12.4 Path-integral Monte Carlo 407to the time-evolution operator. The effect of averaging over all the continuous
paths from(r,τ)to(r′,τ+
τ ), as is to be done when calculating the exact time
evolution, is that the divergences atr,r′ =0 are rounded off. So if we could
find a better approximation to this exact time evolution than the primitive one, we
would not suffer from the divergences any longer. Several such approximations
have been developed [ 22 , 23 ]. They are based either on exact Coulomb potential
solutions (hydrogen atom) or on the cumulant expansion. We consider the latter
approximation in some detail in Problems 12.2 and 12.3; here we shall simply
quote the result:
Vcumulant(r,r′;
τ )=∫
τ0dτ′erf[r(τ′)/√
2 στ′]
r(τ′), (12.87a)where
r(τ′)=r+
τ′
τ(r′−r) and σ(τ′)=
(
τ−τ′)τ′
τ. (12.87b)
The cumulant approximation forVcan be calculated and saved in a tabular form, so
that we can read it into an array at the beginning of the program, and then obtain the
potential for the values needed from this array by interpolation. In fact, for
τfixed,
Vcumulantdepends on the norms of the vectorsrandr′and on the angle between
them. Therefore the table is three-dimensional. We discretiserin, say, 50 steps r
between 0 and some upper limitrmax(which we take equal to 4), and similarly for
r′. For values larger thanrmaxwe simply use the primitive approximation, which
is sufficiently accurate in that case. For the angleθin betweenrandr′we store
cosθ, discretised in 20 steps between−1 and 1 in our table. For actual values
r,r′andu =cosθwe interpolate linearly from the table – see Problem 12.4.
Figure 12.5shows the cumulant potentialV(r=r′,θ=0;
τ=0.2), together
with the Coulomb potential; the rounding effect of the cumulant approximation is
clear. In a path-integral simulation for the hydrogen atom we find a good ground
state distribution, shown in Figure 12.6. For the energy, using the virial estimator
with the original Coulomb potential (which is of course not entirely correct), we
findEG=−0.494±0.014, using
τ=0.2, 100 time slices and 60 000 MC steps
per particle, of which the first 20 000 were removed for equilibration.
Applying the method to helium is done in the same way. Using 150 000 steps
with a chain length of 50 andτ=0.2, the ground state energy is found as 2.93±
0.06 atomic units. Comparing the error with the DMC method, the path-integral
method does not seem to be very efficient, but this is due to the straightforward
implementation. It is possible to improve the PIMC method considerably as will
be described in the next section.
The classical example of a system with interesting behaviour at finite temperature
is dense helium-4. In this case the electrons are not taken into account as independent