Showing papers in "International Journal of Quantum Chemistry in 1970"

Time‐dependent Hartree–Fock calculations in the Pariser–Parr–Pople model. Applications to aniline, azulene and pyridine

Abstract: Propagator or Green's function methods are used to analyze the time-dependent Hartree–Fock model The non-hermitian matrix problem for the time-dependent Hartree–Fock solution is reduced to a problem related only to hermitian matrices Particular attention is given to the calculation of oscillator strength in different approximations The connection between the stability of the Hartree–Fock solution and the solution of the time-dependent Hartree–Fock problem is demonstrated The results of numerical calculations are given for aniline, azulene and pyridine

Molecular MC–SCF calculations

Abstract: A practical method for finding multi-configurational SCF wave functions is proposed The basic equation is equivalent to the Brillouin theorem; comparison with the usual SCF equations obtained through effective hamiltonians gives an interpretation of the offdiagonal Lagrange multipliers Numerical applications to Formaldehyde in a minimum Slater-type orbital basis with four different variational wave functions are reported The molecular orbitals found in these calculations are localized on the chemical bonds The largest contributions to the energy are obtained from π-π and dispersion-type σ-π correlation

A new SCF procedure and its applications to ab initio calculations of the states of the fluorosulphate radical

Abstract: Ab initio calculations using a small Gaussian basis set, including 3d orbitals on the sulphur atom, have been performed on the fluorosulphate radical and the related ions SO3F+ and SO3F−. A new SCF procedure is described and applied to the open shell cases discussed here. The results are compared with recent CNDO calculations and with the experimental transition energies of the radical.

A study of the ground state wave function of carbon monoxide

Abstract: A large configuration-interaction calculation has been performed to determine the wave function, energy, and molecular properties of CO. The most important configurations were used to obtain the natural geminals and their occupation numbers. A pair-energy approach to the correlation energy was attempted with results which differ significantly from the configuration-interaction results.

Ab initio calculations, using a small Gaussian basis set, of the electronic structure of the sulphate ion

Abstract: Ab initio molecular orbital calculations of the electronic structure of the sulphate ion have been performed in which three Gaussian-type functions are used to simulate each member of a minimal basis of Slater-type orbitals. Comparative calculations on H2S show that such a basis excellently reproduces the properties of the valence electrons given by calculations in a Slater basis. The expansion of the basis by the addition of sulphur 3d orbitals results in a large decrease in the molecular energy (1 a.u.) and has a pronounced effect on the ordering and energy of the molecular orbitals. The results of a number of semiempirical schemes are discussed in the light of these results.

Molecular modelling with spherical Gaussians

Abstract: A mathematical model of a molecule is proposed. The electronic structure is described using a molecular orbital wave function constructed from a small number of spherical Gaussians with optimized parameters. The models exhibit the desirable properties of numerical stability, objectivity and transferability. Results are given for CH4, C2H6, cyclo-propane, H2O, CH2O and C2H4. They can be given a chemical interpretation in terms of chemical bonds, lone pairs and atomic cusps.

Critical point, singularities and extrapolations in the helium iso‐electronic sequence

Abstract: The Z-expansion of two-electron systems is analyzed with the Pade technique with emphasis on establishing analytical properties of the function E(Z) formally associated with the power series expansion. The concept of critical point in this connection is stressed. For this sequence it occurs at Zc = 0.911246 with E(Zc) = −0.415184. The structure of E(Z) for Z < Zc is investigated. The use of Pade approximants to extrapolate values of electron affinities is emphasized.

The generalized separated electron pair model. 1. An application to NH3

Abstract: The utility of the separated electron pair (SEP) model (strongly orthogonal geminals) is examined quantitatively, for pyramidal and planar nuclear configurations of the NH3 molecule. The best SEP wave function computed for each species is capable of recovering about half of the correlation energy obtained by a fairly accurate configuration interaction (CI) calculation, (corresponding to roughly 25% of the total molecular correlation energy). It is illustrated that the model can be systematically extended with only a modest effort to yield more accurate results (about 40% of the total correlation energy). The fact that the corrections to the SEP model have a simple physical interpretation suggests that this model may be a useful starting point for “brute force” CI calculations on larger chemical systems.

Optimum hybrid orbitals in localized orbitals

Abstract: Using the criterion for maximizing the projection of localized bond orbitals onto the space spanned by the occupied MO's, a method for constructing hybrid orbitals of a molecule is described. For illustration purposes the method is applied to single-determinant closed shell wave functions, calculated by means of ab initio and semiempirical procedures, for the molecules of methane, acetylene, ethylene, ethane, propylene, butadiene, ammonia and hydrogen cyanide. The predictions of hybridization are briefly discussed.

Study of the electronic structure of alternant radicals by the DODS method

Abstract: The polarization of closed shells in an alternant radical due to the field of its unpaired electron is discussed. It is shown that the Hartree–Fock solution is unstable relative to a small perturbation caused by polarization of closed shells in a long polyene radical. Its ground state is found to be antiferromagnetic with a forbidden zone in the spectra of one-particle excitations.

Valence structures for N2O4. III

Abstract: Further energies for types of N2O4 valence structures (or valence formulae) are reported. The results indicate that two sets of formulae might give satisfactory distributions of the mobile σ-electrons. These are (i) Resonance between covalent and ionic formulae ((B) and (D) of Part II[8]); (ii) A non-paired spatial orbital formula ((F) of Part II [8]). One estimate of the σ-bond order for the NN bond is 0.4–0.5. For this, (i) is to be preferred, because it involves a much lower energy than (ii). But should the order be about 0.7, the energies of (i) and (ii) are similar, and both descriptions are satisfactory. The ionic formula of (i) involves non-localized bonding for the NO moiety. It may be replaced by a formula with localized bonds, and an almost identical energy.

Molecular formation and electron correlation in HeH

Abstract: A one-centre CI wave function for HeH+ reported by Stuart and Matsen for 0.1 ⩽ R ⩽ 5.0 has been analysed in detail from the viewpoint of molecular formation. Further, by means of a natural orbital analysis, it was possible to obtain some measure of the electron correlation contained within such wave functions for various R values. These effects were illustrated by means of a series of difference maps for the electron density. One- and two-particle expectation values were obtained as a function of R. Thus, it was possible to study several aspects of the influence of the proton on the electron charge cloud as we pass from He through to the united atom Li+. The occupation numbers within the natural expansions were compared with those which arise from a similar analysis of a two-centre wave function for HeH+. The “character” of such wave functions for HeH+, and also for He and Li+, were analysed and compared.

Full configuration interaction for the benzyl radical

Abstract: The electronic structure of the benzyl radical in its ground state has been computed using a model Hamiltonian due to Pariser–Parr with full configuration interaction as well as with different truncated configurational sets built on SCF open-shell orbitals. The correlation energy corresponding to this model was found to be equal to –0.929722 eV. With the singly excited configurations only 18% of this energy is taken into account. By extending the basis to include the doubly excited configurations one can account for 94% of the correlation energy. An analysis of the accuracy of the proton hyperfine splitting calculation caused by inaccurate computation of the wave function is given. If only singly and even doubly excited configurations are taken into account one cannot hope to obtain splittings with an accuracy of more than 0.5 g. Inclusion of triply excited configurations lowers this error by one order. In addition, the use of the simple McConnell relation may lead to an error in splitting calculations of no less than 1.5 g.

Self-consistent calculation of excited 1P state wave functions of atoms

Abstract: In an earlier paper (Bibliography [1]) it has been shown that approximate representations of excited states of atoms can be obtained from the study of the singularities in the dynamic polarizability. Starting with these wave functions a more accurate calculation of the excited states in the Hartree–Fock scheme can be made by a perturbation treatment. The resulting wave functions yield significantly improved energy values. The 21P, 31P and 41P states of the He sequence up to C4+ are studied. The expectation values of a number of operators are calculated. The results obtained by the present method compare favourably with other elaborate calculations.

Extended group function calculations for H2O, NH3 and CH4

Abstract: Wave functions expressed as antisymmetrized products of strongly orthogonal geminals have been evaluated for H2O, NH3 and CH4. The geminals have been expressed as linear combinations of 2 × 2 detors constructed with localized SCF-MO's. Several ground state observables have been computed together with the electric polarizabilities and magnetic susceptibilities. In addition, a configuration interaction calculation limited to all possible double group excitations has been carried out.

Hartree–Fock MO–LCAO equations with charge‐dependent atomic integrals

Abstract: The Hartree–Fock equations are derived in the MO-LCAO approximation for the case when the integrals (except overlap integrals) over the atomic orbitals are charge-dependent. It is shown that inclusion of the overlap matrix in the iterative procedure gives equations which are too complicated for the simple model under consideration. The approach is applied to the VESCF method in the PPP scheme.

Solution of the Hartree–Fock problem by expansion onto nested bases

Abstract: A method for solving the Hartree–Fock problem in a finite basis set is derived, which permits each orbital to be expanded in a different basis. If the basis set for each orbital ϕi contains the basis functions for the preceding orbitals, ϕi−1, ϕi−2,… ϕ1, then the ϕi form an orthonormal set. One advantage over the standard Hartree–Fock method is that a different long range behavior for each orbital, as for example is required in the Hartree–Fock-Slater method, can be forced. A calculation on the ground state of beryllium is performed using the nested procedure. Very little energy is lost because of nesting, and the node in the 1s orbital disappears.

A study of dynamic quadrupolar and octupolar excitations and calculations on excited d and f states of atom and ions: He-sequence

Abstract: The quadrupolar and octupolar distortion of the ions in the He-sequence caused by an external electro–magnetic field has been studied by a variation–perturbation method in the Hartree–Fock scheme. For certain frequencies singularities appear in the response of the system to the perturbation. Approximate representations for the excited d and f states have been obtained from a study of these resonances. Such a perturbation calculation has the advantage that representations of the different excited states are obtained independently. The orthogonality to all the lower lying levels of the same symmetry is not required. The only source of inaccuracy implicit in the procedure lies in the improper consideration of the inter-electronic interaction. This is corrected for by an independent calculation, which is again formulated in terms of a perturbation treatment. The resulting wave functions for the excited states are accurate in the Hartree–Fock model. Expectation values of several operators have been calculated with these corrected wave functions.

A multi‐configuration LCAO–MO study for complex unsaturated molecules. I. General theory and its application to the benzene anion

Abstract: A multi-configuration LCAO–MO approach using a π-bond order–bond length linear relation is introduced to predict the geometrical structures for the electronic ground and excited states of unsaturated hydrocarbons. The procedure is designed to include configuration interaction in each iterative computation where the π-electron approximation is employed under the Pariser–Parr type semi-empirical treatment. The π-bond order–bond length relation is determined as rpq = 1.523 – 0.193Ppq, when the bond lengths of ethylene, benzene and naphthalene are used and the groundstate functions including the singly and doubly excited configurations are taken into account to obtain the bond orders Ppq. The iterative calculation is applied to the ground state and the two lowest excited states of the benzene anion in both D6h and D2h molecular geometries. The geometrical structures and the π-electron energies are computed for the ground and excited states of the anion; for the latter, two types of configuration species are used. It is found that the first lowest excited state is not subjected to the Jahn–Teller effect and the calculated excited state energies do not agree with the observed values (c. 1.0 ∼ 2.5 eV higher than the observed values). The latter point is discussed in detail. It is also found that the resultant ground state energy depression due to configuration mixing is not very large and the two types of configuration species used give different CI effects on the energy levels of the two lowest excited states of the anion. Finally, the stabilization energy due to the Jahn–Teller distortion is estimated for the ground state of the anion.

A multi‐configuration LCAO–MO study for complex unsaturated molecules. II. Application to the benzene cation

Abstract: The method of the MC–LCAO–MO approach, described in the preceding paper, is further applied to the benzene cation. Through the iteration process the π-electron energies and the molecular shapes are computed for the ground and two lowest excited states of the cation in both D6h and D2h geometries. A remarkable fact obtained is that a comparatively small variation of the geometrical structure (c. 0.010 – 0.013 A bond length difference) brings about a considerable change of the energy value (c. 0.85 – 1.25 eV). The π-electronic excitation energies obtained from the iteration process are compared with the transition energies calculated from the usual method in which the structures of the excited states are assumed to be the same as the corresponding ground state structures. The difference in the excitation energy between the cation and the anion, and the CI effect on the excited states, are discussed. It is found that the doubly excited configurations play an important role in CI, which is somewhat different from that of the singly excited configurations. The stabilization energy due to the Jahn–Teller distortion is estimated for the ground state of the cation.

Wellenmechanische Behandlung der Protonisierung von F- und CH3-monosubstituierten Äthylenen

Abstract: The systems C2H4F+ and C2H4CH have been investigated by the SCF–MO-P (LCGO) method. It will be shown that the results agree with the empirical rule of Markownikoff. An explanation of the rule of Markownikoff is given by means of the computed results.

Small, simultaneous adjustments of orbital exponents in LCAO–MO–SCF calculations using self‐consistent perturbation theory

Abstract: Self-consistent perturbation theory is introduced to facilitate making small, simultaneous variations in orbital exponents. This is accomplished by interpreting these variations as perturbations on the quantum mechanical system. The minimum-energy condition yields a set of linear equations for the desired exponential corrections.

Fifth-rank molecular polarization tensors

Abstract: Fifth-rank molecular polarization tensors are discussed in terms of their tensor group which includes both the spatial symmetry and the permutation symmetry of the suffixes. This latter may be described by certain four-dimensional point groups, which in the nontotally symmetric cases may be projected into three-dimensional space. The study leads to a determination of the number of independent components of these tensors and a discussion of the types of phenomena which they may be expected to produce.

Self-consistent group calculations on polyatomic molecules V. Molecules with a double or triple bond

Abstract: SCGF calculations are reported for the ground state of ethylene, formaldehyde, acetylene and hydrogen cyanide. A minimum basis of contracted Gaussians was used and optimum hybridization was determined for each of the molecules by systematic variation of the hybridization parameters until the total electronic energy was a minimum. Properties of CH bonds as well as CC, CO and CN σ and π bonds are discussed in some detail. The results show that the assumption of transferable framework integrals β, basic to all semiempirical methods of calculating molecular wave functions, is strictly justified within the SCGF method.

Gaussian overlap approximation in the projected Hartree–Fock method

Abstract: A method for the approximate calculation of matrix elements with respect to projected Hartree–Fock wave functions is proposed. The method is tested on some calculations in the many-parameter AMO method. It is found that the approximation reduces the amount of work, involved in the evaluation of the energy, by a factor of five and that it reproduces the exact values to within a few per cent.

Gaussian molecular orbital calculations of the barrier to internal rotation in the ethyl cation

Abstract: Extended Gaussian orbital basis set calculations have been carried out on an assumed staggered and eclipsed classical geometrical configuration of C2H. The best total energies obtained for these geometries were −78.170692 a.u. and −78.170674 a.u. respectively, corresponding to a barrier to internal rotation of 1.8 × 10−5 a.u. or 11 kcal/mole. An analysis of the charge density matrix indicates that charge is distributed in these molecules in a manner consistent with the concept of hyperconjugation.

Note on the space part of anti-symmetric wave functions in the many-electron problem

Abstract: It is pointed out that if a many-electron antisymmetric wave function is expanded as a sum of spin-product functions, each multiplied by a function of coordinates, the resulting functions of coordinates have many of the same useful features found with the symmetric and antisymmetric functions representing singlet and triplet states in a two-electron system. For finding the energy, or any function of coordinates only, in the approximation in which spin-orbit interaction is neglected, one such function of coordinates can be used, the spins being disregarded. Simple procedures allow one to find matrix components of such operators as S2 and L . S from the functions of coordinates. These procedures are much easier to visualize than the use of projection operators, the permutation group, or other methods in current use. The general procedures are illustrated by application to the three-electron problem of the lithium atom, as treated by Lunell, Kaldor, and Harris, and their application to the contact hyperfine structure is pointed out.