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 $4$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 $m$ in a potential $U(\mbox{$\bf r$})$:

$$i \hbar \frac{\partial \Psi(\mbox{$\bf r$},t)}{\partial t} = \mbox{\rm H} \, \Psi(\mbox{$\bf r$},t)$$ (7.1)

with the energy operator

$$\mbox{\rm H} \equiv - \frac{\hbar^{2}}{2m} \nabla^{2} + \left[U(\mbox{$\bf r$})-E_{T} \right]$$ (7.2)

and a trial energy $E_{T}$.

Define an imaginary time variable $s \equiv it/\hbar$; then

$$\frac{\partial \Psi(\mbox{$\bf r$},s)}{\partial s} = D \nabl... ...eft[ U(\mbox{$\bf r$}) - E_{T} \right] \Psi(\mbox{$\bf r$},s)$$ (7.3)

with $D \equiv \hbar^{2}/2m$.

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

Expand $\Psi$ in eigenfunctions $\Psi_{n}$ of $\mbox{\rm H}$:

• If $E_{T}=E_{0}$ (ground state energy), then all $\Psi_{n}$ except $\Psi_{0}$ will fade out for large ``times'' $s$:
  

  $$\lim_{s \rightarrow \infty} \Psi(\mbox{$\bf r$},s)=\Psi_{0}(\mbox{$\bf r$})$$ (7.4)

  
  

• If $E_{T} > E_{0}$, the total momentary weight $I(s) \equiv \int\Psi(\mbox{$\bf r$},s) \, d\mbox{$\bf r$}$ will grow exponentially in time.
• If $E_{T}<E_{0}$, the integral $I(s)$ decreases exponentially in time.

$\Longrightarrow$ Solve 7.3 for various values of $E_{T}$: find that $E_{T}$ for which $I(s)$ remains stationary. Then $E_{T}=E_{0}$ and $\Psi = \Psi_{0}$.

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

Diffusion part:

$$\frac{\partial n(\mbox{$\bf r$},t)}{\partial t} = D\, \nabla^{2} n(\mbox{$\bf r$},t)$$ (7.5)

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

$$\mbox{$\bf r$}_{i}(t_{n+1})=\mbox{$\bf r$}_{i}(t_{n})+\mbox{$\bf\xi $}_{i}\,,\;\;\;i=1,\dots N$$ (7.6)

where $\xi_{x,y,z}$ are drawn from a Gauss distribution with $\sigma^{2}=2D\, \Delta t$.

Now consider an ensemble of $M$ such $N$-particle systems and write the local distribution density at time $t$ as

$$p(\mbox{$\bf r$},t) \equiv \langle \delta \left[ \mbox{$\bf ... ... \delta \left[ \mbox{$\bf r$}_{i,l}(t)-\mbox{$\bf r$} \right]$$ (7.7)

This is an estimate for the solution $n(\mbox{$\bf r$},t)$ of the diffusion equation $\ref{EQDMC2}$.

Autocatalysis part:

$$\frac{\partial n(\mbox{$\bf r$},t)}{\partial t} = f(\mbox{$\bf r$}) \, n(\mbox{$\bf r$},t)$$ (7.8)

Instead of writing the formal solution as

$$n(\mbox{$\bf r$},t)=n(\mbox{$\bf r$},0)\,exp \left[f(\mbox{$\bf r$})t\right]$$ (7.9)

we use once more a stochastic method.

Consider an ensemble of $M$ systems of $N$ particles each, at fixed positions. Let the number $M$ of systems in the ensemble be allowed to vary:
- Systems that contain many particles located at positions with high values of $f(\mbox{$\bf r$})$ are replicated
- Systems with unfavorable configurations are deleted

Procedure:
Let the ensemble be given at step $t_{n}$.

• For each of the $M(t_{n})$ systems, determine the multiplicity (see equ. 7.9)
  

  $$K_{l}=exp\left[ \sum_{i=1}^{N} f(\mbox{$\bf r$}_{i,l}) \, \Delta t \right]\,, \;\;\;l=1, \dots M(t_{n})$$ (7.10)

  
  

• Replicate the $l$-th system such that on the average $K_{l}$ copies are present. To achieve this, produce first $\mbox{int}(K_{l})-1$ copies ( $\mbox{int}(..)$= next smaller integer) and then, with probability $w \equiv K_{l}-\mbox{int}(K_{l})$, one additional copy. (In practice, draw $\xi$ equidistributed $\in [0,1]$ and check whether $\xi \leq w$.) If $K_{l} < 1$, remove, with probability $1-K_{l}$, the $l$-th system from the ensemble.

Again, the distribution density 7.7 is an estimate of the density at position $\mbox{$\bf r$}$.

Combining the two stochastic techniques for solving the diffusion and autocatalytic equations we obtain the following procedure.

Figure 7.1: Quantum mechanical diffusion Monte Carlo

So far: $\Psi$ must be real and $\ge 0$ $\Longrightarrow$Bosons ( $4$He or similar).

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

Note: The analogy between the wave function $\Psi(\mbox{$\bf r$},t)$ and the local density $n(\mbox{$\bf r$},t)$ is purely formal. It must be distinguished from the physical interpretation of $\vert\Psi (\mbox{$\bf r$})\vert^{2}=prob\{ quantum\;object\;to\; be\; found\; at\; \mbox{$\bf r$}\}$.

Importance sampling DFT:
If the potential $U(\mbox{$\bf r$})$ is highly negative in some region of space, the autocatalytic term may get out of control and must be handled differently:

- Introduce an estimate $\Psi_{T}(\mbox{$\bf r$})$ of the correct solution $\Psi_{0}(\mbox{$\bf r$})$;
- define an auxiliary function

$$f(\mbox{$\bf r$},s) \equiv \Psi_{T}(\mbox{$\bf r$})\,\Psi(\mbox{$\bf r$},s)$$ (7.11)

- insert this in 7.3 to find

$$\frac{\partial f}{\partial s} = D \nabla^{2} f -\left[\frac{... ...abla \cdot \left[ f\,\nabla \ln \vert\Psi_{T}\vert^{2}\right]$$ (7.12)

Since

$$\frac{\mbox{\rm H}\,\Psi_{T}}{\Psi_{T}} \approx E_{0} \approx E_{T}$$ (7.13)

the autocatalytic term is now well-behaved, and the multiplicity $K_{l}$ 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

$$\mbox{$\bf F$}(\mbox{$\bf r$}) \equiv \nabla \, \ln \vert\Psi_{T}(\mbox{$\bf r$})\vert^{2}$$ (7.14)

which creates a flow, or drift. This means that the individual diffusors follow a preferred direction along $\mbox{$\bf F$}(\mbox{$\bf r$})$:

$$\mbox{$\bf r$}_{i,l}(s_{n+1})=\mbox{$\bf r$}_{i,l}(s_{n})+\m... ...i,l}+D \Delta s \, \mbox{$\bf F$}(\mbox{$\bf r$}_{i,l}(s_{n}))$$ (7.15)

The multiplicity $K_{l}$ is now

$$K_{l}=exp\left\{ \left[ \frac{\mbox{\rm H}\Psi_{T}}{\Psi_{T}}-E_{T} \right]\, \Delta s \right\}$$ (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].