Kutzelnigg, Werner ; Mukherjee, Debashis
(2004)
*Irreducible Brillouin conditions and contracted Schrodinger equations for n-electron systems. III. Systems of noninteracting electrons*
Journal of Chemical Physics, 120
(16).
7340_1-7340_10.
ISSN 0021-9606

Full text not available from this repository.

Official URL: http://jcp.aip.org/resource/1/jcpsa6/v120/i16/p734...

Related URL: http://dx.doi.org/10.1063/1.1652465

## Abstract

We analyze the structure and the solutions of the irreducible k-particle Brillouin conditions (IBC_{k}) and the irreducible contracted Schrodinger equations (ICSE_{k}) for an n-electron system without electron interaction. This exercise is very instructive in that it gives one both the perspective and the strategies to be followed in applying the IBC and ICSE to physically realistic systems with electron interaction. The IBC_{1} leads to a Liouville equation for the one-particle density matrix γ_{1} = γ, consistent with our earlier analysis that the IBC_{1} holds both for a pure and an ensemble state. The IBC_{1} or the ICSE_{1} must be solved subject to the constraints imposed by the n-representability condition, which is particularly simple for ν. For a closed-shell state ν is idempotent, i.e., all natural spin orbitals (NSO's) have occupation numbers 0 or 1, and all cumulants λ_{k} with k ≥ 2 vanish. For open-shell states there are NSO's with fractional occupation number, and at the same time nonvanishing elements of λ_{2}, which are related to spin and symmetry coupling. It is often useful to describe an open-shell state by a totally symmetric ensemble state. If one wants to treat a one-particle perturbation by means of perturbation theory, this mainly as a run-up for the study of a two-particle perturbation, one is faced with the problem that the perturbation expansion of the Liouville equation gives information only on the nondiagonal elements (in a basis of the unperturbed states) of ν. There are essentially three possibilities to construct the diagonal elements of ν: (i) to consider the perturbation expansion of the characteristic polynomial of ν, especially the idempotency for closed-shell states, (ii) to rely on the ICSE_{1}, which (at variance with the IBC_{1}) also gives information on the diagonal elements, though not in a very efficient manner, and (iii) to formulate the perturbation theory in terms of a unitary transformation in Fock space. The latter is particularly powerful, especially, when one wishes to study realistic Hamiltonians with a two-body interaction.

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ID Code: | 21919 |

Deposited On: | 23 Nov 2010 09:02 |

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