Methods of composite molecular wave functions. I. Variational principle and multiconfiguration SCF theory

The Silverstone–Stuebing variational principle for the discontinuous wave functions of one-electron systems is generalized for many-electron systems. The variational functional of energy takes real or complex value. The condition that it is real is given. Using the generalized variational principle, a multiconfiguration SCF theory for the composite molecular wave function is formulated. According to the theory, we may divide the whole space into space-filling cells, solve the SCF equations in each cell and build up the wave functions of the system by gathering the wave functions obtained in the cells. For use in the basis-set expansion method, the SCF equations are rewritten as matrix forms in which only one- and two-center integrals appear if an expansion center is located in each cell.     

On the SCF calculation of excited states: Singlet states in the two-electron problem

The problem of determining SCF wave functions for excited electronic states is examined for singlet states of two-electron systems using a Lowdin natural orbital transformation of the full CI wave function. This analysis facilitates the comparison of various SCF methods with one another. The distribution of the full CI states among the natural orbital MCSCF states is obtained for the S states of helium using a modest Gaussian basis set. For SCF methods that are not equivalent to the full CI wave functions, it is shown that the Hartree-Fock plus all single excitation wave functions are equivalent to that of Hartree-Fock plus one single excitation. It is further shown that these wave functions are equivalent to the perfect pair or TCSCF wave functions in which the CI expansion coefficients are restricted to have opposite signs. The case of the natural orbital MCSCF wave function for two orbitals is examined in greater detail. It is shown that the first excited state must always be found on the lower natural orbital MCSCF CI root, thus precluding the use of the Hylleras-Undeim-MacDonald (HUM) theorem in locating this state. It is finally demonstrated that the solution obtained by applying the HUM theorem (minimizing the upper MCSCF CI root with respect to orbital mixing parameters) is an artifact of the MCSCF method and does not correspond to any of the full CI states.     

Ab initio calculations on urea

Multiconfiguration wave functions constructed from contracted Gaussian-lobe functions have been found for the ground and valence-excited states of urea. ICSCF molecular orbitals of the excited states were used as the parent configurations for the CI calculations except for the ¹A₁(π → π*) state. The ¹A₁(π → π*) state used as its parent configuration an orthogonal linear combination of natural orbitals obtained from the second root of a three-configuration SCF calculation. The lowest excited states are predicted to be the n π → π* and π → π* triplet states. The lowest singlet state is predicted to be the n π → π* state with an energy in good agreement with the one known UV band at 7.2 eV. The π → π* singlet state is predicted to be about 1.9 eV higher, contrary to several previous assignments which assumed the lowest band was a π → π* amide resonance band. The predicted ionization energy of 9.0 eV makes this and higher states autoionizing.     

Dependence of valency on geometry for configuration interaction wave functions

Earlier definitions of valencies of atoms, molecules, and molecular orbitals are extended to configuration interaction (CI ) wave functions. Using these definitions, valencies both at equilibrium and nonequilibrium geometries of molecules are calculated at the CI level and compared with non-CI results. CI valency correlation diagrams are obtained. Valency variation with bond length using correlated wave functions is found to behave properly unlike in the case of SCF wave functions.     

A configuration interaction study of the water molecule and interpretation of the walsh diagram

Large scale ab initio SCF and CI calculations are used to investigate the bond angle dependence of the energy and one-electron properties of the water molecule. The SCF wave functions are used to reinvestigate the Hellmann-Feymann force interpretation of Walsh diagrams. The results are much closer to the ideas of Walsh than those obtained by Coulson and Deb.     

Effect of modified virtual orbitals on local Coulomb correlation holes in FCN

Local Coulomb correlation hole distribution functions may be used to assess the extent to which electron correlation effects are present in large scale SCF + CI wave functions. From a set of modified virtual orbitals, ordered according to their interaction with the SCF configuration, we have constructed a limited SCF + CI wave function with improved convergence characteristics with respect to that formed from the canonical virtual orbital set. These wave functions, of the same size yet with different energies, have been used to examine the range and depth of local Coulomb correlation holes in FCN. In all cases, the depth of the local Coulomb hole is no more than 10% or so of that of the corresponding Fermi hole, and the range Fermi correlation is generally less than that of Fermi correlation. This is particularly marked in the high density regions around the nuclei. The significance of our results is discussed in relation to a recent proposal for the incorporation of Coulomb correlation into the local exchange method.     

A configuration interaction study of the spin dipole-dipole parameters for formaldehyde and methylene

The electron spin dipole-dipole contribution to the zero field splitting has been evaluated for the ³A₂ (n → π*) and ³A₁ (π → π*) states of formaldehyde using a CI wave function constructed from contracted Gaussian-lobe functions. The values D = 0.539 cm^?1 and E = 0.031 cm^?1 were obtained for the ³A₂(n → π*) state and D = ?0.588 cm^?1 and E = 0.058 cm^?1 were obtained for the ³A₁ (π → π*) state using the CI wave function constructed from SCF orbitals of the respective parent configurations. An analysis of the effect of CI on the parameters is given for the ³A₂ (n n → π*) state of formaldehyde and the ³B₁ ground state of methylene. Numerical results are given which show that internally consistent self-consistent field orbitals (ICSCF ) are superior to canonical SCF orbitals as a starting point for a CI calculation. Our CI wave function for the ¹A₁ ground state gave an energy of ?114.13658 hartrees which is significantly lower than any previously reported energy calculation. This wave function gives a dipole moment of 2.22 Debye (C⁺O^?) in good agreement with the experimental value of 2.33 ± 0.02 Debye.     

Ab initio MRD–CI calculations on cubane (neutral,carbocation, carboanion) and dissociation of nitrocubanes based on localized orbitals

Ab initio MRD –CI calculations based on localized orbitals were carried out for cubane (neutral, carbocation, carboanion) both in our customary MODPOT basis set and in an all-electron 4–31G basis set. The calculated MRD –CI charge distributions on C1 (the skeletal atom from which the H^? or H⁺ was removed) (ab initio MODPOT neutral 4.221, carbocation 3.796, carboanion 4.282; all-electron 4–31G neutral 6.171, carbocation 5.717, carboanion 6.078) indicate that the + or - charge does not remain localized on C1 but redistributes itself. This has significant implications for preparative reactions of energetically substituted cubanes. The MRD –CI population analyses differ somewhat from the SCF population analyses, especially in the calculated total overlap populations. To investigate this effect on electrostatic molecular potential contour (EMPC ) maps generated from SCF or MRD –CI wave functions, we wrote additional routines to calculate EMPC maps from MRD –CI wave functions. The EMPC maps generated from SCF or MRD –CI wave functions are different if the molecule needs an MRD –CI multideterminant wave function to describe it adequately. The EMPC map is a one-electron property. One-electron properties are derived from the 1-matrix. The 1-matrix is different for SCF or MRD –CI wave functions. Thus, all the one-electron properties (EMPC maps, population analyses, bond deviation indices, etc.) are different when calculated from SCF or MRD –CI wave functions if MRD –CI wave functions are necessary to describe a system properly. We calculate these one-electron properties from the 1-matrix from the final natural orbitals. Our preliminary calculations for the dissociation pathway indicate it takes more energy to dissociate a bond in 1-nitrocubane than in octanitrocubane. Even in their ground electronic states at equilibrium geometry, both 1-nitrocubane and octanitrocubane require MRD –CI wave functions to describe them properly. The c² of the single determinant SCF wave function is only 0.8401 for 1-nitrocubane and 0.8300 for octanitrocubane. There are contributions from skeletal excitations as there are for cubane itself as well as excitations involving the nitrogroup. As the bond in nitrocubane is dissociated to 8.00 bohrs, the c² of the SCF contribution drops to only 0.4606 (1-nitrocubane) and 0.4445 (octanitrocubane). At this C₁? N₁ intermolecular distance, the largest excitations are in the C₁? N₁ bond: (C₁? N₁)² → (C₁? N₁*)², (C₁? N₁) → (C₁? N₁*). We also calculated the first electronically excited state for the dissociation pathway for selected points for both 1-nitrocubane and octanitrocubane.     

The electronic structure and one-electron properties of BH computed from SCF and CI wave functions

The SCF and CI wave functions for BH, obtained in calculations described in detail elsewhere [2], are compared through their electron distributions and electron moments.Taken in part from a Ph.D. thesis submitted to the University of Toronto in 1971.     

Hypervirial SCF treatment for vibrational energy levels of triatomic molecules

The SCF method for the calculation of energy levels of triatomic molecules is applied using the hypervirial perturbative procedure for solving the coupled equations. This treatment allows to obtain in a recursive way the energy corrections and the expectation values required in the SCF treatment, avoiding the explicit calculation of the wave functions. A numerical application is made to the SO₂ and O₃ molecules, comparing our results with those obtained by other methods.     

Scattering of fast electrons and X-rays from molecules: CH4 and C2H2

Self-consistent-field (SCF ) wave functions are used to calculate cross sections for the elastic and inelastic scattering of fast electrons and x-rays from CH₄ and C₂H₂ molecules. The effects of basis set choice and free rotation on these cross sections are investigated. The utility of an approximate scheme to correct SCF inelastic cross sections for the effects of electron correlation is examined. The probability density for the interelectronic distance, or radial intracule density, is obtained and discussed.     

Modification of the Roothaan equations to exclude BSSE from molecular interaction calculations

The Roothaan equations have been modified to compute molecular interactions between weakly bonded systems at the SCF level of theory without the basis set superposition error (BSSE). The increase in complication with respect to the usual SCF algorithm is negligible. Calculation of the SCF energy on large systems, such as nucleic acid pairs, does not pose any computational problem. At the same time, it is shown that a modest change in basis-set quality from 3-21G to 6-31G changes the binding energy by about 50% when computed according to standard SCF “supermolecule” techniques, while remaining practically constant when computed without introducing BSSE. Bader analysis shows that the amount of charge transferred between the interacting units is of the same order of magnitude when performed on standard SCF wave functions and those computed using the new method. The large difference between the corresponding computed energies is thus ascribed to the BSSE. © 1996 John Wiley & Sons, Inc.     

Geometric approximation to nuclear spin–spin coupling constants in the water molecule

The geometric aproximation is used within the framework of triple perturbation theory to evaluate the contributions to nuclear spin–spin coupling constants in the water molecule provided by the Fermi contact, the spin–orbit, and the spin–dipolar interactions. The results, obtained with SCF wave functions expanded over Gaussian basis sets of increasing quality, are compared with corresponding coupled Hartree–Fock estimates. The limits of the geometric approximation to coupling constants are discussed.     

Electron correlation effects in SCF+CI wave functions—Fermi and coulomb correlation holes in HCN

The Coulomb correlation hole distribution function has been computed with respect to various reference centers in the HCN molecule, using standard SCF +CI type wave functions. The extent to which statistical correlation between unlike-spin electrons is introduced into an SCF wave function through the inclusion of configuration interaction has been assessed by an examination of the range and depth of such holes, and compared with the behavior of analogous Fermi distribution functions. Our results show that the range of Fermi correlation is consistently longer than that of the corresponding Coulomb correlation.     

Small-angle electron scattering by formaldehyde and ketene: Effects of electron correlation and chemical binding

The total (elastic plus inelastic) intensities of 51 keV electrons scattered by H₂CO and H₂CCO have been measured over a range of K = (4π/λ) sin(θ/2) = 1–9.5 Å^?1 and compared with the theoretical intensities calculated with SCF and CI wave functions. Significant discrepancies are found between the experimental intensities and the theoretical ones based on the SCF wave functions. Most of the chemical binding and electron correlation effects observed in the total scattered intensities are reproduced by the theoretical intensities based on the CI wave functions calculated with the basis set including polarization functions on all atoms. © 1992 John Wiley & Sons, Inc.     

Modern valence-bond description of the electronic structure of benzocyclobutadiene

The ground state of benzocyclobutadiene, a bicyclic molecule with 8π electrons containing a benzene and a cyclobutadiene ring, is studied by means of modern valence bond (VB) theory in its spin-coupled (SC) form and the complete-active-space self-consistent field (CAS SCF) approach. The CAS SCF wave function is used to optimize the geometry, and SC theory—to obtain a well-correlated and easy to visualize and understand model of the active space hosting the π electrons. It is shown that the π system of the ground state of benzocyclobutadiene can be described with sufficient accuracy as a combination of the π systems of a distorted benzene ring and an isolated double bond. Each of the eight SC orbitals is found to be well-localized about one carbon atom only, with small distortions toward its nearest neighbors. The analysis of the optimal active-space spin-coupling pattern within the SC wave function for benzocyclobutadiene shows unambiguously that this molecule inherits neither the aromatic nor the antiaromatic character of its cyclic components, and it is most appropriate to regard it as nonaromatic. © 1996 John Wiley & Sons, Inc.     

CNDO calculations of diamagnetic shielding in molecules

An approximate procedure for the calculation of diamagnetic shielding in molecules is presented. The method proposed is based on the ‘complete neglect of differential overlap’ (CNDO) molecular wave functions and is formulated according to the zero differential overlap (ZDO) approximation. The results obtained with several CNDO-type wave functions for diatomic and polyatomic molecules are in very good agreement with non-empirical SCF calculations. The ¹⁴N diamagnetic shielding constants in several molecules were computed and some approximations usually adopted in the interpretation of ¹⁴N chemical shifts are critically discussed. It was shown that in some cases the observed ¹⁴N chemical shifts cannot be interpreted solely in terms of the paramagnetic contribution to the shielding constant.     

Semiempirical NDDO calculations with STO-3G and 4-31G basis sets

Possible refinements of semiempirical methods include the use of larger basis sets and of correlated wave functions. These possibilities are investigated in semiempirical NDDO SCF calculations with the STO-3G and 4-31G basis sets, and in correlated calculations at the STO-3G level. The present approach is characterized by the analytical evaluation of all one-center terms and two-electron integrals, and the semiempirical adjustment of the remaining one-electron integrals and the nuclear repulsions. The NDDO SCF results tend to reproduce the correspondingab initio results more closely than experimental data, even if they are parametrized with respect to experiment. The explicit inclusion of electron correlation at the STO-3G level improves the calculated results only slightly.     

Physical nature of the chemical bond II. Valence atomic orbital and energy partitioning studies of linear nitriles

The valence atomic orbitals (VAO 's) of several linear nitriles are determined using non-empirical SCF –LCAO –MO wave functions expanded in a minimal (CN^?, HCN, FCN, C₂N₂), double-zeta (CN^?, HCN), or double-zeta + polarization (HCN) basis of Slater atomic orbitals (AO 's). The molecular energy of each system (except the double-zeta + polarization HCN system) is partitioned according to the procedure of Ruedenberg to obtain numerical values of nitrile C and N atomic and C?N bond components of the energy. In addition, the nitrile results are compared with minimal AO basis results obtained previously by other authors for homonuclear diatomics, diatomic hydrides and H₂O. The numerical data are used to test the internal self-consistency of the various definitions entering the partitioning method, i.e. whether or not analogous quantities assume similar values in chemically similar situations. The analysis of nitrile SCF –MO wave functions in terms of the set of VAO 's characteristic of the system under consideration is shown to be a promising approach to the problem of extracting useful information from the wave functions. In general, numerical results for the nitrile systems studied are fairly consistent with the concepts on which the partitioning method is based: promotion, quasi-classical interaction, sharing penetration, sharing interference and charge transfer. However, the VAO expansions for several energy components need to be investigated further and possibly revised.