One and Two-body Functions for Promotion Calculations

To perform a VMC calculation using an excited state Slater determinant, it is necessary to optimise the Jastrow and functions to minimise the energy/variance of the energy of the wavefunction containing this new determinant. In an attempt to speed up the turn around for performing QMC calculations on excited states, it was decided to experiment with omitting the extra optimisation required to convert the ground state Jastrow and functions to excited state Jastrow and functions. Instead of performing this extra optimisation for each excitation we proceed straight to the DMC calculations. This is possible as long as the trial/guiding wavefunction is of sufficient quality. In the case of DMC calculations this implies two basic requirements;

1. The trial/guiding wavefunction for the excited state must be orthogonal to the ground state wavefunction and all lower energy exact eigenstates or the DMC algorithm will simply propagate out the lowest energy solution.
2. The trial/guiding wavefunction must have an acceptably low variance of the local energy to ensure that the fluctuations in the population of walkers are manageable (see chapters  and for details).

In all the excitation calculations described here, the excited orbitals in the excited state Slater determinants have different values to the ground state orbital they replace, i.e. they are indirect excitations. This ensures that, on the grounds of translational symmetry, all the excited state wavefunctions are orthogonal to the ground state wavefunction, hence satisfying condition (i). Condition (ii) is not so clear cut. In section , where one is adding and removing electrons from the system, the relaxation in the Jastrow and functions between systems with N-1, N and N+1 electrons was significant. This made it necessary to re-optimise the trial/guiding wavefunction for each system. When one promotes an orbital within the Slater determinant as in these calculations, the resulting change in the charge density and hence the change in the Jastrow and functions is not as severe and so it was decided to attempt the DMC calculations for promoted states using the same one- and two-body functions as those in the trial/guiding wavefunction used for the ground state DMC calculation. This will not affect the DMC estimate of the total energy, but it will increase its variance by an amount dependent on the quality of the trial/guiding wavefunction.