Slater-type orbital
Encyclopedia
Slater-type orbitals are functions used as atomic orbital
s in the linear combination of atomic orbitals molecular orbital method
. They are named after the physicist John C. Slater
, who introduced them in 1930.
They possess exponential decay at long range and Kato's cusp condition at short range (when combined as Hydrogen-like functions i.e. the analytical solutions of the stationary Schrödinger for one electron atoms). Unlike the common hydrogen orbitals, STOs have no radial nodes (neither do gaussian-type orbitals).
where
The normalization constant is computed from the integral
Hence
It is common to use the spherical harmonics
depending on the polar coordinates
of the position vector
as the angular part of the Slater orbital.
The radial Laplace operator is split in two differential operators
The first differential operator of the Laplace operator yields
The total Laplace operator yields after applying the second differential operator
the result
Angular dependent derivatives of the spherical harmonics don't depend on the radial function and have to be evaluated separately.
kinetic energy, nuclear attraction and Coulomb repulsion integrals for
placement of the orbital at the center of a single nucleus. Dropping
the normalization factor N, the representation of the orbitals below is
.
The Fourier transform
is
,
where the
are defined by
.
The overlap integral is
of which the normalization integral is a special case. The starlet in the
superscript denotes complex-conjugation
.
The kinetic energy integral is
a sum over three overlap integrals already computed above.
The Coulomb repulsion integral can be evaluated using the Fourier representation
(see above)
which yields
These are either individually calculated with the law of residues
or recursively
as proposed by Cruz et al. (1978).
Analytical ab initio software for poly-atomic molecules has been developed e.g. STOP: a Slater Type Orbital Package in 1996.
SMILES uses analytical expressions when available and Gaussian expansions otherwise. It was first released in 2000.
Various grid integration schemes have been developed, sometimes after analytical work for quadrature (Scrocco). Most famously in the ADF suite of DFT codes.
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 linear combination of atomic orbitals molecular orbital 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...
. They are named after the physicist John C. Slater
John C. Slater
John Clarke Slater was a noted American physicist who made major contributions to the theory of the electronic structure of atoms, molecules and solids. This work is of ongoing importance in chemistry, as well as in many areas of physics. He also made major contributions to microwave electronics....
, who introduced them in 1930.
They possess exponential decay at long range and Kato's cusp condition at short range (when combined as Hydrogen-like functions i.e. the analytical solutions of the stationary Schrödinger for one electron atoms). Unlike the common hydrogen orbitals, STOs have no radial nodes (neither do gaussian-type orbitals).
Definition
STOs have the following radial part:
where
- n is a natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
that plays the role of principal quantum numberPrincipal quantum numberIn atomic physics, the principal quantum symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The principal quantum number can only have positive integer values...
, n = 1,2,..., - N is a normalizing constantNormalizing constantThe concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics.-Definition and examples:In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g.,...
, - r is the distance of the electron from the atomic nucleusAtomic nucleusThe nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
, and -
is a constant related to the effective chargeElectric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
of the nucleus, the nuclear charge being partly shielded by electrons.
The normalization constant is computed from the integral
Hence
It is common to use the spherical harmonics
Spherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...
depending on the polar coordinatesof the position vector
as the angular part of the Slater orbital.Differentials
The first radial derivative of the radial part of a Slater-type orbital isThe radial Laplace operator is split in two differential operators
The first differential operator of the Laplace operator yields
The total Laplace operator yields after applying the second differential operator
the result
Angular dependent derivatives of the spherical harmonics don't depend on the radial function and have to be evaluated separately.
Integrals
The fundamental mathematical properties are those associated with thekinetic energy, nuclear attraction and Coulomb repulsion integrals for
placement of the orbital at the center of a single nucleus. Dropping
the normalization factor N, the representation of the orbitals below is
.The Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
is
,where the
are defined by.The overlap integral is
of which the normalization integral is a special case. The starlet in the
superscript denotes complex-conjugation
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
.
The kinetic energy integral is
a sum over three overlap integrals already computed above.
The Coulomb repulsion integral can be evaluated using the Fourier representation
(see above)
which yields
These are either individually calculated with the law of residues
Methods of contour integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.Contour integration is closely related to the calculus of residues, a methodology of complex analysis....
or recursively
as proposed by Cruz et al. (1978).
STO Software
Slater type orbital (STO) basis functions are used in some quantum chemistry software. The fact that products of two STOs on distinct atoms are more difficult to express than those of Gaussian functions (which give a displaced Gaussian) has led many to expand them in terms of Gaussians.Analytical ab initio software for poly-atomic molecules has been developed e.g. STOP: a Slater Type Orbital Package in 1996.
SMILES uses analytical expressions when available and Gaussian expansions otherwise. It was first released in 2000.
Various grid integration schemes have been developed, sometimes after analytical work for quadrature (Scrocco). Most famously in the ADF suite of DFT codes.
See also
Basis sets used in computational chemistryBasis 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...