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International Journal of Quantum Chemistry

Composite‐system models

Douglas J. Klein, Josep M. Oliva

Published Sep 23, 2010• 3 citations
Publisher

Abstract

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

DOI
10.1002/qua.22830

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Many-electron theory is treated here with much attention focused on the synthesis of the density functional and density matrix theory. This synthesis shows that the exchange–correlation potential, normally obtained as the functional derivative with respect to the electron density of an assumed approximate exchange–correlation energy, can be written exactly in terms of the “near-diagonal” behavior of the fully interacting first-order density matrix and the diagonal of the second-order density matrix, thereby bypassing the functional derivative. Also involved, since the exchange–correlation potential is an “artifact” of reducing the many-electron Schrödinger equation to one-body Slater–Kohn–Sham equations, are the one-body orbitals derived from these latter equations, not just the density thereby obtained. This situation has then motivated a discussion of what constraints are placed on the low-order density matrices of atoms and molecules, described by an assumed spin-independent Hamiltonian, by the requirement that these matrices are constructed from an N-electron wave function which is an eigenfunction of the spin operators S2 and Sz. Some briefer discussion is added on the optimally convergent determinantal expansion of such an N-electron wave function, the relation between Brueckner and Löwdin natural orbitals being commented on as a by-product. In this latter context, it is noted that the exchange-only potential of density functional theory can be represented as a sum of two line integrals, each of which is path-dependent, the sum being, of course, path-independent. But in addition to knowledge of Slater–Kohn–Sham orbitals appearing as in the Harbola–Sahni exchange-only potential (which is one of the path-dependent terms), there is a kinetic correction. This involves Löwdin natural orbitals and their occupation numbers, but only now in the weak coupling limit of the electron–electron interaction, as emphasized in the kinetic “correction” to the Harbola–Sahni formula exhibited by Levy and the writer. Some directions for further work are briefly set out. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 163–176, 1999
This chapter introduces the basic theoretical tools for handling many-body quantum systems. Starting from second quantized operators, we discuss how it is possible to describe the composite wavefunction of multi-particle systems, and discuss representations in various bases. The algebra of Fock states is described for single and multi-mode systems, and how they relate to the eigenstates of the Schrodinger equation. Finally, we describe how interactions between particles can be introduced in a general way, and then describe the most common type of interaction in cold atom systems, the s-wave interaction
Abstract Weyl phase–space representation is derived for the cyclic permutation operators that naturally occur in the quantum theory of exchange interaction. The classical‐like phase–space representation of exchange leads to a special type of coupling between the position and momentum variables. With Gaussian electronic structure basis sets in mind, it is shown that the position–momentum coupling seen in the Weyl representation of the exchange interaction is best described by diffuse Gaussians. A classical phase–space analog of exchange is derived for the Heisenberg–Dirac spin‐Hamiltonian. Novel one‐electron Wigner functions are introduced for description of the two‐electron spin correlations. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates { q i }. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement p i → ( ħ /i)(∂/∂ q i ). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ (∂Ψ/∂ t ) = H Ψ which requires as initial condition the knowledge of the wave function Ψ({ q i }, t ) at some initial time t 0 . (Boundary conditions on Ψ({ q i }, t ) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψ ν ( q ), is a basis for the Hilbert space.
Abstract First, the spin‐free formulation of electronic structure is reviewed within the standard first‐quantized framework, with special attention to its manner of correspondence to the standard antisymmetrized space‐spin (ASspin) formulation. Second, within this first‐quantized framework, note is made of some “discrepancies” in the correspondence, for example, regarding mismatches of symmetry‐related degeneracies. Third, a novel second‐quantized spin‐free formulation is introduced to overcome these “discrepancies,” and so achieve a one‐to‐one correspondence between ASspin wave functions and second‐quantized spin‐free wave functions. Yet further, this second‐quantized spin‐free description enables the identification of electrons as “para‐Fermions,” at least in the nonrelativistic limit of spin‐free interactions. From the point of view of elementary particle physics, a rather complete picture of the behavior of para‐Fermions emerges with electrons as a proxy, and so suggests that in fact para‐Fermions would be difficult to distinguish from ordinary Fermions, at least in this nonrelativistic limit or with but few such particles. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011