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Slater's Rules
Whether we perform a full CI
or only a limited CI, we must be able to
express
in matrix form so that we can diagonalize it and obtain
the eigenvectors and eigenvalues of interest. In this section we discuss
Slater's rules (or the SlaterCondon rules
[9,,10]),
which allow us to express matrix elements
in terms of one and twoelectron
integrals. At the moment, we will express these results in terms
spinorbitals using physicist's notation. The oneelectron integrals are
written as

(tex2html_deferredtex2html_deferred2.tex2html_deferred11i  h  j = _i^*(r_1) h(r_1)
_j(r_1) dr_1
) 
and the twoelectron integrals are written as

(tex2html_deferredtex2html_deferred2.tex2html_deferred12ij   kl = ij  kl  ij  lk
) 
where

(tex2html_deferredtex2html_deferred2.tex2html_deferred13ij  kl =
_i^*(r_1) _j^*(r_2)
1r_12 _k(r_1) _l(r_2) dr_1
dr_2
) 
Before Slater's rules can be used, the two Slater determinants must be
arranged in maximum coincidence. Remember that switching columns in
a determinant introduces a minus sign. For instance, to calculate
,
where we have

(tex2html_deferredtex2html_deferred2.tex2html_deferred14 _1 =  abcd
) 

(tex2html_deferredtex2html_deferred2.tex2html_deferred15 _2 =  crds
) 
then we must first interchange columns of
or
to make the two determinants look as much alike as possible.
For example, we may rearrange
as

(tex2html_deferredtex2html_deferred2.tex2html_deferred16 _2 =  crds =   crsd =  srcd
) 
After the determinants are in maximum coincidence, we see how many spin
orbitals they differ by, and we then use the following rules:
1. Identical Determinants: If the determinants are identical, then

(tex2html_deferredtex2html_deferred2.tex2html_deferred17_1  H  _1 =
_m^N m  h  m + _m>n^N mn   mn
) 
2. Determinants that Differ by One Spin Orbital:

= 

(tex2html_deferredtex2html_deferred2.tex2html_deferred18 _1 &=&  mn ) 

= 



= 


3. Determinants that Differ by Two Spin Orbitals:

= 

(tex2html_deferredtex2html_deferred2.tex2html_deferred19 _1 &=&  mn ) 

= 



= 


4. Determinants that differ by More than Two Spin Orbitals:

= 

(tex2html_deferredtex2html_deferred2.tex2html_deferred20 _1 &=&  mno ) 

= 



= 
0 

The derivation of these rules can be found in Szabo and Ostlund
[1], section 2.3.4 (pp. 7481).
Next: The Variational Theorem
Up: Fundamental Concepts
Previous: The Correlation Energy
C. David Sherrill
20000418