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### Olsen's Full CI Equations

We now turn our attention to the general formulation of the full CI problem in terms of alpha and beta strings. Later on, we will consider how to modify our results for RAS CI's. We begin by describing Olsen's expressions for , which is the action of the Hamiltonian on the CI vector (or matrix) . First we express in second-quantized  form (see section 5).

 (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

 (tex2html_deferredtex2html_deferred6.tex2html_deferred17 Ê_kl = Ê^_kl + Ê^_kl )

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 ijkl, we can permute i and j with k and l. We can also swap and as can easily be verified. This yields equation (9c) from reference [16].
 =

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.

Next: Full CI Algorithm Up: Restricted Active Space CI Previous: Restricted Active Space CI
C. David Sherrill
2000-04-18