Angular dependence of Gaussian-Lobe orbitals. II. Set of axial Gaussian-Lobe orbitals

The topological properties of real spherical harmonic representations on the unit sphere have been found to provide a convenient tool to infer the lobe edifices which mimic these orbitals. The prohibitive number of lobes required in such an approach for l > 2, can be avoided in using only axial Gaussian-lobe orbitals (AGLO ). It is proved that 2l + 1 independent Y_lo-like functions correctly span the relevant Y_lm (m = ?l,l) subspace. The multipolar component analysis of any spatial arrangement of lobes is derived, and allows the optimization of the angular dependence of AGLO S. The cases of d- and f-orbitals are studied in detail and accurate optimized functions are proposed. This method can be easily extended to obtain the atomic orbitals of any azimuthal quantum number l-subspace.     

Expansion of linear combinations of slater-type orbitals in eigenfunctions of the three-dimensional isotropic harmonic oscillator

Several classes of functions related to the Gaussian have been used with success as basis sets for the representation of atomic and molecular orbitals. We have compared the representation of a hydrogen 1s orbital by a sum of Gaussian lobe functions with its expansion in eigenfunctions of the three-dimensional isotropic harmonic oscillator. The lobe functions are shown to achieve better expectation values of the energy, with fewer terms. The lobe functions have the further computational advantage of not containing high powers of the radius. It is concluded that the lobe functions are a superior basis set for use in calculations of the electronic structure of atoms and molecules.     

Spherically symmetric axial Gaussian lobe 3d and 4f orbitals

The axial Gaussian lobe orbital (AGLO ) representations of 3d and 4f orbitals proposed by LeRouzo and Silvi have been angularly optimized to ensure spherical symmetry of filled 3d and 4f shells. The functions have been tested on the hydrogen atom in the presence of high quality s and p basis sets and found to provide excellent minimal Gaussian representations of polarization functions. Exact orbital degeneracy is not obtained within each shell, however. Tabulated values are given to allow arbitrary scaling of the 3d and 4f lobe mimic orbitals.     

Molecular states of atoms. II

Orbital energy parameters, previously obtained from atomic valence state energies, are used in calculating approximate wave functions for their orbitals. The radial factors of these wave functions are expressed as linear combinations of three Gaussian type orbitals with selected exponents, the coefficients being determined by normalisation and reproduction of the kinetic energy and interelectron repulsion parameters. Wave functions of universal form are obtained for the non-transition elements up to xenon. Each calculated s orbital wave function (except 1s) has a radial node, as is appropriate if there is a p orbital in the same shell with none.     

Basis set quality. II. Information theoretic appraisal of various s- orbitals

The expectation values 〈r^k〉 (?2 ? k ? 4, k = 10), values of the charge density ρ(r) at selected points, and coefficients in the MacLaurin expansion of ρ(r) are used to test the quality of 71 orbital basis sets used for the atomic helium Hartree–Fock problem. These tests in position space are complementary to the momentum space tests previously carried out [Int. J. Quantum Chem. 21 , 419 (1982)]. Information theoretic measures with respect to either or both position and momentum space properties are subsequently defined and the orbitals are ranked accordingly. These measures indicate that, for a given orbital, momentum space properties are more poorly predicted than position space ones. Moreover an improvement in the virial ratio does not necessarily lead to a more balanced orbital with respect to position and momentum space properties. Basis sets containing Slater-type orbitals are again found to be rather accurate. The exponentially damped rational function is confirmed to be the outstanding two-parameter unconventional orbital.     

Axial gaussian-lobe orbitals: Optimization of highly angularly dependent functions

Highly angularly dependent axial Gaussian-lobe orbitals (AGLO ) (up to L = 5) are presented. The angular and radial optimizations of these functions have been realized on the ground of theoretical frameworks, previously reported in the literature and somewhat extended here. The numerical difficulties that can appear in the recombination of elementary integrals over lobes are particularly investigated. It is shown that it is necessary to limit the angular accuracy, i.e., the relevant Y_Lo character, in order to preserve the accuracy of atomic integrals. The proposed p, d, f, g, and h AGLOS satisfy this condition, and can be used with confidence in LCAO–MO–SCF calculations. Their advantages, e.g., for the treatment of large symmetrical inorganic systems containing transition metal atoms, are emphasized.     

What do kinetic-energy anisotropies tell us about chemical bonding? I. Diatomic hydrides

The kinetic-energy anisotropies of fifteen diatomic hydrides AH with A = H, Li, Be, B, C, N, O, F, Na, Mg, Al, Si, P, S, Cl are calculated from self-consistent-field wave functions constructed from extended basis sets of Slater-type orbitals. It is found that there is no consistent ordering of the bond-parallel and bond-perpendicular components of the kinetic energy with respect to separated atom values. An analysis of the orbital contributions reveals that nonbonding π orbitals make large contributions to the total kinetic-energy anisotropy. This makes it difficult, if not impossible, to deduce anything about the nature of the chemical bond from the total anisotropy. However, certain dimensionless orbital kinetic-energy anisotropies are useful for interpretative studies because, in free atoms, these quantities have fixed values that depend only on the symmetry of the orbital.     

Momentum space properties of various orbital basis sets used in quantum chemical calculations

Momentum expectation values (p^k), values of the momentum density P(p) at selected points, and coefficients in the MacLaurin and asymptotic expansions of P(p) are used to test the quality of various orbital basis sets previously used for the atomic helium Hartree-Fock problem. The wellknown position-space defects of Gaussian-type orbital expansions are shown to have their momentum-space counterparts. Expansions of even-tempered Slater-type orbitals are found to be rather accurate. The exponentially damped rational function is found to be the outstanding two-parameter unconventional orbital.     

On the defects of molecular symmetry-lobe orbitals

It is proved that two molecular symmetry-lobe orbitals, belonging to different irreducible representations, can have a non-negligible overlap. Using a previously reported multipolar analysis of Gaussian-lobe orbitals (GLOs), it is demonstrated that such defects occur when individual symmetry orbitals (SOs) are both contaminated in a given Y_1m subspace, even if such contaminations are very small. A numerical application illustrates this result in the case of the NH₃ molecule, and it is shown that axial-GLOs allow for the exact cancellation of the symmetry defects.     

Determination of one-centre core integrals from the average energies of atomic configurations

One-center core integrals for valence orbitals are determined from the experimental average energies of neutral atomic configurations from Li through Zn. These values are compared with those estimated from CNDO /1, “INDO /1”, CNDO /2, “INDO /2” and with theoretical values calculated from a pseudo-potential method. The agreement is good between values obtained from neutral atoms and from the psuedo-potential calculation except for the 3d orbitals of the transition elements where the theoretically calculated integrals over single ξ functions are not realistic. These two methods reproduce both term and average configuration energies for the first two rows of atoms; the semiempirical method reliably reproduces them for the third row. The CNDO /1 and INDO /1 methods underestimate atomic energies, while the CNDO /2 and INDO /2 procedures fail rather poorly. The propriety of using core integrals estimated semiempirically in molecular orbital calculations is discussed.     

Radial moment analysis of isovalent hybridization in N2

A radial moment analysis has been performed for the Hartree–Fock molecular orbitals of the nitrogen molecule. The objective of the analysis was to determine the extent of isovalent hybridization in even and odd sigma molecular orbitals. The radial moment analysis for the LC -SCF -AO fragments of the 2σ_g, 2σ_u, and 3σ_g molecular orbitals substantiates Mulliken's earlier conjecture concerning promotion into 3s atomic orbitals for the 3σ_g molecular orbital. The concept of free isovalent hybridization is discussed in terms of the atomic orbital shape defined by the extracted moments.     

Unrestricted CNDO-MO calculations. II. MnO 4 2− and CrO 4 3−

The electronic structures of the tetrahedral molecule ions MnO ₄ ^2– and CrO ₄ ^3– have been investigated within an unrestricted CNDO-MO approximation [Theoret. Chim. Acta (Berl.)20, 317 (1971)]. Calculations assuming the unpaired electron occupies the 3a ₁, 2e, and 4t2 molecular orbitals indicate that the 3a ₁ and2e orbitals have similar orbital energies and that the 4t ₂ orbital is at a higher energy. The experimentally indicated2e orbital for the unpaired electron is obtained with expanded O^1– type atomic orbitals for oxygen and valence metal orbitals of the expanded 3d and plus one ion 4p types. The metal 4s orbitals must be held to the neutral atom type. The optimum valence orbitals above with a slightly contracted 4s type metal orbitals yield the minimum total energy and places the unpaired electron in the 3a ₁ orbital. Since the contracted 4s metal orbital produces results that are not in agreement with experimental data, the method used apparently does not adequately take into account the increased electron-electron repulsions that contracted 4s orbitals produce.     

On the use of spatial symmetry in ab initio calculations. Transformation of the two-electron integrals from atomic orbital to localized molecular orbital basis

Anm ⁵-dependent integral transformation procedure from atomic orbital basis to localized molecular orbitals is described for spatially extended systems with some Abelian symmetry groups. It is shown that exploiting spatial symmetry, the number of non-redundant integrals for normal saturated hydrocarbons can be reduced by a factor of 2.5-3.5, depending on the size of the system and on the basis. Starting from a list of integrals over basis functions in canonical order, the number of multiplications of the four-index transformation is reduced by a factor of 2.8-3.5 as compared to that of Diercksen's algorithm. It is pointed out that even larger reduction can be achieved if negligible integrals over localized molecular orbitals are omitted from the transformation in advance.     

One-Parameter double-zeta atomic functions for the hartree–fock energies of helium isoelectronic sequence

The double-zeta atomic functions are characterized by the nuclear charge z of the two-electron atomic system. The Hartree–Fock total energies and the corresponding orbital energies are calculated using various atomic wave functions for the helium isoelectronic sequence. The expectation values rⁿ of various wave functions are also examined. It is found that the accuracy of our one-parameter double-zeta functions corresponds to the accuracy of the usual five-parameter double-zeta functions.     

Calculation of three-center electric and magnetic multipole moment integrals using translation formulas for Slater-type orbitals

 By the use of translation formulas for the expansion of Slater-type orbitals (STOs) in terms of STOs at a new origin, three-center electric and magnetic multipole moment integrals are expressed in terms of two-center multipole moment integrals for the evaluation of which closed analytical formulas are used. The convergence of the series is tested by calculating concrete cases. Computer results with an accuracy of 10⁻⁷ are obtained for 2^ν– pole electric and magnetic multipole moment integrals for 1≤ν≤5 and for arbitrary values of screening constants of atomic orbitals and internuclear distances. Received: 28 October 1999 / Accepted: 15 February 2000 / Published online: 5 June 2000     

Intrinsic local constituents of molecular electronic wave functions. I. Exact representation of the density matrix in terms of chemically deformed and oriented atomic minimal basis set orbitals

A coherent, intrinsic, basis-set-independent analysis is developed for the invariants of the first-order density matrix of an accurate molecular electronic wavefunction. From the hierarchical ordering of the natural orbitals, the zeroth-order orbital space is deduced, which generates the zeroth-order wavefunction, typically an MCSCF function in the full valence space. It is shown that intrinsically embedded in such wavefunctions are elements that are local in bond regions and elements that are local in atomic regions. Basis-set-independent methods are given that extract and exhibit the intrinsic bond orbitals and the intrinsic minimal-basis quasi-atomic orbitals in terms of which the wavefunction can be exactly constructed. The quasi-atomic orbitals are furthermore oriented by a basis-set independent method (viz. maximization of the sum of the fourth powers of all off-diagonal density matrix elements) so as to exhibit clearly the chemical interactions. The unbiased nature of the method allows for the adaptation of the localized and directed orbitals to changing geometries. Contribution of the Mark S. Gordon 65th Birthday Fegtschrift Issue.     

Molecule intrinsic minimal basis sets. I. Exact resolution of ab initio optimized molecular orbitals in terms of deformed atomic minimal-basis orbitals

A method is presented for expressing the occupied self-consistent-field (SCF) orbitals of a molecule exactly in terms of chemically deformed atomic minimal-basis-set orbitals that deviate as little as possible from free-atom SCF minimal-basis orbitals. The molecular orbitals referred to are the exact SCF orbitals, the free-atom orbitals referred to are the exact atomic SCF orbitals, and the formulation of the deformed "quasiatomic minimal-basis-sets" is independent of the calculational atomic orbital basis used. The resulting resolution of molecular orbitals in terms of quasiatomic minimal basis set orbitals is therefore intrinsic to the exact molecular wave functions. The deformations are analyzed in terms of interatomic contributions. The Mulliken population analysis is formulated in terms of the quasiatomic minimal-basis orbitals. In the virtual SCF orbital space the method leads to a quantitative ab initio formulation of the qualitative model of virtual valence orbitals, which are useful for calculating electron correlation and the interpretation of reactions. The method is applicable to Kohn-Sham density functional theory orbitals and is easily generalized to valence MCSCF orbitals.     

Recursion formulae for calculation of overlap integrals

Multi-ζ Slater-type orbitals are frequently used in molecular orbital calculations. Master formulae and numerical tables are available in literature for overlap integrals between s, p, and d atomic orbitals up to principal quantum number (n) = 3 and for some other selected quantum numbers. However, no master formula or numerical table is available for quantum numbers n = 5 and above and involving ? orbitals. In this article recursion formulae have been presented for the calculation of the overlap integral between any two s, p, d, and ? atomic orbitals formed by a linear combination of Slater-type orbitals. These formulae, when expanded, would give rise to all the master formulae reported in the literature as well as formulae hitherto unreported.