Abstract
We develop a formalism and present an algorithm for optimization of
the trial wave-function used in fixed-node diffusion quantum Monte
Carlo (DMC) methods. The formalism is based on the DMC mixed
estimator of the ground state probability density. We take
advantage of a basic property of the walker configuration
distribution function generated in a DMC calculation, to i)
project-out a multi-determinant expansion of the fixed node ground
state wave-function and ii) to define a cost function that relates
the interacting-ground-state-fixed-node and the
non-interacting-trial wave-functions. We argue in favor of the
conjecture that removing the kink of the fixed-node ground-state
wave-function at the node improves the resulting wave-function nodal
structure. If this conjecture is valid, then the noise in the
fixed-noded wave function resulting from finite sampling would play
a beneficial role, allowing the nodes to adjust towards the ones of
the exact many-body ground state in a simulated annealing-like
process. Based on these conjectures, we propose a method to improve
both single determinant and multi-determinant expressions of the
trial wave-function that can be generalized to other wave-function
forms such as pfaffians. We test the method in a model system where
a near analytical solution can be found. Comparing the DMC
calculations with the exact solutions, we find that the trial
wave-function is systematically improved. The overlap of the
optimized trial wave-function and the exact ground state converges
to 100\% even starting from wave-functions orthogonal to the exact
ground state. Similarly, the DMC total energy and density converges
to the exact solutions for the model. In the optimization process
we find an approximation optimal effective non-interacting
density-functional-like nodal potential whose existence was
predicted in a previous publication [Phys. Rev. B 77 245110
(2008)]. Tests of the method are extended to a model system with a
full Coulomb interaction where we show we can obtain the exact
Kohn-Sham effective potential from the DMC data.
