Mukherjee, Debashis ; Kutzelnigg, Werner
(2001)
*Irreducible Brillouin conditions and contracted Schrodinger equations for n-electron systems. I. The equations satisfied by the density cumulants*
Journal of Chemical Physics, 114
(5).
2047_1-2047_15.
ISSN 0021-9606

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Official URL: http://jcp.aip.org/resource/1/jcpsa6/v114/i5/p2047...

Related URL: http://dx.doi.org/10.1063/1.1337058

## Abstract

Two alternative conditions for the stationarity of the energy expectation value with respect to k-particle excitations are the k-particle Brillouin conditions BC_{k} and the k-particle contracted Schrodinger equations, CSE_{k}. These conditions express the k-particle density matrices γ_{k} in terms of density matrices of higher particle rank. The latter can be eliminated if one expresses the γ_{k} in terms of their cumulants λ_{k}, but this is not sufficient to make the BC_{k} or CSE_{k} separable (extensive), i.e., they are not expressible in terms of only connected diagrams. However, in a formulation based on the recently introduced general normal ordering with respect to arbitrary wave functions, the irreducible counterparts IBC_{k} and ICSE_{k} of the BC_{k} and CSE_{k} can be defined. They are easily evaluated explicitly in terms of the generalized Wick theorem for arbitrary wave functions, and they lead to equations for the direct construction of the cumulants λ_{k}, which are additively separable quantities and which scale linearly with the system size. The IBC_{k} or the ICSE_{k} are necessary conditions for ν and the λ_{k} to represent an exact n-fermionic eigenstate of the given Hamiltonian. To specify the desired state, additional conditions must be satisfied as well, e.g., the partial trace relations which relate λ_{2} to ν and γ_{2}. The particle number and the total spin must be specified and n-representability conditions enter implicitly. While the nondiagonal elements of ν and the λ_{k} are determined by the IBC_{k} or the ICSE_{k}, the additional conditions mainly serve to fix the diagonal elements. A hierarchy of k-particle approximations is defined. It is based on the fact that the expansion in terms of cumulants λ_{k} can be truncated at any particle rank, which would not be possible for the density matrices γ_{k}. For closed-shell states the one-particle approximation agrees with Hartree-Fock.

Item Type: | Article |
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Source: | Copyright of this article belongs to American Institute of Physics. |

ID Code: | 21916 |

Deposited On: | 23 Nov 2010 09:02 |

Last Modified: | 05 Mar 2011 11:58 |

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