Roothaan-Hall equations

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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 ).


  • 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 .

Individual evidence

  1. ^ Roothaan, CCJ: New Developments in Molecular Orbital Theory . In: Reviews of Modern Physics . 23, 1951, pp. 69-89.
  2. ^ Hall, GG: The Molecular Orbital Theory of Chemical Valency. VIII. A Method of Calculating Ionization Potentials . In: Proceedings of the Royal Society A . 205, 1951, pp. 541-552.