University of Southern California Dissertations and Theses

Configuration interaction calculations on the resonance states of HCl- and the excited states of cyclopropane

(USC Thesis Other)

Configuration interaction calculations on the resonance states of HCl<super>-</super> and the excited states of cyclopropane

PDF

Download

Open document

Flip pages

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content CONFIGURATION INTERACTION CALCULATIONS ON THE RESONANCE STATES OF HC1" AND THE EXCITED STATES OF CYCLOPROPANE by Eli sheva Go!dstein A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial F u lfillm e n t of the Requirements fo r the Degree DOCTOR OF PHILOSOPHY (Chemi s t r y ) September 1979 UMI Number: DP21843 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. WwHHtatfoft PBMisMfig UMI DP21843 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 48106 - 1346 ' UNIVERSITY OF SOUTHERN CALIFORNIA T H E G R A D U A T E S C H O O L U N IV E R S IT Y P A R K LO S A N G E L E S , C A L IF O R N IA 9 0 0 0 7 P k h . c ' ? 0 This dissertation, written by ( j (o cX Y ........ under the direction of h.j£Ar?Dissertation Com- cZ . mittee, and approved by all its members, has , been presented to and accepted by The Graduate School, in partial fulfillment of requirements of the degree of D O C T O R O F P H IL O S O P H Y ..................... Dean DISSERTATION COMMITTEE Chairman To Damon, Elon and Danielle fo r t h e ir love and encouragement. i i ACKNOWLEDGEMENTS I wish to express my gratitude to Dr. Gerald Segal, both fo r the learning opportunities he has made available to me over the past fiv e years, and fo r his continuous support and encouragement. I wish to express my thanks to my research group, especially to Dr. Ross Wetmore who made me think of Paldus diagrams when I was inclined to think of o r b it a ls , to Katy Wolf fo r her frie n d s h ip , and to Dr. Dean Liskow, Dr. Jim Diamond and Dr. Saraswathy fo r t h e ir support. F in a lly , I would lik e to express my appreciation to Michele Dea. TABLE OF CONTENTS Page DEDICATION................................................................................................. i i ACKNOWLEDGEMENTS i i i TABLE OF CONTENTS.............................................................................. iv LIST OF TABLES................................................................................ vi LIST OF FIGURES........................................................................ . . v ii ABSTRACT...................................................................................................... ix INTRODUCTION........................................................................ 1 Chapter I. HISTORICAL BACKGROUND .................................................. 9 I I . METHOD......................................................................................... 31 I I I . APPLICATION OF C.I. METHOD.TO ELECTRON- HC1 SCATTERING................................................................ 58 A. P r e l i m i n a r i e s ......................... 58 i ) I n t r o d u c t i o n ............................................ 58 i i ) Summary of Experimental Evidence 63 B. Computational Details .................................... 69 i) Method of Computation........................ 69 i i ) Proof of Resonance Character . . 78 C. In te r p r e ta ti on............................................................ 8 2 i) Explanation of Observation . . . 82 i i ) Electron-po1ar-molecu!ar S c a t t e r i n g ................................................. 88 i v Chapter Page D. Conclusion........................................... 96 E. References................................................................ 98 IV. CYCLOPROPANE.........................................................................117 A . Prel imi nari e s .................................... 117 i) I n t r o d u c t i o n ..........................117 i i ) Summary of Experimental and Theoretical Work . . . . . . . . 120 B. Computational D e t a i l s ................................... 127 C. Discussion of Results . . . . . . . . 138 D. Conclusion....................... 144 E. References ...................................................147 v LIST OF TABLES Table Page 3.1 Gaussian Basis Set . ............................. . . 102 3.2 SCF Results for HC1 (r = 1.2744 A). . . . 103 3.3 Effect of Truncation Upon the C.I. C a l c u l a t i o n s ........................................... 104 3.4 Calculated C.I. Energy Points^9^ (A d ia b a tic ). . ......................................................105 3.5 Summary of Calculated Results (ev) . . . 105 4.1 Cyclopropane Orbital Energies^ ..................... 149 4.2 Comparison of Calculated Results and E x p e r i m e n t ..................................................154 4.3 Summary of C.I. Results fo r All Calculated State s. . . ; . .........................156 4.4 Calculated M C D ........................................................162 4.5 B Term Components . ...................................................164 vi Figure LIST OF FIGURES Page 2.1 P a rtitio n in g of the Hamiltonian M atrix. . 56 3.1 Calculated potential energy curves for HC1 and HC1"............... ... . . . . , . . . . 107 3.2 Radial d is t r ib u t io n function fo r the 2 + scattered electron in the 1 £ state (r=1.2744 A) for the basis set of Table 3 .1 ( I ) , I plus a = .001 s type gaussian ( I I ) , and II plus a = .001 p type gaussian function... . . 109 3.3 Radial d i s t r ib u t i o n function for the 3 2£+ state ( r=1.2744 A) for the basis.set of Table 3.1 ( I ) , I plus a = .001 s type gaussian ( I I ) , and II plus a = .001 p type gaussian f u n c t io n ........................... . .................... 3.4 Radial d is t r ib u t io n function fo r the 3 + scattered electron in the 3 £ O state (r=1.2744 A) fo r the basis set of Table 3.1 ( I ) , I plus a = .001 s type gaussian ( I I ) , and II plus a = .001 p type gaussian f u n c tio n ...................................................... vi i Figure Page 3.5 Radial d i s t r ib u t io n function for the scattered electron in a.root intermediate in energy between 2 2E+ and 3 2Z + state ( r = l .2744 A) for the basis set of Table 3.1 ( I ) , I plus a = .001 s type gaussian ( I I ) , and I I plus a = .001 p type gaussian fu n c tio n ....................................................................115 4.1a. Internal o r b ita ls in cyclopropane, . . . 166 b. External o r b ita ls in cyclopropane . . . 166 c. Geometry of cycl opropane.........................................167 4.2a. -Experimental absorption, MOD and moment a n a l y s i s . ................................................ 169 b. Comparison between calculated and experimental MCD.................................................170 c. Calculated B term component compared to total experimental MCD............................ 171 d. Calculated A term component compared to total experimental MCD.. . . . . . 172 v i i i ABSTRACT Large scale Configuration In tera c tion techniques are used to compute a_b^ in i t i o, wavefuncti ons for several molecular systems in order to derive the oretical estimates fo r t h e ir chemical behaviour. Configuration in te ra c tio n and the s t a b iliz a t io n method are used to compute potential energy curves for the resonant states of HC1” , an important example of electron-polar molecule s catte ring . By using Self- Consistent Field and Cl techniques, resonance states that dissociate to H .~ + Cl and Cl" + H are found as well as those that dissociate to HC1 + e". These curves provide an in te rp re ta tio n of the known experimental observations on this system. In p a rtic u la r explanations are provided on threshold features of v ib ra tio n a l ex c i ta t io n , d is s o c ia tiv e attachment and associative detach ment.without doing rigorous e le ctron-polar molecule scattering computations. To i l l u s t r a t e the true resonant character of the lowest three HC1” states, the basis set was expanded by addition of diffu se functions and a graph of the density functional of each molecular o r b ita l as a function of internuclear distance shown. The three lowest states of HC1 display a high p r o b a b ility of trapping the electron in the area of the ta rg e t, HC1 , and sinosoidal behavior at larger inte rnu clear distances, i x which provides proof of resonant character. To fu r th e r u t i l i z e an e f f i c i e n t Configuration Intera ctio n program the ab i n i t i o wave functions of the excited states of cyclopropane are calculated over the energy range 7-11 eV. Due to the broad and over lapping bands which appear in the op tical spectrum of cyclopropane the eigenvalues of the excited states are not s u f f i c i e n t to id e n t if y unambiguously the nature of the states. Therefore e x c ita tio n energies augmented by o s c i l l a t o r strengths and magnetic moments are reported. Accurate values fo r the MCD parameters A-j/Dq and Bq/Dq are also given. Using these eigenvalues and properties the f i r s t and t h ir d absorption bands are i d e n t if ie d as E‘ states each consisting', of more than one t r a n s it io n . The middle band is i d e n t if ie d as an AJJ state. Dipole forbidden A£ as well as Aj states are reported and may be related to known tra n s itio n s in the trans 1,2-dimethyl cyclopropane spectrum. The lowest t r i p l e t state belonging to the Rydberg band is computed at 7 eV. F in a lly , an ab i n i t i o plo t of the MCD of cyclopropane is reported using band widths s im ila r to those reported experimentally and gaussian line shapes. Close agreement between experimental and ab ijvftio MCD spectra is found. INTRODUCTION I t is more than f i f t y years since the underlying theory of quantum mechanics was introduced by S c h r o e d i n g e r . ^ Although the conceptual framework of quantum chemistry has been well established fo r many years, the success of i t s q u a n tita tiv e ap plicatio n to small systems has been slow. Attempts to calculate th e o r e t ic a lly the electron ic stru cture of molecular systems are nearly as old as the Schroedinger equation. I t is evident that in order to apply the Schroedinger equation with success and e ff ic ie n c y , approximations are necessary. The f i r s t s im p ! if ic a t io n occurs when e le c tron ic and nuclear motions are separated by regarding the nuclei as clamped in fixed positions, as in the Born-Oppenheimer approxima- (2) tio n . The two general approaches towards s i m p l i f i c a tion of the solution of the ele ctron ic Schroedinger equation which have predominated since the early days (3) of the f i e l d are the valence bond theory and (4 ) molecular o r b ita l theory. The l a t t e r appears as a logical extension of Bohr theory with the modification of the one-electron wave function in place of .an electron o r b i t a l , and the additional aspect of having this o r b ita l spread over the e n tire molecule. 1 The basic idea of the M.O. method is to find the approximate wave function for a molecule by assigning each electron a one-electron wave function and the s im p lic it y of this method is best i l l u s t r a t e d by a p plica tions at the atomic le v e l. According to Hartree's (5) s e l f - c o n s i s t e n t - f i e 1d (SCF) model of the atom, the motion of each electron in. the e ffe c tiv e f i e l d of N-l others is governed by the one-electron wave function. Self-consistency of the electron ic charge d is t r ib u t io n with it s own e le c t r o s t a t ic f i e l d leads to a set of coupled i n t e g r o d if f e r e n t ia l equations fo r the N one- p a r tic le wave functions, or atomic o r b it a ls . An improved formalism due to S la te r^ 8^ and F o c k ,^ ^ represents the atomic wave function by a determinant b u i l t of atomic spin o r b ita ls which is consistent with P auli's P r in c i p le . ^ 8^ Application of the va ria tio n a l p rin c ip le to a Slater determinant leads to a set of N coupled equations which are improved by the addition of exchange i n t e r action and are known as the Hartree-Fock equations. The error inherent in the Hartree-Fock method, known as electron c o rre la tio n , arises from the smoothing out of the in te r e le c tr o n ic repulsion into e ffe c tiv e Coulomb and exchange p o te n tia ls . The c o rre la tio n energy accounts fo r roughly a 1% e rror in the total energy but this percentage is magnified in energy differences which are more relevant to experimental q u a ntitie s . A s ig n if ic a n t improvement in computational f a c i l i t y is achieved i f the o r b ita l functions are expanded in terms of a f i n i t e set of basis functions. The integro- d i f f e r e n t i a l equations are then transformed into (9) algebraic equations known as Roothan's equations. This analytic approach makes i t possible to apply the SCF method to molecular systems where the M.O.'s are described as lin e a r combinations-of atomic o r b it a ls . B r i l l o u i n ^ ^ ( 1933 , 1 934) used the Hartree-Fock method to s im p lify the matrix of the many p a r tic le Hamiltonian. He suggested that the H-F solutions fo r a single Slater determinant may be a good s ta r tin g point fo r applying higher order perturbation corrections to the Hamiltonian matrix. Moller and P l e s s e t ^ ^ (1934 ) (12) as well as Nesbet and Epstein ' have also investigated perturbational approaches to the c o rre la tio n energy problem. The major d i f f i c u l t y in applying th e ir theory was r e s t r ic t io n s on the single p a r t ic le function. The improvement of the Hartree-Fock calculations by a superposition of S later determinants is known as Configuration In te ra c tio n . Unfortunately, for even moderate sized molecules (formaldehyde), re lia b le proce dures fo r theoretical description of electron correla tio n require several orders of magnitude more computation than the simple SCF procedure with the same basis. 3 I f a stra ightforw ard C,I. approach is used (this usually include s . a ll single and double e xcitations r e la tiv e to a base c o n fig u ra tio n ), the magnitude of the problem is fa c t o r ia l with N, (where N is the basis 4 s iz e ), which does not compare favorably with N typical of a SCF ca lc u la tio n . Clearly then, the major d i f f i c u l t y with the C.I. procedure is it s size. The number of configurations in a given calculation depends also on the level of accuracy with which one is concerned. I t seems that s ig n if ic a n t corrections to the H-F SCF basis are achieved at the level of single and double excita- (13) tion s. Since i t is not uncommon fo r this number of configurations to reach several thousands, basis set truncation and configuration selection to minimize the size are important. The method of selecting configurations used in this w o r k ^ 4^ is related to methods used by S h a v i t t ^ 5^ and co-workers. The s i m i l a r i t i e s l i e in choosing a group of dominant configurations which form the nucleus of the C.I. The method used here then u t i l i z e s Raleigh- Schroedinger perturbation to rank the remaining configura^ tions according to t h e i r in te ra c tio n with th is nucleus. The main in te ra c tin g configurations are treated exactly and the remaining configurations which are important in t h e ir sum are treated p e r t u r b a t io n a lly . This p a r t it io n in g of the C.I. matrix increases the e ffic ie n c y as well as providing a solution which, is simple to in te r p r e t. In a d ditio n , improved m e thod s^6^ fo r the evaluation of the C.I. matrix elements have made calculations on medium size molecules quite p r a c tic a l. The major work presented here involves application of these SCF and Cl techniques to two problems of chemical in te re s t. Chapter I I I presents a complete discussion of electron scattering from the polar molecule, H C l . ^ ^ Due to the long range in te ra c tio n potential between an electron and a permanent e l e c t r i c dipole, many novel and in te re s tin g features arise. In the case of HC1, the existence of a bound negative ion is not expected since it s dipole moment is less than the c r i t i c a l value (18) shown to bind an electron. However, experimental evidence does indicate the existence of resonant states. The work presented here computes the resonant states of HC1~ that dissociate to H” + Cl as well as Cl" + H and HC1 + e. The Cl method is used to compute potential energy curves which provide explanations and new insights into the various experimental observations. The c a lc u la tions performed here lend support to the novel Cl technique employed and f u l l y e x p lo it its e ffic ie n c y . In Chapter IV, the Cl method is applied to a ring molecule, cyclopropane. Although a wealth of experimental data exists concerning the excited electronic states of cyclopropane, there has been considerable d i f f i c u l t y in 5 the assignment of these states. The absorption spec- (19) trum is in general quite intense but the bands are very broad and overlapping since as many as nine e le c t r o n ic a lly allowed excited states are known to e xist between 8-11 eV. The Magnetic C ircula r D i c h r o i s m ^ ^ spectrum is also d i f f i c u l t to in te r p r e t because the states are crowded together re s u ltin g in s ig n if ic a n t second order perturbations which make the spectral analysis quite d i f f i c u l t . Cyclopropane lends i t s e l f to detailed Cl calculations since a good wave function is necessary to predict accurate molecular properties. S p e c ific a lly , our goal was to find the accurate eigenvalues fo r the excited states of cyclopropane and match these with the respective o s c i l l a t o r strengths and magnetic moments. F in a lly , to insure consistency with experiments, the ab i n i t i o Magnetic C ircular Dichroism and absorption spectra were calculated. Again this work indicates the e ffic ie n c y and accuracy of the Cl method employed even in the case of this f a i r l y large molecule. In the work on cyclopropane, both explanation of experiments and prediction of some states as of yet unobserved are deta i 1ed. 6 REFERENCES 1. E. Schroedinger, Anna]. Physik, 79, 361, 489, 734 (.1926), Phys. Rev. 28, 1049 ( 1 926). 2. N. Born and J.R. Oppenheimer, Ann. Physik, 8j4, 457 ( 1 927) . 3. W. H e itle r and F. London, Z. Physik, £4, 455 (1927). 4. R.S. Mulliken, Phys. Rev. '32, 186 (1928); 32, 761 (1928) , 4J_, 49 ( 1 932) . F. Hund, Z. Physik, 5J_, 753 ( 1928). 5. D.R. Hartree, Proc. Camb. Phil. Soc. 24, 89, 111 (1928). 6. J.C. S later, Phys. Rev. 34, 1293 (1929). 7. V. Fock, Z. Physik, 6£, 126 (1930); Z. Physik, 62, 795 (1930). 8. W. Pauli, Z. Physik, 43, 601 (1927). 9. C.C.J. Roothaan, Rev. Mod. Phys. 2j3 , 69 (1 951 ), 32, 179 (1960). 10. L. B r i l l o u i n , A ctualite s Sci. Ind., No. 71 (1933), No. 159 (1934). 11. C. Moller and M.S. Plesset, Phys. Rev. 46, 618 (1934). 12. P.S. Epstein, Phys. Rev. 28, 695 (1926), R.K. Nesbet, Proc. Roy. Soc. (London), A 230 , 31 2 (.19 55) . 13. A. Pipano and I. S havitt, IJQC 2_t 741 (1968). 7 14. G.A. Segal and R.W. Wetmore, Chem. Phys. Lett. 32, 556 (1975), G.A. Segal, R.W. Wetmore and K. Wolf, Chem. Phys. 30, 296 (1978). 15. I. S ha vitt, The Method of Configuration In te ra c tio n , in Modern Theoretical Chemistry, Vol. I I I . Methods of Electronic Structure Theory, H.F. Schaeffer I I I , ed. Plenum Press, New York and London (1977). 16. R.W. Wetmore and G.A. Segal, Chem. Phys. Lett. 36_, 478 (1 975). 17. E. Goldstein, G.A. Segal and R.W. Wetmore, J. Chem. Phys. 68, 271 (1978). 18. O.H. Crawford, Proc. Phys. Soc. (London) 9J_, 279 ( 1 96 7). 19. H. Basch, M.B. Robin, N.A. Kuebler, C. Baker and D.W. Turner, J. Chem. Phys. 5_]_, 52 ( 1 969 ). 20. A. Gedanken and 0. Schnepp. Chem. Phys. 12_, 341 ( 1 976 ). 8 CHAPTER I HISTORICAL BACKGROUND Ab i n i t i o chemistry re lie s upon the solution of the Schroedinger e q u a t i o n , ^ in which the state function m and the energy Eq are related in the following way: H¥ = Em [1] H, is the Hamiltonian operator describing the total energy fo r the system under in v e s tig a tio n . Neglecting r e l a t i v i s t i c effects as well as s p in -o r b it coupling inte ractio ns (since small chemical systems composed of l i g h t atoms are consi dered), the time-independent Hamiltonian fo r a c o lle c t io n of electrons and nu clei, in atomic units (h = mg = qg = 1) becomes h - - l - l * r + l - + l l - > l ^ + l 1 2MA i 2 A i r Ai A<B r AB i< j r i j [2a] where A, B, sum over the nu clei; i , j sum over the electrons and denotes the nuclear charge. Equation [2a] may be w ritte n as: 2 2 ^ = \ ~ 2M^ + I " ~T + VAi + VAB + Vi j Distance i n 8a.u. is given in bohrs where 1 bohr = 0.52918x10“ cm. The energy unit is the Hartree which equals 27.205 eV. where the f i r s t term is the k in e tic energy of the nuclei, the seond term is the k in e tic energy of the electrons, the t h ir d relates the a ttra c tio n between the nuclei and electrons and the two fin a l terms denote the nuclear and e le ctron ic repulsions respectively. The solution to equation £2] w i l l depend on B, the nuclear coordinate and r, the e le ctron ic coordinate. An extremely useful approximation which f a c i l i t a t e s the solution of equation £2] is the Born-Oppenheimer ( 2 ) approximation. In order to attempt a separation of the wave function into ele ctro n ic and nuclear parts, we w rite the following product ¥(R,r) = n(R)$Cr) When operating on this product with (H-E) and denoting E as e + (E-e), the following equation re su lts: + VA1 + VAB + Vi j ' + I V^/Ma - (E -e )]ri(R ) - \ I jf" tn(R) v j $(r) + 2vA$£r)*vAn(R)]. A In order to achieve complete separation of ele ctro n ic and nuclear coordinates, the la s t term would have to vanish which would indicate that there is no coupling of e le c tro n ic and nuclear motion. The neglect of the 10 coupling terra is a good approximation since the terms are weighted by (1/M^) which becomes small fo r heavier atoms and the terms inside the bracket are of the same (2 3 ) order of magnitude as e le ctro n ic terms. ’ The result is a set of separated equations £-1 I Vi + VA B + Vij - eR]*R<£) - 0 W l - \ I ^ / M a - (E-e(R))]n(R) = o [4] A The subscript R, in equation [3 j means that because of V A f , 4>(r) depends param etrically on the nuclear coordinates. Equation [3] may be solved fo r $ ( r ) , the ele ctro n ic wave function fo r fixed nu clei, and eR, the corresponding e le c tro n ic energy. When $(r) and are calculated at many d if f e r e n t inte rnu clear distances, a potential surface re s u lts . This surface then serves as the e ff e c tiv e potential fo r nuclear motion in the Born-Oppenheimer approximation, and is used to solve fo r n ( R ) . » the nuclear wave function. This work w i l l not attempt to solve equation [4] which is relevant to problems of v ib ra tio n a l analysis or problems of e la s tic scattering of electrons in atoms and molecules, but w i l l concentrate on the solutions of the ele ctron ic part of the problem. I t must be stressed however, that nuclear motion may e ffe c t properties of a molecule which are otherwise considered e le c tro n ic . An example of such an in te rp la y is the Jahn-Teller e ffe c t which causes a break in e lectron ic degeneracy by d is t o r t in g the symmetry of the molecule and is relevant in examining some of the spectroscopic features of (4) cyclopropane. ' The clamped-nuclei approximation may also f a i l in explaining some of the phenomenae associated with binding an. electron to a polar molecule. Non- Born-Oppenheimer rota tion a l degrees of freedom have been shown to e ffe c t s i g n i f i c a n t l y the presence of bound (5) states of the negative ion. Although i t may appear that the Schroedinger equation has.been g reatly s im p lif ie d , the solution of equation [3J is s t i l l very d i f f i c u l t . When $(r) is determined fo r various fixed nuclear configurations of a molecule in { fi ) accord with the v a ria tio n a l p r in c ip le , $(r) must have an expectation value that is always greater than the exact eigenvalue. <$(r)|H j $ ( r)> --------------—- = :<e> > e . [5] <$(r). | $( r ) > exact The best wave function in the v aria tio n a l sense is that for which as many parameters as possible have been optimized to y ie ld the lowest energy. When complete f l e x i b i l i t y in v a ria tio n is allowed, the wave function w i l l be an exact solution of the Schroedinger equation. 1 2 Ine vita b ly however, more approximations to [3] are necessary since the o n l y . f u l l an alytic solution fo r the e lectron ic Hamiltonian is the one electron case. The p a r tic u la r form chosen for $ must be computa t i o n a l l y convenient, simple, and easy to in te r p r e t. H a rtre e ^ ^ f i r s t introduced $ as a simple product of one-electron functions The functions are known as the one-electron spin o r b it a ls . A spin o r b ita l is defined as a product of space coordinate and spin coordinates c .: where Cj may have the values a or 6, the spin functions. Hartree's scheme provides a simple approach to solution of equation (5) since each spin function , ^^(X^) depends on the coordinates of one electron only. There are two major d i f f i c u l t i e s which immediately appear. In order to understand the f i r s t d i f f i c u l t y consider the Hamil tonian fo r the N-electron system as a sum of one- and two-electron parts: &U-, ,x r . .xn) = ip-j (x] )^ 2(x2) . . .xpN(xN) [ 6 ] M M = c* [7] 13 I f i t were possible to ignore the second term in the Hg of equation [ 7 ] , i.e . th.e el ectron-el ectron repulsion term 1 / r ^ . , then the Hamiltonian would simply be the sum of one electron operators and the complete separation of variables in equation [3] would be achieved. This however is not possible since the electron coordinates are in e x tric a b ly mixed in the l / r . . term of the Hamiltonian, The second problem which arises when Hartree's product, £6], is inserted in equation [ 5 ] , is the require- ( 8 ) ment that $ must s a tis fy P a u li’ s p r in c ip le . This adds the r e s t r i c t i o n that $ must be antisymmetric with respect to interchange of the coordinates of any two electrons, as was realized by D i r a c ^ and Hei senberg ^ 0 ^ early in the development of the f i e l d . I f the system contains only two electrons, the anti symmetri zed can be easily found, but in going to larger systems i t becomes d i f f i c u l t to come up with an antisymmetrized $ which is a simple product of space and spin components. The most general solution to this problem is to construct a Slater deterrainant. ^ ^ ^ /NT ^ Cl) (2) T 2 C l). * n( U \b ( n) Y n [8] 14 Having found an antisymmetrized function in the form of the Slater determinant, the conditions for optimization in accordance with equation [3] may now be found. Assuming that the Hamiltonian fo r electron repulsion is replaced ky an average, one-electron operation H., and then trying to fin d the "best approximation" of this type and the best one electron set, ip-j , • * * ^ n 1 s Hartree-Fock (12) method. I t is. assumed that the two-electron term is replaced by an " e ffe c tiv e p o t e n t ia l" , Vp| p> which depends on the a ttra c tio n of the.nuclei and the smeared out average of the N-l other electrons. Since F describes the motion of an electron in the f i e l d of N-l other p a r t ic le s , i t depends on the spatial d i s t r ib u t i o n of these electrons. However, this d is t r ib u t io n is contained in the o r b i t a l s , if> • X T h e r e f o r e one needs to know before constructing a potential V ^ p - (13) Mathematically, the Hartree-Fock equations' are w ritte n as: H e f f e c t i ve C l C . x - J ) = e k ^ k ( . X 1 ). £9] where are the o rb ita l energies. The e ff e c tiv e Hamiltonian consists of N where H is the one-electron operator, 15 h' V 0 ) [ 1 1 ] Thq Fock potential is given by the t e r m , { E J ^ . - K ' where J. is the Coulomb operator defined as J J , i M l ) = ( / (2) dvp) tK (1 ) J 1 J 12 [ 12] and K- is the exchange operator introduced by Fock, The H-F equations are coupled equations since the functions iJ j . are needed in order to calculate the corresponding Hartree-Fock p o te n tia l. The equations may be uncoupled with an i n i t i a l guess and solved for f i r s t improved functions which in turn are used to obtain an improved H-F p o te n tia l. This procedure is continued u n til s e lf consistency is achieved between the o r b ita ls and the potential f i e l d they generate. The re s u ltin g energy expression fo r the single Slater determinant consisting of N electrons is: The f i r s t sum represents the energy of a ll the electrons in the f i e l d of the nuclei alone, and since each o rb ita l K.^.(l) = ( fip. (2)ip. (2) - — dv?)^.(l) J I J i ' t o £ J [13] N N N [14] 16 is doubly occupied, this term is doubled. The second sum represents the ele ctronic in te ra c tio n s . The exchange forces (K. .) tend to keep electrons of the same spin * apart so as to minimize repulsion and therefore exchange always lowers the total energy, E. The Coulombic i n t e g rals, J . . represent potential energies of in te ra c tio n between interpenetrating charge clouds. The solution to the Hartree-Fock in t e g r o - d if f e r e n t ia l (13) equations is carried out numerically; where each o r b ita l is defined by its yalue at a large but f i n i t e number of points. Since the variations here are in the one electron function, the calculations y ie ld the optimum o r b i t a l s , in the sense that they describe the electron in some type of e ffe c tiv e f i e l d . The equations are solyed i t e r a t i v e l y w ith the Hartree-Fock potential from a previous calcu la tio n serving as a guessed value for the next it e r a tio n u n til s e lf consistency is achieved. When this procedure is followed the o r b ita ls are called Hartree-Fock o rb ita ls and they serye as a standard against which less exact calculations can be measured. The more general name fo r the wave function described abpye is the Self Consistent Field wave function. As commonly used, this term refers to a many-electron function which f u l f i l l s two c r i t e r i a . F i r s t , the.form of $scf is the sum of the minimum number of Slater determinants needed in order to obtain a function which has the 17 A A a specified eigenvalues fo r the spin operators Sz> S and which transforms as one of the irre d u c ib le representations of the-point group in question. This sum of Slater determinants is called a configuration. The second requirement of the SCF wave function is that the expectation value of the energy <E> = <$SCF1H1$SCF> < $ SCF l $ S C F > must remain unchanged with respect to a ll f i r s t order variations in $5£p which are consistent with its form. By "form" is meant both the determinantal structure and the nature of the one -p a rticle functions from which the determinant is obtained. The determinantal structure of ^c^p is fixed once the o r b ita l occupancy is given and the spin and space symmetry selected, with the only freedom l e f t in the wave function that of the o r b ita ls themselves. Depending upon the nature of the o r b i t a l s , the SCF technique f a l l s into two categories, numerical and a n a ly tic . The numerical procedure was described above as the o r i g i n a l ’.approach (13) taken by Hartree. This technique is more suitable to atomic systems than molecules. In the an a lytic SCF approach, often called the (14) Hartree-Fock-Roothaan method, the as yet unknown Hartree-Fock function i ^ ( r ) are expanded in terms of 18 a complete basis set xjJCr ): V r) = I xp ( r ) c pk C15] I f this basis set is chosen to consist of atomic o r b it a ls , CA.O.'s), equation [15.] is the fundamental, equation fo r the Linear Combination of Atomic Orbitals approximation. The idea of using atomic o r b ita ls fo r the molecular problem (15) was f i r s t introduced by H e itle r and London in 1928. The extension of the LCAO-MO technique to the SCF proce- (14) dure was f i r s t used by Roothaan in 1951. In the LCAO expansion the A.O.'s are chosen to give a good representation of the atom and th.e molecular o r b ita ls constructed by combining these atomic o r b it a ls on d if f e r e n t centers are delocalized oyer the entire molecule. The functions assigned to d if f e r e n t regions in space since most of the electron density is concentrated around the nu clei, thus the set usually consists of several functions centered on each nucleus. The expan-. s i o n ^ ^ must properly span the e n tire molecule and possess the symmetry of the molecular point group, and the functions x^ a r® chosen to be mutually orthogonal. Expansion of ip in an an alytic basis set is desirable because i t reduces the eigenvalue problem to a matrix algebra problem. In the analytical SCF approach, the o r b ita ls are specified not by t h e ir value but rather by the c o e ffic ie n ts 19 of t h e ir expansion in some f i n i t e basis (x,,K Since the M dimension of (Xy^ i s less than the number of points used to specify a Hartree-Fock o r b i t a l , i t ' s clear that this expansion method may be less accurate and manifest i t s e l f by higher energy than the numerical method. The main advantage of the Hartree-Fock-Roothaan scheme is its s im p lic it y since i t converts coupled i n t e g r o - d i f f e r e n t i a l equations into matrix equations. For atoms, i f the basis is well chosen <E>^ p ^ may approach <E>^p, Selection of a suitable basis involves the most a r b it r a r y decision y et, that of truncating the basis to some f i n i t e size, thereby leaving hopes for an exact solution fa r behind (16) fo r most- systems of chemical in te r e s t. Slater f i r s t suggested the following modified hydrogen-like functions to be used as an atomic basis: S „ ( r , 6 , < J > ) = A r n" 1e " r’ r i{j0 (6,4)) nJlm n y£m Y where ? = (Z-o)/n, Z is the nuclear charge, and a is the screening constant, and n is the p r in c ip le quantum number. Slater basis functions have proven to be of practical value in atoms and lin e a r molecules. In progressing to more complex molecular calculations however, Slater functions become impractical because of the electron repulsion terms which give rise to m u lticenter two electron in te g ra ls . The gaussian basis functions f i r s t (17) suggested by Boys, ' < j> = AnXnYmZ£ e"ar have proven to be e f f i c i e n t as far as m u lti-ce n te r two electron in te g ra ls , in spite of the fact that the gaussian functions may not have the proper asymptotic behaviour for solution of [4J. The main advantage of using Gaussian functions in a va ria tio n a l calcula tion is that they lead to simpler inte gra ls than the Slater type o r b it a ls . Since the product of two gaussians, G^ and Gg centered, on A and B, is a new gaussian, Gg, cen tered on E, the four center problem <G^Gg|G^.Gg> reduces to a two center one <Gg|Gp>. This type of integral is computed much fa s te r than one of the form <S^Sg|$gSp> in the Slater basis. Because the gaussians are less • appropriate fo r describing atomic and molecular systems, many more gaussian functions are needed, as compared to Slaters, to achieve the same level of accuracy. For example, in the Neon atom, 10s and 6p type (1 8 ) gaussians are needed to give the same one-configura- (19) tion energy as 4s and 2p type Slaters. The disad vantage of using gaussians in applications on atoms is magnified since the time consuming step in a Hartree- Fock calcula tion is evaluation of the two-electron in te g ra ls . The number of l a t t e r increases as the fourth power of the number of basis function which as described above is larger fo r gaussians. However, as shown 21 recently, the number of basis functions may be reduced by contracting several gaussians together and tre ating them as one function. Using this contractio n, the number of gaussians can be reduced to the number of Slaters w i t h o u t . serious damage to the calculated energy and Gaussians prove to be more advantageous in both molecules and atomic systems. Because of the ease of calcu la tin g with gaussians and using the contracted functions, a ll the work described here was carried out with gaussian basis sets on each constituent atom. Even i f the basis set chosen is extremely good so that the Hartree-Fock l i m i t is approached, the solution is s t i l l far from the exact energy. The difference between the exact n o n r e l a t i v i s t i c energy and the Hartree-Fock energy is called the c orrelatio n energy and may be quite large even fo r very small systems. Although, the Hartree-Fock approximation yields f a i r l y good results and the c o rre la tio n energy is small compared to Ej_|_p, chemists are not interested in to ta l energies but rather in energy differences which usually are small q u a n titie s . When the difference between, two large qu antitie s is taken, the accuracy of each term becomes very important, and the c o rrela tio n energy is in turn quite important. For example, in calculations on the (21 ) Neon atom, the c o rre la tio n energy is 0.38 a.u. or 0.3% of the total energy which seems to be a small 22 percentage. However, th is corre latio n energy is equal to 10 eV, which is a large chemical quantity. When comparing two closed-shell iso e le c tro n ic systems, the c o rre la tio n energies of these two systems may be nearly equal and, therefore, when one takes the difference between the two Hartree-Fock. re s u lts , the fin a l energy can be excellent due to cancellation of errors. I t is when the energy difference fo r two open-shell systems, is taken that the Hartree-Fock approximation, becomes very poor. This becomes evident in studies on dissocia tion energies or excited e le ctro n ic states. The H-F approximation becomes especially poor fo r potential curves of diatomic molecules because there are additional configuration s, other that the SCF, which should be included in the wave function. For example, the dissocia te 2 ) tion energy Dg fo r 02 1 using a minimum basis SCF is 1.43 eV which d if f e r s from the experimental re s u lt of 5.21 eV s i g n i f i c a n t l y . To contrast t h is , in studies of the dynamics of ion molecule reactions, s p e c i f i c a l l y H.eH+ + H . He + H2+ , the H-F results becomes s u r p ris in g ly r e lia b le . The potential surfaces in the SCF case predict the correct behavior fo r the system and the c o rre latio n energy is found to he constant with geometry. This is due to the .ion-induced dipole a t t r a c tio n , an e ffe c t which is well accounted fo r in the H-F approximation. In Chapter I I I of this work, more evidence 23 fo r the accuracy of the H-f scheme in the case of the dipole f ie ld s w i l l be giyen fo r the in te ra c tio n of HC1 and an electron. In general, i f excellent potential surfaces re s u lt w ithin the H-F approximation, this must be regarded as fo r tu ito u s . To summarize the ideas presented here on the H-F, SCF method, we begin with what exactly is to be varied in this type of c a lcu la tion . This depends upon the form chosen fo r the t r i a l function $. In the Hartree-Fock scheme i t is .th e one-electron functions, where in Roothaan's method i t is the expansion c o e ffic ie n t C. for r the one electron function in terms of some basis. The advantages of the one-configuration wave function is i t s ease of in te rp re ta tio n and i f the is properly optimized i t can be useful in studies on molecular properties s.uch as geometries and dipole moments. Many molecular potential curves near equilibrium are well represented by the SCF potential curve with accuracy close to about 5% fo r values of ca in the a lk a li e h a l i d e s . ^ ^ A good example of dipole moments calculated w ith in the re s tr ic te d H-F approximation is the work of (25) McLean and Yoshimine on closed shell lin e a r molecules. By use of e^xtensiye basis sets, the dipole moments of the XCN type molecules where X denotes a halide were found to he w ithin 5%-9% in e rro r from experiment. Again, as with the geometries, the SCF re sults seem to be 24 especially good fo r one-electron properties although they are known to he quite se nsitive to the p a rtic u la r basis chosen. Second order properties such as the total energy, the two-electron potential energy and others do not give good results in the H-F approximation. Studies of dissocia tion energies, spectroscopic constants wg)(e , (cal cul ated fo r potential surface slices away from the minimum), and electron a f f i n i t i e s , and assignment of excited e le ctron ic states requires going beyond the H-F level of approximation. For closed shell systems, the r e s t r ic t e d Hartree-Fock determinant is correct to " f i r s t order" by v irtu e of B r i l l o u t n ’ s Theorem, but second order errors are not n e g lig ib le . I f c o rre la tio n is included by performing a Configuration In te ra c tio n c a lc u la tio n , single e x c ita tions which enter the wave function by coupling with double e xcitations may cause the major changes in the one- electron properties ['i.e. those properties which depend on the coordinates of one electron, fo r which the dipole operator serves as a typica l example). The H-F approximation f a i l s to y ie ld reasonable molecular o s c i l l a t o r strengths and t y p i c a l l y d if f e r s from e xperi mental re sults by a fa c to r of 2 to 3, The single determinant wave function is adequate fo r the case of a closed shell and fo r some open shell systems. Some examples of the l a t t e r are the cases of 25 a high-spin h a lf open shell which means that all the M electron not in the closed shell sub determinant have spin a (or a ll spin 3) so that S = M/2 eigenfunctions of p S can be obtained from a single determinant. In a special case of the above the single electron outside a closed shell also is well described by the SCF wavefunction. The H-F method ignores the d i f f i c u l t i e s which arise with v a ria tio n of the multi-determinant function. In general, when the spin component of the wave function is combined with the space function to describe a molecular o r b ita l fo r open shall systems, new problems appear. S p e c if ic a lly , fo r the closed shell single determinant, since the Hamiltonian does not contain spin e x p l i c i t l y , the wave function must be a pure spin state which is an 2 eigenfunction of S , the tota l spin operator. The whole idea of spin pairing in doubly occupied o r b ita ls is essential to the Hartree-Fock scheme in order to assure that the Slater determinant r e a lly represents a pure spin state. This no longer holds true fo r unre s t r ic t e d H-F calculations in open shell systems since the Slater determinant becomes mixed with states of higher spin quantum number and a single determinant is no;longer 2 an eigenfunction of S . The most stra ightforw ard procedure beyond the Hartree-Fock level is Configuration In te ra c tio n . The C.I. waye function is a lin e a r combination of Slater 26 determinants b u i l t up from a set of one-electron functions. When the varia tion a l p r in c ip le is applied to the C.I. wave function i t is the c o e ffic ie n ts in terms of a basis of N-electron functions that are varied. This method w i l l be described in detail in Chapter I I . H i s t o r i c a l ly , the f i r s t in vestigatio n of electron c o rrela tio n was undertaken by Hylleraas on the He atom Cl 928-1 930},. Hyleraas introduced three basis methods fo r handling the electron c o rre la tio n problem. The f i r s t , as mentioned above, is the method of superposi tion of con figurations. Since he applied this technique to the two electron case, he used a sum of one electron functions, and yaried the c o e ffic ie n ts of t h e ir expansion. Secondly, he introduced the correlated wave function method in which the electron repulsion term l / r ^ ) is e x p l i c i t l y used in the wave function expansion. This approach came w ith in 0.00046 a.u. of the exact energy of the He atom and seemed to be the most accurate. F in a lly , Hylleraas used the idea of s p l i t t i n g the Is o r b ita l into an inner and outer portion thereby creating an open shell system which relaxes the r e s t r i c t i o n s on the spatial occupations and is s im ila r to the unrestricted H-f techniques. I f the e ffe c tiv e charges used for the two parts of the Is o r b ita l are chosen j u d ic io u s ly , the results fo r He are better in this case than the ap plica tion of C.I. 27 Progress in the f i e l d a fte r Hy1eraas1s work was slow at f i r s t . With the adyent of modern electron ic computers, and th e ir applicatio n to the many body problem in the 1 950's howeyer, deyelopment in the f i e l d was more . rapi d. The advantages and d i f f i c u l t i e s associated with the C. I . method w ill be developed in the next chapter. I t w i l l be shown how the three major problems of a C.I. ca lcu la tio n are overcome, namely generation and selection of the configuration space, construction of the Hamiltonian over spin adopted combinations of determinants and subsequent diagonal iza tio n to obtain the desired eigenvalues and eigenfunctions. IL _ REFERENCES 1. E. Sell roedi nge r , Anna!. Physik, _79 , 361 , 489 , 734 (1926), Phys. Rev. 28, 1049 (1926). 2. N. Born and J.R. Oppenheimer, Ann. Physik, 84, 457 (1927) . 3. W. Kolos, Adv. Quantum Chem. J5, 99 (1970). 4. H. Basch. M.B. Robin, N.A. Kuebler, C. Baker and D.W. Turner, J. Chem. Phys. 5J_, 52 (1969).. 5. W.R. G arrett, J. Chem. Phys. 69_, 2621 ( 1 978). 6. L. Pauling and E.R. Wilson, Introduction to Quantum Meehan i cs, (McGraw-Hill, N.Y., 1 935), p. 180. 7. D.R. Hartree, The Calculation of Atomic Structures (John Wiley and Sons, N.Y., 1957). 8. W. Pauli, Z. Physik, 43^, 601 ( 1 927). 9. P.A.M. Dirac, Proc. Roy. Soc. (London), A112, 661 (1 926 ). 10. W. Heisenberg, Z. Physik, £8, 41 1 (1926), _39 , 499 (1 926), £]_, 239 (1 927). 11. J.C. S la te r, Phys. Rev. £5, 21 (1930). 12. V. Fock, Z. Physik, 61,, 1 26 (1 930), Z. Physik, 62, 795 (1930). 13. V. Fock, M. Petrashen, Physik. Z. Sowejtunion, §_, 368 (1934). 14. C.C.J. Roothaan, Rev. Mod. Phys. 23, 69 (1951). 29 15. W. H e itle r and F. London, Z. Physik, 44, 455 (1927). 16. J.C. S later, Phys. Rev. 36, 57 (1930). 17. S.F. Boys, Proc. Roy. Soc. (London), A200, 542 (1950). 18. S. Huzinaga, J. Chem. Phys. 42^, 1 293 ( 1964). 19. E. Clementi, Table of Atomic Wavefunctions, A Supple ment to IBM, J. Res. Develop. 9_, 2 (1 965). 20. T.H. Dunning, J. Chem. Phys. 5_3 , 2823 (1 970). 21. See H.F. Schaeffer, The Electronic Structure of Atoms and Molecules, Addison-Wes1ey Publishing Co., Reading, Massachusetts, 1972. 22. H.F. Schaeffer, J. Chem. Phys. T4, 2207 (1971). 23. C. Edmiston, J. D o o li t t le , K. Murphy, K.C. Tang, and W. Wilson, J. Chem. Phys. S2 , 341 9 (1970). 24. P.E. Cade and W.M. Huo, J. Chem. Phys. 4_7, 614, 649 (1967). 25. A.D. McLean and M. Yoshimine, J.J.Q.C. 1_5, 313 (1967). 26. E.A. Hylleraas and B. Lendheim, Z. Physik, 6_5, 759 (19 30). CHAPTER II METHOD I t was shown towards the end of Chapter I that the Hartree-Fock approximation is not s u f f i c i e n t l y accurate fo r many problems of chemical in te re s t and that c o r re la tion must be taken into account somehow. One conceptually simple way of going beyond the Hartree-Fock scheme is the use of configuration in te ra c tio n (C .I.) techniques. C.I. is a very general method which may be applied to the statio nary states of atoms and molecules and there fore has a large scope of a p p lic a tio n . In addition to the s im p l i c i t y and gen e ra lity inherent in the C.I. method, there is the additional advantage that in p r in c ip le at lea st, th is method is capable of providing accurate so lu tions. The C.I. wavefunction, ¥, is a lin e a r combination of orthonormal Slater determinants or con figurations, which involves zero, s in g le , double and higher e x c ita tions from a reference determinant and can be represented by the equation V = I c.$. [1] When the C.I. wavefunction is substituted in the e le c tro n ic Schroedinger equation, the expansion c o e f f i - ' c ien ts, c-, are determined to minimize the energy in 31 accordance with the v a ria tio n a l p rin c ip le resulting in the eigenvalue equation E represents the desired eigenvalues and C is th.e correspondi ng eigenvector m a t r i x , The matrix H descrihes the in te ra c tio n between the various configurations. Thu s , describes the interaction, between the determinants $s and H _ e is, composed of one and two electron operators as previously defined in Chapter I equation 17.1. We are interested in r e la t i n g expression [3] back to the one-electron functions • , which form an ortho- normal sret. Thus, the in te ra c tio n between determinants may be s im p lifie d to a sum of in te g ra ls in the one- electron basis. The mismatching of o r b it a ls . , . ^ . , in the determinants results in a generally sparse C.I:, m a trix . The Hamiltonian matrix elements between two configurations., $s and which d i f f e r in t h e i r electron occupation by more than two spin o r b ita ls are id e n t i c a l l y zero. I f Ui- in determinant i> is mismatched w ith \J j. in determinant M, S J $t , the r e s u ltin g one-electron difference matrix element H C = E C [2] 13] n 14] 32 where n is the number of occupied spin o r b i t a l s . For a mismatch of two electrons (i and k in $s and j and 1 in $ ') the matrix element s im p lifie s to H . = ( V . . , , - V . . , . ) [5] st i j k l 11 kj When two determinants are equivalent, the matrix elements become n n H = [ H. . + 7 V - V [6] s s ^ 11 A i i j j i j Ji These diagonal elements of the C.I. matrix w i l l coincide with Roothaan's expressions in Chapter I equation [13]'for closed-shell systems. In order to provide proper electron c o r r e la tio n , a f u l l basis set expansion is needed since the molecular or atomic regions most densely occupied by electrons need a detailed description of th e ir in te ra c tio n . From a p ractical point of view, however, a complete C.I. expan sion is impossible i f such an " i n f i n i t e " basis from an exact SCF calc u la tio n is used. The C.I. method becomes ra p idly impractical even in t h e LCAO approximation as the number of electrons and the dimensions of the A.O. basis increase. For the cyclopropane molecule fo r example, i f 36 M.O.'s are used and a ll 18 valence electrons are considered fo r the = 0 case, the magnitude of the 1 5 determinantal space is closed to 1.5x10 to tal arrange ments. Thus, although in p r in c ip le , the C.I. method is 33 capable of a high degree of accuracy, whether the calcu la tions may be carried out to a good approximation, in an economically fea sible fashion is another question. There is no doubt therefore, that constructing the f u 11 Hamiltonian C.I. matrix for a spin and space symmetrized combination of Slater determinants, although the most stra ig h tfo rw a rd , is too time consuming. This holds true f o r any medium size basis ('v 40 M.O.'s) since the space increases f a c t o r a i l l y with the basis size N. Arguments against the C.I. technique stem from the r e s tr ic tio n s which must be placed on the space and which may re s u lt in s i g n if ic a n t loss of accuracy. Clearly then, the size of the C.I. calcu la tio n presents a major d i f f i c u l t y and truncation of the space is a necessity. One approach to this problem is the truncation of the o r b ita l set. The most common involves truncation of the v ir t u a l o r b ita l space and the r e s t r i c tion of the lowest occupied o r b ita ls to double occupancy. The results of such an approach are very sensitive to the type of o r b ita ls used as a basis fo r the C.I. ca lc u la tio n . For example, i f Natural O rbitals (N.O.'s) rather than SCF-M.O.'s are u s e d , ^ the former are superior fo r the case of o r b ita l tru ncation. Development of Natural ( 2 ) O rbitals came about when a more ra p id ly convergent set of spin o r b it a ls was needed and one which is easier to in te r p r e t than the M.O.'s re s u ltin g from a standard 34 H-F SCF procedure. For a C.I. expansion composed of f Slater determinants, the electron density function is Y = aki *k * i where a ^ is the density matrix. Natural Orbitals reduce y to diagonal form, Y = T b . . 1 v 1 1 1 where b. are known as the occupation numbers and serve as a measure of the importance of an o r b i t a l . Those o r b ita ls whose occupation number, b., is n e g lig ib le may be omitted from the C.I. expansion without s i g n if ic a n t loss of accuracy. Therefore, the Natural O rb ita ls , which are the eigenvectors of y, have shown to provide a superior ( 3) o r b ita l basis fo r correlated wave functions. An example of a method which applies Natural Orbitals to a C.I. procedure is the I t e r a t iv e Natural Orbital (IN O )^^ method. INO is based on the use of molecular o r b ita ls which re s u lt in a on e-particle density matrix which is diagonal fo r the C.I. wave funet ion under ^ The one-electron density fun ction , y , is given as Y = n J[i^*(X(l 1) , X (2 ).. X ( n )] ip[ (X (1 ). . X( n) ] dV (2 ) .. dV (n) where the in te g ra tio n takes place over a ll coordinates but electron 1. The diagonal part of Y, y ( l / l ) is inte rpreted as the p r o b a b ility of find ing electron 1 with spin 1 , at a given p o s itio n , while a ll other N-l positions and spins are a r i b i t r a r y . 35 consideration. Since the o r b ita ls depend on the wave function through. the density matrix and yice versa, th is is an i t e r a t i v e method with alternate diagonalization of the Hamiltonian and density matrices giving the C.I. expansion c o e ffic ie n ts fo r the present ite r a tio n : and fo r the next i t e r a t i o n , respe ctive ly. One of the a t t r a c t i v e points of the INO scheme is that the natural o r b ita ls are defined in the same m u l t i - configuration environment in which they are used and are superior to SCF o r b it a ls which were defined in a scheme which t o t a l l y neglected electron c o r re la tio n . The INO scheme may provide highly accurate Cl wave- functions since as the approximate NO's are obtained a f t e r an i t e r a t i o n they are ordered by th e ir corresponding occupation numbers. This means that the reference con f ig u r a tio n w i l l be constructed from o r b ita ls whose occupa tion numbers are close to two. This " f i r s t natural c onfiguration" w i l l therefore dominate the C.I. expansion to a maximum which insures that a ll singles and doubles r e la t i v e to th is reference state w i l l be extremely e ff e c tive in c o rre la tin g the wave fun ction. The major d is advantage of the NO route is cost because i f any trunca tion is done, this occurs a f t e r several integral t r a n s f o r mations were performed re s u ltin g in excessive labor. Another disadvantage in using N.O.'s is that each state, or root of in te re s t must be extracted separately. 36 Although f t ts clear that N.Q.'s, are superior to M.O.’ s in terms of o r b ita l set truncation and ease_of in t e r p r e t a t io n , N.Q.’ s are found to Be i n f e r i o r to M.O.’ s in a general perturbation approach. According to recent work, done by Shavitt and co-workers the Natural Orbital Hamiltonian was found to have larg er off-diagonal matrix elements than the M.O.’ s. These o f f diagonal elements consist p rim a rily of one- electron k in e tic energy integral con trib u tion s. The o r ig in of the large k in e tic energy in te g ra ls is related to the basic difference between M , 0 s and N.O.’ s. The H-F SCF fo r n electrons produces n/2 doubly occupied o r b ita ls which leads to the single determinant function. The additional set of o r b it a ls in the SCF basis are the y ir t u a l o r b i t a l s , I:n the M,0, basis, this v ir t u a l space does not provide any special c o rre la tio n fo r the inner she lls and therefore the inner shell character is d is tr ib u te d among a ll the v ir t u a l o r b i t a l s . The N,0, basis on the other hand, re s u lts in a clear separation of the v ir t u a l space w ith components of d i f f e r e n t ra dia l extent providing s p e c ific c o rre la tin g o r b ita ls fo r the inner space. These inner shell components, concentrated in a few natural o r b i t a l s w i l l y ie ld large k in e tic energy in te g ra ls and large Hamiltonian matrix elements thereby destroying the inherent rapid convergence of natural o r b it a ls in a pertubative scheme. This problem may be corrected i f these N.O.'s are included in a zero order m atrix, HQ, as is done in the method used here fo r M.O.'s. In a d dition , as the size of the C.I. matrix is increased, the advantage of the N.O.'s over M.O.'s becomes i n s i g n i f ic a n t . For example, when 95% of the c o rre la tio n energy is recovered, the number of single and double configura tions in the M.O. basis is 955 whereas fo r the Natural Orbitals 697 configuration functions were required in calculations on the water molecule. ^ Natural Orbitals do y ie ld accurate wave functions which are superior to M.O.'s as i l l u s t r a t e d in calcu la tions of one-electron properties. Although the work described here employs the M.O. SCF basis exclu sive ly, the extension of the method described here to N.O.'s seems worthwhile. In the SCF basis, truncation of the basis by freezing the inner o r b it a ls in j u s t i f i e d . b y the fact that the inner core is isolated from the chemically active space. Leaving out these inner o r b it a ls s t i l l leads to accurate results because, in general, we are interested in energy differences and these o r b ita ls do not influence the major c o rre la tin g o r b i t a l s . Truncation of the v i r t u a l space is much more a r b itr a r y ( 5) and c le a rly depends on the basis set chosen. Two other, general methods for truncation w i l l be discussed. The f i r s t deals with the levels of e x cita tion 38 required fo r accurate C.I. calcu lation s and is related to the choice of zero order state. The second technique is configuration selection and is extremely important in achieving an e f f i c i e n t method. In some C.I. techniques, the "zero order" state is commonly chosen to be the SCF parent c o n f i g u r a t i o n . ^ This is a rather lim ite d choice and is not the zero order state used in this work. I f one does begin with the unrestricted closed shell SCF wave function then the re s u ltin g configurations are c la s s if ie d as promotions from thejoccupied SCF space to the v i r t u a l o r b it a ls . Clearly then, a ll matrix element between two determinants w i l l vanish i f they d i f f e r by more than two o r b i t a l s . due to the two-electron r e s t r ic t io n s of the Hamiltonian operator. In addition since matrix -elements between the SCF and sing ly excited configurations vanish (9) by B r i l l o u i n ' s theorem, the leading corrections to the closed shell SCF are double e x c ita tio n s . For the open shell case, single e xcita tions become very important and are also s ig n if ic a n t in closed shell systems when the one-electron properties are investigated. In applying a C.I. technique to studies of excited states and potential energy surfaces, the use of the SCF as the generating basis is inadequate since several configurations make an important c on tribu tio n to the wave fun ction. Therefore, in our p a r tic u la r approach, 39 the "zero order" state consists of a number of the most important single and double configurations r e la tiv e to the SCF basis. These configurations w i l l be referred to as reference configurations (RC), or base configurations. These RC1s are i d e n t if ie d in advance by performing a lim ited C.I. calcu la tio n and choosing configurations which have a mixing c o e f f ic ie n t of 0.1 or larger. In our scheme, up to 10 reference configurations may be used in one c a lc u la tio n . Having chosen a set of reference con figuration s, the most s ig n if ic a n t truncation occurs when the C.I. space is r e s tric te d to single and double r e p la c e m e n ts ^ from these RC. This r e s t r i c t i o n reduces the size of the C.I. to a problem of (where N is the bas is s iz e ), which is of manageable proportion. When the C.I. space is generated re la tiv e to these reference function s, select t r i p le s and quadruples r e la t iv e to the SCF configuration are also included. Truncation of the space to ba sically ( 8 ) ju s t singles and doubles is acceptable since Shavitt has,shown that t r i p le s only contributed 0.0004 a.u. of energy in calculations on the BH3 molecule. Since the l a t t e r was performed on the SCF basis, i t is assumed that singles and doubles r e la t iv e to a more general zero order state w i l l re s u lt in wave functions of acceptable accuracy. Truncation of the space may also be achieved by configuration selection. Selection of ind ivid ual 40 configuration is carried out on the basis of th e ir estimated contributions to the wave function in terms of energy or the square of the C.I. c o e ffic ie n t s . One simple approach to estimating second order energy contributions is perturbation theory. The second order contributions to the energy according to Raleigh-Schroedinger perturba tion between the configuration of in te r e s t, $•, and the zero order fu nction, is given by | h. | 2 4E - 10 1 hoo • hH 2 where | h^ Q | represents the second order in te ra c tio n between these two wave functions and h . ^ represents diagonal element of-$^. I f consists of a lin e a r com bination of reference c o nfigurations, the c o e ffic ie n ts may be determined in a small scale C.I. To te s t the importance of the matrix elements h •, representing the in te ra c tio n of the general zero order state and the con f i g u r a t io n , i , are calculated. E ssen tia lly this is the ( 12) method of Shayitt and is a practical approach to configuration selection. In recent years many novel, in d ir e c t C.I. techniques have been developed and w i l l now be b r i e f l y discussed. Buenker and Peyerimhoff^ ^ haye developed a scheme involving configuration selection followed by an energy extrapolation procedure. The configuration selection has 41 M2) been previously used by S h a v it t v 1 and co-workers and involves a "nucleus" of dominant configuration which are diagonalized over the Hamiltonian to y ie ld the zero order fun ction, \jjQ. Clearly this is much more general than the SCF basis alone and provides an excellent s ta rtin g point fo r a C.I. c a lc u la tio n . Buenker and Peyerimhoff then test the importance of all singles and doubles using Raleigh-Schroedinger perturbation theory with the ipQ serving as the zero order fun ction . By subsequent solutions of larger and larger secular equations they are able to extrapolate th e ir results to the f u l l set of singles and doubles. Another important development in the corre latio n (13) problem is Meyer's PNO C.I. technique. Meyer uses paired natural o r b ita ls (PNO) to provide the corre latio n of each occupied SCF o r b i t a l . The advantage of using N.O.'s here is c le a r ly revealed in the rapid convergence of the C.I. expansion as applied to the hydrides LiH through SH. A b r i e f mention must be made of one of the most (14) e f f i c i e n t C.I. methods a tt r ib u t e d to Roos. In this C.I. procedure there is no e x p l i c i t construction of Hamiltonian matrix elements but rather d ire c t construction of the wave function from the one and two electron inte - j gral l i s t . This method cu rren tly lacks g enerality because i t is applicable to cl osed-shel 1 SCF systems only. The sparse nature of the C,I, matrix lends this problem to a perturbational approach. Since in this p a r tic u la r case the e-xact s o lu tio n , ^ exac-t ° f the zero order Hamiltonian is unknown, a 1 1 vari at i on-per tu rb ati on" (15) approach is applicable. Lowdin has shown that the Hamiltonian matrix may be p a rtitio n e d into two blocks, the f i r s t , H_,, contains the zero order functions and a a the second H ^ includes functions which modify Hga and improve the solution to the problem. (16) The p a r t it io n in g of the C.I. matrix used here is based upon the re c o g n iitio n that a ll C.I. problems are characterized by a r e l a t i v e l y compact set of important basis configurations whose combined weight in the fin a l solution of the problem represents greater than 90% of the fin a l eigenvector, and a long " t a i l " of configurations which i n d iv id u a lly contribute l i t t l e to the eigenvector but whose cumulative e ffe c t may be several eV of c o rre la tio n energy. I t is assumed that the important configurations can be recognized and gathered into the upper l e f t hand corner of the C.I. matrix (Haa) and the remaining t a i l functions form the lower rig h t hand block (Hbb), then the s itu a tio n is as depicted in Figure 2.1. The p a rtitio n e d matrix, H3,, is r e l a t i v e l y small (of order 50x50), while the size a a of the Hbb sub-space may be quite large (of order several thousands). The C.I. problem to be solved is: 43 Haa Hab | Ca Hba Hbb K = to [ 6 ] which can be re w ritten as two matrix equations. H C + H , C , = wC ~ a a ~ a ~ a b ~b ~ a ~ba~a + ~bb~b w~b [7] The inversion of the second and s u b s titu tio n into the f i r s t of these leads to: H C ~aa~a Hbb 0)I] 1 H t c = uiC ~ b a ~ a - ~ a [ 8] Equation [8] seems as d i f f i c u l t to solve as equation [6] since we must find the inverse (tol-H ^) ^ and insure that no s i n g l a r i t y occurs ( i . e . no h ^ is e s s e n tia lly degenerate with oi). Therefore, a s in g u la r it y may be avoided by choosing the HQa and blocks in such a manner that the roots of in t e r e s t , , l i e fa r from the h ^ . By p a r t it io n in g the C.I. matrix as in equation [6 ], and by insuring that the roots l i e fa r from the h ^ , the matrix ( w I - H ^ ) is made to be diagonally dominant. The p a r t it io n in g technique has insured that all the highly in te ra c tiv e and important configurations are placed in Hafl. This implies that the t a i l configurations are only s l i g h t l y in te ra c tiv e and therefore would be expected to have small o f f diagonal elements with the a block c o n fig u ra tio n s . 44 Having made the matri-X diagonally dominated, i t may he wri tten as: ° U ) - f> where D(w) is the diagonal m atrix: D = ( p i - hbbI) [9] and /©/represents the o f f diagonal elements: # = ( H b b - h bb I ) 1 1 0 ] Then, the term Cwl •- Ht,b is expanded to represent a series of o f f diagonal terms about, the inverse diagonal, D" 1 : (jot - H j ^ r 1 = I T 1 + D^OfT1 + D"10D’' 10.D"1 + ••• H U When this expansion is substituted in equation [8J the re s u lt is K ca + HbaD~1 Cu)Eb °a + H jjD"1 (uJOD^1 Cwl' Hb Ca + . . . " ^>Ca 1 1 2] This may be rew ritten in a more compact and perhaps more meaningful way: IK + V(m)]c = we 1 1 3 J < V < v The y(jjj) term in equation [13j shows the e ff e c t of the " t a i l " configurations from the submatrix b., on the energy, 45 to, and on the core block H . The "V matrix" has terms a a identical with B r i 11ouin-Wigner (BW) perturbation terms. Thus, the term HabD' ’ <“ > Hba represents a second order BW correction to the zero order fun ction, H = a , and the successive terms in the V matrix a « represent th ir d and higher orders of corrections. The drawback here is that the V matrix depends on the energy, to, and solution must be achieved by it e r a t io n . Therefore, with each i t e r a t i o n for each root of in te re s t, the contraction over the matrix H ^ , which has dimensions of 16,000 configurations fo r the HC1" molecule and 32,000 fo r the work on cyclopropane, must be repeated so that the term Haj)D~'*Hb is found. Clearly, we need to get around this d i f f i c u l t y . Since the energy appears only in the diagonal elements D ~ ^ ( t o ) , expansion of the diagonal about a guess value fo r the root, o > , seems reasonable. Since all the important configurations are in the a block, diagonal-ization of Haa to give a to value presents a good guess fo r the fin a l energies. The difference between to and the true value to, (to -to) o o is Ato. The expansion is performed about to in terms o f Ato: D-"1 (to) • D ' V ) l (D_1(u. Ua))" [14] ° n = 0 0 46 resulting in a [15] where the successive orders of the V matrix are repre sented by: The d i f f i c u l t task now is to decide exactly where the V matrix expansion should be truncated. I t is clear that we are not interested in re tainin g terms such as accomplished and the total Hamiltonian matrix needs to be calculated. Truncation at the second order le v e l, to the most e f f i c i e n t method. Truncation at this level was shown to d i f f e r by only 2 kcal from calculations R estrictin g the expansion to second order also insures that only d ire c t in te ractio n s are considered between a core of dominant configurations and the t a i l . The great advantage of this approach is that the o f f diagonal matrix elements of need not be evaluated, V<0) ~aa aa = Hab[D-'(a.0) + o '1 (Mo)0D-1(U >o)]Hba [ 16 ] D ^ (w)0Q7 ^ (o j ) , since then no savings has been i.e . at D""*(o) ) w i l l be the most economical and leads performed using th ir d order and higher terms. ( 16 ) 47 thus greatly reducing th.e amount of work necessary to carry out the c a lc u la tio n s . Accuracy at the second order level is guaranteed by choosing extremely general zeroth order functions i . e . the 50 dimensional H „ block. The a a expansion in £15] w i l l converge ra p id ly . W e have insured that the terms Cool-h^^) are r e la t i v e l y large and powers of t h e ir reciprocals w i l l f a l l o f f ra p id ly , while Ato is small. In practice, expansion through f i f t h order is s u f f i c i e n t for a ll cases. This procedure enables us to fin d several roots of in te re s t (provided they l i e s u f f i c i e n t l y close to one another) in one c a lc u la tio n . The Haa block is chosen by using Raleigh Schroedinger p e rtu r bation theory to test the in te ra c tio n between any con fig u ra tio n and the "seed" ( i . e . most important) configura tion s. The maximum dimension of the a block is set to 50 and i t has been shown^10,1^ that the method is capable of high accuracy. The method described here is ( O \ s im ila r to the method of Gershgorn and S havitt. Our method has reduced the C.I. problem from order N^, the typical dimension fo r singles and doubles, to a computation lin e a r in the number of configurations and therefore proportional to the square of the configura tion l i s t as fa r as C.I. matrix elements. The p a r t it io n in g technique may be summarized as follows: Configuration chosen in a previous, smal1 C.I. are gathered to form the nucleus portion of the core 48 block.. Tliis nucleus lias dimension 1-10 configurations. The nucleus w i l l generally contain a ll the reference configurations and in addition any state which exhibits strong mixing with the root of in te re s t. States which are close in energy to the root of in te re s t must also be "forced" into this nucleus portion of the core block Haa> to insure that no s in g u la r it y occurs. Then the in te ra c tio n of the nucleus with the single and double replacements r e la tiv e to some reference configuration is calculated to form part of Hab' Those functions which mix s i g n i f i c a n t l y with the nucleus (according to Raleigh-Schroedinger perturbation the ory), w i l l complete the core block which has typical dimensions of 50. Con fig u ra tio n s which in te ra c t to less than 10~6 a.u. are dropped at this stage. The rest of the H ^ matrix is calculated and then the core block, H is diagonalized ct a to provide a guess value, ai , fo r the diagonal expansion. The V matrix is calculated next by contraction of Ha^ onto the diagonal D(.w0). F in a lly , the core and potential problem is solved by i t e r a t i o n . No more than 2-4 i t e r a tion cycles are needed per root of in te r e s t. The second most challenging problem of a C.I. calcu la tion is the generation of matrix elements. This step is always time consuming because generation of C.I. matrix elements over spin eigenfunctions becomes extremely d i f f i c u l t in systems of many electrons. The work of 49 Segal and W e tm o re ^ ^ *^ ^ has s im p lifie d this problem to an e f f i c i e n t and compact formalism, applicable to con fig u ra tio n s with 10 open s he lls. To s ummard ze the e n tire :mefhod a$' applied; to . H Q1 and cyclopropane: 1. Careful choice of the basis set to insure that an unbiased calcula tion may then be carried out. This means, in general, attempting to find a basis in which dissociation energies, electron a f f i n i t i e s or ordering of excited e le c tro n ic states are w ith in acceptable accuracy. 2. An SCF ca lcu la tion is performed using the Gaussian-70 p a c k a g e ^ ^ or B i g m o l l i ^ ^ on the ground state configuration. In calculations on HC1", Gaussian-70 was used. Bigmolli may be used when d functions augment the basis set e ith e r in the Rydberg space (as fo r cyclo propane) or for p o la riz a tio n . In a ddition, Bigmolli is capable of handling more than six p r im itiv e gaussians per contracted function. This program may be used for Dunning basis of the type 3s, 2p contracted from Hunzinaga 1 s ^ (9s,5p) fo r f i r s t row atoms since the 3s consists of 7, 2 and one Gaussian respectively. 3. The SCF integrals are then transformed to the M.O. basis. In general, fo r the f i r s t row atoms or basis sets consisting of less than 40 A.O.'s the e n tire basis is transformed. 50 4. A small-scale singles and doubles C.I. is performed in order to q u a l i t a t i v e l y understand the nature and ordering of the states of in te re s t and to select reference configurations. 5. With the above information available the "nucleus" of each state of in te r e s t is determined. This augments the base configuration by addition of 9 single and double replacements. The calculations reported here include a ll singles and doubles with respect to the base configuration. However, often in one c a lc u la tio n , m u ltip le base configurations are used so that selected t r i p l e and quadruples with respect to the ground state SCF configuration are also included. I f n roots are required in a c a lc u la tio n , the (ri+.l) energeti ca 1 ly lowest configurations w i l l be chosen for the nucleus. In c alcu la tion on the excited states of cyclopropane, as many as 9 C.I. base configurations are used to generate all singles and doubles. The re s u ltin g 10 roots are thus obtained simultaneously since they are close lying and therefore w ith in a reasonable radius of conver gence. In typical ground state computations, single..and; double replacements r e la t iv e to the SCF base and the next highest configuration of the same symmetry, were gene ra te d . In the case of calculations on HC1 , selection of a Haa block equal to 50 configurations is not necessary. 51 When the size of the H .aa block was lim ite d to twenty conf i gurati ons ,... the same 1 eve! of accuracy as the f u l l problem is obtained. Therefore, since the H . block is a a smaller, the H ^ matrix enters the calculations more economically since only inte ractio ns of twenty configura tions with the t a i l need to be considered. Instead of ca lcu la tin g the e n tire s t r i p (50x16,000), the thinner s t r i p was calculated. Differences between the f u l l 50x50 core block plus 50x16,000 H ^ were found to vary by only 0.0:2eV from a truncated c a lc u la tio n . Details on this technique are given in Chapter I I I . I t must be noted that in general, the s t r i p must be at least as wide as the nucleus portion of the core C.I. This w i l l insure that a ll possible inte ractio ns between the nuclear functions and the t a i l are included. The f u l l Hajj s t r i p and are presented in Figure 2.1 where i t becomes clear that the f u l l s t r i p must have a width of 50 while the truncated s t r i p has a width equal to the nucleus (at least 10). In the following chapters, the method outlined here is employed to a rrive at accurate results fo r disso cia tion processes and excited ele ctro n ic states. The e ffic ie n c y of the method w i l l be i l l u s t r a t e d f i r s t in the calculation of several surfaces of HC1 and HC1~. The accuracy of the results enables us to explain present experimental results involving HC1~ resonances. 52 In application of tfie method to cyclopropane, the ordering of excited states results in c l a r i f i c a t i o n of experimental re sults. The wave functions calculated are used to predict properties such as o s c i l l a t o r strengths and magnetic moments q u a n t it a t iv e ly , and provide comparison with experimental results as well as an explanation of observations on this unique molecule. REFERENCES 1. I. S h a v itt, B.J. Rosenberg, and S. P a l a l i k i t , IJQC , 10 , 30 ( 1 976 ). 2. P.O. Lowdin, Phys. Rev. 97, 1474 (1955). 3. D. Ter Haar, Rep. Prog. Phys. 14, 304 (1961). 4. C.F. Bender and E.R. Davidson, J. Phys. Chem. 70, 2675 (.1 965); Phys. Rev. 183, 21 (1969 ). 5. R.M. Stevens, J. Chem. Phys. 1]_, 2086 ( 1974). 6. A. Pipano and I. S havitt, IJQC 2, 741 (1968). 7. P. Claverie, S. Diner and J.P. Malrieu, IJQC 1, 75 (1967). 8. Z. Gershgorn and I. S ha vitt, IJQC 2_, 751 ( 1968). 9. L. B r i l l o u i n , Actual i te-s? Sci. Indust. No. 159 (T934). 10. G.A. Segal, R.W. Wetmore and K. Wolf, Chem. Phys. 30 , 296 ( 1 978) . 11. R.J. Buenker and S.D. Peyerimhoff, Theoret, Chim. Acta. 15, 33 (1974); R.J. Buenker and S.D. Peyerim h o ff, Theoret. Chim. Acta. H , 21 7 (1 975). 12. I. S ha vitt, The Method of Configuration In te ra c tio n , in Modern Theoretical' Chemistry, Vol. I I I . Methods, of. :FI;ec.troni'C Strulct.urie TJieo.ry;,. H.F. Schaeffer, T J I , ed. Plenum Press, New York -and. London ( 1977). 13. W. Meyer, IJQC 15, 10 (1971); W. Meyer, Theoret.. Chim. Acta 35, 272 (1974) . 14. B. Roos, Chem. Phys. Le tt. ]_5, 1 53 ( 1 972). 54 15. P.O. Lowdin, J. Math. Phys. 3 , 969 (J 9,621. 16. G.A. Segal and R.W. Wetmore, Chem. Phys. Lett. 32, 556 (1975). 17. R.W. Wetmore and G.A. Segal, Chem. Phys. Lett. 36, 478 (1975). 18. W.J. Hehre, W.A. Lathan, R. D it c h f ie ld , M.D. Newton, J.A. Pople, Gaussian 70, program #236 , QCPE, Indiana U niversity. 19. R.C. R a ffe n e tti, J. Chem. Phys. 58, 4452 (1973). 20. S. Huzinaga, J. Chem. Phys. 42, 1293 (1965). Figure 2.1. P a rtitio n in g of the Hamiltonian matrix. 56 Typical Dimensions nucleus Dominant Interactions Haa Hob = Hba Core C o re /T a il Interactions 40 -50 Hbb T ail Hba full space 57 CHAPTER III. APPLICATION OF C.I. METHOD TO ELECTRON-HC1 SCATTERING A. Prelimi naries i) Introduction The in te ra c tio n of low energy electrons with mole cules is of primary in te r e s t in b io lo g ica l systems, (1-3) astrophysics, gas discharge and ra diatio n chemistry. When an electron collides with an isolated molecule or atom several events may occur. Examples of such events are io n iz a tio n , molecular fragmentation, e x c ita tio n of vib ra tio n a l and ro ta tio n al states and negative ion forma tion . Recent advances in experimental techniques in v o l ving electron impact and electron transmission experi- (4-5) ments, have increased s i g n i f i c a n t l y the range of application of low energy electron scattering . Some recent applications include v ib ra tio n a l e x c ita tio n by electron impact in gas lasers to achieve population ( 6 ) inversion 1 and detailed studies of the e le c tron ic stru cture of atoms and m o l e c u l e s . ^ These sophisticated experiments have also been applied to the study of compound states or resonances, and the various decay ( 8 ) channels accessible to them. ' In recent years, the topic of electron capture by molecules which possess a dipole moment has also attracted considerable a tte n tio n . (9) The present study extends previous work on the 58 in te re s tin g e ffe c ts of the d ip olar f i e l d on resonances. Resonances have important implications in a s tro physics, in the ion chemistry of the earth's lower ionosphere , ^ 9^ ^ in the study of electronegative gases in e le c t r ic a l discharge and in the chemistry of ion-atom c o llis io n s . Consider the reaction, H~ + H H2 + e" which proceeds via a compound state (H^- ). This reaction controls the loss of H" in the solar photo sphere which in turn determines the solar opacity in ( 2 ) certain regions of the spectrum. ' Resonances also play an important role in s o lid - s ta te and plasma physics. In the sol i d - s t a t e , ^ 9^ c o m p o u n d states formed by in te ra c tio n of low-energy electrons and im purities in metals (such as Cr or Mn) are known to dominate the scattering cross sections. In laboratory plasmas, the recombination process between the electron and ion proceeds via a resonance before i t decays to a bound H i s t o r i c a l l y , the study of resonance states of (15) atoms and molecules began with the Auger e ffe c t: M + hv M* M+ + e” where M* is often a resonance state. This same process has been long recognized in molecular spectroscopy as predissociation in which there is an overlapping of discrete and continuous states. Other early observations 59 of resonances in the vacuum U.V. were made by K r u g e r ^ ^ who showed that emission lines in He were due to tra n s itio n s between the continuum and bound states. The most s ig n if ic a n t the ore tical progress in this f i e l d has been made when the resonance model, long known to physicists , was applied to electron-molecule scat tering in the 1960's. A b r ie f summary of the model and the general c ha ra c te ris tic s of resonances follows. Compound states (resonances) are formed when an electron is incident upon a molecule and is then temporarily captured in the neighborhood of the target. Resonances of th is type have life tim e s ranging from 10- ^ to 10~10 seconds where t = — (x is the l i f e t i m e , r is the resonance'width) r Physical evidence for the existence of resonances at low incident energies is provided by sharp peaks in the scattering cross section of atoms and molecules. These sharp peaks may be accompanied by large proba b i l i t i e s fo r e x c ita tio n of the target in which the incoming electron gives up a large fra c tio n of its i n i t i a l energy. Possible decay channels fo r resonances range from v ib ra tion al and ro ta tion a l e x c ita tio n , AB(v,j) + er ■ + AB“ A B ( v 1 , j 1 ) + e” e le c tro n ic e x c ita tio n , 60 AB + e -*■ AB AB* + e and disso cia tive attachement AB + e” AB" A~+ B or B’ + A Clearly t h e n , ; i f the intermediate state, AB", is a resonance, ( i . e . i t has a high p ro b a b ility of occurring) then the stru cture of this state w i l l dominate the decay processes mentioned above. This is true fo r e le c tro n ic e x c ita tio n near threshold and fo r vibra tion al e x c ita tio n since a major portion of the cross section fo r these processes proceeds via a compound state. The cross section of disso cia tive attachment on the other hand, w i l l depend on the lif e t im e of the resonances and on the rate at which i t dissociates in to a negative ion and a neutral a t o m . ^ ^ I t is convenient to c la s s ify resonances into several categories according to the states of the target (19) molecule. I f the electron is trapped in the potential of the ground state of the target molecule, the compound state is called a shape or single p a r t ic le resonance. This name refers to the shape of the potential which is responsible fo r trapping the electron. Shape resonances occur at low energies (below 10 eV) and e x h ib it a l i f e time of 10 ^ to 10 ^ seconds and decay via vibra tion al e x c ita tio n or disso cia tive attachment. Since vibra tion a l -1 3 -1 4 periods are of order 10 - 10 sec, the resonances -1 5 o f - 1i fetime 10 sec are. short re la tiv e to a v ib ra tio n a l 61 period and t y p i c a l ly possess a broad peak of approximately 2/3 eV. I f the resonances are longer than a vib ra tio n a l period, the compound state may display vib ra tio n al structure as observed by sharp spikes where the v ib ra tio n a l levels of the resonances are located. The resonant states of HC1" studied here belong in this category. The second type of compound state is called a core- excited resonance since the target is in an e le c t r o n ic a lly excited state. Often, the physical process involves an incoming electron which excites a Rydberg electron on the target to form a temporary negative-ion complex consisting of two Rydberg electrons moving in the f i e l d of the p ositive ion core. The core excited resonances are divided into Feshbach (Type I) and Type II resonances. The former l i e below the "parent" (ta rge t) with life tim e s -1 2 -1 3 of 10 - 10 sec. Feshbach resonances usually involve the binding of pairs of Rydberg electrons to the positive ion core ( fo r example: HC1* + 2 electrons) and since decay to the parent molecule is e n e rg e tic a lly forbidden they are f a i r l y stable. Type I I resonances l i e above the excited parent state and are s im ila r to shape resonances. The l a t t e r w i l l not be observed in the e la s tic channel since they commonly decay to the excited parent state. Since Type II states are not as sharp as Type I, they w i l l consist of a va rie ty of £- type waves. The Fe-shbach. states of HC1" were not investigated in the 62 present study but are expected to l i e between 9.2 and 9.6 eV above the ground state of HCl.^2^ W e turn now to a detailed discussion of the known experimental results of low energy electron s cattering on the HC1 molecule. i i ) Summary of Experimental Evidence In recent years electron scattering from polar molecules has received considerable experimental a tte n tio n , often with results which are puzzling with respect to theories of an electron bound in the f i e l d of a (9) (21) dipole or in tu ite d potential energy surfaces. This work was designed to provide an a b - i n i t i o the ore tical analysis of the resonance states of HCl", fo r which a wealth of experimental electron scattering data e xists. A f u l l discussion of the i nte rpretation of the data on the basis of these calculated potential curves w i l l be presented in the next section. W e begin by presenting a detailed description of the experiments. The relevant experiments are: (a) Dissociative attachment: (1 ) e“ + HCl ■ + Cl" + H (below 1 eV) (2) e" + HCl + H" + Cl (6-10 eV) The onset of Cl" formation is v e r tic a l with a threshold at 0.7 eV and a maximum at 0.84 eV.^22^ Most i n t e r e s t i n g l y , 1ow-amp]itude o s c illa t o r y structure with an energy spacing of approximately 0.3 eV (.roughly the 63 stre tching frequency of HCll are superimposed on the (23) higher energy side of the cross section. Abouaf and T e i1e t - B i 11ey f i t this structure by a series of s tr a ig h t lines of d i f f e r i n g po sitive slopes which i n t e r sect thus causing a slope d is c o n tin u ity at the v ib ra tio n a l energies of MCI. In dissociative attachment leading to H.", two peaks are observed. The f i r s t one has a steep, possibly v e r tic a l onset at 6.9 eV and the second, where onset overlaps the high energy part of the f i r s t peak, has a maximum at 9.2 eV and is not seen to rise v e r t i c a lly . (-24^ (b) Associative detachment: Cl" + H + HC1 ( v , j ) + e“ In general, the electron is detached from a negative ion with the energy i t obtains from exothermic bond forma tion (_HC1) as well as the energy brought into c o l l i s i o n by CT and H. Various reactions may take place, depending upon the s p e c ific potential surface of i n t e r action. I f Cl" and H were approaching along a repulsive curve, then s cattering with no reaction would be observed. However, i f the in te ra c tio n potential is a t t r a c t iv e and cuts the HC1 curve at some geometry, then associative detachment occurs. I f an excited state of HC1" lies above HC1 a ft e r the crossing occurs, autodetachment to HC1 + e” is exothermic and fast compared to c o llis io n time and the p r o b a b ility of associative detachment is very 64 high. I f HC1 were bound below HC1, unless the crossing point were somewhere below the Cl” + H l i m i t , no associa tiv e detachment would be observed. The l i m i t , Cl” + H lie s only 0.7 eV above HC1. Furthermore, since the process occurs at a fast rate with no temperature (25) dependence, no s ig n if ic a n t barriers are expected to exist fo r formation of HC1” from Cl + H. F in a lly , as Cl" approaches H, the curve can s p l i t into various curves which represent d if f e r e n t states of HC1” . I f HC1" autodetaches into a repulsive state of H .C 1 i t w i l l re sult in the products H + Cl + e“ i f there is s u f f ic i e n t k in e tic energy of approach. (c) Vibrational e x c ita tio n : e" + HC1 e" + HC1 (v , J ) In each f in a l v ib ra tio n a l state, v = 1 , 2 . . . a -1 5 2 large (10 cm ) sharply peaked cross section is seen at threshold. This cross section has a second maximum of larger width around 2.5 eV and then decreases mono- (26) to n ic a lly above 3 eV. The cross sections are is o tro p ic in the regions of the peaks, in d ic a tin g that they are not caused by ele ctron -dip ole long range in te ra c tio n s , i . e . by d ire c t v ib ra tion al e x c ita tio n . I f i t were caused by the dipolar f i e l d one would expect s, p and d waves and not an is o tr o p ic d i s t r ib u t i o n . This is due to the fact that d ire c t forces w i l l in general be observed as a forward peaked cross section due to the contribu tion ________________ 55 of many p a rtia l wayes. A closer range potential however, yie ld s an is o tro p ic cross section which is sharply peaked and is due to few p a rtia l waves, ( i . e . the s wave dominates the cross section). (d) In the e la s t ic channel, measuring the current d e rivative at 180°, a peak is seen at Esca^--j- - w (HC1), where u)g is the HC1 v ib ra tio n a l constant; at Escat t = 2toe a peak which is ten times smaller is seen and (27) nothing is c le a rly seen at 3w , 4w , etc. ' c c Since no simple explanation based on the existence of several potential curves seemed to exist p r io r to the present calcu la tion of the potential energy surfaces, a ll previous in te rp re ta tio n s of the experimental (212328) data * * invoke complex dynamic arguments based upon approximate cross section expressionsand a paramet rised resonance V(R) to explain these processes. (21) In both papers by Fiquet-Fayard ' and Crawford and (28) Koch which d i f f e r in t h e ir explanation of the phenomena of d isso cia tive attachment to Cl~ + H, claims are made of good f i t s of parametrised HC1" potential curves to the data. These curves d i f f e r , even q u a l i t a t i v e l y , from each other and both d i f f e r q u a n t it a tiv e ly from the computed ab i n i t i o ones presented here. Craw ford and.Koch o ff e r a d ire c t s cattering model and a distorted-wave calculation to explain dissociative attachment. They were led to th is model by the assumption 66 th a t, since the lowest state dissociating to H + Cl" is of £ symmetry, (1) no angular momentum barriers could e x is t and (2) without an angular momentum b a rrie r no resonance can occur. W e find both these arguments unsupported by our computations. HC1 does not present an is o tro p ic s-symmetry f i e l d in a ll directions and r e fle c tio n resonances can e x is t over a well even when q no b a rrie r is present. (For example, 2 S state of He). In p a r t ic u la r , as reported below, s t a b i l iz a t i o n finds several low-energy £ resonances. The object in studying the resonances of HCl" was not ju s t to obtain highly accurate wave functions for one or two states near t h e i r equilibrium bond lengths, but rather to get a q u a l i t a t i v e l y and q u a n t it a tiv e ly accurate description of the potential curves of the many states of in te re s t. S p e c if ic a lly , we sought to: (1) correlate the various molecular states with the asymptotic states of the constituent atoms; (2) confirm the presence of experimentally observed states, predict those states as yet unobserved and include t h e ir symme tr ie s and positions; (3) explain the nature of the states in terms of simple configuration in te ra c tio n mixing terms. In order to assure proper dissociation fo r a ll states of in t e r e s t , the configuration in te ra c tio n approach is essential. A' SCF approach alone was not considered since we are interested not only in the lowest states of 67 each symmetry, but seyeral states of E+ and n symmetry. The corre latio n energy is not constant fo r these excited states. The SCF approach is also inadequate because of the improper dissociation behavior of the one-configura- ti.on wave function . These inadequacies indicate the necessity of more sophisticated treatment such as C.I. In carrying out the configuration in te ra c tio n calcu la tions i t is of utmost importance to tre a t a ll the elec tro nic states e quivale ntly. This w i l l insure proper s p l i t t i n g s among the various excited states and the equivalent treatment of the ground state configuration w i l l give r e lia b le re sults. I t is the experimental facts mentioned above that we seek to in te r p r e t via the extensive calculations on the ground state of HC1 and the resonant states of HCl" reported here. The fundamental approach to the calcula- (29) tion of resonances is the s t a b i l iz a t i o n method' ' using square integrable (gaussian) functions and Configuration In te ra c tio n . In this case, a possible solution to the C.I. equations is an attempt to represent the l i m i t HCl plus an electron at i n f i n i t e separation w ith in a basis set of functions of f i n i t e extent, so care must be taken that any stable root is indeed a resonance and not an a r t i f a c t of this type. This, as well as the c r i t e r i a used to i d e n t i f y stable roots w ith in a coarsely grained basis set, w i l l be discussed below. 68 B. Computational Details i) Method of Computation The atomic o r b ita l 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 simultaneously represent HCl and HCl" at a range of in te rnuclear distances. The Dunning basis of contracted gaussian functions chosen was the ( lls / 7 p ) p rim itiv e gaussian basis o f . Huzinaga( ^ ^ for (31% Cl contracted to 6s and 4p fu n c tio n s .' y In addition to these functions, uncontracted gaussian Rydberg s type functions with exponents of .023 and .004 were added to Cl as well as p type functions of exponents .049, .018, and .005. A p type p o la riz a tio n function of exponent 1.0 was added to hydrogen. The e n tire atomic o r b ita l basis is given in Table 3.1. Some calculations with additional s and/or p type functions of exponent .001 were also carried out fo r the purpose of proving s t a b i l i t y and the r e a l i t y of the calculated resonances as w i l l be discussed below. The computational approach u t i l i z e d here was the c alcu la tion of the SCF wave function fo r HCl w ith in this basis set. This resulted in a C.I. basis set of 34 molecular o r b ita ls which was truncated to 30 functions by dropping the two lowest and two highest energy molecular o r b i t a l s . A complete l i s t of the M.0. eigen values and the energy of HCl at i t s eq uilibriu m i n t e r nuclear distance are given in Table I I . This may be 69 compared to the near hartree-Fock HCl energy of -460. 1 1 85 (32) a.u. obtained by McLean and Yoshimine. ' ' The basic approach to the configuration i n t e r action step fo r any given state was the selection of one or more seed configurations to represent the state of in te re s t and the generation of all unique single and double hold p a rtic le e.xcitations r e la tiv e to the seed configura tions to form the e n tire C.I. space. In some cases, th is level of configuration in te ra c tio n results in very large C.I. matrices, the largest being of order 16,699 2 4 * spin functions fo r the £ states of HCl . The method of solution of the configuration in te ra ctio n problem was a p a r t it io n in g method, a detailed discussion of which has been presented in Chapter I I . In calculations on HCl and HCl” , i t became apparent that the f u l l computational machinery described in Chapter II was not re a lly necessary fo r th is case and that the calculations could be done even mo re simply. In electron scattering from HCl, the scattered electron is r e la t i v e l y uncorrelated from the electrons of the target molecule. The s cattering events are rather long range since they are dominated by the e l e c t r i c dipole of HCl. This has a number of im plications fo r the cal- c u l a t i o n : 1. The molecular o b tia ls of HCl and the v ir tu a l space generated by the corresponding SCF calculation 70 form an adequate basis set for the C.I. calculations of HCl". Comparative calculations using this basis and the SCF o r b ita ls fo r HCl" showed no s ig n if ic a n t difference at the level of C.I. used here. The v irtu a l o r b ita ls of an SCF calcula tion on HCl are eigenfunctions of the f u l l n-electron HCl potential including its dominant feature, the permanent dipole moment. Since the scattering is long range, the target molecule is r e la t i v e l y l i t t l e perturbed by the scattering event. As a re s u lt of t h i s , the v ir t u a l o r b ita ls of HCl prove to be close to the natural o r b ita ls of HCl" fo r the scat tered ele ctron, as judged from the r e l a t i v e l y heavy weight of a single configuration in the fin a l C.I. re su lts. 2. Using this basis set at any given inte rnu clear distance, the c o rre la tio n energy of the Rydberg type states of HCl" (defined as re la tiv e to the energy of the seed configuration obtained by populating the appropriate v ir t u a l o r b ita l of HCl by a single electron) is constant to 0.1 eV or less since i t is ju s t the c orrelation energy of HCl at th is in te rnu clear distance, the core of HCl molecular o r b it a ls heing always f u l l y occupied in the negative ion states discussed here. O Thus at 1.2744 A, the c o rre la tio n energy of HCl 2 + was calculated to be 2.781 eV while that of the 1 2 state of HCl” was 2.795 eV and the 2 22 + state was 2 + 2.804 eV. The 3 2 state, which is more valence l i k e , 71 showed 4.06 eV of correla tio n energy. Sim ilar results O obtain u n til the 2-3 A range where the Cl is becoming more lik e Cl" and the o rb ita ls representing the scattered O electron begin to r a d ia lly contract. At 2.0 A, the c o rre la tio n energy calculated for HCl was 3.40 eV while that fo r 1 was 3.61 eV and the 2 ^E+. state showed 3.51 eV. The one or two "seed" configurations fo r the states considered were expanded by one to nine additional crucial configurations. This set of nuclear configura- tions is depicted in Figure 2.1 by the small block in the uppermost l e f t hand corner of the matrix and most of the fin a l weight of the wave function resides in this nucleus. I t was found that using th is M.O. basis set _ 3 and a threshold of 1x10 a.u. as the second order Rayleigh Schroedinger perturbation energy fo r inclusion in the core block, a block of dimension 20 was s u f f ic i e n t to obtain accuracy to about 0.02 eV when compared to the f u l l 50 dimensional block. This was true for most cases except the atoms Cl and Cl". In a d d itio n , i f the fin a l C.I. wave function is mainly made up of the few configura tions gathered in the upper l e f t hand corder of Hga, the matrix elements in the s t r i p Hab (Figure 3.1) enter the calcu la tion to s i g n if ic a n t l y lower order than does the remainder of the Hab block. Setting the remainder to zero proved not to change the calc u la tio n in most cases w ithin the accuracy of the overall scheme. Since this truncation implies less work in constructing the i n t e r action matrix Hab, a 20 dimensional core block and the truncation of Hab was used everywhere except as noted. E s s e n tia lly , a core block of order 20 is s u f f i c i e n t to describe HCl in this basis set. The e ffe c t of the use of a core block of 20 and truncation is shown in Table 3.3. Using that portion of this basis set which is centered upon Cl, the calculated Electron A f f i n i t y of Cl (33) is 3.3 eV where the experimental value of 3.6 eV. This level of accuracy was achieved by including all single and double excita tio ns from the SCF fo r Cl and Cl” through the n = 2 level and the f u l l 50 dimensional a^ block. D fo r HCl was calculated to be 4.1 eV, using a f u l l ( 34) c a lc u la tio n , where the experimental value is 4.4 eV. We have thus achieved balancing errors of 0.3 eV fo r the negative ion and neutral lim it s (H + Cl) so that i t is reasonable to assume that the HCl” curves are c o rre c tly spaced re la tiv e to HCl. The calculated energy surfaces of HCl and HCl" are presented in Figure 3.2 and Table 3.4. Full C.I. calcu- O O la tio ns were carried out at 0.9 A, 1.2 744 A ( e q u i l i - f O C ) ° O o o brium) 1.5 A, 1.7 A, 2.0 A, 3.0 A and i n f i n i t e separation. The ground state HCl calculations were performed with a configuration space chosen from a ll single and double hole p a r t ic le excitations from the 73 1 + closed shell SCF base plus trie next lowest energy Z state (7+10). This generated a total space of 6,643 confi gurati ons. Each HCl” state was then described by adding an extra electron to the HCl closed shell configuration and including a ll single and double excitations from this base. I t should be realized that the resonances of HCl p rim a rily arise from its dipolar f i e l d . As the bond length is increased, this potential tends to zero. Some of the resonant states then merge into continuum functions and become impossible to follow in the C.I. c a lc u la tio n , a s itu a tio n represented by dotted lines in Figure 3.2. This is an important consideration in in te rp r e tin g elec tron scattering from polar molecules. I f one attempts to i n t u i t the scattering states by c o rre la tio n from the separated atom l i m i t s , one w i l l miss states which are physically important but which e x is t only in the region of the equilibrium inte rnu clea r distance due to the new / or 1 potential f ie ld s formed there. 2 + The 1 Z state was generated through occupation of M.O. 10, an s type Rydberg function. At r 0 , this configuration represents 9 71 of the total wave function while the total energy correction from i t due to C.I. toals 2.795 eV. At longer distances th is state merge 2 1 smoothly into the l i m i t H( S) + Cl ( S). The calculated potential energy curve lies 0.28 eV above this l i m i t at 74 o 1.7 A. I t therefore e xhibits an a c t i v i t a t i o n energy for associative attachment, a conclusion in disagreement (25) with the experimental results which place an upper l i m i t of 0.1 eV on any such b a r r ie r . I t is our feeling that th is maximum, which arises at the SCF stage, repre sents inaccuracy in the c a lc u la tio n . I t is probably due to gaussian basis set inadequacy since i t persists with a f u l l 50 dimensional a ^ block c a lc u la tio n . This seems p a r t i c u l a r l y reasonable when one notes that the errors in the lim it s are of order 0.3 eV and that the calculated onset of Cl” production, i f the b a rrie r is ignored, is 0.83 eV, in close agreement with experi- ( 2 2 ) ment. This onset would, of course, also be v e rtic a l in agreement with the experimental observations. While this state is of Z symmetry, its radial d is tr ib u tio n shows strong concentration at the hydrogen end of the molecule.(Figure 3.3). I t would therefore show a strong con tribu tio n from higher p a rtia l waves and i t would be expected to have a s ig n if ic a n t lif e t im e due to the angular momentum barriers associated with these c o n t r i butions. 2 + The 2 E state results from the occupation of M.O. 13 at r g , e s s e n tia lly a p symmetry Rydberg function. Although this configuration has a weight of 97% in the C.I. wave function at e q u ilibriu m , the vector becomes mixed at longer distances and we conclude that i t merges 75 into a continuum funotion as previously discussed. This is shown as dotted lines in Figure 3.2 in the region.where no calculations were carried out. Scattering from i t would have a large p wave component indica tin g a s ig n if ic a n t 2 + lif e t im e . The state lies only 0.19 eV above the 1 E state of r g , and these two states , which are of the same symmetry but c le a rly not two cuts through the same wavepacket, w i l l in te r fe r e in such a way as to produce f 2 6 1 the v ib ra tio n a l stru cture seen at low impact energies. This w i l l be discussed in d e t a i l i n the next section. 2 + The 3 £ state is a much more r a d ia lly contracted state which results from the occupation of M.O. 18, p r in c ip a lly an s type function polarized toward hydrogen. I t also is quite pure at r g , the prime configuration comprising 86% of the C.I. vector, but becomes confused at larger r. The minimum of th is state lie s 2.6 eV above HCl, corresponding to the broad resonance beginning at (2 6 1 2.6 in the vibra tion a l e x c ita tio n spectrum. The higher energy region is s l i g h t l y more complex. The l i m i t H" (IS) + Cl ( 2f?) w i l l give rise to 2£ + and 2 II states of HCl . These arise from the HCl core plus the n MO p a ir. 11 and 12 at r . The resultan t repulsive 2n state rises smoothly and lie s 9.28 eV above HCl at r g , in close agreement with the gaussian shaped peak in H~ production whose maximum lie s at 9.2 eV. Since this is a r a d i a l l y extended sta te, labor was saved in the 76 calculations by carrying out a C.I. calcula tion between a ll possible single e xcita tions from the parent configura tion and subtracting the HCl c o rre la tio n energy at each internuclear distance. As previously discussed, this procedure seems to be f u l l y j u s t i f i e d by the physics of the problem. 2 + At long internuclear distances, the 4 2 state arises from the same l i m i t and is not fa r s p l i t from 2 the H state. I t is , however, crossed by a number of 2 + ° E states at distances shorter than 1.5 A. When the adiabatic contour is drawn, carrying out calculations 2 in the same approximation as used f o r the n state, 2 + 2 + the s itu a tio n shown as 4 E and 5 E is found. H 2 + production from the 4 E state would have a v e rtic a l onset. Its maximum lie s 6.78 eV above HCl which should ( R) be compared with experimentally observed onset, 6.9 eV. The calcula tion on these states seem, therefore, to be in excellent agreement with the experimental observations. The comparison between the calculated and experimentally observed onsets, maxima, etc. is summarized in Table 3.5. I t must be emphasized that other roots were found between the states we have discussed. These could be continuum functions or another s lic e through the wave- packets of the states we have discussed since each of these have, in r e a l i t y , some s ig n if ic a n t width. One of these w i l l be discussed in the next section but we have 77 lim ite d ourselves to presenting the data on those states which are stable.and which seem to account for the exp eri mental data. i:i) Proof of Resonance Character The C.I. calculations described here u t i l i z e d as M.O. basis ^derived from SCF calculations on HCl. The resonances tend to be due to the trapping of an electron w ithin the e l e c t r i c dipolar f i e l d of HCl at r e la t i v e l y long range, so that the target is l i t t l e perturbed during the resonance process. As a re s u lt of t h is , the v irtu a l o r b ita ls of HCl, which represent an additional electron in the f i e l d of a 11 the electrons of HCl, prove to be rather good approximations to the natural o r b ita ls of the scattered electron. This is seen from the extremely heavy weight of the "seed" configuration in the fin a l C.I. vectors. ( 37) In the o rig in a l s t a b i l iz a t i o n method, heavy weights of a single or a few configurations were used as a c r i t e r i o n to id e n t i f y stable roots. These calcu la tio n s, however, were carried out on molecules without permanent e le c t r ic dipole moments so that there were no dominating f i e l d s , and the conditions found fo r HCl, i . e . a series of rather pure C.I. vectors representing successive Rydberg states in the f i e l d of a dipole, did not obtain. In the case of dipolar molecules lik e HCl, p u rity of the 78 C.I. vector is a rather useless c r i t e r i o n for s t a b i l i t y . Energy s t a b i l i t y with v a ria tio n of the basis set is, however, a useful c r i t e r i o n . One must be c areful, however, that one is not attempting, w ith in a square integrable basis set, to represent HCl plus an electron at i n f i n i t y . With a s u ita b ly f l e x i b le basis set, this s itu a tio n can appear to be e n e rg etically stable, p a r t ic u l a r l y since i t would be e s s e n tia lly e n e rg e tic a lly degene rate with HCl, a condition s im ila r to that shown by the 2 + 1 S state. In order to test this and to demonstrate that the lower energy states we have i d e n t if ie d as resonances are indeed resonances and not computational a r t i f a c t s , we have chosen to take advantage of the re la tiv e p u rity of the C.I. vectors, which indicate that the M.O. used fo r the scattered electron is a f a i r approximation to the natural o r b ita l for that electron, and graph the density functional of th is M.O. as a function of distance fo r various states using increasingly f 1exible basis sets. M.O.'s which are attempting to place the scattered electron at i n f i n i t y should increasingly show this behaviour with increasing f l e x i b i l i t y and longer range basis, functions ava ilable. Continuum functions without resonance character -should.appear.increasingly sinusoidal, while resoances should show a high p r o b a b ility of the 79 electron being in the area of the target with sinusoidal behavior at larger r. The increase of f l e x i b i l i t y necessary was provided by adding to the basis set of Table 3.1 f i r s t an s o r b ita l centered on Cl of exponent .001, then adding to this basts a pff o r b ita l of the same exponent. Figure 2 + 3.2 shows the behavior of the 1 2 disso cia tive attach ment state of HCl" at the equilibrium distance of HCl under th is treatment. Increasing f l e x i b i l i t y (bottom to top] results in the -main density peak moving to shorter r and residing behind the H side. The maximum yalue of this peak remains r e l a t i v e l y constant but, c le a r ly , sinusoidal behavior is being b u i l t at longer distances. This is the classic i n t u i t i v e picture of a resonance. 2 + Sim ilar behavior is observed fo r the 2 E mimic state, although this is a p^ type Rydberg and exhibits a doubly peaked behavior [Figure 3.3). The M.O. repre senting the broad resonance is highly peaked in a ll basis sets and the peak is much narrower than in the other states. [Figure 3.4). In contrast to th is behavior, consider Figure 3.5 which shows a state which lie s at an energy between the mimic and broad resonance states. This may be an attempt, w ith in the basis set used, to represent sinusoidal behavior everywhere, i . e . a non re sonant state or i t 80 2 + may be another cut through the wavepacket of the 2 E state with additional continuum behavior. These c a l culations cannot resolve this question, but there is 2 + a clear change in character upon reaching the 3 E state, Fi gure 3.5. In none of these plots is there any in dica tion of an attempt to represent an electron at i n f i n i t y . Rather, those states i d e n t if ie d as resonances appear to be ju s t that. They are, of course, e n e rg e tic a lly stable to w ith in the numerical accuracy of the c a lc u la tio n , about .05 eV 2 + while the dimension of the C.I. fo r the E states rises to 23,193 spinfunctions for the largest basis set. 2 + Additional roots appear below the 1 E state with the addition of gaussian o r b it a ls of exponent .001. These 2 + may be fu rth e r slices through the wavepacket of the 1 E state or may be attempts to place the scattered electron at i n f i n i t y . The lowest of these is only .05 eV above HCl and we have not pursued the point fu rth e r. Furthermore, the lowest state of HCl" is vafia - O t i o n a l l y correct beyond about 1.6 A since this is the lowest state of the HCl + e“ system. This state connects 2 + smoothly with the 1 E state w ith in the HCl potential w e ll. These states appear to be well behaved and to be true resonances, not computational a r t i f a c t s . 81 C. In te rp reta tio n i) Explanation of Observation Given the states in Figure 3.2 the explanations of the observations are q u a l i t a t i v e l y quite simple. The very broad resonances in vibra tion al e x c ita tio n with a maximum at about 2.6 eV is accounted fo r by the th ir d — 2 -f" (highest in energy) HCl" 3 £ curve. The unusual breadth of this well implies a very close v ibra tio n a l spacing. This in turn, considering the resolution of the e xperi ment and the broadening of the levels due to f i n i t e life tim e s and subsequent overlapping of vib ra tio n al peaks, explains why no structure is observed. The symmetry accounts fo r the isotopic nature of the angular d i s t r ib u t io n of electrons scattered via this state. The lowest HCl" 1 2£+ state is a t t r a c t iv e in spite of the small b a r r ie r calculated. This b a rrie r is considered to be an erro r due to an inadequate basis set especially in a crossing region. r The three higher resonance states are involved in d isso cia tive attachement (DA) to H- . The l i m i t H~(^S) + Cl(^P) gives ris e to and states of HCl.. We note that the E state lie s above the state at larger R and that the n state has the correct energy (9.2 ^ 9.3 eV) to explain the second ’peak mentioned in the Dissociative Attachment to H". Since the state is repulsive, the Gaussian shape of the second peak is 82 2 + - ' explained. The highest 2 state of HCl which goes to H“ and which arises from the same mol e c u !a r-o rb ita l occupations as the n state (HCl core IT) lie s at a higher energy in the Franck-Condon region of HCl thah can explain the experimental peak at 6.9 eV. 2 + There is , however, a 2 state with a minimum at 5.8 eV which is due to an electron in the dipole f i e l d 2 + of HCl and this state crosses the highest computed 2 state. The adiabatic contour results in a 2 state that is disso cia tive with a b a r r ie r at 6.8 eV. This of course ex plains the v e r tic a l rise in the cr^( H~) at 6.9 eV. The closeness of the 2 b a r r ie r and the II state accounts fo r the overlap of the aD^(H” ) peak. The v e rtic a l onset in the d issocia tive attachment cross section (Cl") is simply explained by the a t t r a c t iv e 2 nature of the lowest 2 state of the HCl . More in te re s tin g is the observation that the short-range part of the curve runs nearly p a ra lle l to and close to the 2 ^2+ state of HCl". This l a t t e r "mimic" state, in the well region, has a curvature and Rg value very close to that of the ground state of HCl, being e f f e c t iv e ly due to the a tt r a c tio n of the HCl dipolar f i e l d fo r the sca tterin g electron. I t therefore has the f i r s t few v ib ra tio n a l spacings as in HCl. Now since the f i r s t 2 and second HCl states have the same 2 symmetry and since th e ir curves at short R run p a ra lle l to and close 83 to each other, a predissociative type of interference is to be expected on the high-energy side of the aDA(Cl") peak. This in te ra c tio n w i l l give rise to interference with a period near that of the v ib ra tio n a l period of the ("mimic") second HCl" state which in turn is close to the period of HCl. (38) Cusps and sudden changes in resonance width ' due to the opening of HCl vib ra tio n al channels could also appear superimposed in the o s c illa t io n s and cause the slope d is c o n tin u itie s that Abouaf and T e i l l e t - (23) B i l l y can parametrise into t h e ir f i t s of experimental data at these channel energies. I t is also to be noted that in the following we shall show that the predissocia ting "mimic" state also shows up in the v ib ra tio n a l e x c ita tio n experiments of Rohr and L i n d e r . T h e t o t a l i t y of cusps plus p redisso cia tion, coupled with our calcu la tio n s, c e r ta in ly gives the simplest and most cohesive explanation of the experimental observations. The process is much lik e that in d isso cia tive attach ment in H .2 where a s im ila r interference is seen at about 12-13 eV and explained by H.S. T a y l o r . T h e fact (23) that the structure in HCl is not a typica l i n t e r ference pattern as i t is in H2 could be due to d if f e r e n t phases and amplitudes fo r the competing processes in the d i f f e r e n t systems. 84 Lastly the sharp peak- in th„e ytb.ration.al eixcitation threshold must be explained. To understand t h is , i t is to be noted again that the second H.C1" curye has a curvature (in the HCl well) quite s im ila r to HCl" and that of HCl. The v ib ra tio n a l wave functions with d i f f e r e n t v in H.C1 and HCl" are nearly orthogonal. The amplitude fo r v ib ra tio n a l e x c ita tio n from v^(HCl) to v^(HCl) is a two-step process. The matrix elements in this process go from v^(HCl) vr (.H.Cl") ■ + v^(HCl) and therefore the cross section would be small unless v r = vfS where is the i n i t i a l state, v f is the resonant state and is the fin a l state. The reason is that fo r a ll v and v* the v,- m atrix-el ement r f i r fa c to r w i l l have a small (order A; A < 1) v ib ra tio n a l c o n trib u tio n . The second fa ctor from v r -► w i l l be small (order a) unless v r = vf where the fa c to r w i l l be of order u n ity . Hence for vf f v^, the cross section 4 2 order is A , while for v r = the order is A . In r e la t iv e magnitude, the v. vr -* (v^, = v ) is therefore much stronger than the ■ + v r ■ + (v^ f vr ) process and only one HCl' leyel should be observed in the v- to HCl t r a n s it io n . The observed level would be v r (HCD = v^(HCl). Since the two curves HCl" and HCl are displayed in energy only by at most 0.32 ± 0.06 eV and the HCl” curve lie s above HCl, the single peak should h.e observed around 0.32 ± 0.06 eV above the threshold 85 fo r v^(HCl) in the process. This is what is observed. To account fo r the large size of the cross -1 5 2 section (10 cm ) is problematic. No explanation for th is was offered in the previous in te r p r e ta t io n of this data since one lacked the proper curves and could not calculate the e le c tro n ic part of the t r a n s it io n matrix elements fo r HCl -* HCl" HCl. W e have not done this e ith e r but believe, because of the long-range electron- polar-molecule nature of the resonance that the ele ctro n ic amplitude w i l l be high and w i l l , when done c o r re c tly , (27) give the correct magnitude. I t should be realized ' that s l i g h t displacements of RQ(HC1) from R (HCl") can C C ' greatly enhance the tr a n s it io n amplitude. Our curves' do not have the same Rg , only s im ila r R0. The 2s nature of the second HCl" curve and the HCl ground state is consistent with the is o tro p ic nature of the threshold peak in vib ra tio n a l e x c ita tio n . As to the e la s t ic data, since our HCl” "mimic" state is displaced from HCl to w ithin 0.05 eV of u>e(HCl) and since 0.05 eV is w ith in our e rror (to say nothing of the width of the resonance), one can only say that the HCl vibra tion a l levels s ta r tin g with v = 2 overlap those of HCl". This, plus the above Franck-Con don arguments, (27) explain why, in e la s t ic s c a tte rin g , Burrow ' sees an e ffe c t ten times larger at the v = 0 energy of HCl" (which is the v = 1 of HCl) than at t h e v = 1 energy of 86 HC1“ (which is the v = 2 of HCl). I t is not incon ceivable tha t, because of this "overlap of le v e ls " , resonant and cusp phenomena are both active in the "peaks" of th is measurement. ? Since the lowest E state of HCl is found to be approximately 0.18 eV above the ground state of HCl, - (251 no stable HCl is expected to be observed in measuring thermal-electron-attachment rate constants. In these calculations the size of the c o e ffic ie n ts in the C.I. expansion and an inve stig atio n of the natural o r b ita ls are used to pick the stable roots. In drawing the curve at each R, there is a choice of whether to draw the diabatic or adiabatic curves. Generally, i f the c o l l i s i o n proceeds i n f i n i t e l y slowly and there is f i n i t e in te ra c tio n between the two states, then the adiabatic curve is f o l l o w e d . ^ Conversely, a c o l l i s i o n occurring with a f i n i t e v e lo c ity and no in te ra c tio n at the crossing point indicates that the dia batic curves may -(38) be drawn. In H£V ' the diabatic curves not only f i t experiment but were the only stable roots at each R. Here, in HCl many more stable roots occur (because of the "states that do not e x is t at i n f i n i t e R" concept) and a choice must be made between adiabatic or diabatic (41 ) curves. Mandl proved that fo r resonances, resonant roots avoid each other when both th e ir energies and widths are of s im ila r values. For our s itu a tio n this 87 is supportive of our adiabatic choice in H ,C 1 and the diabatic ones in H^. In the former th e.stable roots that are near in energy a ll have (because they are stable) f i n i t e and s im ila r widths. In the l a t t e r the lack of stable roots implies that the crossing states have zero lif e t im e and hence dicta te the dia batic choice. The adiabatic curves were chosen because of this argument and because they agree with the experimental data. A d d itio n a lly , by doing a rough calculation using the Landau-Zener formalism, the t r a n s it io n p r o b a b ility for diabatic t r a n s it io n was found to be extremely small. This is sensible since the states involved have con siderable in te ra c tio n at or near the ’'crossing point" and should follow the adiabatic p o te n tia l. ( i f ) Electron-polar-m olecular scattering I t is to be noted that our c a lc u la tio n s , which include a ll induced and d ire c t forces, do not give rise to any true binding of the electron to HCl.^40^ The controversy over the role of the dipole poten t i a l observed in experiments on e le c tro n -po la r molecules scattering has sparked much discussion. The most in te re s tin g feature of these experiments is that w ith in 0.5 eV of threshold, the observed cross-sections in vibra tion a l e x c ita tio n are larger by a fa c to r of 10-100 than estimates with Born's approximation. In other words 88 these cross sections cannot be explained as being solely due to c o llis io n a l momentum tra n s fe r. In attempting to explain these observations, theoreticians have pursued two c o n f lic t in g arguments. (42) One group argues that due to the long range character of the dipole f i e l d the Born Approximation should s t i l l be somewhat v a lid . The role of the dipole f i e l d is of utmost importance since a ll polar molecules with large dipoles have extremely sharp peaks at the vib ra tio n a l threshold. In a ddition, this group argues that the is o tro p ic nature of the peaks indicates s- wave sca tte ring . Shape resonances imply that the electron is trapped in a c e n trifu g a l b a r rie r . Therefore, s-wave s cattering cannot be associated with shape resonances. The calcula tion performed by this group are based on the idea that the electron inte racts with the long-range t a i l of the dipole. Using these arguments, Herzenberg and Dube are able to reproduce the main threshold features of v ib ra tio n a l e x c ita tio n in e-HCl scattering and by including the dipole potential they attempt to remove the discrepancy between experiment and Born's Approximation. ( 36 ) The opposing argument ' focuses on the role of the dipole potential in supporting resonant states of the electron in the f i e l d of the molecule. This group argues that vib ra tio n al resonances wiht o s c illa t o r y 89 stru cture have been observed for both polar and nonpolar molecules ( ^ - f o r example), and the sharp peaks are more generally explained via the resonant process. This implies that the d is t o r t io n of the incoming electron wave by the potential well of the target molecule is extreme. This d is t o r t io n invalidates the a p p l i c a b i l i t y of the Born approximation. The resonances arisin g in e-HCl scattering are typical of processes which are dominated by very few p a rtia l waves and the angular depen dence of the scatte ring cross section is refle cte d by the symmetry of the resonant state. In general, the isotropy of the state does not imply that a shape resonance is impossible. The problem of ac tu a lly 'binding an electron to a fix e d , f i n i t e dipole has also a ttra c te d , considerable the oretical a tte n tio n . I t is of in te re s t because i f a general resonant sta te , AB", is bound re la tiv e to the molecule AB, then a stable negative ion would be observed. (43) (9 ) Levy-Leblond ' and la t e r Crawford have considered electron scattering by a point dipole. They calcu la ted the minimum dipole moment required to bind an electron and found y . = 1.63 Debye units mm J This led to the prediction that i f a molecule has a dipole moment greater than 1.63 D (the " c r i t i c a l " dipole) then its negative ion would have a positive 90 electron a f f i n i t y and would be bound. A b r ie f summary of this theory w i l l now be presented. The point dipole potential energy is given by: V = cos 6 [1] (r ) where y is the e l e c t r i c dipole moment of the polar molecule, r is the distance of the electron from the center of mass and 6 is the angle between y and r. By using this dipole p o te n tia l, one can w rite Schroedin- ger's equation fo r an electron in the f i e l d of the dipole as f 0 2 m e y cos 6 2 m E / ^ j v T - p [2] ? 2 m E 2 m ey 2y l e t t i n g k = ------*— and a = ------5------- =------- where a is a dimension!ess dipole moment and aQ is the Bohr radius, we have: [ v 2 + + k2j = 0 £33 r This equation is separable in spherical coordinates, i K r , 6 , <j>) = R ( . r ) 0 ( . 0 ) $ ( < { > ) [4J using this t r i a l solution in equation [ 3 j we obtain + in2 \ * ( .4 .) = 0 £ 5 ] dcj) v 91 «2 + k 2 . B dr^ r dr r R(r) = 0 [6] ^ (sin 0 -nr) + m 0— - a cos 0 - BVx 6(0) = 0 sin 0 d 0 d 0 - 2 q s i n 0 [7] where m is an integer and B a separation constant. For m= 0, the solution of equation [7] are the Legendre Polynomials therefore 9(0) = I P^ (cos 6) [8] Substitu tin g [8] into [7] (° b i - 1 + u u + 1 > ■ B]bi + “ I b* +i = 0 I = 0,1 ,2.. [9] In the l i m i t of E -*■ 0, equation [6] becomes, f-7 + TJ7 - -7* R(r) ' 0 C10] dr r We seek a s tationa ry solution ( i . e . one that has a f i n i t e number of nodes) fo r equation [10] in the l i m i t that E 0. Following, Landau and L i f s h i t z [44], the t r i a l solution for equation [10] is : R(r) ^ r s ; s u b s titu tin g in equation [10] we get, s(s + 1) - B = 0 [11] 92 Two solutions are in te re s tin g at this point re su lting in (a) solutions with B £ 1/4 have the c h a ra c te ris tic s of decreasing most ra pid ly at large r and have the form R( r) = TTiT' (b) solutions with B > 1/4 which have an i n f i n i t e number of zeros and have the form R(r) a r “ 1/2 cos (B - 1 /4 ) 1 /2 . I t turns out that in case (a) there must be at the most one node while case (b) has an i n f i n i t e number of nodes. For solution (a), B = -1/4, equation [9] becomes: 'tr,} b - i +^ u + 1 ) + 1 ^ t ^ j ‘w ° [ 12] where a • is the minimum dipole. When li m i t s below mi n B = 1/4 are examined, i . e . fo r B _ < 1/4, no bound states in the f i e l d of the dipole are found and y < ucri- ^ ca-j • For y > ^ c r i t i c a l there 1 s an i n f i n i t e number of bound states (negative energy le v e ls ) , in accord with conditions above B = 1/4. This may be explained as follows: A p a r tic le with B < 1/4 has no negative levels and fo r B > 1/4 there are an i n f i n i t e number of negative energy levels. There fore the wave function fo r the state with E = 0 is the 93 same as B < _ 1/4, i . e . i t has no zero values at f i n i t e distances and corresponds to the lowest le v e l. Therefore the state with B = -1/4 yields the minimum dipole since at that point (E 0) the electron in the f i e l d of the dipole forming a system AB" becomes degenerate with the molecul eO AB. Various techniques have been used in order to c a l culate a . from a recursion r e la t io n , such as equation mi n : n [12], For d i f f e r e n t £ 1s one gets an i n f i n i t e set of homogeneous equations, t h e ir solutions being equivalent to finding an eigenvalue of an i n f i n i t e matrix. From th is matrix a . and so y . is determined, min min R e strictin g ourselves to the f i r s t 20 or so terms ( 20x 20 determinant), a . =1.2787 or y . =1.54296 A.U. min min From this theory one may expect the negative ion state of HF to be bound since the dipole of HF, 1,82 D is larger than the c r i t i c a l dipole whereas that of HC1 is smaller (1.2 D). However, theoretical calculations - (45) on LiCl have been shown to d i f f e r s i g n i f i c a n t l y from the p o in t-d ip ole model calculations thereby making fu r th e r accurate computations of an electron in the f i e l d of the dipole very important. I t is in te re s tin g to compare the calculations per formed on HC1~ to previous work done on HF". The lowest 94 H + Cl” curve computed here d if f e r s greatly from that computed by Bondybey et a l . ^ ^ and Michels et a l . ^ ^ for H + F” . Even though H + F“ may d i f f e r from H + Cl” we now know that the way Bondybey et a l . carried out that computation was in c o rre c t. For R0 less than the R where HF" crosses HF, no recognition was made of the non-stationary nature of the problem. Furthermore, i t seems that the basis set chosen by Bondybey et a l . was not f l e x i b le enough to describe the HF” system c o rre c tly in that i t was chosen to l i m i t his ca lcu la tio n to the diabatic picture e x p l i c i t l y om itting long-range binding in the dip olar f i e l d . Weiss and Krauss^*^ as well as (49) 2 + Hartmann et al. ' found the lowest E of HF to be s l i g h t l y a t t r a c t iv e into the autodetaching region and t h e ir calcu la tio n agrees q u a l i t a t i v e l y with our lowest HC1" curve. More im portantly, because a ll the above computations on HF must have basis set deficiencie s, J o r d a n ^ 1 ^ has computed an HF” electron a f f i n i t y which is -4 greater than 10 eV. This f i t s well with Crawford's (9) p re d ic tio n ' that HF must be e le c t r o n ic a lly stable, since the dipole moment of HF of 1.82 D is larger (as opposed to HC1 where i t is smaller) than a c r i t i c a l dipole fo r binding of an electron to a dipole f i e l d . HF" w i l l not be found experimentally according to Jordan since i t s zero-point ro ta tio n al constant is greater than its e le ctro n ic binding which causes a tota l HF” energy 95 higher than that of HF. I t must be pointed out that some of the states d is cussed here represent resonances due to the a t t r a c tio n of an electron by the dip ola r f i e l d of HC1. This e ffe c t gives rise to a series of states of each symmetry in the region of Rg which lead to the l i m i t of H + Cl + e“ . This series w i l l generally be characterized by an i n creasingly diffu se character of the resonant electron wave function with increasing energy. In Figure 3.1, we have, fo r the purpose of c l a r i t y , only depicted those three states of this nature which appear to be physically important to the in te r p r e ta t io n of the available experi mental data on this system. D. Conclusion The c alcu la tion of the potential energy surfaces presented in Figure 3.1 are in close agreement with the experimental observations detailed in the i n t r o duction with deviations from experiment amounting to no more than a few tenths of an eV. I t would therefore appear that the computational methods which we have employed were f u l l y e ff e c tiv e as applied to this problem and that the resonances which we have calculated corres pond to physical r e a l i t y . The presence of a permanent dipole moment in HC1 gives ris e to resonances which cannot he in tu ite d from the separated atom lim it s and i t 96 would appear that calculations of this type are essential to the in te rp re ta tio n of electron scattering from polar molecules. In ad d ition , this C.I. approach is accurate and e f f i c i e n t enough .for future studies of scattering in the f i e l d of the dipole. I t would be in te re s tin g to carry out a n - i n i t i o calculations on HF" and H^O" which have dipoles above the " c r i t i c a l " dipole defined pre viously. One would e-xpect to calculate bound negative ion states fo r some of these resonances. Since th is C.I. method is extremely e f f i c i e n t much can be learned about the potential energy surfaces of resonance states in the dipole f i e l d and valuable comparisons may be carried out between these C.I. method and the more tedious scattering calcu la tion s. 97 £;. References 1. S.W. Massey, Electronic and Ionic Impact Phenomena, 2nd ed. (Clarendon, Oxford, 1969), Vol. I I . 2. F.C. Fehsenfeld, E.E. Fergeson, and A .I. Schmeltekopf, J. Chem. Phys. 45, 1844 (1966). 3. L.G. Christophorou, Atomic and Molecular Radiation Physics (W iley-Intersciences, Lon don, 1971). 4. L. SanChe and G.J. Schulz, J. Chem. Phys. 58^, 479 ( 1973 ) . 5. C.E. Kuyatt and J.A. Simpson, Rev. S c i. In s t. 38, 103 (1967); R.I. H a ll, A. Chutjian and S. Trajmar, J. Phys. B. 6, L264 (1 973). 6 . C.K.N. Patel, App. Phys. Le tt. 6_, 1 2 (1965); Lasers, ed. A.K. Levine (1968); P.K. Tien, D. McNair, H.L. Hodges, Phys. Rev. Le tt. 1_2 , 30 (1 964). 7. See for example: D.C. Cartwright, W.J. Hunt, W. Williams, S. Trajamar.and W.A. Goddard, Phys. Rev. A, 8 , 2436 (1973) . 8. G.J. Schulz, Reviews of Mod. Phys. 4_5, 378 (1973). 9. R.F. W allis, R. Herman a n d H .W .M iln e s , J. Mol. Spectrosc. 4, 51.(1960); Crawford, O.H., Dalgarno, A. and Hays, P.B., Molec. Phys. ]_3 , 1 81 (1 967); Crawford, O.H., Proc. Phys. Soc. 91_, 279 (1967). 10. R.S. N a rc is i, B u ll. Amer. Phys. Soc. J_5, 578 (1970). 11. C. Ergensoy, G.H. Vineyard and A. Englert, Phys. Rev. 133, A595 (1964) . 12. J. Friedel, M e ta llic Solid S o lu tio n s , edited by J. Friedel and A. Guinier, 1963, New York: W.A. Benjamin Inc. 13. A. Burgess, Astrophys. J. 139, 776 (1964). 14. Atomic and Molecular Processes, edited by D.R. Bates, 1963, Academic Press, New York and London. 15. P. Auger, J. Phys. Radium £, 205 (1925). 16. P.G. Kruger, Phys. Rev. 3£, 855 (1930). 17. H.S.W. Massey, Negative Ions, 2nd e d itio n , Cambridge, University Press, London and New York (1950). 18. G.J. Schulz, Rev. of Modern Phys. £5, 423 (1973). 19. G.J. Schulz, Rev. of Modern Phys. 45, 210 (1973). 20. D. Spence and T. Noguchi, J. Chem. Phys. £3,505 (1978). 21. F. Figuet-Fayard , J. Phys, B: Atom. Mol. Phys. 7, 810 (1974a); Vacuum, 24 , 533 (,1974b). 22. J.P. Z-i.esel , I. Nenner and G.J. Schulz, J.. Chem. Plly s * '§1? 1943 Cl 975). 23. R. Abouaf and D. T e i1l e t - B i 1l y , J. Phys. B: Atom. Molec. Phys. £0 , 2261 ( 1 977). 24. R. Azria, L. Roussier, T. Paineau, M. Tronc, Rev. Phys. Appl . £, 469 ( 19.74), 25. C.J. Howard, F.C. Fehsenfeld. and M. McFarland, J. Chem. Phys. £0, 5068 (1974). 26. K. Rohr, F. Linder, J. Phys. B: Atom. Molec. Phys. 9, 2 521 (1976). 99 27. P. Burrow.,: J. Phys. B: Atom. Molec. Phys. 7 _ , L385 ( 1974) . 28. 0. H. Crawford, B.J.D. Koch, J. Chem. Phys. 1JL» 4512 (1974). 29., H.S. Taylor and A. Hazi , Phys. Rev. A, 1_4, 2071 T (1976); H.S. Taylor, J.K. Williams and I. Eliezer, J. Chem. Phys. 47, 2165 (1967). 30. S. Huzinaga, J. Chem. Phys. 5£, 1371 (1969). 31. T.H. Dunning and P.J. Hays, "Modern Theoretical Chemistry," Vol. 3, H.F. Schaefer, ed., Plenum Press, New York (1977). 32. A.D. McLean; and M. Yoshimine, "Tables of Linear Molecular Wave Functions", Supplement to IBM Journal of Research and Development (1967). 33. R.S. B erry: and C.W. Riemann, J. Chem. Phys. 38, 1540 (1963). 34. G. Herzberg, "Spectra of Diatomic Molecules", Van Norstrand, New York (1950). 35. L.E. Sutton,.ed, "Interatom ic Distances", Special Publication No*. 11, The Chemical Society, London (1958). 36. H.S. Taylor, E. Goldstein, ,G.A. Segal, J. Phys. B: Atom, and Molec. Phys. 37. J.T. Dowell and T.E. Sharp, Phys. Rev. 166, 124 (1968). 100 38. M. Tronc, F. Fiquet-Fayard, C. Schermann and T .I. H all, J. Phys. B: Atom. Molec. Phys. ]_0, 305 ( 1977). 39. H.S. Taylor, Adv. Chem. Phys. 1_8, 91 ( 1970). 40. F.J. Davis, R.N. Compton: and D.R. Nelson, J. Chem. Phys. 59, 2324 (1973). 41. F. Mandl , Proc. Phys. Soc. '90, 913 (1 967). 42. L. Dube and A. Herzberg, Phys. Rev. Le tt. 3J3, 820 (1977). 43. J.M. Levy-Leblond, Phys. Rev. J_53 , 1 (1967). 44. L.D. Landau and E.M. L i f s c h i t z , Quantum Mechanics (Pergamon Press, London), 1958, sec. 35. 45. K.D. Jordan and W. Luken, J. Chem. Phys. 64, 2760 (1976),. 46. V. Bondybey, P.K. Pearson and H.F. Schaefer, J. Chem. PhysV '57, 1 1 23 (1972). 47. H.H. Michels, R.E. Harris and J.C. Browne, J. Chem. Phys. 48, 2421 (1968). 48. A.W. Weiss and M. Krauss, J. Chem. Phys. 52^, 4363 (1 970.). 49. W.N. Hartmann, T.L. G ilb e rt, K.A. Kaiser and A. C. Wahl, Phys. Rev. B, 2 , 1 140 (1 970). 50. K.D. Jordan and J.J, Wendoloski, J. Chem. Phys. 66, 4968 (19.77). 101 Table 3.1. Gaussian Basis Set Chiorine Hydrogen [8s] «s [2s] 40850.0 0.002532 13.36 0.032828 6179.0 0.019207 2.013 0.231204 1425.0 0.095257 0.4538 0.817226 409.2 0.345589 0.1233 1.000000 135.5 0.129056 . 50.13 0.648511 20.21 0.275487 S [lp] 6.283 1.000000 1.000000 1.000000 ' 2.460 1.000000 0.5271 1.000000 0.1884 1.000000 0.023 1.000000 0.004 1.000000 S [7p] 240.8 0.014595 56.56- 0.099047 17.85 0.330462 6.350 0.682874 6.350 0.561785 2.403 1.351901 0.6410 1.000000 0.1838 1.000000 0.0490 1.000000 0.0180 1.000000 0.0050 1.000000 102 Table 3.2. SCF Results fo r H C 1 ( r = 1.2744 A) M.O. Symmetry Eigenvalue (a.u.) 1 < -104.8584 2 z! - 10.5847 3 z+ - 8.0500 4 n - 8.0472 5 - 8.0472 6 - 1.1284 7 z+ - 0.6213 8 n - 0.4792 9 n. - 0.4792 10 i + 0.0050 1 1 n 0.0198 12 n+ 0.0198 13 2 . 0.0122 14 E 0.0491 15 n 0.0570 16 0.0570 17 E, 0.0650 18 E, 0.1850 19 E 0.2066 20 n 0.2206 21 1 1 1 0.2206 22 0.7375 23 E 0.9426 24 E 1.0329 25 n' 1.0769 26 n 1.0769 27 n 1.9756 28 1.9756 29 E 2.3637 30 n 6.6193 31 n+ 6.6193 32 z! 6.7652 33 j* 9.4528 34 223.4754 Total Energy: -460.0518 a.u. Electronic Energy : -467.0941 a.u. Nuclear Energy: 7.0423 a.u. 103 Table 3.3. Effect of Truncation Upon the C.I. Calculations Full C .I. (ev) Trucated C .I. (ev) Haa " 50 H , = 20, H\ only aa 5 ab O HC1. ( r = l.2744 A) -12518.484 -12518.484 HC1" ( r = l.2744 A) -12518.340 -12518.361 Cl" -12504.017 -12504.051 Cl -12500.727 -12500.919 E.A. (Cl) 3.3 3.1 De (HCl) 4.1 3.9 104 Table 3.4. Calculated C.1. Energy P o in ts ^ (Adiabati c) 0.9 A 1.2744 A 1.5 A 1.7 A 2.0 A 3.0 A C O H C 1 ¥ 5.12 0.00 0.55 1.46 2.46 3.98 .4,16 hcT i V 5,26 0.12 0.58 1.11 0.95 0.99 0.83 22E+ 5,45 0.31 0.83 — 2.69 — — 32Z+ 8,03 4.38 3.26 2.97 2.59 — 4V ( b ) 5.17 6.78 — 5.50 3.58 3.38 2n(b) 9.28 7.78 — 5.8 3.62 3.38 52£+(b) 9.56 7.01 -- 7.78 — -- (a) (b) Relative C.I. of to HC1 at equilibrium , E = 12518.484 ev. 75 single excitation only plus correlation correction as noted in te xt. 105 Table 3.5. Summary o f Calculated Results (ev) Observed ■ k Calculated 2 + H C 1 1 l > 0.7 ev 0.12 2 + H C 1 3 £ 2.6-3.0 2.6 2 + H C 1 4 I 6.9 6.78 HCl" 2n 9.2 9.28 CT + H 0.86 0.83 Cl + H" 3.68 3.39 Electron a f f in it y Cl 3.63 3.3 Dissociation Energy HCl 4.43 4.1 * Relative to HCl at equilibrium , E = 12518.484 ev 106 Figure 3.1. Calculated potential energy curves for HCl and HCl". 107 E(eV) .HCR4 Z ) H~+ Cl HCl ( ' I ) H+cr oo 2.0 2.2 2.4 2.6 2.8 3.0 1 .4 o Figure 3.2 Radial d is t r ib u t io n function fo r 2 + the scattered electron in the 1 E state (r=1.2744 A) for the basis set of Table 3.1 ■ (I), I plus a = .001 s type gaussian ( I I ) , and II plus a = .001 p type gaussian function. 109 .02 V .02 .02 20 24 24 20 C l-H o Figure 3.3. Radi a.i. d is t r ib u t io n function fo r the 3 V state ( r = l .2 744 A) for the basis set of Table 3.1 ( I ) , I plus a = .001 s type gaussian ( I I ) , and I I plus a = .001 p type gaussian function. 111 cm C O C M O O C M C D O r o c m K > CM tO C M o o o l/hzJ ' / 11 2 Figure 3.4. Radial d i s t r ib u t io n function fo r 3 + the scattered electron in the 3' E state (r=1.2744 A) for the basis set of Table 3.1 ( I ) , I plus a = .001 s type gaussian ( I I ) , and II plus a = .001 p type gaussian f u n c tio n . 113 20 24 Cl-H r (A) Figure 3. Radial d is t r ib u t io n function f o r the scattered electron in a root 2 + intermediate in energy between 2 E and 3 ^E+ state ( r = l . 27.44 A) fo r the basis set of Table 3.1 ( I ) , I plus a = .001 s type gaussian ( I I ) , and II plus a = .001 p type gaussian function. 115 91L .04 .03 .02 04 03 C \J 02 C J 04 03 .02 24 20 "8 20 24 Cl-H 1 ' CHAPTER IV CYCLOPROPANE A. Preliminaries i) Introduction The in te rp r e ta tio n of the unusual physical and chemical properties of cyclopropane has presented a challenge to experimental and the oretical chemists for many years. Of special in te re s t is that fact that cyclo propane undergoes addition in the presence of bromine whereas cyclobutane and higher homologues y ie ld s u b s titu tion products. S im ila rly , cyclopropane reacts with HBr to y ie ld propyl bromide while the higher homologues do not react at all . ^ ^ This presents some evidence that cyclopropane behaves more lik e an o le fin than a saturated alkane. The a b i l i t y of cyclopropane to p a rtic ip a te in con jugation is evidenced in systems where the cyclopropyl ring is attached to a carbonyl group and behaves lik e ( 2 ) a, 3 unsaturated carbonyl systems. Strong evidence in favor of conjugation are the observed spectroscopic (3) s h if t s to lower frequency upon s u b s titu tio n . The far U.V. absorption spectrum starts at wavelengths very much longer than those at which typical a C-C electrons absorb, in d ic a tin g that the electrons in cyclopropane are less t i g h t l y bound than in the regular alkanes. Clearly then,. elu cidation of the U.V. spectrum of cyclopropane is of great in te r e s t, i f only to better understand the chemical behaviour of this molecule. Cyclopropane is the smallest c y c lic hydrocarbon and i t s three-membered ring is known to have considerable s te r ic s tr a in . This s tra in f a c i l i t a t e s the opening of the ring to e ith e r nucleo p hilic or e le c t r o p h i1ic reagents. (4 5) Early the oretical studies * have been focused on understanding the charge d i s t r ib u t i o n and reaction mechanisms of this molecule. L i t t l e work has been done on it s excited e le ctro n ic states. Recent in te re s t in cyclopropane has assumed a new ( 6) d ire c tio n . This in te re s t is d i r e c t l y related to the growth of m u ltip le IR photon e x c ita tio n processes. I.R. laser chemistry is unique since the energy can be deposited and maintained in a p a r tic u la r v ib ra tio n a l mode. A selective reaction involving this mode may then be induced. Cyclopropane is especially suited fo r these investigatio ns since energy deposited into d if f e r e n t vib ra tio n a l modes yie lds d if f e r e n t products and may unravel a wealth of information about reaction mechanisms. This work concentrates on studying the e le ctro n ic spectrum of cyclopropane. I t is best to begin with a simple, q u a lit a t iv e description of the bonding in this molecule. A model for bonding in cyclopropane must be consistent 118 with the observations described above as well as a H-C-H bond angle of 1 1 8 ° . ^ I f the carbons in cyclopropane 2 are assumed to approach sp h y b rid iz a tio n , the carbon atoms are expected to resemble ethylene which explains why these two molecules have s im ila r conjugation proper ties and spectroscopic s h i f t s . 2 Two of the sp hybrids on each carbon are assumed to be directed towards the hydrogens and form a localized 2 C-H bond. The th ird sp hybrid is directed into the center of the ring to form the in-plane " in te r n a l" ( 8 ) delocalized system. The internal system (see Figure 4.1a), gives rise to molecular o r b ita ls a-j , and the degenerate set e ‘ . The remaining p o r b ita l on each carbon is perpendicular to the bisector as in Figure 4.1b, forming an external system. This consists of a£ (out of phase) and a set of e' bonding o r b i t a l s , w ith in the D3h symmetry of the molecule. As a re s u lt of this basic o r b ita l s tru c tu re , early attempts at theoretical calculations of the excited state spectrum encountered an extremely dense manifold of e le ctro n ic states. This is re flected in the absorption experiments on cyclopropane, where extremely broad and overlapping bands are observed. A b r ie f summary of experimental and theo re tica l work performed on cyclopropane f o l 1ows. 119 i i ) Summary of Experimental and Theoretical Work The absorption spectrum of cyclopropane consists of ( 9) three broad bands. 1 The f i r s t band extends from 60,000 to 65,000 cm- ^ (7.44-8.18 eV) with a maximum at 63,000 cm~^ (7.8 eV) and an observed o s c i l l a t o r strength of 0.12. The second band extends from 68,500-70 ,150 cm- ^ ( 8.5-8.7 eV) with o s c i l l a t o r strength of 0.04 and a maximum at 70,000 cm~^ ( 8.68 eV). F in a lly the t h ir d and most intense band ( f = 0.7) has a maximum at 83,000 cm~^ (10.3 eV) and extends over an energy region 9.2-10.5 eV. The f i r s t and th ir d of these bands have discrete structure superimposed on the broad background. The lowest band disappears e n t ir e ly in the solid phase spectrum whereas the sharp structure on the t h ir d band also disappears confirming the Rydberg character of these tra n s itio n s . A weak band observed at 80,000 cm- ^ by S a n d o r f y ^ ^ may also _ ] belong to a Rydberg series. The bands at 70,000 cm (8.7 eV) and 83,000 cm” ^ (10.29 eV) are id e n t i f i e d as valence tra n s itio n s on the basis of comparing gas-phase to condensed phase spectra. I t must be emphasized however that the lower energy t r a n s it io n (8.7 eV) may have e ith er Rydberg or valence character since the Condensed phase studies on this band are by no means conclusive. F in a lly , a very weak system composed of twelve, sharp bands was observed between 6.56 and 7.1 e V . ^ ’ ^ ^ The integrated - 4 in te n s ity of this system was found to be 10 of the 1 20 in te n s ity of the other e le c tro n ic t r a n s itio n s , c le a rly belonging to e ith e r a spin or symmetry forbidden t r a n s i tio n . A considerable amount of additional information about the excited states of this molecule can be obtained from the magnetic c ir c u la r dichroism (MCD) spectrum since the ubiquity of degenerate excited states renders cyclo propane a p a r t ic u la r ly favorable subject fo r study by this technique. This spectrum has been recorded by (13) Gedanken and Schnepp. The nuclear motion averaged dichroism exhibited by a molecule fo r e x c ita tio n from a non-degenerate ground state, as in cyclopropane, A, to an excited state, J under the influence of a magnetic f i e l d , is to a reasonable (14) approximation, given by' ' — = -1.002xl0 ' 2 C-Af’ + Bf°] [1] v wh ere A(A-J) = - \ I , I m{<A|m|JA>x<JA, |m|A>[<JA| y |J A ,>] . A , A [ 2 ] \ <A | m j JA>x< JA | m | K|<> •< | y | A>/Ru)ka + I <A | m | Ja>x<Kj< | m | A> •< JA | y | K^>/Ro)Kj [3] 121 The subscripts A* k denote the ind ivid ual members of the possibly degenerate excited states, m is the dipole moment, y the magnetic dipole moment and f° is a lin e shape fu nction , while f 1 is its de riva tive with respect to frequency. The A(A+J) term arises from the f i r s t order Zeeman s p l i t t i n g of the members of a degenerate excited state and vanishes i f the excited state possesses no magnetic moment. An important feature of this term is that i t appears m u ltip lie d by the f i r s t d e riv a tiv e of the line shape function with respect to frequency. Therefore, the appearance of dispersive lin e shapes in the MCD of cyclopropane is diagnostic for a degenerate excited state. Whereas the A-term exists only fo r degenerate states, the B-term is generally present fo r a ll tra n s itio n s and is due to a second order Zeeman e ff e c t. The B term arises from the mixing of excited states with each other through the perturbation of the magnetic f i e l d . In eq. 3, the i n f i n i t e sum runs over a ll e le ctro n ic states. Therefore, unless very few states dominate the mixing process, usually fo r the reason that they be very close to each other in energy, the oretical analysis through, standard perturbation theory w i l l not be possible. Fortunately, the f i r s t sum in the B term w i l l generally be small r e la t iv e to A terms due to the denominators I f , however, other excited states 122 l i e close to the state of in te re s t J and are magnetically coupled to i t as well as e l e c t r i c a l l y coupled to the ground state , the second term in eq. 3 can lead to sizeable B terms. This, as we shall see, is c e rta in ly the case of cyclopropane. The observed MCD spectrum in the 175-135 nm region is shown in Figure 4.2a. I t exhibits two broad bands, the f i r s t of which appears to represent a strong A term centered about 7.87 eV and the second a B term with maximum dichroism at 8.63 eV. A high resolution study of the f i r s t band reveals that troughs in the MCD corres pond to peaks in the absorption. Therefore there is strong evidence fo r the existence of two tra n s itio n s whose MCD have the opposite signs. Gedanken and Schnepp have f i t this spectrum using moment analysis and gaussian lin e shapes as the sum of two A and two B terms and attempted assignments on the basis of this two state analysis. Moment analysis has been applied to many simple spectra and seems to y ie ld valuable information. However, when the bands are broad and overlapping as in cyclopro-: pane, problems may arise due to the model dependence of the method. I f a ll the co n trib u tin g tra n s itio n s are not included in calculations of the parameters, moment analysis may y ie ld erroneous re sults. 123 To augment the information obtained from the MCD and absorption spectrum, the o s c i l l a t o r strengths and t h e o r e t ic a lly calculated. The o s c i l l a t o r strengths iy\y ( f } the theoretical measure of in te n s ity obtained from the tr a n s itio n moment. The three forms for this " f" value are: f ( r ) , f (V) and f ( r * V ) . The dipole length formula, f ( r ) is f ( r ) - f AE!<*0 |m|,,e)<c. te d > |2 where m is the e l e c t r i c dipole operator m = e I r i and r^ are the ele ctron 's coordinates, iji .(ground state) and 4^ are the wave functions with eigenvalues r e.xcited 3 En and E„v , so that AE is (ErtV .. , - E ). 0 excited excited o Because of the re la tio n s h ip [H ,r] = V, ^ I <^0 Ir l^exci ted> I = 3AE I <^0 1 V1 ^exci ted> I * This is known as the dipole v e lo c ity formula. The "mixed form" fo r f, f ( r * V ) = |- <ib I r I xb .. I V I . .> ' 3 o' '^excited r o' 1 excited has the advantage of being less sensitive to correla tio n errors in e ith e r ground or excited state than the dipole v e lo c ity or dipole length formula since energy does not appear. 124 While much of the previous the o re tical work attempts the calc u la tio n of the eigenvalues and o s c i l l a t o r strengths of these t r a n s itio n s , there has been no previous c alc u la tion of the magnetic moments and MCD parameters fo r this molecule. This.would involve calculations of the matrix elements of the wavefunction over the magnetic moment operator: r X V : fo r example in the x d ire c tio n : ( r * V ) x = Mx = < \ | y £ - ^ l \ , xcued> Theoretical studies previously performed have trie d to c l a r i f y the spectrum of cyclopropane. Probably the best of these is the early work of Buenker and Peyerim- (15) h off. By use of a b - i n i t i o SCF and lim ite d Cl, they study the spectrum of cyclopropane but neglect to include Rydberg functions in t h e ir basis set, concluding that the m ajority of observed bands in cyclopropane are due to non v e r tic a l t r a n s itio n s . I t must, however, be said that the exclusion of Rydberg functions probably led them to fundamentally flawed re s u lts . Their findings indicate that most of the higher e le c tro n ic states posses a wide angle geometry r a d ic a lly d i f f e r e n t from the near 50° C-C-C angle found at the equi1ibriurn geometry. Semi-empi ri cal methods have been used by Meyer^®- and C l a r k ^ 7^ with lim ite d success.. More recently, (18) Meyer et a l . augmented the CNDO/2 calculations with an a ll valence Cl and achieve good results fo r the f i r s t band. D.T. Clark a c tu a lly finds seven s in g le t - s in g le t tra n s itio n s in the energy range 6.6-9.7-eV. He f a i l s to explain the lowest and highest energy bands. His assignment of the weak system at 6.7 eV to A.j + Aj, ( i . e . symmetry forbidden) c o n f lic t s with the spin-forbidden t r a n s itio n A-j-^E' assigned by Meyer^*^ and Brown and K r i s h n a . ^ ^ F in a lly , Basch Robin et al. used SCF calculations (with no Rydberg functions) and repeatedly fou n d ,th eir states to be 15,000-25,000 cm- ^ higher than experiment. The idea that theo re tical studies without Rydbergs are unable to explain spectra is supported by the case of the ethylene molecule. Exclusion of the Rydbergs there results in erroneous results and diffu se o r b it a ls are absolutely necessary to properly describe the ele ctro n ic s tructure of this o l e f i n , even fo r the lowest energy s in g le t e_xcited state. I t can be said in general that the assignment of the excited states for cyclopropane is complicated by the broad and overlapping band stru cture s. Serious ambiguities e.xist when experimental and theoretical o s c i l l a t o r strengths are compared fo r a ll three bands. Therefore, we have carried out extensive a b - i n i t i o configuration in te ra c tio n calculations directed towards the elucidation of the spectra of cyclopropane are performed. By using accurate wave functions, the eigenvalues, o s c i l l a t o r 126 strengths and magnetic moment are calculated. An ab i n i t i o MCD spectrum is also calculated and is shown to be in close agreement with the experimental plo t. The next section d e ta ils the computational techniques. B. Computational Details (19) The geometry of cyclopropane was assumed to be of D ^ symmetry with r ^ = 1.5138 A, r ^ = 1 .0820 A, <CCC = 60° and <<HCH = 116.2°. The coordinate system used is shown in Figure 4.1c. The A.O. basis chosen was a (9s,6p) carbon basis contracted to [4s,2p] as defined by Dunning. Dunning's (5s) hydrogen basis contracted to a single func tion was used with a scale factor of 1.414. A set of s,p and d single gaussian functions of exponent 0.02 were set at the o r ig in of the coordinate system at the center of the molecule. I t is therefore to be expected that these calculations should be accurate through n = 3 Rydberg states. The SCF o r b ita l energies are shown in Table 4,1 . This basis set generates 46 molecular o r b i t a l s . Of these, 40 were considered in the configuration in te ra c tio n c a lc u la tio n ; the three carbon Is molecular o r b it a ls held as a frozen core and the three highest energy v ir tu a l o r b i t a l s , t h e ir antibonding complement, were ignored. 127 The configuration in te ra c tio n c a lc u la tio n s , which range up to a Cl basis of 32,542 spin eigenfunctions, were carried out using the perturbational configuration in te ra c tio n techniques which we have described else- (2 1 22 23) where. ’ ’ ■ The m ajo rity of the configuration in te ra c tio n calculations which we report here were carried out by generating a ll single and double hole p a r tic le e xcita tions of the proper symmetry from 10 base configurations belonging to the same irre d u c ib le repre- sentation. The excited states IE ', 2E', 3 E ' , 4 E1 and 5E’ , for instance, were generated in a single c a lc u la tio n , the appropriate base configuration for a ll of them being included in the generating set. This procedure implies a very large configuration space, but guarantees that the calculated state energies represent proper upper hounds. Adoption of th is approach, with our present computer programs, forced us to r e s t r i c t the molecular o r b ita ls u t i l i z e d in these calculations to the space between 3e( and 8e ', MO ! s 11 to 34. These re sults were then checked by carrying out s im ila r calculations over the f u l l M O space fo r these states, taken one at a time an.d generating. a ll single and double hole p a r t ic le e x c i tatio n from only the 2-4 important contributions fo r any p a rtic u la r state. This procedure samples a d iffe r e n t portion of the f u l l Cl space, presumably including a 128 larg er portion of the space opposite to a p a rtic u la r state, but is on shakier grounds with respect to upper bound r e s t r ic t io n s . In any case, l i t t l e difference was found between the two sets of calculations and we have preferred to report the results of the smaller M O space calculations as the calculations of preference. The perturbational Cl approach employed u t i l i z e s a 50 dimensional zero order function. This is a fixed dimension in present computer codes, although a second, more f l e x i b l e code is presently being w r itte n . A zero order function of 50 terms cannot acceptably span the E' space in cyclopropane beyond 6 E *. W e therefore, report results from the second type o f c a lc u la tio n , i . e . a single state computation, for the states reported to be found above 10 eV. In our view, these few results are not on as firm ground as the states below this demarcation lin e . In these c a lc u la tio n s , the 50 dimensional zero order function comprised about 95% of the modulus squared weight of the wayefunction. The perturbational correction was therefore about 5% of the total eigenfunction fo r each state. Since this was so, a ll re q u is ite matrix elements o s c tla to r strengths and magnetic moments were computed using only the 50 dimensional portions of the wavefunction , om itting the perturbati onal t a i l . I t is es ti.matdd ;;that this procedure, while obviously cost e f f i c i e n t , w i l l 129 en tail no more than a 10% e rro r in our computed values, a leyel of accuracy f u l l y consistent with the accuracy of the experimental data and other errors in the c alc u la tion . The calculated y e r tic a l tr a n s itio n energies and o s c i l l a t o r strengths for cyclopropane are given in Table 4. 2 „ an d the nature of these states is reported in Table 4.3. As w i l l be discussed in the next section, these results are strongly at variance with the assignments of the MCD in reference 9, d i f f e r i n g p r in c ip a lly in that fa r more states are found to be of important to the MCD than were assumed by Gedanken and Schnepp. in view of this difference and because i t is not obvious, on inspection, that the states in Table 4.2 w i l l in te rfe r e to y ie ld the observed spectrum, we have chosen to reconstruct the actual spectrum. Before proceeding with a detailed description' of the calculated spectrum, the method of moments and all relevant MCD parameters w i l l be discussed. Following (14) the notation of Stephens, Mowery and Shatz, - the method of moments involves the integrated properties of absorption and MCD bands. The n moment of the molar e x tin c tio n c o e f f ic ie n t e, fo r a t ra n s itio n A+J, is given by 14] 1 30' where v° is the mean absorption frequency. From equation (4), the zeroth moment fo r absorption about v° is < E > 0 " / v dV t 5 ] and - = 1.089x10^ D f° [ 6] v o L J where Dq is the dipole strength, D = if3 <A | m | J >*<J, |m |A> [ 6a] O A A In the case of cyclopropane the matrix element <A|m|J> = m, 2 becomes, fo r a degenerate state, Dq = 2m , with units 2 of debye „ f° in equation [ 6] is the normalized band shape Jf° dv = 1 Therefore equation [5] s im p lifie s to <e> = ■ f 1 .089x102 D_ f°dv = 1 .089.X102 Dn 1 7J 0 J o 0 S im ila rly , for the MCD band, i f Ac is defined as the di. f ference Ae = £l e f t " £ri ght then 131 <Ae>n > dv (v - w ° !n [ 8] I t should be noted here that a frequently used e xp e ri mental parameter is the molecular e l l i p t i c ! t y - per gauss, EeJ . This q u a n tity may be related to Ac by [e]m = m 3300 W e prefer to use.Ac since this is the quantity which w i l l provide the comparison between theory and experiment. For the zeroth moment in MCD, <Ae> = dv £9] 0 J V and <Ae> * -1.0Q2.xl0~2 B £10] o o where Bq ) is given by equation £3], with units 2 -1 3 dehye /era , where 3 is bohr magneton. For the f i r s t moment in MCD, we have <Ae>-j = -1 o002xl0" 2 {A1 + [11] where A-j is defined in equation £ 2] and has units of 2 3 debye . where - x ,,t relates to energy differences between ground AA and excited state poten tial surfaces. 132 I f v° is defined so that <e (A->-J) >-j = 0 [ 12] then the f i r s t moment fo r dipole strength becomes, D1 = v° Dq(A-*J) and therefore, BAA ' = B dXA' [13] which means that B is independent of x ^ ■ and Bl =-v° Bg [14] Equation [10] may therefore be s im p lifie d to <Ae>] = -1 .002xl0" 2 A-, [15] These experimental parameters are then used to find the ra tio s , A-j/Dq = -1.072xl04 < A c > i / < e > q and B g / D g = -1.072x10^ < A e > g / < c > Q F in a lly we may express the difference in l e f t and r ig h t e x tin c tio n c o e ffic ie n ts as a sum of the f i r s t and second moments as in equation [ 1] . 133 The values which are extracted from the moment analysis must be related to the th e o r e t ic a lly calculated values. The quantity which may be re adily computed is the magnetic moment of a t r a n s i t i o n , Ai ' D o - - 7 l , lm C’ 6] A jA where p is the magnetic moment operator defined on page 125. The dipole strength, Dg is simply found by squaring the t r a n s itio n moment and summing over degenera cies (equation [ 6a ]) . I t is therefore possible to th e o r e t ic a lly compute A-j , Dg or t h e ir quotient, A-j/Dg and compare these to experimental values. S im ila rly , the the o re tical value fo r the Bg term is found from equation [ 3 ] . These values are used in p lo ttin g the MCD spectrum by taking th e ir product with the appropriate lin e shape. The lin e shapes f ' and f° are described using Gaussian band shapes: f o = 1,. exp £_ (v-x.Q) 2/A 2] [17] 7 T ’ A and fg 5 5 12(.v0-v)/Tr1^ 2A3] exp - ( v - v q) 2/A 2 with f f ° dv = 1 and Jfg vdy = - 1. 1 34 Calculation of the Bq term become necessary when i t is of the same order of magnitude as A -^ since both these terms are additive in t h e i r con tribu tion to Ae. A rough estimate fo r a comparison between these two terms is provided by (BQ x A v ^ 2)> where Av^ 2 i s tlie band half width at h a lf height. In the case of cyclopropane fo r (13) example, the BQ term extracted from moment analysis -4 2 - 1 fo r the f i r s t band is 6x 10 3 debye /cm , the band h a lf width at h a lf height is 1 500 cm-1 and the A-j term 2 is found to be 2.22 3 debye . The B term is of the same order of magnitude as the A term which indicates the necessity of carrying out a calcu la tion which includes th is second order term. I t is clear from Figure 4.2c and 4.2d,which show the calculated A and B terms separately, that the B term dominates the spectrum, especially when the states l i e closely together. Q u a lita tiv e ly , the B term in cyclopropane is possible i f an A^Cz) dipole allowed state, lie s close to an E ' ; (x,y) allowed state. These two excited states may be cojpled by the magnetic f i e l d (y , y ) to give a sizeable * y c o n tribu tion to the second term in equation [3 ], The coupling of ^A£ with 3E‘ in the region 8.7 eV dominates the MCD in that region, for example. The coupling of the A£ state with other E' states is found to be due to the larger energy separations, i . e . the term increases. 135 I t is found in c alculating the B term that the f i r s t term in the summation is r e l a t i v e l y small since the denominator involves energy differences of more than 7 eV and the term Rw^ reduces the size of the f i r s t term s i g n i f i c a n t l y . For example, the magnetic dipole allowed E"(xz,yz) state must be considered in 'th e f i r s t sum. Coupling via the e l e c t r i c dipole f i e l d with the E1 state, is allowed: <Aj|m|E' >x<E' |m|E"> *<EM|v|A|>/RwE , When matrix elements of this type are calcula ted, they are found to be nearly two orders of magnitude smaller than the second summation in the B term and may be neglected. Other possible contributions due to the second order perturbations are the interactio ns between the E' states themselves. Since E' is an e l e c t r i c dipole allowed state i t may couple through the p z component of the magnetic f i e l d to a neighboring E'. S ig n ific a n t B term contributions were found between IE' and 2 E' . The calcu lation s of the A term are more s tra ig h t forward and involve calculations of the degenerate state (_E'■) magnetic moments. I t should be noted that although, the magnitude of the magnetic moments is s ig n if ic a n t fo r the I E ', 3E1 and 5E' ex cited states, there is only a small con tribu tion of the A-| term to the ’ 136 total MCD. This is due to the weak dipole strength ODq) of these states. For lack of anything b e tte r, the band widths, A, were taken to be those found by Gedanken and Schnepp of the 157.5 nm and 144.5 nm bands, 3270 and 2000 cm” ^ r e s p e c t i v e l y . The procedure followed was to use the calculated wave functions fo r each state to calculate the re q uisite matrix elements over m and y. These were inserted, using computed energies , into equation £2] and,[3] to find (A+J) and B(A-*J). The sum over states in B(A-*J) was lim ite d to the states computed, i . e . those in Table 4.3, on the assumption that the B term, i f large, is dominated by those states close to J and magnetically coupled to i t . Under this r e s t r i c t i o n , computed B terms obey w ithin this subset of states, the sum rule that B terms must sum to zero. The re sultan t and B terms were f i t t e d to the lin e shape factors given by equation [7] and [ 8] with IE' and 2E1 having a width of 3270 cm"^ and all others having a width of 2000 c m " \ The resultan t terms were then summed at each f r e quency and the spectrum plo tted . The computed MCD is given in Figures 4 , 2b,c,d. I t must be emphasized however that these plots involve considerable interference of terms and are quite sensitive to the assumptions made as to lin e widths and tr a n s it io n frequencies. The figures displayed, which use computed frequencies and probably improper widths, are close to an optimal f i t to experiment, given our computed matrix elements. C. Discussion of Results The lowest frequency band of cyclopropane is extremely broad and appears both in absorption and MCD. Previous d i f f i c u l t i e s in assignment of this band were due to the general b e l ie f that only one t r a n s it io n was involved. Based on this assumption, the observed magnetic moment has a value A-j/Dq = 0.53 while that calculated ( 12) by use of simple LCAO techniques has A -j / Dq = 0.16. In addition to this discrepancy, the previously calculated o s c i l l a t o r strength fo r the f i r s t band was found to be ( 9 ) O.Q4v - which d if f e r s g re a tly from the observed value of 0. 12. All workers have agreed however that the f i r s t band should be assigned to an E' state. There are three experimental factors which must be considered here: the in te n s ity of 0.12, the MCD, A-j/Dq values of 0.53 (large) and the Rydberg nature of the who!e band. Our results explain a ll three observations. Th.e lowest t r a n s it io n , I E 1 involves e x cita tio n from the 3e' o r b ita l into the 3s Rydberg composed of a lin e a r 2 combination of s, dz and dx 2+y 2 diffu s e functions. Proof of the diffu se nature of this o r b ita l is provided ? 2 by the large radial extent <r > = 94.4 a.u . 138 This t r a n s it io n occurs at 7,61 eV and is quite weak but the MCD is very strong with a calculated value fo r A1/ Dq 0.57 (see Table 4.4) which is in very close agreement with the value of 0.53 observed from moment analysis. This is typical of A^/Dq values fo r t r a n s i tions involving one degenerate o r b ita l (3e!) and one nondegenerate o r b ita l (3s Rydberg). Assuming that the matrix elements are f a i r l y large fo r the t r a n s itio n 3e( 3s, then the resu 1 t i ng magnetic moment should be r e l a t i v e l y large. In e x c ita tio n s of the type e ' x e a 1 though the magnitude of the individual matrix elements, may be large, they often have opposite phases and cancel, re s u ltin g in smaller magnetic moments. The C.T. c o e ffic ie n ts fo r other states mixing with 1E! are shown in Table 4.3. The second t r a n s i t i o n , 2E* occurs at 8.11 eV and carries an o s c i l l a t o r strength of 0.12 accounting fo r the overall in t e n s it y of the f i r s t band. The 2E’ state is a Rydberg state invo lving e x c ita tion from 3e' to the 3p Rydberg. The radial extent of 2 the p and p o r b it a ls is proof of difuseness, <x > = -x - y 2 2 2 <y > = 43.3 and <r > = 72.6 au . The magnetic moment for th is state CA-j/Dq) is smaller than fo r the I E ', -0.172, as expected from an e ' x e 1 e x c ita tio n . Clearly then, band I is due to two t r a n s it io n s , both of Rydberg character, one invo lving the 3s and one the 3p Rydberg o r b i t a l s . In a d d itio n , the, perturbation between 1E! 139 and 2E1 is s ig n if ic a n t re s u ltin g in a p ositive B term fo r IE' and a negative one for 2E‘ . Our assignment also explains the o s c illa t io n s observed in the high resolution MCD and c le a rly this is due to two A terms of opposite sign con tribu ting to the same band. Furthermore, the low energy IE' state has a smaller A-j term than the 2E' state because of it s low Dq value. This explains the rather skewed A term in which the A-| value fo r I E 1 is close to h a lf of that of the 2E‘ state. The dipole forbidden t r a n s it io n s , 1 Aj and 1 A£ o r ig in a tin g from the lowest e'xe' e x c ita tio n (3e' 4e') are also very important in veryfing the nature of the f i r s t band.. The CD spectra of trans 1 ,2 -dimethylcyclo- propane shows a weak system near 200 nm which has been assigned to an e le c t r ic dipole and magnetic dipole fo r - (13) bidden t r a n s i t i o n . ' Since the methyl groups repre sent a r e l a t i v e l y minor perturbation of the C^Hg ring, the spectrum of t r a n s - 1, 2-dimethyl cyclopropane is expec ted to be s im ila r to that of C^Hg but with some s h ifts 1 1 to the red. Therefore the A -J and A£ states of cyclo propane may very well be those analogous to the lowest A -*■ A t r a n s it io n in the methylated molecule and our results are consistent with this assignment, although more the oretical work, must be carried out before this may be asserted with any c e r ta in ty . 140 The absorption spectrum shows the second band to span from 8.3-8.9 eV. I t appears that there is some doubt about the valence character of this state but i t is known to be dipole allowed with f - 0.04. Two previous assignments have been A|-V'E?, where e 1 valence o r b ita ls (9 ) are involved but the calculated in te n s ity of 0.4 is at variance with experimental observations. The other (13) assignment'' involves the dipole allowed A£ state. Moment analysis on th is band shows a very weak magnetic moment (A-|/Dq = 0.07), which is close to zero as may be expected of a non-degenerate state. However, Gedanken and Schnepp must also consider an e 'x e 1 tr a n s itio n which also may y ie ld a r e l a t i v e l y small magnetic moment. Our calculations find the f i r s t A^ state to l i e at 8.7 eV, with a weak in t e n s i t y , f = 0.007, invo lvin g a t r a n s itio n from 3e! -» • 2e" which is a 3dx z ’ 3c *yZ Rydberg 2 2 tr a n s itio n of radial extent <r > = 9 0 . 8 au . No s i g n i f i cant mixing with the higher valence A!) states is found as c le a rly shown in Table 4.3. This would id e n t i f y the principal c o n trib u to r to the MCD peak in this region (8.7 eV). as a B . term which arises from the close proximity to the VA£ state of the- next state, 3E' at 8.85 eV. The next excited A£ state has both Rydberg and valence character ( f = 0.241), mixing le" 4e'p and le" 5e! d„ ,d ~ 9 Rydbergs w i t h . l e " -» • 6e* valence., xy x^-y^ This ZA£ l i es at much higher energies, 10.58 eV. 141 Our ca lculations show a 3E' state of weak in te n s ity to l i e at 8.85 eV and th is state has a considerable A1/ Dq term, +0.582 which is of s im i1 a r - in t e n s it y as the f i r s t large A-j/Dq term c a lc u la te d .. This state contributes l i t t l e to the to ta l MCD of the middle band because i t has a weak in t e n s it y . The sums of the calculated o s c i l l a t o r strengths of 1A£ and 3E1 is at best 0.01 which is of the same order of magnitude as experiment (small) and implies a weak t r a n s it io n . Although the calculated MCD spectrum matches experiment closely fo r the second band, a question s t i l l remains as to the character of this t r a n s it io n . Our results c le a rly indicate a Rydberg tr a n s itio n while experimental data in the condensed phase hints at a valence t r a n s it io n . One possible explanation suggested by R o b in ^ 0^ is that the next band which is quite intense, dominates the spectrum under condensed phase conditions and masks the Rydberg character of this weak t r a n s it io n . The second p o s s i b i l i t y is that a portion of the in te n s ity of this band is due to some vibronic i nte racti on. The weak band system observed by Sandorfy is calcu lated to l i e at 9.92 eV (5E1 ) agreeing with the observed data as fa r as the energy value goes. This e xcita tio n inyolyes, a tr a n s itio n from 3e' to 6a-j, has Rydberg mixed with, yalence character and a magnetic moment of 0.618. 142 The th ir d band consisting of a most intense t r a n s i tion (0.7) with a maximum at 10.3 eV and with some Rydberg character observed along the low energy side has been previously assigned to the tra n s itio n 3e' -> ■ la,-,* All the o re tica l values fo r this state have been 2-3 eV higher than experiment. Our calculations using two generating functions and including a ll singles and doubles out of the 2 bases show that a Rydberg component lie s at 10.28 eV whereas the valence 3e’ ■ + la£ lie s at 11.58 eV with o s c i l l a t o r strength of 0.8. The Rydberg component is due to ex c ita tio n from le" 2a£ involving C-H out of plane components and is quite intense ( f = 0.16). I t is probable that our high energies fo r this t h ir d band.are due to t h e ir proximity to the f i r s t io n iz a tio n l i m i t as well as the many other roots of E' symmetry which appear below this state. In a d d itio n , the 2 A£ state appearing in this region and mixing in with the 3e' -> • la^ also seems to contribute some in t e n s it y (0.24) (see Table 4.2). Our assignment of this band is in agreement with previous the ore tica l studies, but in ad ditio n , our results c le a rly i d e n t i f y the Rydberg tr a n s it io n superimposed on this intense valence band. The sum of the o s c i l l a t o r strengths in the 10.3-10.5 eV region is 0.40, which is weaker than observed exp erim e n ta lly. This is probably due to the larger error invo lved:in computations of the wave- function as compared to the lower states. S p e c ific a 1l y , the perturbation correction may be as high as 11. 6% for the 2A£ and the 6 E * states re s u ltin g in an uncertainty of 23.2% in the properties. F in a lly , calculations were performed to id e n t if y Kuebler observe a complex band system in this region by using a two photon experiment. They assign the band to argue that a strong Jahn-Teller d is t o r t io n dominates this system and deforms the ring stru cture in this low energy region. Since our calculations have already accounted for the 3s Rydberg state at 7.61 eV, and since the lowest hr> state lie s at 8.07 eM, we f e l t that the low lying t r a n s it io n , of weak i n t e n s it y , must belong to a spin forbidden state. W e found the 3s Rydberg t r i p l e t state, 3 I E ’ to l i e at 7.1 eV above the ground state and we therefore believe the band system observed recently by Robin maybe due to this spin-forbidden tr a n s it io n . D. Conclusion This in ve stig atio n c le a rly demonstrates the importance Of a b - i n i t i o calculations as applied to MCD and absorption spectra. Our study is able to elucidate many of the uncertainties in the e le c tro n ic assignments of the excited ates of cyclopropane. By using accurate wave functions, the tr a n s itio n between 51,000-56,000 cm"^. Robin and the t r a n s itio n e' to 3s with overall symmetry E1, in -5 spite of the weakness of this band (c = 10 ). They 144 the A and B terms were calculated re s u ltin g in an ab- i n i t o MCD spectrum. The differences between our results and experimental moment analysis stems e s s e n tia lly from differences in the number of states under a p a r tic u la r absorption band. Moment analysis appears to depend d i r e c t l y upon a p a r tic u la r model of excited states present and therefore may not always expand on the information found in absorption spectra. In general, there is quite a b i t of information which may be th e o r e t ic a lly calculated and then matched to the MCD. The magnetic moment, related to A -j / D q , and the second order perturbation terms, Bq/Dq, can be most valuable in the assignment of excited states. In systems where degeneracies due to high symmetry e x is t, a study such as this is extremely h e lp fu l. Other systems which, lend themselves to s im ila r studies are benzene, the higher c y c lic alkanes, and work is now underway on the extremely in te re s tin g molecule, allene. The same techniques as those used here may be applied to systems which are expected to possess large magnetic moments due to high symmetry, but are found to lack these properties. Studies of this kind may expand our understanding of systems which are Jahn-Teller s p l i t , or undergo some other sort of d is t o r t io n . There is much savings in find ing whether a molecule is indeed d isto rted in the excited state, by c alculating magnetic 145 moments as opposed to complete potential surfaces. Clearly then, continuing a b - i n i t i o studies of MCD bands would be most valuable to both theoretician and exp eri mental i s t . 146 E. References 1. Fuson and Gilman, Organic Chemistry, Wiley & Sons, 19 38, Vol . I . 2. R. Robinson, J. Chem. Soc. 109 , 1042 ( 1 916), Kohler and Conant, J. Am. Chem. Soc. 39.» 1404 (1917) . 3. Rogers and Roberts, J. Am. Chem. Soc. £ 8, 843 (1946). 4. R. Bonaccorsi, E. Scrocco and J. Tomasi, J. Chem. Phys. 52, 5270 (1970). 5. R.M. Stevens, E. Switkes, E.A. Laws and W.N. Lipscomb, J. Am. Chem. Soc. 9_3, 2603 (1971 ). 6. R.B. Hall and A. Kaldor, J. Chem. Phys. 70, 4027 (1979). 7. A.D. Walsh, Nature 759, 167, 712 (1 942). 8. R.D, Brown and B.G. Krishna, J. Chem. Phys. 45, 1482 (1966). 9. H.. Basch, M.B. Robin, N.A. Kuebler, C. Baker, and JD,W .. Turner, 0. Chem. Phys. 5J[, 52 (1969), 10. M.B. Robin, Higher Excited States of Polyatomic Molecules X, 1975, Academic Press, New York. 11. B.M. Robin a.nd N.A. Kuebler, J. Chem. Phys. 69, 806 (.1 978) . 12. P. Wagner and A.B.F. Duncan, JChem. ; Phys. 2J_, 516 (1 9.5 31. 147 13. A. Gedanken and 0. Schnepp, Chem. Phys. 1_2, 341 (1 9 76 ). 14. P.J. Stephens, Ann. Rev. Phys. Chem. 2 _ 5 _ , 201 (1974); P.J. Stephens, R.L. Mowery and P.N. Schatz, J. Chem. Phys. 5_5 , 224 (1971 ) . 15. R.J. Bu.enker and S.D. Peyerimhoff, J. Phys. Chem. 73, 1299 (1969). 16. A.Y. Meyer, Theoret. Chim. Acta 2_2, 271 ( 1 971 ). 17. D.T. Clark, Theoret. Chim. Acta 10, 111 (1968). 18. A.Y. Meyer and R. Pasternak, Theoret. Chim. Acta 4_5 , 45 (1977). 19. C.J. F r itc h le , J r . , Acta. Cryst. 2£, 34 (1966). 20. T.D. Dunning, J r . , J. Chem. Phys. 5_3, 282 (1970). 21. G.A. Segal and R.W. Wetmore, Chem. Phys. Lett. 32, 556 (19.75). 22. R»W. Wetmore and G.A. Segal, Chem. Phys. Le tt. 36, 478 (1975). 23. G.A. Segal, R.W. Wetmore and k. Wolf, Chem. Phys. 30, 296 (1978). Table 4.1. Cyclopropane O rbital Energies Symmetry Designation Eigenvalue Orbital Description r 2 a x2 2 y 2 z °3h C 2v 4 2a] al -1.134849 (c1s+C2s+£3S) c" c 5 2e' 6 al b2 -0.81533 C " H Px’ Py 7 la" -0.674835 C-H pz 8 3a] bl -0.619241 "in te rn a l" sp2 C-H 9 le" 10 bl a2 -0.51215 tt C-H (out of plane) n 3e1 12 b2 al -0.41345 C-C "external" pv ,p.. a y 149 Table 4.1. (continued) Symmetry Desgination Eigenvalue Orbital Description r 2 a 2 X 2 y 2 z °3h C 2v 13 4a-j al 0.026705 3s Rydberg, - d2 -d x + y 94.4 28.8 28.8 36.7 14 2a£ bl 0.0511639 Pz out o f plane Rydberg 68.9 13.7 13.7 41.6 15 4e1 16 al b2 0.054109 2 sp "in te rn a l" p ,p Rydberg x y 72.6 72.6 43.3 14.5 14.5 43.3 14.8 14.8 17 5a '- , al 0.065534 3dz2 Rydberg, d2 , +d x +y 87.5 24.6 24.6 38.4 18 2e" 19 bl a2 0.07021809 out o f plane dxz>dyZ Rydberg 90.8 90.8 38.9 12.9 12.9 38.9 38.9 38.9 20 5e' 21 b2 al 0.072293 d , d „ Rydberg xy x2_y 2 91.4 91.4 39.1 39.1 39.1 39.1 13.1 13.1 22 6a-| cn n al 0,140304 Mixed Rydberg + valence 59.2 16.4 16.4 26.5 Table 4.1. (continued) Symmetry Designation Eigenvalue Orbital Description 2 a r 2 X 2 y 2 z °3h C 2v 23 la£ —^ 0.2817669 2 "externa" sp , p^,Py valence 14.7 6.6 6.6 1.5 24 6e 1 25 al b2 0.2990946 2 "in te rn a l" sp , p p valence xH y 26 7aV al 0.4395364 a C-H 27 3a£ bl 0.5070971 C - Pz 28 7e' 29 b2 al 0.5157376 "external" C-C P jP x y 30 3e" 31 bl a2 0.5258 C-H 32 2a^ -- 0 . 70395 C-C CJ1 Table 4.1 (continued) Symmetry Designation °3h C 2v Eigenvalue Orbi tal Descri ption 2 a r 2 X 2 y 2 z 33 4a£ b] 0.73966 C-H 34 b9 8e' '*■ 35 a-j 0.7752 C-C 36 8a -J a1 .83976 a C-H 37 a, 9e' 1 38 b2 .86770, C-C "in te rn a l" 39 9a] a] .97423 2 sp C-C + o C-H 40 b. 4e" 1 41 a2 1.1643 C-H cn ro Table 4.1 (.contin Symmetry Designation Eigenvalue °3h C 2v 42 a, lOe * ' 1.375 43 b2 Total SCF Energy: -117.009975973 n (a) in units of a.u. Orbi tal Description sp^ C-C "in te rn a l" cn co 2 a Table 4.2. Comparison of Calculated Results and Experiment/) Allowed States Primary Excitations Calculated excitation energy in ev f r f V f r*V Experiment IE' 3e' + 4a‘ ! 12+13, 11+13 7.61 0.009 0.013 0.011 ; CM 0 0 1 2E1 3e‘ + 4e' 12+15, 11+16 11+15, 12+16 8.11 0.127 0.101 0.113 f = 0.12 1A£ 3e' + 2e“ 11+19, 12+18 8.7 0.007 0.001 0.005 3E1 3e‘ + 5a1' 11+17, 12+17 8.85 0.002 0.001 0.0016 8.3 - 8.93 f = 0.04 4E' 3e' + 5e' 12+20, 11+21 11+20, 12+21 9.08 0.002 0.001 0.001 154 Allowed Primary Table 4.2. (conti Calculated excitations nued) V7 States Excitations energy in ev f r f V Experiment 5E' 3e' + 6aj 11+22, 12+22 9.92 0.001 0.004 0.002 6E1 3e1 + la^ 9+14, 10+14 11+23, 12+23 10.28 0.168 0.152 0.164 W eak Rydberg at 9.9 ev 9.17 - 10.5 Maximum: 10.3 Valence with 2A£ le" + 4e1 9+15,, 10+16 10.58 0.241 0.216 0.234 Rydberg Super imposed f = 0.7 L U r - ^ 1 55 3e' + laJ> 11+23, 12+23 11.9 1.0 0.88 0.878 Table 4.3. Summary of C.I. Results fo r A ll Calculated States Excited State Main Contributions C.I. C oefficient Eigenvalue ? 1 1 (a) 11 -K 13 -0.85 7.10 ev 1 1 -± 22 -0.39 12 -> 15 -0.14 (b) 12 13 0.85 12 22 0.39 1 1 16. 0.14 1 1E' 3e1 4a] (a) 1 1 13 0.90 7.61 ev 3s Rydberg 1 1 -> 15 0.15 12 -* 16 0.07 1 1 -> 17 0.14 1 1 22 0.30 (b) 12 13 0.90 12 11 -> 15 16 -0.15 0.07 ’ 12 17 0.14 12 - 4 - 22 0.30 1 56 -Table 4.3. (continued) Excited State Main Contributions C.I. C oefficient Eigenvalue i \ 3e1 -> 4e‘ 1 1 - » ■ 15 -0.65 8.07 ev 12 16 0.70 12 -> 25 - 0.11 11 -¥ 24 0.11 i 12 -¥ 15 0.71 8.08 ev 1 1 - > ■ 16 0.64 12 -> 24 -0.095 11 25 0.095 2E‘ 3e* - * ■ 4er (a) 12 13 -0.19 8.11 ev •' s 3p *x,y Rydberg 12 * > ■ 15 -0.62 1 1 16 0.69 12 17 -0.08 (b) 11 - > • 13 -0.19 1 1 15 0.69 12 -> 16 0.62 C n 11 -> 17 -0.08 158 Table 4.3 Excited State Main Contributions 1 h '2 3e' + 2e" 12 + 19 3dv7 ^ dw Rydberg 1 1 + 18 (a) 12 + 13 12 + 17 12 + 22 3e‘ + 2e" 3e1 + 5a] 3dz2 Rydberg (b) 11 + 13 11 + 17 11+22 1 E" 3e1 + 2e" (a) 12 + 18 (b) 11 + 1 9 (continued) C .I. C oefficient' Eigenvalue. -0.96 8.7 ev -0.95 0.12 0.95 0.10 0.12 0.95 0.10 ■0.96 8.79 ev 8.85 8.9 ev 0.96 Table 4.3. Excited State Main Contributions 4 E' 3e' £ 5e' dxy ’ d 2 2 J x -y (a) Rydberg (b) 2 h'z 3e‘ ■+'5e1 2 Aj 3e' + 5e' 5 E* 3e' •> 6a * Mixed Valence t Rydberg (a) (b) 20 21 1 -> • 21 2 - 20 1 -> 21 2 s- 20 1 » 20 2 21 1 - 13 1 17 1 + 22 2 -> 13 2 17 2 + 22 (continued) C .I. C oefficient -0.67 0.70 -0.67 0.69 -0.69 0.68 0.70 0.66 0.34 0.18 - 0.88 -0.34 -0.18 0.88 Eigenvalue 9.08 ev 9.09 ev 9.19 ev 9.92 ev Table 4.3 (continued) Excited State Main Contributions C .I. C oefficient Eigenvalue 6 E* e"- + a2" 10 + 14 0.96 10.28 ev (c j 3 pz Rydberg 1 1 -> 23 0.11 2v 3e* la£ 2 9 + 15 -0.64 10.58 ev Rydberg + Valence 10 + 16 -0.72 9 + 13 0.087 10 + 20 0.13 9 + 24 0.11 9 + 21 0.13 10 + 25 0.11 2 E" (a) 9-5-16 0.67 10.62 ev 10 + 16 0.72 (b) 10 + 15 0.67 9 + 15 -0.72 C T l O Table 4.3 Excited State Main Contributions 7 E' 3e' -> ~ \a 'z 11 23 C 2v e" ^ a2 (3pz R ydberg) io h 9 -» • 19 (continued) C .I. C oe fficient' 0.95 0.23 O.oO Eigenvalue 11.5 ev Table 4.4. Calculated M C D " * v - - V D oa eigenvalue Bg D S a “ A1 IE' .573 7161 2.8xl0" 4 0.309 .177 2E1 -.172 81 IT -2.4xl0 " 4 3.89 -.670 1A£ — 817 5.3xl0-4 0.05 — 3E1 0.582 8:85 -3.8x1 O ' 4 0.067 .039 4E' 0.10 9:1 -l.lxlO " 4 0.066 .0066 5E1 0.618 9T92 .. -8x10" 5 0.068 .042 Matrix elements used here: (a) (b) -A-j/Dg is the calculated value - 1/2< \p (rxV)z jipx> (units* 3) 2 -1 BQ is m iu n its of 6 debye /cm (c) 2 D g has units of debye C T > ro (d) A -j has units o f g <A]Im x IE x> “ m D. = 2 m 2 a - J- a1 ~ 1 1 1 u. cn CO Table 4.4 (continued) debye2 Table 4.5. B Term Components IE' 2E1 1AJ (8.11 ev) (8.7 ev) 3E1 4E1 1 5E' (7.61 ev) (8.85 ev) (9.08 ev) (9.92 ev) IE' — 2.8xl0" 4 1.3xl0~6 0.06x10'5 0.0 0.0 2E' - 2. 8x l0"4 0.0 3.2x10“ 5 0. 8x10"5 0.0 1^2 -1.3xlO" 6 0.00 3.6xl0" 4 9.3x10" 5 8x10"5 3E1 -0.06xl0~5 -3.2x10“ 5 -3 .6xl0 ' 4 — IxlO' 5 0.0 4E1 o • o -0.8xl0 -5 -9.3x10-5 -IxlO ' 5 -- .5 0.07x10 5E1 a> -F * 0.0 0.0 - 8x10-5 -0.07x10"5 -- Figure 4. a. internal o r b ita ls in cyclopropane. b. External o r b ita ls in cyclopropane. c. Geometry o f cyclopropane. 165 y (c) 167 Fi gure 4.2a. Experimental absorption, MCD and moment analysis. b. Comparison between calculated and experimental MCD. c . Calculated B term component compared to tb .ta l experimental MCD. d. Calculated A term component compared to to ta l experimental MCD. 168 A€x 1 0 * M. C. 0 . of Cyclopropane -.5 EXPERIMENTAL RESULTS TOTAL FIT ___ -10 5 0 0 0 30 00 2000 1 0 0 0 174 170 155 WAVELENGTH (nm | 150 165 WAVELENGTH A €xlO S + 1 0 COMPONENTS (A TERM) COMPONENTS (B TERM) - 5 -10 I/O 155 165 ISO 1 6 0 140 135 WAVELENGTH inm) (a) 169 •E MCD SPEiCTRH 0 F.XPF.R IMF.NTRL MCD O t- 1 . 0 Oo cr cr. O + + + - 152. 00 -1411.00 ' -130,00 NRN0MET.E-RS - 1 2 0 .0 0 -112.00 -96.00 -176.00 -163.00 -160.00 * MCD SPECTRq O EXPERIMENTAL o o < \i 0 CD C O ' in x X Oo , ,o X ' o •J ( / " > a a.o . L l! o a : h- _i !+ + ' + O 0 o 0 0 0 + # 0, * + & ■ 0 + & 0 + 3 0 + 0 + P H 0 + 0 +++ 0 0 0 0 0 + - + 0 0 0 O 0 0 + 0 0 + 0 0 + + + + + + + + + + +,,+ + + + + + + + + + + + + + MCO I “r ~l I rr I pi I — I - T I r I -18*1.00 -176.00 -168.00 -160.00 -152.00 -UUl.OO -136.00 -120.00 -120.00 -112.00 -104.00 -96.00 NANOMETERS MCO SPECTRA 0 EXPER f M S. M -O L M t U m Oo Q. o _J a : i— © +, ( \ l -176.00 -163.00 -160.00 -136,00 -128.00 - 112.00 -104.00 -96.00 NANOMETERS

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

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

PDF

CO2 Laser-induced chemistry of C23CN

PDF

Competitive and sequential absorption and dissociation mechanisms in the infrared fragmentation of polyatomics

PDF

Calculations of electrostatic interactions in proteins

PDF

Brillouin scattering in molecular crystals: Sym-trichlorobenzene

PDF

Application of nonlinear spectroscopic techniques to the studies of molecular electronic states and reaction dynamics.

PDF

Electric field induced spectra of carbon disulfide and its dipole moment in an excited state

PDF

Ab initio calculations of vibrational absorption and vibrational circular dichroism

PDF

A synthetic model approach to H-bonded oxyhemoglobin

PDF

Biochemical and genetic studies on Escherichia coli DNA polymerase II, a member of group B ``alpha-like'' DNA polymerases

PDF

Application of many-body Green's function formalism to the elastic scattering and bound state properties of helium atom

PDF

A study of the interactions of polar gases with solid proteins and some simple organic compounds

PDF

Brillouin Scattering In Molecular Crystals: Sym-Trichlorobenzene

PDF

A highly efficient and general method for the chemical synthesis of beta,gamma-fluoro- and difluoromethylene analogs of purine and pyrimidine 5'-ribo- and deoxyribonucleotides

PDF

Crystallographic studies on the interaction of nucleosides and nucleotides with transition metals / by

PDF

Absolute configuration studies of molecules bearing chiral methylene groups

PDF

A study of the double sulphates of some rare-earth elements with sodium, potassium, ammonium and thallium

PDF

A study of the reactivity between tetraphosphorus trisulfide, P₄S₃, and iridium and other transition-metal complexes

PDF

Green'S Function Technique In Atomic And Molecular Physics To Special Consideration Of Electron Scattering Problems In The Generalized Random Phase Approximation (Grpa)

PDF

A chemical study of the glycogens formed after administration of glucose, fructose, and galactose

Asset Metadata

Creator Goldstein, Elisheva (author)

Core Title Configuration interaction calculations on the resonance states of HCl- and the excited states of cyclopropane

Degree Doctor of Philosophy

Degree Program Chemistry

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

Tag chemistry, general,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Segal, Gerald (committee chair), Dows, D.A. (committee member), Goodman, Myron (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c17-629353

Unique identifier UC11355041

Identifier DP21843.pdf (filename),usctheses-c17-629353 (legacy record id)

Legacy Identifier DP21843.pdf

Dmrecord 629353

Document Type Dissertation

Rights Goldstein, Elisheva

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA