Doubly excited determinants

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. 

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 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. 

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. 

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... 

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)). 

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 ... 

It is instructive to discuss the MP2-R12 method [37] before going into more involved CC-R12. As in MP2, the wave function of MP2-R12 (IT1)) is a linear combination of the reference HF determinant ( o)) and doubly excited determinants produced by the action of a two-electron excitation operator T ... 

As for the denominator of Eq. (7.40), from inspection of Eq. (7.43), a(0) for each doubly excited determinant will differ from that for the ground state only by including in the sum the energies of the virtual orbitals into which excitation has occurred and excluding the energies of the two orbitals from which excitation has taken place. Thus, the full expression for the second-order energy correction is... 

Naturally, if one wants to improve convergence, one can proceed to higher orders in perturbation theory (note, however, that even at infinite order, there is no guarantee of convergence when a finite basis set has been used). At third order, it is still true that only matrix elements involving doubly excited determinants need be evaluated, so MP3 is not too much more expensive than MP2. A fair body of empirical evidence, however, suggests... 

CISD configuration interaction with singly and doubly excited determinants only DRE Dewar resonance energy... 

Instead, practical methods involve a subset of possible Slater determinants, especially those in which two electrons are moved from the orbitals they occupy in the HF wavefunction into empty orbitals. These doubly excited determinants provide a description of the physical effect missing in HF theory, correlation between the motions of different electrons. Single and triple excitations are also included in some correlated ab initio methods. Different methods use different techniques to decide which determinants to include, and all these methods are computationally more expensive than HF theory, in some cases considerably more. Single-reference correlated methods start from the HF wavefunction and include various excited determinants. Important methods in inorganic chemistry include Mpller-Plesset perturbation theory (MP2), coupled cluster theory with single and double excitations (CCSD), and a modified form of CCSD that also accounts approximately for triple excitations, CCSD(T). 

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

For H2O with the above 6-31G(d) basis there are 30 x 10 CSFs N = 10, A/ = 19) with the larger 6-31 lG(2d,2p) basis there are 106 x 10 CSFs N = 10, M = 41). For H2C=CH2 with the 6-31G(d) basis there are 334 x 10 2 CSFs N = 16, M = 38). for small molecules. As the molecule gets larger the CISD method recovers less and less of the correlation energy, as discussed 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 Cl... 

In order to achieve a high accuracy, it would seem desirable to explicitly include 4.11 Methods Involving Interelectronic Distances terms in the wave functions which are linear in the interelectronic distance. This is the idea in the R12 methods developed by Kutzelnigg and co-workers.-" The first order The necessity for going beyond the HF approximation is the fact that electrons are correction to the HF wave function only involves doubly excited determinants (eqs. further apart than described by the product of their orbitd densities, i.e. their motions (4 35) (4.37)). In R12 methods additional terms are included which essentially are ... 

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