The approximation (3.18) of the Green's function
has first order accuracy in the timestep. High order approximations
can be used. One of the possibilities to gain second order accuracy
is to use the formula
(3.92)
The result for the energy in the DMC algorithm depends on the value
of the timestep used. The exact ground-state energy is obtained by
extrapolating the results to the zero timestep. Approximation
(3.92) for the evaluation operator leads to a
quadratic dependence of the energy on the timestep. The result of
such a calculation is presented in Fig. 3.3. In this
respect the use of a quadratic algorithm, such as (), is preferable because for small timestep the
results are less sensitive to the choice of the timestep and with a
judicious choice one does not need to extrapolate.
Figure 3.3:
Hard spheres at
.
Dependence of the energy on the time step.
On one side the timestep has to be small so that the approximation
in the Green's function is good, on the other side the larger is
the timestep the faster the phase space is explored and less number
of iterations are needed for the same statistical accuracy.
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