Wave Functions for Small Molecules based on Linear Combinations of Atomic Orbitals By R. G. Clark* and E. Theal Stewart CHEMISTRY DEPARTMENT UNIVERSITY OF DUNDEE SCOTLAND 1 Atomic Orbitals in Molecular Wave Functions That molecules are made up of atoms is not in itself a good reason for construct- ing molecular wave functions from atomic wave functions. The LCAO (linear combinations of atomic orbitals) procedure has no fundamental wave-mechanical basis and it can thus be justified in a quantitative sense only by the evidence of a large number of uniformly successful calculations on a variety of molecules. It is only very recently that this evidence has become available. Because of the difficulty caused by the number and the complexity of the electron-repulsion integrals for all but the smallest molecules the accumulation of the required information has had to await the development of large-scale electronic computing techniques.Until the late nineteen-fifties most LCAO calculations were restricted to two widely different groups of molecules on the one hand there were strict and accurate calculations on a few exceedingly small molecules and on the other there were Huckel (or modified Hiickel) calculations on a great number of large w-orbital systems. The former were too unrepresentative to be generally informa- tive the latter too heavily loaded with experimental data. It is obvious from the Bibliography in Section 6 that great progress has been made during the last decade in bridging the gap between the two groups of mol- ecules. It is now known not only that LCAO calculations can give molecular energies correct to within ca.0.5 % but that reasonable estimates can be made of spectroscopic constants and various molecular properties dependent on elec- tronic charge distribution. Computations on strict wave-mechanical principles have been carried out on molecules having nearly seventy electrons without recourse to experimental data. The work is constantly being extended but the main conclusions are now clear so this seems an appropriate time to review the subject for chemists in general and to indicate both what has been achieved and what has not. Several earlier accounts have been given for specialist readers the latest and most comprehen- sive being by Krauss.l An admirably clear summary of the position in 1959 was provided by Allen and Karo,l and we take this as our starting point.*Present address Department of Computing Science University of Stirling Scotland. l M. Krauss 'Compendium of Ab Znitio Calculations of Molecular Energies and Properties' Technical Note 438 National Bureau of Standards Washington 1967. L. C. Allen and A. M. Karo Rev. Mod. Phys. 1960 32 275. 95 LCAO Wave Functions for Small Molecules 2 Molecular-orbital Wave Functions If as in the majority of cases considered in this Review N molecular orbitals xl x2 . . . XN are used to describe the ground state of a 2N-electron molecule there is only one way of distributing the spin factors u and p and the complete wave function is thus the single determinant3 where xi signifies xiu and xi signifies xtp. The determinant (1) is obtained by imposing the Pauli antisymmetry requirement on the much simpler Schrodinger function (2) Because it is formulated by ignoring electron-repulsion terms in the Hamilton- ian operator the simple product (2) cannot be an exact solution of the relevant Schrodinger energy equation however accurate the individual molecular orbitals may be; the same is true of the determinant (1).If all the molecular orbitals have their best possible forms (and not merely the best forms obtainable by varying a small number of parameters such as LCAO coefficients or orbital exponents) the wave function (1) is described as a Hartree-Fock or self-consistent-field wave function. The difference between the Hartree-Fock energy for an atom or molecule and the energy obtained from an exact wave function with the same Hamiltonian operator is known (conveniently but for reasons which transcend the basic postulates of wave mechanics) as the correlation energy.The correlation energy of a molecule is almost always larger than the sum of the correlation energies of the component atom^.^ The wave functions discussed in this Review though very often (and very improperly) describedas Hartree-Fock or self-consistent-field wave functions are in fact just LCAO wave functions in which each molecular orbital x is written as a linear combination of atomic orbitals $a $b . . . $m (3) The rn atomic orbitals in any calculation are known as the basis set. The greater the value of my the more nearly an LCAO wave function approaches the Hartree- Fock wave function. The atomic orbitals are assigned the usual spherical polar form3 # = ROO where the angular factors @ and @ are exactly the same as for the hydrogen atom Xi(1) Xa(3) zd4) X N ( N - 1) xN(N) X = ca#a -k Cb#b + - - + cm#m and R is given by R =p-le-C;t (4) C.A. Coulson and E. T. Stewart in ‘The Chemistry of Alkenes’ ed. S. Patai Interscience New York 1964 ch. 1. 4 R. K. Nesbet Adv. Chem. Phys. 1965,9 321. 96 Clark and Stewart (as in a Slater5 orbital). For each atom in each molecule the ‘orbital exponent’ 5 depends not only on the quantum numbers n and I but also (because molecules lack the characteristic spherical symmetry of atoms) on the quantum number rn. Optimising the molecular wave functions subject to the restrictions imposed by the forms (3) and (4) requires the minimisation of the electronic energy with respect to the coefficients c in the molecular orbitals (3) and the exponential parameters < in the atomic orbitals (4).The variation of the coefficients is straightforward in principle and is normally effected by a method of successive approximation devised by Roothaan;’ the variation process is always carried to completion. On the other hand optimisation with respect to all the atomic- orbital exponential parameters [ in a molecular-orbital wave function is a very tedious process which can only be carried out by trial and error; it is usually left incomplete or omitted altogether. The values of f used in a molecular-orbital wave function are often those appropriate to variational calculations on the free atoms or simply those specified by Slater’s Rules? As a half measure the members of a set of arbitrarily chosen orbital exponents may all be multiplied by the same scale factor.If the energy is minimised with respect to the scale factor the variation principle and the virial theorem can be satisfied simultaneously.8 As far as computation is concerned an LCAO wave function can usually be improved more efficiently by extending the basis set than by optimising the atomic orbital exponents. The larger the basis set the less the improvement that remains to be brought about by varying the atomic-orbital exponents. Of course the ‘chemical’ interpretation of a wave function is simplest when the basis set is as small as possible. A method of extending the basis set which is directly related to the choice of orbital exponents involves the use of Clementi’s ‘double-zeta’ atomic orbitals,@ in which the simple exponential factor exp(- f:r) in the radial function (4) is replaced by a linear combination of two exponential factors exp(- c1r) and exp(- &r).Optimum values of and la are known for atomic orbitals with 2 < 36. These same exponents are used in a molecular calculation but the linear- combination coefficients are included in the variation process applied to the coefficients in equation (3). If a molecular wave function based on a single configuration of molecular orbitals is insufficiently accurate for its purpose the spin-orbital product (2) must be replaced by a linear combination of spin-orbitals products i.e. the determi- nant (1) must be replaced by a linear combination of determinants Y = C l Y + C2Y2 + . . (5) 6 J. C. Slater ‘Quantum Theory of Atomic Structure’ McGraw-Hill New York vol.1 1960. J. C. Slater ‘Electronic Structure of Molecules’ McGraw-Hill New York 1963; W. N. Lipscomb Adv. Magn. Resonance 1966,2 137. 7 C. C. J. Roothaan Rev. Mod. Phys. (a) 1951,23,69; (b) 1960 32 179. P.-0. Lowdin J. Mol. Spectroscopy 1959 3 46; A. D. McLean J. Chem. Phys. 1964,40 2774. 0 E. Clementi J. Chern. Phys. 1964,40,1944; E. Clementi R. Matcha and A. Veillard. ibid. 1967,47 1865. 97 LCAO Wave Functions for Small Molecules Each spin-orbital product or determinant represents one configuration. Varia- tional techniques for determining the coefficients Cl C2 . . . have been described by Clementi.lo The purely mathematical process of building up a wave function by linear combination of single-configuration determinants is known (mis- leadingly but almost universally) as configuration 'interaction'.Even a singleconfiguration wave function may have the form (5) if there are several equivalent ways of assigning spin factors to the molecular orbitals each determinant corresponding to one assignment. This will be the case if (as in most excited states) there are unpaired molecular orbital^.^ In these circumstances the linear -combination coefficients are found by using the Schrodinger spin opera- tors; the variation principle is not required for determining the coefficients C, C, . . . in (9 but the form of (5) complicates the algebraic procedures76 used in determining the coefficients ca Cb . . . in (3). 3 Types of Basis Function Although this Review is concerned with the construction of molecular orbitals from atomic orbitals we must mention in passing other types of basis function which are at present of considerable importance.Even with the extensive computing resources now available the accurate evaluation of the three- and four-centre integrals which arise in LCAO calcula- tions on polyatomic molecules still takes a considerable time. These are electron- repulsion integrals in which the four atomic orbitals in the integrands are functions of co-ordinates having three or four different origins (at the various nuclei). The difficulty is greatest for non-linear molecules which accounts for the otherwise surprising number of unusual linear molecules listed in Section 6. The evaluation of individual three- or four-centre integrals can be greatly simplified by replacing atomic orbitals of the Slater type (4) by Gaussian func- tions,11,12 in which the exponents are - cr2 instead of - (r.Unfortunately to obtain wave functions of comparable accuracy requires about five times as many Gaussian functions as Slater functions; i.e. ca. 54 (= 625) times as many integrals have to be evaluated. Nevertheless Gaussian basis sets have been used with considerable success for quite large molecules (e.g. formyl fluoride,13 benzene,14 pyridine,15 naphthalene16) and even to calculate energy surfaces for the NH3-HCl reacti0n.l' 'Gaussian lobe' functions utilise the relative ease of evaluating multicentre integrals in another way. In this method (Allen12) angularly dependent atomic lo E. Clementi J. Chem. Phys. 1967,46,3842; A. Veillard and E. Clementi Theor. Chim. Acta 1967 7 133.l1 I. G. Csizmadia M. C. Harrison J. W. Moskowitz and B. T. Sutcliffe Theor. Chim. Acta 1966 6 191. 12 L. C. Allen Internat. J. Quantum Chem. 1967 1 S39. l3 I. 0. Csizmadia M. C. Harrison and B. T. Sutcliffe Theor. Chim. Actu 1966 6 217. l4 J. M. Schulman and J. W. Moskowitz J . G e m . Phys. 1965,43 3287. 15 E. Clementi J . Chem. Phys. 1967 46 473 1. 16 R. J. Buenker and S. D. Peyerimhoff Chem. Phys. Letters 1969 3 37. 1' E. Clementi J . Chem. Phys. 1967,46 3851; 1967 47,2323; E. Clementi and J. N. Gayles ibid. 1967 47 3837. 98 Clark and Stewart orbitals are simulated by suitable positioning of groups of Gaussian ‘lobes’ (purely radial Gaussian functions). The problem of evaluating multicentre integrals can be avoided altogether by the use of single-centre basis sets.18 For example a wave function for hydrogen fluoridels can be constructed from F orbitals only and a wave function for methaneao from C orbitals only.In a similar way a wave function for acetylenea1 can be constructed from orbitals centred on the two C atoms. As would be expected single-centre calculations require very large basis sets and they are of limited application. As procedures for the evaluation of multicentre integrals are continuing to improve,22 it seems likely that atomic orbitals will eventually displace Gaussian functions from routine molecular calculations. The recent publication by Stevenson and Lips~omb~~ of LCAO wave functions for ScH,NH and TiH3F is a notable step in this direction. 4 Carbon Monoxide In this Section we illustrate the information that may be obtained from LCAO calculations by reference to carbon monoxide one of the molecules which have been studied most intensively.A. Wave Functions.-The CO molecule has fourteen electrons so at least seven atomic orbitals must be combined to construct the seven molecular orbitals (or fourteen molecular spin-orbitals) required for the simplest type of ground-state wave function. The sevenatomic orbitals must obviouslyinclude Is 2s,2pZ (=2po) from each atom. It is impossible to choose any other 2p orbital without being committed to both 2pz and 2pU (which are degenerate) from each atom; so a minimum basis set comprises ten atomic orbitals. These give ten linearly inde- pendent molecular orbitals instead of the seven required for a wavefunction of type (1). The three molecular orbitals which are superfluous in the ground state (‘virtual) orbitals) can be used in excited states;7a but strictly a separate minimis- ation of all atomic-orbital coefficients should be carried out for each excited state.7b As shown in Section 6 minimum-basis-set calculations have been performed by using atomic-orbital exponents determined both by Slater’s Rules and by atomic variational calculations and by Sahni et aLa5 and H U O ~ ~ using atomic- orbital exponents varied in the molecular calculation.Ne~bet,~‘ H U O ~ ~ and l8 B. D. Joshi J . Chem. Phys. 1967,46 875. R. Moccia J. Chem. Phys. 1964 40 2164. 2o D. M. Bishop Mol. Phys. 1963 6 305. 21 J. R. Hoyland J. Chem. Phys. 1968,48 5736. 28 R. E. Christoffersen and K. Ruedenberg J. Chem. Phys. 1968 49 4285; L. S. Salmon F. W. Birss and K. Ruedenberg ibid.1968 49 4293; D. M. Silver and K. Ruedenberg ibid. 1968 49 4301 4306 (and references cited therein). 24 B. J. Ransil Rev. Mod. Phys. 1960 32 239 245. 26 R. C. Sahni C D. La Budde and B. C. Sawhney Trans. Faraduy SOC. 1966 62 1933. 26 W. M. Huo J. Chem. Phys. 1965,43 624. 27 R. K. Nesbet J . Chem. Phys. 1964 40 3619. P. E. Stevenson and W. N. Lipscomb J . Chem. Phys. 1969,50 3306. 99 LCAO Wave Functions for Small Molecules Yoshimine and McLean2* have carried out calculations with a variety of extended basis sets. Extension of the basis set has been found (as is generally the case) to be more effective than optimising orbital exponents2* or using configuration inter- action with a small basis set. With a minimum basis set a single-determinant function of type (1) gives 99.09% of the observed molecular energy and a 14- term function of type (5) gives only 0.04 % more.2B Single-determinant extended- basis-set calculations give ca.99.5 %. The molecular orbitals comprising wave function are reproduced in Table 1. They are of three symmetry types a m and nu. It is obvious that the la and 20 molecular orbitals are virtually unchanged atomic orbitals la NN Is0 and 20% ISC. This is because the two 1s orbitals are concentrated so much about their respective nuclei that they do not overlap appreciably either with each other or with the other atomic orbitals in the system. This is clear also from the orbital energies the la and 20 molecular orbital energies are not significantly different from the corresponding orbital energies in other molecules containing the same atoms or indeed from the 1s orbital energies in the free atoms.It will be noted that there is no lack of combination between 2s and 2pz orbitals in all the higher-energy (T orbitals; hybridisation is a feature of all properly optimised LCAO wave functions and is not restricted to particular electronic or geometrical configurations. The sharp quantum distinction between s andpa orbitals which depends upon the spherical symmetry of atoms is lost in molecules. The ratio between 2s and 2pz coefficients in Table 1 varies from one orbital to another; the fixed ratio which is found in simple valence-bond functions is not determined variationally and has no relevance in molecular-orbital wave functions . Wave functions similar to that in Table 1 have been discussed by Coulson and Stewart in more detail than can be given here.B. Dissociation Energy.-Because the correlation energy of a molecule almost always exceeds the sum of the correlation energies of its atoms Hartree-Fock calculations tend to underestimate dissociation energies sometimes very grossly. The same is usually true of single-configuration LCAO approximations to Hartree-Fock calculations as is shown for CO in Table 2 It happens in the results quoted that the better wave function gives the better dissociation energy but this is not a general feature. Extending a basis set may either increase or decrease a calculated dissociation energy depending on whether the improvement in the molecular energy is greater or less than the improvement in the sum of the atomic energies. It is very well knowns that one of the principal disadvantages of molecular- orbital wave functions is that they correspond to charged instead of neutral atomic dissociation products.It is possible to estimate the correlation energy of a molecule by assuming it to be approximately the same as that of the Hartree- a* M. Yoshimine and A. D. McLean Internat. J. Quantum Chem. 1967,1 S313. 2Q S. Fraga and B. J. Rand J. Chem. Phys. 1962,36 1127. 100 Table 1 Molecular-orbital wave function for the ground state of carbon monoxidea Configuration :* la22a230240217;221~~2502 Internuclear distance = 2.132 B? = 1.1282 A Molecular Coefficients of carbon orbitals Coeficients of oxygen orbitals: Orbital A A Orbital* f \ I \ energy§ 1s 2s 2PzlT 2Pz 2Pv Is 2s 2P4 2Px 2PY (Ht) 10 - 0.0002 0.0069 0.0063 - 0.9960 - 0.0202 - 0.0058 - 20,706 2a 0.9964 0.0171 0-0059 - 0.0002 - 0.0054 - 0.0007 - 11.353 3a - 0.1152 0.2401 0.1687 - 0.2148 0.7588 0.2232 - 1.499 40 0.1468 - 0.5383 - 0.0668 0.1263 0.6529 - 0.6350 - 0.732 SU - 0.1403 0.7579 - 0.5658 0.0022 0.0366 - 0.4379 - 0.481 60 - 0.0916 0.9694 1.2509 0.1197 - 1.1289 - 0.9415 0,932 IT 0-4686 0.7712 - 0.583 1% 0-4686 0.7712 - 0.583 27nr 0.9225 - 0.6897 0-261 2% 0.9225 - 0.6897 0.261 Molecular energy - 112.344 H (caIcuIated); - 113.377 H (observed).a Ref. 24. * The numerals preceding the symbols u and r are not quantum numbers but merely serial numbers indicating the order of orbital energies for each sym- metry type. The molecular orbitals 60 27rz and 2vrY ('virtual orbitals') are not relevant in the ground state. 7 The positive z axes point from the nuclei towards each other.f The orbital energy E associated with a particular spin-orbital consists of (i) the kinetic energy of the spin-orbital (ii) the potential energy of attraction between the spin-orbital and all the nuclei (iii) the potential energy of repulsion between the spin-orbital and all the other spin-orbitals in the system. For tl a formal definition see refs. 3 and 6. x c s 0 t 1 B (= Bohr) = 0.052917 nm; 1 H (= Hartree) = 4-3594 aJ (atomic units for infinite nuclear mass). $' Atomic-orbital exponents as specified by Slater's Rules. 9 s % w LCAO Wave Functions jor Small Molecules Fock ionic dissociation As the correlation energies of atoms and ions with 2 d 30 are ~ C I I O W ~ ~ ~ this provides a means of improving calculated molecular energies and hence calculated dissociation energies.32 Nesbet2' has estimated the correlation energy of C2+ + 02- to be 3.18 ev higher than that of C + 0. If a correction of this amount is added to the better of the two calculated dissociation energies (7.84 ev) quoted in Table 2 the adjusted value (11.02 ev) very closely approaches the observed dissociation energy (1 1.24 ev). Other methods of adjustment have been s u g g e ~ t e d . ~ ~ ~ ~ Table 2 Molecular properties of the ground state of carbon monoxide Molecular energy (H) Error in calculated molecular Dissociation energy (ev) Internuclear distance at energy minimum (B) Spectroscopic constants me (cm-l) UeXe (cm-l) Be (cm-l) a e (cm-l) energy (%I k (los dyne cm2) Dipole moment (D) Quadrupole coupling constant cmz) Electric field gradient at oxygen nucleus (atomic units) Calculated with minimum basis set - 112.344' 0.91 5-38' 2~182~ 2398~4~ 8~989~ 1.8419b 0.01 1 3 b - 0.730" - 0.15" Calculated with extended basis setd - 112.786 0.52 7.84 2.08 1 243 1 11.69 2.027 0.01 525 0.274 23.86 - 0.0214 - 0.679 Observede - 113.377 1 1 *242 2.1 32 21 69.8 13.295 1.9313 0.01 75 19.02 - 0.118 0.01 63 - 0.64 a Ref.24; b Ref. 25; C Ref. 49(a); Ref. 26; e Quoted in ref. 26. C. Spectroscopic Constants.-By performing an LCAO calculation for a range of internuclear distances it is possible to express the molecular energy as a function of internuclear distance and so to calculate spectroscopic constants. Table 2 shows that the results are by no means perfect as would be expected if only 3oR. K. Nesbet J. Chem. Phys. 1962,36 1518. *l E. Clementi and A.Veillard J. Chern. Phys. 1966 44 3050. aaE. Clementi J. Chem. Phys. 1963 38 2780; 1963 39 487. 33 K. Carlson and P. Skancke J. Chern. Phys. 1964,40,613; V. McKoy ibid. 1965,42,2232; F. Grimaldi ibid. 1965 43 S59. s4 C. Hollister and 0. Sinanoglu J. Amer. Chem. Soc. 1966 88 13. 102 Clark and Stewart because of the faulty estimation of dissociation energy. McLeana6 and Schwende- manS6 have discussed the various sources of error. A more subtle though not necessarily more accurate method of calculating spectroscopic constants involves the use of the Hellmann-Feynman theorema7 (which applies to genuine Hartree-Fock wave functionsa8 as well as to exact wave functions). D. Electronic Charge Distribution.-There is good reason to believe that the electronic charge distribution corresponding to a Hartree-Fock wave function should not differ significantly from the true d i s t r i b u t i ~ n .~ ~ ~ ~ ~ It may be hoped that the same is true of an LCAO wave function derived by applying the variation principle to a sufficiently large basis set (though it must be remembered that the quality of a wave function is not always measured very sensitively by its energy4I). To illustrate the effect of molecule formation on electronic charge distribution we show in the Figure how the charge distribution in the carbon monoxide molecule (as from an extended-basis-set wave functionzs) differs from that in a hypothetical system consisting of a carbon atom and an oxygen atom at the same internuclear separation. According to earlier views based on less adequate evidence than is now available the formation of a molecule was believed to be accompanied by an increase in electronic charge between the nuclei and a corresponding decrease beyond the nuclei.It is clear however from the Figure and from similar diagrams for many other diatomic (homonuclear and heteronuclear) that there is (i) an increase in electronic charge around the internuclear axis in the region between the nuclei (ii) a sharp decrease immedi- ately around each nucleus and (iii) an increase beyond each nucleus. E. Dipole Moment.-The fact that an enlargement of the basis set does not neces- sarily improve the agreement between calculated and experimental values of molecular properties (other than molecular energies) is shown strikingly in the case of the dipole moment of carbon monoxide (Table 2).With the minimum basis set the absolute value is much too large whereas with various extended sets26-28 the sign is wrong. It has taken a 200-term configuration-interaction calculationq4 to produce a value (- 0.17 D) which gives reasonable agreement between theory and experiment. It should be pointed out that the dipole moment of carbon monoxide is unusually small and depends very sensitively on inter- nuclear distance; for these reasons it would perhaps be wrong to attach too much 36 A. D. McLean J. Chem. Phys. 1964 40 243. 36 R. H. Schwendeman J. Chem. Phys. 1966,44,2115. 37 L. Salem J. Chem. Phys. 1963 38 1227; J. Goodisman ibid. 1963 39 2397; R. H. Schwendeman ibid. 1966,44 556; D. P. Chong Theor. Chim. Acta 1968 11 205. 38 R. E. Stanton J. Chem. Phys. 1962 36 1298.39 G. G. Hall Phil. Mag. 1961,6 249; A h . Quantum Chem. 1964 1 241. 40 J. Gerratt Ann. Reports(A) 1968 65 3. 41 J. Goodisman J. Chem. Phys. 1963 38 304. 43 R. F. W. Bader and A. K. Chandra Canad. J. Chem. 1968,46,953. 44 F. Grimaldi A. Lecourt and C. Moser Internat. J. Quantum Chem. 1967 1 S153. R. F. W. Bader and A. D. Bandrauk J. Chem. Phys. 1968,49 1653 1666. 103 LCAO Wave Functions for Small Molecules 4 0 / \ \ ‘. Figure Electron density diflerence map for carbon monoxide. The horizontal line represents the internuclear axis and its intersections with the vertical lines mark the positions of the carbon (left) and oxygen (right) nuclei. At each point A p = pco - ( Contours are plotted for I APT= 0,O.OOl. 0.01 0.1 e B - ~ . Unbroken lines dotted lines and doshed lines represent contours on which A is respectively zero positive and negative.In each zone I A p I decreases outwards front the internuclear axis. (All dorted lines and dashed lines form closed loops they are shown incomplete in regions where on the scale of the diagram they merge into the aa’jacent unbroken lines.) [Redrawn from a more detailed map by Bader and Bandrauk ref. 42.1 + PO). where p is the electron density. significance to the discrepancies associated with the single-configuration wave- functions. F. Electrical and Magnetic Properties.-Other quantities which have been calculated for the CO molecule include magnetic sus~eptibility,~~ magnetic shielding,4s rotational magnetic polari~abilities,~~ and quadrupole 45 M. Karplus and H. J. Kolker J. Chem. Phys. 1963,38 1263; J .R. de la Vega D. Ziobro and H. F. Hameka Physica 1967 37 265. 46 C. W. Kern and W. N . Lipscomb J. Chem. Phys. 1962 37 260. 47 J. R. de la Vega and H. F. Hameka J. Chem. Phys. 1967,47 1834. 48 M. Karplus and H. J. Kolker J. Chem. Phys. 1963 39 2011 ; J. M. O’Hare and R. P. Hurst ibid. 1967 46 2356; A. D. McLean and M. Yoshimine ibid. 1967 46 3682. 104 Clark and Stewart coupling constant.49 Several mefhods6O have been devised for calculating these quantities with LCAO wave functions. Lipscomb61 and Gerratt 40 have provided comprehensive reviews of the subject and have shown that reasonable agreement with experiment is usually obtained. As far as multipole moments are concerned both theoretical calculations and experimental determinations appear to be beset by considerable difficulties.62 G.Excited States.-From a basis set of m atomic orbitals m linearly independent and orthogonal molecular orbitals are obtained automatically when the electronic energy is minimised with respect to the atomic-orbital coefficients. A 2N-electron ground-state wave function in which all molecular orbitals are paired requires only the N molecular orbitals of lowest orbital energy and thus the (m - N ) higher-energy orbitals are superfluous (‘virtual’ orbitals). The simplest way of constructing wave functions for excited states is to replace one or more ground- state molecular orbitals by ‘virtual’ orbitals. This method has been used with a variety of basis sets by Lefebvre-Brion Moser and NesbeP3 to calculate ‘potential-energy’ curves spectroscopic constants dipole moments and dipole- moment derivatives for a number of low-lying excited states of CO.A more thorough-going application of the variation principle requires that the atomic-orbital coefficients should be optimised afresh for each excited state. This procedure which involves a surprising increase in technical difficulty has been used by H u o ~ ~ for CO (extended basis set) and by Sahni and S a ~ h n e y ~ ~ for CO+ (minimum basis set). Calculations on excited states always give less satisfactory agreement with experiment than do calculations on ground states. 5 Improved and Adjusted Wave Functions A. Configuration Interaction.-As pointed out in Section 4 single-determinant wave functions do not correspond to neutral dissociation products. In the case of certain molecules as Das Wahl and others660 57 have shown for Li, F, and NaF this difficulty can be overcome by adding to the wave function a second determin- ant based on another molecular configuration.The variation principle ensures of (a) J. W. Richardson Rev. Mod. Phys. 1960,32,461; (b) H. Lefebvre-Brion C. M. Moser M. Karplus Rev. Mod. Phys. 1960,32,455; D. F. Tuan S. T. Epstein and J. 0. Hirsch- R. K. Nesbet and M. Yamazaki J. Chem. Phys. 1963,38,2311. felder J. Chem. Phys. 1966,44 431. 61 W. N. Lipscomb Adv. Magn. Resonance 1966,2 137. 63 D. E. Stogryn and A. P. Stogryn MoZ. Phys. 1966,11 371. sB H. Lefebvre-Brion C. M. Moser and R. K. Nesbet J. MoZ. Spectroscopy 1964,13 418; R. K. Nesbet J. Chem. Phys. 1965,43,4403. 64 W. M. HUO J. Chem. Phys. 1966,45 1554. O6 R. C. Sahni and B. C. Sawhney Trans. Faraday SOC.1967 63 1. 66 G. Das and A. C. Wahl J. Chem. Phys. 1966,44,87; G. Das ibid. 1967,46,1568; G. Das and A. C. Wahl ibid. 1967 47 2934; B. Levy ibid. 1968 48 1994. 67 A. C. Wahl P. J. Bertoncini 0. Das and T. L. Gilbert Internat. J . Quantum Chem. 1967 1 S123. 105 LCAO Wave Functions for Small Molecules course that this reduces the calculated molecular energy. Calculated spectro- scopic quantities are also improved. The change is most striking in Fz for which the double-configuration wave function unlike the simple LCAO wave func- tion,68 does not give a negative dissociation energy. The nature of the double-configuration wave function is illustrated most simply by reference to Liz for which the single-determinant function is based on the configuration lagz lauz 2og2. If this is combined with a second determinant based on the configuration lag2 lou2 2au2 and the linear-combination coefficient (a function of internuclear distance) is determined by applying the variation principle the resultingwave function [of the type ( 5 ) in Section 21 becomes awavefunction for two neutral Li atoms (Is2 2s) as R 4 co .Exactly analogous wave functions have long been known for the hydrogen m01ecule.~ (In H2 a linear combination of ogz and uu2 gives the correct behaviour on dissociation; agz by itself does not.) Wahl et aL6’ have given a general survey of configuration-interaction wave functions for diatomic molecules. B. Open-shell Wave Functions.-Another way of improving single-configurat ion wave functions is to avoid the orbital pairing indicated in equations (1) and (2).General procedures of this type have been devised by L o ~ d i n ~ ~ Goddard,60 and Kaldor.61 C. Constrained Wave Functions.-In the integrals from which molecular properties are calculated the integrands vary considerably from one point to another in the co-ordinate space. The manner in which different regions of a molecule are weighted in the integration depends upon the nature of the Schrodinger operator representing the molecular property. For some properties the regions nearest the nuclei are the most important for others the regions furthest away. This is why even a very flexible wave function may if optimised with respect to the energy give disappointingly imprecise values for other quantities; whereas a poor wave function is almost certain to give poor results a good wave function (in the varia- tional sense) will not necessarily give good results.For this reason early success in the precise calculation of a wide range of properties seems unlikely to be achieved merely by making variational wave functions more and more complicated. Mukherji and Karplus,62 in an altogether different approach have argued that if quite a simple energy-optimised wave function is adjusted so as to give the observed numerical values for some molecular properties the adjusted wave function might be expected to give good values for other properties represented by integrals in which the same regions of the molecule are important. They re-varied some of the LCAO coeficients in Rand’s minimum-basis-set wave function2* for HF subject to the constraint that the 68 A. C. Wahl J.Chem. Phys. 1964,41,2600. 69 P.-0. Lowdin Phys. Rev. 1955 97 1509. W. A. Goddard Phys. Rev. 1967 157 73 81 ; J. Chem. Phys. 1968,48,450 5337. U. Kaldor J. Chem. Phys. 1968,48 835. eaA. Mukherji and M. Karplus J. Chem. Phys. 1962 38 44. 106 Clark and Stewart relevant integrals should give the experimental values for the dipole moment and the deuteron quadrupole coupling constant. This reduced the errors in the calculated diamagnetic and paramagnetic susceptibilities by more than half. The adjustment in the previously optimised LCAO coefficients necessarily raised the calculated energy but by a mere 0404%. Variants and extensions of the procedure followed by Mukherji and Karplus have been formulated by other authors.6a 6 Bibliography To demonstrate the scope and the extent of strict wave-mechanical calculations on LCAO wave functions we give in Tables 3-8 a list of the relevant publications in the period from 1960 to mid-1969 on systems of four or more electrons.It is clear that in addition to energy a very wide range of molecular properties can be calculated from LCAO wave functions (in some cases not very precisely as yet). We include in these Tables many calculations which have been superceded by others of greater complexity. This is partly for the reason given at the end of Section 5 but mainly because the effects of varying orbital exponents extending basis sets and using multiconfiguration wave functions can be judged only by consideration of a substantial collection of numerical examples. We have made this collection as complete as possible. It seems likely that in the next few years calculations will be carried out on polyatomic molecules much more complicated than those we list and that work on diatomic molecules will normally go beyond the single-configuration LCAO approximation.Y. Rasiel and D. R. Whitman J. Chem. Phys. 1965,42,2124; D. P. Chong and Y. Rasiel ibid. 1966,44 1819; W. B. Brown ibid. 1966.44,567; C. P. Yue and D. P. Chong Theor. Chim. Ada 1968,12,431; S. Fraga and F. W. Birss ibid. 1966,5,398; S. Fraga and G. Malli ibid. 1966,5 446. Notes on Tables 3-6. In the columns headed ‘Basis set’ the first letter denotes the size M = minimum; E = extended. S . . . . . determined arbitrarily e.g. by Slater’s Rules; A . . . . . optimised for the free atom; P . . . . . partially optimised for the molecule; M . . . .. completely optimised for the molecule. If the second letter is E the wave function is formulated in elliptical co-ordinates. The numbers in parentheses refer for each atom to the number of radially distinct orbitals of each symmetry type in the order (0 n a) the order of the atoms matching that in the chemical formula. For non-planar molecules only the total number of atomic orbitals on each nucleus is listed. The letter C following the parentheses indicates a ‘configuration-interaction’ calculation. In the case of FH for example MS(3,1)(1,0)C denotes a multiconfiguration wave function using the following minimum basis set of atomic orbitals with the exponents determined by Slater’s Rules The second letter if S A P or M refers to the atomic-orbital exponents F(u) IS 2 ~ 2pz (= 2 p ~ ) ; F(?T) 2pz 2py (not radially distinct); H(u) 1s; H(n) nil.107 LCAO Wave Functions for Small Molecules For a wave function in elliptical co-ordinates only one set of numbers is required to specify the orbital symmetry types. (Such wave functions do not come strictly within the scope of this Review but they are listed for comparative purposes.) If a calculation is carried out for more than one geometrical configuration (more than one set of bond lengths and bond angles) the number is listed under the heading ‘Additional calculations’. In the same column are noted calculations on excited states (e) and calculations on positive ions (i). Letters in parentheses in the Reference columns denote papers which are concerned with excited-state wave functions and do not give details of the corresponding ground-state wave functions.Such reference letters in parentheses are given quite arbitrarily together with the first entry for each molecule. In almost all cases the internuclear distances (R) for which the molecular energies (E) and the dipole moments ( p ) have been calculated are those determined experimentally. (To save space molecular dimensions are not given in Table 6.) The sign of the dipole moment is positive if the lighter nucleus is at the positive end. Table 3 Molecular energies (E) o j homonuclear diatomic molecules* Basis set MM(3,O) MM(3 ,O)C EP( 15,3)C EP(17,6)C M M (3 90) MM(3,O)C E~(16,8)C EE( 10,8)C MM(391) MM(3,l)C MS(3,U MM(3,1) MM(3,1) MM(3,l)C W 5 2 ) W 7 3 ) EM(20,6) MS(3,l) MM(3,l) MM(3,l)C EM( 18,lO) EP( 18,lO)C ES(6,3) - E (HI 14.842 14-852 14.899 14.903 29-058 29.105 29.220 49.145 75.224 75.319 108.574 108.634 108634 108-661 108.785 108.971 108.993 149.092 197.877 197.956 198.768 198.838 679.166 R (B) 5-05 1 5.05 1 5.25 5-07 3.78 3.78 4.5 3.0 2.3475 2.3475 2.0675 2.068 2-1 2.068 2-068 2.068 2.068 2-28 17 2.68 2.68 2.68 2.68 3-58 Additional calculations 8 10 1 O,e e,i 53,i e,i 5 16,i e,i 6 Reference a b d a b e f 44 b g(0-9) a h b i k g a b 1 m C i C *LCAO wave functions and potential-energy curves have been calculated for Hear Nen and Ar (T.L. Gilbert and A. C. Wahl J. Chem. Phys. 1967,47,3425). a B. J. Ransil Rev. Mod. Phys. 1960,32,239,245; b S . Fraga and B. J. Ransil J. Chem. Phys. 1962,36 1127; C G. Das and A. C. Wahl J. Chem. Phys. 1966,44 87; G . Das J. Chem. Phys. 1967 46 1568; e C. F. Bender and E.R. Davidson J. Chem. Phys. 1967 47 4972; f C. F. Bender and E. R. Davidson J. Chem. Phys. 1967,46,3313; 9 R. C. Sahni and E. J. de Lorenzo J. Chem. Phys. 1965 42 3612; h R. C. Sahni and B. C . Sawhney Internat. J. 108 Clark and Stewart Quantum Chem. 1967 1 251; * J. W. Richardson J. Chem. Phys. 1961 35 1829; 1 R. K. Nesbet J. Chem. Phys. 1964,40 3619; k P. E. Cade K. D. Sales and A. C. Wahl J. Chem. Phys. 1966,44 1973; 1 A. C. Wahl J. Chem. Phys. I964,41,2600;m D. B. Boyd and W. N. Lipscomb J. Chem. Phys. 1967,46,910; n R. K. Nesbet and P. F. Fougere J. Chem. Phys. 1966,44,285; G. Verhaegen ibid. 1968,49,4696; 0 G. Verhaegen W. G. Richards and C. M. Moser J. Chem. Phys. 1967 47 2595.; P R. K. Nesbet J. Chem. Phys. 1965,43,4403; 4 H. Lefebvre-Brion and C. M. Moser J . Chem.Phys. 1965 43 1394. Table 4 Molecular energies (E) and dipole moments (p) of diatomic hydrides Molecule Basis set LiH BeH+ BeH BH CH+ CH NH OH OH- FH NeH+ NaH MgH AlH SiH - E R p Additional Refer- (H) (B) (D) calculations ence 7.970 7.984 7.987 7.987 8.006 8.017 8.039 8.041 8.061 14.836 15.153 15.221 25.075 25.090 25.131 37-859 38,279 54-325 54.345 54,978 75.421 75.41 8 99.536 99.564 99.991 100-057 lOO.058 100.070 100.071 100.257 128628 3-01 5 3.01 5 3.02 3-015 3.015 3-02 3.2 2.99 3.01 5 2.68 2.538 2.538 2.329 2.329 2-336 2-34 2.124 1 *976 1 -976 1.961 1.8342 1.781 1.733 1-733 1.733 1-7328 1.7328 1.7328 1.7328 1 -733 1.83 - 5.92 - 5-57 - 6.035 - 6.002 16,i - 6.04 - 5.89 3 4 - 5.96 2 - 5.965 6 - 0.282 16,i - 0.07 e 1.58 1-53 1.733 16,i 1-57 16,i* 2.01 2.06 1.627 16,i* 1.78 16,i 3.353 17 1.44 1.3 2.009 3 1 *827 1.984 6,i 1.942 6,i 1 *934 1 -649 6 6 162.393 3.566 - 6-962 15,i 200.157 3.271 - 1.516 15,i 242.463 3.114 0.17 15,i 289.436 2.874 0-302 15,i* LCAO Wave Functions for Small Molecules Moleule Basis set -E (H) PH EM(12,6)(4,2) 341.293 SH EM(12,6)(4,2) 398,102 SH- EM(12,6)(4,2) 398.146 CIH EA(9,4)(3,1) 459.804 EM(12,6)(4,2) 460-1 10 EP(17,10)(6,3) 460.1 12 2.708 2.551 2.5 12 2.4085 2-4087 2.4087 p Additional (D) calculations 0.538 15,i* 0.861 15,i 3.546 15 1.387 3,i 1.197 15,i 1,215 Refer- ence * Electron affinities calculated by similar calculations on negative ions (P.E. Cade Proc. Phys. SOC. 1967 91 842). a B. J. Ransil Rev. Mod. Phys. 1963,32,239,245; S . Fraga and B. J. Ransil J. Chem. Phys. 1962,36,1127; C J. R. Hoyland J. Chem. Phys. 1967,47,1556; d P.E. Cade and W. M . Huo J. Chem. Phys. 1967,47,614; e P . Linder Theor. Chim. Acta 1966,5 336; f R. K. Nesbet and S. L. Kahalas J. Chem. Phys. 1963 39 529; 9 F. E. Harris and H. S. Taylor Physicu 1964 30 105; h D. D. Ebbing J. Chem. Phys. 1962,36,1361; C. F. Bender and E. R. Davidson J. Phys. Chem. 1966,70,2675; F. Jenc Coll. Czech. Chem. Comm. 1963,28,2064.; k A. C. H. Chan and E. R. Davidson J. Chem. Phys. 1968,49,727; P. E. Cade J. Chem. Phys. 1967 47 2390; R. K. Nesbet Rev. Mod. Phys. 1960,32,272; 0 E. Clementi J. Chem. Phys. 1962,36,33;p A. D. McLean and M. Yoshimine J. Chem. Phys. 1967,47 3256; Q C. F. Bender and E. R. Davidson J. Chem. Phys. 1967,47 360; r S. Peyerimhoff J. Chem. Phys. 1965,43,998; 8 P. E. Cade and W. M. Huo J. Chem. Phys. 1967,47 649; t R. K. Nesbet J.Chem. Phys. 1964,41 1W;U H. S. Taylor J. Chem. Phys. 1963 39 3382; C. F. Bender and E. R. Davidson ibid. 1968,49,4222; R. E. Brown and H. Shull Znternat. J. Quantum Chem. 1968,2 663; W. M. Huo J. Chem. Phys. 1968 49,1482; W C. F. Bender and E. R. Davidson J. Chem. Phys. 1968,49,4989; W. G. Richards and R. C. Wilson Trans. Faraday SOC. 1968 64 1729. R. K. Nesbet J. Chem. Phys. 1962,36 1518; Table 5 Molecular energies (E) and dbole moments (p) of heteronuclear diatomic molecules Molecule Basis set - E R p Additional Refer- A lkali-metal halides (HI (B) (I)) calculations ence LiF LiCl LiBr NaF NaCl NaBr KF KCI RbF MA(3,1)(3,1) MP(3,1)(3,1 )C EP(6,2)(9,4) EP(7,3)(11,6) EP(8,3)( 1497) EP(7,3)( 18,10,2) EP(13,6)(10,5) EP(13,6)(14,7) EP( 1 1,5)(18,10,2) EP(18,8)(9,5) EP( 1 5,7)( 1 2,6) EP(21,11,2)(8,4) 106.381 2.85 106.412 2-85 106.989 2.8877 106.992 2.8877 467.055 3.825 2579-89 4-0655 261.379 3.628 621 -457 4.4609 2734.29 4.728 698.685 4.1035 1058.76 5.039 3037.77 4.3653 3-43 6.297 13 6.3 7 7.256 9 10 - 8.367 9 9.101 8 7 5 a b d e f g h f i f f C 110 Molecule Basis set Group I1 compounds Be0 EP(5,2)(8,4) EP(6,2)(10,5) BeS ES(5,2)(13,6) EP(10,4)( 12,6) MgO EP( 14,7)( 12,6) CaO EP(l5,7)(12,6) SrO EP(20,10,2)(7,3) Group V compounds NF MM(3,1)(3,1) PN EP(14,7)(11,6) PO ES(6,3)(3,1) Transition-metal compounds ScO EP(8,4,2)(3,2,1) ScF EP(10,5,2)(4,3) TiN EP(8,4,2)(3,2) Ti0 EP(8,4,2)(3,2) vo EP(8,4,2)(3,2) W10,5,2)(4,3) 89.428 2.676 89.448 2.4377 7.29 89.454 2-4377 7.35 412.097 274-386 3,3052 9.18 751.559 3.4412 11.48 3206.23 3.6283 10-2 78.717 2.421 - 1.430 123.604 2.385 - 1.96 123.676 2.385 - 1.13 124.140 2.385 - 0.668 124.166 2.391 - 0.945 124.167 2.391 - 0.88 341.483 3.126 1.34 1 1 1.956 2.075 112.326 2.132 112.344 2-132 112.392 2.132 1 12.396 2- 132 112.759 2.132 1 12.786 2.1 32 112.789 2.132 91.927 2.18 435.330 2.9 363.852 2.854 - 0.592 - 0.730 0.0872 0.397 0-274 0.28 - 1.84 1.6 3-68 153.205 2.45 395.185 2.818 3.23 414,137 2.738 - 0.7 833.096 3.05 - 2.6 858.545 3.31 - 4-64 901.127 3.00 - 3-55 921.542 3.0618 - 2.863 922.498 2.91 - 5.93 1015.89 2.91 - 3.61 Clark and Stewart Additional Refer- calculations ence 4,e 10 9 8 7 7 e e 5 8 7 7 57,i 72 5 7 7 3 7 13,e,i 7 1 3,e e 2,e 3,e e 3 4 LCAO Wave Functions for Small Molecules B.J. Ransil Rev. Mod. Phys. 1960,32,239,245; S . Fraga and B. J. Rand J. Chem. Phys. 1962,36 1127; C A.D. McLean J. Chem. Phys. 1963,39,2653; M. Yoshimine and A. D. McLean Internat. J. Quantum Chem. 1967 1 S313; e R. L. Matcha J. Chem. Phys. 1967 47 4595; f A. D. McLean and M. Yoshimine IBM J. Research and Development 1968 12 206; * R. L. Matcha J. Chem. Phys. 1967 47 5295; h R. L. Matcha J. Chem. Phys. 1968 48,335; R. L. Matcha J. Chem. Phys. 1968,49,1264; f G. Verhaegen and W. G. Richards J. Chem. Phys. 1966,45,1828; k M. Yoshimine J. Chem. Phys. 1964,40,2970; 1 G. Verhaegen and W. G. Richards Proc. Phys. SOC. 1967,90 579; J. L. Masse and M. Btirlocher Helv. Chim. Acta. 1964,47 314;n R. K. Nesbet J. Chem. Phys. 1964 40 3619; OW. M. HUO J. Chem. Phys. 1965 43 624; p R. C. Sahni and B. C. Sawhney Trans. Faraday SOC. 1967 63 1; Q H. Brion and C. M. Moser J. Chem. Phys. 1960,32 1194; r R.C. Sahni C. D. La Budde and B. C. Sawhney Trans. Faraday SOC. 1966,62 1933; 8 R. Bonaccorsi C. Petron- golo E. Scrocco and J. Tomasi J. Chem. Phys. 1968 48 1500; t W. G. Richards Trans. Faraday SOC. 1967 63 257;u R. C. Sahni Trans. Faraday SOC. 1967,63 801; V D. B. Boyd and W. N. Lipscomb J. Chem. Phys. 1967,46,910; W K. D. Carlson E. Ludena and C. M. Moser J. Chem. Phys. 1965 43 2408; 2 K. D. Carlson and C. M. Moser J. Chem. Phys. 1967 46 3 5 ; Y K. D. Carlson C. R. Claydon and C. M. Moser J. Chem. Phys. 1967 46 4963; K. D. Carlson and R. K. Nesbet J. Chem. Phys. 1964,41,1051 ;aa K. D. Carlson and C. M. Moser J. Chem. Phys. 1966,44,3259; bb W. M. Huo K. F. Freed and W. Klemperer J. Chem. Phys. 1967,46,3556; CC W. G. Richards G. Verhaegen and C. M. Moser J. Chem. Phys. 1966,45 3226; dd G.Verhaegen W. G. Richards and C. M. Moser J. Chem. Phys. 1967 46 160; ee H. Lefebvre-Brion and C. M. Moser J. Mol. Spectroscopy 1965 15 21 1 ; ff P. Merryman C. M. Moser and R. K. Nesbet J. Chem. Phys. 1960,32,631; H. Lefebvre- Brion C. M. Moser and R. K. Nesbet ibid. 1960,33 931; 1961,34 1950; 1961,35 1702; J. Mol. Spectroscopy 1964 13 418; W. M. Huo J. Chem. Phys. 1966 45 1554; 88 K. D. Carlson and C. M. Moser J. Phys. Chem. 1963,67 2644. Table 6 Molecular energies (E) of polyatomic molecules Molecule Basis set - E (H) 26-338 26-352 36-907 52-678 52-715 100-730 38-904 40.1 14 40- 128 40-181 40.205 77.834 77.876 79.069 79.098 290.5 1 9 Additional Reference calculations a * b a b d & 13,e 4 a9.f 3 g 3 h i a(@?) 7,e,i i 2 a,k 2 I m Clark and Stewart Molecule Basis set 76.544 76.61 7 76.678 76.854 115.583 83.731 175.724 535.767 168.578 92.547 92.590 92.9 15 183.982 184.657 166.459 167-270 167.076 191.780 489-91 1 55 1 -825 1 13.088 113.165 150.844 186.843 187.076 187.723 508.492 5 10.3 3 1 1 13.450 1 13.427 Additional Reference calculations n a e,i 0 P 4 4 r S S S 12 t lf a S V P t S t S S S W X e Y n 4 S z aa bb,cc bb,dd S 113 LCAO Wave Functions for Small Molecules Molecule Basis set 56.005 56499 34 1 a309 162.542 162.705 163.224 183.757 202.901 203.108 203.174 203 -9 8 6 227.708 488.77 75.681 76.005 150.157 150.223 223.479 272.425 273.526 397-842 198.283 199.393 199.573 Additional Reference calculations a 2,i ee IT w t gg t 9 S W gg hh ii hh ii kk 11 8 mm 7 nn hh hh ii m W 00 21 S PP PP B-H distance treated as variational parameter.114 Clark and Stewart W. E. Palke and W. N. Lipscomb J. Amer. Chem. SOC. 1966,88,2384; W. E. Palke and W. N. Lipscomb J. Chem. Phys. 1966,45 3948; C R. A. Hegstrom W. E. Palke and W. N. Lipscomb J. Chem. Phys. 1967 46 920; W. E. Palke and W. N. Lipscomb J. Chem. Phys. 1966,45 3945;e J. M. Foster and S. F. Boys Rev. Mod. Phys. 1960,32 305;f J. Sinai J. Chem. Phys. 1963,39,1575; g R. M. Pitzer J. Chem. Phys. 1967,46,4871 ; h B. J. Womick J. Chem. Phys. 1964 40 2860; G. P. Arrighini C. Guidotti M. Maestro R. Moccia and 0. Salvetti J. Chem. Phys. 1968,49 2224; J U. Kaldor and I. Shavitt J. Chem. Phys. 1968,48 191; k R. M. Pitzer and W. N. Lipscomb J. Chem. Phys. 1963 39 1995; R. M. Pitzer J . Chem. Phys. 1967 47 965; m F. P. Boer and W. N. Lipscomb J. Chem. Phys. 1969,50,989;n A.D. McLean J. Chem. Phys. 1960,32,1595; 0 M. G. Griffith and L. Good- man J. Chem. Phys. 1967 47 4494; P A. D. McLean and M. Yoshimine IBM J. Research and Development 1968,12,206; Q M. D. Newton and W. N. Lipscomb J. Amer. Chem. SOC. 1967 89 4261; 'A. Veillard J. Chem. Phys. 1968 48 1994; 8 M. Yoshimine and A. D. McLean Internat. J. Quantum Chem. 1967 1 S313; R. Bonaccorsi C. Petrongolo E. Scrocco and J. Tomasi J. Chem. Phys. 1968 48 1500;UA. D. McLean J. Chem. Phys. 1962,37,627; v E. Clementi and A. D. McLean J. Chem. Phys. 1962,36,563; E. Clementi J. Chem. Phys. 1961 34 1468; 2 E. Clementi and A. D. McLean J. Chem. Phys. 1962,36 45; Y E. Clementi J. Amer. Chem. SOC. 1961 83 4501; A. D. McLean J. Chem. Phys. 1963 38 1347; a@ E. Clementi J. Chem. Phys. 1962 36 750; bb M. D.Newton and W. E. Palke J. Chem. Phys. 1966,45 2329; Cc J. M. Foster and S . F. Boys Rev. Mod. Phys. 1960 32,303; S . Aung R. M. Pitzer and S. I. Chan J. Chem. Phys. 1966,45,3457; dd P. L. Good- friend F. W. Birss and A. B. F. Duncan Rev. Mod. Phys. 1960 32 307; ee U. Kaldor and I. Shavitt J. Chem. Phys. l966,45,888;ffD. B. Boyd and W. N. Lipscomb J. Chem. Phys. 1967,46,910; gg E. Clementi and A. D. McLean J. Chem. Phys. 1963,39,323; hh C. Petron- golo E. Scrocco and J. Tomasi J. Chem. Phys. 1968,48,407; f( R. Bonaccorsi C. Petron- golo E. Scrocco and J. Tomasi J. Chem. Phys. 1968 48 1497; jj D. B. Boyd and W. N. Lipscomb J. Chem. Phys. 1968 48 4968; kk J. Andriessen Chem. Phys. Letters 1969 3 257; S. Aung R. M. Pitzer and S. I. Chan J . Chem. Phys. 1968,49,2071; mm U. Kaldor and I.Shavitt J. Chem. Phys. 1966,44 1823;nn W. E. Palke and R. M. Pitzer J. Chem. Phys. 1967 46 3948; ooE. Clementi and A. D. McLean J. Chem. Phys. 1962 36 745; *P P. E. Stevenson and W. N. Lipscomb J. Chem. Phys. 1969,50,3306; QQ T. H. Dunning and V. McKoy J. Chem. Phys. 1967,47 1735. Tables 7 and 8 on the following pages list papers in which various mole- cular properties are calculated from previously published wave functions. 115 Table 7 Literature references to calculations of various molecular quantities Diatomic molecules Molecule LiH BeH BH CH NH OH FH NaH,MgH,AIH,SH,SiH,PH ClH Liz Be2 B&2 c2 N2 F2 LiF Be0 BeF MgF BN BF co NO Electron density a-f c-f c-f b-f,k,n c CYg c 0 0 e,f,i-m e,Li i,i f-. e-kP e,f,h,k,ss e,f,i-l h d e-hP d-h P Magnetic Polar isabilit ies properties q-Y w,kk,ll u-x,z,aa bb U s-w,cc,dd dd kk 11 s-v,ee,fl kk,N V f l u,v,aa,ii t-v,ii,jj 11 kk,ll,mm b 0 2 8 Quadrupole Spectroscopic coupling constant quantities fb and/or dipole 3 9 moment nn,oo vv-xx 6' 3 ww x x $ ww ww aaa bbb ww ww UII,CCC a P.Politzer and R. E. Brown J. Chem. Phys. 1966,45,451; b R. F. W. Bader and W. H. Henneker J. Amer. Chem. SOC. 1966,88,280; C R. F. W. Bader I. Keaveny and P. E. Cade J. Chem. Phys. 1967,47,3381; d G. Doggett J. Chem. SOC. (A). 1969,229; e V. Magnasco and A. Pefico J. Chem. Phys. 1967,47,971 ; f B. J. Ransil and S. Fraga J. Chem. Phys. 1961,34,727; 8 E. Clementi and H. Clementi J. Chem. Phys. 1962,36,2824; R. F. W. Bader and A. D. Bandrank J. Chem. Phys. 1968,49 1653 1666; f A. C. Wahl Science 1966,151,961 ; f R. F. W. Bader W. H.Henneker ana P. E. Cade J. Chem. Phys. 1967 46 3341; k B. J. Ransil and J. J. Sinai J. Chem. Phys. 1967 46 4050; W. D. Lyon and J. 0. Hirschfelder J. Chem. Phys. 1967,46 1788; R. F. W. Bader and A. K. Chandra Canad. J. Chem. 1968,46,953; n C. W. Kern and M. Karplus J. Chem. Phys. 1964,40 1374; 196543,2926; O P. E. Cade R. F. W. Bader W. H. Henneker and I. Keaveny J. Chem. Phys. 1969,50,5313; P E. R. Davidson J. Chem. Phys. 1967,46,3320; Q M. Karplus and H. J. Kolker J. Chem. Phys. 1961,35,2235; C. W. Kern and W. N. Lipscomb Phys. Rev. Letters 1961,7,19; R . M. Stevens R. M. Piker. and W. N. Lipscomb J. Chem. Phys. 1963,38 550; J. R. de la Vega and H. F. Hameka ibid. 1964,40 1929; J. Gruninger and H. F. Hameka Chem. Phys. Letters 1967 1 14; G. P. Arrighini F. Grossi and M. Maestro Theor.Chim. Acta 1966 5 266; H. J. Kolker and M. Karplus J. Chem. Phys. 1964,41 1259; t M. Karplus and H. J. Kolker J. Chem. Phys. 1963. 38 1263; U C. W. Kern and W. N. Lipscomb J. Chem. Phys. 1962,37,260; 1) J. R. de la Vega and H. F. Hameka J. Chem. Phys. 1967.47 1834; J. R. de la Vega Y. Fang and H. F. Hameka Physica 1967,36,577; R. A. Hegstrom and W. N. Lipscomb J. Chem. Phys. 1967.46 1594; Y R. M. Stevens and W. N. Lipscomb J. Chem. Phys. 1964,40,2238; R. M. Stevens and W. N. Lipscomb J. Chem. Phys. 1965,42,3666; aa R. A. Hegstrom and W. N. Lipscomb J. Chem. Phys. 1966 45,2378; 1968,48,835; bb D. S. Bartow and J. W. Richardson J. Chem. Phys. 1965,42,4018; Cc T. P. Das and M. Karplus J. Chem. Phys. 1962,36 2275; R. P. Hurst M. Karplus and T. P. Das ibid. 1962.36,2786; H. Hamano H.Kim and H. F. Hameka Physica 1963,29,111; D. Zeroka and H. F. Hameka J. Chem. Phys. 1966,45,300; Y. 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LCAO Wave Functions for Small Molecules Table 8 Literature re ferenccs to calculations of various molecular quantities Polyatomic molecules Molecule Electron Magnetic density properties 0 P C S C C BPHI CHI CaH C2Hd HCN CaNa GIG co2 HCHO 0 sco NH3 Q N3- h N2O S H2O H202 HF2- h FCN m n CICN HCCCN OCN- SCN- n C2H4 a C. W. Kern and W. N. Lipscomb J. Chem. Phys. 1962 37 275; b J. L. Sinai J. Chem. Phys. 1964 40 3596; C G. P. Arrighini M. Maestro and R. Moccia Chem. Phys. Letters 1967,1,242; J.Chem. Phys. 1968,49,882; d T. Caves and M. Karplus J. Chem. Phys. 1966 45 1670; E. Clementi and H. Clementi J. Chem. Phys. 1962 36 2824; f A. D. McLean B. J. Ransil and R. S. Mulliken J. Chem. Phys. 1960,32 1873.17 C. Barbier and G. Berthier Internat. J. Quantum Chem. 1967 1 657; R. H. Pritchard and C. W. Kern J. Arner. Chem. Soc. 1969,91,1631; fi C. W. Kern and M. Karplus J. Chern. Phys. 1965,42,1062; { E. A. G. Armour and A. J. Stone Proc. Roy. SOC. 1967 A 302,25; 0. J. Sovers M. Karplus and C. W. Kern J. Chem. Phys. 1966,45 3895; k R. E. Wyatt and R. G. Pam J. Chem. Phys. 1965,43 S217; 1966,44 1529; 0. J. Sovers C. W. Kern R. M. Pitzer and M. Karplus ibid. 1968 49 2592; Z L. Burnelle Theor. Chim. Actu 1964 2 177; J. B. Moffatt and H. E. Popkie Internat. J. Quunturn Chem.1968,2 565. n R. Bonaccorsi E. Scrocco and J. Tomasi J. Chem. Phys. 1969 50 2940; 0 W. H. F. Flygare J. M. Pochan G. I. Kerbey T. Caves M. Karplus S. Aung R. M. Pitzer and S. I. Chan J. Chem. Phys. 1966,45 2793; P A. D. McLean and M. Yoshimine J. Chem. Phys. 1967 46 3682; 0 C. W. Kern J. Chem. Phys. 1967 46 4543; r M. P. Melrose and R. G. Parr Theor. Chim. Acta 1967 8 150; A. D. McLean and M. Yoshimine J. Chem. Phys. 1966,45 3676. k r r Quadrupole Internal rotation coupling or inversion constant barrier d h i i,n 118