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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 quantum mechanics, an atomic orbital () is a Function (mathematics), function describing the location and Matter wave, wave-like behavior of an electron in an atom. This function describes an electron's Charge density, charge distribution a ...
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 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 mat ...
, although they are not a standard eigenvalue problem because they are nonlinear: :\mathbf \mathbf = \mathbf \mathbf \mathbf where F is the
Fock matrix The Fock matrix is defined by the Fock operator. In its general form the Fock operator writes: :\hat F(i) = \hat h(i)+\sum_^ hat J_j(i)-\hat K_j(i)/math> Where ''i'' runs over the total ''N'' spin orbitals. In the closed-shell case, it can be si ...
(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. The general principle for orbital overlap is that, the greater the overlap between ...
of the basis functions, and \epsilon 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 are a family of methods for converting a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear c ...
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 method is named after Douglas ...


References

Quantum chemistry {{quantum-chemistry-stub