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Roothaan equations

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Title: Roothaan equations  
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Subject: Hartree–Fock method, Restricted open-shell Hartree–Fock, Unrestricted Hartree–Fock, George G. Hall, Fock matrix
Collection: Quantum Chemistry
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Roothaan equations

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 orbitals or atomic orbitals, respectively, are doubly occupied. This is generally called Restricted Hartree–Fock theory.

The method was developed independently by

  1. ^ Frank Jensen, Introduction to Computational Chemistry, John Wiley and Sons, 1999, pp. 65–69, ISBN 0-471-98085-4
  2. ^ Roothaan, C. C. J. (1951). "New Developments in Molecular Orbital Theory". Reviews of Modern Physics 23: 69–89.  
  3. ^ Hall, G. G. (1951). "The Molecular Orbital Theory of Chemical Valency. VIII. A Method of Calculating Ionization Potentials".  


See also

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.

where F is the so-called Fock matrix (which depends on the coefficients C due to electron-electron 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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