________________________ : ___________________________________ 1 43 ? represented by the tt positive ion core plus an electron occupying successively higher Rydberg t t M.O.'s. Its minimum lie s 13.75 eV above the HF ground states at about 2.0 bohr separation. This state is l i k e l y to be one of the ^£+ states observed by DiLonardo and Douglas between 12.9 and 14.4 eV. Although the perturbations observed in the higher vibrational levels of the B 1 ^ E+ state may be a res u lt of the interaction of the B state with the 3^E+ , i t is unlikely that this could be the case. Figure V-l shows the crossing of the 1 ^E+ and the 3^e+ to occur at about 13.7 eV. This is s ig n if ic a n t ly higher than the energy where the perturbations beg i n (12.9 eV). Even assuming the calculated potential curves to be too high by a few tenths of an eV, i t does not seem reasonable that t h e 3 \ +is responsible for the perturbations in the region of the v=27 level of the B^E+ state. The l^A HF state is identical in orbital occupation to the 3^E+ state ju st discussed. The two states p a ra llel one another and the minimum of the l^A at 13.66 eV is only s l i g h t l y lower than that of the 3^E+ state. To our knowledge, this state has not yet been detected experi.-i men t a l l y . According to Figure V-l , the 11 tt HF state is repulsive. This state corresponds to the dissociation continuum reported by DiLonardo and Douglas to occur in the region of about 1400 A (8.9 eV).® At short _______________________^ 144 internuclear distances, the 1 ^-rr is the f i r s t member of an + 2 s Rydberg series leading to the ionization l i m i t HF ( i t ) . The energy of this state at its minimum,together with the location of the ionization 1 irtiiit, leads to a quantum defect of 1 . 5 5 , which seems reasonable for an s Rydberg series. At longer distances, this state, lik e the B 2^£+ state, becomes valence in character. The I^ tt, together with the 1^E+ HF ground state leads to the l i m i t H(^S) + F(^P). The 2^t t is probably the f i r s t member of a p Rydberg + 2 series leading to the ionization potential HF ( t t ) . The minimum energy of this state given the minimum energy of the ionization l i m i t results in a quantum defect of 0.74, a value not unexpected for a p Rydberg series. Its dominant contributions again come from configurations 4 . representing an HF core and one electron occupying higher energy Z+ M.O.'s. The minimum of this state lie s at 1.978 bohr, 13.40 eV above the HF ground state. We calculate an m of 0.26 eV. DiLonardo and Douglas speculate that the cause of the perturbations above the v=26 vibrational level in the B^£+ state are due to a crossing of the B state by 3 a t t state. They observe this perturbation at 12.9 eV. The resultsof our calculations support DiLonardo and Douglas' i n t e r p r e t a t i o n . the 2 " * t t crosses the B state at 3 about 13.4 eV according to Figure V -l. Assuming the t t state lie s a few tenths of an eV below the 2 ^ t t , i t would indeed be in the proper energy range to explain the ______________________________ 145 perturbations. Indeed, there appear to be no singlet states in the energy region of the perturbations. Our calculated curve for the 21 state lie s about 0.25 eV 6 above the experimental value for Tg of 13.15 eV. The 2^t t state might also be the state observed by O Salama and Hasted at 12.82 eV. Since our ab i n i t i o procedure generally produces results that are too high by a few tenths of an eV when compared with experimental values, this is c e rta in ly possible. The location of our calculated state , however, seems to agree more closely with the experimental data of DiLonardo and Douglas who placed the observed t t state s l i g h t l y higher in energy. The 3^ t t state is probably the second member of the s Rydberg series leading to the H F+ ( ^ t t ) ionization l i m i t . Its dominant contributions to the final wavefunction i l l u s t r a t e this fa c t. At the calculated minimum of 1.977 bohr, the state lies 13.96 eV above the ground state of HF. Through a three point parabola f i t , w e calculate an of 0.37 eV. This state has apparently not been observed experim en tally. We have calculated the potential curves for the ground state and six excited states of HF below 14 eV. The I ^ tt, and the 3^t t in the Frank’ Condon Region are l i k e l y to be successively higher members of a Rydberg s series + 2 1 leading to the f i r s t ionization l i m i t HF ( t t ) . The 2 t t is probably the f i r s t member of a p Rydberg series leading 146 to the same l i m i t . The 21! * state :at short - internuclear distances is the f i r s t member of a Rydberg s series + 2 + leading to the excited ionization l i m i t HF ( E ). At longer internuclear separations, this state is valence in character and leads to the l i m i t H+ + F". The 3^E+ and the 1A states are each the f i r s t member of a Rydberg p i t , series leading to the ionization l i m i t HF-(^it). Our calculated potential curves agree in all major respects with the experimental results of DiLonardo and C * 7 Douglas ’ and find some disagreement with the results O of Salama and Hasted. We are convinced that no states other than those we have claculated l i e within the energy range of about 11 to 14 eV. We do not find the Rydberg series reported by Salama and Hasted at 11.72 eV. In the course of this study, we found i t useful to perform what are probably the f i r s t configuration in t e r - + 2 action calculations on the HF ( t t ) state. We find its minimum at 1.92 bohr at an energy 16.05 eV above the ground state of HF (1^E+ ). We calculate an a>e of 3211 cm” 1 (0.398 eV) through a three point f i t to a parabola. These calculated data agree exceptionally well with the experimental data which place the state at 16.05 eV withi.re-= 1 .94 bohr, with an o o e of 3016 cm" 1 (0.37 eV) . 10 HF~ . For convenience, we w ill present the results of the 1 47 results of HF potential curves for two separate energy regions: 9 to 12 eV, and 12 to 14 eV. F i r s t , however, a few general comments on the HF" state calculations w ill be u s e fu l. Each of the HF~ states has a parent HF excited state and a grandparent HF+ state. The grandparent state is readily id e n tifie d by the "hole" in the normally occupied M.O.'s. Although in the'discussion of the states of HF, we presented results only for singlet states we can presume that each has a corresponding t r i p l e t state where the electrons occupy the same spatial M.O.'s but have unpaired spins. By adding an extra electron to one of the other M.O.'s of each HF state, we can produce an HF" Feshbach state. Thus each singlet HF state is the parent of one HF” state , and each t r i p l e t HF state is the parent of one HF" state. For every singlet state con sidered in the last section, there are therefore two corresponding HF" states, one of which has a singlet HF parent and one of which has a t r i p l e t HF parent. Because of this "doublying" e f f e c t , the spectrum of the HF" states is extremely dense in the region of 9 to 14 eV. In addi tio n , most of the HF" states that l i e in this energy range are repulsive at a ll internuclear distances. This is not unexpected, p a r t ic u l a r ly at lower energies, where the binding energy to the grandparent positive ion core is very large. 148 We have calculated the potential curves of a ll HF" Feshbach states between 9 and 14 eV that are formed when an extra electron is added to the calculated singlet HF excited states and th e ir corresponding t r i p l e t states. We do not present the detailed potential curves of the states that are repulsive except where they are important to the in te rpreta tion of the experimental data. 9-12 eV. Table V- 6 presents the calculated energy results for the HF- potential curves that are displayed in Figure V-2 for this energy range. Table V-7 l i s t s the configura tions used as bases in the calculations for excitation generation together with th e ir fin a l percentage weighting in the wavefunction for various pertinent internuclear di stances. The 7^E+ , the 8 ^E + and the 9^£ + HF" states are all 2 + - repulsive Feshbach states. The 6 £ HF state which 2 + crosses these repulsive E states is a resonance HF state comprised of a f u l l y occupied HF core with the extra electron occupying a valence E+ M.O. The states in this energy region are of in te res t solely because of the dip observed at 10.05 eV in the transmission experiment 4 of Mathur and Hasted. 2 + The 6 E HF valence state is formed by occupation of the HF core with the extra electron in M.O. #17. We have not attempted to examine the c h a racteristics of this ' 149 state further since i t is important only because i t 2 + crosses the other E Feshbach states. 2 + The 7 E HF state is formed by adding two electrons ■f 2 + + to the normally unoccupied E M.O.'s of the E HF core. 2 + At longer distances this state crosses the 5 E state - 2 which leads, together with an HF t t state to the lim its H“ (^s) + F(^p). A detailed discussion of the 5 and 7 2 + E states was presented in Chapter IV. At short in te r - 2 + nuclear distances, the 7 E HF state descends rapidly 2 + and crosses the 8 and 9 E states at about 2 bohr. These l a t t e r states are formed by t h e a d d i 11 o n_ of two electrons 2 + to the normally unoccupied t t M.O.s of the t t HF core. 2 + The parent of the 7 E state is the t r i p l e t corresponding in orbital occupation to the B 2^E+ HF state. The 2 + parent of the 8 E state is the t r i p l e t corresponding to the 3^E+ HF state while the parent of the 9^E+ HF state is the 3^E + state of HF. The grandparent of all 2 + three states is the t t HF ionization l i m i t . The l e f t hand side of Table V- 6 displays the diabatic 2 + curves for the four E HF states. On the rig h t hand side, the adiabats that might produce the observed dip in transmission are i l l u s t r a t e d . We have shown the 2 + 2 + 8 E state to cross the 9 and 7 E states d ia b a tic a lly at about 2.0 bohr. At approximately 2.2 bohr, the 2 + 2 + 8 E state begins to in te ra c t with the 6 E : . I t follows 2 + the 6 E state curve for a short distance, then dissociates 1 50 2 + and a d i a b a t i c a 1 Ty follows the curve of the 8 E state. Whether or not the complex adiabatic interaction of 2 + the four E states a c tu a lly occurs must remain specula tiv e . The calculated energy where the interaction occurs 4 agrees with the experimental value of 10.05 eV. I t also appears from the adiabats of Figure V- 6 that only one vibrational level is excited before the dissociative channel is open, which also confirms the experimental data. Although this type of interaction might explain the dip in transmission observed by Mathur and Hasted, there is at least one strong reason for i t to be u n lik e ly . The main problem with the assignment of this avoided crossing as the experimentally observed feature is that i t would occur at about 2.2 bohr. This is well outside the Franck Condon region, and i t seems reasonable to assume that i t would therefore not be detected in a transmission experiment. Thus, although we do not believe the crossing to correspond to the experimental data, we have raised the p o s s ib ility here for completeness. 12 to 14 eV. In Table V-8 , we present data for three potential curves shown in Figure V-3 for the HF- states within this energy range. Table V-9 l i s t s the generating configuration bases and th e ir percentage weightings in the fin a l wave- funtion at 1.732, 2.0, and 2.5 bohr. As discussed e a r l i e r , there are numerous HF- states in this energy region. 1 51 I t is noteworthy indeed, that only one of these states, 2 the 6 i t , appears to be bound, and only two other states, 2 the 5 and 7 i t cross i t . The details of most of the other states are not presented since they complicate the spec trum and are not relevant to the explanation of the experimental data. 2 The 5, 6 , and 7 t t HF states of Figure V-3 are formed by adding two electrons to higher energy E+ M.O.'s + 2 of the HF ( ‘ i t ) core. The grandparent of the three t t 2 + states is therefore the t t HF ionization l i m i t . The 2 1 parent of the 5 t t is the 3 t t HF state shown in Figure 2 2 V -l; the parents of the 6 i t and 7 u states are a singlet and a t r i p l e t state of HF lying at about 15 eV. Although the singlet HF state is not shown in Figure V - l , i t is the 5^tt state of HF. Table V-10 gives the energy and weighting of the configurations contributing most 2 strongly to the final wavefunction of the 5 t t HF state and its parent, the 3^tt HF state. Table V -ll provides 2 the same information for the 6 and 7 t t HF states and the 2 1 7 t t parent, the 5 t t HF state. Id e n t if ic a t io n and assignment of the HF~ Feshbach states is an extremely complicated exercise for two primary reasons. F i r s t , i t is frequently d i f f i c u l t to follow a negative ion state from point to point across the potential curve because of the density of states of the same symmetry. Second, parentage assignment is often not straightforward both because we have only obtained results for the singlet HF states and because these HF states are themselves interacting with other neutral excited states across the surface. Given the information of Tables V-10 and V - l l , however, we w ill attempt to c l a r i f y the techniques we used for id e n t i f i c a t i o n . We emphasize that these methods are somewhat q u a lit a tiv e in nature. 2 The 5 t t HF state described in Table V-10 is one 2 - of the two HF States that crosses the bound 6 t t HF state. At all three internuclear distances, its largest contribution to the fin a l wave function arises from the 2 2 2 3 2 configuration la 2a 3a 1 t t 4a . This contribution, however, diminishes as we move to longer internuclear distances. A lt e r n a t iv e ly , at the equilibrium internuclear distance, the contribution from the las t lis te d configuration, 2 2 2 3 la 2a 3a 4 tt 8 a is minimal, but becomes s in g ific a n t at 2.5 bohr. We thus conclude that the state is losing the character of the f i r s t configuration and picking up the character of the la s t configuration as we move across the surface. One of the methods we use to "recognize" this state at the three distqnces is by noting its largest c o n t r i buting contributions. We have just seen, however, that the large contributions change somewhat from point to point. The other method we use to id e n tify the state at _______________________________________________________________________ 153 each point is by the constancy in the signs of the spin functions of the configurations. From 1.732 bohr to 2 . 0 bohr, the signs of the th ird configuration and one spin function of the f i f t h configuration of the 2 5 t t HF State have changed. I t may be that the f i r s t spin function of the third configuration has "gone through zero" between the calculated points. I t is more obvious that the second spin functions of the th ird and f i f t h configurations have indeed "gone through zero". These become larger and opposite in sign between 1.732 and 2.0 bohr. At 1.732 bohr, the 3^ t t HF state has its largest 2 2 2 3 contribution from the configuration la 2a 3a 4 tt 4a. At longer distances, the weight of this configuration diminishes while the contribution of the configuration 2 2 ^ 3 1 la 2a 3a 1 t t 7a increases. The 3 i t HF state interacts strongly with another state, the 2^-rr HF state shown in Figure V - l. The largest contribution to this l a t t e r state at 1.732 bohr is the configuration that attains 2 increasing weight in the 3 i t HF state. Parentage assignment of Feshbach states is extremely q u a lit a tiv e in nature. I t is important to emphasize here that the HF parent state and the correspond!*ng HF" state need not retain identical character across the potential surface. One must keep in mind that the HF states i n t e r act with other HF states, while the HF" states in te ra c t _____________154 with other HF states of d iffe r e n t character. The parentage assignment is based largely on the character of the states of 1.732 bohr. In addition, there is one other important id e ntifyin g c h a ra c te ris tic . The signs of the spin functions of a p a rtic u la r HF" states are e ither in phase or out of phase within a p a rtic u la r spin function and should id e ally mimic those of the parent HF state at all internuclear distances. In conjunction with this observation, i t should be kept in mind that the sign of the f i r s t spin function of a p a rt ic u la r state is a r b it ra ry and i t is only the signs of the other configura tions r e la t iv e to the sign of the f i r s t configuration that is important. We observe that the largest contributions to the 2 wavefunction of the 5 t t HF state are the configurations 2 2 2 3 la 2a 3a 1 i t 4a na, where n _ > 4. The parent state of the 2 5 t t state should thus be expected to have a large c o n tri- 2 2 2 3 bution from the configuration la 2a 3a 1 t t 4a. Indeed, we note that the 3^t t HF. state f u l f i l l s this requirement. I t is prim arily on this basis that we believe the 3^ tt 2 HF state to be the parent of the 5 t t HF state. In Table V-11, we show the energy and the contribution 2 to the fin a l wavefunction of the 6 t t HF state, the bound 2 state, and the 7 t t HF state which crosses i t . We i l l u s t r a t e the same data for the assigned parent 5 ^ tt HF state. We think i t probable that the parent of the 1 55 ? ■ *1 6 t t HF“ state is the t r i p l e t counterpart to the 5 t t HF state simply because t r i p l e t states generally l i e lower in energy. At the equilibrium internuclear distance, the 5 ^ tt 2 2 2 3 HF state is dominated by the configuration la 2a 3a 4 tt 8 a. The same configuration, with the extra electron occupying the 4a M.O. also dominates the two HF“ states. We note also that the r e la t iv e signs of the spin functions of the HF- states mimic those of the parent 5^ t t HF state reasonably w ell. Because of the heavy mixing among the ? 5, 6 , and 7 i t states, however, this requirement does not hold exactly. Another s ig n ific a n t observation is that the r e la t iv e signs of the spin function within a p a rtic u la r 2 configuration are in phase for the 6 t t HF state and out 2 of phase for the 7 i t HF state. In our experience, this is the expected behavior for two HF- states arising from the same single t and t r i p l e t parent HF state. 2 I t was already noted that the character of the 5 t t HF" state shown in Table V-10 changes between 1.732 bohr and 2.0 bohr. The contribution of the f i r s t configuration decreases while that of the la s t configuration 2 2 2 la 2a 3a 4TT4a8a increases. Exactly the reverse behavior 2 is observed for the 6 and 7 t t HF states. This can be understood readily with the aid of Figure V-3. Both the 2 2 5 and 7 t t states cross the 6 t t bound HF state just beyond 2.0 bohr. I t is therefore expected that all three states, _________ . _________________ 156 which are of the same symmetry, begin to in te rac t strongly at 2.0 bohr. This is indeed the reason for the change in character of all three HF“ states between 1.732 and 2 . 0 bohr. The Feshbach state observed experimental 1y by Spence 3 2 and Noguchi at 12.825 eV c le a rly corresponds to the 6 tt state of HF“ which has a calculated energy of 12.795 eV at 2.0 bohr. The reported perturbation on the f i r s t v ib r a tional level of the bound state is undoubtedly due to the 2 9 interaction of the 6 i t HF state with the 6 and HF 2 2 states. At 2.0 bohr, the 5 t t and 7 t t states l i e about 2 0.2 and 0.4 eV above the 6 t t states respectively. The vibrational spacing of the state observed by Spence and 3 Noguchi is reported to be 0.355 eV. I t is therefore reasonable to assume that one or both of the interacting states are responsible for the perturbation. Through a three^-point parabola f i t , we have calculated a vibrational 2 spacing of 0.61 eV for the 6 tt HF state. This is much larger than the experimental value of 0.355 EV. We believe the poor agreement with experiment to result from the wide spacing of the three calculated points. The good agreement with experimental obtained for some of the HF states, could be because the calculated energy points were more closely space; i t might also be that the agreement was simply fo rtu ito u s . In any case, we do not believe 2 that the 6 t t HF state is the Feshbach state observed by 1 57 1 4 Mathur and Hasted at 12.388 eV. The authors themselves admit that the vibrational spacing of the state, 0.132 eV, does not agree with the vibrational spacing of the grandparent HF+ ( ^tt} state. Our calculated spacing is much higher than 0.132 eV, and at least is closer to the experimental value reported by Spence and Noguchi. In summary, we find ourselves in good agreement with the HF~ experimental results of Spence and Noguchi, but in disagreement with those of Mathur and Hasted. We do not believe the complicated set of adiabats' displayed in Figure V-2 explains the resonance observed by Mathur and Hasted at 10.05 eV. We do not find the a ttra c tiv e state reported to occur at 12.388 eV. On the other hand, we do find an a t tra c tiv e state in the region of 12.8 eV with character!stics that agree in all major respects with the results of Spence and Noguchi. I t should be noted that we also find disagreement with the HF results of Salama and Hasted which were apparently obtained in the same laboratory as the HF~ results of Mathur and Hasted. Binding Energies. As discussed e a r l i e r , the binding energy of a negative ion state is defined as the difference between the experi mentally observed energy of the negative ion state and the energy of the positive ion state , or the ionization l i m i t . The bindvng 'energies of two 3sa^ electrons to 1 58 the positive ion core have been determined for several diatomic molecules. They have a ll been found to be about 4 eV. The binding energy of the 6 t t HF state given in Figure V-3 is approximately 3.3 eV, calculated at the energy minimum. This binding energy is lower than that of any of the resonances of the diatomic molecules given by Schulz in reference 19. This implies that the two outer electrons in our calculated state are held less strongly to the positive ion core than they are in the other diatomic molecules that have been investigated. The3s0 g M.O. in these, diatomic molecules corresponds to the 4a M.O. in .HF" at short internuclear distances. In HF", one of the configurations contributing to the final wavefunction is c e rta in ly that of the position ion core with two electrons occupying the 4a M.O. However, the contributions to the fin a l wavefunction also include 2 other configurations. The parent state of the HF 6 tt Feshbach state is the th ird member of an s Rydberg series 2 + (n = 5) leading to the t t HF ionization p o te n tia l. We can therefore assume th at, on average, at least one of the additional electrons in the HF" state occupies an M.O. for which n=5. That this electron is held more loosely to the core than would be an electron in a 3sa y M.O. is reasonable. We thus expect the binding energy to be less than for the diatomic molecules reported by Schulz. ________________________________________________________________________159 The electron a f f i n i t y of a detected resonance can be calculated by taking the difference in energy of the HF- state and the corresponding HF parent state. The 2 3 parent of the 6 t t HF state is the 5 i HF state which we did not calculate. However, asuming the t r i p l e t to be a few tenths of an eV below the corresponding singlet state, we would expect an electron a f f i n i t y of about 2 eV at 2.0 bohr. Conclusi ons We have reported the potential curves for the ground state of HF, several excited states of HF, and some of the Feshbach states of HF". Our calculated potential curves for the states of HF, though s li g h t ly high in energy, give good agreement with the experimental observations 6 7 of DiLonardo and Douglas 5 and poor agreement with those O of Salama and Hasted. The results of the HF" calculations c le a rly i l l u s t r a t e the u t i l i t y of applying ab i n i t i o techniques to the study of negative ion states. We id e n tify a complicated set of adiabats in the region of 10 eV which could conceivably be the cause of an observed dip in transmission 4 observed by Mathur and Hasted. Because of the location of the crossing of the states on the potential curves (> 2 . 0 bohr), however, we believe that the ' interaction is too fa r from the Franck Condon Region to be detected. v • . 160 Neither do we find an a t t r a c t i v e HF" state in the energy range of 12.388 eV also reported by Mathur and Hasted.^ In contrast,, we do id e n tify a Feshbach state at 12.8 eV with ch aracteristics that agree in a ll major respects 3 with a state detected by Spence and Noguchi. 161 References 1. G.J. Schulz, Rev. Mod. Phys. £5, 424 (1973). 2. G.J. Schulz, Rev. Mod. Phys. 45, 378 (1973). 3. D. Spence and T. Noguchi, J. Chem. Phys. 6_3, 505 (1975). 4. D. Mathur and J.B. Hasted, Chem. Phys. .34 , 29 (1 978). 5. J.W.C. Johns and R.F. Barrow, Proc. Roy. Soc. (London), A 251, 504 (1959). 6 . D. DiLonardo and A.E. Douglas, Can. J. Phys. J51_, 434 (1 973) . 7. D. DiLonardo and A.E. Douglas, J. Chem. Phys. 56, 5185 (1972). 8 . A. Salama and J.B. Hasted, J. Phys. B: Atom. Molec. Phys. 9, L333 (1976). 9. H.J. Lempka, T.R. Passmore, and W.C. Price, Proc. Roy. Soc. A, 304, 53 (1968). 10. J. Berkowtiz, Chem. Phys. Lett. JJ_, 21 (1971). 11. S. Huzinaga, J. Chem. Phys. £2, 1293 (1965). 12. T.H. Dunning and P.J. Hays, "Modern Theoretical Chemistry", Vol,ume 3, H.F. Shaefer, Ed., Plenum Press, New York, N.Y. (1977). 13. A.D. McLean and M. Yoshimine, J. Chem. Phys. £7, 3 256 (1967). 14. E. Goldstein, G.A. Segal and R.W. Wetmore, J. Chem. Phys. 6 8 , 271 (1978). 162 15. G.A. Segal and R.W. Wetmore, Chem. Phys. L ett. 32 , 556 (1975). 16. R.W. Wetmore and G.A. Segal, Chem. Phys. Lett. 36 , 478 (1975). 17. G.A. Segal, R.W. Wetmore, and K. Wolf, Chem. Phys. 30; 269 (1977). 18. J. Berkowitz, W.A. Chuptea, P.M. Guyon, J.H. Holloway, and R. Spohr, J. Chem. Phys. j[4 , 5165 (1971). 19. G.J. Schulz, Rev. Mod. Phys. 45, 423 (1973). 163 Table V-l Gaussian Basis Set Fluorine Hydrogen t * [5s] I s [2 s] 9995.0 0.001166 13.36 0.032828 1506.0 0.008870 2.013 0.231024 350.3 0.042380 0.4538 0.817226 104.1 0.142929 34.84 0.355372 0.1233 1 .0 0 0 0 0 0 1 2 .2 2 0.462085 4.369 0.140848 [lp ] 1 2 .2 2 -0.148452 1 .0 0 0 0 0 0 1 . 000000 1.208 1.05527 0.3634 1 .0 0 0 0 0 0 0.036 1 .0 0 0 0 0 0 0.0066 1 .0 0 0 0 0 0 £ P [5p] 44.36 0.020876 10.08 0.130107 2.996 0.396166 0.9383 0.620404 0.2733 1 .0 0 0 0 0 0 0.074 1 .0 0 0 0 0 0 0.029 1 .0 0 0 0 0 0 0.0054 1 .0 0 0 0 0 0 I d [Id ] 1.15 1 .0 0 0 0 0 0 164 Table V-2 SCF Results for HF (R = 1.732 bohr) M.O. Text Notation Symmetry Eigenvalue (a.u.) 1 1 a Z+ -26.29879 2 2 a Z+ -1.60169 3 3a Z+ -0.76886 4 1 7T TT -0.65063 5 7T -0.65063 6 4a Z+ 0.00696 7 2tr TT 0.01335 8 TT 0.01335 9 5a Z+ 0.01511 10 6 a z + 0.06850 11 3 tt IT 0.07965 12 IT 0.07965 13 7a z + 0.09187 14 8a z + 0.26703 15 4 tt TT 0.30504 16 TT 0.30504 17 9a z + 0.32973 18 10a z + 0.92540 19 11a z + 1 .33003 20 5 tt TT 1.34772 21 TT 1.34772 22 1 2a z + 1.71036 23 6 tt TT ‘l .84189 24 TT 1.84189 25 13a z + 2.75879 26 1A A 2.91399 27 A 2.91399 165 Table V-2 (continued) M.O. Text Notation Symmetry Eigenvalue (a.u.) 28 , t t 3.35095 7ir 29 t t 3.35095 30 14a Z+ 4.17831 31 15a ' Z+ 5.65648 Total Energy: -100.04905 a.u. Nuclear Repulsion Energy = 5.19630 a.u. Total Electronic Energy = -105.24535 a.u. 1 66 Table V-3 SCF Molecular Orbital Designation M.O. # 1.732 4.000 Order Symmetry Order Symmetry 1 la Z+ la z+ 2 2a z+ 2a E+ 3 3a E+ 1 TT T T 4 5 17 T T T T T 3a T T E+ 6 4a E+ 4a E+ 7 2tt T T 5 a E+ 8 9 5a T T z+ 2tt T T T T 10 6a E+ 6a z+ n 3t t T T 3t t T T 12 TT TT 13 7a E+ 7a E+ 14 8a . E+ 8a s + 15 4t T TT 4t t TT 16 TT TT 17 9a E+ 9a + E 18 10a E+ 10a E+ 19 11a E+ 11a E+ 20 5t t TT 5t t TT 21 TT IT 22 12a E+ 12a E+ 23 6# TT 6 tt TT 24 t t ' TT 25 13a E+ 13 a E+ 26 1A A 7 tt TT 27 A TT 167 Table V-3 (continued) M.O. # 1.732 4.000 Order Symmetry Order Symmetry 28 ? 7 r t t 14a :;e+ 29 J ' 1A '* 30 14a E A 31 15 I + 15a E+ 168 Table V-4 Calculated Cl Energy Points for H F a State 1.5 1.732 2 .0 Internuclear Distance (bohr) 2.3 2.5 3.0 3.8 4.0 4.2 5.0 7.0 1 V 0.761 0 0.336 — 2.046 3.624 5.187 5.384 5.552 5.922 6.015 2 V - 14.482b 13.637b 13.021 10.658 10,617 10.634 1 2 . 5C 3 V - 13.843 13.746 — 14.779 - - ; — — — 1 'a - 13.739 13.661 — 14.566 — — ; — — — 1 - 10.797 9.586 8.519 7.017 — — — 2 1, - 13.524 13.398 13.614 --; — — -- 3 1, - 14.220 13.961 14.411 -- — — — a Relative to H F at equilibrium, E = -2727.430 eV. b Determined by fittin g a parabola to the calculated values at 3.8, 4.0, and 4.2 bohr. ^ Adjusted to f i t experimental data. C T > lO Generating Table V-5 Bases for HF Potential Curves State 1.732 bohr Configuration and Weighting 2.3 bohr ( « 4.0 bohr i V 2 2 2 4 la 2a^3a 1tth {•99) (.9 9 ) 1a22a2lir43o2 (.74) la 22a23alT/hoa (.00) ( . 00) * ? 2 4 la 2a I t t 3a4a (.06) 2 2 4 2 la 2a I t t 10a (.00) (.00) T 20 2, 4 . 2 la 2a I t t 4a (.17) 2V 2 2 4 la 2a 3alu 4a (-2 3 ) 2 2 4 2 la 2a I tt 3a (.18) la 22a23alTr^6a --- (.4 9 ) la 22a2l 7 T^ 3a4a (.59) la 22a23alir^7a -- (.14) t 20 2, 4, 2 la 2a In 4a (.18) la22a23al-rr^8a --- (.06) 3 V la 22a23a2l7T32 7 T (.3 4 ) la 22a23a2l7r33 T r (.6 2 ) 1 ’ tt la 22a23a2lTT 34a (.17) (.14) la22a23a2i;.7r36a (.38) (.48) la 22a23a2lT T 3 7a ( . 22) (.10) *^J O la 22a23a2l 7r38a (.17) (.2 3 ) Table V-5 (continued) Configuration and Weighting State 1.732 bohr 2.3 bohr 4.0 bohr z1. lo 22023o2lir34o ( .0 2 ) (.44) 1o22023o21it360 (.2 2 ) (.0 1 ) 1o22o23o21ti3 70 (.5 5 ) (.1 2 ) l 0 22023o2lu 38a (.1 3 ) (.3 5 ) 3 1* l 0 22023o2lir34o (.7 2 ) (.1 8 ) l 0 22a2302l7r36a (.0 1 ) (.0 4 ) 1o22o23o21tt3 7o (.1 2 ) ( .6 7 ) 1o22023o21t t38o (.0 8 ) (.0 1 ) l'a l 0 22023a2ln 3 2iT (.3 0 ) l 0 2202302lrt33T T ( .6 5 ) Calculated Cl Table V-6 Energy Points for HF * " — 9 to 12 eV a State T .732 Internuclear Distance (bohr) 2.0 2.5 4.0 s Y — 8.811 10.324 — 72I + 13.601 11.328 8.716 5.809 822+ 11.887 11.258 9.379 6.362 92Z+ 12.657 11.521 9.923 6.506 a Relative to HF at equilibrium, E = -2727.430. 1 72 Table V-7 Generating Bases for HF” Potential Curves — 9 to 12 eV Configuration and Weighting {% ) State 1.732 bohr 2.5 bohr 4.0 bohr 62£+ — l 2? 23 21 4q I la c-a 3g Itt yg \ la^2a2l7T^6a9a ( .90) . 0 0 ) — 72E+ 1 a22o23a2l if^4c27r (.48) 1 a22a23a2l T T 34a2a ( .54) 2 2 3 ? la 2a Itt 3a 4a2n (.83) la22a23a2lT T 34a3iT (.36) la22a23a2liT24a3a ( . 0 0 ) la^2a2lTT^3a24a3-rr ( . 0 0 ) 8 2E+ lq22a23a2lTr34a2Tr (.59) la 22a23a2lTr24a27r ( . 6 6 ) la 22a2lT T 33a24a2ir ( . 8 6 ) 1 c22a23a2l T T 34a27r (.15) la22a23a2liT24a3Tr ( .0 1 ) 1 a22a2l 7 T 33a24a3-rr ( . 0 1 ) 9 2Z+ la 22a23a l7r44a2 (.23) la22a23alTr44a2 ( .24) Ia 22a2l7r43a4a2 (.15) 1 a22a23al T r4‘4a6a (.32) Ia22a23al7r44a6a ( .34) 1 a22a2l T T 43a4a6a (.45) 1 q22a23alTr44a7a ( . 22) la 22a23aliT44a7a ( .06) Ia 22a2lu43a4a7a ( . 0 0 ) lo 22a23al 7r4 4a8a ( . 20) Ia 22a23al7r4 4a8a ( •12) la 22a2lT T 43a4a9a ( . 0 0 ) 1 73 Table V- 8 Calculated Cl Energy Points for HF- — 12 to 14 eV a Internuclear Distance (bohr) State 1.732 2.0 2.5 HF" 5 2 tt 1 3 . 4 1 8 1 3 . 0 1 6 HF" 6 2 t t 1 4 . 2 0 3 1 2 . 7 9 5 HF" 7 2 tt 1 4 . 4 9 8 1 3 . 2 2 9 a Relative to HF at equilibrium, E = -2727.430 eV. 1 74 1 2 . 6 4 2 1 4 . 7 2 2 1 1 . 9 9 4 Generating Bases for HF' Table-:-V-9 Potential Curves - 12 to 14 eV State 1 .7 3 2 bohr Configuration and Weighting {% ) 2.5 bohr 5 2 tt 2 2 2 3 ? l a 2a 3a Itt 4a (.59) l a 22a23a21ir34a2 ( .3 4 ) l a 22a23a2l7T34a6a (.09) 1 a 22a23a2l n34a6a ( . 1 0) 2 2 2 3 l a 2a 3a Itt 4a7a ( . 12) l a 22a23a2ln 34a7a ( .0 3 ) 1 a 22a23a2l7r34a8a ( . 0 0) I a 22a23a2l7r34a8a ( .4 5 ) 6 2 tt l a 22a23a2lir34a2 (.09) 2 7 7 3 2 l a 2a 3a In 4a ( .0 6 ) • % „ 2 2 2 3 la * 2 a £3a£lTT°4a6a ( . 10) 1 a 22a23a2l n34a6a ( .1 8 ) l a 22a23a2lTr34a7a ( . 0 1 ) 1a 22a23a2ln 34a7a ( . 0 2 ) 2 2 2 3 l a 2a 3a Itt 4a8a (.4 8 ) l a 22a23o2ln 34a8a ( . 2 2) 7 2 tt l a 22a23a2lTr34a2 ( . 0 1) , 2 , 2 , 2, 3 . 2 l a 2a 3a I tt 4a / ( . 0 0) l a 22a23a2lTi34a6a (.1 4 ) l a 22a23a2ln 34a6a (.0 8 ) l a 22a23a2lfr34a7a ( . 0 0) l a 22a23a2ln 34a7a ( . 0 1 ) I a 22a23a2l7r34a8a (.7 2 ) l a 22a23a2ln 34a8a (.5 6 ) • Table V-10 2 Energy and Configuration Weighting of the 5 tt H F State and the 3^u H F State State Configuration 1.732 bohr Configuration Weighting 2 .0 bohr 2.5 bohr HF" 5 2 tt 2 2 2 3 2 la 2c 3a Itt 4c -0.77 -0.60 -0.58 2 2 2 3 Ta 2a 3a Itt 4a5a -0.05 -0.18 - 0 .0 0 -0.13 -0.04 +0.01 2 2 2 3 la 2a 3a In 4a6a -0.14 +0.28 -0.13 -0.26 +0.01 -0.28 2 2 2 3 la 2a 3a I tt 4c7a +0.17 +0.11 +0 .1 0 +0.31 +0.16 +0.13 2 2 2 3 la 2a 3a In 4c8a - 0.01 -0.50 -0.36 . +0.01 -0.26 -0.56 (13.418 eV) (13.016 eV) (12.642 eV) • v j CTl Table V-10 (continued) State Configuration 1.732 bohr Configuration Weighting 2 .0 bohr 2.5 bohr H F 3^t t 2 2 2 3 la 2c 3a I t t 4a -0.85 -0.61 -0.43 la 22a23a2lT T 3 5a +0.01 -0.14 - 0.21 la 22a23a2liT36a +0.09 -0.24 - 0.21 , 20 20 2, 3.. la 2a 3a I t t 7a -0.35 - 0 .6 8 -0.82 , 20 20 2. 30 la 2a 3a In 8a +0.29 +0.21 +0.08 (14.220 eV) (13.961 eV) (14.411 eV) ^4 Table V-ll ? Energy and Configuration Weighting of the 4 and 5 i t H F States and the 5^t t H F State State Configuration 1.732 bohr Configuration Weighting 2 .0 bohr 2.5 bohr HF" 6 2 t t 2 2 2 3 2 la 2° 3a In 4a +0.30 -0.40 -0.24 la 22a23a2lTr34a5a - 0 .0 0 - 0 .1 2 +0.04 - 0 .0 2 -0.04 -0.15 la 22a23a2l7T 34a6a -0.26 -0.28 -0.16 -0.18 -0.33 -0.39 la 22a23a2lT T 34a7a -0.04 +0.13 -0.07 - 0 .1 0 +0.13 +0.13 la 22a23a2lT T 34a8a +0.49 +0.25 +0.27 +0.48 +0.19 +0.38 ■ , 2 n 20 2, 3C 2 la 2a 3a In 6a +0.39 +0.47 +0.53 1 a22a23a2l 7 T 36a7a :+ 0 .15 +0.19 +0.25 +0.25 +0.32 +0.05 (14.203 eV) (12.795 eV) (14.722 eV) m m m t 00 Table V-ll (continued) Configuration Weighting State i , Configuration 1.732 bohr 2 .0 bohr 2.5 bohr HF" 7 2 tt 2 2 2 3 2 la 2a 3a i t t 4a +0.09 +0.32 +0.03 2 2 2 3 la 2a 3a I t t 4a5a -0.07 - 0 .1 0 -0.17 ■ +0 .0 2 +0.05 +0.03 2 2 2 3 la 2a 3a In 4a6a +0.33 +0.33 +0.27 -0.19 -0.23 -0.09 2 2 2 3 la 2a 3a I t t 4a7a -0.05 -0.09 -0.08 - 0 .0 0 -0.03 +0.03 2 2 2 3 la 2a 3a I tt 4a8a - 0 .6 6 -0.45 -0.65 +0.54 +0.57 +0.38 (14.498 eV) (13.229 eV) (11.994 eV) 1 79 Table V-11 (continued) State Configuration 1.732 bohr Configuration Weighting 2 .0 bohr 2.5 bohr ■ * H F 5 ^ 2 2 2 3 la 2a 3a I t t 4a -0.18 -0.35 -0.42 2 2 2 3 la 2a 3a I t t 5a +0.13 +0.09 +0.09 2 2 2 3 la 2a 3a In 6a +0.59 +0 .6 6 +0.65 la 22a23a2l 7 T 3 7a -0.09 -0.13 =0.03 la 22a23a2l7T 38a -0.74 -0.61 -0.58 (15.693 eV) (14,933 eV) (15.086 eV) 00 o Figure V-1 Calculated Potential Energy Curves for HF 181 Energy (eV) ro no O + oj oj j o b cr m O Ol o 05 o -> 1 b i Calculated for Figure V-2 Potential Energy Curves HF" - - 9 to 12 eV 183 DIABATIC ADIABATIC 2 .0 3.0 4.0 R(bohr) Figure V-3 Calculated Potential Energy Curves for HF" - - 12 to 14 eV 185 98 L Energy (eV) N > c r OJ

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Creator Wolf, Kathleen (author)

Core Title Configuration interaction calculations on the resonance states of HF- the excited states of HF, and the Feschbach states of HF-

Degree Doctor of Philosophy

Degree Program Chemistry

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag chemistry, inorganic,OAI-PMH Harvest

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Advisor Segal, Gerald (committee chair), Dows, D.A. (committee member), Dunn, Arnold (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c17-635688

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