Triply excited determinants

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. 

EOMCCSD(T) calculations, while facilitating the open-shell implementation of the CR-EOMCCSD(T) method employing the restricted open-shell Hartree-Fock (ROHE) orbitals [59]. Indeed, the use of spin-orbital energy differences (ca -f -f Cc — — ej — e ) instead of the complete form of the diagonal matrix elements of involving triply excited determinants to... 

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...

The formula for the second-order correction to the wave function (eq, (4.39)) contains products of the type ( y H >()(, H o)- The o is the HF determinant and the bracket can only be non-zero if, is a doubly excited determinant. This means that the first bracket only can be non-zero if is either a singly, doubly, triply or quadruply excited determinant (H is a two-electron operator). The second-order wave function allows calculation of the fourth- and fifth-order energies, these terms therefore have contributions from determinants which are singly, doubly, triply or excited. The computational cost of the fourth-order energy without th contribution from the triply excited determinants, MP4(SDQ), increases as M , while the triples contribution increases as M . MP4 is still a computationally feasible model for many molecular. systems, requiring a time similar to CTSD. Tn... 

The eigenvalues, loj, are the transition energies. Since only single excitations are used in the CIS approximation, this approach fails in the description of excited states with significant contributions from doubly and triply excited determinants, 4>, 4>. ... 

F+V 0 ). Similar in the B-matrix one obtains additional matrix elements between two singly excited determinants, ( F + V l), and between two doubly excited determinants, ( F - - F [see Exercise 10.5]. This implies that in SOPPA more electron correlation is included in the ground state in terms of triply excited determinants, but also that the excited states are correlated by the admixture of doubly excited determinants. 

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