A basis set in theoretical and computational chemistry is a set of functions (called basis functions) which are combined in linear combinations (generally as part of a quantum chemical calculation) to create molecular orbitals. For convenience these functions are typically atomic orbitals centered on atoms, but can theoretically be any function; plane waves are frequently used in materials calculations.
Introduction
In modern computational chemistry, quantum chemical calculations are typically performed using a finite set of basis functions. In these cases, the wavefunctions of the system in question are 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 hydrogenlike 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.^{[1]}
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). These atomic orbitals are well described with Slatertype orbitals (STOs), as STOs decay exponentially with distance from the nuclei, accurately describing the longrange overlap between atoms, and reach a maximum at zero, well describing the charge and spin at the nucleus. STOs are computationally difficult and it was later realized by Frank Boys that these Slatertype 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 Gaussiantype 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 second period of the periodic system (Li  Ne) would have a basis set of five functions (two s functions and three p functions).
A dpolarization function added to a p orbital^{[2]}
The most common addition to minimal basis sets is probably the addition of polarization functions, denoted (in the names of basis sets developed by Pople) 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 pfunction 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, dtype functions can be added to a basis set with valence p orbitals, and ffunctions to a basis set with dtype orbitals, and so on. Another, more precise notation indicates exactly which and how many functions are added to the basis set, such as (d, p).
Another common addition to basis sets is the addition of diffuse functions, denoted in Popletype sets by a plus sign, +, and in Dunningtype sets by "aug" (from "augmented"). 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 STOnG, 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 researchquality publication, but are much cheaper than their larger counterparts. Commonly used minimal basis sets of this type are:

STO3G

STO4G

STO6G

STO3G*  Polarized version of STO3G
There are several other minimum basis sets that have been used such as the MidiX basis sets.
Splitvalence 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, quadruplezeta, and so on, basis sets (zeta, ζ, was commonly used to represent the exponent of an STO basis function^{[3]}). Since the different orbitals of the split have different spatial extents, the combination allows the electron density to adjust its spatial extent appropriate to the particular molecular environment. Minimum basis sets are fixed and are unable to adjust to different molecular environments.
Pople basis sets
The notation for the splitvalence basis sets arising from the group of John Pople is typically XYZg.^{[4]} 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 splitvalence doublezeta basis set. Splitvalence triple and quadruplezeta basis sets are also used, denoted as XYZWg, XYZWVg, etc. Here is a list of commonly used splitvalence basis sets of this type:

321G

321G*  Polarized

321+G  Diffuse functions

321+G*  With polarization and diffuse functions

421G

431G

621G

631G

631G*

631+G*

631G(3df, 3pd)

6311G

6311G*

6311+G*
The 631G* basis set (defined for the atoms H through Zn) is a valence doublezeta polarized basis set that adds to the 631G set six dtype CartesianGaussian polarization functions on each of the atoms Li through Ca and ten ftype Cartesian Gaussian polarization functions on each of the atoms Sc through Zn.
Correlationconsistent basis sets
Some of the most widely used basis sets are those developed by Dunning and coworkers,^{[5]} since they are designed to converge systematically to the completebasisset (CBS) limit using empirical extrapolation techniques. For first and secondrow atoms, the basis sets are ccpVNZ where N=D,T,Q,5,6,... (D=double, T=triples, etc.). The 'ccp', stands for 'correlationconsistent polarized' and the 'V' indicates they are valenceonly basis sets. They include successively larger shells of polarization (correlating) functions (d, f, g, etc.). More recently these 'correlationconsistent polarized' basis sets have become widely used and are the current state of the art for correlated or postHartree–Fock calculations. Examples of these are:

ccpVDZ  Doublezeta

ccpVTZ  Triplezeta

ccpVQZ  Quadruplezeta

ccpV5Z  Quintuplezeta, etc.

augccpVDZ, etc.  Augmented versions of the preceding basis sets with added diffuse functions.
For period3 atoms (AlAr), additional functions are necessary; these are the ccpV(N+d)Z basis sets. Even larger atoms may employ pseudopotential basis sets, ccpVNZPP, or relativisticcontracted DouglasKroll basis sets, ccpVNZDK.
These basis sets can be augmented with core functions for geometric and nuclear property calculations, and with diffuse functions for electronic excitedstate calculations, electric field property calculations, and longrange 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 energetic property, although care must be taken when extrapolating energy differences as the individual energy components may converge at different rates.

HHe

LiNe

NaAr

ccpVDZ

[2s1p] → 5 func.

[3s2p1d] → 14 func.

[4s3p1d] → 18 func.

ccpVTZ

[3s2p1d] → 14 func.

[4s3p2d1f] → 30 func.

[5s4p2d1f] → 34 func.

ccpVQZ

[4s3p2d1f] → 30 func.

[5s4p3d2f1g] → 55 func.

[6s5p3d2f1g] → 59 func.

To understand how to get the number of functions take the ccpVDZ basis set for H: There are two s (L = 0) orbitals and one p (L = 1) orbital that has 3 components along the zaxis (m_{L} = 1,0,1) corresponding to p_{x}, p_{y} and p_{z}. Thus, five spatial orbitals in total. Note that each orbital can hold two electrons of opposite spin.
For example, Ar [1s, 2s, 2p, 3s, 3p] has 3 s orbitals (L=0) and 2 sets of p orbitals (L=1). Using ccpVDZ, orbitals are [1s, 2s, 2p, 3s, 3s', 3p, 3p', 3d'] (where ' represents the added in polarisation orbitals), with 4 s orbitals, 3 sets of p orbitals and 1 set of d orbitals.
Other splitvalence basis sets
Some other splitvalence (SV) basis sets are :

SV(P)

SVP

DZV  Valence doublezeta

TZV  Valence triplezeta

TZVPP  Valence triplezeta plus polarization

QZVPP  Valence quadruplezeta plus polarization
Planewave basis sets
In addition to localized basis sets, planewave basis sets can also be used in quantumchemical simulations. Typically, a finite number of planewave 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 planewave basis functions than with their localized counterparts.
In practice, planewave 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 planewave basis set unless a very high energy cutoff, and therefore small wavelength, is used. This combined method of a planewave basis set with a core pseudopotential is often abbreviated as a PSPW calculation.
Furthermore, as all functions in the basis are mutually orthogonal and are not associated with any particular atom, planewave basis sets do not exhibit basisset superposition error. However, they are less well suited to gasphase calculations. Using Fast Fourier Transforms, one can work with planewave 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 planewave basis is that it is guaranteed to converge in a smooth, monotonic manner to the target wavefunction, while there is only a guarantee of monotonic convergence for all Gaussiantype basis sets when used in variational calculations. (An exception to the latter point is the correlation consistent basis sets.) The properties of the Fourier Transform allow a vector representing the gradient of the total energy with respect to the planewave coefficients to be calculated with a computational effort that scales as NPW*ln(NPW) where NPW is the number of planewaves. When this property is combined with separable pseudopotentials of the KleinmanBylander type and preconditioned conjugate gradient solution techniques, the dynamic simulation of periodic problems containing hundreds of atoms becomes possible.
Realspace basis sets
On the same principle as the plane waves but in real space, there are basis sets whose functions are centered on a uniform mesh in real space. This is the case for the finite difference, the functions sinc or wavelets. In the case of the latter, it is possible to have an adaptive mesh closer to the nucleus using the scaling properties of wavelets. These methods use functions that are localized which allow the development of order N methods.
See also
References

^ Roman M. Balabin (2010). "Intramolecular basis set superposition error as a measure of basis set incompleteness: Can one reach the basis set limit without extrapolation?".

^ Errol G. Lewars. Computational Chemistry: Introduction to the Theory and Applications of Molecular and Quantum Mechanics (1st ed.). Springer.

^

^ Ditchfield, R; Hehre, W.J; Pople, J. A. (1971). "Self‐Consistent Molecular‐Orbital Methods. IX. An Extended Gaussian‐Type Basis for Molecular‐Orbital Studies of Organic Molecules". J. Chem. Phys. 54 (2): 724–728.

^ Dunning, Thomas H. (1989). "Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen". J. Chem. Phys. 90 (2): 1007–1023.
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. pp. 461–466.

Cramer, Christopher J. (2002). Essentials of Computational Chemistry. Chichester: John Wiley & Sons, Ltd. pp. 154–168.

Jensen, Frank (1999). Introduction to Computational Chemistry. John Wiley and Sons. pp. 150–176.

Leach, Andrew R. (1996). Molecular Modelling: Principles and Applications. Singapore: Longman. pp. 68–77.

Hehre, Warren J.. (2003). A Guide to Molecular Mechanics and Quantum Chemical Calculations. Irvine, California: Wavefunction, Inc. pp. 40–47.

http://www.chem.swin.edu.au/modules/mod8/basis1.html

Moran, Damian; Simmonett, Andrew C.; Leach, Franklin E.; Allen, Wesley D.; Schleyer, Paul v. R.; Schaefer, Henry F. (2006). "Popular Theoretical Methods Predict Benzene and Arenes To Be Nonplanar". Journal of the American Chemical Society 128 (29): 9342–3.
External links

ChemViz  Basis Sets Lab Activity

EMSL Basis Set Exchange
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