In order to develop a computationally tractable model, the number of excited determinants in the CI expansion (eq. (4.2)) must be reduced.Truncating the excitation level at one (CI with Singles (CIS)) does not give any improvement over the HF result as all matrix elements between the HF wave function and singly excited determinants are zero. CIS is equal to HF for the ground state energy, although higher roots from the secular equations may be used as approximations to excited states. It has already been mentioned that only doubly excited determinants have matrix elements with the HF wave function different from zero, thus the lowest CI level that gives an improvement over the HF result is to include only doubly excited states, yielding the CI with Doubles (CID) model. Compared with the number of doubly excited determinants, there are relatively few singly excited determinants (see for example Table 4.1), and including these gives the CISD method. Computationally, this is only a marginal increase in effort over CID. Although the singly excited determinants have zero matrix elements with the HF reference, they enter the wave function indirectly as they have non-zero matrix elements with the doubly excited determinants. In the large basis set limit the CISD method scales as M basis.
The next level in improvement is inclusion of the triply excited determinants, giving the CISDT method, which is an Mbasis method. Taking into account also quadruply excited determinants yields the CISDTQ method which is an M b0asis method. As shown below, the CISDTQ model in general gives results close to the full CI limit, but even truncating the excitation level at four produces so many configurations that it can only be applied to small molecules and small basis sets. The only CI method that is generally applicable for a large variety of systems is CISD. For computationally feasible systems (i.e. medium-size molecules and basis sets), it typically recovers 80-90% of the available correlation energy. The percentage is highest for small molecules; as the molecule gets larger the CISD method recovers less and less of the correlation energy, which is discussed in more detail in Section 4.5.
Since only doubly excited determinants have non-zero matrix elements with the HF state, these are the most important. This may be illustrated by considering a full CI calculation for the Ne atom in a [5s4p3d] basis, where the ls-electrons are omitted from the correlation treatment.4 The contribution to the full CI wave function from each level of excitation is given in Table 4.2.
The weight is the sum of a2t coefficients at the given excitation level, eq. (4.2). The CI method determines the coefficients from the variational principle, thus Table 4.2 shows that the doubly excited determinants are by far the most important in terms of energy. The singly excited determinants are the second most important, followed by the quadruples and triples. Excitations higher than four make only very small
Table 4.2 Weights of excited configurations for the neon atom
Was this article helpful?