Asymptotic formulas for Coulomb and hybrid integrals with slater type functions

Asymptotic expressions are obtained for two-center Coulomb and hybrid integrals with Slater functions. Results of numerical test of these expressions are presented. The use of asymptotic formulas in LCAO-MO calculations of electronic molecular structure substantially reduces the computation time. V.I. Vernadskii Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 35, No. 2, pp. 12–15, March–April, 1994. Translated by L. Chernomorskaya     

A new method to calculate overlap integrals of vibrational wave functions in the theory of vibronic spectra of polyatomic molecules

An approximate method to calculate overlap integrals of vibrational wave functions of combined electron states is proposed. It uses reducibility of the general transformation of normal coordinates and quasiorthogonality of the Dushinsky matrix. Simple analytical expressions and convenient recurrent relations for the desired integrals of Franck-Condon and Herzberg-Teller types are found. The errors of calculation are of the same order as those of the existing “accurate” methods (≤5%), and the speed of calculation is higher by more than two orders. The study was carried out under financial support of the Russian Fundamental Research Fund (93-02-3405). K. A. Timiryazev Moscow Agricultural Academy. Translated fromZhurnal Strukturnoi Khimii, Vol. 35, No. 2, pp. 16–23, March–April, 1994. Translated by L. Chernomorskaya     

New Method for calculating effective charges on atoms in molecules, clusters, and solids

A procedure for calculating effective charges on atoms in an arbitrary polyatomic system has been developed on the basis of the one-electron DV cluster method. Effective charges are calculated as three-dimensional integrals of electron density over the space around atomic nuclei in the object under investigation. An original procedure is suggested to assign the charge density to a particular center at every point of space. Translated fromZhurnal Struktumoi Khimii, vol. 39, no. 6, pp. 1134–1140, November–December, 1998.     

Role of water structure in thermodynamics of nonpolar solvation

The SSOZ (site-site Ornstein-Zernike) equation with an original closure condition for liquid molecular systems is used to calculate thermodynamic functions of noble gas solvation in water. Water is modeled by two close sets of atom-atomic potential functions. The calculations indicate that the chemical solvation potential is strongly sensitive to water structure. A comparison with experiment is given. Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 37, No. 4, pp. 736–741, July–August, 1996. Translated by L. Smolina     

Integral invariants for high-symmetry open-shell molecules.

A numerical procedure for expanding electron repulsion integrals 〈mm|nn〉 on degenerate molecular orbitals of γ symmetry (γ=e, t, g, h) into integral invariants (reduced matrix elements) H^k(γ, γ) is suggested. The latter are analogous in their sense to Slater-Condon parameters F^k(l, l) for atoms with an electronic configuration l^N. The method is applicable to nonlinear molecules of arbitrary symmetries, including “not readily reducible” groups. G. K. Boreskov Institute of Catalysis, Siberian Branch, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 39, No. 1, pp. 3–17, January–February, 1998. This work was supported by RFFR grants No. 96-03-01167 and 96-03-34035.     

Improved method for calculating vibronic spectra of complex molecules

The previously suggested approximate method for calculating the overlap integrals of vibrational wave functions is considerably improved for the purpose of maximally accurate calculation of excitation-induced mixing of normal coordinates. A general formula is obtained for all types of overlap integrals as a finite power series of the potential surface shift parameters; the coefficients are derivatives of the corresponding generating Junctions represented as polynomials of the shift vector elements of the normal coordinates and the mixing matrix. The spectra of model molecules of decatetraene and tetra- and hexadecaheptaene were calculated using the expressions derived in this work and a semiempirical parametric method for determination of excitation-induced changes in the potential surface of molecules. The calculations confirmed the high efficiency of both the parametric method and the new technique. Translated fromZhumal Strukturnoi Khimii, Vol. 38, No. 2, pp. 240–247, March–April, 1997.     

On numerical solution of the integral Hartree-Fock equations for molecules by the method of MO LCAO in the momentum space

To develop a numerical solution of mentioned equations the method of factorized projection of integral operator kernel is applied. All matrix elements of the method are calculated analytically, being expressed in terms of two types of standard integrals: the overlap integrals and one-electron Coulomb integrals. To calculate the integrals we used the O(4)-symmetry of hydrogen-like atomic orbitals as well as operational technique of differentiation with respect to scalar and vector parameters.     

Semiempirical parametric method in the theory of vibronic spectra of polyatomic molecules

A semiempirical parametric method for calculating the vibrational structure of the electronic spectra of polyatomic molecules is developed; the method is based on the adiabatic molecular model and uses a single parametric system for all excited states. Within the model approach, simplified analytical expressions for potential surface variation during molecular excitation are derived; the expressions include the principal terms according to the order of magnitude. The first and second derivatives of Coulomb and resonance one-electron integrals with respect to natural coordinates in a basis set of hybrid atomic orbitals are used as parameters. It is shown that the parameters possess distinct locality, are transferable in molecular series, and may be easily ranked according to absolute values; describing a molecular model requires few most significant parameters. Excitation-induced variations of potential surfaces and absorption spectra of some molecules (butadiene, hexatriene, octatetraene) are calculated using only two parameters, which are the same for all molecules. The results of calculations are in good agreement with the experimental data. Supported by RFFR grant No. 95-03-08808. V.I. Vernadskii Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences. Translated fromZhumal Struckturnoi Khimii, Vol. 37, No. 3, pp. 419–431, May–June, 1996.     

Double exchange in polynuclear mixed-valence clusters

A method for calculating the energies of the tunnel states of mixed-valence, clusters is proposed. It is based on the secondary quantization technique and allows one to obtain analytical expressions for all matrix elements that appear in double exchange theory as well as to determine the microstructures of the parameters of this theory. The energy spectra of trimeric systems are calculated with allowance made for the three-center transfer integrals, and the magnetic properties of these systems are investigated. Moldova State University. Translated fromZhurnal Struktumoi Khimii, Vol. 36, No. 4, pp. 644–653, July–August, 1995. Translated by I. Izvekova     

Evaluation of two-center,two- and three-electron integrals involving correlation factors over Slater-type orbitals. II. Kinetic and potential energy integrals and examples of numerical results

This article is concerned with the construction of the general algorithm for evaluating two-center, two- and three-electron integrals occurring in matrix elements of one-electron operators in the basis of variational correlated functions. This problem has been solved here in prolate spherical coordinates, using the modified and extended form of the Neumann expansion of the interelectronic distance function r^k_ij derived in Part I of this series for k = ?1, 0, 1, 2. This work expands the method proposed by one of us in the preceding paper for integrals of the types mentioned above. The results of numerical calculations for different types of the two- and three-electron integrals are presented. The problem of convergence of the proposed procedures used is also discussed.     

Calculation of the one-electron two-center integrals with STOS using recurrence-based algorithms

We present a new algorithm for the calculation of two-center one-electron integrals with STOS based on two sets of recurrence relations. The first one enables us to calculate any of these integrals in terms of a few “basic integrals.” Furthermore, these basic integrals are written in terms of certain auxiliary functions which can be obtained through the second set of relations from only two starting values. We give also simple formulas for these starting quantities. Finally, the numerical stability, accuracy, and speed of the different steps of the algorithm are analyzed.     

Ab initio quantum chemical calculations of enolized and deprotonated forms of Β-diketones and their derivatives

The data of ab initio quantum chemical calculations of Β-diketones and their anions performed in split valence-shell bases are presented. These compounds are of great interest as ligands in numerous chelate complexes. The electronic structures of substituted enolized Β-diketones containing the period III elements S and Cl are compared. It is shown that using the Huzinaga pseudopotential for describing core electronic shells makes it possible to adequately reproduce the main peculiarities of the structure of the outer electronic layers at a qualitative level. The basis used substantial affects on the calculated values of atomic charges, bond orders, and dipole moments, whereas the one-electron energies are less sensitive to the choice of the basis. Translated fromZhurnal Struktumoi Khimii, Vol.40, No. 2, pp. 234–241, March–April, 1999.     

Inclusion of correlations in He, Be, and Ne by the configuration interaction method

The influence of various approximations used to include correlations in He, Be, and Ne atoms on correlation energies and on the structure of correlation functions is analyzed. The effects of three-and four-electron correlations are evaluated. For He, Be, and Ne atoms, the correlation energies were found to equal 99.99, 100.11, and 97.9% of the experimental value, respectively. A method of obtaining a one-electron function basis is proposed to ensure the best convergence of correlation energies as the number of included configurations increases. Translated fromZhurnal Struktumoi Khimii, Vol. 39, No. 6, pp. 1001–1012, November–December, 1998.     

Evaluation of one-electron integrals for arbitrary operators V(r) over Cartesian Gaussians: Application to inverse-square distance and Yukawa operators

A general procedure is presented for generating one-electron integrals over any arbitrary potential operator that is a function of radial distance only. The procedure outlines that for a nucleus centered at point C integrals over Cartesian Gaussians can be written as linear combinations of 1-D integrals. These Cartesian Gaussian functions are expressed in a compact form involving easily computed auxiliary functions. It is well known that integrals over the Coulomb operator can be expressed in terms of F_n(T) integrals, where By means of a substitution for F_n(T) by other simple functions, algorithms that form integrals over an arbitrary function can be generated. Formation of such integrals is accomplished with minor editing of existing code based on the McMurchie–Davidson formalism. Further, the method is applied using the inverse-square distance and Yukawa potential operators V(r) over Cartesian Gaussian functions. Thus, the proposed methodology covers a large class of one-electron integrals necessary for theoretical studies of molecular systems by ab initio calculations. Finally, by virtue of the procedure's recursive nature it provides us with an efficient scheme of computing the proposed class of one-electron integrals. © 1993 John Wiley & Sons, Inc.     

Fragment calculation of electronic structures of polyatomic molecules in the ground state. II. A method of delocalized states of fragments

A fragment method is proposed to calculate the electronic structures of polyatomic molecules in the ground state. Localization and delocalization of the electronic states of molecular fragments are calculated simultaneously. The compact formulation of the method allows algorithmically efficient calculations of the electronic structures of interacting molecular fragments as well as of the whole molecules. V. I. Vernadskii Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences. Translated fromZhurnal Struktumoi Khimii, Vol. 36, No. 3, pp. 395–400, May–June, 1995. Translated by I. Izvekova     

Nonlinear transformation methods for accelerating the convergence of Coulomb integrals over exponential type functions

It is well known that in any ab initio molecular orbital (MO) calculation, the major task involves the computation of molecular integrals, among which the computation of Coulomb integrals are the most frequently encountered. As the molecular system gets larger, computation of these integrals becomes one of the most laborious and time consuming steps in molecular systems calculation. Improvement of the computational methods of molecular integrals would be indispensable to a further development in computational studies of large molecular systems. The atomic orbital basis functions chosen in the present work are Slater type functions. These functions can be expressed as finite linear combinations of B functions which are suitable to apply the Fourier transform method. The difficulties of the numerical evaluation of the analytic expressions of the integrals of interest arise mainly from the presence of highly oscillatory semi-infinite integrals. In this work, we present a generalized algorithm based on the nonlinear transformation of Sidi, for a precise and fast numerical evaluation of molecular integrals over Slater type functions and over B functions. Numerical results obtained for the three-center two-electron Coulomb and hybrid integrals over B functions and over Slater type functions. Comparisons with numerical results obtained using alternatives approaches and an existing code are listed.     

Unified analytical treatment of one-electron two-center integrals with noninteger n Slater-type orbitals

A unified treatment of one-electron two-center integrals over noninteger n Slater-type orbitals is described. Using an appropriate prolate spheroidal coordinate system with the two atomic centers as foci, all the molecular integrals are expressed by a single analytical formula which can be readily and compactly programmed. The analysis of the numerical performance of the computational algorithm is also presented. Received: 1 April 1999 / Accepted: 2 July 1999 / Published online: 2 November 1999     

Symmetry-adapted integrals over many-electron basis functions and operators

We consider integrals over symmetry-adapted basis functions that involve the coordinates of more than one electron. We focus on basis functions that can be written as products of one-electron functions and (say) a two-electron function. We show first that the two-electron parts of the basis functions can be absorbed into the operator, resulting in an integral over only one-electron basis functions, but a more complicated many-electron operator. We then prove a general formula for expressing such integrals in terms of symmetry-distinct integrals only. Received: 16 June 2000 / Accepted: 10 July 2000 / Published online: 19 January 2001     

Dispersion interaction of conjugated molecules: One- and many-electron models

A scheme for calculating van der Waals constants C₆, C₈, and C₁₀ in terms of the full configuration interaction for π-shells is suggested. The required imaginary frequency dynamic polarizabilities are also investigated with one-electron schemes. The variational perturbation theory of the Hartree-Fock method considerably exaggerates the constant C_2l the conjugated variant of perturbation theory is preferable. The additive scheme for estimating the Σ-contribution allows one to calculate the energy of dispersion attraction in real conjugated systems, as demonstrated by semiempirical estimates of evaporation heat in butadiene, benzene, naphthalene, and phenylacetylene. Translated fromZhurnal Strukturnoi Khimii, Vol. 38, No. 6, pp. 1029–1037, November–December, 1997. Original article submitted December 14, 1996.