Computational Physics

406 Quantum Monte Carlo methods

is that 1/βand〈K〉/(NM)are both large, but their difference is small. Hermanet al.
[20]have proposed a different estimator for the energy, given by
〈
E
N

〉

β

=

〈

1

M

M∑− 1

m= 0

[

V(Rm)+

1

2

Rm·∇RmV(Rm)

]〉

. (12.84)

This is called thevirial energy estimator, and it will be considered in Problem 12.6.
The virial estimator is not always superior to the direct expression, as was
observed by Singer and Smith for Lennard–Jones systems[ 21 ]; this is presum-
ably due to the steepness of the Lennard–Jones potential causing large fluctuations
in the virial.

12.4.2 Applications

We check the PIMC method for the harmonic oscillator in one dimension. We have
only one particle per time slice. The particles all move in a ‘background potential’,
which is the harmonic oscillator potential, and particles in neighbouring slices are
coupled by the kinetic, harmonic coupling. The partition function reads

Z=

∫

dx 0 ...dxM− 1 exp

{

−

β

M

M∑− 1

m= 0

[

(xm−xm+ 1 )^2

2 

β^2

+

1

2

xm^2

]}

. (12.85)

We have usedβ=10 andM=100. Thirty thousand MCS were performed, of
which the first two thousand were deleted to reach equilibrium. The maximum dis-
placement was tuned to yield an acceptance rate of about 0.5. The spacing between
the energy levels of the harmonic oscillator is 1; thereforeβ=10 corresponds
to large temperature. We find for the energyE=0.51±0.02, in agreement with
the exact ground state energy of 1/2. The ground state amplitude can also be
determined, and it is found to match the exact form|ψ(x)|^2 =e−x
2
very well.
The next application is the hydrogen atom. This turns out to be less successful,
just as in the case of the diffusion MC method. The reason is again that writing
the time-evolution operator as the product of the exponentials of the kinetic and
potential energies is not justified when the electron approaches the nucleus, as the
Coulomb potential diverges there – CBH commutators therefore diverge too. The
use of guide functions is not possible in PIMC, so we have to think of something
else. The solution lies in the fact that theexacttime-evolution operator over a time
slice tdoes not diverge atr=0; we suffer from divergences because we have
used the so-calledprimitive approximation

T(r→r′;
τ )=

1

( 2 π

τ)^3 /^2

exp[−(r−r′)/( 2 

τ )]exp{−

τ[V(r)+V(r′)]/ 2 }

(12.86)