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Evaluating the Kinetic Energy

The single-particle kinetic energy operator for electron i is


The expected kinetic energy of electron i is therefore


This quantity is obtained using a Monte Carlo integration as described above. The Metropolis algorithm is used to sample the probability distribution tex2html_wrap_inline6245 , where tex2html_wrap_inline5833 is the wavefunction described in the previous section, and the estimator


is accumulated over the simulation to give the kinetic energy of electron i.

The calculation of tex2html_wrap_inline6251 is actually performed in two parts due of the form of the wavefunction being used. The trial wavefunction involves exponentials of the functions u(r) and which make it convenient to deal with logarithms of the wavefunction rather than differentiating the wavefunction directly. Defining






If one considers the trial wavefunction in Eq.(gif), introduced in the previous section, then




tex2html_wrap_inline6255 and tex2html_wrap_inline6257 are calculated from these equations at each step in the random walk. The kinetic energy as given by Eq.(gif), is also calculated at each step and averages of all three quantities are found at the end of the simulation. The consistency of these three is checked using Green's relation, which shows that


for all properly sampled wavefunctions. This consistency check is extremely useful when debugging a QMC code. If either the first or second derivative of the wavefunction has been calculated incorrectly, this will immediately show up in this consistency check and it is often clear which of the derivatives is being evaluated wrongly. The variances of tex2html_wrap_inline6259 and tex2html_wrap_inline6261 are both much greater[26, 23] than the variance of the kinetic energy as given by Eq.(gif), therefore it is this quantity which is used to estimate the kinetic energy in Monte Carlo calculations.

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996