C. David Sherrill

School of Chemistry and Biochemistry

Georgia Institute of Technology

December 1996

My thesis [1] already discusses the size extensivity problem of truncated configuration interaction methods. Here I will derive the famous Davison correction [2]; my approach follows that of Meissner [3], filling in some of the details. An alternative derivation is also found in Szabo and Ostlund [4]. Additionally, Duch and Diercksen provide a very nice review of size extensivity corrections.

Consider the CID method for *m* identical noninteracting two-electron
systems. Since they are noninteracting, we will write the total
wavefunction as a product of wavefunctions for the individual monomers
(cf. Szabo and Ostlund, [4] p. 269). The ground state
wavefunction for monomer *k* will be written as determinant
,
while the
doubly-excited wavefunction is
.
(Assume that singles are
noninteracting). Hence the total wavefunction is (eq. A1 of Meissner)

(1) |

where of course

Now we diagonalize the Hamiltonian matrix, subtracting out the SCF energy
from the diagonal. The determinantal equation is

(2) |

where

x |
= | ||

= | |||

= | (3) |

and

y |
= | ||

= | |||

= | |||

= | (4) |

The secular equation yields

(5) |

and from the eigenvalue equation,

It is reasonable to expect

(7) |

Stopping at three terms, we obtain

Now recall that the correlation energy should be proportional to the number of monomers in the system,

(9) |

Combining this equation with eq. (6) yields

(10) |

We need to cancel the term

(11) |

which is the ``renormalized'' Davidson correction. If , this is very close to the traditional Davidson correction,

(12) |