CASINO manual - Theory of Condensed Matter

the energy becomes roughly constant (at around 500 moves in this case). This splits the graph quite
neatly into an equilibration phase and a statistics-accumulation phase. When we average energies to
produce the final energy and error bar, we should only include those moves between 500 and the end
of the run. More detailed information about choosing the number of moves for equilibration is given
in Sec. 24.3.
To see DMC in action, go to ~/CASINO/examples/molecule/h2/RHF/dmc/ and let’s calculate the
DMC energy of a hydrogen molecule. The input files should already be set up correctly (though as
before you may need to gunzip the gwfn.data file). We see that an equilibrated VMC run is set to
go for vmc nstep= 1000 moves in order to produce vmc nconfig write= 1000 configurations or
walkers 10 . For DMC equilibration, we run for dmc equil nstep= 2000 moves, and for the statistics
accumulation we run here for dmc stats nstep= 100000 moves (note that unlike vmc nstep, these
are per processor quantities, since in DMC the parallelization is done over configurations, not steps.
Study the definition of these keywords carefully in Sec. 7.3). The dmc stats nblock keyword is set
to 5; this does not mean we do 5 blocks of 100000 moves each; rather, that we do 100000 moves and we
checkpoint the data (write to output etc.) 5 times over the course of that calculation, i.e., every 20000
moves. The dmc target weight parameter is set to 1000.0—the same as vmc nconfig write, but
note that they are allowed to differ. The dtdmc parameter is the DMC time step—note that it is
much smaller than the VMC time step (it needs to be small because the DMC Green’s function is
only exact in the limit that the time step goes to zero). A typical value in a DMC simulation with
all-electron ions might be 0.003, while a typical value in a simulation with pseudopotentials might be
0.02. Note that for accurate work, you need to consider extrapolating your DMC results to zero time
step (see the utility extrapolate tau, described in Sec. 10). Now type runqmc.
When casino has finished running, type reblock in the directory containing the dmc.hist file. reblock
will read these files and then starting asking you questions. How many initial lines of data do you
wish to discard? Usually the same as the number of lines of equilibration data—which is printed
out—though look at the ‘graphdmc’ plot to be sure. Find the move number where the green line in
the plot becomes approximately constant (e.g., 501 in the NiO case). Then, what units do you want
the answer in? Whatever you want. reblock will then show you the correlation-time analysis of the
error bar, and start the reblocking analysis, where you need to choose a block length. Choose a value
where the ‘Std err of mean’ column starts to plateau (again, the plot reblock utility can help you
visualize this). A value in the thousands might be typical; the value increases as you reduce the time
step. reblock will then print out the final DMC energies and error bars, together with an analysis of
the population fluctuations, effective time steps and acceptance ratios.
In the case of molecular hydrogen, for my short 5-minute run the final energy is −1.174322 ±
0.00018 a.u. The exact energy is −1.1744757 a.u., which is within our reblocked error bar, so this
result is pretty good without paying particular attention to getting it absolutely right—this is due to
the system being exactly solvable in DMC due to the lack of nodes in the wave function.
6.5 How to perform QMC calculations for periodic systems
Suppose you wish to use QMC to study condensed matter rather than isolated atoms or molecules.
For example, you might be interested in the bulk properties of a crystalline solid. Obviously, in a
QMC simulation you can only study a small ‘simulation cell’ of such a crystal. Hopefully this cell will
be sufficiently large that you are able to obtain the bulk properties of the material. Fortunately the
boundary conditions on the simulation cell can be chosen to maximise the rate of convergence of the
bulk properties with respect to the size of the cell. Periodic boundary conditions are a highly efficient
and convenient choice, and are ubiquitous in computational studies of condensed matter.
According to Bloch’s theorem, single-particle orbitals in a periodic crystal can be written as ψ k (r) =
u k (r) exp(ik · r), where u k (r) has the periodicity of the primitive cell and k lies in the first Brillouin
zone of the primitive cell. The allowed Bloch vectors k are determined by the boundary conditions on
the simulation supercell. Under periodic boundary conditions the allowed k are the reciprocal lattice
points of the supercell. Under so-called twisted boundary conditions the allowed k are the reciprocal
10 What is a typical number of DMC walkers? We usually say ‘around 1000’. Of course this is a bit like the length of a
piece of string. Nevertheless, for most systems with most wave functions on most computers it ensures that populationcontrol
bias—see later—is completely negligible, but still lets you gather data for a sufficiently long period in imaginary
time that you can see a plateau on your reblock.plot without heroic computational effort. If one has to give a single
number for the target weight, clearly 1000 configurations is as sensible a suggestion as any. Obviously if you want to
run on 20000 cores or whatever then you need a larger number of configurations.
23