CASINO manual - Theory of Condensed Matter

• In Movie Settings choose Trajectory. The movie can be saved in the AVI or MPEG format.
Choose by clicking on Format and tick the preferred format. Then check whether the name of
the temporary directory suggested is right (this is where the RGB files are created). Note that
this directory should be free of RGB files belonging to other users. If this has to be changed
then click on the Set working directory button and browse for the directory.
• Type in the name of the movie in the box provided. Click on the Make Movie button.
• The movie will be displayed in the Open GL Display screen. The .mpg or .avi movie file
will be produced in the working directory being specified. They have to be viewed with other
viewers, for example mpeg play for .mpg files.
For an example movie made with vmd (a casino VMC simulation of cyclohexane) see www.tcm.phy.
cam.ac.uk/~mdt26/downloads/cyclohexane2.mpg.
11.2.2 JMOL
jmol is a free, open source molecule viewer. It supports computers running Windows, Mac OS X and
Linux/Unix systems. Jmol can be downloaded from jmol.sourceforge.net/
• Type jmol. Click on File→ Open. Browse for the file movie.out and click Open.
• Click on Display and untick the box for Bonds.
• Click on Extras → Animate. An animation tool bar will appear. To start the movie click on
the ‘play’ symbol.
12 Detailed information: the VMC method
12.1 Evaluating expectation values
The expectation value of the Hamiltonian Ĥ with respect to the trial wave function Ψ can be written
as
∫ 〈Ĥ〉 = EL (R)|Ψ(R)| 2 dR
∫
|Ψ(R)|2 dR , (11)
−1
where E L (R) = Ψ (R)Ĥ(R)Ψ(R) is the local energy. We can evaluate this expectation value by
using the Metropolis algorithm [24] to generate a sequence of configurations R distributed according
to |Ψ(R)| 2 and averaging the corresponding local energies.
12.2 The sampling algorithm
The implementation of VMC in casino involves making trial moves, whether of a single electron or
of the entire configuration, and accepting or rejecting the moves in accordance with the Metropolis
algorithm. The Metropolis transition probability density is Gaussian of variance τ, where τ is the
VMC time step (dtvmc).
In the electron-by-electron algorithm, each step consists in proposing individual moves for each of the
electrons and subject each move to a separate accept/reject step. In the configuration-by-configuration
algorithm one configuration move is proposed per step, and is accepted or rejected as a whole.
Another difference in the casino implementation of these two methods is that in the configuration-byconfiguration
the local energy (and all other expectation values) are evaluated both before and after
the move, and it is the average of the two, weighted by the acceptance probability, that enters the
accumulation arrays. In the electron-by-electron algorithm we only evaluate the energy after having
moved the configuration.
The configuration-by-configuration algorithm has the disadvantage of suffering from long correlation
times, which makes it in practice more expensive than the electron-by-electron algorithm in virtually
all cases. Thus we will restrict our discussion below to the electron-by-electron algorithm.
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