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Symmetry properties of the configuration interaction space in relation to one‐ and two‐particle operators: The splitting theorem

### Abstract

The configuration interaction (CI) space *X* _{ n } built upon *n* electrons moving over 2*n* orthonormalized orbitals χ_{ i } is considered. It is shown that the space *X* _{ n } splits into two complementary subspaces*X* ^{+} _{ n } and *X* ^{−} _{ n } having special properties: each state Ψ^{+}∈*X* ^{+} _{ n } and Ψ^{−}∈*X* ^{−} _{ n } is ‘‘alternantlike’’ in the sense that it has a uniform charge density distribution over all orbitals χ_{ i } and vanishing bond‐orders between all orbitals of the same parity. In addition, matrix elements Γ(*i* *j*;*k* *l*) of a two‐particle density matrix vanish whenever four distinct orbitals are involved and there is an odd number of orbitals of the same parity. Further, Γ(*i* *j*;*l* *j*)=γ(*i* *l*)/4 ( *j*≠*i*,*l*), whenever (*i*) and (*l*) are of different parity. This last relation shows the connection between a two‐particle (Γ) and a one‐particle (γ) density matrix. ‘‘Elementary’’ alternant and antialternant operators are identified. These operators connect either only the states in the same subspace, or only the states in different subspaces, and each one‐ and two‐particle symmetric operator can be represented by their linear combination. Alternant Hamiltonians, which can be represented as linear combinations of elementary alternant operators, have alternantlike eigenstates. It is also shown that each symmetric Hamiltonian possessing alternantlike eigenstates can be represented as such a linear combination. In particular, the PPP Hamiltonian describing an alternant hydrocarbon system is such a case. Complementary subspaces*X* ^{+} _{ n } and *X* ^{−} _{ n } can be explicitly constructed using the so‐called regular resonance structures (RRS’s) which are normalized determinants containing mutually disjunct bond orbitals. Expressions for the derivation of matrix elements of one‐ and two‐particle operators between different RRS’s are also derived.

© 1984 American Institute of Physics

Received 18 October 1983
Accepted 10 February 1984

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/content/aip/journal/jmp/25/9/10.1063/1.526483

1984-09-01

2016-09-30

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