Gaussian orbital
Encyclopedia
In computational chemistry
and molecular physics
, Gaussian orbitals (also known as Gaussian type orbitals, GTOs or Gaussians) are function
s used as atomic orbital
s in the LCAO method
for the computation of electron orbital
s in molecule
s and numerous properties that depend on these.
in molecular quantum chemical calculations is the 'Gaussian Product Theorem', which guarantees that the product of two GTOs centered on two different atoms is a finite sum of Gaussians centered on a point along the axis connecting them. In this manner, four-center integrals can be reduced to finite sums of two-center integrals, and in a next step to finite sums of one-center integrals. The speedup by 4--5 orders of magnitude compared to Slater orbitals more than outweighs the extra cost entailed by the larger number of basis functions generally required in a Gaussian calculation.
For reasons of convenience, many Gaussian integral evaluation programs work in a basis of Cartesian Gaussians even when spherical Gaussians are requested: the 'contaminants' are deleted a posteriori.
Slater radial orbitals read
Gaussian primitives for radial orbitals read
where
is normalisation constant.
The database determines shapes of (atom-centered) basis functions (in fact radial part of these function) optimised for one or the other criteria.
The shapes are described as tables of factors and coefficients. The factors and coefficient define Gaussian primitives, which needs to be contracted (summed over) to determine a radial orbital. The basis functions possess a usual radial-angular decomposition
where
is spherical harmonic functions, 
and 
are angular momentum and projection of angular momentum quantum numbers, 
are spherical coordinates. The function 
is the radial orbital we want to determine.
Radial orbital
is a sum of Gaussian primitives with an angular momentum factor in front of expression
where
and 
are listed in the tables of EMSL portal, the radial coordinate 
must be in atomic units (see Bohr radius
) and
are normalisation factors to ensure the norm of Gaussian primitives 
is one
A closed form expression for normalisation factors
can be obtained using Gaussian integrals
Particular values of the normalisation factor for different
are
and Hehre (1978) developed a local coordinate method. Obara and Saika introduced efficient recursion relations in 1985, which was followed by the development of other important recurrence relations. Gill and Pople (1990) introduced a 'prism' algorithm which allowed efficient use of 20 different calculation paths.
Computational 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...
and molecular physics
Molecular physics
Molecular physics is the study of the physical properties of molecules, the chemical bonds between atoms as well as the molecular dynamics. Its most important experimental techniques are the various types of spectroscopy...
, Gaussian orbitals (also known as Gaussian type orbitals, GTOs or Gaussians) are function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s used as 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 the LCAO method
Linear combination of atomic orbitals molecular orbital method
A linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry. In quantum mechanics, electron configurations of atoms are described as wavefunctions...
for the computation of electron orbital
Electron orbital
An electron orbital may refer to:* An atomic orbital, describing the behaviour of an electron in an atom* A molecular orbital, describing the behaviour of an electron in a molecule- See also :...
s in molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
s and numerous properties that depend on these.
Rationale
The principal reason for the use of Gaussian basis functionsBasis 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. Otherwise, the...
in molecular quantum chemical calculations is the 'Gaussian Product Theorem', which guarantees that the product of two GTOs centered on two different atoms is a finite sum of Gaussians centered on a point along the axis connecting them. In this manner, four-center integrals can be reduced to finite sums of two-center integrals, and in a next step to finite sums of one-center integrals. The speedup by 4--5 orders of magnitude compared to Slater orbitals more than outweighs the extra cost entailed by the larger number of basis functions generally required in a Gaussian calculation.
For reasons of convenience, many Gaussian integral evaluation programs work in a basis of Cartesian Gaussians even when spherical Gaussians are requested: the 'contaminants' are deleted a posteriori.
Slater radial orbitals read
Gaussian primitives for radial orbitals read
-
,
where
is normalisation constant.Using tables of Gaussian orbitals
There is a large database of Gaussian orbitals EMSL portal.The database determines shapes of (atom-centered) basis functions (in fact radial part of these function) optimised for one or the other criteria.
The shapes are described as tables of factors and coefficients. The factors and coefficient define Gaussian primitives, which needs to be contracted (summed over) to determine a radial orbital. The basis functions possess a usual radial-angular decomposition
-
,
where
is spherical harmonic functions, and are angular momentum and projection of angular momentum quantum numbers, are spherical coordinates. The function is the radial orbital we want to determine.Radial orbital
is a sum of Gaussian primitives with an angular momentum factor in front of expression
-
,
where
and are listed in the tables of EMSL portal, the radial coordinate must be in atomic units (see Bohr radiusBohr radius
The Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom...
) and
are normalisation factors to ensure the norm of Gaussian primitives is one
A closed form expression for normalisation factors
can be obtained using Gaussian integrals
Particular values of the normalisation factor for different
are
Molecular integrals
Molecular integrals over Cartesian Gaussian functions were first proposed by Boys in 1950. Since then much work has been done to speed up the evaluation of these integrals which are the slowest part of many quantum chemical calculations. McMurchie and Davidson (1978) introduced Hermite Gaussian functions to take advantage of differential relations. 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...
and Hehre (1978) developed a local coordinate method. Obara and Saika introduced efficient recursion relations in 1985, which was followed by the development of other important recurrence relations. Gill and Pople (1990) introduced a 'prism' algorithm which allowed efficient use of 20 different calculation paths.
The POLYATOM System
The POLYATOM System was the first package for ab initio calculations using Gaussian orbitals that was applied to a wide variety of molecules. It was developed in Slater's Solid State and Molecular Theory Group (SSMTG) at MIT using the resources of the Cooperative Computing Laboratory. The mathematical infrastructure and operational software were developed by Imre Csizmadia, Malcolm Harrison, Jules Moskowitz and Brian Sutcliffe.External links
- A visualization of all common and uncommon atomic orbitals, from 1s to 7g (Note that the radial part of the expressions given corresponds to Slater orbitals rather than Gaussians. The angular parts, and hence their shapes as displayed in figures, are the same as those of spherical Gaussians.)