The Cost of Configuration Interaction Methods

Earlier we compared the cost of nonrelativistic and relativistic methods for perturbation and coupled-cluster calculations, and also the cost of transforming the integrals. We now turn to the cost of Cl methods, and consider the case of a full Cl, which is the basis of the complete active space (CAS) SCF method, and of a singles and doubles Cl (SDCI) calculation, which is often used for dynamic correlation from a given active space. 

Eor this purpose we use the determinant basis introduced earlier (chapter 9), given in terms of A and B strings, and consider all possible determinants that may be constructed from a given set of Kramers pairs. We group these determinants into subsets with a given value of Na and Nb, characterized by Mk = — Ng). 

In the nonrelativistic case, the Cl expansion is limited to a given Mk = Ms value due to spin factorization. In this case the length of a full Cl expansion, in the absence of symmetry, is ), where n is the number of active orbitals. In the Kramers- 

The lengths of the A-particle bases for relativistic and nonrelativistic calculations for the case of 12 electrons in 12 active orbitals is presented in table 12.1, which clearly demonstrates the size problems connected with the relativistic full Cl expansion. In the best possible case, for Ms = 0, the relativistic full Cl vector is only three times the length of the nonrelativistic vector. 

The immediate consequence of this increased number of determinants is an increase of the size of the Hamiltonian matrix. Eor the = 0 case, the relativistic Hamiltonian matrix has nine times as many elements as in the nonrelativistic Hamiltonian matrix. We would expect the work in multiplying the Hamiltonian matrix by a vector (the time-consuming step in modern Cl procedures, as discussed in the previous section) to increase by a factor of 36 9 from the size of the Hamiltonian matrix and 4 from the complex arithmetic. 

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