392 Quantum Monte Carlo methods
The FP-diffusion term will be used to diffuse the walkers, whereas the ‘potential’
EL(R)−ETis used in a branching process. By writing out all the terms on the left
and right hand sides of Eq. (12.58), it can be checked that this equation reduces to
the imaginary time-dependent Schrödinger equation (12.33).
The procedure is now a combination of the Fokker–Planck VMC and of the
DMC method without guide function: we let the walkers diffuse just as in the
Fokker–Planck VMC method, with a transition probability
T
τ(Rn→Rn+ 1 )=1
√
2 π
τexp{−[Rn+ 1 −Rn−F(Rn)
τ/ 2 ]^2 /( 2
τ )}.
(12.60)
Then branching is performed, according to the valueq=exp{−
τ[EL(R)−ET]}.
What do we gain by this method? We avoid problems of the kind encountered above
with strongly varying potentials. The role ofVin standard DMC is now taken over
byEL(R), which is (we hope) rather flat. IfT(R)were anexacteigenstate, thenEL
would be independent ofR.IfTis a reasonable approximation to the ground state,
thenEL(R)is reasonably flat, and the method will be reliable. It is clear now why the
cusp conditions are so important: they guarantee that the trial function converges
to the exact solution in those regions where the potential diverges strongly. These
are the points that cause problems. The method using trial – or guide – functions
was introduced by Kalos [8] and is commonly calledimportance sampling Monte
Carlo.
We can again correct for the time step error using a Metropolis procedure, just
as we did for VMC in Section 12.2.5. Note thatGis not symmetric, so we must use
the generalised Metropolis method in order to guarantee detailed balance (see also
the variational Fokker–Planck simulation). A trial displacement is accepted with
probability
min(
1,
T
τ(R′→R)ρ(R′)
T
τ(R→R′)ρ(R))
(12.61)
and rejected otherwise.
With importance sampling, the algorithm reads:
Put the walkers at random positions in configurational space;
REPEAT
FOR all walkers DO
Shift walker from its positionRto a new positionR′
by first moving it over a distanceF
τ/2 and then
adding a random displacement according to the
transition probability (12.24);
Accept the move with a probability given by (12.61);
IF Accepted THEN