Computational Physics

80 The Hartree–Fock method

Table 4.5.Correlation energies in atomic

units forH 2 andH 2 O.

Molecule CI Exact

H 2 −0.039 69 −0.0409

H 2 O −0.298 −0.37

DatatakenfromRef.[6].

Table 4.6. Bond lengths in atomic units for

H 2 andH 2 O.

Molecule HF CI Exact Experiment

H 2 1.385 1.396 1.401

H 2 O 1.776 1.800 1.809

Data takenfromRef. [6]. Exact results were

obtained by variational calculus[18]. Experi-

mentalresultsarefromRef.[19].

within the Born–Oppenheimer approximation) by a variational method using basis
functions depending on the positions of both electrons [18]. The CI results are
excellent for both cases.
In CI, the spin-orbitals from which the Slater determinants are built are the
eigenstates of the Fock operator as determined self-consistently for the ground state.
As only a restricted number of determinants can be taken into account, the dimension
of the subspace spanned by the Slater determinants is quite limited. If, within these
Slater determinants, the orbitals are allowed to vary by relaxing the ground state
coefficients of the basis functions, this subspace can be increased considerably. In
the so-called multi-configuration self-consistent field theory (MCSCF), this process
is carried out, but because of the large amount of variation possible this leads to
a huge numerical problem. Finally, perturbative analysis allows correlation effects
to be taken into account [ 5 , 6 ].

Exercises

4.1 In this problem we show that the large masses of the nuclei compared with those of

the electrons lead to the Born–Oppenheimer approximation.

The wave functionof a collection of electrons and nuclei depends on the

positionsRnof the nuclei andriof the electrons (we neglect the spin-degrees of