Third-order Many-body Perturbative Calculations

Davidson, in The World of Quantum Chemistry Proceedings of the First International Congress on Quantum Chemistry, ed. R. Daudel and B. Pullman, D. Reidel, 1974. 

In the integral sorting phase of the calculation, the following lists of integrals are created  

The two remaining particle states can then be summed over to give g K = jS Fj yKAC IC v AKy-iIC 11, KA  

Finally, the third-order hole-particle energy is given by 

Similar algorithms have been devised to evaluate the other third-order and the second-order energy components. 

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Third-order calculations are considered in the next section. This is followed by a brief discussion of the computation of higher-order terms and of the evaluation of bubble diagrams which are required when molecular properties are calculated or when a reference function other than the closed-shell Hartree-Fock function is employed. The impact of the new generation of computers, which have vector processing capabilities, on many-body perturbative calculations is discussed very briefly in the final section. 

M. G. Sheppard and K. F. Freed, Effective valence shell Hamiltonian calculations using third-order quasi-degenerate many-body perturbation theory. J. Chem. Phys. 75, 4507 (1981). 

Fig. 5. Magnitude of the basis set truncation error in calculations of electron correlation energies for some closed-shell diatomic molecules. S indicates the calculations performed using smaller ba sets, and L designates calculations with larger basis sets, (i), (ii) and (iii) denote many-body perturbation theory calculations of the correlation energy through second, third and fourth order, respectively.

The vertical excitation energies for the a A B A, C - X transitions at re = 1.9614 ao (the experimental value for the equilibrium internuclear distance in the NH molecule) were calculated by the many-body perturbation theory of second and third order (H" study) [9]. 

The first choice seems to be more natural since, H() being invariant, the partitioning scheme remains untouched of Moeller-Plesset type. The price to be paid for this principal simplicity, however, is high in calculational details, as the well-developed, systematic many-body graphical algorithms are not applicable if the unperturbed eigenfunctions bear a complicated structure. In a series of papers [44-48], Pulay and Ssebo developed formulas for the second- and third -and fourth-order perturbative corrections with localized orbitals using a CEPA-... 

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