Reduced Density Matrices for Valence Bond Wavefunctions. III. Density Matrices for the 4×4 Square Planar Net

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Abstract
One‐ and two‐particle reduced density matrices have been calculated for a ``resonating'' Heitler‐London wavefunction, in a system which is a finite model of a two‐dimensional square planar lattice structure (4×4 net of one‐electron atoms with Born‐von Karman boundary conditions). The Heitler‐London wavefunction describes the electronic ground state of a system with a single narrow half‐filled band (the Mott insulator). Various measures of electron correlation (pair correlation functions, etc.) are defined and calculated. The intent of these calculations is to provide ``empirical'' guidelines for many‐body methods for calculation of Green's functions for one and two particles (reduced density matrices are initial values for the respective Green's functions). Results show that the Heitler‐London description is incompatible with several assumptions and widely used models for the Mott insulator. In particular it is found that (a) pair correlations in the Heitler‐London ground state remain local even when ``resonance'' of an extremely large number of valence‐bond structures occurs; the long range antiferromagnetic ordering obtained in several models of the Mott insulator (the ``two‐sublattice'' or ``different bands for different spins'' models) does not occur. (b) The density matrix for the Heitler‐London state does not exhibit a BCS type ``pair condensation'', as has been conjectured to describe the Mott insulator in some recent work. Our results strongly suggest that if the Heitler‐London description is appropriate, the ``Hubbard Hamiltonian'', which includes only intraatomic pair interactions, does not offer an adequate model of the Mott insulator.

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