A DMC Algorithm

The algorithm used in the DMC calculations reported in this thesis for solving Eq.() is illustrated in figure . It can be divided into the following steps:

  
Figure: Flow chart illustrating the DMC algorithm.

1. Initialise a set of configurations (typically 100-500), with configurations distributed according to a probability density , given by . These are generated by choosing configurations produced in a VMC run. The configurations are taken sufficiently far apart in the VMC run so as to ensure their independence, using the procedure described in section .
2. Each configuration in the list is taken in turn. The electrons in the configuration are then moved, one at a time, by letting them diffuse independently for a time . If the current electron is the electron in configuration j, it is moved according to

   

   where is the three-dimensional coordinate of the electron being moved, and is a three-dimensional Gaussian random variable with a mean of zero and a variance of . The time step is chosen to be small enough so that the time step error[30], introduced by the approximation to the Green's function in Eq.(), is less than the statistical uncertainty. Usually is chosen so that the rate of acceptance of moves is .

3. The fixed-node approximation is now applied. We check to see whether the move has caused the walker to cross the nodal surface. If this has occurred we reject the move and go to the next electron on the list.
4. Now calculate a weight for the move,

   

   The move is then accepted with probability

   

5. After all the electrons in the configuration have been moved the time associated with this configuration is advanced by one time step, .
6. Calculate the branching probability for configuration j, using:

   

   Then make

   

   copies of the walker, where is a uniform random number in the range [0,1].

7. If then weight and any other accumulated quantities by the branching probability, . This branching probability is a measure of the importance weight of the walker.
8. is an adjustable energy used to stabilise the population of walkers. is recalculated according to

    

   Where is a parameter adjusted so as to control population fluctuations. is the number of configurations at the present time and is the target number of configurations.

9. Repeat steps (2) through to (8) for all the configurations for a certain number of steps. Note that the total number of steps is divided up into blocks. The number of steps in each block is chosen such that the averages accumulated over the blocks of steps are statistically independent. This block length usually corresponds to diffusing each walker for a time of order a few atomic units.
10. Calculate the weighted mean of and all other desired averages. It is not necessary to use the more complicated scheme for updating the averages (section ) used in VMC as the probability of accepting a given move is much higher (99%) due to the smaller time step used in DMC calculations.
11. Use the cumulative estimate of to update the trial energy . We do this according to , where is the block average of the energy. Note, the initial value of is usually taken as the VMC energy for the same system.
12. Renormalise the number of walkers to ; do this by randomly removing walkers ( ), or creating copies of existing walkers ( ), where is the desired number of walkers.
13. Repeat steps (2) to (12) until there is no longer a detectable trend in the block average of , i.e. the components of excited eigenstates in the wavefunction have effectively decayed away as shown in Eq.(). At this point the steady-state has been reached.
14. Repeat steps (2) through to (12) until the variance of the cumulative averages has reached the desired level.

At the end of a DMC calculation the average values of , , and should be the same to within error bars.

The output from an example DMC calculation is shown in figure . It illustrates the initial exponential decay in the energy, taking place over the first 150 or so time steps, as the contribution to the distribution, f, from the excited states decays with time. When the graph levels out, the energy is oscillating due to statistical noise. Averages for observable quantities would be taken over the final 800 steps.

 

It is useful to note that if the weights in step (4) are set to

as in VMC, the drift term is removed, and the branching probabilities in step (7) are set to unity for all configurations, the algorithm reduces back to the VMC algorithm already described. This not only highlights the many similarities between the two algorithms, but also provides a useful method of checking whether a DMC algorithm is working correctly as it is can then be directly compared with known VMC results.