On bound states in two-body systems

The application of the Σ-separation method to the calculation of multicenter two-electron molecular integrals with Slater-type basis functions is reported. The approach is based on the approximation of a scalar component of the two-center atomic density by a two-center expansion over Slater-type functions. A least-squares fit was used to determine the coefficients of the expansion. The angular multipliers of the atomic density were treated exactly. It is shown that this approach can serve as a sufficiently accurate and fast algorithm for the calculation of multicenter two-electron molecular integrals with Slater-type basis functions. © 1995 John Wiley & Sons, Inc.     

Configuration interaction study of H2 and H3 systems

The configuration interaction method has been applied to the H₂ and H₃ systems. The effect of increasing the size of the atomic Slater-type orbital basis has been studied. A minimization procedure with respect to orbital exponents has been carried out.     

Numerically Integrable Confined Basis Functions for Band Structure Calculations

Numerical atomic and Slater-type radial functions are matched with polynomials which vanish at a predetermined radius. Outside this radius the functions are identically equal to zero. Such functions with a different number of derivatives continuous at the cutoff and matching radii were used for band structure calculations of copper. The utilization of Gauss quadratures, constructed without accounting for discontinuity of the basis functions, was shown to ensure the required accuracy of numerical integration for matrix element calculations with the basis functions that have three lowest derivatives discontinuous at the cutoff and matching radii. In the case of matching radius variations over a wide range, the quality of the basis set is virtually independent of this value. Increasing the cutoff radius improves the basis quality. Moreover, a basis limit is reached even at the cutoff radius smaller than a two-fold distance between the nearest neighbors. The resulting basis set is not inferior in quality to conventional bases of atomic or Slater-type functions and, in contrast to conventional functions, requires no special techniques of numerical integration. The proposed basis set enables a reduction of computational time by a factor of 2 approximately.     

Generalized exponential functions applied to atomic calculations

The use of a generalized exponential function r ^v?1 exp(?ζr ^μ ) as a radial basis function in atomic calculations is studied with our special interest in the variationally optimum value of the parameter μ, since special cases of μ = 1 and μ = 2 correspond respectively to the radial parts of commonly-used Slater-type and Gaussian-type functions. Roothaan-Hartree-Fock calculations are performed for ground-state neutral atoms with atomic number Z = 2–54, singly-charged cations with Z = 3–55, and anions with Z = 1–53 within the single-zeta (or minimal basis) framework. For all the species examined, the optimtum μ values are found to be smaller than unity and increase towards unity as the atomic number increases. The present results support the use of Slater-type functions when μ is restricted to be an integer, but suggest from the variational point of view that even the exponential decay of Slater-type functions is too “strong” within the single-zeta approximation.     

A vibrational circular dichroism implementation within a Slater-type-orbital based density functional framework and its application to hexa- and hepta-helicenes

We describe the implementation of the rotational strengths for vibrational circular dichroism (VCD) in the Slater-type orbital based Amsterdam Density Functional (ADF) package. We show that our implementation, which makes use of analytical derivative techniques and London atomic orbitals, yields origin independent rotational strengths. The basis set dependence in the particular case of Slater-type basis functions is also discussed. It turns out that the triple zeta STO basis sets with one set of polarization functions (TZP) are adequate for VCD calculations. The origin- dependence of the atomic axial tensors is checked by a distributed origin gauge implementation. The distributed and common origin gauge implementations yield virtually identical atomic axial tensors with the Slater-type basis sets employed here, proving that our implementation yields origin independent rotational strengths. We verify the implementation for a set of benchmark molecules, for which the dependence of the VCD spectra on the particular choice of the exchange–correlation functional is studied. The pure functionals BP86 and OLYP show a particularly good performance. Then, we apply this approach to study the VCD spectra of hexa- and hepta- helicenes. In particular we focus on relationships between the sign of the rotational strengths of the two helicenes. Electronic supplementary material The online version of this article (doi: ) contains supplementary material, which is available to authorized users.     

Numerical MC SCF and CI calculations of the ground-state potential energy curve of N2

The numerical procedure of McCullough is used in calculations of Hartree-Fock and MC SCF wavefunctions for ground state of N₂. The latter are derived using the complete set of 18 spin and symmetry adapted configurations in the space of MOs that arise from 2p atomic orbitals. An increase in dissociation energy of 0.17 eV is observed when compared to MC SCF calculations in a large basis of Slater-type functions and the same set of configurations. Integrals involving the numerical MC SCF MOs are used in CI calculations in which substitutions involving the 1s and 2s electrons are included. The increase in dissociation energy due to numerical versus basis set valence CI is 0.08 eV. Spectroscopic constants and molecular quadrupole moments are reported.     

Implementation of atomic basis set composed of 1s Gaussian and 1s Slater-type orbitals to carry out quantum mechanics molecular calculations

A mixed atomic basis set formed with ls Slater-type orbitals and 1s floating spherical Gaussian orbitals is implemented. Evaluation of multicenter integrals is carried out using a method based on expansion of binary products of atomic basis functions in terms of a complete basis set, and a systematic analysis is performed. The proposed algorithm is very stable and furnishes fairly good results for total energy and geometry. An LCAO-SCF test calculation is carried out on LiH. The trends observed show that there are some combinations of mixed orbitals that are appropriate to describe the system. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 604–609, 1999     

Atomic SCF loge localized wave functions

We report in this work SCF atomic calculations for Li, Be, B, C, and Ne using a basis set of completely loge-localized functions. For these second row atoms the total volume R³ was partitioned into a spherical loge of radius R and its volume complement. The loge-localized basis functions were constructed as a product of Slater-type orbitals and a cut-off factor. The energy values obtained differ significantly from the Hartree–Fock ones indicating that the delocalization effects—not included in these calculations—are important.     

A practical solution to the “unknown normalization” problem

We analyze the theoretical basis of a procedure to determine an unknown normalization factor in discretized wave functions, which we have successfully used in a series of calculations of resonance widths for atomic and molecular systems. By reducing this determination to that of a suitable interpolation function for the energy eigenvalues, the problem is easily solved when atomic basis sets are chosen according to simple rules. Illustrations of our procedure are presented for atomic, molecular, and model systems; renormalized wave functions are compared with the exact ones for these model systems. The resulting method of renormalized continuum wave functions has a wide range of application in the study of long-lived quasibound states (predissociation, autoionization, photoionization, unimolecular reactions, etc.).     

Weighted small gaussian expansions of Slater-type atomic orbitals

Small gaussian expansions of Slater-type atomic orbitals are generated under a wide range of radial weighting conditions by full least-squares procedures, and the quality of the wave functions obtained is examined for a particular expectation value, electron density at the nucleus.     

Basis set selections for relativistic self-consistent field calculations

Practical methods of generating reliable and economic basis sets for relativistic self-consistent fields (RSCF) calculations are developed. Large component basis sets are generated from constrained optimizations of exponents in the nonrelativistic atomic calculations for light atoms. For heavy atoms, large component basis sets for inner core orbitals are generated by fitting numerical atomic spinors of Dirac-Hartree-Fock calculations with appropriate number of Slater-type functions. Small component basis sets are obtained by using the kinetic balance condition and other computational criteria. With judicious selections of the basis sets, virtual orbitals in RSCF calculations become very similar to those in nonrelativistic calculations, implying that relativistic virtual orbitals can be used in electron correlation calculations in the same manner as the conventional nonrelativistic virtual orbitals. It is also evident that the Koopmans' theorem is also valid in RSCF results.     

Even-tempered Slater-type orbitals revisited: from hydrogen to krypton

Even-tempered Slater-type orbital basis sets were developed in 1973, based on total atomic energy optimization. Here, we revisit ET STOs and propose new sets based on past experience and recent computational studies. From preliminary atomic and molecular tests, these sets are shown to be very well balanced and to perform, at lower cost, almost as well as a very large (close to complete) basis set.     

Double- and triple-zeta Slater-type basis sets with common exponents

Double- and triple-zeta basis sets of Slater-type functions (STFs) are developed for the 17 atoms from He to Ar. For computational economy, the exponents of STFs corresponding to the same atomic subshell are restricted to be common. Instead, the principal quantum numbers of the STFs are thoroughly optimized within the framework of integer values to reduce the energy loss due to the common exponent restriction. Received: 10 November 1999 / Accepted: 25 January 2000 / Published online: 19 April 2000     

X-ray and electron scattering factors for many-electron atomic system

Analytical expressions are developed for the x-ray and electron scattering factors for a many-electron atomic system when the single configuration wave function of the system is written as a sum of Slater determinants of spin orbitals. The radial part of the orbital is expanded in terms of Slater-type orbitals (STO 's). The expressions so developed have been used to calculate the coherent and incoherent x-ray and electron scattering factors and intensities for all the neutral atoms up to krypton (Z = 36) and for some positive and negative ions of chemical interest. The results obtained are used to test the value of Hartree–Fock wave functions for the evaluation of “one-electron properties” of many-electron atomic systems.     

Even-tempered Roothaan-Hartree-Fock wave functions for the third- and fourth-row atoms

Summary Roothaan-Hartree-Fock wave functions composed of 12s8p6d, 12s10p6d, and 12s10p8d even-tempered (ET) Slater-type functions (STFs), respectively, are reported for the atoms K-Zn, Ga-Kr, and Rb-Xe in their ground state. Despite the limited variational freedom in the Et method, the resultant atomic energies are found to compare well with fully-optimized wave functions of similar sizes. In particular, the present ET results reproduce almost completely the fully-optimized Sekiya-Tatewaki energies with the same basis set size for the atoms K-Zn. All the present energies are also lower than the Clementi-Roetti ones with slightly smaller but fully-optimized basis sets. A generalized even-tempered scheme is suggested and shown to give good results for Xe.     

Inclusion of relativistic effects into ZDO methods. IV. Relativistic CNDO/1

A relativistic, four-component version of the CNDO (complete neglect of differential overlap) method is introduced. This method spans the class of zero-differential-overlap approximation and it utilizes a nonempirical parametrization based on results of atomic Dirac-Fock calculations. The spin-orbit splitting is included implicitly using the relativistic basis set which distinguishes Slater-type functions of the lower and higher component spinors. The method is applicable to closed-shell as well as to open-shell systems. The actual version used is the R -CNDO /1_χ with a variable scaling approach. Applications to molecular geometries, ionization potentials, and energies of spin-orbit splitting are demonstrated for AH, AH₂, AH₃, MH, InX, and HgI₂ molecules at the self-consistent-field level as well as in the second order of the many-body perturbation theory.     

STOP: A slater-type orbital package for molecular electronic structure determination

A general ab initio package using Slater-type atomic orbitals is presented. This package, called STOP, uses the one-center two-range expansion method to evaluate the multicenter electronic integrals. Thoroughly optimized numerical techniques, in particular, convergence accelerators and suitable Gauss quadratures, are used in the algorithms which provide accurate numerical values for all these integrals. STOP thus provides wavefunctions for general molecular structures at the self-consistent field level for the first time over a Slater-type orbital basis. © 1996 John Wiley & Sons, Inc.     

Linear augmented Slater-type orbital method for free standing clusters

We have developed a Scalable Linear Augmented Slater-Type Orbital (LASTO) method for electronic-structure calculations on free-standing atomic clusters. As with other linear methods we solve the Schr?dinger equation using a mixed basis set consisting of numerical functions inside atom-centered spheres and matched onto tail functions outside. The tail functions are Slater-type orbitals, which are localized, exponentially decaying functions. To solve the Poisson equation between spheres, we use a finite difference method replacing the rapidly varying charge density inside the spheres with a smoothed density with the same multipole moments. We use multigrid techniques on the mesh, which yields the Coulomb potential on the spheres and in turn defines the potential inside via a Dirichlet problem. To solve the linear eigen-problem, we use ScaLAPACK, a well-developed package to solve large eigensystems with dense matrices. We have tested the method on small clusters of palladium.     

Comparison of RHF,NDDO, and MOM molecular one-electron expectation values calculated using weighted and unweighted STO–NG (ω) basis functions

The effect of basis functions on molecular one-electron property expectation values calculated by approximate methods is examined using weighted and unweighted least-squares Gaussian-type orbital function expansions of Slater-type orbital functions.     

On the effectiveness of exponential type orbitals with hyperbolic cosine functions in atomic calculations

Unconventional basis functions, constructed from exponential type orbitals (ETOs) with hyperbolic cosine functions, are applied to Roothaan-Hartree-Fock calculations of atoms within the minimal basis sets framework. The most popular ETOs, Slater type orbitals, B functions and \(\psi ^{(\alpha ^*)}\) functions with \(\alpha ^*=2\), and two types of hyperbolic cosine functions, \(\cosh (\beta r)\) and \(\cosh (\beta r+\gamma )\), are used in this work. The performance of the present basis functions is investigated and compared to the conventional double-zeta Slater-type basis set and numerical Hartree-Fock results. The improvement in the atomic energies clearly demonstrates how the accuracy increases when we move from ETO to ETO with hyperbolic cosine basis functions. The resulting improved minimal basis sets can also be useful in molecular calculations.