Gaussian Basis Sets - Overview

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.

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.

Terminology

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.

• Exponents - Exponents are the prefactors multiplying r ² in the exponential part of the Gaussian. Exponents are often represented by the Greek character zeta. A small exponent, e.g. 0.01 will produce a diffuse function, whereas a large exponent, e.g. 10,000, makes for a function which is very tight about the origin (usually an atom).

• Contraction Coefficients - Compared to the radial solutions of the hydrogen atom that are described in most undergraduate chemistry texts, individual Gaussians have the wrong behavior at the origin and die off too rapidly at large distances. However, by combining several individual Gaussians (sometimes referred to as "primitives" or "Gaussian primitives" into fixed linear combinations of functions known as "contracted" functions, it is possible to mimic the shape of the hydrogenic functions, which fall off as e-(zeta*r) with distance. Because this reduces the number of variational degrees of freedom, the cost of the calculation, especially post Hartree-Fock calculations, such as perturbation theory or configuration interaction, are reduced with only a modest impact on the accuracy.

  There are two types of contracted basis functions in widespread use today. In "segemented" contractions a given primitive function appears in no more than one contracted function. Two examples of segmented contractions are the STO-3G and the Dunning DZ basis sets, the latter of which is shown here for the carbon atom.

  

  Segmented contractions became more difficult to construct as you went further down the periodic table. This led to the development of another way of contracting Gaussian primitives.

  The second category of contracted basis function is referred to as a "general" contraction. In general contractions a given primitive can appear in more than one contracted function. Examples of generally contracted basis sets are the NASA Ames Atomic Natural Orbital (ANO) and the Roos ANO basis sets. An example from the latter family is shown to the right. Generally contracted basis sets have a reputation for being more time consuming to use than their segmented counterparts. This is partially due to the relative scarcity of integral programs which are designed to handle these basis functions efficiently. Although it is possible to use general contractions with any integral program, there will be a penalty to pay with some codes. For example, if there are 10 generally contracted s-symmetry basis functions defined in terms of 20 Gaussian primitives, some codes will interpret this situation as a calculation over 10 x 20 = 200 primitives.

  Finally, some basis sets possess characteristics of both contraction styles. For the sake of simplicity, these sets are normally referred to as generally contracted basis sets. The new correlation consistent basis set family developed by T. H. Dunning, Jr. and co-workers is an example of a hybrid contraction style.

  With this last category of contraction it is sometimes possible to reformat the basis set so that the performance penalty you're forced to pay for using generally contracted functions in programs that weren't designed for them is minimized. In the Ecce Basis Set Tool we refer to this reformatting as "optimizing" the general contraction.

  

• Polarization Functions - Gaussians of a higher symmetry than the ground state occupied atomic orbitals are referred to as polarization functions. For example, since the first row elements (Li - Ne) possess occupied s and p atomic orbitals, d, f, g, etc. functions would be classified as polarization functions. The qualitative importance of polarization functions is that they permit the molecular wavefunction more flexibility to distort away from spherical symmetry in the neighborhood of each atom.

  The general shapes of the spherical harmonic forms of the basis functions up through l = 3 (f functions) is shown here.

  

  From a practical perspective, the first polarization functions (e.g. a set of d functions on carbon) are the most important additions one can make to the basis set beyond the valence s and p functions. At the Hartree-Fock level of theory, most properties converge to the complete basis set limit relatively quickly with the addition of more polarization functions. However, at the correlated level of theory the convergence is typically much slower, so that many higher l functions are needed in order to reach the complete basis set limit. In particularly difficult cases, such as the dissociation energy of N2, basis sets containing d and f polarization functions still underestimate the true value of De by more than 10 kcal/mol.

  As is evident from these pictures, as the l value increases, so does the number of angular nodes (places where the orbital changes sign).

  

  The l = 4 (g) functions have 4 such nodal planes. Although g functions are not shown, you can imagine that their 3D shapes have gotten to be quite complex and there are a lot of them. Because of the large number of g functions and the fact that integrals over g functions are time consuming to compute, relatively few polyatomic calculations are performed with these functions.

  Some programs can perform calculations with two different forms of the higher l value Gaussians. There are l*(l+1)/2 + (l+1) Cartesian Gaussians, but only 2*l+1 Spherical Harmonic functions for a given value of l. Thus, for l greater than 1 (p functions) there are more Cartesian than Spherical components. The relative numbers of Cartesian and Spherical Gaussians are:
  d functions (l=2) 6 Cart. and 5 Spher.,
  f functions (l=3) 10 Cart. and 7 Spher.,
  g functions (l=4) 15 Cart. and 9 Spher.,
  h functions (l=5) 21 Cart. and 11 Spher.

  

  A wide range of special polarization sets have been designed for various properties or types of calculations.

• Orbital Basis Sets - In the Ecce Basis Set Tool we distinguish between "Orbital", "Effective Core Potential" (ECP) and the auxiliary "DFT Fitting" basis sets that are used in some density functional calculations. As the name implies, "orbital" basis sets are used to expand the molecular orbitals of a chemical system. This category of basis set has received a great deal of attention over the past 30 years, resulting in a very large number of orbital sets in the chemistry literature.

• Effective Core Basis Sets - These basis sets are designed for use in calculations that replace the inner core electrons, e.g. 1s electrons in carbon, with special projectors that prevent variational collapse of the remaining electrons. Typically, when ECPs are published there are accompanying special basis sets that are designed to work with the ECPs. However, in principal it should be possible to use any orbital basis set (minus the core functions) with any ECP.

• Fitting Basis Sets - In order to obtain efficiency in density functional calculations, auxilliary uncontracted basis sets are used to fit the electron (i.e. charge) density and the exchange-correlation functional. The area of DFT fitting basis sets is still in its infancy. Thus, there are relatively few such sets available in the literature.