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Molecular Physics

The relationship between formal scattering theory and the density matrix

L. Stanton

Published Jan 1, 1971• 7 citations
Publisher

Abstract

A study has been carried out on the relationship between formal scattering theory and the density matrix formalism. The density operator is developed in terms of the scattering operator S and the observable square modulus matrix elements of S are shown to be equivalent to elements of the density matrix. The Møller wave operators are similarly treated and subsequently used in obtaining the density matrix expression for the transition matrix. Finally, using the latter, it is shown that the hierarchy of approximations to the density matrix yields equivalent results to those obtained from the Born series.

DOI
10.1080/00268977100100621

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A study has been carried out on the relationship between formal scattering theory and the density matrix formalism. The density operator is developed in terms of the scattering operator S and the observable square modulus matrix elements of S are shown to be equivalent to elements of the density matrix. The Møller wave operators are similarly treated and subsequently used in obtaining the density matrix expression for the transition matrix. Finally, using the latter, it is shown that the hierarchy of approximations to the density matrix yields equivalent results to those obtained from the Born series.
A tetradic scattering matrix formalism is proposed as a comprehensive framework for describing scattering and relaxation processes of both stationary and non-stationary systems. In the latter case (realized by the scattering of the molecular-maser beam) ordinary dyadic T-matrix formalism involving on-the-energy-shell scattering amplitudes is inadequate, the introduction of off-the-shell elements being required. Expressions are given relating the tetradic T-matrix elements to various scattering and attenuation rates (or cross sections). The relation to the tetradic level-shift operator in relaxation (linear response) theory is discussed. Photon absorption is presented as the successive occurrence of a scattering and a relaxation process, from the point of view of this formalism.
The time-dependent theory of scattering is reformulated. In §1, it is developed on the basis of a new limiting process which is self-consistent, and the equivalence is shown of Heisenberg's S-matrix and Dyson's one, when the total Hamiltonian permits the existence of bound states. In §2, a theory of scattering of wave packet is proposed in conformity with the physical picture. The damping factor e−ε|t| is derived from the amplitude of the wave packet. In §3, the rearrangement scattering is treated on the basis of the wave packet formalism.
The relationship between densities and density matrices is explored in the case of a finite-basis-set expansion. The space of one-electron density matrices can be divided into two orthogonal subspaces with elements in one of them in one-to-one correspondence with densities. A component in the other does not contribute to the density. The set of densities is convex but there may be densities which cannot be obtained from a density matrix. The matrix of a local potential has a component only in the first subspace, and any such matrix can be obtained from a local potential. It is possible for Hamiltonian matrices differing by the matrix of a local potential to have a common ground-state eigenvector, so a Hohenberg-Kohn theorem cannot always be established. When it can, the explicit local potential with a given ground-state density can be formally obtained when appropriate conditions are satisfied. The details of the decomposition of the space of matrices and of subsequent developments depend on linear-dependency relationships among basis-set products, and are thus basis-set dependent.
An efficient approach to calculate approximate pure-state and transition reduced density matrices in the framework of the multireference relativistic Fock-space coupled cluster (FS CC) theory is proposed. The method is based on the effective operator formalism and consists of the direct substitution of the FS CC Ansatz for a wave operator into the effective operator expression with the subsequent truncation of expansion at the terms quadratic in cluster amplitudes. The final density matrix is defined by active-space density matrices of different ranks ‘dressed’ with contributions from cluster operators. The method gives a connected expression for pure-state density matrices, provided that the intermediate normalisation condition is fulfilled. Moreover, under some additional assumptions, the connectivity can also be ensured for calculated transition property matrix elements and natural transition spinors. The developed technique allows for fast and accurate calculations of one-particle reduced density matrices for a wide range of electronic states. A pilot application of the new technique to construct averaged atomic natural orbital (ANO) basis sets for fully relativistic electronic structure calculations is presented.