Basis sets for molecular calculations

Variational calculations of the H-atom are performed with different types of Gaussian basis sets, (a) Huzinaga basis sets of 1s-Gaussians, (b) ns basis sets with n = 1, 2, 3, 4 ⋯ and optimized common exponents for each set, (c) ns basis sets with n = 1, 3, 5 ⋯ (d) ns basis sets with n = 1, 2, 3, 5, 7 ⋯ (e) ns basis sets with n = 1, 2, 3, 4 ⋯ but with all orbital exponents optimized. The number of basis functions in a set is N . For the sets (a) the errors of the energy (upper or lower bound) and of the expectation values r k , k = −2, −1, ⋯ 5 appear to converge as ≈ exp (— b √ N ), for sets (b) and (e) as ≈exp (— bN ), for set (c) the error of the upper bound converges as ≈N −2 and for set (d) as ≈ N −4 . For medium accuracy the Huzinaga sets are surprisingly good. They are poorly suited only for representing the cusp at the nucleus and the variance of the energy. For high accuracy basis sets (b) and (e) are superior. They represent the cusp correctly. For them the energy variance converges rapidly to zero, and all other quantities converge finally much faster. Basis sets with only even powers of r (or with just 2 s included) and a common (optimized) orbital exponent are, although formally “complete in the energy space”, of no practical use. Basis sets (e) yield by far the fastest convergence. The results are analyzed and their consequences are discussed.

Gaussian basis sets and the nuclear cusp problem

The expectation value of the electronic energy of a n-electron system may be written as the ratio of two integrals, 

$$E = / $$

integration being taken over the 3n spatial and n spin coordinates of the electrons. Procedures for integration over the spin variables are not the concern of the present work, and we proceed under the assumption that the spin integrations of the energy expectation value have been completed. We are thus left with the evaluation of two integrals over the 3n spatial coordinates. A rather widely used procedure is to construct the trial form of the many electron wavefunction, Ψ, from a set of functions of the coordinates of one electron. We will denote the basis set of one electron functions by {Φ}. Notice that each basis function defines a set of n building blocks to be used in the construction of the wavefunction, since each function may be written in the coordinates of any of the n electrons.

An Introduction to Molecular Integral Evaluation

The expectation values 〈rk〉 (−2 ⩽ k ⩽ 4, k = 10), values of the charge density ρ(r) at selected points, and coefficients in the MacLaurin expansion of ρ(r) are used to test the quality of 71 orbital basis sets used for the atomic helium Hartree–Fock problem. These tests in position space are complementary to the momentum space tests previously carried out [Int. J. Quantum Chem. 21, 419 (1982)]. Information theoretic measures with respect to either or both position and momentum space properties are subsequently defined and the orbitals are ranked accordingly. These measures indicate that, for a given orbital, momentum space properties are more poorly predicted than position space ones. Moreover an improvement in the virial ratio does not necessarily lead to a more balanced orbital with respect to position and momentum space properties. Basis sets containing Slater-type orbitals are again found to be rather accurate. The exponentially damped rational function is confirmed to be the outstanding two-parameter unconventional orbital.

Basis set quality. II. Information theoretic appraisal of various s- orbitals†

The history and evolution of Gaussian basis sets in molecular electronic structure calculations are reviewed, from the original proposals by McWeeny and Boys to the current high-quality generally-contracted basis sets of the atomic natural orbital and correlation-consistent types. Various explorations of modified forms, generalizations, and alternatives are mentioned, but the focus is on the mainstream developments whose origin can largely be traced to the work of Reeves and his students. The introduction of various standard sequences of basis sets, such as those of Pople and coworkers, with well-documented and extensive test results, and the wide distribution of easy-to-use computer codes, such as the GAUSSIAN series, have contributed greatly to the explosion in the use of ab initio calculations of molecular structure in the chemistry research community.

The History and Evolution of Gaussian Basis Sets

The present paper describes the first part of an investigation into the possibility of finding a set of simple functions for the construction of molecular wave functions which satisfies two essential requirements: that the number of basis functions required to produce a given accuracy be comparable to the number of Slater functions required to give the same accuracy, and that the molecular multicentre integrals should be considerably easier to evaluate than those over Slater functions. It is shown that a basis made up of 1s gaussians supplemented by a single ‘cusp function’ of the form may be expected to satisfy the requirements for the s-orbitals in atoms other than hydrogen. Calculations of the ground-state orbital approximations of the helium and beryllium atoms show that an inner-shell 1s orbital can be satisfactorily described by the cusp function and two gaussians, whilst subsequent s-orbitals should require not more than two or three gaussians per orbital to produce better than ‘double-zeta’ accuracy.

Stephen Sykes

Papers

Cusped-gaussian wave functions