The Roothaan equations are a representation of the Hartree–Fock equation in a non orthonormal basis set which can be of Gaussiantype or Slatertype. It applies to closedshell molecules or atoms where all molecular orbitals or atomic orbitals, respectively, are doubly occupied. This is generally called Restricted Hartree–Fock theory.
The method was developed independently by

^ Frank Jensen, Introduction to Computational Chemistry, John Wiley and Sons, 1999, pp. 65–69, ISBN 0471980854

^ Roothaan, C. C. J. (1951). "New Developments in Molecular Orbital Theory". Reviews of Modern Physics 23: 69–89.

^ Hall, G. G. (1951). "The Molecular Orbital Theory of Chemical Valency. VIII. A Method of Calculating Ionization Potentials".
References
See also
In contrast to the Hartree–Fock equations  which are integrodifferential equations  the Roothaan–Hall equations have a matrixform. Therefore, they can be solved using standard techniques.
where F is the socalled Fock matrix (which depends on the coefficients C due to electronelectron interactions), C is a matrix of coefficients, S is the overlap matrix 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 applied to the Hartree–Fock equation using a particular basis set.

\mathbf{F} \mathbf{C} = \mathbf{S} \mathbf{C} \mathbf{\epsilon}
, although they are not a standard eigenvalue problem because they are nonlinear:
generalized eigenvalue problem The Roothaan equations can be written in a form resembling ^{[3]}^{[2]}[1]
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