Gaussian basis sets for molecular applications

The choice of basis set in quantum chemical calculations can have a huge impact on the quality of the results, especially for correlated ab initio methods. This article provides an overview of the development of Gaussian basis sets for molecular calculations, with a focus on four popular families of modern atom‐centered, energy‐optimized bases: atomic natural orbital, correlation consistent, polarization consistent, and def2. The terminology used for describing basis sets is briefly covered, along with an overview of the auxiliary basis sets used in a number of integral approximation techniques and an outlook on possible future directions of basis set design. © 2012 Wiley Periodicals, Inc.     

High-quality Gaussian basis sets for fourth-row atoms

Summary Energy-optimized Gaussian basis sets of triple-zeta quality for the atoms Rb-Xe have been derived. Two series of basis sets are developed; (24s 16p 10d) and (26s 16p 10d) sets which we expand to 13d and 19p functions as the 4d and 5p shells become occupied. For the atoms lighter than Cd, the (24s 16p 10d) sets with triple-zeta valence distributions are higher in energy than the corresponding double-zeta distribution. To ensure a triple-zeta distribution and a global energy minimum the (26s 16p 10d) sets were derived. Total atomic energies from the largest basis sets are between 198 and 284E _H above the numerical Hartree-Fock energies.     

Optimization of Gaussian basis sets for Dirac-Hartree-Fock calculations

We investigate the optimization of Gaussian basis sets for relativistic calculations within the framework of the restricted Dirac-Hartree-Fock (DHF) method for atoms. We compare results for Rn of nonrelativistic and relativistic basis set optimizations with a finite nuclear-size. Optimization of separate sets for each spin-orbit component shows that the basis set demands for the lower j component are greater than for the higher j component. In particular, the p _1/2 set requires almost as many functions as the s _1/2 set. This implies that for the development of basis sets for heavy atoms, the symmetry type for which a given number of functions is selected should be based on j, not on l, as has been the case in most molecular calculations performed to date.     

Molecular integrals over generalized gaussian basis sets

A method is given for obtaining the common molecular integrals over generalized gaussian functions: The present algorithms are expected to be more efficient than those given in earlier work by the same author.     

Relativistic Gaussian basis sets for radon through plutonium

Summary Relativistic Gaussian basis sets of neutral atoms Rn-Pu and ions Th⁺⁴, U⁺³ and Pu⁺³ in the configurations of average energies are presented. The exponent parameters of the basis sets are determined by least-squares fitting to the numerical Dirac-Fock wave functions. The total energies obtained are within 0.155 a.u. of the Dirac-Fock limits and the qualities of the basis sets are between double-zeta and triple-zeta in the valence parts. Using the exponent parameters the Breit interaction energies have been calculated by perturbation theory and the self-consistent field treatment.     

Medium-size Gaussian basis sets for hydrogen through argon

Summary Medium-sized Gaussian basis sets are reoptimized for the ground states of the atoms from hydrogen through argon. The composition of these basis sets is (4s), (5s), and (6s) for H and He, (9s5p) and (12s7p) for the atoms Li to Ne, and (12s8p) and (12s9p) for the atoms Na to Ar. Basis sets for the² P states of Li and Na, and the³ P states of Be and Mg are also constructed since they are useful in molecular calculations. In all cases, our energies are lower than those obtained previously with Gaussian basis sets of the same size.     

Locality of exchange matrices for common Gaussian basis sets

This article investigates a new approach to local approximations to exchange. When a finite basis set is introduced, any operator may be regarded as equivalent to a local operator if its matrix can be reproduced as the matrix of a multiplicative function operator. The expectation values of such operators can be determined from the electron density. Any matrix can be divided into local and nonlocal components, in a way dependent on linear dependencies among basis-set products. A measure of locality is provided by the ratio of the norm of the local component to that of the whole matrix. The local contribution to an expectation value can also be compared with the total. The self-consistent field exchange matrices and their expectation values for atoms Li through Ne and for LiH and HF with several Cartesian Gaussian basis sets were investigated in this way, and for the atoms, the exchange supermatrix was also examined. It has been found in all cases that the matrices and expectation values are more than 92% local; most are more than 99% local. © 1997 John Wiley & Sons, Inc.     

A general formulation for the efficient evaluation of n-electron integrals over products of Gaussian charge distributions with Gaussian geminal functions

In this work, we present a general formulation for the evaluation of many-electron integrals which arise when traditional one particle expansions are augmented with explicitly correlated Gaussian geminal functions. The integrand is expressed as a product of charge distributions, one for each electron, multiplied by one or more Gaussian geminal factors. Our formulation begins by focusing on the quadratic form that arises in the general n-electron integral. Using the Rys polynomial method for the evaluation of potential energy integrals, we derive a general formula for the evaluation of any n-electron integral. This general expression contains four parameters ω, θ, v, and h, which can be evaluated by an examination of the general quadratic form. Our analysis contains general expressions for any n-electron integral over s-type functions as well as the recursion needed to build up arbitrary angular momentum. The general recursion relation requires at most n + 1 terms for any n-electron integral. To illustrate the general method, we develop explicit expressions for the evaluation of two, three, and four particle electron repulsion integrals as well as two and three particle overlap and nuclear attraction integrals. We conclude our exposition with a discussion of a preliminary computational implementation as well as general computational requirements. Implementation on parallel computers is briefly discussed.     

Cubic-grid Gaussian basis sets for electron scattering calculations IV

Summary An analytical formula has been derived for averaging the differential cross section for electron scattering with respect to isotropic target molecule orientation. It may be applied to any type ofT-matrix element k ^out|T|k ⁱⁿ in which the plane-wave functionsk ^out andk ⁱⁿ are expanded in a set ofs-type Gaussian functions. The formula for averaging was tested against results obtained by Monte-Carlo-type calculations and against experimental data for elastic electron scattering by the H₂ molecule.     

Optimal use of the recurrence relations for the evaluation of molecular integrals over Cartesian Gaussian basis functions

We consider the tree search problem for the recurrence relation that appears in the evaluation of molecular integrals over Cartesian Gaussian basis functions. A systematic way of performing tree search is shown. By applying the result of tree searching to the LRL2 method of Lindh, Ryu, and Liu (LRL) (J. Chem. Phys., 95 , 5889 1991), which is an auxiliary function-based method, we obtain significant reductions of the floating point operations (FLOPS) counts in the K⁴ region. The resulting FLOPS counts in the K⁴ region are comparable up to [dd|dd] angular momentum cases to the LRL1 method of LRL, currently the method requiring least FLOPS for [dd|dd] and higher angular momentum basis functions. For [ff|ff], [gg|gg], [hh|hh], and [ii|ii] cases, the required FLOPS are 24, 40, 51, and 59%, respectively, less than the LRL1 method in the K⁴ region. These are the best FLOPS counts available in the literature for high angular momentum cases. Also, there will be no overhead in either the K² or K⁰ region in implementing the present scheme. This should lead to more efficient codes of integral evaluations for higher angular momentum cases than any other existing codes. © 1993 John Wiley & Sons, Inc.     

Efficient evaluation of one-center three-electron Gaussian integrals

 An algorithm is presented for the efficient evaluation of two types of one-center three-electron Gaussian integrals. These integrals are required to avoid the resolution-of-identity (RI) approximation in explicitly correlated linear R12 methods. Without the RI approximation, it is possible to enforce rigorously the strong orthogonality of the second-order M?ller–Plesset R12 ansatz. A test calculation is performed using atomic Gaussian-type orbitals of the neon atom. Received: 21 November 2000 / Accepted: 6 April 2001 / Published online: 9 August 2001     

Gaussian basis sets for transition metals of the second series

A Gaussian basis set consisting of (15s, 9p, 8d) Gaussian functions has been optimized for the transition metal atoms of the second series (fourth-row atoms).     

Efficient evaluation of Coulomb integrals in a mixed Gaussian and plane‐wave basis

An efficient way of calculation is presented for matrix elements between two plane waves interacting with a molecular Coulombic field. In concurrence with the absolute value of the momentum transfer vector, K = k₁ ? k ₂ , the most effective method of calculation is selected. The case of K = 0 requires special treatment. For 0 < |K| ≤ 0.3, it is profitable to evaluate the integrals by means of the multipole expansion, and for |K| > 0.3 the density fitting can be applied. For the large |K| the electronic part of the integral is much smaller than the nuclear part and the integral may be approximated by the nuclear contribution only. Some examples for testing the accuracy and time saved are presented. The primary purpose of this paper is to accelerate electron scattering calculations, but it also may be profitable for the electronic structure theory in attempts to use mixed Gaussian and plane‐wave basis sets. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Evaluation of Gaussian basis sets in ab initio

Ab initio wave functions for NH₃ have been calculated using the STONG program, where N is 2 to 6, and with Gaussian 70, employing the 4–31 and 6–31 basis sets. Electric field gradients at the N atom were computed from the gaussian basis orbitals, and, with STONG, also from the equivalent STO's. Results from the two split valence shell Gaussian 70 calculations are in good agreement with values from extended SCF and SCFCI calculations. Because of adequate agreement with more extended calculations, the use of the 4–31 basis set is chosen to calculate field gradients in molecules consisting of five or more second row atoms. It should be adequate for the prediction of the region for searching for new ¹⁴N, ¹⁷O and ²H NQR spectra. An improved value of the electric quadrupole moment for the ¹⁴N nucleus is proposed.     

On the use of Gaussian shell type basis orbitals for single-center expansions. I. Evaluation of integrals

Formulas are derived for all Hamiltonian integrals required for molecular computations using a novel basis for single-center expansions. The basis orbitals depend exponentially upon α(r ? ρ)² where r and ρ are the distance from center to electron and to a variationally scaled spherical shell, respectively. Comparisons are made between these so-called Gaussian shell orbitals (GSO ) and the conventional GTO and STO bases for single-center calculations. A preliminary comparison on H using a single GSO , a non-integer STO , and a GTO gives the optimized energies: ?0.51089 a.u., ?0.50504 a.u., and ?0.50422 a.u., respectively.     

A study on universal Gaussian basis sets for first-row atoms

Analysis of various optimum and non-optimum Gaussian basis sets for firstrow elements have indicated that with a minimum increase of the basis set size and without loss of accuracy of the calculated total energy, a single universal Gaussian basis set may replace individually optimized Gaussian basis sets for a series of atoms. Such a universal Gaussian basis set may substantially reduce the computational work required for the calculation of molecular integrals in ab initio MO calculations.