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International Journal of Quantum Chemistry

Size‐extensivity in multireference many‐body perturbation theories: A direct comparison between single‐reference and multireference perturbation theories in the nondegenerate case

Leszek Meissner, K. Jankowski

Published Dec 1, 1989• 6 citations
Publisher

Abstract

Abstract Linkage properties of the diagrammatic representation of the energies obtained in the multireference many‐body perturbation calculations with respect to the incompleteness or completeness of the model space are discussed. The case of not completely degenerate model space is considered for which a comparison with the standard single‐reference many‐body perturbation expansion is possible. The Hose–Kaldor type of graphical representation of the perturbation expansion for the effective Hamiltonian is used in this comparison. It is shown that for an incomplete model space the perturbation expansion is not size‐extensive. In this case, for a truncated expansion of the effective Hamiltonian, the energies obtained by diagonalization of the effective Hamiltonian matrix are represented by both linked and unlinked irreducible contributions. The unlinked ones do not appear when the complete model space is used.

DOI
10.1002/qua.560360604

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Abstract Linkage properties of the diagrammatic representation of the energies obtained in the multireference many‐body perturbation calculations with respect to the incompleteness or completeness of the model space are discussed. The case of not completely degenerate model space is considered for which a comparison with the standard single‐reference many‐body perturbation expansion is possible. The Hose–Kaldor type of graphical representation of the perturbation expansion for the effective Hamiltonian is used in this comparison. It is shown that for an incomplete model space the perturbation expansion is not size‐extensive. In this case, for a truncated expansion of the effective Hamiltonian, the energies obtained by diagonalization of the effective Hamiltonian matrix are represented by both linked and unlinked irreducible contributions. The unlinked ones do not appear when the complete model space is used.
A size-extensive multidimensional many-body theory is developed from an order-expanded van Vleck transformation. This provides an effective Hamiltonian in a model space consisting of a set of determinants whose zeroth-order energies may be nondegenerate. Expressions for the effective Hamiltonian in terms of the perturbation and a set of resolvents generalized from the Rayleigh–Schrödinger form are given. Perturbative evaluation of the resultant formulas via diagrammatic expansion is illustrated and discussed. The diagrams required through second order for a model space consisting of a Hartree–Fock solution plus selected singly and doubly excited determinants are presented, and their relation to those employed in the method of Hose and Kaldor is discussed.
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
Effective Hamiltonians and effective operators produce, respectively, exact energies and matrix elements of a time-independent operator A for a finite number of eigenstates of a time-independent Hamiltonian H. We obtain degenerate and quasidegenerate perturbative expressions for the particularly useful canonical effective operator ÂC through second order in perturbation theory. The corresponding ÂC diagrammatic expressions are derived for the case where ÂC acts in a complete finite space. Our first order results have been used previously for ab initio computations of dipole and transition dipole moments in diatomic hydrides and for testing the assumptions in semiempirical methods for dipole properties. A brief discussion is also provided on the computational labors required by first and second order ÂC many-body calculations, the derivation of ÂC diagrams when ÂC acts in an incomplete finite space, and on the derivation of diagrammatic rules for ÂC in arbitrary perturbation order.
Abstract The multireference, state‐specific, second‐order, Brillouin–Wigner perturbation theory is presented as a robust approach to the electron correlation problem for systems demanding the use of a multireference function. A posteriori modifications are made which, in the case of a single reference function, recover the well known formula of second‐order many‐body perturbation theory (MBPT2), i.e. Møller–Plesset (MP2) theory, and in the multireference case can be shown to be equivalent to state‐specific multireference Rayleigh–Schrödinger‐like perturbation theory. It is shown that multireference many‐body Brillouin–Wigner perturbation theory when restricted to second‐order is a true many‐body theory, that is, it is a theory, which has the property of extensivity. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007
Abstract In reply to Kaldor's [Int. J. Quantum Chem. XX, XXX (1985)] criticism of our study of simple four‐electron models, in which the degree of quasi‐degeneracy can be continuously varied, by the finite‐order nondegenerate many‐body perturbation theory, we examine in more detail a simple two‐state model that the used to substantiate his claim that “the low order sum of the perturbation series is not very meaningful” in view of its divergence. It is shown that in contrast to Kaldor's claim, the partitioning used increases the radius of convergence of the considered perturbation series and is in principle capable to make it convergent. It is also shown that the convergence of the series is not very essential and that even divergent series can provide useful estimate of the exact result, particularly when the resummation techniques, such as Padè approximants or continued fractions, are employed. Finally, the shortcomings of the existing multi‐reference perturbation approaches, which Kaldor advocates, are pointed out.
Many-body perturbation theory (MBPT) based on Green’s functions and Feynman diagrams provides a fundamental theoretical framework for various ab initio computational approaches in molecular and materials science, including the random phase approximation (RPA) and GW approximation. Unfortunately, this perturbation expansion often fails in systems with strong multireference characters. Extending diagrammatic MBPT to the multireference case is highly nontrivial and remains largely unexplored, primarily due to the breakdown of Wick’s theorem. In this work, we develop a diagrammatic multireference generalization of MBPT for computing correlation energies of strongly correlated systems, by using the cumulant expansion of many-body Green’s functions in place of Wick’s theorem. This theoretical framework bridges the gap between MBPT in condensed matter physics and multireference perturbation theories (MRPT) in quantum chemistry, which had been almost exclusively formulated within time-independent wave function frameworks prior to this work. Our formulation enables the explicit incorporation of strong correlation effects from the outset as in MRPT, while treating residual weak interactions through a generalized diagrammatic perturbation expansion as in MBPT. As a concrete demonstration, we formulate a multireference (MR) extension of the standard single-reference (SR) RPA by systematically resumming generalized ring diagrams, which naturally leads to a unified set of equations applicable to both SR and MR cases. Benchmark calculations on prototypical molecular systems reveal that MR-RPA successfully resolves the well-known failure of SR-RPA in strongly correlated systems. This theoretical advancement paves the way for advancing ab initio computational methods through diagrammatic resummation techniques in future.