Gaussian Basis Sets - Overview

Ethylene From the point of view of ab initio (first principles) electronic structure methods, a basis set is simply a collection of functions, whose members are typically associated with one or more of the atoms in a molecule. When people say that they are "using the 3-21G basis on ethylene" they really mean that they're performing a calculation with the appropriate carbon and hydrogen 3-21G basis functions ("3-21G" is the just the name given to this basis set family by scientists who originally developed it) positioned at the two carbons and four hydrogens in C2H4, for a total of 26 functions.

Basis sets are a mathematical convenience because the quantum mechanical equations which describe the behavior of electrons in molecules are most easily solved by expanding the wavefunction or density in terms of a finite set. Only in specialized cases, such as diatomic molecules, has it proven computationally feasible to forego the use of basis sets in favor of fully numerical techniques.

N2 disassocation energy Along with the sophistication of the approach used in describing the correlated motions of the electrons in a molecule, basis sets represent one of the two primary user-selectable input parameters for ab initio programs such as Gaussian, GAMESS and NWChem. A poorly chosen basis set will typically lead to large inaccuracies in the computed results or, in some cases, qualitatively incorrect findings. A simple example is the dissociation energy of N2. The experimental value of De is 228 kcal/mol, whereas small basis set RHF predicts a value of 39 kcal/mol. Larger basis sets, used with highly correlated methods can come within 1 - 2 kcal/mol of experiment. Another example is the hydronium cation, H3O+, which has a pyramidal shape like ammonia. With small basis sets this molecule is incorrectly predicted to be flat.

Some basis sets consist of relatively few functions. For example, the STO-3G basis has only one function per occupied atomic orbital (1s, 2s, 2px, 2py, 2pz). Others have a large number of functions of different symmetries (e.g. s, p, d, f...). Basis sets which are too small may lack the flexibility to describe the basic physics of a problem and can produce qualitatively misleading results, with no hint of trouble. Likewise, overly large basis sets may waste many hours of computer time.

Gaussian graph


Over the years theoretical chemists have used a variety of different functional forms as basis functions. Some of the earliest calculations were done with exponential functions that mimicked the atomic hydrogen orbitals. However, for practical purposes, nearly all of today's ab initio calculations on polyatomic molecules use Cartesian Gaussians of the form:

g(r) = N*(x l)*(y m)*(zeta n)*exp(-zeta*r 2),

where N is a normalization constant which insures that the square of the Gaussian gives a value of 1.0 when integrated over all space, (l,m,n) are integer powers of the electron's Cartesian coordinates ranging from 0 to some small positive value, and zeta is an exponent which helps determine the radial size of the function. The variable r represents the distance of the electron from the origin of the Gaussian.

Functions with L = l+m+n = 0 are spherically symmetric about the origin and are known as "s" functions. Similary, the three functions corresponding to l+m+n = 1 are the p(x), p(y), p(z) functions, etc. The Cartesian Gaussians possess six functions with l+m+n = 2, from which the five spherical components, d(xy), d(xz), d(yz), d(xx-yy) and d(2zz-xx-yy), can be constructed. The remaining function is of spherical symmetry and is customarily deleted. As the total L value increases, the difference in the number of Cartesian and spherical components increases. Many electronic structure programs are able to handle either form.

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Revised: November 3, 2002