DMC algorithm

If the wavefunction of the system $f({\bf R}, t)$ is real and positive, as it happens in case of ground state of a bose system, it can be treated as population density distribution^3.1

$$f({\bf R}, t) = \sum\limits_{i=1}^{N_W} C \delta ({\bf R - R_i}(t)),$$ (3.26)

here $C$ is a positive constant, ${\bf R_i}(t)$ are coordinates of a population element (so called walker) in $3N$-dimensional configuration space, $f({\bf R}, t)\,{\bf dR}$ gives the probability to find a walker at time $t$ in vicinity ${\bf dR}$ of point ${\bf R}$. Let us now interpret the action of the each of the three terms of the Hamiltonian (3.15) on the population distribution or, being the same, the action of the corresponding Green's functions (3.21, 3.23, 3.25). In terms of Markov Chains the Green's function is the $G({\bf R},{\bf R'},t)$ is the transition matrix which determines the evolution of the distribution (see eq.(3.14)). The first term means diffusion of each of the walkers in configuration space

$${\bf R^{(1)}}(t+\triangle t) = {\bf R}(t) + {\bf\chi},$$ (3.27)

here ${\bf\chi}$ is a random value from a gaussian distribution $\exp(-\chi^2/(4 D \triangle t))$. The second term describes the action of the drift force, which guides the walkers to places in the configuration space, where the trial wavefunction is maximal. This is the way how importance sampling acts in this algorithm.

$${\bf R^{(2)}}(t+\triangle t) = {\bf R}(t) + \triangle t D F({\bf R})$$ (3.28)

The corresponding Green's functions of these two steps (3.21 - 3.23) are normalized to one $\int G({\bf x,x'},t)\,{\bf dx} = 1$. The normalization of wavefunction $f$ is then conserved meaning that the number of walkers remains constant. The third term is the branching term

$$f^{(3)}({\bf R}, t+ \triangle t) = \exp\left(-(E_L({\bf R})-E) \triangle t \right) f({\bf R}, t)$$ (3.29)

Here the corresponding Green's function (3.25) is no longer normalized and, when the quantity in the exponent in is negative (i.e. large values of local energy), then the density of population decreases and vice-versa.