Computational Physics - Department of Physics

468 14 Quantum Monte Carlo Methods

for general systems with more than one electron. We use the Born-Oppenheimer approxima-
tion in our discussion of atomic and molecular systems.
The first term of Eq. (14.12) is the nucleus-electron potential and the second term is
the electron-electron potential. The inter-electronic potential is the main problem in atomic
physics. Because of this term, the Hamiltonian cannot be separated into one-particle parts,
and the problem must be solved as a whole. A common approximation is to regard the ef-
fects of the electron-electron interactions either as averaged over the domain or by means of
introducing a density functional. Popular methods in this direction are Hartree-Fock theory
and Density Functional theory. These approaches are actually very efficient, and about99%
or more of the electronic energies are obtained for most Hartree-Fock calculations. Other ob-
servables are usually obtained to an accuracy of about 90 −95%(ref. [94]). We discuss these
methods in chapter 15, where also systems with more than two electrons are discussed in
more detail. Here we limit ourselves to systems with at most two electrons. Relevant systems
are neutral helium with two electrons, the hydrogen molecule or two electrons confined in a
two-dimensional harmonic oscillator trap.

14.5.2The Hydrogen Atom

The spatial Schrödinger equation for the three-dimensional hydrogen atom can be solved in
a closed form, see for example Ref. [93] for details. To achieve this, we rewrite the equation
in terms of spherical coordinates using

x=rsinθcosφ,

y=rsinθsinφ,

and
z=rcosθ.

The reason we introduce spherical coordinates is due to the spherical symmetry of the
Coulomb potential
e^2
4 π ε 0 r

=

e^2

4 π ε 0

√

x^2 +y^2 +z^2

,

where we have usedr=

√

x^2 +y^2 +z^2. It is not possible to find a separable solution of the type

ψ(x,y,z) =ψ(x)ψ(y)ψ(z).

as we can with the harmonic oscillator in three dimensions. However, with spherical coordi-
nates we can find a solution of the form

ψ(r,θ,φ) =R(r)P(θ)F(φ) =RPF.

These three coordinates yield in turn three quantum numberswhich determine the energy of
the system. We obtain three sets of ordinary second-order differential equations [93],

1

F

∂^2 F

∂ φ^2

=−C^2 φ,

Crsin^2 (θ)P+sin(θ)

∂

∂ θ(sin(θ)

∂P

∂ θ) =C

2

φP,

and