Epstein-Nesbet

As was shown by Claverie et al.46) this denominator shift leads to the same expression for k0 as does the Epstein-Nesbet partitioning of the Hamiltonian. 

The sum of the perturbation expansion to infinite order is, of course, independent of the choice of the zero-order operator. Assuming that the perturbation series have converged, the model, or Moller-Plesset, and shifted, or Epstein-Nesbet, perturbation series will give identical results at sufficiently high order. 

Epstein-Nesbet, perturbation scheme is used. All of the quadruple-excitation terms may be written in the form... 

Arrays in which the calculated energy components will be stored are set to zero, etotal will contain the total energy whilst etwo will contain the two-body component, iz = 1 corresponds to the Hartree-Fock model zero order hamiltonian, that is the Moller-Plesset expansion whereas iz = 2 identifies the shifted denominator scheme which uses the Epstein-Nesbet zero order hamiltonian. ediag will be used to store the diagonal components. These energies are stored in the common block ptres together with the orbital energy (eorb (60)). 

This program handles both the Moller-Plesset and the Epstein-Nesbet perturbation series. For the Moller-Plesset expansion, the denominators do not depend on the spin case and are given by... 

The third order ring energy component for the perturbation series corresponding to the Epstein-Nesbet zero-order hamiltonian is given by... 

Now the required products of denominator factors corresponding to the Epstein-Nesbet perturbation expansion are formed for each of the possible spin cases. Code to check for vanishing denominators is included so as to avoid overflow. A vanishing denominator causes an error condition via a goto 902. 

For single reference perturbation theory, there is a choice of reference hamil-the Moller-Plesset and Epstein-Nesbet zero-order hamiltonians were two choices considered in the early literature (see, for example, Ref. 54). 

The denominator factors are now required. They are stored in the arrays dl, d2, d3, d4, d5 corresponding to the different spin cases labelled by // = 1, 2, 3,4, 5. This program handles both the Mpller-Plesset and the Epstein-Nesbet perturbation series. For the Mpller-Plesset expansion, the denominators do not depend on the spin case and are given by... 

The elements of the arrays fl, f2, f3, f4, f5 can now be updated for both the Mpller-Plesset expansion and the Epstein-Nesbet series. 

Dabcd CCSD), respectively, whereas the CR-CC(2,3) and other CR-CCOm/i. mg) schemes discussed here rely on the Epstein-Nesbet-type denominators, such as 7) (CCSD), although we could obviously consider alternative forms of these denominators as well, as implied by the above considerations. Just like CCSD(2)y and CCSD(2), the CR-CC(2,3) and CR-CC(2,4) methods are rigorously size extensive. This has been illustrated numerically in Ref. [46]. 

Mdller-Plesset (27) (MP) and Epstein-Nesbet (28) (EN). The resulting PT expansions will be based on a single spin-free spatial configuration and will be formulated using the techniques that characterize CAUGA based CC theories. We shall thus refer to the resulting formalism as the UGA... 

The series of values presented in table 3a originate from CASPT2 calculations carried out in the frame of Moller-Plesset and Epstein-Nesbet second-order perturbation theories bracketing the total energy upwards and downwards by 0.02-0.05 a.u.. 

The data of Table 3b have been computed by replacing the active space of the CASPT2 treatment by the CIPSI multireference space of the MRPT2 method documented in section 2.1. The selection of the components of the reference space by this technique enables us to reduce the gap between the Moller-Plesset and Epstein-Nesbet total energies to less than 0.01 a.u., and so to improve our evaluation of energy balances, disregarding a possible overestimation of the Moller-Plesset values due to intruder states, as it is the case in the ScNC system. 

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