The generalized Slater-Condon rules

In this notation, fi j) stands for the one-electron spin-orbital Xn,i,m,m, as a function of the space and spin coordinates of electron j. We now let S be the matrix of overlap integrals between the isoenergetic configurations 

Then the first generahzed Slater-Condon rule states that the sealar produet of the two configurations is given by 

The second generalized Slater-Condon rule states that if 

The third generalized Slater-Condon rule states that for a two-electron operator V  

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Having used the generalized Sturmian method to calculate the wave function for an A-electron atom, we are in a position to derive both the corresponding density distribution and the first-order density matrix [36-52], However, because we cannot assume orthonormality between the one-electron spin-orbitals of different configurations, it is necessary to use expressions analogous to the generalized Slater-Condon rules. If we let... 

If the number of orbitals increases, the first quantized derivation presented above becomes more and more involved. As the size of the determinant increases, one should apply the general Slater-Condon rules, or rederive them similarly as was done for the Hiickel energy expression (Sect. 6.4). On the other hand, the general result of Eq. (6.23) is valid for any number of electrons and orbitals. 

Then (as is shown in Ref. [20]) the first generalized Slater-Condon rule can be... 

In using equation (44), it is important to remember that in order for the properties of the first-order density matrix to be correct, it is necessary that = 1. This normahzation can be achieved by means of the first generalized Slater-Condon rule, equation (35). 

The averaging of SCF energy expressions to impose symmetry and equivalence restrictions is a straightforward, if sometimes tedious, application of the Slater-Condon rules for matrix elements between determinants of orthonormal orbitals. This matter is discussed in detail elsewhere. The most general SCF programs can handle energy expressions of the form... 

The calculation of expectation values of operators over the wavefunction, expanded in terms of these determinants, involves the expansion of each determinant in terms of the N expansion terms followed by the spatial coordinate and spin integrations. This procedure is simplified when the spatial orbitals are chosen to be orthonormal. This results in the set of Slater Condon rules for the evaluation of one- and two-electron operators. A particularly compact representation of the algebra associated with the manipulation of determinantal expansions is the method of second quantization or the occupation number representation . This is discussed in detail in several textbooks and review articles - - , to which the reader is referred for more detail. An especially entertaining presentation of second quantization is given by Mattuck . The usefulness of this approach is that it allows quite general algebraic manipulations to be performed on operator expressions. These formal manipulations are more cumbersome to perform in the wavefunction approach. It should be stressed, however, that these approaches are equivalent in content, if not in style, and lead to identical results and computational procedures. 

Fast computers led the interest of many researchers to general many-electron systems like Cl expansions based on an orbital description and Slater determinants. The main advantage of these methods is the reduction of n-electron Hamiltonian matrix elements to one- and two-electron integrals, as stated in the Slater-Condon rules, but also showing a slow convergence. There are two sources of the slow convergence of the Cl expansion. (1) The combinatorial problem . For an n-electron system and a basis of m spin-free one-electron functions the number of... 

Matrix elements between determinantal wave functions can be evaluated by the so-called Slater (or Slater-Condon) rules. We shall not derive the Slater rules in general, but in some particular cases the first and second quantization-based derivations will be compared. 

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