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.     

Overlap integrals of B functions

Two different methods for the evaluation of overlap integrals of B functions with different scaling parameters are analyzed critically. The first method consists of an infinite series expansion in terms of overlap integrals with equal scaling parameters [14]. The second method consists of an integral representation for the overlap integral which has to be evaluated numerically. Bhattacharya and Dhabal [13] recommend the use of Gauss-Legendre quadrature for this purpose. However, we show that Gauss-Jacobi quadrature gives better results, in particular for larger quantum number. We also show that the convergence of the infinite series can be improved if suitable convergence accelerators are applied. Since an internal error analysis can be done quite easily in the case of an infinite series even if it is accelerated, whereas it is very costly in the case of Gauss quadratures, the infinite series is probably more efficient than the integral representation. Overlap integrals of all commonly occurring exponentially declining basis functions such as Slater-type functions, can be expressed by finite sums of overlap integrals of B functions, because these basis functions can be represented by linear combinations of B functions.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

Gaussian and finite-element Coulomb method for the fast evaluation of Coulomb integrals

The authors propose a new linear-scaling method for the fast evaluation of Coulomb integrals with Gaussian basis functions called the Gaussian and finite-element Coulomb (GFC) method. In this method, the Coulomb potential is expanded in a basis of mixed Gaussian and finite-element auxiliary functions that express the core and smooth Coulomb potentials, respectively. Coulomb integrals can be evaluated by three-center one-electron overlap integrals among two Gaussian basis functions and one mixed auxiliary function. Thus, the computational cost and scaling for large molecules are drastically reduced. Several applications to molecular systems show that the GFC method is more efficient than the analytical integration approach that requires four-center two-electron repulsion integrals. The GFC method realizes a near linear scaling for both one-dimensional alanine alpha-helix chains and three-dimensional diamond pieces.     

Big data analysis of ab Initio molecular integrals in the neglect of diatomic differential overlap approximation

Most modern semiempirical quantum-chemical (SQC) methods are based on the neglect of diatomic differential overlap (NDDO) approximation to ab initio molecular integrals. Here, we check the validity of this approximation by computing all relevant integrals for 32 typical organic molecules using Gaussian-type orbitals and various basis sets (from valence-only minimal to all-electron triple-ζ basis sets) covering in total more than 15.6 million one-electron (1-e) and 10.3 billion two-electron (2-e) integrals. The integrals are calculated in the nonorthogonal atomic basis and then transformed by symmetric orthogonalization to the Löwdin basis. In the case of the 1-e integrals, we find strong orthogonalization effects that need to be included in SQC models, for example, by strategies such as those adopted in the available OMx methods. For the valence-only minimal basis, we confirm that the 2-e Coulomb integrals in the Löwdin basis are quantitatively close to their counterparts in the atomic basis and that the 2-e exchange integrals can be safely neglected in line with the NDDO approximation. For larger all-electron basis sets, there are strong multishell orthogonalization effects that lead to more irregular patterns in the transformed 2-e integrals and thus cast doubt on the validity of the NDDO approximation for extended basis sets. Focusing on the valence-only minimal basis, we find that some of the NDDO-neglected integrals are reduced but remain sizable after the transformation to the Löwdin basis; this is true for the two-center 2-e hybrid integrals, the three-center 1-e nuclear attraction integrals, and the corresponding three-center 2-e hybrid integrals. We consider a scheme with a valence-only minimal basis that includes such terms as a possible strategy to go beyond the NDDO integral approximation in attempts to improve SQC methods. © 2018 Wiley Periodicals, Inc.     

One-center expansion for pseudopotential matrix elements

Semilocal pseudopotential operators can be expressed as a linear combination of nonlocal (projection) operators. Pseudopotential operator integrals over a molecular basis set are therefore reduced to linear combinations of overlap integrals products. Molecular calculations indicate that sufficient precision can be achieved with a limited number of nonlocal operators. Analytic derivatives of pseudopotential integrals are easily deduced and implemented in a standard quantum chemistry program.     

Angular overlap model parameterisation of Anderson''s superexchange theory. I. A quantification of Goodenough-Kanamori rules

A parameterisation of Anderson's exchange formulas on the basis of an extension of the angular overlap model (AOM) is proposed. Transfer integrals are expressed in terms of the metal-metal and metal-ligand bonding parameters, which can be estimated from independent spectroscopic studies, or calculated using solid state expressions. Analytical expressions for the transfer integrals between various d-orbitals, appropriate for cubic crystal lattices, comprising octahedra sharing common vertices and common edges serve as a quantification of the Goodenough-Kanamori rules. On the basis of the present parameterisation we give an explanation of the “exchange integral versus bond-distance” dependence. Some potential applications of the model are briefly discussed.     

Molecular symmetry and ab initio calculations: IV. Symmetry-matrix and symmetry-supermatrix in calculations of two-electron repulsion integrals

The product of two Gaussians having different centers is itself a one-center Gaussian, thus multicenter integrals with a Cartesian Gaussian basis can be reduced to one-center integrals. Recurrence relations for overlap integrals and electron repulsion integrals (ERIs) are derived at these centers. The calculations of overlap integrals and ERIs are carried out step by step from the highest symmetry case (one center) to required cases (different centers) by using the translation of Cartesian Gaussians. Full exploitation of symmetry in calculation processes can result in optimal use of these recurrence relations. Compared with the recently published algorithms, based on the recurrence relations derived by Obara and Saika [J. Chem. Phys., 84 , 3963 (1986)], the floating point operations (FLOPs) for ERI calculations (having four different centers) can be reduced by a factor of ca. 2. A significant extra saving in calculations and storage can be obtained if atoms, linear, or planar molecules are discussed. © 1997 John Wiley & Sons, Inc.     

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     

Overlap Integrals Between Irregular Solid Harmonics and STOs Via the Fourier Transform Methods

In this paper, a unified analytical and numerical treatment of overlap integrals between Slater type orbitals (STOs) and irregular Solid Harmonics (ISH) with different screening parameters is presented via the Fourier transform method. Fourier transform of STOs is probably the simplest to express of overlap integrals. Consequently, it is relatively easy to express the Fourier integral representations of the overlap integrals as finite sums and infinite series of STOs, ISHs, Gegenbauer, and Gaunt coefficients. The another mathematical tools except for Fourier transform have used partial-fraction decomposition and Taylor expansions of rational functions. Our approach leads to considerable simplification of the derivation of the previously known analytical representations for the overlap integrals between STOs and ISHs with different screening parameters. These overlap integrals have also been calculated for extremely large quantum numbers using Gegenbauer, Clebsch-Gordan and Binomial coefficients. The accuracy of the numerical results is quite high for the quantum numbers of Slater functions, irregular solid harmonic functions and for arbitrary values of internuclear distances and screening parameters of atomic orbitals.     

Numerical calculation of overlap and kinetic integrals in prolate spheroidal coordinates. II

The efficient algorithm calculating the overlap and the kinetic integrals for the numerical atomic orbitals is presented. On the basis of the prolate spheroidal coordinates, the overlap and the kinetic integral are reduced to the integral over the rectangular domain. The integration over the rectangular domain is performed by the adaptive integration scheme. The developed algorithm is applied to calculate the integrals for the pairs of hydrogen and gallium eigenfunctions. It is demonstrated that high accuracy can be obtained for small number of integrand evaluations what guarantees the efficiency of the presented algorithm. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Calculation of overlap integrals over Slater-type orbitals using translational and rotational transformations for spherical harmonics

Using translation and rotation formulas for spherical harmonics the finite sums through the basic overlap integrals and spherical harmonics are derived for the arbitrary overlap integrals over Slater-type orbitals (STOs). The recurrence relations for the evaluation of basic overlap integrals have been established recently [Guseinov II, Mamedov BA (1999) J Mol Struct (THEOCHEM) 465:1]. By the use of the derived expressions the overlap integrals can be calculated most efficiently and accurately, especially for large quantum numbers of STOs. Received: 2 May 2000 / Accepted: 31 May 2000 / Published online: 11 September 2000     

Energy and energy gradient matrix elements with N-particle explicitly correlated complex Gaussian basis functions with L=1

In this work we consider explicitly correlated complex Gaussian basis functions for expanding the wave function of an N-particle system with the L=1 total orbital angular momentum. We derive analytical expressions for various matrix elements with these basis functions including the overlap, kinetic energy, and potential energy (Coulomb interaction) matrix elements, as well as matrix elements of other quantities. The derivatives of the overlap, kinetic, and potential energy integrals with respect to the Gaussian exponential parameters are also derived and used to calculate the energy gradient. All the derivations are performed using the formalism of the matrix differential calculus that facilitates a way of expressing the integrals in an elegant matrix form, which is convenient for the theoretical analysis and the computer implementation. The new method is tested in calculations of two systems: the lowest P state of the beryllium atom and the bound P state of the positronium molecule (with the negative parity). Both calculations yielded new, lowest-to-date, variational upper bounds, while the number of basis functions used was significantly smaller than in previous studies. It was possible to accomplish this due to the use of the analytic energy gradient in the minimization of the variational energy.     

Calculation of electric multipole moment integrals with the different screening parameters via the Fourier transform method

Three‐center electric multipole moment integrals over Slater‐type orbitals (STOs) can be evaluated by translating the orbitals on one center to the other and reducing the system to an expansion of two‐center integrals. These are then evaluated using Fourier transforms. The resulting expression depends on the overlap integrals that can be evaluated with the greatest ease. They involve expressions for STO with different screening parameters that are known analytically. This work gives the overall expressions analytically in a compact form, based on Gegenbauer polynomials. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Auxiliary functions for molecular integrals with Slater‐type orbitals. II. Gauss transform methods

The Gauss transform of Slater‐type orbitals is used to express several types of molecular integrals involving these functions in terms of simple auxiliary functions. After reviewing this transform and the way it can be combined with the shift operator technique, a master formula for overlap integrals is derived and used to obtain multipolar moments associated to fragments of two‐center distributions and overlaps of derivatives of Slater functions. Moreover, it is proved that integrals involving two‐center distributions and irregular harmonics placed at arbitrary points (which determine the electrostatic potential, field and field gradient, as well as higher order derivatives of the potential) can be expressed in terms of auxiliary functions of the same type as those appearing in the overlap. The recurrence relations and series expansions of these functions are thoroughly studied, and algorithms for their calculation are presented. The usefulness and efficiency of this procedure are tested by developing two independent codes: one for the derivatives of the overlap integrals with respect to the centers of the functions, and another for derivatives of the potential (electrostatic field, field gradient, and so forth) at arbitrary points. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Novel methods for calculating the fine vibronic spectra of polyatomic molecules

Novel approximate methods for calculating the vibrational structure of the electronic spectra of polyatomic molecules—a method for the direct calculation of the overlap integrals of vibrational wave functions for the electronic states involved in a transition and a variational method for the solution of the vibrational problem for the excited states—are discussed. The methods are based on the consideration of the displacement and entanglement of normal coordinates, the quasiorthogonality of the Dushinsky transformation, and the classification of the states by total vibrational quantum numbers. Matrix perturbation theory is employed. It is shown that the accuracy of these methods compares well with the accuracy of the available “exact” techniques (the errors are ∼1 cm⁻¹ for frequency and 10% for relative intensity). At the same time, calculations by the new methods are performed more than two orders of magnitude faster than by the previously known methods. K. A. Timiryazev Agricultural Academy, Moscow. Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 2, pp. 217–230, March–April, 1995. Translated by I. Izvekova     

Theoretical investigation of the magnetic interactions of Ni9 complexes

On the basis of density-functional theory (DFT) calculations, a theoretical analysis of the exchange interactions in Ni9L2(O2CMe)8{(2-py)2CO2}4, was performed, where L is a bridging ligand, OH- (1) or N3- (2). Each magnetic interaction between the Ni spin centers is analyzed for 1 and 2 in terms of exchange integrals (J values), orbital overlap integrals (T values) and natural orbitals. It was found that a J3 interaction, which is a magnetic interaction via the bridging ligand orbitals, mainly controls the whole magnetic properties, and the dominant interaction is a sigma-type orbital interaction between Ni dz2 orbitals. Further investigations on the magnetostructural correlations are performed on the J3 interactions using simplest Ni-L-Ni models. These models reproduced the magnetic interactions qualitatively well not only for the Ni9 complexes but also for other inorganic complexes. Strong correlations have been found between the magnetic orbital overlaps (T values) and the Ni-L-Ni angle. These results revealed that the difference of the magnetic properties between OH- and N3- is caused by the orbital overlap integral (T values) of the sigma-type J3 interaction pathway. The magnetic interactions are also discussed from a Hubbard model by evaluating the transfer integral (t) and on-site Coulomb integrals (U), in relation to the Heisenberg picture.     

Quantum and semiclassical theories for nonadiabatic transitions based on overlap integrals related to fast degrees of freedom

Alternative treatments of quantum and semiclassical theories for nonadiabatic dynamics are presented. These treatments require no derivative couplings and instead are based on overlap integrals between eigenstates corresponding to fast degrees of freedom, such as electronic states. Derived from mathematical transformations of the Schr?dinger equation, the theories describe nonlocal characteristics of nonadiabatic transitions. The idea that overlap integrals can be used for nonadiabatic transitions stems from an article by Johnson and Levine [Chem. Phys. Lett. 13, 168 (1972)]. Furthermore, overlap integrals in path-integral form have been recently made available by Schmidt and Tully [J. Chem. Phys. 127, 094103 (2007)] to analyze nonadiabatic effects in thermal equilibrium systems. The present paper expands this idea to dynamic problems presented in path-integral form that involve nonadiabatic semiclassical propagators. Applications to one-dimensional nonadiabatic transitions have provided excellent results, thereby verifying the procedure. In principle these theories that are presented can be applied to multidimensional systems, although numerical costs could be quite expensive.     

Calculation of molecular electric and magnetic multipole moment integrals of integer and noninteger n Slater orbitals using overlap integrals

Closed formulas are established for the magnetic multipole moment integrals of integer and noninteger n Slater‐type orbitals (ISTOs and NISTOs) in terms of electric multipole moment integrals for which the analytic expressions through the overlap integrals with ISTOs and NISTOs are derived. The overlap integrals are evaluated by the use of auxiliary functions. Using the derived expressions the multipole moment integrals, and therefore the electric and magnetic properties of molecules, can be evaluated most efficiently and accurately. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

配位多面体的群重叠积分

本文讨论了配位多面体的群重叠积分计算问题, 给出了这种群重叠积分的一般形式以及它们满足的一些关系, 用第一类点群将群重叠积分的计算化简, 由此定义了有关的几何参数并研究了它们的性质, 以正六面体为例说明了这些参数的具体计算。     

A study of the weinhold-wang bounds to overlap integrals

The Weinhold-Wang extension of the Braun-Rebane formula for bounds to overlap integrals has been investigated numerically for the first and second excited s states of the hydrogen atom. The effect of the choice of basis sets is demonstrated with particular emphasis on the difference between a complete set and an incomplete set of expansion functions