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Basis set (chemistry)

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. Additionally, basis sets composed of sets of plane waves down to a cutoff wavelength are often used, especially in calculations involving systems with periodic boundary conditions.

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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. If the finite basis is expanded towards an infinite complete set of functions, calculations using such a basis set are said to approach the basis set limit.

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 by Frank Boys 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.

A minimum basis set is one in which, on each atom in the molecule, a single basis function is used for each orbital in a Hartree-Fock calculation on the free atom. However, for atoms such as lithium, basis functions of p type are added to the basis functions corresponding to the 1s and 2s orbitals of the free atom. For example, each atom in the first row of the periodic system (Li - Ne) would have a basis set of five functions (two s functions and three p functions).

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.

Minimal basis sets

The most common minimal basis set is STO-nG, where n is an integer. This n 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. Commonly used minimal basis sets of this type are:

STO-3G
STO-4G
STO-6G
STO-3G* - Polarized version of STO-3G

There are several other minimum basis sets that have been used such as the MidiX basis sets.

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). Basis sets in which there are multiple basis functions corresponding to each valence atomic orbital, are called valence double, triple, or quadruple-zeta basis sets. Since the different orbitals of the split have different spatial extents, the combination allows the electron density in adjust its spatial extend appropriate to the particular molecular environment. Minimum basis sets are fixed and are unable to adjust to different molecular environments. Basis sets in which there are multiple basis functions corresponding to each atomic orbital, including both valence orbitals and core orbitals are called double, triple, or quadruple-zeta basis sets.

Pople basis sets

The notation for the split-valence basis sets arising from the group of John Pople is typically X-YZg. In this case, X represents the number of 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 of this type:

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*

Correlation consistent basis sets

Some of the most widely used basis sets are those developed by Dunning and coworkers, since they are designed to converge systematically to the complete basis set (CBS) limit using extrapolation techniques. For first- and second-row atoms, the basis sets are cc-pVNZ where N=D,T,Q,5,6,... (D=double, T=triples, etc.). The 'cc-p', stands for 'correlation consistent polarized' and the 'V' indicates they are valence only basis sets. They include successively larger shells of polarization (correlating) functions (d, f, g, etc.). More recently these 'correlation consistent polarized' basis sets have become widely used and are the current state of the art for correlated or post Hartree-Fock calculations. Examples of these are:

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

For third-row atoms, additional functions are necessary; these are the cc-pV(N+d)Z basis sets. Even large atoms require the cc-pVNZ-PP and cc-pVNZ-DK families of basis sets, where PP and DK stand for pseudopotential and Douglas-Kroll, respectively.

These basis sets can be augmented with core functions for geometric and nuclear property calculations, and with diffuse functions for electronic excited-state calculations, electric field property calculations, and long-range interactions, such as van der Waals forces. A recipe for constructing additional augmented functions exists; as many as five augmented functions have been used in second hyperpolarizability calculations in the literature. Because of the rigorous construction of these basis sets, extrapolation can be done for almost any property.

Other split valence basis sets

Other split valence basis sets often have rather generic names such as:-

SV(P)
SVP
DZV
TZV
TZVPP - Valence triple-zeta plus polarisation
QZVPP - Valence quadruple-zeta plus polarisation

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. In practice, plane wave basis sets are often used in combination with an 'effective core potential' or pseudopotential, so that the plane waves are only used to describe the valence charge density. This is because core electrons tend to be concentrated very close to the atomic nuclei, resulting in large wavefunction and density gradients near the nuclei which are not easily described by a plane wave basis set unless a very high energy cutoff, (and therefore small plane wavelength), is used. This combined method of a plane wave basis set with a core pseudopotential is often abbreviated as a PSPW calculation. 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. Using Fast Fourier Transforms, one can work with plane wave basis sets in reciprocal space in which not only the aforementioned integrals, such as the kinetic energy, but also derivatives are computationally less demanding to be carried out. Another important advantage of a plane wave basis is that it is guaranteed to converge to the target wave function while there is no such guarantee for Gaussian type basis sets.

References

All the many basis sets discussed here along with others are discussed in the references below which themselves give references to the original journal articles:

Levine, Ira N. (1991). Quantum Chemistry. Englewood Cliffs, New jersey: Prentice Hall, 461 - 466. ISBN 0-205-12770-3.
Cramer, Christopher J. (2002). Essentials of Computational Chemistry. Chichester: John Wiley & Sons, Ltd., 154 - 168. ISBN 0-471-48552-7.
Jensen, Frank (1999). Introduction to Computational Chemistry. Chichester, England: John Wiley and Sons, 150 - 176. ISBN 0471984156.
Leach, Andrew R. (1996). Molecular Modelling: Principles and Applications. Singapore: Longman, 68 - 77. ISBN 0-582-23933-8.
http://www.chem.swin.edu.au/modules/mod8/basis1.html
Davidson, Ernest; Feller, David (1986). "Basis set selection for molecular calculations". Chemical Reviews 86 (4): 681 - 696.

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Basis_set_(chemistry)". A list of authors is available in Wikipedia.