Roothaan equations
Encyclopedia
The Roothaan equations are a representation of the Hartree-Fock
equation in a non orthonormal basis set
which can be of Gaussian-type
or Slater-type
. It applies to closed-shell molecules or atoms where all molecular orbital
s or atomic orbital
s, respectively, are doubly occupied. This is generally called Restricted Hartree-Fock theory.
The method was developed independently by Clemens C. J. Roothaan
and George G. Hall
in 1951, and is thus sometimes called the Roothaan-Hall equations. The Roothaan equations can be written in a form resembling generalized eigenvalue problem, although they are not a standard eigenvalue problem because they are nonlinear:
where F is the so-called Fock matrix
(which depends on the coefficients C due to electron-electron interactions), C is a matrix of coefficients, S is the overlap matrix
of the basis functions, and
is the (diagonal, by convention) matrix of orbital energies. In the case of an orthonormalised basis set the overlap matrix, S, reduces to the identity matrix. These equations are essentially a special case of a Galerkin method
applied to the Hartree-Fock equation using a particular basis set.
In contrast to the Hartree-Fock
equations - which are integro-differential equations - the Roothaan-Hall equations have a matrix-from. Therefore they can be solved using standard techniques.
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....
equation in a non orthonormal basis set
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. Otherwise, the...
which can be of Gaussian-type
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...
or Slater-type
Slater-type orbital
Slater-type orbitals are functions used as atomic orbitals in the linear combination of atomic orbitals molecular orbital method. They are named after the physicist John C. Slater, who introduced them in 1930....
. It applies to closed-shell molecules or atoms where all 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 or 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, respectively, are doubly occupied. This is generally called Restricted Hartree-Fock theory.
The method was developed independently by Clemens C. J. Roothaan
Clemens C. J. Roothaan
Clemens C.J. Roothaan was born in 1918 in Nijmegen, the Netherlands. He enrolled TU Delft in 1935 to study electrical engineering. During World War II he was first detained as a prisoner of war camp, later he was sent for a year to a concentration camp in Vught...
and George G. Hall
George G. Hall
George Garfield Hall , is an applied mathematician and scientist of distinction, known for original work and contributions to the field of Quantum chemistry....
in 1951, and is thus sometimes called the Roothaan-Hall equations. The Roothaan equations can be written in a form resembling generalized eigenvalue problem, although they are not a standard eigenvalue problem because they are nonlinear:
where F is the so-called Fock matrix
Fock matrix
In the Hartree-Fock method of quantum mechanics, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of basis vectors....
(which depends on the coefficients C due to electron-electron interactions), C is a matrix of coefficients, S is the overlap matrix
Overlap matrix
The overlap matrix is a square matrix, used in quantum chemistry to describe the inter-relationship of a set of basis vectors of a quantum system. In particular, if the vectors are orthogonal to one another, the overlap matrix will be diagonal. In addition, if the basis vectors form an...
of the basis functions, and
is the (diagonal, by convention) matrix of orbital energies. In the case of an orthonormalised basis set the overlap matrix, S, reduces to the identity matrix. These equations are essentially a special case of a Galerkin methodGalerkin method
In mathematics, in the area of numerical analysis, Galerkin methods are a class of methods for converting a continuous operator problem to a discrete problem. In principle, it is the equivalent of applying the method of variation of parameters to a function space, by converting the equation to a...
applied to the Hartree-Fock equation using a particular basis set.
In contrast to the 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....
equations - which are integro-differential equations - the Roothaan-Hall equations have a matrix-from. Therefore they can be solved using standard techniques.