# Roothaan-Hall equations

The **Roothaan-Hall equations** are a variant of the Hartree-Fock equations in a non-orthonormal basis. They are used in quantum chemistry to calculate the properties of atoms and molecules . It is customary for the preparation of equations orbitals of Gauss - or Slater-type to use. Its application is limited to atoms with a closed electron shell, that is, each orbital is completely occupied by two electrons. This case is also called Restricted Hartree-Fock theory .

The method was developed independently by Clemens CJ Roothaan and George G. Hall in 1951 . The Roothaan-Hall equations are written in the form of a generalized eigenvalue problem:

Here are

- F the Fock matrix (which depends on C due to the electron-electron interaction)
- C is the matrix of the LCAO coefficients
- S the overlap matrix of the non-orthogonal basis functions
- the (by convention diagonal) matrix of orbital energies.

In the case of an orthogonal basis set, the matrix S is reduced to the identity matrix.

Since the Fock matrix depends on the LCAO coefficients, the Roothan-Hall equations are generally solved iteratively ( self-consistent field method ).

## literature

- Attila Szabo, Neil S. Ostlund:
*Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory*. McGraw-Hill, New York 1989, ISBN 0-07-062739-8 . - Frank Jensen, Introduction to Computational Chemistry, John Wiley and Sons, 1999, pp. 65-69, ISBN 0-471-98085-4 .