The Condon-Slater Rules

In the Hartree-Fock approximation, the wave function of an atom (or molecule) is a Slater determinant or a linear combination of a few Slater determinants [for example, Eq. (10.44)]. A configuration-interaction wave function such as (11.17) is a linear combination of many Slater determinants. To evaluate the energy and other properties of atoms and molecules using Hartree-Fock or configuration-interaction wave functions, we must be able to evaluate integrals of the form (H b D), where D and D are Slater determinants of orthonormal spin-orbiteils and B is an operator. 

Each spin-orbital m, is a product of a spatial orbital 0i and a spin function (Ti, where o- is either a or /3. We have m, = 0,o-, and (M,(l) My(l)) = 8, where (M,(l) My(l)) involves a sum over the spin coordinate of electron 1 and an integration over its spatial coordinates. If u, and My have different spin functions, then (10.12) ensures the orthogonality of M, and Uj if m, and My have the same spin function, their orthogonality is due to the orthogonality of the spatietl orbitals and 6j. 

Condon and Slater showed that the n-electron integral (Z) 5 Z)) can be reduced to sums of certain one- and two-electron integrals. The derivation of these Condon-Slater formulas uses the determinant expression of Problem 8.18 together with the orthonormality of the spin-orbitals. (See Parr, pp. 23-27 for the derivation.) Table 11.3 gives the Condon-Slater formulas. 

In Table 11.3, each matrix element of involves summation over the spin coordinates of electrons 1 and 2 and integration over the full range of the spatial coordinates of electrons 1 and 2. Each matrix element of fi involves summation over the spin coordinate of electron 1 and integration over its spatial coordinates. The variables in the sums and definite integrals are dummy variables. 

Let us apply these equations to evaluate D H D), where H is the Hamiltonian of an n-electron atom with spin rbit interaction neglected and D is a Slater determinant of n spin-orbitals. We have H = S,i ft + S, Sy i gij, where / = - Ze lri 

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A somewhat special case is the matrix element between the HF determinant and a singly excited CSF. The Condon-Slater rules applied to this situation dictate that... 

Figure 7.4 Structure of the CI matrix as blocked by classes of determinants. The HF block is the (1,1) position, the matrix elements between the HF and singly excited determinants are zero by Brillouin s theorem, and between the HF and triply excited determinants are zero by the Condon-Slater rules. In a system of reasonable size, remaining regions of the matrix become increasingly sparse, but the number of determinants in each block grows to be extremely large. Thus, the (1,1) eigenvalue is most affected by the doubles, then by the singles, then by the triples, etc...

The matrix elements r o are quite straightforward to evaluate. Before leaving them, however, it is worthwhile to make some qualitative observations about them. First, the Condon-Slater rules dictate that for the one-electron operator r, the only matrix elements that survive are those between determinants differing by at most two electronic orbitals. Thus, only absorptions generating singly or doubly excited states are allowed. 

The second approach is to truncate the expansion at some level of excitation. By Brillouin s theorem, the single excited configurations will not mix with the HF reference. By the Condon-Slater rules, this leaves the doubles configurations as the most important for including in the Cl expansion. Thus, the smallest reasonable truncated Cl wavefunction includes the reference and all doubles configurations (CID) ... 

The Condon-Slater rules give the values of one- and two-electron integrals involving Slater determinants and can be used to evaluate properties (such as the energy) for a wave function that is a linear combination of Slater determinants. 

Use the Condon-Slater rules to prove the orthonormality of two n-electron Slater determinants of orthonormal spin-orbitals. 

Consider the matrix elements (i/> ° 4>o) in (15.86), where o is a closed-shell single determinant. One finds (Szabo and Ostlund, Section 6.5) that this integral vanishes for all singly excited i/ J s that is, 4> i 4>o) = 0 for all i and a. Also,(i/>f i H I>o) vanishes for all whose excitation level is three or higher. This follows from the Condon-Slater rules (Table 11.3). Hence, we need consider only doubly excited to find Also, the same reasoning applied to Eq. (9.27) shows that 1/ 0 , the first-order correction to the wave function, contains only doubly excited i/f s. 

Condon-Slater rules (Table 11.3) show that the matrix elements of H between Slater determinants differing by four spin-orbitals are zero. Similar use of the Condon-Slater rules gives for the integral on the left of (15.100) (Problem 15.59)... 

The terms involving /i are hydrogenlike energies, and their sum equals in Eq. (10.49). The remaining terms equal in Eq. (10.51). As noted at the beginning of Section 9.4, (0) + (i) equals the variational integral D H D), so the Condon-Slater rules have been checked in this case. 

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