Excited determinants

Consider now the case where an electron with a spin is moved from orbital i to orbital a. The first S-type determinant in Figure 4.1 is of this type. Alternatively, the electron with /3 spin could be moved from orbital i to orbital a. Both of these excited determinants will have an value of 0, but neither are eigenfunctions of the operator. The difference and sum of these two determinants describe a singlet state and the 5 = 0 component of a triplet (which depends on the exact definition of the 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 disappearance of matrix elements between the HF reference and singly excited states is known as Brillouins theorem. The HF reference state therefore only has nonzero matrix elements with doubly excited determinants, and the full Cl matrix acquires a block diagonal structure. 

The number of excited determinants thus grows factorially with the size of the basis set. Many of these excited determinants will of course have different spin multiplicity (triplet, quintet etc. states for a singlet HF determinant), and can therefore be left out in the calculation. Generating only the singlet CSFs, the number of configurations at each excitation level is shown in Table 4.1. 

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 Cl... 

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. 

The full Cl for the states involves only two configurations, the reference HF and the doubly excited determinant. 

The dissociation problem is solved in the case of a full Cl wave function. As seen from eq. (4.19), the ionic term can be made to disappear by setting ai = —no- The full Cl wave function generates the lowest possible energy (within the limitations of the chosen basis set) at all distances, with the optimum weights of the HF and doubly excited determinants determined by the variational principle. In the general case of a polyatomic molecule and a large basis set, correct dissociation of all bonds can be achieved if the Cl wave function contains all determinants generated by a full Cl in the valence orbital space. The latter corresponds to a full Cl if a minimum basis is employed, but is much smaller than a full Cl if an extended basis is used. 

In developing perturbation theory it was assumed that the solutions to the unpermrbed problem formed a complete set. This is general means that there must be an infinite number of functions, which is impossible in actual calculations. The lowest energy solution to the unperturbed problem is the HF wave function, additional higher energy solutions are excited Slater determinants, analogously to the Cl method. When a finite basis set is employed it is only possible to generate a finite number of excited determinants. The expansion of the many-electron wave function is therefore truncated. 

The formula for the first-order correction to the wave function (eq. (4.37)) similarly only contains contributions from doubly excited determinants. Since knowledge of the first-order wave function allows calculation of the energy up to third order (In - - 1 = 3, eq. (4.34)), it is immediately clear that the third-order energy also only contains contributions from doubly excited determinants. Qualitatively speaking, the MP2 contribution describes the correlation between pairs of electrons while MP3 describes the interaction between pairs. The formula for calculating this contribution is somewhat... 

As shown in Table 4.2, the most important contribution to the energy in a Cl procedure comes from doubly excited determinants. This is also shown by the perturbation expansion, the second- and third-order energy corrections only involve doubles. At fourth order the singles, triples and quadruples enter the expansion for the first time. This is again consistent with Table 4.2, which shows that these types of excitation are of similar importance. 

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)). 

The notation (tp jtl +. ..) indicates that several other terms involving permutations of the indices are omitted. Multiplying eq. (4.50) with a doubly excited determinant gives... 

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. 

By including the doubly excited determinant, built from the antibonding MO, the amount of covalent and ionic terms may be varied, and be determined completely by the variational principle (eq. (4.19)). 

Just as the single determinant MO wave function may be improved by including excited determinants, the simple VB-HL function may also be improved by adding terms which correspond to higher energy configurations for the fragments, in this case ionic structures. 

Compared to the overlap of the undistorted atomic orbitals used in the HL wave function, which is just 5ab. it is seen that the overlap is increased (c is positive), i.e. the orbitals distort so that they overlap better in order to make a bond. Although the distortion is fairly small (a few %) this effectively eliminates the need for including ionic VB terms. When c is variationally optimized, the MO-CI, VB-HL and VB-CF wave functions (eqs. (7.4), (7.7) and (7.8)) are all completely equivalent. The MO approach incorporates the flexibility in terms of an excited determinant, the VB-FIL in terms of ionic structures, and the VB-CF in terms of distorted atomic orbitals. 

In the MO-CI language, the correct dissociation of a single bond requires addition of a second doubly excited determinant to the wave function. The VB-CF wave function, on the other hand, dissociates smoothly to the correct limit, the VB orbitals simply reverting to their pure atomic shapes, and the overlap disappearing. 

The only nonvanishing matrix elements HKL associated with the SCF determinant will thus be the diagonal element, which is identical with the Hartree-Fock energy, and the interaction elements with the doubly excited determinants... 

Doubly excited determinants with respect to Jo are obtained if two SMO s are replaced by virtual SMO s ... 

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. 

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