Derivative operators and forward walking in DMC

Using Diffusion Monte Carlo (DMC) to compute estimators other than the energy leads to a "mixed estimator"- an average that combines the trial wavefunction and the exact ground state (actually the fixed node ground state) - not what we really want. However, the number of children of each walker after a long time is proportional to the ratio of the fixed node ground state to the trial wavefunction. This leads to the forward walking procedure for obtaining correct averages over the fixed node ground state. See Casulleras and Boronat ( PRB 52, 3654) for examples, references, and implementation details.

Numerical values for derivative operators, such as the kinetic energy, are computed by applying the operator to the trial wavefunction. The forward walking procedure only corrects the distribution in the average, not the values of the estimator. (we really want the derivative applied to the exact wavefunction, not the trial wavefunction).

Computing the kinetic energy (and the pressure) by naively doing forward walking on the kinetic energy is incorrect. For the kinetic energy, we can use the total energy and the forward walked potential energy to obtain the kinetic energy.