Computational Physics - Department of Physics

466 14 Quantum Monte Carlo Methods

expectation value of the local energy into a kinetic energy part and a potential energy part.
The expectation value of the kinetic energy is

−

∫

∫dRΨT∗(R)∇^2 ΨT(R)

dRΨT∗(R)ΨT(R)

, (14.8)

and we could be tempted to compute, if the wave function obeysspherical symmetry, just the
second derivative with respect to one coordinate axis and then multiply by three. This will
most likely increase the variance, and should be avoided, even if the final expectation values
are similar. For quantum mechanical systems, as discussed below, the exact wave function
leads to a variance which is exactly zero.
Another shortcut we could think of is to transform the numerator in the latter equation to
∫
dRΨT∗(R)∇^2 ΨT(R) =−

∫

dR(∇ΨT∗(R))(∇ΨT(R)), (14.9)

using integration by parts and the relation
∫
dR∇(ΨT∗(R)∇ΨT(R)) = 0 ,

where we have used the fact that the wave function is zero atR=±∞. This relation can in
turn be rewritten through integration by parts to
∫
dR(∇ΨT∗(R))(∇ΨT(R))+

∫

dRΨT∗(R)∇^2 ΨT(R)) = 0.

The right-hand side of Eq. (14.9) involves only first derivatives. However, in case the wave
function is the exact one, or rather close to the exact one, the left-hand side yields just a
constant times the wave function squared, implying zero variance. The rhs does not and may
therefore increase the variance.
If we use integration by parts for the harmonic oscillator case, the new local energy is

EL(x) =x^2 ( 1 +α^4 ),

and the variance

σ^2 =

(α^4 + 1 )^2

2 α^4

,

which is larger than the variance of Eq. (14.7).

14.5 Variational Monte Carlo for atoms

The Hamiltonian for anN-electron atomic system consists of two terms

Hˆ(R) =Tˆ(R)+Vˆ(R), (14.10)

the kinetic and the potential energy operator. HereR={r 1 ,r 2 ,...rN}represents the spatial
and spin degrees of freedom associated with the different particles. The classical kinetic
energy

T=P

2

2 M

+

N

∑

j= 1

p^2 j

2 m

,

is transformed to the quantum mechanical kinetic energy operator by operator substitution
of the momentum (pk→−ih ̄∂/∂xk)