Cubic-grid Gaussian basis sets for electron scattering calculations. I. Definition and construction

We propose a new type of Gaussian basis sets for use in calculations of electron scattering by molecules. Instead of locating the basis-set functions on the atomic centers of the target molecule, we place primitive s-type Gaussians at the positions of a cubic lattice with a regular grid. The grid and the Gaussian exponent are fixed so as to give the best representation of the plane-wave function. Plane-wave functions and Green functions obtained by means of the cubic-grid basis set are tested graphically against exact functions and functions expressed by means of a conventional Gaussian basis set. © 1995 John Wiley & Sons, Inc.     

Spherically symmetric axial Gaussian lobe 3d and 4f orbitals

The axial Gaussian lobe orbital (AGLO ) representations of 3d and 4f orbitals proposed by LeRouzo and Silvi have been angularly optimized to ensure spherical symmetry of filled 3d and 4f shells. The functions have been tested on the hydrogen atom in the presence of high quality s and p basis sets and found to provide excellent minimal Gaussian representations of polarization functions. Exact orbital degeneracy is not obtained within each shell, however. Tabulated values are given to allow arbitrary scaling of the 3d and 4f lobe mimic orbitals.     

Notes on hydrogenic Gaussian lobe functions

The Gaussian lobe representation of 2p and 3d_xy functions of a one-electron atom has been reconsidered. Contrary to earlier results, it has been found that the minimum energy associated with the simplest case of these functions is reached when the expansion centers coincide with the nucleus. This also seems to be true for representations with larger expansion lengths. Under similar conditions, Whitten's octahedral representation of the 2p function gives a better energy value than Poshusta's tetrahedral one.     

A simplified representation of the potential produced by Gaussian charge distributions

A procedure is presented which allows a more economical representation of the potential produced by orbital charge distributions in which the orbitals are expanded in terms of a finite set of polynomial Gaussian functions. The basic idea is that the products of pairs of Gaussian basis functions, on which the charge distributions are expanded, are expressed in terms of a new basis set of optimally chosen single Gaussian functions. Such a procedure has been tested in a particular case and a few possible applications have been suggested.     

Properties of pyridine and pyrazine from ab initio molecular orbital wave functions

Various molecular properties have been calculated for pyridine and pyrazine from Gaussian lobe ab initio SCF molecular wave functions. Values are compared with available experimental data. In general, agreement is satisfactory with the exception of the rather sensitive asymmetry parameter of the quadrupole coupling tensor. The distributions of total electronic charge, and of selected molecular orbitals have been displayed as plots of the charge density contours in two dimensions.     

Application of an ion-fitted pseudopotential to HF,H2O,NH3, BeO,and HCl in a Gaussian lobe basis

The core potentials for atoms of atomic numer 1–18 fitted to ion spectra by Chang, Habitz, Pittel, and Schwarz have been extended to the molecular case in a Gaussian lobe basis by using a six-Gaussian (6G-POT ) representation for the exponential factors of the atomic core potentials. In a (9s/5p/1d) basis the 6G-POT one-electron energies, dipole moments, and Mulliken charges are improved over a one-Gaussian potential form for HF, NH₃, and H₂O; BeO also yields good agreement within 2.6% of the experimental bond length. For HCl, the core potential shows larger errors in the dipole moment (7%) and one-electron eigenvalues (2%), but a 75% saving in computer time is realized for HCl compared with only about 35% for first-row systems using the 6G-POT core potentials. Analytical expressions are given to extend the 6G-POT method up to s, p, d, f, and g valence shells.     

Dependence of approximate ab initio molecular loge sizes on the quality of basis functions

Size and shape parameters for the core, bonding, and lone electron pairs of the ten-electron hydrides (CH₄, NH₃, H₂O, HF) were determined from ab initio MO wave functions using various Gaussian basis sets. The fundamental features of approximate electron pair loge representation are somewhat more sensitive to the quality of the basis functions than the molecular total energy. The total size of the molecular electron distribution is less affected by basis set variations than its components: the core, bonding, and lone pair sizes. There is an apparent tendency to “preserve” the total size of molecular distribution.     

Core correlating basis functions for elements 31–118

Gaussian functions for correlation of all core shells of elements from Z = 31 to Z = 118 have been optimized in relativistic singles and doubles CI calculations, performed on the shell of highest angular momentum for each principal quantum number. The SCF functions were derived from the double-zeta, triple-zeta, and quadruple-zeta basis sets previously optimized by the author. Only those Gaussian functions that are not represented in the SCF basis sets were optimized. The functions are available from the Dirac program web site, .     

A formulation of the quantum-mechanical many-body problem in terms of hyperspherical coordinates

Hyperspherical coordinates are used to construct a matrix representation of a general N-particle Hamiltonian in the case where the interaction is electrostatic. The Yukawa interaction can be treated similarly, as is shown in an appendix. The basis functions used to construct the matrix representation of H are mononomials inthe 3N coordinates of the particles, multiplied by functions of the hyperradius. Methods for transforming from this representation to a symmetry-adapted representation are also discussed.     

From linear combinations to integrals: A new approach to the basis function problem

A new formalism is presented, based upon the finite element method, that permits a dual representation of orbitals in terms of exponential or Gaussian functions as both an integral over the space of exponential parameters and as a linear combination of basis functions. The method has been implemented for the atomic Hartree–Fock problem using exponential functions and test calculations made for atoms ranging from B to Cl. Accurate and consistent results can be obtained for a variety of atoms in a simple way using computational schemes that are systematic and hierarchic in nature. The new formalism is promising for any method where the calculation of integrals is not a major problem, such as some approaches of the density functional method and the pseudospectral formulation of ab initio methods. © 1992 by John Wiley & Sons, Inc.     

The expansion of hydrogen states in Gaussian orbitals

The convergence properties of Gaussian orbitals are studied by considering a very simple system, the hydrogen atom. We have variationally optimized even-tempered basis sets containing up to 60 s functions for the ground state and the first excited S state of the hydrogen atom, to an accuracy of 10^–15E_h. In addition, we have freely optimized the exponents in basis sets containing up to 12 Gaussians. We have studied the convergence of the total energy, the kinetic energy, the extent of the atom as measured by r², and the Fermi-contact interaction at the nucleus in these basis sets as well as in basis sets augmented with additional diffuse or steep functions.     

Intramolecular interaction effects and the structure of N2OS and N2O2: An Ab initio study

An ab initio study of O?N? N?S with full geometry optimization has been carried out to corroborate the presence of an interaction between the terminal atoms in this type of structure, which, in O?N? N?O, apparently stabilizes the cis conformer. Using the unscaled 4–31G basis set with a full set of d functions on the sulfur, there is a potential minimun at the trans but not the cis geometry. A gauche conformer with a torsional angle of 77.2° is the most stable. With N₂O₂ this basis set gives potential minima at both the cis and trans geometries, but the trans conformer is slightly more stable, contrary to experiment and the results of (7,3) basis-set calculations reported in the literature in which Gaussian lobe functions were employed. Using a (9,5) basis set there is no longer a potential minimum at the cis geometry, and a gauche structure is more stable than the cis conformer as in the case of N₂OS with the less-extended basis set. Force constants (harmonic and anharmonic), compliance constants, relaxed force constants, and interaction-displacement coordinates for both molecules are compared for key structural elements.     

Gaussian basis sets for calculation of spin densities in first-row atoms

Summary The suitability of Gaussian basis sets for ab initio calculation of Fermi contact spin densities is established by application to the prototype first-row atoms B-F having open shell p electrons. Small multiconfiguration self-consistent-field wave functions are used to describe relevant spin and orbital polarization effects. Basis sets are evaluated by comparing the results to highly precise numerical grid calculations previously carried out with the same wave function models. It is found that modest contracted Gaussian basis sets developed primarily for Hartree-Fock calculations can give semiquantitative results if augmented by diffuse functions and if further uncontracted in the outer core-inner valence region.     

Two-electron integral evaluation for uncontracted geometrical-type Gaussian functions

A new algorithm for efficient evaluation of two-electron repulsion integrals (ERIs) using uncontracted geometrical-type Gaussian basis functions is presented. Integrals are evaluated by the Habitz and Clementi method. The use of uncontracted geometrical basis sets allows grouping of basis functions into shells (s, sp, spd, or spdf) and processing of integrals in blocks (shell quartets). By utilizing information common to a block of integrals, this method achieves high efficiency. This technique has been incorporated into the KGNMOL molecular interaction program. Representative timings for a number of molecules with different basis sets are presented. The new code is found to be significantly faster than the previous program. For ERIs involving only s and p functions, the new algorithm is a factor of two faster than previously. The new program is also found to be competitive when compared with other standard molecular packages, such as HONDO-8 and Gaussian 86.     

Study of the quality of Gaussian basis sets for carbon and silicon: Calculations on methane and silane

A number of Gaussian basis sets for carbon and silicon have been examined in terms of the one-electron properties of methane and silane. The convergence of the properties to their limiting values is not monotonic but, in general, a representation that involves five Gaussian functions per occupied atomic orbital on the heavy atom is sufficient to closely approach the limits. A relationship between the sizes and partitioned electronic energies is shown to hold to a good approximation for the Boys spatially localized molecular orbitals employed in this study.