Epstein-Nesbet perturbation

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... 

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... 

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. 

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... 

On the one hand, the Moller-Plesset partitioning of into and V is not unique and therefore the different orders of perturbation theory are also not uniquely defined. Various other choices of V were proposed " but they all led to different variants of the Epstein-Nesbet perturbation theory with a shifted denominator. This procedure also seems to be feasible for infinite systems, so there is hope that in the future more than 70 to 75% of the correlation energy will be obtained even in the second order. 

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

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 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.. 

Note that if matrix D is approximated by only its diagonal part, we recover the Epstein-Nesbet second-order energy. Unlike E orrCEN), eorr(I MBPT(oo)) is exact to third order in perturbation theory. In fourth order, it is missing contributions due to single, triple, and quadruple excitations. 

Double perturbation theory using Epstein-Nesbet partition. 

However, even though thanks to parametrization the MNDO method partially takes account of the electron correlation energy, the common drawback of a one-determinant approximation, namely, its inability to correctly evaluate the bond-breaking reactions, is not thereby removed as was pointed out, for example, in Ref. [70] using as an illustration a number of dissociation reactions. In such cases, a more complete inclusion of the correlation energy can remedy the situation. In this connection, Thiel suggested a MNDOC scheme [71] in which the correlation corrections are calculated explicitly in the second order of perturbation from the Epstein-Nesbet formula [72, 73] ... 

Nakano, H. (1993a). MCSCF reference quasidegenerate perturbation theory with Epstein-Nesbet partitioning. The Journal of Chemical Physies, 99, 7983-7992. 

BWPT = Brillouin-Wigner perturbation theory EN = Epstein-Nesbet FCI = full Cl MBPT = many-body perturbation theory MRS PT = multireference state perturbation theory PT = perturbation theory RSPT = Rayleigh-SchrSdinger perturbation theory SRS PT = single-reference state perturbation theory. 

A related perturbation scheme is based on the Epstein-Nesbet partitioning of the Hamiltonian, where the zero-order operator contains those parts of the Hamiltonian that conserve the spin-orbital occupations... 

In addition to our earlier work (9,54), we wish to point out that we can also calculate the first-order coefficients by relying entirely on perturbation theory. Of course, the result will very much depend on the way we partition the Hamiltonian H into the unperturbed part Hq and the perturbation W, H -Hq+W. Since our zero-order wave function is assumed to represent a general multi-configurational Cl wave function, it is easier to employ the Epstein-Nesbet (EN) type perturbation theory. For this purpose we choose the unperturbed Hamiltonian Hq as follow... 

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