Diffusion Monte Carlo

Diffusion Monte Carlo (DMC) represents imaginary-time projection by diffusing and branching walkers. For a positive ground-state problem it can systematically improve a trial amplitude. Fermionic antisymmetry requires additional treatment; the widely used fixed-node constraint introduces an approximation whose error is not removed by more walkers or smaller time steps.

Required background. Ground-state projection explains spectral filtering, and VMC defines the trial function and local energy.

Diffusion and branching from imaginary time

Write $H=-D\nabla^2+V$, where $D=\hbar^2/(2m)$ for equal-mass Cartesian coordinates. Use an imaginary-time parameter $\tau$ with units of inverse energy:

$$\partial_\tau\psi =D\nabla^2\psi-(V-E_{\rm ref})\psi.$$

The diffusion term produces Gaussian displacements of covariance $2D\Delta\tau I$. The potential term changes weights or offspring counts. $E_{\rm ref}$ controls the overall population scale rather than the physical eigenfunctions.

With a positive importance function $\psi_T$ and $f=\psi_T\psi$,

$$\partial_\tau f =D\nabla^2f-\nabla\cdot(bf) -(E_L-E_{\rm ref})f, \qquad b=2D\nabla\log\psi_T.$$

This is drift-diffusion plus branching. Deriving the divergence form before coding prevents a missing drift-divergence contribution from being mistaken for a harmless convention.

A practical short-time step

Move a walker using $x'=x+b(x)\Delta\tau+\sqrt{2D\Delta\tau}\,\eta$, with standard normal $\eta$, and apply an appropriate short-time branching factor. A symmetric local-energy factor is commonly based on the average of $E_L(x)$ and $E_L(x')$. The detailed drift proposal, accept/reject correction, branch rule, and treatment near nodes determine the actual time-step error order; do not assume all DMC implementations are quadratic.

Population control correlates walkers. Resampling a finite ensemble can bias energy estimates, especially when local-energy fluctuations are large. Independent runs and blocking are necessary, and population size must be varied independently from time step.

An exact guiding-function benchmark

For the dimensionless oscillator, $D=1/2$ and $\psi_T=e^{-x^2/2}$. Then $b=-x$ and $E_L=1/2$. With $E_{\rm ref}=1/2$, branching weights are constant. The continuous drift-diffusion process is Ornstein–Uhlenbeck and has stationary variance $1/2$.

Euler drift discretization gives $x'=(1-\Delta\tau)x+\sqrt{\Delta\tau}\eta$. Its stationary variance is

$$v=\frac{\Delta\tau}{1-(1-\Delta\tau)^2} =\frac1{2-\Delta\tau},$$

for $0<\Delta\tau<2$. Thus the reported local energy can remain exactly $1/2$ while the sampled position distribution has a time-step error. Checking $\langle x^2\rangle$ exposes what energy alone misses.

Fermions and estimator limitations

A fixed-node calculation constrains walkers to nodal regions of a trial antisymmetric wavefunction. For a local real Hamiltonian under the usual fixed-node variational assumptions, the projected energy is an upper bound determined by those nodes. Improving amplitudes within fixed nodes does not remove nodal bias. Nonlocal pseudopotentials require a specified treatment and can alter variational properties.

DMC naturally samples a mixed distribution proportional to $\psi_T\psi_0$. The local-energy mean yields the projected energy, but a general operator not commuting with $H$ has a mixed-estimator bias. The extrapolation $2\langle O\rangle_{\rm mixed}-\langle O\rangle_{\rm VMC}$ cancels leading trial-state error only under a small-error expansion; it is not an exact identity.

Exercises

Which limits must be varied to distinguish an accurate projected energy from an accurate fermion ground energy?

Solution

Converge projection/equilibration time, production statistics, time step, population size, and spatial or basis approximations. Separately improve the nodal or phase constraint. The first set can converge precisely to a constrained answer that still differs from the fermion ground state. Also validate the estimator used for each non-energy observable.

References

• Foulkes, W. M. C., Lubos Mitas, R. J. Needs, and G. Rajagopal, “Quantum Monte Carlo simulations of solids,” Reviews of Modern Physics 73, 33–83 (2001), doi:10.1103/RevModPhys.73.33 — variational and diffusion Monte Carlo with error analysis.