Integrals Irs and Jrs used in the computation of magnetic quantities by means of the McWeeny coupled perturbation method

We have shown that integrals I_rs and J_rs which occur in coupled Hartree–Fock perturbation, on a basis of gauge invariant atomic orbitals with the London approximation and neglect differential overlap, can be reduced, by appropriate transformations, to the overlap integral type. The computational program of S_rs, I_rs, and J_rs integrals is elaborated for Slater-type atomic orbitals. The process proposed presents a double advantage: it is extended over the entire Periodical Table and does not use the analytical formulas of Mulliken.     

Automatic geometry optimization for molecules with d orbitals. I. General analytic formulas for derivatives of slater orbitals

General analytic formulas for derivatives of real Slater orbitals with respect to the Cartesian coordinates x and y have been derived. The application of this result to molecular geometry optimization or to the construction of a force constant matrix is briefly discussed.     

General expressions for monocenter repulsion integrals in a basis of real atomic orbitals

General expressions for monocenter electron repulsion integrals in a basis of real atomic orbitals are derived in terms of the radial integrals R. The final expressions for these integrals can be classified into five main classes which are characterized by the angular part of the real atomic orbitals. For a basis of real s, p, d, and f AO's the total number of monocenter repulsion integrals is 65536, from which 6652 are different from zero. The nonzero integrals can be classified into 430 groups which contain integrals of equal value.     

Construction of basis sets of hybrid atomic orbitals for describing the nonequivalent bonds of an atom with three neighboring atoms (sp 2 andsp 3 hybridization)

The paper describes methods for obtaining basis sets of sp² and sp³ hybrid atomic orbitals to describe nonequivalent chemical bonds in organic molecules with heteroatoms. Being affine, these basis sets present an alternative to the conventional basis sets of Cartesian atomic orbitals. Unlike the latter, the hybrid atomic orbitals are invariant to spatial rotations of molecules, thus providing the rotational invariance of the results of any semiempirical quantum chemical calculations. The construction of the basis sets of hybrid functions for atoms surrounded with three neighboring atoms (planar and nonplanar configurations) is considered in detail. V. I. Vernadskii Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 36. No. 6, pp. 963–968, November–December, 1995. Translated by I. Izvekova     

Construction of basis sets of hybrid atomic orbitals for describing the nonequivalent bonds of an atom with four neighboring atoms (sp 3 hybridization)

Methods for constructing the basis sets of sp³ hybrid atomic orbitals for atoms surrounded with four neighboring atoms are discussed in detail. The results obtained are generalized as a theorem of sp³ hybridization. A general method is proposed for constructing basis sets of orthogonal sp³ hybrid atomic orbitals for atoms with distorted tetrahedral environments. V. I. Vernadskii Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 6, pp.969–975, November–December, 1995. Translated by I. Izvekova     

Exact formulas for overlap integrals of Slater-type orbitals with equal screening constants

Exact formulas for 147 overlap integrals between Slater-type orbitals with equal screening constants are presented in the most simplified form. This represents all combinations of orbitals with quantum numbers: 1 ≤ N ≤ 5, 0 ≤ L ≤ 3, and M ≤ L. The formulas are automatically generated by computer using the “C-matrix” single-center expansion method. There are no limitations to the applicability of this method to orbitals of higher quantum numbers.     

Complex two- and four-index transformation over two-component kramers-degenerate spinors

A two- and four-index integral transformation over complex molecular spinors expanded in a basis of real atomic spin orbitals is presented. The well-known N⁵ algorithm for the non-relativistic case has been extended to the relativistic case and implemented. The incorporation of spatial double-group symmetry is discussed.     

Use of Recursion and Analytical Relations in Evaluation of Hypergeometric Functions Arising in Multicenter Integrals with Noninteger <Emphasis Type="Italic">n</Emphasis> Slater Type Orbitals

In this work we present the new recursion and analytical relations for the calculation of hypergeometric functions F(1,b;c;z) occurring in multicenter integrals of noninteger n Slater type orbitals. The formulas obtained are numerically stable for 0 < z < 1 and all integer and noninteger values of parameters b and c The Author cordially congratulates Prof. I.I. Guseinov on his 70th birthday     

Relativistic molecular orbitals for the double group D3h

Relativistic symmetry orbitals are given for the double group D_3h. For atomic orbitals at the symmetry center a general expression is presented. The atomic orbitals of the s, p_½, and p_3/2 variety outside the center are also considered. The representation matrices are given in explicit form.     

Theory of complete orthonormal relativistic vector wave function sets and Slater type relativistic vector orbitals in coordinate,momentum and four-dimensional spaces

The new analytical relations for the relativistic vector wave functions and Slater type relativistic vector orbitals in coordinate, momentum and four-dimensional spaces are derived using the properties of quasirelativistic vector spherical harmonics introduced by the author in previous paper (I.I. Guseinov, J. Math. Chem., 44, 197 (2008)) and complete orthonormal scalar basis sets of nonrelativistic ψ ^α -exponential type orbitals (ψ ^α -ETO), -momentum space orbitals (-MSO) and z ^α -hyperspherical harmonics (z ^α -HSH). The 6-component relativistic vector wave functions obtained are complete without the inclusion of the continuum. The relativistic vector wave function sets and Slater type relativistic vector orbitals are expressed through the corresponding quasirelativistic vector wave functions and Slater vector orbitals, respectively. The analytical formulas are also derived for overlap integrals over Slater type relativistic vector orbitals with the same screening constants in coordinate space.     

Computation of overlap integrals over Slater-type orbitals using auxiliary functions

Analytical expressions through the binomial coefficients and recursive relations are derived for the expansion coefficients of overlap integrals in terms of a product of well-known auxiliary functions A_k and B_k. These formulas are especially useful for the calculation of overlap integrals for large quantum numbers. Accuracy of the computer results is satisfactory for the values of quantum numbers up to 50 and for the arbitrary values of screening constants of atomic orbitals and internuclear distances. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 199–204, 1998     

Slater-orbital molecular integrals by numerical Fourier-transform methods. II. Four-center integrals over is orbitals

The four-center nonplanar electron repulsion integrals over 1s Slater-type atomic orbitals are considered by a numerical Fourier-transform method. It is shown that the highly oscillating integrand appearing in the Fourier inversion formula could be successfully treated by using Tchebyscheff quadrature. The resulting formulas are thoroughly discussed with particular emphasis on their numerical features and convergence properties. It follows that the aforementioned integrals may be calculated with a good accuracy with a moderate amount of computing time.     

The use of population localised orbitals to interpret molecular orbital calculations

An efficient and general method is derived to calculate population localised molecular orbitals (LMO's) from a given SCF eigenvector matrix, by reduction to an eigenvalue problem. Applications to both localised molecules (NH₃ and C₂H₂) and delocalised ones (B₂H₆, C₆H₆ and butadiene) are discussed in some detail. It is shown that unequal occupation of atomic energy levels leads to non-orthogonal LMO's. The consequences of non-orthogonal atomic hybrid orbitals are discussed, formulas for their overlap in terms of atomic occupation numbers are derived and it is shown that the occupation numbers are connected to LMO atomic orbital coefficients by various sum rules.     

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.     

The Fock Matrix Analysis for Atomic Orbitals in Molecular Orbitals II. The Electronic Structure of N2 Molecules

Weinhold's natural hybrid orbitals can be chosen as the molecular adapted atomic orbitals to build the canonical molecular orbitals of N₂ molecules. The molecular Fock matrix expanded in the natural hybrid orbitals can reveal deeper insight of the electronic structure and reaction of the N₂ molecule. For example, the magnitude of F_ab can signify the bonding character of the paired electrons as well as the diradical character of the unpaired electrons for both σ‐ and π‐types. Discarding the concept of the overlap between non‐orthogonal atomic orbitals, the different orbitals for different spins in the unrestricted Hartree‐Fock wavefunction reveal that there are three pairs of opposite spin density flows between two atoms, which proceed until the bonding molecular orbitals form.     

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.     

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.     

Gaussian orbitals optimized for lower bounds of hydrogenic atoms

Various optimization criteria are compared for the hydrogen atom to find orbitals which improve lower bounds computed from the Weinstein, Temple, and Stevenson-Crawford formulas. Minimization of squared energy deviation, “variance,” is recommended because the resulting lower bound orbitals give excellent lower bounds, converge to the exact wave function, are relatively easy to optimize, and are insensitive to the estimated energy eigenvalue. New linear combinations of Gaussian orbitals which minimize the variance are presented for the 1s, 2s, 2p, 3s, 3p, and 3d orbitals. These orbitals are compared with previous linear combinations with regard to their expectation values and local properties.     

Validity of <Emphasis Type="Italic">B</Emphasis>-splines as a universal basis set for atomic Hartree–Fock–Roothaan calculations

The validity of B-splines as a universal basis set for atomic Hartree–Fock–Roothaan calculations is studied. In order to accomplish our aim, the ground-state energies of neutral atoms He–Xe, cations Li ⁺–Xe ⁺, and anions H ^-–I ^- with the nuclear charge Z=54 are calculated by the Hartree–Fock–Roothaan method with the B-spline sets. All radial functions of the atoms and singly charged ions are expanded by common B-spline sets regardless of atomic systems and symmetries of atomic orbitals. The energies obtained by the best B-spline set are in excellent agreement with ten-digit numerical Hartree–Fock results.     

On the choice and definition of atomic-orbital bases. I. General considerations; promotion and hybridization; electric dipole moments

After a brief discussion of the physical significance of the choice of the basis in molecular calculations, the nature and definition of an atomic-orbital basis for use in limited calculations is discussed, in view of the possibility of replacing, say, ordinary 2s and 2p Slater orbitals by appropriate hybridized-promoted atomic orbitals. It is indicated that, if the orbitals must be defined in connection with a given interpretation scheme for the behavior of molecules, hybridization and promotion may be necessary. The two kinds of conditions one may wish to impose on a restricted atomic-orbital set are explicitly considered. The first is that the atomic orbitals should be hybrids directed along the bonds and at the same time satisfy the maximum overlap criterion; the other is the requirement that the atomic orbitals should be such that the electric dipole moment of a polyatomic molecule described in terms of a semiempirical bond-orbital scheme should be expressed as the dipole moment of the system of bond charges located at the nuclei. The latter condition is treated in detail, showing that it implies a cancellation of atomic and overlap moments. The equations defining the atomic orbitals satisfying the condition in question are given. In the course of the mathematical treatment some general results concerning the expression of the dipole moment of a molecule and the definition of net atomic charges are given, showing that, for systems where overlap integrals are low, the atomic populations can be taken as sums of the squares of the coefficients of orthogonalized atomic orbitals. Applications of the results will be presented in part II.