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
In chemistry, a molecular orbital is a mathematical function describing the location and 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 ...
s or
atomic orbital
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any spe ...
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 (August 29, 1918 – June 17, 2019) was a Dutch physicist and chemist known for his development of the self-consistent field theory of molecular structure.
Biography
Roothaan was born in Nijmegen. He enrolled TU Delft in 19 ...
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
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matri ...
, although they are not a standard eigenvalue problem because they are nonlinear:
:
where F is the
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.
It is most often formed in computational chemistry when ...
(which depends on the coefficients C due to electron-electron interactions), C is a matrix of coefficients, S is the
overlap matrix
In chemical bonds, an orbital overlap is the concentration of orbitals on adjacent atoms in the same regions of space. Orbital overlap can lead to bond formation. Linus Pauling explained the importance of orbital overlap in the molecular bond 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 method
In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete proble ...
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-form. Therefore, they can be solved using standard techniques.
See also
*
Hartree–Fock method
In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.
The Hartree–Fock method often a ...
References
Quantum chemistry
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