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Re-writing Equation 3.2.40 in matrix notation gives

*S* is now written in terms of an upper triangular matrix using
Choleski decomposition,

If we then define a vector *d* through *Uc* = *d*, we can rewrite
Equation 3.2.41 as a standard eigenvalue problem

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(37) |

Evaluation of *U* and *U*^{-1} is an *O*(*N*^{3}) problem, and the
eigenvalues are then found using a Householder scheme. Since the
occupied states only represent a small fraction of the Hamiltonian
eigenstates, the required eigenvectors are most efficiently found by
inverse iteration since only the required states are calculated.
Nonetheless this is normally the time dominant step, scaling as
*O*(*N*^{3}).

*Chris Ewels*

*11/13/1997*