Basis set (chemistry)

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A basis set in chemistry is a set of functions used to create the molecular orbitals, which are expanded as a linear combination of such functions with the weights or coefficients to be determined. Usually these functions are atomic orbitals, in that they are centered on atoms, but functions centered in bonds or lone pairs have been used as have pairs of functions centered in the two lobes of a p orbital.

1 Introduction
2 Localized Gaussian basis sets
3 Plane wave basis sets
4 See also
5 External links

[edit] Introduction

In modern computational chemistry, quantum chemical calculations are typically performed within a finite set of basis functions. In these cases, the wavefunctions under consideration are all represented as vectors, the components of which correspond to coefficients in a linear combination of the basis functions in the basis set used. The operators are then represented as matrices, (rank two tensors), in this finite basis. In this article, basis function and atomic orbital are sometimes used interchangeably, although it should be noted that these basis functions are usually not actually the exact atomic orbitals, even for the corresponding hydrogen-like atoms, due to approximations and simplifications of their analytic formulas.

When molecular calculations are performed, it is common to use a basis composed of a finite number of atomic orbitals, centered at each atomic nucleus within the molecule (linear combination of atomic orbitals ansatz). Initially, these atomic orbitals were typically Slater orbitals, which corresponded to a set of functions which decayed exponentially with distance from the nuclei. Later, it was realized that these Slater-type orbitals could in turn be approximated as linear combinations of Gaussian orbitals instead. Because it is easier to calculate overlap and other integrals with Gaussian basis functions, this led to huge computational savings (see John Pople).

Today, there are hundreds of basis sets composed of Gaussian-type orbitals (GTOs). The smallest of these are called minimal basis sets, and they are typically composed of the minimum number of basis functions required to represent all of the electrons on each atom. The largest of these can contain literally dozens to hundreds of basis functions on each atom.

The most common addition to minimal basis sets is probably the addition of polarization functions, denoted by an asterisk, *. Two asterisks, **, indicate that polarization functions are also added to light atoms (hydrogen and helium). These are auxiliary functions with one additional node. For example, the only basis function located on a hydrogen atom in a minimal basis set would be a function approximating the 1s atomic orbital. When polarization is added to this basis set, a p-function is also added to the basis set. This adds some additional needed flexibility within the basis set, effectively allowing molecular orbitals involving the hydrogen atoms to be more asymmetric about the hydrogen nucleus. This is an important result when considering accurate representations of bonding between atoms, because the very presence of the bonded atom makes the energetic environment of the electrons spherically asymmetric. Similarly, d-type functions can be added to a basis set with valence p orbitals, and f-functions to a basis set with d-type orbitals, and so on. Another, more precise notation indicates exactly which and how many functions are added to the basis set, such as (p, d).

Another common addition to basis sets is the addition of diffuse functions, denoted by a plus sign, +. Two plus signs indicate that diffuse functions are also added to light atoms (hydrogen and helium). These are very shallow Gaussian basis functions, which more accurately represent the "tail" portion of the atomic orbitals, which are distant from the atomic nuclei. These additional basis functions can be important when considering anions and other large, "soft" molecular systems.

[edit] Localized Gaussian basis sets

[edit] Minimal basis sets

A common naming convention for minimal basis sets is STO-XG, where X is an integer. This X value represents the number of Gaussian primitive functions comprising a single basis function. In these basis sets, the same number of Gaussian primitives comprise core and valence orbitals. Minimal basis sets typically give rough results that are insufficient for research-quality publication, but are much cheaper than their larger counterparts. Here is a list of commonly used minimal basis sets:

STO-2G
STO-3G
STO-6G
STO-3G* - Polarized version of STO-3G

[edit] Split-valence basis sets

During most molecular bonding, it is the valence electrons which principally take part in the bonding. In recognition of this fact, it is common to represent valence orbitals by more than one basis function, (each of which can in turn be composed of a fixed linear combination of primitive Gaussian functions). The notation for these split-valence basis sets is typically X-YZg. In this case, X represents the number primitive Gaussians comprising each core atomic orbital basis function. The Y and Z indicate that the valence orbitals are composed of two basis functions each, the first one composed of a linear combination of Y primitive Gaussian functions, the other composed of a linear combination of Z primitive Gaussian functions. In this case, the presence of two numbers after the hyphens implies that this basis set is a split-valence double-zeta basis set. Split-valence triple- and quadruple-zeta basis sets are also used, denoted as X-YZWg, X-YZWVg, etc. Here is a list of commonly used split-valence basis sets:

3-21g
3-21g* - Polarized
3-21+g - Diffuse functions
3-21+g* - With polarization and diffuse functions
6-31g
6-31g*
6-31+g*
6-31g(3df, 3pd)
6-311g
6-311g*
6-311+g*
SV(P)
SVP

[edit] Double, triple, quadruple zeta basis sets

Basis sets in which there are multiple basis functions corresponding to each atomic orbital, including both valence orbitals and core orbitals or just the valence orbitals, are called double, triple, or quadruple-zeta basis sets. Here is a list of commonly used multiple zeta basis sets:

cc-pVDZ - Double-zeta
cc-pVTZ - Triple-zeta
cc-pVQZ - Quadruple-zeta
cc-pV5Z - Quintuple-zeta, etc.
aug-cc-pVDZ, etc. - Augmented versions of the preceding basis sets with added diffuse functions
TZVPP - Triple-zeta
QZVPP - Quadruple-zeta

The 'cc-p' at the beginning of some of the above basis sets stands for 'correlation consistent polarized' basis sets. They are double/triple/quadruple/quintuple-zeta for the valence orbitals only {the 'V' stands for valence) and include successively larger shells of polarization (correlating) functions (d, f, g, etc.) that can yield convergence of the electronic energy to the complete basis set limit. These basis sets are the current state of the art for correlated or post Hartree-Fock calculations.

[edit] Plane wave basis sets

In addition to localized basis sets, plane wave basis sets can also be used in quantum chemical simulations. Typically, a finite number of plane wave functions are used, below a specific cutoff energy which is chosen for a certain calculation. These basis sets are popular in calculations involving periodic boundary conditions. Certain integrals and operations are much easier to code and carry out with plane wave basis functions, than with their localized counterparts; furthermore, as all functions in the basis are mutually orthogonal, plane wave basis sets do not exhibit basis set superposition error. However, they are less well suited to gas-phase calculations.