Equations for linked clusters and the energy variational principle

R. F. Bishop, J. Arponen, Erkki Pajanne

J. Arponen

Isaiah Shavitt, Rodney J. Bartlett

Leo P. Kadanoff, Mahito Kohmoto

The ground state wavefunction of an infinitely large, spatially homogeneous system is evaluated in terms of wavefunctions for linked clusters of points within the system. The linked cluster wavefunction is shown to obey a nonlinear form of the Schrodinger equation, with the unlinked pieces providing the nonlinearity. A variational principle for the ground state energy is derived.

R. Brout

The general form of the energy of the ground state of a many-fermion system is shown to be exactly of the form proposed by Brueckner and Bethe, without approximation. In a variational treatment, if the trial wave function is picked containing only pair correlations, together with all possible unlinked pairs, it is described by a two-body excitation matrix $〈{m}_{1}{m}_{2}|A|{p}_{1}{p}_{2}〉$. Variation of this matrix in the Ritz-Rayleigh principle yields a set of integral equations of the scattering type for the matrix $A$. Hole-state energies are given self-consistently in terms of the matrix $A$, but particle-state energies are Hartree-Fock energies. This may be corrected for by widely enlarging the class of terms admitted into the wave function. If the approximation is then made of omitting a class of terms, defined as cross-linked clusters in $〈\ensuremath{\psi}|H|\ensuremath{\psi}〉$, the particle-state energies are easily renormalized. Variation then leads to an infinite hierarchy of integral equations.

Andrei Derevianko