Olsen's Full CI Equations

(tex2html_deferredtex2html_deferred6.tex2html_deferred16 H = _kl^n h_kl Ê_kl + 12 _ijkl^n (ij|kl) ( Ê_ij Ê_kl - _jk Ê_il ) ) |

where is the shift operator

Again, is given by

(tex2html_deferredtex2html_deferred6.tex2html_deferred18 (I_, I_) = _J_, J_ (J_) (J_) | H | (I_) (I_) C(J_, J_) ) |

As we will proceed to show, can be split up into three terms: one involving only beta components of the linear group generators (), one involving only alpha components of the generators (), and one involving mixtures of the two (). Inserting equation (6.16) into equation (6.18) yields

= | (tex2html_deferredtex2html_deferred6.tex2html_deferred19(I_, I_) &=& _J_, J_ (J_) (J_) | _kl^n h_kl Ê_kl ) | ||

+ |

Now expanding the shift operators according to equation (6.17),
we write
as a sum of three terms

(tex2html_deferredtex2html_deferred6.tex2html_deferred20(I_, I_) = _1(I_, I_) + _2(I_, I_) + _3(I_, I_) ) |

where

= | (tex2html_deferredtex2html_deferred6.tex2html_deferred21_1(I_, I_) &=& _J_, J_ (J_) (J_) | _kl^n h_kl Ê^_kl ) | ||

+ |

and

= | (tex2html_deferredtex2html_deferred6.tex2html_deferred22_2(I_, I_) &=& _J_, J_ (J_) (J_) | _kl^n h_kl Ê^_kl ) | ||

+ |

and

(tex2html_deferredtex2html_deferred6.tex2html_deferred23_3(I_, I_) = _J_, J_ (J_) (J_) | 12 _ijkl^n (ij|kl) ( Ê^_ij Ê^_kl + Ê^_ij Ê^_kl ) | (I_) (I_) C(J_, J_) ) |

Obviously, these expressions can be simplified further.
First, we observe that the expression for
contains no operators. Therefore,
unless
.
Using this fact, and integrating out the
part, we obtain

(tex2html_deferredtex2html_deferred6.tex2html_deferred24_1(I_, I_) = _J_ (J_) | _kl^n h_kl Ê^_kl + 12 _ijkl^n (ij|kl) ( Ê^_ij Ê^_kl - _jk Ê^_il ) | (I_) C(I_, J_) ) |

Taking out the Kronecker delta term and rearranging, we have

= | (tex2html_deferredtex2html_deferred6.tex2html_deferred25_1(I_, I_) &=& _J_ _kl^n h_kl (J_) | Ê^_kl | (I_) C(I_, J_) ) | ||

- | |||

+ |

Now the second term can be combined with the first to give equation (9a) of reference [16].

= | (tex2html_deferredtex2html_deferred6.tex2html_deferred26_1(I_, I_) &=& _J_ _kl^n (J_) | Ê^_kl | (I_) [ h_kl - 12 _j^n (kj|jl) ] C(I_, J_) ) | ||

+ |

Similarly,
can be simplified to

= | (tex2html_deferredtex2html_deferred6.tex2html_deferred27_2(I_, I_) &=& _J_ _kl^n (J_) | Ê^_kl | (I_) [ h_kl - 12 _j^n (kj|jl) ] C(J_, I_) ) | ||

+ |

For efficient implementation, it is convenient to precompute the
quantities

(tex2html_deferredtex2html_deferred6.tex2html_deferred28h'_kl = h_kl - 12 _j^n (kj|jl) ) |

Finally, we simplify . It may be rewritten as

= | |||

+ |

Since we sum over all

= |

Thus we have written the action of the Hamiltonian on the current CI vector in terms of alpha and beta strings and alpha and beta shift operators. The product is written as a sum of three terms: the first () involves only beta shift operators, the second () involves only alpha shift operators, and the third () involves both alpha and beta shift operators. Note that, except for the factor , is independent of so that the algorithm for computing is vectorizable (see below). Analogous results hold for . This situation does not obtain in the computation of , however, which is the rate-limiting step.