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Spectrum Folding

It is theoretically possible to use the fact that the variance of the local energy of an eigenstate of the Hamiltonian is zero to calculate excitation energies within QMC. The zero variance property of the ground state has already been utilised within the variance minimisation procedure described in chapter gif. In that case, the zero variance provides a useful lower bound to the quantity being minimised, namely the variance.

It is possible to extend this principle to search for the excited eigenstates of the same Hamiltonian. An ensemble of independent configurations were sampled from the ground state wavefunction using the procedure described in chapter gif. The variance of the local energy of this ensemble was then evaluated using the following expression,


where tex2html_wrap_inline8329 is the energy about which the variance is to be evaluated. The variance about tex2html_wrap_inline8329 was then minimised with respect to the variational parameters in the wavefunction for a series of values of tex2html_wrap_inline8329 (i.e. scan over the energy range). The hope was that around each eigenvalue of the Hamiltonian, the variance should decrease considerably hence indicating the presence of such an eigenvalue. This is represented schematically in figure gif.

Figure: Schematic representation of the relationship between the local energy and its variance.

Unfortunately, we found that any small changes in the variance due to changes we made in the value of tex2html_wrap_inline8329 were not distinguishable from the statistical noise present. To make this procedure effective, a considerably larger number of configurations would be required than are normally used in the variance minimisation procedure. This makes the technique prohibitively expensive.

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996