Two-center Overlap Integrals Research Articles

COMPUTATION OF MULTICENTER NUCLEAR-ATTRACTION INTEGRALS OF INTEGER AND NONINTEGER n SLATER ORBITALS USING AUXILIARY FUNCTIONS

A unified treatment of multicenter nuclear-attraction integrals with integer n and noninteger n* Slater-type orbitals (ISTOs and NISTOs) is described. Using different sets of series expansion formulas for NISTOs and their two-center distributions in terms of ISTOs at a displaced center obtained by one of the authors, the three-center nuclear-attraction integrals over NISTOs are expressed through the products of overlap and two-center nuclear-attraction integrals. The two-center overlap and nuclear-attraction integrals are calculated by the use of well-known auxiliary functions Aσ and Bk. Accuracy of the computer results is quite high for quantum numbers, screening constants, and location of orbitals.

On the Evaluation of Two-Center Overlap Integrals over Integer and Noninteger n-Slater-Type Orbitals

In this study, two-center overlap integrals over Slater-type orbitals (STOs) with integer and noninteger principal quantum numbers in unaligned coordinate systems have been calculated using formulas for overlap integrals in aligned coordinate systems obtained by the author's previous work (T. Özdoǧan and M. Orbay, Int. J. Quant. Chem. 87 (2002) 15). The obtained results for integer case have been found to be in excellent agreement with the prior literature. On the other hand, to the best of authors knowledge, because of the scarcity of the literatures on the use of noninteger n-STOs in unaligned coordinate systems, the results for noninteger case have been tested with the limit of integer case, and good agreement has been obtained too.

On the calculation of arbitrary multielectron molecular integrals over Slater-type orbitals using recurrence relations for overlap integrals. IV. Use of recurrence relations for basic two-center overlap and hybrid integrals

On the calculation of arbitrary multielectron molecular integrals over Slater-type orbitals using recurrence relations for overlap integrals. IV. Use of recurrence relations for basic two-center overlap and hybrid integrals

Computation of molecular integrals over Slater-type orbitals. VIII. Calculation of overlap integrals with different screening parameters using series expansion formulas for Slater-type orbitals

Using expansion formulas for the Slater-type orbitals (STOs) obtained by the one of authors [Phys. Rev. A 31 (1985) 2851] the two-center overlap integrals with different screening parameters are expressed through the overlap integrals with the same screening parameter. The series representation of overlap integrals obtained in this paper is useful for small differences of the exponential parameters. For large differences of the exponential parameters the convergence of the infinite series becomes rather slow.

Two-center nonexchange integrals over Slater orbitals

We have used computer algebra (CA) to construct general formulas for the two-center overlap, resonance, Coulomb and hybrid integrals over Slater orbitals (STOs). Individual formulas then were produced for the Coulomb integrals containing all combinations of K, L and M shell orbitals. The numerical evaluation of these formulas is very rapid and allows unrestricted accuracy. Tests give complete agreement with the output of numerical programs written by other workers. Computer algebra eliminates terms of equal magnitude and opposite sign that can cause catastrophic loss of accuracy in a purely numerical computation. Strong parallelization is possible in the construction and evaluation of the formulas. Their availability opens up the prospect of carrying symbolic computation into the eigenvalue finding stage of quantum chemistry. The present calculations also highlight several needs and opportunities in the field of computer algebra.

Computation of molecular integrals over Slater type orbitals I. Calculations of overlap integrals using recurrence relations

A new algorithm is presented for the calculation of two-center overlap integrals over Slater type orbitals, based on sets of recurrence relations and analytical formulas. The recurrence relations enable us to calculate some of overlap integrals by the use of which other overlap integrals are calculated analytically. The recurrence relations obtained in this work for these `basic overlap integrals' are especially useful for the calculation of any overlap integral for large quantum numbers. An accuracy of the computer results is satisfactory for the values of principal quantum numbers of Slater functions up to 50, and for the arbitrary values of screening constants of atomic orbitals and internuclear distances.

Comprehensive strategy for the calculation of overlap integrals with Slater‐type orbitals

A strategy is presented for the calculation of two-center overlap integrals over Slater-type orbitals. Displaced orbitals are expanded in spherical harmonics with Löwdin α-functions as coefficients. The exponentials in the α-functions are expanded, leading to representation in terms of stored E and F matrices. For a given precision, the number of terms needed for each orbital for a specified harmonic, and its displacement multiplied by its screening constant, is predetermined and stored. A survey of these data is presented. The one-dimensional integration needed for the overlap is done by Gauss-Legendre numerical integration over the interior region and analytically over the exterior. Complete stability is achieved and excellent results obtained. Implications for all multicenter molecular integrals are apparent. © 1997 John Wiley & Sons, Inc.

Spherically symmetrical properties of expansion coefficients for translation of spherical harmonics

A theorem regarding the angular dependence has been established for the expansion coefficients for translation of spherical harmonics (SH): If we add the products of all the translation coefficients with the same ℓ values, but different m's, the result is independent of orientation. The spherically symmetrical properties of the translation coefficients K ℓ m,ℓ' m' for SH obtained in the present paper and, the rotation coefficients T λ ℓ m,ℓ' m' the two-center overlap integrals over arbitrary atomic Orbitals S nℓ m, n'ℓ' m' and the translation coefficients V N nℓ m, n'ℓ' m' for Slater-type orbitals (STO's) given recently by the author [I.I. Guseinov, J. Mol. Struct. (Theochem), 343 (1995) 173] are the same. The analytical formulas also have been derived for translation coefficients of SH in terms of binomial coefficients. The final results are especially useful for machine computations of arbitrary multi-electron molecular integrals for which the series expansion formulas have recently been derived by the author.

Spherically symmetrical properties of two-center overlap integrals over arbitrary atomic orbitals and translation coefficients for Slater-type orbitals

The orthogonality relations are derived for the rotation coefficients of two-center overlap integrals over arbitrary atomic orbitals (AAOs) and expansion coefficients for translation of Slater-type orbitals (STOs). Using these formulas, a very interesting theorem regarding the angular dependence is established. If we add the products of all the overlap integrals or all the translation coefficients with the same n and l values, but different m values, the result is independent of orientation. The final results are of a simple structure and are, therefore, especially useful for machine computations of multielectron multicenter molecular integrals by expanding one- and two-center electron charge density over STOs in terms of STOs about a new center.

Spherically symmetrical properties of two-center overlap integrals over arbitrary atomic orbitals and translation coefficients for Slater-type orbitals

The orthogonality relations are derived for the rotation coefficients of two-center overlap integrals over arbitrary atomic orbitals (AAOs) and expansion coefficients for translation of Slater-type orbitals (STOs). Using these formulas, a very interesting theorem regarding the angular dependence is established. If we add the products of all the overlap integrals or all the translation coefficients with the same n and l values, but different m values, the result is independent of orientation. The final results are of a simple structure and are, therefore, especially useful for machine computations of multielectron multicenter molecular integrals by expanding one- and two-center electron charge density over STOs in terms of STOs about a new center.

Two-center overlap integrals over Slater-type orbitals constrained to a spherical integration volume: Analytical expressions.

A special type of two-center overlap integral is considered in which the integration does not extend over the whole space, but is constrained to the interior or exterior of a sphere about one of the centers of the orbitals. It is shown that exact, closed analytical expressions can be derived for this kind of two-center integral over Slater-type orbitals. These analytical expressions are of a very simple analytical structure and allow for a rapid integral evaluation. By applying the angular momentum recurrences which we proposed recently [W. Hierse and P. M. Oppeneer, J. Chem. Phys. 99, 1278 (1993)], the evaluation performance can be further accelerated. In the frequency occurring case of nearly equal scaling parameters the analytical integral expressions are totally stable, but in the (pathological) case of one vanishing scaling parameter the integral over the interior volume is not stable. Exact integration formulas for the latter limiting case are also given.

Calculation of two-center integrals between Slater-type orbitals

Two-center overlap, nuclear-attraction and kinetic energy integrals between Slater-type orbitals (STOs) centered on two different points are calculated. The expansion of STO in terms of spherical harmonics centered on an atom displaced from the first one is used. Coefficients of the expansion are stored in an adjustable array (its size depends on quantum numbers of the STO involved) in the first step of the calculation. In the second step a simple and closed formula is used for obtaining the required type of integral between STOs (with quantum numbers in the range for which calculations in the first step have been performed).

Unified kernel function approach to two-center integrations in quantum-chemical calculations

We present a unified kernel function treatment of the two-center integrals over atomic orbitals appearing in quantum chemical calculations on the basis of density functional theory. By using the Fourier transform method, we obtain analytical expressions for the kernel functions, from which actual integral values can be evaluated through twofold integration. As our only assumption is that the angular dependences of the atomic basis functions is given by spherical harmonics, our kernel functions are applicable to almost arbitrary types of analytically given or numerically tabulated atomic radial functions. In addition, we give some interesting new recurrence relations holding between parts of kernel functions, which enable an entirely recursive treatment of the angular momentum arguments. When applied to analytically given atomic basis functions, which contain a factor rν+l, a qualitatively new and general set of recurrence relations for auxiliary functions of two-center integrals and, in some cases, of the two-center integrals themselves, results. As an example, two-center overlap integrals over Slater-type orbitals are calculated recursively for higher angular momentum quantum numbers and compared with the results of a direct calculation. We expect the present recurrence relations to be superior to other existing recurrences with respect to angular momentum quantum numbers.

Programs for the evaluation of overlap integrals with B functions

Programs for the evaluation of overlap integrals and some related integrals with certain exponential-type orbitals (ETO's), the B functions [E. Filter and E.O. Steinborn, Phys. Rev. A 18 (1978) 1] are presented. B functions possess a very simple Fourier transform which results in relatively compact general formulas for molecular integrals derivable using the Fourier transform method. In addition, molecular integrals for the more common Slater-type orbitals (STO's) and other ETO's can be written as finite linear combinations of integrals with B functions because these functions span the space of ETO's. The programs are based on a number of different formulas and representations for the overlap integrals. Simple finite expressions are used in the one-center case, and in the two-center case with equal exponential parameters. In the two-center case, the so-called Jacobi polynomial representation can only be used for largely different exponential parameters. Otherwise, a one-dimensional integral representation [H.P. Trivedi and E.O. Steinborn, Phys. Rev. A 27 (1983) 670] for the two-center overlap integral of B functions with different exponential parameters is used. The numerical properties of this integral representation, and special quadrature methods to deal with them, have been discussed recently [H.H.H. Homeier and E.O. Steinborn, Int. J. Quantum Chem. 42 (1992) 761]. These results show that these Möbius-transformation based quadrature rules, which are well suited for the numerical integration of functions possessing a sharp peak at or near one boundary of integration [H.H.H. Homeier and E.O. Steinborn, J. Comput. Phys. 87 (1990) 61], are superior to several other methods and can be used to calculate the overlap integrals reliably and economically.

Recent progress on representations for Coulomb integrals of exponential-type orbitals

We present some recent work on Coulomb integrals with certain exponential-type orbitals (ETOs), the B functions (E. Filter and E.O. Steinborn, Phys. Rev. A, 18 (1978) 1). The main advantage of B functions is that they possess a very simple Fourier transform (E.J. Weniger and E.O. Steinborn, J. Chem Phys., 78 (1983) 6121), which results in relatively compact general formulas for molecular integrals derivable using the Fourier transform method. Because, in addition, these functions can be taken as a basis of the space of ETOs, molecular integrals for the more common Slater-type orbitals and other ETOs can be written as finite linear combinations of integrals with B functions. Various representations for Coulomb integrals of B functions with equal exponential parameters and three integral representations of Coulomb integrals of B functions with different exponential parameters are given, which could be derived recently. The integral representations are one-dimensional and contain more simple molecular integrals as part of the integrand. Upon insertion of various analytic representations of the latter integrals by finite sums a large number of different integral representations for the Coulomb integrals emerge. The structure of one of the integral representations is completely analogous to a one-dimensional integral representation (H.P. Trivedi and E.O. Steinborn, Phys. Rev. A, 27 (1983) 670) for the two-center overlap integral of B functions with different exponential parameters. Further, a Jacobi polynomial representation for the Coulomb integrals of B functions with different exponential parameters is discussed. Besides finite sums of functions regular at the origin this representation contains irregular solid harmonics and distributional contributions which are displayed explicitly.

Generation of a two-center overlap integral over Slater orbitals of higher principal quantum numbers.

The expressions for two-center overlap integrals between angular s, p, and d Slater orbitals of arbitrary, higher principal quantum number are explicitly listed. The expressions obtained are extremely compact and independent of the coordinate system. It is further shown that the numerical values of the integrals obtained in this way are free from any numerical instability.

New formula for integrals in semiempirical molecular orbital methods: Part 1. General theory and results for two-center overlap integrals and their first and second derivatives with respect to Cartesian nuclear displacements

A very simple method for deriving formulas for integrals over Slater—Zener type atomic orbitals is given. Essentially, the method obtains integrals over orbitals having higher n and l quantum numbers from the integral over 1 s orbitals by repeated application of operators involving the orbital exponents and the positions of the orbital centers. The method is made practicable by using a symbolic algebra expert system to derive the formulas. Formulas for overlap integrals (and their derivatives) over 1 s, 2 s, and 2 p atomic orbitals are given directly in Cartesian coordinates, thus avoiding the transformation from a local to a molecular coordinate system.

Evaluation of two-center overlap and nuclear-attraction integrals for Slater-type orbitals.

General formulas are derived for the two-center overlap and nuclear-attraction integrals with the arbitrary location and screening constants of Slater-type orbitals. The final results are expressed in terms of the spherical harmonics and the usual ${A}_{k}$ and ${B}_{k}$ functions. Useful formulas have also been established for the radial parts of overlap and nuclear attraction integrals in the form of polynomials in p=\ensuremath{\zeta}R.

Fourier transform of a two-center product of exponential-type orbitals. Application to one- and two-electron multicenter integrals

A new formula for the Fourier transform (FT) of a two-center RBF (reduced Bessel function) charge distribution permitting partial-wave analysis is derived with the use of Feynman's identity. This formula is valid for all quantum numbers. It is also independent of the orientation of the coordinate axes. A new representation for the two-center overlap integral (to which the kinetic energy, two-center attraction and Coulomb repulsion integrals can be readily reduced) and for the three-center attraction integral is obtained with the help of the FT. It is stable for all values of the orbital exponents. The method developed by Graovac et al. for computing repulsion integrals for $s$ states is generalized to include all states. Numerical test values of several one- and two-electron integrals are also reported. A strategy which should enhance the efficiency of computation by making maximal use of the FT's already computed is suggested.

Electronic energy levels of cinnabar (α-HgS)

We have calculated the electronic energy levels of trigonal mercury sulfide in the semiempirical tight-binding scheme. The atomic levels $3s$, $3p$ of sulfur and $6s$, $6p$ of mercury constitute the starting basis set. The computed two-center overlap integrals are subsequently scaled by a common multiplicative factor to reproduce the experimental optical gap. The effect of the atomic virtual level $4s$ of sulfur on the conduction bands is also taken into account. The calculated valence-band structure agrees well with the ultraviolet-photoelectron-spectroscopy (UPS) data. The complete band structure is consistent with the absorption measurements which indicate an indirect absorption edge immediately followed by the direct one (at the zone center). This picture can also explain the dependence of the absorption coefficient on the polarization of the exciting light in terms of the symmetry of the levels and the selection rules.