14.5 Variational Monte Carlo for atoms 475
monic oscillator are some of the few cases where we can use a trial wave function proportional
to the exact one. These two systems offer some of the few examples where we can find an
exact solution to the problem. In most cases of interest, we do not knowa priorithe exact
Table 14.5Result for ground state energy of the hydrogen atom as function of the variational parameterα.
The exact result is forα= 1 with an energyE=− 1 / 2. The variableNis the number of Monte Carlo samples. In
this calculation we fixedN= 100000 and a step length of 4 Bohr radii was used in order to obtain an acceptance
of≈50%.
α 〈H〉 σ^2
5.00000E-01 -3.76740E-01 6.10503E-02
6.00000E-01 -4.21744E-01 5.22322E-02
7.00000E-01 -4.57759E-01 4.51201E-02
8.00000E-01 -4.81461E-01 3.05736E-02
9.00000E-01 -4.95899E-01 8.20497E-03
1.00000E-00 -5.00000E-01 0.00000E+00
1.10000E+00 -4.93738E-01 1.16989E-02
1.20000E+00 -4.75563E-01 8.85899E-02
1.30000E+00 -4.54341E-01 1.45171E-01
1.40000E+00 -4.13220E-01 3.14113E-01
1.50000E+00 -3.72241E-01 5.45568E-01
wave function, or how to make a good trial wave function. In essentially all real problems a
large amount of CPU time and numerical experimenting is needed in order to ascertain the
validity of a Monte Carlo estimate. The next examples deal with such problems.
14.5.4The Helium Atom
Most physical problems of interest in atomic, molecular andsolid state physics consist of
many interacting electrons and ions. The total number of particlesNis usually sufficiently
large that an exact solution cannot be found. Controlled andwell understood approximations
are sought to reduce the complexity to a tractable level. Once the equations are solved, a
large number of properties may be calculated from the wave function. Errors or approxima-
tions made in obtaining the wave function will be manifest inany property derived from the
wave function. Where high accuracy is required, considerable attention must be paid to the
derivation of the wave function and any approximations made.
The helium atom consists of two electrons and a nucleus with chargeZ= 2. In setting
up the Hamiltonian of this system, we need to account for the repulsion between the two
electrons as well. A common and very reasonable approximation used in the solution of of the
Schrödinger equation for systems of interacting electronsand ions is the Born-Oppenheimer
approximation discussed above. But even this simplified electronic Hamiltonian remains very
difficult to solve. No closed-form solutions exist for general systems with more than one
electron.
To set up the problem, we start by labelling the distance between electron 1 and the nu-
cleus asr 1. Similarly we haver 2 for electron 2. The contribution to the potential energy due
to the attraction from the nucleus is
−
2 ke^2
r 1−
2 ke^2
r 2,
and if we add the repulsion arising from the two interacting electrons, we obtain the potential
energy