Basis set (chemistry)
Encyclopedia
A basis set in chemistry
is a set of functions used to create the molecular orbital
s, which are expanded as a linear combination of such functions with the weights or coefficients to be determined. Usually these functions are atomic orbital
s, in that they are centered on atoms. Otherwise, the functions are centered on bonds or lone pairs. Pairs of functions centered in the two lobes of a p orbital have also been used. Additionally, basis sets composed of sets of plane wave
s down to a cutoff wavelength are often used, especially in calculations involving systems with periodic boundary conditions
.
, quantum chemical
calculations are typically performed within a finite set of basis function
s. In these cases, the wavefunction
s 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 orbital
s, 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 orbital
s 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 second period 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 (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 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 in Pople-type sets by a plus sign, +, and in Dunning-type 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.
, 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:
There are several other minimum basis sets that have been used such as the MidiX basis sets.
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:
The 6-31G* basis set (defined for the atoms H through Zn) is a valence double-zeta polarized basis set that adds to the 6-31G set six d-type Cartesian-Gaussian polarization functions on each of the atoms Li through Ca and ten f-type Cartesian Gaussian polarization functions on each of the atoms Sc through Zn.
calculations. Examples of these are:
For period-3 atoms (Al-Ar), additional functions are necessary; these are the cc-pV(N+d)Z basis sets. Even larger atoms may employ pseudopotential basis sets, cc-pVNZ-PP, or relativistic-contracted Douglas-Kroll basis sets, cc-pVNZ-DK.
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 energetic property, although care must be taken when extrapolating energy differences as the individual energy components may converge at different rates.
To understand how to get the number of functions take the cc-pVDZ basis set for H:
There are two s (L = 0) orbitals and one p (L = 1) orbital that has 3 components along the z-axis (mL = -1,0,1) corresponding to px, py and pz. Thus, five spatial orbitals in total. Note that each orbital can hold two electrons of opposite spin.
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 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 in a smooth, monotonic manner to the target wavefunction, while there is only a guarantee of monotonic convergence for all Gaussian-type basis sets when used in variational calculations. (An exception to the latter point is the correlation consistent basis sets.)
, 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.
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
is a set of functions used to create the molecular orbital
Molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The term "orbital" was first...
s, which are expanded as a linear combination of such functions with the weights or coefficients to be determined. Usually these functions are atomic orbital
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
s, in that they are centered on atoms. Otherwise, the functions are centered on bonds or lone pairs. Pairs of functions centered in the two lobes of a p orbital have also been used. Additionally, basis sets composed of sets of plane wave
Plane wave
In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....
s down to a cutoff wavelength are often used, especially in calculations involving systems with periodic boundary conditions
Periodic boundary conditions
In mathematical models and computer simulations, periodic boundary conditions are a set of boundary conditions that are often used to simulate a large system by modelling a small part that is far from its edge...
.
Introduction
In modern computational chemistryComputational chemistry
Computational chemistry is a branch of chemistry that uses principles of computer science to assist in solving chemical problems. It uses the results of theoretical chemistry, incorporated into efficient computer programs, to calculate the structures and properties of molecules and solids...
, quantum chemical
Quantum chemistry
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
calculations are typically performed within a finite set of basis function
Basis function
In mathematics, a basis function is an element of a particular basis for a function space. Every continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis...
s. In these cases, the wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
s 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
Operator (physics)
In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
are then represented as matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
, (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 orbital
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
s, centered at each atomic nucleus within the molecule (linear combination of atomic orbitals ansatz
Ansatz
Ansatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-Definition:...
). 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
S. Francis Boys
Samuel Francis Boys FRS was born in Pudsey, Yorkshire, England. He was educated at the Grammar School in Pudsey and then at Imperial College, London. He graduated in Chemistry in 1932. He did his Ph.D...
that these Slater-type orbitals could in turn be approximated as linear combinations of Gaussian orbital
Gaussian orbital
In computational chemistry and molecular physics, Gaussian orbitals are functions used as atomic orbitals in the LCAO method for the computation of electron orbitals in molecules and numerous properties that depend on these.- Rationale :The principal reason for the use of Gaussian basis functions...
s instead. Because it is easier to calculate overlap and other integrals with Gaussian basis functions, this led to huge computational savings (see John Pople
John Pople
Sir John Anthony Pople, KBE, FRS, was a Nobel-Prize winning theoretical chemist. Born in Burnham-on-Sea, Somerset, England, he attended Bristol Grammar School. He won a scholarship to Trinity College, Cambridge in 1943. He received his B. A. in 1946. Between 1945 and 1947 he worked at the Bristol...
).
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
Hartree-Fock
In computational physics and chemistry, the Hartree–Fock method is an approximate method for the determination of the ground-state wave function and ground-state energy of a quantum many-body system....
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).
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
Polarization
Polarization is a property of certain types of waves that describes the orientation of their oscillations. Electromagnetic waves, such as light, and gravitational waves exhibit polarization; acoustic waves in a gas or liquid do not have polarization because the direction of vibration and...
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 in Pople-type sets by a plus sign, +, and in Dunning-type 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 STO-nGSTO-nG basis sets
STO-nG basis sets are minimal basis sets, where n primitive Gaussian orbitals are fitted to a single Slater-type orbital . n originally took the values 2 - 6. They were first proposed by John Pople. A minimum basis set is where only sufficient orbitals are used to contain all the electrons in the...
, 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, quadruple-zeta, and so on, basis sets. 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 split-valence basis sets arising from the group of John PopleJohn Pople
Sir John Anthony Pople, KBE, FRS, was a Nobel-Prize winning theoretical chemist. Born in Burnham-on-Sea, Somerset, England, he attended Bristol Grammar School. He won a scholarship to Trinity College, Cambridge in 1943. He received his B. A. in 1946. Between 1945 and 1947 he worked at the Bristol...
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
- 4-21G
- 4-31G
- 6-21G
- 6-31G
- 6-31G*
- 6-31+G*
- 6-31G(3df, 3pd)
- 6-311G
- 6-311G*
- 6-311+G*
The 6-31G* basis set (defined for the atoms H through Zn) is a valence double-zeta polarized basis set that adds to the 6-31G set six d-type Cartesian-Gaussian polarization functions on each of the atoms Li through Ca and ten f-type Cartesian Gaussian polarization functions on each of the atoms Sc through Zn.
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 empirical 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-FockPost-Hartree-Fock
In computational chemistry, post-Hartree–Fock methods are the set of methods developed to improve on the Hartree–Fock , or self-consistent field method...
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 period-3 atoms (Al-Ar), additional functions are necessary; these are the cc-pV(N+d)Z basis sets. Even larger atoms may employ pseudopotential basis sets, cc-pVNZ-PP, or relativistic-contracted Douglas-Kroll basis sets, cc-pVNZ-DK.
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 energetic property, although care must be taken when extrapolating energy differences as the individual energy components may converge at different rates.
| H-He | Li-Ne | Na-Ar | |
|---|---|---|---|
| cc-pVDZ | [2s1p] → 5 func. | [3s2p1d] → 14 func. | [4s3p2d] → 23 func. |
| cc-pVTZ | [3s2p1d] → 14 func. | [4s3p2d1f] → 30 func. | [5s4p3d1f] → 39 func. |
| cc-pVQZ | [4s3p2d1f] → 30 func. | [5s4p3d2f1g] → 55 func. | [6s5p4d2f1g] → 64 func. |
To understand how to get the number of functions take the cc-pVDZ basis set for H:
There are two s (L = 0) orbitals and one p (L = 1) orbital that has 3 components along the z-axis (mL = -1,0,1) corresponding to px, py and pz. Thus, five spatial orbitals in total. Note that each orbital can hold two electrons of opposite spin.
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 polarization
- QZVPP - Valence quadruple-zeta plus polarization
Plane-wave basis sets
In addition to localized basis sets, plane-wavePlane wave
In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....
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
Periodic boundary conditions
In mathematical models and computer simulations, periodic boundary conditions are a set of boundary conditions that are often used to simulate a large system by modelling a small part that is far from its edge...
. 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
Pseudopotential
In physics, a pseudopotential or effective potential is used as an approximation for the simplified description of complex systems. Applications include atomic physics and neutron scattering.- Atomic physics :...
, 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 wavelength, is used. This combined method of a plane-wave basis set with a core pseudopotential
Pseudopotential
In physics, a pseudopotential or effective potential is used as an approximation for the simplified description of complex systems. Applications include atomic physics and neutron scattering.- Atomic physics :...
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
Basis set superposition error
In quantum chemistry, calculations of molecular properties are susceptible to basis set superposition error if they use finite basis sets. As the atoms of interacting molecules approach one another, their basis functions overlap...
. 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 in a smooth, monotonic manner to the target wavefunction, while there is only a guarantee of monotonic convergence for all Gaussian-type basis sets when used in variational calculations. (An exception to the latter point is the correlation consistent basis sets.)
Real-space 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 differenceFinite difference
A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
, 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
- Basis set superposition errorBasis set superposition errorIn quantum chemistry, calculations of molecular properties are susceptible to basis set superposition error if they use finite basis sets. As the atoms of interacting molecules approach one another, their basis functions overlap...
- Angular momentumAngular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
- Atomic orbitals
- Molecular orbitals
- List of quantum chemistry and solid state physics software