qmc.rst

Quantum Monte Carlo: a 1 min introduction

Quantum Monte Carlo simulations rely on the variational principle:

$$E = \frac{\int \Psi^*_\theta(R) \; H \; \Psi_\theta(R) dR}{\int |\Psi_\theta(R)|^2} \geq E_0$$

where Ψ_θ(R) is the wave function of the system computed for the atomic and electronic positions R, and with variational parameters θ, H is the Hamiltonian of the system given by:

$$H = -\frac{1}{2}\sum_i \Delta_i + \sum_{i>j} \frac{1}{|r_i-r_j|} - \sum_{i\alpha} \frac{Z_\alpha}{|r_i-R_\alpha|} - \sum_{\alpha>\beta}\frac{Z_\alpha Z_\beta}{|R_\alpha-R_\beta|}$$

where Δ_i is the Laplacian w.r.t the i-th electron, r_i is the position of the i-th electron, R_α the position of the α-th atom and Z_α its atomic number. QMC simulations express this integral as:

E = ∫ρ(R)E_L(R)dR ≥ E₀

with:

$$\rho(R) = \frac{|\Psi_\theta(R)|^2}{\int |\Psi_\theta(R)|^2 dR}$$

reprensent the denisty associated with the wave function, and:

$$E_L(R) = \frac{H\Psi_\theta(R)}{\Psi_\theta(R)}$$

are the so called local energies of the system. QMC simulation then approximated the total energy as:

$$E \approx \frac{1}{M}\sum_i^M \frac{H\Psi_\theta(R_i)}{\Psi_\theta(R_i)}$$

where R_i are samples of the density ρ for example obtained via Metropolis Hasting sampling. QMC simulations rely then on the optimization of the variational parameters of the wave function, θ, to minimize the value of the total energy of the system.