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International Journal of Quantum Chemistry

On the use of spin graphs for spin adapting many‐body perturbation theory

Atri Mukhopadhyay

Published Jun 1, 1984• 3 citations
Publisher

Abstract

Abstract In connection with spin adaptation in many‐body perturbation theory, this paper reexamines the use of spin graphs in view of a Hugenholtz‐like representation where both the orbital and the spin parts coexist. Together with the idea of essentially distinct diagrams, this representation leads to an economic handling of spin adaptation. As a side issue, the appropriate generalization of the Epstein–Nesbet partitioning for spin‐adapted perturbation theory is obtained. Compact formulas up to fourth order of the ground‐state energy, and up to third order for excitation energies and ionization potentials are given.

DOI
10.1002/qua.560250605

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Abstract In connection with spin adaptation in many‐body perturbation theory, this paper reexamines the use of spin graphs in view of a Hugenholtz‐like representation where both the orbital and the spin parts coexist. Together with the idea of essentially distinct diagrams, this representation leads to an economic handling of spin adaptation. As a side issue, the appropriate generalization of the Epstein–Nesbet partitioning for spin‐adapted perturbation theory is obtained. Compact formulas up to fourth order of the ground‐state energy, and up to third order for excitation energies and ionization potentials are given.
Abstract This paper shows that the spin‐shift formalism developed in B. T. Pickup and A. Mukhopadhyay [Int. J. Quantum Chem. 26, 101 (1984)] supports a one‐component diagrammatics which has a systematics akin to that in the spin‐orbital many‐body theory. The diagrams are neither Goldstone nor Yutsis type, and characterize the chain U(2R) ⊃ U(R)⊗SU(2) on which the spin‐shift formalism is based. Accordingly, while the lines in such diagrams are labeled by the orbital indices, the diagram structure adequately reflects the irreducible representation of the group U(R). In this sense the paper presents a unitary group approach to the natural generalization of the usual many‐body theory for the spin‐adapted cases. A set of very simple rules is derived; their similarity with the corresponding rules in the ordinary many‐body theory and practical utility are discussed in connection with (a) matrix elements over many‐electron spin states and (b) closed‐ and open‐shell many‐body perturbation theory. A possibility of integral‐driven many‐body perturbation theory for open‐shells is indicated. Connections of this formalism with others are also discussed.
Abstract Multipartitioning multireference many‐body perturbation theory (Zaitevskii and Malrieu, Chem. Phys. Lett. 1995, 233, 597) is investigated with regard to symmetry and size‐extensivity. We show that the spin‐adapted formulation suffers from spatial symmetry breaking and propose a general symmetry‐conserving zero‐order Hamiltonian. We analyze size‐extensivity of various partitionings at the third order and find that extensivity holds if one‐particle quantities in the zero‐order Hamiltonian are properly chosen. In particular, third order of the spin‐adapted and general symmetry‐adapted theory prove to be extensive. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009
We present a low-scaling diagrammatic Monte Carlo approach to molecular correlation energies. Using combinatorial graph theory to encode many-body Hugenholtz diagrams, we sample the M\o{}ller-Plesset (MPn\mathrm{MP}nMPn) perturbation series, obtaining accurate correlation energies up to n=5n=5n=5, with quadratic scaling in the number of basis functions. Our technique reduces the computational complexity of the molecular many-fermion correlation problem, opening up the possibility of low-scaling, accurate stochastic computations for a wide class of many-body systems described by Hugenholtz diagrams.
In these lectures I shall give a review of the non‐relativistic many‐body perturbation theory (MBPT). In the first section a brief summary will be given of the Rayleigh‐Schrödinger perturbation theory for non‐degenerate and degenerate systems, and in the second section the second quantization and the graphical representation and introduced. In the third section the diagrammatic expansions are developed for closed‐ and open‐shell systems, and in the fourth section, finally, the procedure is generalized to the all‐order and coupled‐cluster approaches.
The second-order multireference perturbation theory employing multiple partitioning of the many-electron Hamiltonian into a zero-order part and a perturbation is formulated in terms of many-body diagrams. The essential difference from the standard diagrammatic technique of Hose and Kaldor concerns the rules of evaluation of energy denominators which take into account the dependence of the Hamiltonian partitioning on the bra and ket determinantal vectors of a given matrix element, as well as the presence of several two-particle terms in zero-order operators. The novel formulation naturally gives rise to a “sum-over-orbital” procedure of correlation calculations on molecular electronic states, particularly efficient in treating the problems with large number of correlated electrons and extensive one-electron bases. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 395–401, 1999
Abstract We present a new algorithm to evaluate the grand potential at high and finite temperatures using many-body perturbation theory. This algorithm enables us to calculate the contribution of any Hugenholtz or Feynman vacuum diagrams and formulate the results as a sum of divided differences. Additionally, the proposed method is applicable to any interaction in any dimension, allowing us to calculate thermodynamic quantities efficiently at any given temperature, particularly at high temperatures.Furthermore, we apply this algorithm to the Heisenberg spin-1/2 XXZ chain. We obtain all coefficients of the high-temperature expansion of the free energy and susceptibility per site of this model up to the sixth order.