Variational Monte Carlo (VMC)

VCM calculations use a trial function $ \Psi(\alpha, \mathbf R,\mathbf
R_I)$ whose parameters $ \alpha_{ }$ are determined from the minimisation of the average energy or its variance:

$\displaystyle E(\alpha)=\frac{\langle \Psi(\alpha)\vert H \vert\Psi(\alpha) \rangle} {\langle \Psi(\alpha)\vert \Psi(\alpha) \rangle}  .$ (1)

A typical trial function for an electronic system is the product between the Slater determinants of one particle orbitals (OPO), obtained from a DFT calculation, multiplied by a function that describes the particle correlations.

$\displaystyle \Psi(\alpha, \mathbf R, \mathbf R_{I})$ $\displaystyle =e^{J(\alpha,\mathbf R,\mathbf R_I)}D_{\uparrow}(\mathbf r_1,\dots,\mathbf r_{N_\uparrow})$ (2)
  $\displaystyle \times D_{\downarrow}(\mathbf r_{N_\uparrow+1\dots,\mathbf r_{N_e}})  ,$ (3)

where $ D_{\uparrow,\downarrow}$ are the Slater determinants of the electrons with spin up($ \uparrow$) or down( $ \downarrow$). The simplest Jastrow factor $ J(\alpha, \mathbf R, \mathbf R_I)$ is a sum of two-electron functions

$\displaystyle J(\alpha, \mathbf R, \mathbf R_I)=-\sum_{\substack{i>j  \sigma_i,\sigma_j}} u_{\sigma_i,\sigma_j} (\alpha,\vert r_i-r_j\vert)$ (4)

where u is taken from the homogeneous electron gas. Over the years more elaborate extensions of $ J(\alpha, \mathbf R, \mathbf R_I)$ have been developed in order to include three electron correlations and electron ion correlations [4].

Many studies have shown that VMC calculations recover up to $ 70-90\%$ of the correlation energy but the remaining contributions are hard to conquer because the results cannot be improved systematically in the VMC framework. As a matter of fact the main use of VMC calculations in CASINO is to generate the configuration set needed for the Diffusion Monte Carlo calculation.