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The '''Roothaan equations''' are a representation of the [[Hartree-Fock]] equation in a non orthonormal [[Basis set (chemistry)|basis set]] which can be of [[Gaussian orbital|Gaussian-type]] or [[Slater-type orbital|Slater-type]]. It applies to closed-shell molecules or atoms where all [[molecular orbital]]s or [[atomic orbital]]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''.<ref>Frank Jensen, Introduction to Computational Chemistry, John Wiley and Sons, 1999, pg 65 - 69, ISBN 0 471 98055</ref><ref>{{cite journal |doi= 10.1103/RevModPhys.23.69 |title= New Developments in Molecular Orbital Theory |year= 1951 |author= Roothaan, C. C. J. |journal= Reviews of Modern Physics |volume= 23 |pages= 69–89 |bibcode=1951RvMP...23...69R}}</ref><ref>{{cite journal |doi= 10.1098/rspa.1951.0048 |title= The Molecular Orbital Theory of Chemical Valency. VIII. A Method of Calculating Ionization Potentials |year= 1951 |author= Hall, G. G. |journal= [[Proceedings of the Royal Society A]] |volume= 205 |pages= 541–552|bibcode = 1951RSPSA.205..541H }}</ref> The Roothaan equations can be written in a form resembling [[generalized eigenvalue problem]], although they are not a standard eigenvalue problem because they are nonlinear:


:<math>\mathbf{F} \mathbf{C} = \mathbf{S} \mathbf{C} \mathbf{\epsilon}</math>
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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 <math>\epsilon</math> 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.
 
In contrast to the [[Hartree-Fock]] equations - which are integro-differential equations - the Roothaan-Hall equations have a matrix-from. Therefore they can be solved using standard techniques.
 
==See also==
* [[Hartree-Fock]]
 
==References==
<references/>
 
[[Category:Quantum chemistry]]
 
{{quantum-stub}}

Latest revision as of 00:53, 15 April 2014


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