Diffusion Monte Carlo (DMC)

The DMC method is based on the observation that the Schrodinger equation in imaginary time ($ t=i\tau$),

$\displaystyle -\frac{\partial \Psi(\mathbf R, \tau)}{\partial \tau}= -\frac{1}{2}\nabla^2\Psi(\mathbf R, \tau)+(V(\mathbf R)-E_T)\Psi(\mathbf R, \tau)  .$ (5)

is a diffusion equation plus a branching/annihilation term given by the potential. If the parameter $ E_T$ is tuned to the groundstate energy the trial wavefunction is projected to the ground state of the Hamiltonian. The anti-symmetric nature of the electronic wavefunction poses serious problems for the numerical solution of Eq ([*]). A way out of to this difficulty is to introduce the so called fixed-node approximation, in which a modification of Eq ([*]) is used, that projects the wavefunction onto the ground state wavefunction which has the same nodal surface of a trial wavefunction. Practical computations show that this is a very good approximation.