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International Journal of Quantum Chemistry

Composite particles in quantum chemistry: From two‐electron bonds to cold atoms

Tamás Zoboki, Péter Jeszenszki, Péter R. Śurján

Published Apr 27, 2012• 6 citations
Publisher

Abstract

Abstract We review the problem of forming composite particles from elementary electrons. Simple pairs are treated first, followed by linear combinations including geminal type wave functions. The antisymmetrized product of strongly orthogonal geminals (APSG) theory can also be developed in this language, and the use of APSG as a reference state is discussed. The theory forming composite particles from more than two electrons is less elaborated, although this idea is routinely used in approximate models in the current literature. © 2012 Wiley Periodicals, Inc.

DOI
10.1002/qua.24125

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Abstract We review the problem of forming composite particles from elementary electrons. Simple pairs are treated first, followed by linear combinations including geminal type wave functions. The antisymmetrized product of strongly orthogonal geminals (APSG) theory can also be developed in this language, and the use of APSG as a reference state is discussed. The theory forming composite particles from more than two electrons is less elaborated, although this idea is routinely used in approximate models in the current literature. © 2012 Wiley Periodicals, Inc.
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In this review, we discuss the recent progress in developing geminal-based theories for challenging problems in quantum chemistry. Specifically, we focus on the antisymmetrized geminal power, generalized valence bond, antisymmetrized product of strongly orthogonal geminals, singlet-type orthogonal geminals, the antisymmetric product of 1-reference orbital geminal, also known as the pair coupled cluster doubles ansatz, and geminals constructed from Richardson-Gaudin states. Furthermore, we review various corrections to account for the missing dynamical correlation effects in geminal models and possible extensions to target electronically excited states and open-shell species. Finally, we discuss some numerical examples and present-day challenges for geminal-based models, including a quantitative and qualitative analysis of wave functions, and software availability.
We present a formalism for the description of many-particle systems when the particles are themselves composites of several elementary constituents. We investigate specifically the case of composites made up of two fermions, the method being easily extended to more general composites. The formalism, although its development is guided by the knowledge that there are fermion constituents in the composites, is given entirely in the composite language and requires no reference to these underlying fermions. Thus, the procedure maintains all required symmetries at every step. Composite field operators satisfying all symmetries are defined, these fields obeying neither Bose nor Fermi commutation rules, but more complicated ones. The many-composite dynamics is studied via the Green's functions formed from composite operator products. A Hartree-Fock-like approximation is introduced and leads to the generation of composite wave functions describing the distortion of the composite internal states by the presence of other composites in the system.
Abstract The methods of quantum chemistry and solid state theory to solve the many‐body problem are reviewed. We start with the definitions of reduced density matrices, their properties (contraction sum rules, spectral resolutions, cumulant expansion, and N‐representability), and their determining equations (contracted Schrödinger equations), and we summarize recent extensions and generalizations of the traditional quantum chemical methods, of the density functional theory, and of the quasiparticle theory: from finite to extended systems (incremental method), from density‐to‐density matrix (density matrix functional theory), from weak to strong correlation (dynamical mean field theory), from homogeneous (Kimball–Overhauser approach) to inhomogeneous and finite systems. Measures of the correlation strength are discussed. The cumulant two‐body reduced density matrix proves to be a key quantity. Its spectral resolution contains geminals, possibly the solutions of an approximate effective two‐body equation, and we sketch the idea of how its contraction sum rule can be used for a variational treatment. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
Abstract Composite many‐electron systems are considered in terms of the wave‐functions for the different component subsystems (e.g., atoms). In this weak‐interaction limit, there may be degrees of freedom, which arise from that of the degeneracies of the wave‐functions for each of the isolated subsystems, and which then are describable in terms of an effective Hamiltonian for the couplings and liftings of these degeneracies in the composite system. The representation of an ordinary Schrödinger Hamiltonian in such a 0‐order model space is developed in the context of an example of a (possibly macroscopically large) set of H atoms interacting with one another, to obtain a Heisenberg spin‐Hamiltonian, which is formally “complete” in not neglecting any higher‐order permutations. At the same time, the interactions are hierarchically ordered to manifest the more important ones first, all in a “linked” size‐consistent manner, avoiding various earlier‐encountered “catastrophes.” A classical electrostatic interpretation of the higher order exchange matrix elements is made, and some novel spin‐coupling dependences are found with a decay like an inverse power‐law range (rather than exponential). © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010