Nuclear attraction integrals in the homogeneously charged sphere model of the atomic nucleus

Formulas are presented for molecular nuclear attraction integrals over Gaussian-type functions. They are not complicated, which indicates that the homogeneously charged sphere model can be adopted in molecular calculations.     

Computation of one and two electron spin-orbit integrals

Methods for the computation of one- and two-electron spin-orbit integrals over Gaussian-type basis functions are presented. We show that existing nuclear-attraction and electron-repulsion integral codes can be readily adapted for the efficient evaluation of spin-orbit integrals; in particular, one can take advantage of recent advances in the computation of derivative integrals. Recurrence relations for the nuclear attraction integrals are also developed.     

Choosing GTO basis sets for periodic HF calculations

We investigate numerical linear dependencies of Gaussian-type orbital basis sets employed in the framework of the Hartree-Fock self-consistent field method for periodic structures, which so far have hampered the use of extended basis sets for non-ionic crystals. These linear dependencies occur when diffuse basis functions are included in a basis set in an uncontrolled manner. We use the condition number of the overlap matrix to lead us in the construction of extended basis sets for periodic structures which avoid numerical linear dependencies. Extended basis sets of high quality are optimized for a number of periodic structures (fcc He, α-Be, α-BN, and B1 NaF) with respect to the energy of the constituent atoms or ions. The results obtained with our basis sets, which do not require reoptimization in the crystal environment, compare favorably with those obtained with other extended basis sets reported in the literature. Received: 20 July 1998/Accepted: 21 August 1998 / Published online: 19 October 1998     

MIDI! basis set for silicon, bromine, and iodine

The MIDI! basis set is extended to three new atoms: silicon, bromine, and iodine. The basis functions for these heteroatoms are developed from the standard 3-21G basis set by adding one Gaussian-type d subshell to each Si, Br, or I atom. The exponents of the d functions are optimized to minimize errors in the geometries and charge distributions that these basis functions yield when they are used in Hartree-Fock calculations with all atoms represented by the MIDI! basis. The MIDI! basis is defined to use five spherical d functions in a d subshell. We present a detailed comparison of such calculations to calculations employing six Cartesian d functions in each d subshell; these studies show that 5D and 6D options give very similar results for molecular geometries and dipole moments, not only for compounds containing Si, Br, and I but also for compounds containing N, O, F, P, S, and Cl. The MIDI! basis set is also tested successfully for hypervalent Si compounds. Received: 7 January 1998 / Accepted: 7 January 1998     

Molecular integrals of Breit interaction over Laguerre Gaussian-type functions

The Breit interaction consists of two interactions of the magnetic and the retardation terms. The molecular integrals involving the former interaction are shown to be reduced to the familiar integrals of the Coulomb repulsion. The integrals involving the latter interaction are formulated over the Laguerre Gaussian-type functions and some advantages of the formulation are discussed. Bayman's gradient formula of the solid spherical harmonic is generalized to include the gradient of the homogeneous solid spherical harmonic.     

On the evaluation of derivatives of Gaussian integrals

Summary We show that by a suitable change of variables, the derivatives of molecular integrals over Gaussian-type functions required for analytic energy derivatives can be evaluated with significantly less computational effort than current formulations. The reduction in effort increases with the order of differentiation.Dedicated to Prof. Klaus Ruedenberg     

Photoionization cross sections of H2(+) and H2 with complex Gaussian-type basis functions optimized for the frequency-dependent polarizabilities

Within the framework of the complex basis function method, the photoionization cross sections of H(2)(+) and H(2) were calculated based on the variational principle for the frequency-dependent polarizabilities. In these calculations, complex orbital exponents of Gaussian-type basis functions for the final state continuum wavefunctions were fully optimized for each photon energy with the numerical Newton-Raphson method. In most cases, the use of only one or two complex Gaussian-type basis functions was enough to obtain excellent agreement with previous high precision calculations and available experimental results. However, there were a few cases, in which the use of complex basis functions having various angular momentum quantum numbers was crucial to obtain the accurate results. The behavior of the complex orbital exponents as a function of photon energy was discussed in relation to the scaling relation and the effective charge for photoelectron. The success of this method implies the effectiveness of the optimization of orbital exponents to reduce the number of basis functions and shows the possibility to calculate photoionization cross sections of general molecules using only Gaussian-type basis functions.     

Contracted polarization functions for the atoms Ca, Ga–Kr, Sr, and In–Xe

 Contracted Gaussian-type function sets are proposed for polarization functions of the atoms Ga–Kr and In–Xe. We also report polarization functions for Ca and Sr. A segmented contraction scheme is used for its compactness and computational efficiency. The contraction coefficients and orbital exponents are fully optimized to minimize the deviation from accurate atomic natural orbitals. The present polarization functions yield more than 99% of atomic correlation energies predicted by accurate natural orbitals of the same size. Received: 23 February 2001 / Accepted: 19 April 2001 / Published online: 13 June 2001     

Multicenter and multiparticle integrals for explicitly correlated cartesian gaussian-type functions

An analytical derivation of multicenter and multiparticle integrals for explicitly correlated Cartesian Gaussian-type cluster functions is demonstrated. The evaluation method is based on the application of raising operators that transform spherical cluster Gaussian functions into Cartesian Gaussian functions.     

Contracted Gaussian-type basis functions revisited. III. Atoms K through Kr

Six minimal basis sets of contracted Gaussian-type functions (GTFs) are developed for the third-row atoms K through Kr. The smallest and largest sets for transition metal atoms are (3333/33/3) and (8433/84/8), respectively, where a slash distinguishes the s, p, and d symmetries and single-digit figures in the parentheses denote the numbers of primitive GTFs. The two largest sets, (7433/74/7) and (8433/84/8), surpass the (62111111/33111/311) set of Schaefer et al. in the associated total energies. Our (8433/84/8) set is also superior to their (842111/631/411) set. The quality of the present basis sets is tested by self-consistent field (SCF) and configuration interaction (CI) calculations on the Cu₂ molecule. As the accuracy of the basis set increases, SCF calculations show a decrease in the dissociation energy and an increase in the equilibrium internuclear distance. The same tendencies are found in the results of CI calculations with and without a Davidson correction. All the present basis sets are freely available at the internet address: http://202.35.198.41/∼htatewak/. Received: 17 June 1998 / Accepted: 4 August 1998 / Published online: 23 November 1998     

Full variational molecular orbital method: Application to the positron-molecule complexes

Optimal Gaussian-type orbital (GTO) basis sets of positron and electron in positron-molecule complexes are proposed by using the full variational treatment of molecular orbital (FVMO) method. The analytical expression for the energy gradient with respect to parameters of positronic and electronic GTO such as the orbital exponents, the orbital centers, and the linear combination of atomic orbital (LCAO) coefficients, is derived. Wave functions obtained by the FVMO method include the effect of electronic or positronic orbital relaxation explicitly and satisfy the virial and Hellmann–Feynman theorems completely. We have demonstrated the optimization of each orbital exponent in various positron-atomic and anion systems, and estimated the positron affinity (PA) as the difference between their energies. Our PA obtained with small basis set is in good agreement with the numerical Hartree–Fock result. We have calculated the OH⁻ and [OH⁻; e⁺] species as the positron-molecular system by the FVMO method. This result shows that the positronic basis set not only becomes more diffuse but also moves toward the oxygen atom. Moreover, we have applied this method to determine both the nuclear and electronic wave functions of LiH and LiD molecules simultaneously, and obtained the isotopic effect directly. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 491–501, 1998     

Application of explicitly correlated Gaussian functions for calculations of the ground state of the beryllium atom

The electronic energy of atoms and molecules may be evaluated accurately by the use of wave functions where the interelectronic distances are explicitly present. In particular, explicitly correlated Gaussian-type functions make these types of calculations feasible and computationally tractable even for more extended systems. The resulting multielectron integrals may be reduced to standard one- and two-electron integrals that are readily evaluated. Initial calculations have been made for the Be atom where all four electrons were correlated at the same time. The preliminary results show that accurate results may be obtained. © 1993 John Wiley & Sons, Inc.     

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.     

Computation of some new two-electron Gaussian integrals

Summary The evaluation of a new form of two-electron integrals is required if the interelectronic distancer ₁₂ is used as a variable in then-electron functions of electron correlation methods. The McMurchie-Davidson algorithm for the generation of molecular integrals over Gaussian-type functions is ideally suited to this. The new Gaussian integrals are formed from Hermite integrals overr ₁₂ (rather than 1/r ₁₂) by standard techniques. The Hermite integrals overr ₁₂ itself are generated by a simple procedure with negligible computational effort. The key results are discussed in the context of general recursion formulas. On leave from: Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, W-4630 Bochum, Germany     

Contracted polarization functions for the atoms helium through neon

Contracted Gaussian-type function sets are proposed for polarization functions of the atoms helium through neon, with the exception of lithium. A segmented contraction scheme is used for its compactness and computational efficiency. The contraction coefficients and orbital exponents are fully optimized to minimize the difference from accurate atomic natural orbitals. The present polarization functions yield more than 99% of atomic correlation energies predicted by accurate natural orbitals of the same size. Received: 16 April 1997 / Accepted: 26 June 1997     

The Gaussian and augmented-plane-wave density functional method for ab initio molecular dynamics simulations

A new algorithm for density-functional-theory-based ab initio molecular dynamics simulations is presented. The Kohn–Sham orbitals are expanded in Gaussian-type functions and an augmented-plane-wave-type approach is used to represent the electronic density. This extends previous work of ours where the density was expanded only in plane waves. We describe the total density in a smooth extended part which we represent in plane waves as in our previous work and parts localised close to the nuclei which are expanded in Gaussians. Using this representation of the charge we show how the localised and extended part can be treated separately, achieving a computational cost for the calculation of the Kohn–Sham matrix that scales with the system size N as O(NlogN). Furthermore, we are able to reduce drastically the size of the plane-wave basis. In addition, we introduce a multiple-cutoff method that improves considerably the performance of this approach. Finally, we demonstrate with a series of numerical examples the accuracy and efficiency of the new algorithm, both for electronic structure calculations and for ab initio molecular dynamics simulations. Received: 15 December 1998 /Accepted: 18 February 1999 /Published online: 14 July 1999     

Evaluation of GTO molecular integrals

Methods for evaluating the function F_m(t) occurring in molecular integrals over Gaussian-type orbitals are reviewed and extended. Formulas based on Bessel function and continued-fraction expansions are analyzed. The recommended evaluation procedures, embodied in a portable computer program, involve Padé approximations for various argument intervals and use recursion in m. The program is economical in storage requirements and faster than those in current use.     

The energetic effects of p,d, and f Gaussian polarization functions on closed-shell AHn oxygen and sulfur hydrides

A series of LCAO -MO -SCF calculations, using various basis sets of Gaussian-type functions, has been made in order to study the effects of p, d, and f polarization functions for a 10-electron isoelectronic series of oxygen hydrides and for an 18-electron isoelectronic series of sulfur hydrides. Conclusions from these results suggest that meaningful proton affinities cannot be calculated without the inclusion of a d function on the heavy atom and a p function on the hydrogen atoms.     

Contracted Gaussian-type basis functions revisited II. Atoms Na through Ar

We report five minimal-type contracted Gaussian-type function (CGTF) basis sets of the second-row atoms, Na – Ar, for molecular applications. Three of the present CGTF sets are revised versions of those given by Huzinaga and co-workers and the other two are newly developed for more accurate calculations. Practical utility and improved reliability of the present basis sets, augmented by polarization functions, are confirmed by test calculations on the P atom and P₂ molecule both at the self-consistent field (SCF) and configuration interaction (CI) levels. Received: 10 February 1997 / Accepted: 23 April 1997     

Molecular orbital studies of conformers and the barrier to internal rotation in 1,3-butadiene

The potential curve for rotation around the central bond in 1,3-butadiene has been estimated by ab initio calculations using Gaussian-type basis functions. The calculations, which also include limited geometry variation during rotation, suggest that in the SCF approximation the second stable form of the molecule is a gauche conformation rather than a cis. The predicted energy difference between the planar trans ground state and the stable gauche form is 2.7 kcal/mole and the barrier to internal rotation is found to be 6.0 kcal/mole using a (9,5) basis for carbon and 4s functions on hydrogen.