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7.1 Diffusion Monte Carlo (DMC)

First suggested in the forties (see [MEYERS 56]); rediscovered in the eighties ([CEPERLEY 80]).

Originally applied to the ground state of a bosonic system such as He [KALOS 74,WHITLOCK 79]. Later: extended to fermions and excited states ([BARNETT 86,CEPERLEY 88]); see also [CEPERLEY 96].

Time-dependent Schroedinger equation for a particle of mass in a potential : (7.1)

with the energy operator (7.2)

and a trial energy .

Define an imaginary time variable ; then (7.3)

with . Diffusion with autocatalysis! Visualize as describing the density of bacteria diffusing in a fluid with locally varying nutrient concentration.

Expand in eigenfunctions of :
• If (ground state energy), then all except will fade out for large times'' : (7.4)

• If , the total momentary weight will grow exponentially in time.
• If , the integral decreases exponentially in time. Solve 7.3 for various values of : find that for which remains stationary. Then and .

How to solve 7.3? Consider the diffusion and the autocatalysis parts of the equation separately.

Diffusion part: (7.5)

May be solved by a random walk procedure: Start Brownian walkers (7.6)

where are drawn from a Gauss distribution with .

Now consider an ensemble of such -particle systems and write the local distribution density at time as (7.7)

This is an estimate for the solution of the diffusion equation .

Autocatalysis part: (7.8)

Instead of writing the formal solution as (7.9)

we use once more a stochastic method.

Consider an ensemble of systems of particles each, at fixed positions. Let the number of systems in the ensemble be allowed to vary:
- Systems that contain many particles located at positions with high values of are replicated
- Systems with unfavorable configurations are deleted

Procedure:
Let the ensemble be given at step .
• For each of the systems, determine the multiplicity (see equ. 7.9) (7.10)

• Replicate the -th system such that on the average copies are present. To achieve this, produce first copies ( = next smaller integer) and then, with probability , one additional copy. (In practice, draw equidistributed and check whether .) If , remove, with probability , the -th system from the ensemble.
Again, the distribution density 7.7 is an estimate of the density at position .

Combining the two stochastic techniques for solving the diffusion and autocatalytic equations we obtain the following procedure. So far: must be real and  Bosons ( He or similar).

Generalization for fermions: fixed node and released node approximation [CEPERLEY 88].

Note: The analogy between the wave function and the local density is purely formal. It must be distinguished from the physical interpretation of .

Importance sampling DFT:
If the potential is highly negative in some region of space, the autocatalytic term may get out of control and must be handled differently:

- Introduce an estimate of the correct solution ;
- define an auxiliary function (7.11)

- insert this in 7.3 to find (7.12)

Since (7.13)

the autocatalytic term is now well-behaved, and the multiplicity will remain bounded.

Visualisation of equ. 7.12: The new term looks like an advective contribution. In the image of a diffusing and multiplying bacterial strain there is now an additional driving force (7.14)

which creates a flow, or drift. This means that the individual diffusors follow a preferred direction along : (7.15)

The multiplicity is now (7.16)

Green's function Monte Carlo (GFMC):
Another formulation of the DMC procedure [SKINNER 85].

Recent literature: See [CEPERLEY 96] and web sites [CEPERLEY WWW] or [CAVENDISH WWW].   Next: 7.2 Path Integral Monte Up: 7. Quantum Mechanical Simulation Previous: 7. Quantum Mechanical Simulation
Franz J. Vesely Oct 2005