Variational Monte Carlo

Considering the Born-Oppenheimer approximation, a molecule with n_el electrons and N_atoms nuclei can be described by the time-independent Schrödinger equation

with the Hamiltonian

By r=(r1,…,rn↑,…,rnel)∈R3×nel we denote the set of electron positions divided into n_↑ spin-up and n_↓ = n_el − n_↑ spin-down electrons. The solution to the electronic Schrödinger equation ψ needs to fulfill the anti-symmetry property, i.e. ψ(Pr)=−ψ(r) for any permutation P of two electrons of the same spin. Finding the groundstate wavefunction of a system corresponds to finding the solution to Eq. (6), with the lowest eigenvalue E₀. Using the Rayleigh-Ritz principle, an approximate solution can be found through minimization of the loss

using a parameterized trial wavefunction ψ_θ. The expectation value in Eq. (8) is computed by drawing samples r from the unnormalized probability distribution ψθ2(r) using Markov Chain Monte Carlo (MCMC). The application of the Hamiltonian to the wavefunction can be computed using automatic differentiation and the loss is minimized using gradient based minimization. A full calculation typically consists of three steps:

To obtain a single wavefunction for a dataset of multiple geometries or compounds, only minimal changes are required. During supervised and variational optimization, for each gradient step we pick one geometry from the dataset. We pick geometries either in a round-robin fashion, or based on the last computed energy variance for that geometry. We run the Metropolis Hastings algorithm³⁵ for that geometry to draw electron positions r and then evaluate energies and gradients. For each geometry we keep a distinct set of electron samples r.