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Quantum chemical calculations on hf and some related molecules Bruce, Robert Emerson 1972

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QUANTUM CHEMICAL CALCULATIONS ON HF AND SOKE RELATED MOLECULES by ROBERT EMERSON BRUCE B.Sc., University of British Columbia, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE ln the Department of Chemistry We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1972 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver 8, Canada Date -ii-Abstract This thesis reports some quantum chemical calcu lations directed at elucidating principles useful for refining calculations of electron distribution and other properties for complex molecules. In this work calcu lations have been made with the valence bond and mole cular orbital methods using minimum basis sets of Slater-type orbitals on the ground states of the molecules HP and HO, and on states of HF+ corresponding to the ioni zation of either a Is electron or a ZpfC electron from fluorine in HF. Calculations have been made for mole cular energies, bond lengths, force constants, dipole moments, and electron distributions as given by Mulll-ken population analysis. For HF, the perfect pairing model with molecule-optimized exponents yields molecular energies about 6 kcal./mole lower than the comparable molecular orbital calculations; the dipole moment calculated by the per fect pairing method is 0.3 D. closer to the experimen tal value (1.82 D.) than that calculated by the molecu lar orbital method. The HF equilibrium bond length and force constants are calculated to a reasonable degree of accuracy with the two methods, although the first ionization potentials seem to be better calculated by the molecular orbital method either by Koopman's Theorem -Ii i-or by taking the difference between the energies of the two states. The calculations reported in this thesis show clearly that in general free atom exponents are not re liable for calculating molecular properties, and this is important for calculations on larger molecules which most frequently use basis functions appropriate to free atoms. As part of a programme for finding ways of op timizing exponents relatively inexpensively, for use with more complex molecules, an approximation due to Lowdin, for overlap charge distributions ln electron repulsion integrals, was tested. The results reported In this thesis show that the method has promise in pro viding a way of initially optimizing exponents prior to the actual calculation wherein all integrals are evalu ated exactly. -iv-Table of Contents page Abstract ii Table of Contents iv List of Tables v Acknowledgement vii Chapter One - Introduction 1 The Molecular Orbital Method % 6 The Valence Bond Method 12 Aims of the Thesis 21 Chapter Two - Calculations on HF, HF+ and HO 2^ Basis Functions 27 Valence Bond and Molecular Orbital Wave Functions 31 Computational Details 36 Chapter Three - Results and Discussion UrZ Atomic Orbital Exponents 51 Molecular Energies 55 Bond Lengths and Force Constants 6l Electron Distributions 6*4-Concluding Remarks 71 Bibliography 76 -V-Llst of Tables Tables page 1 Valence bond configurations for the state of HP lb 2 Results of some previous calculations of mo lecular properties for HFt^ 25 3 Results of some previous calculations of mo lecular properties of HF+.J,. and H047J. 26 k Zero-order wave functions in equation (36) for the 1I state of HF 33 5 Orbital exponents and molecular properties for different wave functions of HF at the experi mental bond distance (1.733 a.u.) 4 3 6 Variation parameters and Mulliken populations for different wave functions of HF at the ex perimental bond distance (1.733 a.u.) 7 Orbital exponents and molecular properties for different wave functions of HF at calculated equilibrium bond distances ^5 8 Variation parameters and Mulliken populations for different wave functions of HF at calcula ted equilibrium bond distances 46 9 Orbital exponents and molecular properties for different wave functions of HF+2^ at HF expe rimental bond distance (1.733 a.u.) k7 -vl-Tables page 10 Variation parameters and Kulliken populations for different wave functions of HF+a^. at HF ex perimental bond distance (1.733 a.u.) 4-8 11 Orbital exponents and molecular properties for a series of wave functions for HF*^ and H0aff» ^9 12 Variation parameters and Kulliken populations for a series of wave functions for HF+^«- and H0a 50 Acknowledgement I would like to gratefully acknowledge the assis tance and encouragement which I received from Dr. K. A. R. Mitchell throughout the course of this study. His guidance, advice and many helpful discussions were in valuable both for the research and the preparation of this thesis. I also wish to thank J. K. Wannop for the preparation of the manuscript, and my parents for their constant support. Chapter One Introduction Quantum mechanics is important in chemistry for several reasons. In the most fundamental sense, it provides, in principle, the means of determining the oretically all the properties of molecules, either by the time-dependent or the time-independent Schrodinger equation,* and, given the properties of individual molecules and the interaction energies between them, statistical mechanics allows predictions to be made for macroscopic collections of molecules. That the possibilities for exact quantum mechanical calcula tions on individual molecules are somewhat limited, can be assessed by noting that agreement between the ory and experiment for the binding energy of the sim plest neutral molecule, H?, has only recently been -2-reached.2 Thus, for molecular systems of general in terest to the chemist, theoretical treatments must be based on some degree of approximation. Molecular properties in organic and inorganic chemistry are often discussed in terms of electron distributions,^1'' and ln this vein Platt^ has argued that a theory of chemistry is primarily a theory of e-lectron density. Early quantum mechanical calculations on atoms and molecules, and experimental studies, espe cially in structural chemistry, have led to quantum chemical concepts such as orbitals, ionic character, hybridization, and electron pair bonds. These con cepts are freely used in discussing electron density in molecules,^'^ although density distributions can rarely be obtained directly by experiment. Another use of quantum mechanics in chemistry has evolved with the development, during the last two or three decades, of experimental techniques, such as nu clear magnetic resonance, electron spin resonance, nu clear quadrupole resonance, Mossbauer spectroscopy and photoelectron spectroscopy, which are now widely used by chemists ln attempting to gain an improved under standing of chemical bonding. Quantum mechanics has been employed in this context, both for elucidating the basic physics of these experiments, and for developing approximate computational schemes from which calculated -3-molecular properties can be compared with experimental values. This provides important Information for asses sing the validity of the models of electron density and chemical bonding used by chemists. Two major approaches have been developed for ap proximate calculations on molecules, and these are the molecular orbital method and the valence bond method. The former has been more generally used, mainly because it has been considered to be computationally simpler. Nevertheless, recent developments have led to efficient computational schemes for valence bond calculations, and, moreover, attempts are now being made to develop o semi-empirical schemes with this method. Also it has been known for some time that calculations using the perfect pairing model, such as that proposed by Hurley, Lennard-Jones and Pople,^ which represents an extension to polyatomic molecules of the Heitier-London calcula tion on Hg,*^ can give better molecular energies than the corresponding molecular orbital calculations.1* This improvement occurs because electron motions are better correlated in Heitler-London type wave functions than in molecular orbital wave functions. The use fulness of perfect pairing wave functions in polyatomics is closely related to the usefulness of the concept of hybridization, which is itself dependent on the pro perties of atomic orbitals in molecules. The behaviour of atomic orbitals in molecules is of general interest, but it is also of particular Importance for studying molecules containing the heavier atoms (such as those of the second row of the periodic table and beyond, in cluding transition metals) for which the details of chemical bonding have not yet been established unam biguously in a number of Important cases.^"^ Large basis set calculations on these molecules would seem to be impractical in the near future, and the alterna tive is to attempt to make reasonable calculations of molecular properties by using well chosen restricted basis sets of atomic orbitals. In any event, large basis set calculations are difficult to interpret in terms of quantum chemical concepts,an example being 1 ft Kulllke^s suggestion that the increase in bond length observed on ionizing a TT electron ln many dia tomic hydrides indicates a degree of Tt bonding in these molecules, and therefore the Involvement of 2prr atomic orbitals on hydrogen. Although large basis set calculations have been performed for diatomic hydrides, including up to 3d orbitals on hydrogen ln the basis 19 set, 7 the chemical significance of hydrogen 2p7r orbi tals in bonding has not been determined. In discussing the valence bond and molecular or bital methods of molecular calculations, one starts with the time-independent Schrodinger equation -5-where H is the Hamiltonian operator, E is the energy of the system, and V is the state function. In the nonre-lativistic approximation, the Hamiltonian operator can be written as for a collection of N electrons and S nuclei, where the first term represents the summed kinetic energies of the nuclei, the second term represents the summed kinetic e-nergies of the electrons, the third term represents the attraction energy between the electrons and the nuclei, and the fourth and fifth terms represent respectively the nuclear-nuclear repulsions and the electron-electron re pulsions. In molecular calculations the Born-Oppenheimer approximation20 is frequently made. Physically this ap proximation consists of regarding the motions of the nuc lei in a molecule as insignificantly small in comparison to the motions of the electrons, and this is dependent on the masses of the nuclei being very much greater than the masses of the electrons. Thus one regards the nuclei as remaining essentially at rest relative to the motions of the electrons. Using the Born-Oppenheimer approximation, therefore, the wave function is approximated as a function of the electron co-ordinates only, the nuclei being regard ed as stationary. Then the electron motions are contained -6-in the electronic wave function, V , which is obtained in principle by solving the equation K% = , (3) where ^.i ^ A»l >* A<« AV In the Born-Oppenheimer approximation, E is Eg plus the nuclear-nuclear repulsion energy. The molecular orbital and valence bond methods pro vide schemes for writing down approximate forms of the e-lectronic wave function, and for calculating the approx corresponding electronic energies according to (in the Dirac notation) t e * (5) The approximate electronic wave functions are obtained ac-21 cording to the variation principle; by which the best: wave function is selected according to the criterion of minimum energy. The Molecular Orbital Method The molecular orbital method originated from studies 22 2^ by Hund and Mulliken made within a few years of the formulation of quantum mechanics, and this method repre sents the direct extension to molecules of the atomic -7-2^ 25 orbital method for atoms. ' For singlet states of a molecule containing 2N electrons, the electronic wave function in molecular orbital theory is approximated by a single determinant as in where only the diagonal elements of the determinant are defined explicitly. The determinantal form of equa tion (6) is convenient for ensuring consistency with 26 _i. the antisymmetry principle? (2N)~2 is the normaliza tion factor. Each molecular orbital is doubly occupied by electrons of opposite spin, fi spin being indicated in equation (6) by a bar over the molecular orbital. The molecular orbitals are one-electron functions which extend over the whole molecule and they can be defined to be that set of orthonorraal functions, satisfying the conditions which minimize the electronic energy of the system according to F Ofc. fHe f %.>  H'~ (tJt> ' (8) where H is the electronic Hamiltonian defined in e equation (M . In earlier work on atoms, orbitals were given in -8-numerical formj in practical applications to molecules, however, they are usually expanded following the pro cedure reviewed by Roothaan2? over a set of basis func tions as in *. • (9) Eoothaan's procedure consists of determining, by the variation theorem, the coefficients in equation (9) In order to specify the molecular orbitals. Often the basis functions ^ in equation (9) may be identified as atomic orbitals. In practice, a linear combination of atomic orbitals represents an approximation to a mo lecular orbital wave function because only a restricted number of atomic orbitals are included in the basis set, although, in principle, one may approach as close to the limit as desired. The atomic orbitals in equation (9) may be centred on only one atom in a molecule, how ever, the convergence to minimum energy is then slow, and, with modern computing facilities, this approxima-tion seems to be of only limited value. The energy of the determinantal wave function in equation (6), where the one-electron orbitals satisfy the conditions in equation (?), can be expressed2^ as i j where -9-Hii-CM-W-ziliO (ID gives the contribution of one electron in ^ to the total electronic energy. The term in equation (11) in volving the Laplaclan operator represents the kinetic energy of one electron in Y^i the second term represents the attraction between an electron in and the nuclei. In equation (10), J1j and represent, respectively, Coulomb and exchange electron repulsion integrals de fined as and *->r[[Vi(1)*i<Vifc*jM*i(i) driJei < (13) The integrals J±y and Kij» can readily be ex panded in terms of the basis orbitals in equation (9)» and, following Eoothaan's procedure, a self-consistent field calculation allows the determination of the co efficients, c^, in equation (9) given the molecular Integrals over the basis orbitals. For open shell systems more than one determinant can be written for a given configuration, and the de terminants must be combined according to the approp riate electronic state in order to obtain approximations -10-to the total wave function of the system. A simple ex ample is the triplet state of a two-electron system for which the total electronic wave function is set up in terms of the orbitals and ^2 as , (1*0 sz.-l 1 A detailed discussion of the molecular orbital method for open shell systems has been given by Roothaan.-^ Molecular orbital calculations using the Roothaan procedure and evaluating all molecular integrals with out approximation become excessively expensive as the number of electrons in the molecule and the size of the basis set increase. The greater computational ef fort and expense is due in part to the number of elec tron-electron repulsion integrals to be evaluated, which increases as approximately the fourth power of the basis set.-^* Also, for large basis sets, it is often found that more time is required to evaluate integrals involving higher members of the basis set than to e-valuate integrals involving the lower members of the set. These factors have led to the development of a number of approximate molecular orbital methods suitable -11-for application on a routine basis to molecules which are too complex to be readily treated using the more complete methods. In these approximate molecular orbital methods one attempts to make judicious approximations which will simplify the computations so that properties of fairly large molecules can be calculated without either imposing concepts such as preconceived bonding schemes, or eliminating established physical features such as the relative energy levels of atomic orbitals. One development has been to incorporate empirical date Into a model such as is done in the Huckel method-^*33 developed for jr electrons in organic systems and ex tended to include all the valence electronsThis method does not explicitly Include electron-electron repulsions, but by relating Huckel's Coulomb Integrals to valence ionization potentials, and expressing the resonance integrals in terms of the Coulomb and overlap integrals, Hoffmann-^ has discussed charge distributions and conformation energies of a large number of hydro carbons, and similar methods have been applied to many Inorganic molecules. 36,37 Less drastic approximations are made in the Com plete Neglect of Differential Overlap and related me thods which are discussed in a recent book by Pople and Beveridge^ and also in a book edited by Sinanoglu -12-and Wiberg.39 In these methods, emphasis is placed on the valence electrons, and electron repulsion In tegrals are Included, but approximations are made such as Atomic spectral data are again incorporated in these methods, but a guiding principle is that they are for mulated so that the calculated results are invariant to the rotation of axes. This property is required phy sically, but is not shown by the extended Huckel method. Many applications have been made to the calculation of molecular energies, molecular geometries, charge distri butions, ionization potentials, and nuclear magnetic resonance parametersand these methods have been established as providing a reasonable balance be tween computational expense and worthwhile calculations of molecular properties. The Valence Bond Method (15) and (16) Historically, the valence bond theory provided -13-the first method for molecular calculations, and this theory originated from the work of Heitler, London, Slater, and Pauling.^ In this method one assumes a set of basis functions for a molecule, and these func tions are most frequently identified as atomic orbitals. In the most complete form of the valence bond method, combinations of determinantal functions are written down for all possible ways of accommodating the elecfc trons in the various atomic orbital functions in ac-cordance with both the Pauli principle, and with the symmetry of the particular electronic state for which the wave function is being expressed. The determinan tal functions are defined by the various valence bond configurations for a given electronic state. As an illustrative example, all the valence bond configu rations are listed in Table 1 for the state of HP using a basis set of the Is atomic orbital at hydrogen, and the Is, 2s, 2pO% and 2p*r atomic orbitals at fluorine. The ground state wave function Is then obtained by a free mixing of the zero-order wave functions corres ponding to all the configurations as ln where c^ Is the linear mixing coefficient, and is the appropriate combination of determinantal functions for the i^n valence bond configuration. As examples, -14-Table 1. Valence bond configurations for the ^ state of HF 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Is2 2s2 77£2 73^2 cr h Is2 2s2 77^2 7T22 CT2 Is2 2s2 irt2 irz2 h2 2 2 2 2 Is 2s 77^ ^ <rr2 & h Is2 2s TT2 Tf2 <r h2 2 2 9 9 Is 2s^ -77^ 7r-2 <r^ h^ 2 2 2 2 Is 2s 7T1 7T2 cr- h 2 2? 2 is 2s 7r± ir2 * h ., „ 2 2 2 .2 Is 2s ir. <TC7 cr h 2 21 2 2 9 2s 7^ TT <r h The symbols Is, 2s, ^, 7r^, *r, h refer respectively to Is, 2s, 2p-Tlt 2pir2, 2pcr functions at F and the Is func tion at H, -15-the specific forms of the unnormalized zero-order wave functions for the first two configurations in Table 1 are t and f z | is U2*ZZ Tr^tr^ir-rl m (19) The linear coefficients in equation (17) and the corres ponding energies are obtained by application of the 12 variation principle. The technique is well known, and involves solving the secular equation DET JM-.J - Sii £ {-- o (20) for the energies, and Z I ci (Hi3 - ES.-j)-* (21) j » for the coefficients. The matrix elements In equations (20) and (21) are defined as H;j * $y*Ht Vi U (22) and ) •* (23) When all the configurations formed from a given basis set are mixed, as in equation (17), the valence bond method is equivalent to a complete configuration interaction calculation in terms of molecular orbitals expanded over the same basis.^ Clearly, as the size of a basis set is increased, more configurations can be formed with the appropriate symmetry, and, in the limit, the energies obtained by the valence bond method con verge to the eigenvalues for a complete solution of the Schrodinger equation in the Born-Oppenheimer approxi mation. In less accurate applications of the method, however, it turns out that some configurations can be disregarded. A valence bond study by Harris and Michels^ on HF for a range of bond lengths, has shown that mixing the six most important configurations leads to a cal culated molecular energy only 0.35 kcal./mole above that obtained from the mixing of all eleven configurations. The five configurations which may be neglected, with only small error, either correspond to charge distri butions in the sense H""-F+, which is contrary to che mical experience based on the concept of electronega tivity, or correspond to configurations involving ex citation of electrons from the fluorine Is core. Si milarly, calculations by Maglagan and Schnuelle^ for BeHg show, for the particular case of using free atom exponents and a Be-H distance of 1.3^ A, that the effect of neglecting those valence bond configurations, where Be is more negatively charged than H,- and neglecting -17-also those configurations where Be is non-bonding, raises the total energy by only 1,2 kcal./mole. These consi derations indicate ways of selecting for approximate cal culations those valence bond configurations which are most significant to a given basis set. A consequence of such a selection Is an appreciable saving In computational effort and expense, and this becomes more important for larger molecules. A more restricted form of the valence bond method is that which involves perfect pairing,^ and this method is usually based on hybrid rather than natural atomic orbitals. In this approach, electron pair bonds are constructed between orbitals in a molecule, and the to tal electronic wave function is given in terms of the determinantal functions appropriate to the various spin couplings for the electron pair bonds. As an ex ample of this approach, one can consider BeHg, for which two electron pair bonds are assumed to be formed from the overlap of the Is orbitals (designated 11 and 12 ) on each hydrogen with the appropriate directed hybrid functions h^ and h2 at the Be atom. For this example, the unnormalized perfect pairing wave function is written as - l^x^'-^M^J -lAj./.AIWM.V* 1,(24) where one electron pair bond is formed by the overlap -18-of 11 and hlt and the other bond is formed by the overlap of 12 and hg. Two interpretations can be given to Vpp.^ In the first, ^pp involves only neutral configurations at Be, and h^ and h2 are the digonal hybrids formed from the 2s and 2p«" atomic orbitals at Be. The second interpretation allows for ionic character in the BeH bonds by expressing h^ and h2 as suitable combinations of the Is orbitals at H and the digonal hybrids as ln h« = N(d. + kl«) 1 1 1 (25) h2 = N(d2 + kl2) , where k is a measure of the ionic character of the bond and may be taken as a variation parameter} N is the nor malization factor. In general, within the perfect pairing model, for a closed-shell molecule with n electron-pair bonds, there will be 2n determinants in the electronic wave function. Hurley, Lennard-Jones, and Pople have showr? that, provided that the orbitals involved in any ele-tron pair bond are orthogonal to all other orbitals in a molecule, a comparatively simple expression can be written in closed form for the electronic energy cor responding to the perfect pairing wave function of the molecule. The evaluation of this expression is related to the methods which are discussed ln chapter two for evaluating the Hamiltonian and overlap matrix elements -19-in equations (22) and (23). A useful criterion for different methods for mo lecular calculations is provided by the agreement between calculated and experimental molecular properties. Cal culations performed using the valence bond and molecular orbital methods with comparable basis sets, enable com parisons to be made between the two methods. Detailed comparisons of the valence bond and mole-cular orbital methods have been made for Hg. ' Using a minimum basis set of Slater-type orbitals with energy-48 optimized exponents, the valence bond method predicts an equilibrium bond length of 0.743 A and a binding energy of 85.94 kcal./mole. The experimental values are 0.741 A and 109.98 kcal./mole respectively, while the corresponding quantities from the molecular orbital me thod** are 0.732 A and 80.27 kcal./mole with a minimum basis set. The molecular orbital method becomes much less reliable as the bond distance Is increased, and this is associated with the overemphasis of ionic con tributions in the molecular orbital method 5^' Karo and Olsen^0 have compared the molecular orbital and valence bond methods for the ground state (*£ ) of LiH using a basis set of numerical Is, 2s, and 2p orbitals at Li and a Slater Is orbital at H. At the equilibrium bond length of 1,56 A, both methods predict a dipole moment of 6.05 which may be compared with the experimental -20-val\ie of 5.88 D.;-3 and at this bond distance, the va lence bond method gives a molecular energy which Is o.4 kcal./mole lower than that given by the molecular orbital method. Again, as the internuclear distance increases, the molecular orbital method becomes relatively less Q reliable. Maglagan and Schnuelle have noted that the valence bond method generally gives lower molecular energies than the molecular orbital method for mole cules ln which the model of electron pair bonds is frequently used. By contrast, the molecular orbital me thod is comparatively better for delocalized systems such as benzene, although, as noted previously, with sufficient refinement the two methods merge. A de tailed comparison has recently been made by Mitchell and Thirunamachandran^ for BeHg employing a basis set of Slater-type orbitals with energy-optimized exponents. The molecular energy calculated with the perfect pair ing model is 15 kcal./mole below that for the molecu lar orbital method, and the calculated Be-H distances are 1.35 A and 1.37 A for the perfect pairing and mo lecular orbital methods respectively. A further in teresting comparison between the molecular orbital and valence bond methods has been made by Harrison and Allen-* for the ground state and two low excited states *A^ and *B^ of CH^. Using a basis set of Gaussian lobe functions, these authors calculate the bond angles of -21-the ^3±, 1A1, and 1B1 states to be 131°. 111°, and 15^° according to the molecular orbital method, and using the valence bond approach the corresponding angles are 138°, 108°, and 148° which are to be compared with ex perimental equilibrium bond angles of ~135°» 103°. and 140° for the three states.^3 Finally,, the 1A1-1B1 ver tical transition is 0.96 kcal./mole closer to the exper imental value using the valence bond method. Aims of the Thesis This study follows recent calculations by Mitchell and Thirunamachandran^»^ on BeH^, in which the mole cular orbital and valence bond methods have been compared using a minimum basis set of energy-optimized Slater-type orbitals. The work on BeRg indicates that the per fect pairing model provides a very good approximation for this molecule, and further comparative calculations are necessary in order to provide basic information with which to assess these methods on a wider basis. HF was chosen as a suitable molecule for continuing this work because the computational effort is reduced for di atomic hydrides, and the large electronegativity dif--22-ference between H and P contrasts with the much smal ler difference in electronegativities between neigh bouring atoms in BeH2. Although many calculations have already been made on HF using large basis sets, 7*-J it still seems worthwhile to make a direct comparison of the different models for molecular calculations asing an energy-optimized minimum basis set of Slater-type orbitals, ln part because the large basis set calculations are difficult to interpret in terms of quantum chemical concepts. Connected with the perfect pairing model is the concept of hybridization, and ln this work, attempts are made to compare the atomic orbital hybridization at P in HF and HF+, and also to compare with the hybridization at 0 in the diatomic KO which Is isoelectronic with HF+, Calculations on HF+ were made for states in which a Is or a 2ptf elec tron has been ionized from fluorine in the neutral mo lecule . Since minimum basis set calculations are performed in the various semi-empirical molecular orbital methods currently being used,-^'39 and because these methods have been designed especially for evaluating properties of related series of molecules, it seems necessary, in order to gain a better understanding of these semi-em pirical methods, to know Just how well minimum basis set calculations are able, In principle, to give useful -23-calculated values of properties such as bond lengths, ionization potentials, dipole moments, and molecular energies. This is especially so for more complicated molecules, such as those containing an atom of the se cond row of the Periodic Table or beyond, for which computational expense most often requires calculations to be carried out using restricted basis sets. It is hoped that the calculations on HF, HF+, and HO reported in this thesis can contribute both to a better understanding of the reliability of minimum basis set computations for calculating molecular properties of basic interest to the chemist, and to a better a-wareness of the value and limitations of the perfect pairing model and the use of hybridization In molecular wave function calculations. Finally, a preliminary attempt is made to assess the value of an approximation, proposed by Lowdin,^^ for simplifying molecular integrals for the purpose of optimizing atomic orbital exponents. The need for suitable integral approximations is great est for more complex molecules, but it is necessary to assess the applicability of any approximations by comparing results obtained using the approximations with those from the more complete calculations and also with experimental results, and this indicates ln part the reason for this study on HF. -24-Chapter Two Calculations on HF, HF+, and HO In principle, quantum chemical calculations on molecules can give direct information about electronic distribution by squaring the electronic wave function? some assessment of the reliability of actual calcula tions may be obtained by comparing calculated molecular properties with those measured experimentally. In Tables 2 and 3 results are collected from a number of recent calculations on HF£^ , HF"^. , and H0i;r, which are of special interest to this thesis. The main part of this chapter describes the details of the calculations which have been carried out during this work, and the results in Tables 2 and 3 provide a reference for asses sing the results which are discussed in chapter three. Table 2. Results of some previous calculations of molecular properties for HFj Wave function Molecular properties Model Basis set Ref. Energies (a-u-L H-F Distance (a.u.) Dipole Moments (D.) Ionization Potl. (ltr) (eV) Force Constants •10^(dynes/cm) mo min. STO free atom exp. 57 -99.4915 1.733 1.12 mo min. STO opt. exp. 58 -99.5361 1.733 1.44 vb min. STO opt. exp. 59 -99.563* 1.860 mo + ci min. STO opt. exp. 58 -99.5640 1.733 1.30 mo ext. GTO opt. atom exp. 60 -99.8873 1.7^3 14.00 9.60 mo ext. GTO opt. exp. 55 -100.0622 1.733 mo ext. STO opt. exp. 19 -100.0708 1.696 14.6* 11.22 Experimental -100.5271 1.7238 1.8195 16.06 9.657 * The value is taken from a graph **1 a.u.= 27.205e.V. Table 3. Results of some previous calculations of molecular properties of EF+ifr and H04fl. Wave function H0'-a Molecular properties Model Basis' set Ref. Energies (a.u.) Bond dis tance (a.u.) Dipole Moments (D.) Ionization Potl.U )(eV) Force Constants •lo5(dynes/cm) mo ext. GTO opt. atom exp. 60 -75.2872 1.813 11.7 vb H.-Fock a.o. 61 -75.325 1.80 mo H.-Fock a.o. 61 -75.327 2.00 mo ext. STO opt.exp. 19 -75.^208 1.80 11.3* 7.79 Experimental -75.778 1.834 1.66 13.2 9.216 EFlr it mo ext. GTO opt. atom exp. 60 -99.373** 1.96 mo ext. STO opt. exp 62 -99.53* 1.85 14.45 Experimental -27-Basls Functions As seen in Table 2, a number of different types of functions have been used as bases for molecular cal culations. In principle, any set of functions which is complete may be used to expand a molecular wave function} in practice the choice of functions is determined by the computational effort required to obtain a desired level of convergence. Most often the basis functions used in molecular calculations are related in some way to atomic orbital functions. Traditionally, the Hartree-Fock method for atoms leads to atomic orbitals which are in numerical form, however, these tabulated functions are too unwieldy to be useful ln molecular computations. In more recent work analytic expressions are used to repre sent atomic orbitals and these analytic atomic orbitals are expressed as where represents the appropriate spherical harmonics (listed for example by Pauling and Wilson*^ )t arK3 the Rt (r) are some type of radial function. One commonly used representation of the radial functions Is that gi ven by Slater^ and expressed as % - r"'"1- txF(-<ir)9 (27) -28-where n^ and are the principal quantum number and orbital exponent respectively; is a normalization factor. This form was proposed by Slater in order to approximate, in a simple way, the radial functions from the Hartree-Fock calculations for atomic orbitals. The radial function in equation (2?), when combined with a spherical harmonic defines a Slater-type orbital. Sla ter-type orbitals have no radial nodes, however, they do converge efficiently in atomic and molecular calcul ations. An alternative radial function is the Gaussian function, conveniently expressed as Rn£ - N-r^*. <up (-*.r*) . (28) The use of Gaussian functions was first proposed by 64 Boys, because integrals required in molecular calcul ations are more easily obtained using Gaussian functions than Slater functions. This advantage must be weighed against the fact that these functions provide much slow er convergence for molecular energies than do Slater-type orbitals; in fact it requires about half the number of Slater basis functions as Gaussian functions to ob tain a given molecular energy. In the present study, the interest centres mainly on the use of minimum basis set calculations and asses sing the possibilities for calculating properties of large molecules; therefore Slater-type radial functions -29-combined with the appropriate spherical harmonics are used as basis sets. To overcome the lack of orthogo nality which occurs for pure Slater-type orbitals of the same symmetry on the same centre, orthogonalized Slater-type orbitals are constructed according to the Schmidt orthogonalization procedure.^ por orbitals on the same centre this procedure consists of reinsta ting the radial nodes which were neglected in the ap proximation implicit In equation (27). In the general case, to Schmidt orthogonalize a function ^ to another function one "takes *i.Hfc., = *«• " 5 , (29) where S ' • (30) When normalized, ^or^n0g defines the function P^ which has been Schmidt orthogonalized to The Slater functions used as a basis set for the study of HF consist of a Is function at H and Is, 2s, 2p<r and the degenerate pair of 2p;r functions at F. The formulation of the energy expressions is simplified if all valence orbital functions are orthogonalized to the Fls core, and this is achieved by Schmidt orthogonalizing both His and F2s individually to Fls. In the perfect pairing model of KF there is one -30-electron pair bond constructed from the overlap of the Is atomic orbital at H with a hybrid formed from the 2s and 2p«r atomic orbitals at F, The second hybrid formed from these two atomic orbitals at F is regarded as a non-bonding orbital, and is doubly occupied. These two hyb rids may be expressed as d1 = sina 2s + cos <* 2per- (31) and d2 = sln/3 2s» + cos ,3 2pa~/ (32) where d^ is the bonding hybrid and d2 is the non-bond ing hybrid. The primes in the expression for d2 allow for the possibility that the radial functions used for the bonding hybrid could be different from those used for the non-bonding hybrid» ^ Is a variation parameter which determines the mixing of the atomic orbitals ln d^, and is chosen so as to ensure that d^ and d«> are orthogonal by taking tan/9 = -Spp / (Sss»tan<* ), (33) where Sgg is the overlap integral between the 2s and 2s' functions and S ^ is the overlap between 2p and PP 2p«. -31-Valence Bond and Molecular Orbital Wave Functions In the perfect pairing model the ground state wave function of HF can be written in unnormalized form as a combination of two determinants as in where there is a single electron pair bond between the hybrid d^ and the orbital combination designated h in equation (3*0s Is, ff^t and rr^ refer to doubly occupied non-bonding orbitals at F. Allowance Is made in equation (34) for the possibility of ionic character in the H-F <r bond, expected since F is more electronegative than H, by forming the electron pair bond between the hybrid d^ and a linear combination of the Is function at H with d^ defined as = ( sin Y . a + cos Y - <J t ) , (35) In equation (35). a Is the H^g atomic orbital orthogo nalized to the F core, and y is a variation parameter X s which gives a measure of the ionic character of the H-F a- bond. This formulation corresponds to the second in terpretation given to on page 18. When d^, dg» and h, as given by equations (3l)» (32), and (35) are substituted into equation (3k), Ypp can be ex panded as -32-^P-JL (36) where each V c6rresponds to a valence bond configu ration expressed In terms of the orthogonalized Sla ter-type orbitals. The fifteen which can be formed from the expansion of V , are listed in Table 4. The PP coefficients c^ in equation (36) are then determined by the values of the parameters , /3 , and Y . For the special case that 2s == 2s1 and 2ps 2p', corresponding to the minimum basis set situation, only three of the fifteen configurations ln Table 4 are different, and in this case the wave function obtained by solving the secular equation for these three structures is equi valent to the perfect pairing wave function in equa tion (34). These perfect pairing and valence bond wave functions for the ground state of HF are to be compared with the corresponding molecular orbital wave functions discussed below; howver, consideration is first given, within the perfect pairing framework, to some doublet states of HF+ which are of interest, in interpreting results from photoelectron spectroscopy^ and electron spectroscopy for chemical analysis^ on the neutral HF molecule. Two states of HF+ are considered in an approximate way. The first is a 25i state -33-Table If. Zero-order wave functions in equation (36) for the 1X. state of HF 1. V |lsfsn*Jr*ff^g-s 2s'2s2s| 2. *2= |lslS jr^n^/Tgff^S '2T'2p2pl 3. Jlslsfr^TTg^2? 2p»2s2s| 4. 11 s lis/r2 2 p1 fp^pSipl 5. |lsls'7r1fr^'n'2^22s "2s"'2sa| + J1 sIsTT2^2s' 2s1 a2s | 6. V |l sis n^fr^ t^n^s1 2s~'2paf + |lsls/T1F1 ^2^22s ' 2"s*a2Pl 7. Jl s 1 s ^7?^ ^2^2 2 s »2s"»2s2p] + | lslsP'177jfr2J7^2s'2s12p2s| 8. V 11 s 1 s n2*~2 2 p 2p12sa| + [lsrsTr1fr^7r2?fj2p,2p'a2s | 9. V llslsT^ff^trg^2? 2p»2pa| + (lsls 7T1lf^7r2?f22pl2p,a2p| 10. *io- 11 si s ?JV':22p 2^282^1 + |lsls7r1fr^7r2^2p«2p-'2p2sj 11. llsrs/r.;F"^ojf~2s 1 112 2 2p~'2s2s| +11 s Is rt TT^ tr2"F2 2 p12s12 s 2s | 12.t12= 11 si s •7r^7r^/77'2"^22s '2p»2p2p| + |lsrs^^rr2^2p,2s",2p2p| 13.V13= 11 s Is ^^i^^ 2 s' 2p»2s2p| +1 lsls ^^7r2^2P'2sT2s2p| + 11 S1 S ff^TT^ JTg^ 2 S 2p'2p2si + |lsrsV1rr^r2^2p,'2s12p2s| Islsi^rF^r^7^?23 2p»2sa|+ \ lsisr17r^7r27^2p,2Fl2sa| + (lsls^/F^r^fl^s' ^a^l + |l sis ^^^2^2?1 2s' a2s| 15.*15= llslSTT^^n^TT^S 1 2p~'2pa| + 11 s Is ^ 7^ ^2^2 2 p1 2~sT 2 pa/ + |lslsr^ff^rr2^2s 1 2p~'a2p| + [lsfs^tr^ /r2l7^2pl2s",a2p| Symbols are a$ in the text. -34-obtained by the ionization of an electron from the P1 orbital in the neutral HFj the second is a "ustate ob tained by removing an electron from a 2pTr atomic orbital at F. In the perfect pairing approximation, the appro-priate wave functions for these two states of HFT may be expressed in unnormalized form as %r * I isir^r^i^Xlri^U | jsn-jT,.ir,F^X^TI (37) and = I is is Bi^ Klt | + I is is ff, ff* ff* J JI + 11» fi ffi ^ JxUj + I IsH jr,^ 7TAd **XK I , (38) where S in both cases. The wave function in equation (38) also represents an approximation within the perfect pairing model for the ground state of the diatomic spe<* cies HO which is isoelectronic with HF*. Wave functions in the molecular orbital approach have been constructed so as to enable as close a compa rison as possible with the valence bond and perfect pairing wave functions discussed above. Consequently, the molecular orbitals were constructed from only the valence basis orbitals. The molecular orbitals which would involve mainly the inner basis orbitals, were sim ply taken to be core atomic orbitals as was done in the perfect pairing calculations. On this basis, the electronic wave function for the ground state of HF in -35-the molecular orbital approximation can be expressed as (39) where and are respectively the bonding and es sentially non-bonding molecular orbitals, defined as irt = Ns ( s«* S • a + coi & • <k 4) (24.0) and a- = t4, (i'^e-a + cos 6 (s/'» £^tt e*»£0),,, 6 (41) in such a way as to be useful for discussing hybri dization in this model. In equations (40) and (41) Nj and Nt are normalization factors; h and £ (along with <*) are variation parameters, and £ is fixed byi the condition that <r^ and <r^ are mutually orthogonal. Since the F^g core orbital has the same symmetry as and tTg, a free mixing of this core orbital must lead to a slight lowering of energy by the variation prin ciple. The corresponding approximate wave functions in the molecular orbital approximation for the 2- and ^TT electronic states of HF formed by removing either an F^g or an F2prr electron from HF are expressed as and -36-Comrjutatlonal Details Given the electronic wave function of a molecule and the electronic Hamiltonian, the electronic energy of the system is obtained by equation (5). When the determinantal wave functions defined in the previous section are substituted into equation (5). the expression for the electronic energy involves functions of the type and where ^ and <Kj are many-electron determinantal func tions. The general procedures for the evaluation of these many-electron matrix elements In terms of coef ficients and one and. two-electron molecular integrals 68 have been given by Lowdin and are fully discussed by 69 Slater. 7 The latter treatment was followed in this work with all expressions for the overlap and Hamilto--37-nian matrix elements being evaluated by hand by deter mining the appropriate coefficients, in terms of the overlap Integrals, for all the molecular integrals oc curring for the particular basis set. Computer pro grammes were written to sum all these contributions and consideration is now given to the methods employed to obtain the various molecular integrals. The basis set, as described above, becomes con taminated by the orthogonalization procedures, however, the one and two-electron molecular integrals over the basis orbitals are readily expanded in terms of one and two-electron integrals involving only Slater-type orbitals. A procedure due to Magnusson and ZauH?0 pro vides a convenient way of obtaining those electron-elec tron repulsion integrals which Involve a charge distri bution on a single centre such as where a and b are the two nuclei and ^ to are Slater-type orbitals on the indicated centres. This procedure involves expressing the integral in (46) as where ^(if^fg) *S tJie P°tential due to *ne charge dis tribution t/1S|//2a centred on a. Potentials of this type for Slater-type orbitals have been tabulated by Magnus--38-70 son and Zauli/ and some extensions and corrections 71 have been reported by Mitchell.' When using polar co-ordinates ( r, 9 , <P ) at each centre, the integral in (47) involves six variables? however, by intro ducing the elliptical co-ordinates ^ and v, defined as and y = jr ( - r b ) , (49) where R is the inter-nuclear distance, the integral in (47) can readily be expressed in terms of the three variables ju. t v, and <f> , where ^ measures the angle of rotation about the inter-nuclear axis a-b. Also for the known form of the Slater-type orbitals and the potential V, the Integration over <p is trivial and can be done an alytically. Thus the evaluation of the integral in (47) requires Integration over the two co-ordinates ^ and v, and to cover all space the respective ranges are 1 to °° and -1 to +1. The evaluation of integrals of the type in (46) was performed using two-dimensional Gaussian quadrature?2 with Legendre polynomials of order 16; and as an illustration of this technique, an integral involving a single variable with limits p and q is given in this approximation by -39-ffth)l..l£ I f(x, ?-' • VU) , (50) where x^ is the i^n root of the Legendre polynomial of order n, and a^ are tabulated constants associated with each x^. The form of equation (50) can be directly ex tended to any number of variables. One advantage of this approach for the evaluation of the two-electron integrals of the type in (46) is that all the one-electron integrals can be obtained by the same methods at the same time. The overlap and nuclear attrac tion integrals respectively, written in general as < fi I (51) and <*il$l*i> (52) represent special cases of the Integral in (47). Fur thermore, the kinetic energy integrals Ul-X^K-) (53) can be expressed in terms of overlap integrals as shown by Roothaan''7^ Who gave the expression -40-for the effect of the kinetic energy operator on a Slater-type orbital represented by ( nlm ) with expo nent cH • The method used in going from (46) to (47) is not applicable in a convenient way for evaluating the electron-electron repulsion exchange integrals of the type <WIV.1), (55) Exact numerical values of these integrals were obtained by using a computer programme written by Pitzer, Wright 74 and Barnett' and translated into Fortran IV by Mitchell. Since these Integrals were much the most time consuming, an approximation proposed by Lowdin-^ was also used to obtain values of the integrals. Lowdin's approximation consists of expressing the charge distribution V^Y^ as - s^xwr^KCW)], (56) where S^ is the overlap integral between ^ and W^, and and A2 are determined by the condition that the dipole moments of the charge distributions on the right and left hand sides of (56) are equal. Substitution of (56) into (55) yields -41-and the right hand side now involves integrals which can be evaluated by the numerical method discussed a-bove. Secular equations for wave functions of the type in equation (36) were solved with computer programmes from Quantum Chemistry Programme Exchange. ?5»76 For the molecular orbital and perfect pairing calculations, the molecular energies were minimized by varying the relevant mixing parameters by making successive five point per variable grid searches until the energy con verged to the fifth decimal place (energies in atomic units). The optimum orbital exponents were obtained by varying the individual exponents in turn until self-con sistency was achieved in the exponent values to two de cimal places. The bond distances corresponding to mi nimum energies for the various wave functions were ob tained by determining the orbital exponents for minimum energy for a series of bond distances, and then inter polating exponents linearly and calculating energies for the intermediate lengths, thereby allowing estima tion of the equilibrium distance. -42-Chapter Three Results and Discussion Using the wave functions and procedures described in chapter two, a series of calculations have been made for HF, HF+, and HO in their ground states, and also for HF+ in the 2£ state obtained on ionizing a fluorine core Is electron from HF, Computations have been made using molecule-optimized exponents for the Slater-type functions, and the resulting wave functions, molecular energies, one-electron energies, Mulliken populations, dipole moments, H-F bond distances and force constants are reported in Tables 5-12, Included in these tables are comparative results obtained from calculations using free atom exponents.7? Table 5. Orbital exponents and molecular properties for different wave functions of HF at the experimental bond distance (1.733 a.u.) Orbital exponents Molecular properties ^ Wave function His F2s F2pcr F2p7T Energy (a.u.) Dipole Moment (D) Ionization potentials (l<r)(eV) (lir) (eV) pp a 1.34 2.55 2.60 2.49 -99.5450 1.73 713.29 12.39 PP b 1.38 2.56 2.59 2.49 -99.5449 1.77 PP c 1.00 2.56 2.55 2.55 -99.4956 1.44 712.32 12.45 mo a 1.32 2.56 2.67 2.50 -99.5355 1.44 714.07 12.66 mo b 1.36 2.56 2.63 2.49 -99.5346 1.55 mo c 1.00 2.56 2.55 2.55 -99.4908 1.12 714.12 13.32 All Fls exponents have been optimized at 8,65. * All properties have been calculated exactly, a - All orbital exponents have been optimized completely. b - Orbital exponents have been optimized with the Lowdin approximation in (56). c - Free atom exponents have been used. Table 6. Variation parameters and Mulliken populations for different wave functions of HF at the experimental bond distance (1.733 a.u.) Variation parameters Mulliken populations Wave function sin <* sin JT sin 5 sin^ sin 6 His F2s F2p<r pp a 0.1203 0.93^ 0.685 1.999 1.316 PP b 0.1281 0.9313 0.679 1.999 1.322 pp c 0.0875 0.9^38 0.702 2.000 1.298 mo a 0.4000 0.6000 0.0344 -0.1970 0.773 1.9^4 1.284 mo b 0.2797 0.5922 -0.0422 -0.1488 0.7^ 1.9^5 1.312 mo c 0.309^ 0.6250 -0.0219 -0.2042 0.781 1.959 1.260 a, b, cf - are as in Table 5. Table 7. Orbital exponents and molecular properties for different wave functions of HF at calculated equilibrium bond distances Orbital exponents Molecular properties Wave function His F2s F2po- F2pjr Energy (a.u.) Bond length (a.u.) Dipole Moment(D.) Force constant pp a 1.33 2.55 2.60 2.49 -99.5456 1.77 1.72 a. 3 PP b 1.35 2.55 2.58 2.49 -99.5410 1.79 I.83 7.1 pp c 1.00 2.56 2.55 2.55 -99.5088 1.93 1.31 7.1 mG a 1.31 2.56 2.66 2.50 -99.5356 1.75 1.41 8.5 mo b 1.35 2.56 2.62 2.49 -99.5263 1.76 1.71 7.6 mo c 1.00 2.56 2.55 2.55 -99.5021 1.93k 0.89 7.2 All Fls exponents have been optimized at 8.65. a - All orbital exponents have been optimized exactly. b - The Lowdln approximation in (56) has been used for calculating exponents and molecular properties, c - Free atom exponents have been used and molecular properties have been cal culated exactly. Table 8. Variation parameters and Mulliken populations for different wave functions of HP at calculated equilibrium bond distances Variation parameters Mulliken ] populations Wave function sin <* sin if sin b sin ^ sin 6 His F2s F2p<r PP a 0.1188 0.9378 0.694 1.999 1.306 PP b 0.1376 0.9313 0.682 1.999 1.319 PP c 0.0700 0.9613 0.769 2.000 1.232 mo a 0.4938 0.5875 0.1000 -0.2301 0.777 1,9^8 1.275 mo b 0.3125 0.5750 -0.0313 -0.1428 0.721 1.936 1.3^3 mo c 0.3375 0.6563 0.0187 -0.2185 0.860 1.965 1.175 a, b, c, - are as in Table 7. Table 9. Orbital exponents and molecular properties for different wave functions of HF* at HF experimental bond distance (1,733 a.u.) Orbital exponents Molecular properties Wave function His Fls F2s F2p<r- F2pfl" Energy (a.u.) Dipole Moment (D.) PP a 1.49 8.97 2.77 3.03 2.97 -74.1245 2.70 pp b 1.50 8.97 2.77 3.02 2.97 -74.1245 2.73 PP c 1.00 8.65 2.56 2.55 2.55 -73.5303 3.80 mo a 1.49 8.97 2.77 3.03 2.96 -74.1103 2.81 mo b 1.50 8.97 2.77 3.02 2.96 -74.1102 2.80 mo c 1.00 8.65 2.56 2.55 2.55 -73.5316 3.83 a, b, cr are as in Table 7. Table 10. Variation parameters and Mulliken populations for different wave functions of HF \ at HF experimental bond distance (1.733 a, ,u.) Variation parameters Mulliken populations Wave function sin o< sin * sin & sin % sin e. His F2s F2po-PP a 0.1625 0.7938 0.466 1.999 1.536 PP b 0.1625 0.7888 0.459 1.999 1.5^2 PP c 0.0000 0.3719 0.146 2.000 1.854 mo a 0.4125 0.4000 0.0000 -0.1157 0.452 1.951 1.597 mo b 0.3625 0.4031 -0.0187 -0.1024 0.451 1.95^ 1.595 mo c 0.0469 0.1719 -0.0500 -0.0555 0.108 2.033 1.859 a, b, c,- are as in Table 7. Table 11. Orbital exponents and molecular properties for a series of wave functions for HF lt and H0_ Orbital exponents Molecular properties Wave function His Fls F2s F2p<r F2pT Energy (a.u.) Bond length (a.u,) Dipole Moment(D.) HF pp a 1.48 8.64 2.63 2.74 2.67 -99.1983 1.73 2.44 b 1.48 8.64 2.63 2.73 2.67 -99.1983 1.73 2.44 c 1.00 8.65 2.56 2.55 2.55 -99.0989 1.73 2.79 d 1.43 8.64 2.63 2.71 2.68 -99.2031 1.84 2.55 HF mo a 1.47 8.64 2.63 2.74 2.67 -99.1835 1.73 2.46 b 1.47 8.64 2.63 2.74 2.67 -99.1835 1.73 2.46 c 1.00 8.65 2.56 2.55 2.55 -99.0900 1.73 2.87 H0*pp b 1.28 7.66 2.24 2.27 2.18 -75.1154 I.83 1.16 * Calculation for HO ** 1.73 is a fixed value for the bond, length, a, b, c, - are as in Table 7. d. - The Lowdin approximation has been used, for calculating exponents and molecular properties. Table 12. Variation parameters and Mulliken populations for a series of wave functions for and HO Variation parameters Mulliken populations Wave function sin <* sin r sin b sin ^ sin £ His F2s F2p<r-PP a 0.1560 0.8391 0.525 1.999 1.476 PP b 0.1563 0.8375 0.523 1.999 1.478 PP c O.0656 0.7313 0.384 2.000 1.616 pp d 0.1594 0.8438 0.53^ 1.999 1.467 mo a 0.0734 0.4563 -0.1391 -0.0311 0.530 1.948 1.522 mo b 0.0734 0.4563 -0.1391 -0.0311 0.530 1.948 1.522 mo c 0.29H 0.3625 -0.0438 -0.1115 0.373 1.982 1.645 PP b* 0.1500 0.9881 0.863 1.999 1.139 * Calculation for HO. a, h, c, - are as in Table 7. d - is as in Table 11. -51-Atomlc Orbital Extxments The first choice of variables for molecular cal culations with a basis set of Slater-type orbitals con cerns the selection of appropriate orbital exponents. In semi-empirical schemes, free atom exponents are usu ally used,3^»39 however, examination of the optimized exponent values in Tables 5» 7» 9» and 11 shows that in certain cases the exponent values are considerably mo dified from atomic values; and this indicates that in general the choice of suitable exponent values is not a trivial one. Looking first at the exponent values for HF in Tables 5 and 7, a significant change has occurred in the His exponent, from 1.00 for the free atom value to the optimized value of about 1.32, depending on the particular wave function. As the distance of maximum probability for a Slater-type orbital is given by • , (58) where n is the principal quantum number and °< is the orbital exponent, an increase in an exponent value cor responds to a contraction of the Slater-type orbital. The His orbital appears to be contracted ln HF com pared with the free H atom, and the contraction can be related, In part at least, to a transfer of charge from -52-H to F expected by electronegativity arguments and shown by the Kulllken populations in Tables 6 and 8. This charge transfer results in H becoming positively charged and the electronic density at H is in conse quence held more tightly. The optimized exponents for the Fls, F2s, and F2pfl" orbitals experience only small changes from the free atom values, although an increase of around 0.1 is shown by the F2p<?~ orbital. Previous experience^6,78,79 jiag ±nftic8Lte& that exponent values often tend to increase by this amount for orbitals in volved in bonding, and this can be related to the Vi-rial Theorem.^0 When highly polarizable excited orbi tals are involved in bonding however, the changes in 81 exponent values may be large. x Similar exponent va lues are found from both the perfect pairing and mole cular orbital calculations for HF, the greatest dif ference for the calculated equilibrium bond length is 0.06 for F2p*r. The results for HF and HF+Jr show that the Fls exponent is not sensitive to changes in the valence shell electronic structure, and this is expected for a core orbital which has a very low polarizabllity, As shown in Table 9 however, on ionizing an Fls elec tron from HF this exponent value is increased very significantly. In general, for this ionization all the exponents are increased and this corresponds to a -53-contraction of the atomic orbitals which is expected since the remaining electrons will be held more tightly in the positively charged species, as has been noted in previous calculations on C2H2 and C2H2+ by Goodman and 8? Griffith, although these workers did not optimize the His exponent which they fixed at 1.20. In HF"tt the Fls exponent now has a value of 8.97 which is very close to the value (9.00) obtained by Slater's rules.^ The His exponent in HF+a^ is 0.5 larger than the free atom value, and 0,2 larger than that for HF. Again, this increase for HF+j^ can be associated with the large transfer of electronic charge from H to F. It may be noted that the effect of this charge transfer is that HF+a^ approximates to the situation represented by H+-F where the electron distribution at F tends towards spherical symmetry. This is reflected in the F2p(j-and F2pn- exponents being more nearly equal than, for example, in HF, The optimum exponent values in the perfect pair ing and molecular orbital models are nearly identical, the greatest difference being 0,01, and again this can by rationalized by the tendency to approach H+-F, The optimum exponent values for HFift., listed in Table 11, are intermediate between the exponent values for HF and those for KFI^ , With the same doubly occu pied Fls core, the Fls exponent for HF"^^ has the value of 8,65 equal both to that for the free atom, and that -54-for HF. The contraction of the His orbital in HF+r is less than that in HF*^ but greater than the His con traction in HF. The results ln Tables 5-12 give evidence that calculations of molecular properties such as bond lengths and dipole moments with minimum basis sets are much im proved if molecule-optimized exponents are used rather than free atom exponents. In applying this result to more complex molecules, it will be necessary to have con venient and relatively inexpensive methods for optimizing Slater orbital exponents, and with this in mind consider ation has been given to obtaining optimal exponents when the time-consuming electron repulsion integrals involving two two-centre charge distributions are evaluated using the approximation due to Lowdin in equation (56). The first point to note is that in all cases in Tables 5t 7» 9» and 11, the optimum exponents obtained using the Low-din approximation are quite similar to the values from e-xact calculations and therefore are rather different from free atom values; in the cases of HF"^ and the a-greement is very close. For these cases the contraction in the His orbitals, as reflected in the large His expo nent values, reduces the numerical values of the two-centre exchange Integrals with the consequence that the errors in troduced by the approximation are reduced also. Likewise, for these two states of HF+, the molecular properties cal--55-culated with the Lowdin approximation are very similar to those from the exact calculations, and even for neu tral HF the errors introduced are not large considering the saving in computation time. This suggests it could be advantageous to investigate further in this context. Molecular Energies As noted in Tables 2 and 3, (with the exception of HF+3£ ), molecular energies lower than the values repor ted in Tables 5t 7t and 11 have been given previously for the molecular species of interest here. However, the intention in the present work is to restrict the ba sis sets to forms which have applications to more com plex molecules, and consequently the results obtained will be discussed more in relation to similar calcula tions, rather than to those with the extended basis sets noted in Tables 2 and 3. The molecular energies for HF at the calculated e-qullibrium bond distances listed in Table 7» show that the perfect pairing model gives a molecular energy (-99.5^56 a.u.) which is 6.27 kcal./mole lower than that from the molecular orbital model (-99.5356 a.u.) for a minimum basis set of Slater-type orbitals with energy-optimized exponents. Using the Lowdin approx--56-imation for the two-centre exchange integrals in the way described above, the perfect pairing wave function gives a molecular energy (-99.5410 a.u.) 9.22 kcal./mole lower than the molecular orbital calculation (-99.5263 a.u.), and this perfect pairing energy is 2.88 kcal./mole higher than that obtained when all Integrals are evaluated ex actly. Using free atom exponents?? and evaluating all integrals exactly, the molecular energy for HF for the perfect pairing wave function (-99.5088 a.u.) is 3.20 kcal./mole lower than that for the molecular orbital wave function (-99. 5021 a.u.) but it is 23.07 kcal./mole above that obtained with energy-optimized exponents. Similarly, the energy for HF calculated with the mole cular orbital model is 21.00 kcal./mole higher when free atom exponents are used Instead of molecule-optimized exponents. Thus for the three different sets of calcu lations in Table 7, the perfect pairing model yields lower energies than the corresponding molecular orbital calculations, and the use of free atom exponents gives energies more than 20 kcal./mole higher than the ener gies obtained with molecule-optimized exponents. As expected, results in Table 5 show similar trends for calculations on HF with the bond length fixed at the experimental value (1.733 a.u.). Previously, Hansil^? has used a minimum basis set of Slater-type orbitals for a molecular orbital calcula--57-tlon on KP at the experimental bond distance and repor ted an energy of -99.4785 a»u» using orbital exponents obtained from Slater's rules (Kls=1.00, Fls=8.70, F2s=2.60, F2p=2.60). An energy 0.0006 a.u. (O.38 kcal./mole) higher than Ransil's energy vras obtained with the computer programme used in this study for the same values of bond length and orbital exponents. This difference is attributed to the use in this work of the pure Is atomic orbital at F in the molecular orbital wave function in equation (39). By the variation prin ciple this constraint must raise the energy compared with the case when all atomic orbitals of the same sym metry type are freely mixed to form the molecular orbi tals . When the same values are given separately to the F2s and the F2pff" exponents in the hybrids d^ and dgt the perfect pairing wave function in equation (3^) corres-ponds to a free mixing of the configurations (F2s) (Hls)1(F2p<r)1t (F2s)2(F2p<r)2, and (F2p<r)2 (His)1 (F2S)1, ( omitting the common core (Fls)2(F2p7T'1)2(F2piT1)2). Pre viously, Silk and Murrell,59 using a minimum basis set of Slater-type orbitals, mixed these three configura tions In combination with three more configurations cor responding to E_F+. Silk and Kurrell'gave the same ex ponent values for the F2pc" and F2pfl" orbitals and calcu lated an equilibrium bond length of 1.86 a.u. in only -58-fair agreement with the experimental value (1.733 a.u.). Molecular energies for the perfect pairing wave function in equation (34) can not be compared directly with Silk and Murrell's because they mixed more configurations, but at 1.4 a.u. where they find the H~P+ configurations to contribute only slightly, the programmes used in this work give a molecular energy (-99.472 5 a.u.) 1.25 kcal./mole higher than their published value (-99.4745 a.u.), using their exponents. As for HF, calculations on HF",^ using optimized orbital exponents give lower molecular energies for the perfect pairing wave function than for the molecular or bital model. When exponents are optimized for exact calculation of all molecular Integrals, the perfect pairing energy is 8.91 kcal./mole lower than the mole cular orbital energy. As noted above, the use of the Lowdin approximation yields exponents in close agreement with those from the exact calculation, and the increase in energy in using Lowdin exponents is only 0.07 kcal./mole. A very great difference occurs when free atom exponents are used; in the perfect pairing model the molecular energy is then 373 kcal./mole above the value obtained with molecule-optimized exponents and the corresponding value for the molecular orbital model is 363 kcal./mole. This large difference between energies obtained using free atom exponents and those obtained -59-with optimized exponents, emphasizes that electronic re laxation must be included in calculations of E.S.C.A. energies. As noted in the section on orbital exponents, however, this study does Indicate that it may be possible to use Slater's rules^ or something similar?? for esti mating exponents when the positive ion is formed by the removal of a core electron. The comparison of perfect pairing and molecular orbital energies for HP+Jff. is similar to that reported above for HF and HF4"^ , and the details are to be found in Table 11. The interest in the energy of HFin this work is mainly in relation to calculating the first ionization potential of HF with a minimum basis set of Slater-type orbitals. The energies are presented in Table 5 for the io nizing of an Fls electron (ltr) or an F2p7T electron (iff) as calculated assuming no reorganization of the remain ing electrons ( this is usually referred to as Koopman' Theorem^) . A value of 13.32 e.V, is calculated for the 1 vr ionization potential with free atom exponents, and a bond length of 1.733 a.u,, and this is to be compared 84 with the experimental value of 16,06 e.V, by photoe lectron spectroscopy. Using Koopmans1 Theorem and ex ponents given by Slater's rules, Pople and Beveridge^ report a value of 12.65 e.V. with the molecular orbital theory, and with Slater-type orbitals optimized for HF -60-(Table 5) calculated values of 12.66 and 12.39 e.V. are obtained for the molecular orbital and perfect pairing models respectively. In principle, an improved calcula tion of the vertical ionization potential is made by taking the differences between the molecular energies of HF and HF+J/r for the H-F bond length, but surprising ly the first ionization potential calculated this way has a value in less good agreement for both the perfect pairing and the molecular orbital model using either optimized or free atom exponents , than the value ob tained with Koopmans' Theorem. Thus at the experimental bond length (1.733 a.u.) energy differences between Ta bles 5 and 11 give a value of 10.90 e.V. from the mole cular orbital method using free atom exponents. The reasons for the less good agreement in taking the dif ferences between the state energies is not clear, al though presumably It is related in part to the restricted form of the basis set. As noted already, the differences between using free atom exponents and molecule-optimized exponents are much larger for calculating the energy of ionizing an Fls electron. Using Koopmans* Theorem, this ionization potential is calculated to be 71^.12 e.V. when using free atom exponents and the molecular orbital method. This value can be compared with the values of 706.26 e.V, and 691.57 e.V, obtained respectively with free atom -61-exponents and molecule-optimized exponents when dif ferences in the molecular orbital energies for HF and HF+j£ in Tables 5 and 9 are taken. Unfortunately, these numbers can not be compared with experiment since HF does not seem to have been studied by E.S.C.A. yet. Bond Lengths and Force Constants Equilibrium bond lengths have been calculated u-sing the perfect pairing and molecular orbital wave functions for various sets of orbital exponents, and the method for obtaining the equilibrium distance has been described on page M. The results in Table 7 show that the molecular orbital method using molecule-optimized exponents gives an equilibrium bond distance of 1.75 a.u. which is only 0.02 a.u. (0.01 A) longer than the experimental value of 1.733 a.u.$3 The perfect pairing method gives a calculated value of 1.77 a.u.. Using exponent values obtained with the Lowdin approximation, the calculated equilibrium bond lengths are within 0.02 a.u. (0.01 A) of the best calculated values, and there fore are in reasonable agreement with experiment. It is significant that the calculations with free atom expo nents give bond lengths in poor agreement with experi--62-mental values. Thus using free atom exponents, the perfect pairing model predicts an equilibrium bond length 0.20 a.u. greater than that obtained using op timized exponents. Likewise, the molecular orbital cal culation gives a bond distance 0.18 a.u. larger with free atom exponents than that obtained with molecule-op timized exponents. Thus for these calculations, the error in calculated bond lengths using free atom expo nents is an order of magnitude greater than the error introduced using exponent values optimized with Low-din's approximation. As discussed above, and shown in Table 11, the optimum orbital exponent values for HF+X^. calculated with and without Lowdin's approximation are very nearly equal. Therefore it seems reasonable to calculate the equilibrium bond length of HF+, by using the Lowdin approximation for optimizing exponents at different bond distances. The calculated equilibrium value of 1.84 a.u, is close to the value (1.85 a.u.) calculated 6? recently by Richards and Raftery, although appreciably lower than the value of 1.96 a.u, quoted by Fople^0 u-sing a basis set of Gaussian functions derived from a-tomlc wave functions. No experimental value for the equilibrium distance in HF"^ is presently available. The Lowdin approximation has also been used for a pre liminary calculation on the equilibrium distance in HO -63-for which the experimental value is I.83 a.u. ^ in the 2/T state. Using the Lowdin approximation, the opti mized exponent values at 1.83 a.u. are Hls=1.28, 01s=7.66, 02s=2.24, 02p^-=2.2?, and 02ptf=2.l8 and with these ex ponents the calculated equilibrium distance is 1.91 a.u,, 0.08 a.u, longer than the experimental value. As there is less charge transfer from H in EO compared with EF"£j. the Lowdin approximation may be less reliable for HO, .It is possible, therefore, that a calculation with all integrals evaluated exactly would give an improved va lue for the equilibrium bond distance in HO, Like bond lengths, values of force constants are often used to give information about the nature of the bonding. The calculation of a force constant depends on calculating molecular energies as a function of the displacement from equilibrium, and in the harmonic ap-2Q / \ proximation, 7 the stretching force constant (k) for a diatomic molecule is obtained from £ = T (r ' r*)X (59) where E is the molecular energy calculated at a bond length r, and re is the calculated equilibrium bond distance. The force constant, therefore, is readily determined from a plot of E versus (r-re) . Using molecule-optimized exponents, HF stretching force constants equal to 8,3 x 10^ dynes/cm and 8,5 x 10^ -64-dynes/cm are obtained from the perfect pairing and mo lecular orbital models respectively, and these values are to be compared with an experimental value of 9.66 x 1CK dynes/cm. The correspondence to the experimen tal value is less good by about 1 x 10^ dynes/cm when free atom exponents are used to calculate the force constant. Electron Distributions The calculated charge distributions in the mole cules of interest in this work are determined by the values of the variation parameters <*• , * , 5 and ^ in e-quations (31)» (35)»(40), and (4l). Experimentally, in formation relating to charge distributions is obtained by measurements of dipole moments and of the higher mo-8 5 86 ments, ' and these moments may be calculated from molecular wave functions. Thus, for a molecule, the dipole moment, which is a vector quantity, is given by ^ , -*<f/fc|r<> (6o) for a state function f; r is a sum of the electron po sition vectors. For diatomic hydrides with cylindrical symmetry about the internuclear axis, the dipole moment is directed along this axis with magnitude -65-(61) where ^ is the electronic wave function, z is the sum of components along the internuclear axis of electron positions, r^ and are respectively the position and charge of the i^n nucleus. Another convenient measure of electron distribu tions which is used frequently for molecular wave func tions expressed as a basis of atomic orbital functions On is provided by the population analysis due to Mulliken. In the molecular orbital model, when the l^n molecular orbital is expressed as % •• 1 C« ^ . (62) the total electron population of in the linear com bination of atomic orbitals - molecular orbital method, is given by PJ*° * 1 "i {CiJ 4- £ C,-u C;YSUV ] (63) where The summation over i is over all occupied molecular or bitals and n^ is the occupation number. Implicit in equation (63) is that the overlap charge distribution has been partitioned equally between the two centres -66-involved. An equivalent population analysis for va lence bond wave functions is obtained according: to the o o following procedure:00 - f , <65) where the zer-order wave function V'; corresponds to a configuration with occupancy n (i) for the atomic or bital fiu. Then the total electron population in fiu in the valence bond method is given by i ( i*i ) where V itiWi). (67) Some of the variation parameters in equations (35) and (40) provide measures of electron distributions. Thus for the perfect pairing model an Increase in sin * indicates an increase in the charge at H; sin 5 equal to 1.00 implying no charge transfer while sin*equal to 0.0 corresponds to transfer of one electron from H. Simi larly in the molecular orbital model sin S is a measure of the charge at H in the bonding molecular orbital. Quantitatively, as the values of either sin * or sin $ decrease, one may expect the His orbital population to decrease and correspondingly the dipole moment to in crease. In both the molecular orbital and perfect pair--67-ing models, as used in this work, sin <* is a measure of the sp hybridization at F. As sin o<" increases, the hy brid designated d^ has more F2s character, and corres pondingly the hybrid designated dg has less F2s charac ter. The trends in these various measures of electron distribution will now be examined for the different mo lecular wave functions. Looking first at the results in Tables 7 and 8 for the calculated HF equilibrium distances, the agree ment to 0.01 in the values of sin Jf or sin 5 using the exactly-optimized and the Lowdin-optimized exponents is reflected in the His populations being similar for either set of exponents. The His populations are, how ever, slightly higher for the molecular orbital model (0.78) than for the perfect pairing model (0.69); and this is consistent with the calculated dipole moment being higher for the perfect pairing calculation (1.72 D.) than for the molecular orbital model (l.4l D.), The experimentally-measured dipole moment of HF is 1.82 D..^9 With free atom exponents, the charge redistri bution on formation of HF is calculated to be less, and this is reflected in the lower calculated values of the dipole moment, being 1.31 D» and 0.89 D. for the per fect pairing and molecular orbital calculations respec tively. Similar trends in results are found for the calculations at experimental distance of 1.733 a.u.. -68-The values of sin <*. in Tables 6 and 8 indicate that the perfect pairing model is consistent with some what less sp hybridization than the molecular orbital method, and although it is well known that the concept of hybridization is not necessary in the molecular or-12 bital theory, results of comparing values of sin «* are consistent with the Mulliken populations on the P2s orbital being 0.05 greater in the perfect pairing cal culation than in the molecular orbital calculation. Nevertheless, the hybridization at F in HF is small in both models, as expected from the average 2s to 2p pro motion energy, which from atomic spectral data is es-12 timated to be 20.8 e.V. The results obtained are consistent with the bonding hybrid d^ being essentially F2p<r, and therefore, the non-bonding hybrid designated d2 being mainly F2s in character. Even though the sp hybridization at F seems to be small, the hybrids have been looked at in a different way for the purpose of molecular calculations. This ex tension involved assigning one orbital exponent to d^ and a different exponent to d2 without regard to the basis Slater-type orbitals; that is the Slater exponents are selected such that c< is = (68) -69-and \ with a. 5 ^ <=< -?s' . (70) With these basis functions, the energy was completely minimized for the perfect pairing wave function. The optimized exponents were found to be Kls=1.35» Fls=8,65» F2pjr=2.49, d1=2.62 and d2=2.55. The optimum exponents for d^ and d2 are within 0.02 of the optimum values of F2p<r and F2s in Table ?, however, this approach results in a perfect pairing energy of -99.5^59 a.u. which is 0.19 kcal./mole lower than the previous best perfect pairing energy, and a calculated equilibrium H-F distance of 1.75 a.u. which is 0.02a.u. better than the perfect pairing calculation using the more conventional basis of atomic orbitals. Assigning exponents to hybrids ra ther than natural atomic orbitals might be expected to give greater improvements in the calculations of pro perties of molecules ln which the hybridization of a-tomic orbitals is suggested to occur to a greater ex tent than at F in HF, An interesting observation from all the results is that with the single exception of the molecular or bital calculation for HF* atomic orbital hybridiza--70-tion at P is less when free atom exponents are used in stead of molecule-optimized exponents. This suggests that one should perhaps be cautious in deducing conclu sions regarding possibilities of hybridization purely from considerations related to data for free atoms, al though it may be noted that in the limit of complete transfer of electronic charge from H to F, the hybri dization picture becomes irrelevant. It is not too clear at present, but the fact that the molecular orbi tal calculation of HF+ajr is out of line may be related to this consideration. Another odd feature of this calculation is that the Mulliken population for F2s is indicated, to be greater than 2. This result is asso ciated with the equal partitioning of overlap charge between the two centres in the Mulliken analysis. This is not realistic and is well known to yield negative 90 populations in some cases. The results in Tables 10 and 12 show that large modifications occur in the calculated electron distri butions for HF*^ and HF+a7r when free atom exponents are used instead of molecule-optimized exponents. How ever, when the Lowdin approximation is used for deter mining orbital exponents for these molecular species, the Mulliken populations and dipole moments are in close agreement with those from the exact calculations. As expected, the charge transfer from H to F increases in -71-the series HF, K?+r , HF*^ , and this is reflected in the values for the Kulliken populations and the calcu lated dipole moments. Also, as noted above, the His ex ponent tends to increase with this charge transfer. A further comparison with HF+llr is provided by a preliminary calculation on H0ijr with the perfect pair ing model and utilizing the Lowdin approximation. In this case it turns out that sin ^ is close to unity and, correspondingly, the His electron population is 0.86. This is to be expected because of the much lower elec tronegativity of 0 compared with F in the species be ing considered. The smaller charge transfer compared with F results in a lower dipole moment; our calculated value of 1.16 D, is to be compared with the experimental value of 1.66 D.91 for HO. Concluding Remarks The interest in finding useful methods for deriving wave functions with applications to complex molecules stems from many considerations, but without doubt there is currently much interest in developing quantum chemical methods with applications to molecular systems as di verse as those of biological Interest^ and those present in structures of the solid state.9-2 Inevitably, methods -72-with application to complex systems must first be tested on simpler molecules. On the whole, the molecular orbi tal method has proved most useful in applications to com plex systems so far, 38,39 but recent advances in compu tational techniques have indicated the feasibility of ma king valence bond calculations on a more routine basis to polyatomic molecules ,8»^5 and it has been known for some time that calculations in the perfect pairing mo del can be formulated readily with comparatively simple expressions for molecular energies. 9»69 Probably in the future, wave functions for complex molecules will be written so as to represent some hybrid of the molecular orbital and perfect pairing schemes, such as is built into many conventional bonding models (eg. the localized and delocalized components of the electronic structure of benzene). In part, this thesis has been directed at comparing the molecular orbital and perfect pairing mo dels for some simple diatomic hydrides, and this repre sents an extension of the recent work by Kitchell and 46 Thirunamac hand ran on BerL,. We can conclude that for minimum basis set calculations, both the perfect pairing and molecular orbital models provide reasonable accounts of a number of basic properties of HP, HF*^ and HF+j^. More specifically, calculations of molecular energies using wave functions in the perfect pairing framework are between 6 and 10 kcal/mole lower than those from -73-corresponding calculations using molecular orbital wave functions. Equilibrium bond lengths calculated with both the perfect pairing and molecular orbital mo dels are within 0.04 a.u. (0.02 A) of the experimental value for HF; where comparison with experimental data is possible dipole moments seem to be better predicted with the perfect pairing model, although, force con stants and ionization potentials are calculated with si milar reliabilities with the perfect pairing and mole cular orbital methods. To some extent this contrasts with the situation for BeHg, where the perfect pairing model seems to be preferable to the molecular orbital model; in part this contrast can be associated with the greater electronegativity difference between H and F? thus in the limit of H+F*" both the perfect pairing and the molecular orbital models would merge with the ionic model. The comparisons made here between calculations with free atom exponents and those with molecule-opti mized exponents show that free atom exponent values can not in general be considered appropriate for calculations which attempt to evaluate properties such as bond lengths and dipole moments. Often semi-empirical methods3^ give the same exponent values to the different 2p orbitals, which, in a molecular environment, are often inequlva-lent. This restriction is made in part for simplicity -74-and in part to maintain rotational lnvariance. Ne vertheless, it should be noted that this constraint does introduce error compared with the situation where the symmetrically different 2p orbitals have different exponents. It may be noted that in this work, the per fect pairing calculations for HF gave much better cal culations of the H-F equilibrium distance than that re-ported by Silk and Murrell, 7 even though these workers used a larger set of basis configurations for their va lence bond wave functions. The difference in bond length seems to be due to the allowance in this work of diffe rent 2pr and 2pJT exponents, whereas Silk and Murrell constrained theirs to have a single value. For this ba sis set with Slater-type orbitals it seems therefore that the choice of orbital exponents is quite crucial for estimating molecular properties. Further studies of the changes in exponents from free atom values are neces sary to enable reasonable predictions of suitable ex ponents for larger molecules. The determination of molecule-optimized exponent values by means of exact calculations does, however, become very expensive for large molecules, and for this reason it is important that there are approximate schemes available for limiting the computational effort. One way Is to approximate the molecular integrals by dras tically limiting the number that need to be calculated -75-exactly. 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