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Energy Minimization

Energy minimization is the most common optimization strategy in modern QMC. It aims to find the parameters that minimize the expectation value of the Hamiltonian.

Stochastic Reconfiguration (SR)

Stochastic Reconfiguration (SR) is a robust energy minimization method similar to the natural gradient method. It is selected with method = 'sr_n'.

Key Parameters

  • sr_tau: The time step (step size) for the parameter update. Typical values are 0.01 - 0.1.
  • sr_eps: A small regularization parameter to stabilize the inversion of the overlap matrix.
  • sr_adiag: Diagonal shift for the overlap matrix (similar to sr_eps).

Usage Example

%module optwf
    ioptwf      1
    method      'sr_n'
    nopt_iter   20
    sr_tau      0.05
%endmodule

Comparison with Variance Minimization

While variance minimization is robust, there are strong motivations for optimizing the energy directly:

  1. Primary Goal: One typically seeks the lowest energy in a VMC or DMC calculation, rather than the lowest variance.
  2. Parameters: Variance minimization is highly effective for Jastrow coefficients. However, for determinantal coefficients (coefficients of determinants, orbital expansions, csf coefficients), it can take many iterations and get stuck in local minima. Most authors use variance minimization primarily for Jastrow parameters.
  3. Observables: For a given trial wave function form, energy-minimized wave functions on average yield more accurate values of other expectation values.
  4. Forces: The Hellmann-Feynman theorem can be better exploited with energy-minimized wave functions to compute forces on nuclei.

Despite these points, variance minimization remains a cornerstone technique, particularly for the stable optimization of Jastrow factors.