Slater determinant triply

In the configuration interaction method, the wavefunction is expanded in a basis of Slater determinants. The Hartree-Fock determinant, noted is taken as a reference zeroth-order wavefunction. Slater determinants corresponding to excited configurations are generated by swapping occupied MOs V a with virtual (unoccupied) MOs ipr and can be classified with respect to the number of excited electrons. Singly excited Slater determinants are noted 3, doubly excited Slater determinants triply excited Slater determinants X so on. The configuration interaction wavefunction then reads... 

How are the additional determinants beyond the HF constructed With N electrons and M basis functions, solution of the Roothaan-Hall equations for the RHF case will yield N/2 occupied MOs and M — N/2 unoccupied (virtual) MOs. Except for a minimum basis, there will always be more virtual than occupied MOs. A Slater detemfinant is determined by N/2 spatial MOs multiplied by two spin functions to yield N spinorbitals. By replacing MOs which are occupied in the HF determinant by MOs which are unoccupied, a whole series of determinants may be generated. These can be denoted according to how many occupied HF MOs have been replaced by unoccupied MOs, i.e. Slater determinants which are singly, doubly, triply, quadruply etc. excited relative to the HF determinant, up to a maximum of N excited electrons. These... 

This is perhaps the easiest method to understand. It is based on the variational principle (Appendix B), analogous to the HF method. The trial wave function is written as a linear combination of determinants with the expansion coefficients determined by requiring that the energy should be a minimum (or at least stationary), a procedure known as Configuration Interaction (Cl). The MOs used for building the excited Slater determinants are taken from a Hartree-Fock calculation and held fixed. Subscripts S, D, T etc. indicate determinants which are singly, doubly, triply etc. excited relative to the... 

Let us continue throwing away determinants. This time, however, we have to make a compromise i.e.. some of the Slater determinants are arbitrarily considered not to be important (which will worsen the results, if they are rejected). Which of the determinants should be considered as not important The general opinion in quantum chemistry is that the multiple excitations are less and less important (when the multiplicity increases). If we take only the singly, doubly, triply, and quadruply excited determinants, the number of determinants will reduce to 25000 and we will obtain 99% of the approximate correlation energy defined above. If we take the singly and doubly excited determinants only, there are only 360 of them, and 94% of the correlation effect is obtained. This is why this Cl Singles and Doubles (CISD) method is used so often. 

The simplest way of truncating the complete set of electronic configurations is to include those Slater determinants in the Cl expansion that differ by one, two, three, and so on spin orbitals compared to some reference configuration Oq. Truncated Cl expansions then are linear combinations of the reference determinant and aU singly, doubly, triply, and so on excited configurations. For instance, the Cl singles doubles (CISD) wave functions can be written as... 

Furthermore, this idea can be extended to doubly, triply etc. excited states. In contrast to existing SCF methods for hole states, we achieve the effect of the excitation (or ionization) of electrons by using orthogonality constraints imposed on the orbitals of the doubly excited state s Slater determinant. For example, for description of excitations from cp p. and (pQi ground state orbitals, we require a fulfillment of conditions... 

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...

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