Computational Physics - Department of Physics

14.5 Variational Monte Carlo for atoms 477

The energy becomes then
(
̂h 1 ψa(r 1 )

)

ψb(r 2 )+

(

̂h 2 ψb(r 2 )

)

ψa(r 1 ) = (Ea+Eb)ψa(r 1 )ψb(r 2 ),

yielding

Eab=Z^2 E 0

(

1

n^2 a

+

1

n^2 b

)

.

If we insertZ= 2 and assume that the ground state is determined by two electrons in the
lowest-lying hydrogen orbit withna=nb= 1 , the energy becomes

Eab= 8 E 0 =− 108 .8 eV,

while the experimental value is− 78. 8 eV. Clearly, this discrepancy is essentially due to our
omission of the repulsion arising from the interaction of two electrons.

14.5.4.1 Choice of trial wave function

The choice of trial wave function is critical in variationalMonte Carlo calculations. How to
choose it is however a highly non-trivial task. All observables are evaluated with respect to
the probability distribution

P(R) =

|ψT(R)|^2

∫

|ψT(R)|^2 dR

.

generated by the trial wave function. The trial wave function must approximate an exact
eigenstate in order that accurate results are to be obtained. Improved trial wave functions
also improve the importance sampling, reducing the cost of obtaining a certain statistical
accuracy.
Quantum Monte Carlo methods are able to exploit trial wave functions of arbitrary forms.
Any wave function that is physical and for which the value, the gradient and the laplacian of
the wave function may be efficiently computed can be used. Thepower of Quantum Monte
Carlo methods lies in the flexibility of the form of the trial wave function.
It is important that the trial wave function satisfies as manyknown properties of the exact
wave function as possible. A good trial wave function shouldexhibit much of the same fea-
tures as does the exact wave function. Especially, it shouldbe well-defined at the origin, that
isΨ(|R|= 0 ) 6 = 0 , and its derivative at the origin should also be well-defined. One possible
guideline in choosing the trial wave function is the use of constraints about the behavior of
the wave function when the distance between one electron andthe nucleus or two electrons
approaches zero. These constraints are the so-called “cuspconditions” and are related to the
derivatives of the wave function.
To see this, let us single out one of the electrons in the helium atom and assume that this
electron is close to the nucleus, i.e.,r 1 → 0. We assume also that the two electrons are far
from each other and thatr 26 = 0. The local energy can then be written as

EL(R) =

1

ψT(R)

HψT(R) =

1

ψT(R)

(

−

1

2

∇^21 −

Z

r 1

)

ψT(R)+finite terms.

Writing out the kinetic energy term in the spherical coordinates of electron 1 , we arrive at
the following expression for the local energy

EL(R) =

1

RT(r 1 )

(

−

1

2

d^2

dr^21

−

1

r 1

d

dr 1

−

Z

r 1

)

RT(r 1 )+finite terms,