Computational Physics

12.2 The variational Monte Carlo method 377

The same can be done for the variance with the result

var(E)v=

( 1 − 4 α^2 )^2

32 α^2

. (12.8)

The Monte Carlo results match the analytical values as is clear from the table. Also
inTable 12.1we show results for the hydrogen atom with the Hamiltonian

H=−

1

2

∇^2 −

1

r

. (12.9)

The exact ground state with energyE=− 1 /2 is given as e−r; we take variational
trial functions of the form e−αr, so that the ground state is again incorporated in the
variational subspace. Although we could consider the present problem as a one-
dimensional one by using the spherical symmetry of the potential and the ground
state wave function, we shall treat it here as a fully three-dimensional problem to
illustrate the general approach. For this case, the analytical values of the average
local energy and variance can also be calculated. This is left as an exercise for the
reader.
The local energy is given by

EL(r)=−

1

r

−

1

2

α

(

α−

2

r

)

. (12.10)

It is seen fromTable 12.1that the energy is minimal at the ground state and that its
variance vanishes there too.
Finally we consider the helium atom, which we have already studied extensively
inChapters 4and 5. Constructing good trial functions is a problem on its own –
here we shall use the form:

ψ(r 1 ,r 2 )=e−^2 r^1 e−^2 r^2 er^12 /[^2 (^1 +αr^12 )] (12.11)

wherer 12 =|r 1 −r 2 |. This function consists of a product of two atomic one-electron
orbitals and a correlation term. The local energy now has the form:

EL(r 1 ,r 2 )=− 4 +(rˆ 1 −rˆ 2 )·(r 1 −r 2 )

1

r 12 ( 1 +αr 12 )^2

−

1

r 12 ( 1 +αr 12 )^3

−

1

4 ( 1 +αr 12 )^4

+

1

r 12

(12.12)

Withrˆwe denote a unit vector alongr, andr 12 is the distance between the two
electrons. Energies and variances are also displayed in Table 12.1. The variance
does not have a sharp minimum for the same value ofαas the energy. The reason
is that most of the variance is due to the trial wave function not being exact, even
for the best value ofα. The optimum value of the energy,−2.878 1±0.000 5,
should be compared with the Hartree-Fock value of−2.861 7 a.u. and the DFT value