Variational Monte Carlo (VMC)¶

Variational Monte Carlo (VMC) is a quantum Monte Carlo method that evaluates the expectation value of the Hamiltonian (energy) and other properties for a given trial wavefunction.

Overview¶

In VMC, the energy is calculated as:

\[ E_V = \frac{\int \Psi_T^*(\mathbf{R}) \hat{H} \Psi_T(\mathbf{R}) d\mathbf{R}}{\int |\Psi_T(\mathbf{R})|^2 d\mathbf{R}} = \int P(\mathbf{R}) E_L(\mathbf{R}) d\mathbf{R} \]

where: - \(\Psi_T(\mathbf{R})\) is the trial wavefunction. - \(P(\mathbf{R}) = \frac{|\Psi_T(\mathbf{R})|^2}{\int |\Psi_T(\mathbf{R})|^2 d\mathbf{R}}\) is the probability distribution sampled by the Metropolis algorithm. - \(E_L(\mathbf{R}) = \frac{\hat{H} \Psi_T(\mathbf{R})}{\Psi_T(\mathbf{R})}\) is the local energy.

Key Features¶

Zero-Variance Principle: If \(\Psi_T\) is the exact eigenstate, \(E_L(\mathbf{R})\) is constant (the eigenenergy), and the statistical error vanishes.
Optimization: VMC is used to optimize the parameters of the trial wavefunction (Jastrow, orbitals) by minimizing the energy or variance.
Walker Generation: VMC generates configurations (walkers) distributed according to \(|\Psi_T|^2\), which are used as starting points for DMC calculations.

Running VMC¶

VMC calculations are controlled by the blocking_vmc module in the input file.

%module blocking_vmc
    vmc_nstep     20
    vmc_nblk      200
    vmc_nblkeq    1
    vmc_nconf_new 100
%endmodule

See Input Keywords for details on configuration.