Slater Functions Research Articles

Calculation of 3-Center One-Electron Integrals \(\left\langle {AB|r_c^{ - 1} } \right\rangle \) and Coulomb Repulsion Matrix Elements \(\left\langle {AB|CC'} \right\rangle \) in an Exponential Type Spherical AO Basis

A new method for calculating 3-center one-electron integrals and matrix elements of electron interaction of \(\left\langle {AB|CC'} \right\rangle \) type in MO LCAO theory, where A, B, and C are the centers of exponential type AO including Slater and hydrogen-like basis functions, has been developed. Integrals of this type are reduced to double integrals over the square with edge 1. This provides the grounds for effective calculation of quantum-chemical 3-center integrals in a basis of exponential type spherical functions.

Per‐Olov Löwdin curriculum vitae

Per‐Olov Löwdin curriculum vitae

Calculation of nuclear‐attraction and modified overlap integrals using Gegenbauer coefficients*

AbstractThe modified overlap and nuclear‐attraction integrals over Slater‐type orbitals are calculated for extremely large quantum numbers using Gegenbauer coefficients and basic overlap integrals. The accuracy of the results is quite high for the principal quantum numbers of Slater functions and for arbitrary values of internuclear distances and screening constants of atomic orbitals. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2002

Basis Functions with Polynomial Tails for Band Structure Calculations

The electronic structure of a copper crystal was calculated in a basis of standard atomic and Slater functions (BSAF) and in a basis of functions with polynomial tails (BFPT). The accuracy of BFPT calculations even in their simplest version is comparable to that of BSAF calculations. The calculation demands less computing time because the new basis identically turns zero all matrix elements between the functions of the atoms that are wider apart than a certain specified distance.

Four‐center integrals for Gaussian and exponential functions

The shift operator technique is used for deriving, in a unified manner, the master formulas for the four-center repulsion integrals involving Gaussian (GTO), Slater (STO), and Bessel (BTO) basis functions. Moreover, for the two classes of exponential-type functions (ETO), i.e., STO and BTO, we give the expressions corresponding to both the Gauss and Fourier transforms. From the comparison of the master formulas of GTO and ETO, we conclude that ETO can perform more efficiently than GTO, and we remark the points where the effort must be focused to carry out this possibility. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 16–28, 2001

Computation of molecular integrals over Slater-type orbitals. III. Calculation of multicenter nuclear-attraction integrals using recurrence relations for overlap integrals

Using recurrence relations for basic overlap integrals two- and three-center nuclear-attraction integrals are calculated for extremely large quantum numbers. The accuracy of the results is quite high for the principal quantum numbers of Slater functions and for the arbitrary values of internuclear distances and screening constants of atomic orbitals.

Master formulas for two- and three-center one-electron integrals involving Cartesian GTO, STO, and BTO

As a first application of the shift operators method we derive master formulas for the two- and three-center one-electron integrals involving Gaussians, Slater, and Bessel basis functions. All these formulas have a common structure consisting in linear combinations of polynomials of differences of nuclear coordinates. Whereas the polynomials are independent of the type (GTO, BTO, or STO) of basis functions, the coefficients depend on both the class of integral (overlap, kinetic energy, nuclear attraction) and the type of basis functions. We present the general expression of polynomials and coefficients as well as the recurrence relations for both the polynomials and the whole integrals. Finally, we remark on the formal and computational advantages of this approach. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 83–93, 2000

Coupled-channel calculations of ionization and excitation cross sections for ion-helium collisions

Using the semiclassical approximation we calculate cross sections for excitation, single and double ionization of helium in collisions with , and in the energy range between and . The method is the solution of time-dependent coupled-channel equations for the amplitudes of two-electron basis states. The basis states are obtained by diagonalizing the Hamiltonian of the helium atom in a finite basis of antisymmetrized products of hydrogen-like Slater functions. The projectile-electron interaction takes the retardation effects into account. The results are compared with experimental data and other theoretical approaches.

Multiple one-center expansions of charge distributions associated with Slater orbitals

Starting from an integral representation of the charge distribution associated with pairs of Slater functions (STOs), exact and approximate multiple one-center expansions are derived. A numerical quadrature based on the conservation of the long-range potential is presented and discussed. Moreover, its performance for integral evaluation is tested. From these preliminary results, we conclude that highly accurate calculations of molecular integrals with STOs, involving a computation cost similar to that of Gaussians, are an actual possibility.

Basisless variational method of solving Hartree-Fock equations

V. A. Popov and A. V. Vekman UDC 539.186 Hartree-Fock equations (HFE) for atoms and ions can be solved by numerical iteration methods in basis sets of Slater and Ganssian functions (see, e.g., [1-3]). The ideas employed in these works are also applicable to HFE for molecules [4]. However, for molecules with complex electronic structures, we often encounter the problem of Hartree-Fock instability [5]. Divergent branches of solutions in eigenvalue problems for electrons in complex crystals are especially troublesome. It is therefore important to seek new methods of solving HFE. We got interested in the method of solving the Shrdinger equation presented in [6], because it possesses cubic convergence, is a basisless method, and allows generalization for systems of nonlinear equations such as HFE:

Slater Functions for Y to Cd Atoms by the Distance between Subspaces

Slater functions for the atoms Y-Cd have been formulated by the distance between subspaces method. Basis sets proposed here are single- and double-zeta size and have been constructed using numerical Hartree-Fock functions as reference. A comparative study with Clementi and Roetti basis sets of the same size has been carried out, obtaining a uniform criterion for the behavior of the series of atoms Y-Cd when the number of d electrons is varied. The new basis sets provide a better simulation of some atomic properties and appear to be appropriate for molecular and solid state calculations.

Basis set superposition errors for Slater vs. gaussian basis functions in H-bond interactions

The quality of the SCF description of the interactions in a small set of H-bonded adducts (OH 2 … F −, HOH … F −, the water dimer, HF … HF) produced by Slater basis functions is examined, paying particular attention to the basis set superposition error (BSSE), affecting equilibrium distances and interaction energies, and to the effect of counterpoise corrections (CP) on those values. Interaction energies and equilibrium distances are then compared to the results previously obtained using a variety of common gaussian basis sets (STO-3G, MINI-1, 6-31G * *), besides a basis set especially designed to be practically free from BSSE. The Slater basis sets employed are both minimal (best atom zeta (BAZ)) and extended (double zetas with polarization functions added (DZP)). The performance of the BAZ basis set for the neutral dimers is disappointing, since it is analogous to that exhibited by the STO-3G basis set, both at the SCF and CP corrected levels. For the anionic adducts, the BAZ behavior is comparable to that obtained with the use of the MINI-1 basis set, especially at the CP corrected level. On the contrary, the DZP description of the neutral dimers is better than that produced by the extended gaussian basis sets with polarization functions (6-31G * *) at the SCF level and, moreover, when CP corrected, it is able to hold the comparison with the basis set especially designed to be practically free from BSSE, even though each heavy atom is described with ten basis functions less in the outer shells and only one more in the inner shell. For the anionic systems the DZP basis set is slightly worse than for the neutral dimers, in particular for the linear adduct.

Molecular integrals with Slater basis. V. Recurrence algorithm for the exchange integrals

The numerical and analytical procedures, used for calculating exchange integrals with Slater functions, fail to give high accuracy for large quantum numbers and some values of the parameters. We propose here an algorithmic approach based on recurrence relations which start from some simple functions. An extensive study in which exponents, interatomic distances, and quantum numbers were varied proves that the new procedure is fully reliable. Finally, the cost of the procedure is analyzed, including the effect of the number of terms needed to attain a given accuracy.

The equivalence of explicitly correlated Slater and Gaussian functions in variational quantum chemistry computations: The ground state of H 2

It is demonstrated that variational calculations based on explicitly correlated Gaussian functions for the hydrogen molecule in its ground state lead to energies of the same level of accuracy as those based on the Kolos-Wolniewicz functions. The energies obtained are the lowest reported so far, in contrast to the well-known bad asymptotic properties of Gaussian functions.

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.

Multicenter one-electron integrals with slater functions of the operatorr l+2kYl, m(r)

Multicenter one-electron integrals with slater functions of the operatorr l+2kYl, m(r)

Regularization method and molecular integrals with singular exponential functions

The regularization principle, which is based on the concept of linearly independent singular functions, makes it possible to calculate many important types of molecular matrix elements arising in the variational LCAO-MO-SCF scheme. This is done using a direct approach that employs reduction of these elements to finite sums of convergent and divergent one-electron integrals. A universal algorithm is developed to calculate two-center one-electron molecular integrals involving both singular and ordinary Slater functions. The numerical stability of the algorithms and the accuracy of the integral calculation are analyzed, and numerical estimates are given.

A simplified E-separation method: multicenter integrals and test calculations of small molecules

A new approximate method of evaluating multicenter molecular integrals with Slater functions has been developed with the help of a simplified version of the Σ-separation approach. With respect to calculational effort and computation time the proposed method is approximately equivalent to the Mulliken approximation and simple variants of the Ruedenberg approximation. However, the method ensures greater accuracy compared to these earlier approaches. The advantage of the Σ-approximation is especially important in the evaluation of integrals with d and f functions. A new simplified nonempirical variant of the Hartree-Fock-Roothaan method has been developed within the general framework of the Σ-separation approach. The balance between certain types of one- and two-electron integrals is shown to exert a substantial influence on the overall accuracy of the method. The practical value of the approach is illustrated by calculations of total energies, dipole moments and force constants.

Study of relativistic effects in atoms and molecules by the kinetically balanced LCAO approach: Ground state of hydrogen and of hydrogenic atoms in Slater and Gaussian basis functions

Study of relativistic effects in atoms and molecules by the kinetically balanced LCAO approach: Ground state of hydrogen and of hydrogenic atoms in Slater and Gaussian basis functions

Long‐Range multicenter integrals with slater functions: Gauss transform‐based methods

AbstractThe separation of the short‐ and long‐range terms in the potentials generated by pairs of Slater functions is reformulated in the context of the Gauss transform method. Analytic expressions of the long‐range potentials (in closed form) are derived for equal exponents and generalized (as expansion series) for different exponents. Additionally, the representation of these potentials from small sets of charges or lowest‐order multipoles is examined, paying special attention to their values and optimal positions. Finally, numerical tests of the formal developments are presented. It is concluded that the long‐range three‐ and four‐center integrals can be calculated with high accuracy in a simple and relatively inexpensive manner. © John Wiley & Sons, Inc.