7.1 Diffusion Monte Carlo (DMC)

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) |

(7.2) |

Define an imaginary time variable ; then

with .

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.

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

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

(7.6) |

Now consider an

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

(7.8) |

we use once more a

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

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.

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:

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) |

Since

(7.13) |

Visualisation of equ. 7.12: The new term looks like an

(7.14) |

(7.15) |

(7.16) |

Another formulation of the DMC procedure [SKINNER 85].

See also: