C. David Sherrill

School of Chemistry and Biochemistry

Georgia Institute of Technology

September 1996

At times there appears to be a confusion in the literature regarding
the computational scaling of the configuration interaction method.
For example, Gershgorn and Shavitt claim [1] that CI
scales as
,
where *n* is the number of orbitals,
and *m* is the excitation level (e.g., 2 for CISD). However, it is
commonly held that CISD actually scales as
;
for
CISDTQ, various scalings are given, but the most common one is
.
This would seem to imply a general scaling rule
of
,
in contradiction to the former rule.
These notes explain the scaling of the CI method and demonstrate that,
depending on circumstances, either limit may be appropriate. For the
present purposes, it will suffice to focus simply on Slater
determinants rather than other possible *N*-electron basis functions,
such as configuration state functions (CSF's).

For present purposes, it is sufficient to work with spin orbitals.
Typically, the dimension of the CI space is dominated by determinants
with the maximum excitation level, *m*. Thus,

(1) |

with

(2) |

where

Each element of a trial vector
**b**_{i} need only be
multiplied by the nonzero elements of
**H**. The Hamiltonian
will connect a maximally excited determinant with other maximally
excited determinants and with other determinants having excitation
level
.
The number of interacting determinants is
roughly

(3) |

which we further approximate as

(4) |

Each element in

(5) |

Except for full CI, we typically expect

(6) |

These results show that CISD and CISDTQ will scale as and , respectively, for typical cases, and as and for very large numbers of orbitals or a small number of active electrons.

For very high levels of excitation, including full CI, the number of
interacting matrix elements for a given determinant becomes
approximately *N*^{2} *n*^{2}, so that the computational cost becomes
roughly

(7) |

where we have replaced the term