Singly excited determinants

The matrix element between the HF and a singly excited determinant is a matrix element of the Fock operator between two different MOs (eq. (3.36)). 

The weight is the sum of coefficients at the given excitation level, eq. (4.2). The Cl 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, then follow the quadruples and triples. Excitations higher than 4 make only very small contributions, although there are actually many more of these highly excited determinants than the triples and quadruples, as illustrated in Table 4,1. 

The relative importance of tlie different excitations may qualitatively be understood by noting tliat the doubles provide electron correlation for electron pairs, Quadruply excited determinants are important as they primarily correspond to products of double excitations. The singly excited determinants allow inclusion of multi-reference charactei in the wave function, i.e. they allow the orbitals to relax . Although the HF orbitals are optimum for the single determinant wave function, that is no longer the case when man) determinants are included. The triply excited determinants are doubly excited relative tc the singles, and can then be viewed as providing correlation for the multi-reference part of the Cl wave function. 

Only equation for the amplitudes is obtained by multiplying the Schrddinger equation (4.50) from the left by a singly excited determinant ( ) and integrating. 

The CCSD energy is given by the general CC equation (4.53), and amplitude equations are derived by multiplying (4.50) with a singly excited determinant and integrating (analogously to eq. (4.54)). 

Since the singly excited determinants effectively relax the orbitals in a CCSD calculation, non-canonical HF orbitals can also be used in coupled cluster methods. This allows for example the use of open-shell singlet states (which require two Slater determinants) as reference for a coupled cluster calculation. 

The simplest description of an excited state is the orbital picture where one electron has been moved from an occupied to an unoccupied orbital, i.e. an S-type determinant as illustrated in Figure 4.1. The lowest level of theory for a qualitative description of excited states is therefore a Cl including only the singly excited determinants, denoted CIS. CIS gives wave functions of roughly HF quality for excited states, since no orbital optimization is involved. For valence excited states, for example those arising from excitations between rr-orbitals in an unsaturated system, this may be a reasonable description. There are, however, normally also quite low-lying states which essentially correspond to a double excitation, and those require at least inclusion of the doubles as well, i.e. CISD. 

The wave function /lo constructed from SCF orbitals is "so good that it cannot be improved by the inclusion of singly excited determinants. The main effect of increasing the number of singly excited determinants in the Cl problem will be a better description of the excited state levels. 

Configuration interaction (Cl) is conceptually the simplest procedure for improving on the Hartree-Fock approximation. Consider the determinant formed from the n lowest-energy occupied spin orbitals this determinant is o) and represents the appropriate SCF reference state. In addition, consider the determinants formed by promoting one electron from an orbital k to an orbital v that is unoccupied in these are the singly excited determinants ). Similarly, consider doubly excited (k, v,t) determinants and so on up to n-tuply excited determinants. Then use these many-electron wavefimctions in an expansion describing the Cl many-electron wavefunction [Pg.13]

H is the Hamiltonian operator and the numbering of the CSFs is arbitrary, but for convenience we will take I l = I hf and then all singly excited determinants, all doubly excited, etc. Solving the secular equation is equivalent to diagonalizing H, and permits determination of the CI coefficients associated with each energy. While this is presented without derivation, the formalism is entirely analogous to that used to develop Eq. (4.21). 

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...

Let us return, however, to singly excited determinants. While, like triples, they fail to interact with the ground state (although in this case because of Brillouin s theorem), they too mix with doubles and thus can have some influence on the lowest eigenvalue. In this instance, there are sufficiently few singles compared to doubles that it does not make the problem significantly more difficult to include them, and this level of theory is known as CISD. 

The HF determinant A single-excited determinant A doubly-excited determinant... 

Hie simplest approach to obtaining excitation energies theoretically is to introduce configuration interaction with singly excited determinants. This Table 5... 

To get more insight into the various effects seen in table 2, the different influences of the excitations on Ahave to be studied. Going from a S-CI to a SD-CI treatment, the double excitations can influence Ais0 in two ways [20]. A direct effect arises from the coefficients of the double excitations themselves, which are not contained in the S-CI wavefunction. A second influence of the double excitations on Aiao is of a more indirect nature. Due to interactions within the SD-CI Hamilton matrix between configurations already included in the S-CI and the double excitations, the coefficients of the RHF determinant and of the singly excited determinants obtained by a SD-CI treatment differ from those obtained by the S-CI treatment. From these differences in... 

Anntbpr curnmnnlv used method is Quadratic CISD fOClSD). It was originally than MP methods. Since the singly excited determinants allow the MOs to relax in order... 

The leading term of H in Eq. [122] is simply the electronic Hamiltonian itself. For its contribution to the T x amplitude equation, we must therefore evaluate matrix elements of between singly excited determinants and o,... 

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