Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

In general, there can be many equilibrium points, either stable (a minimum:
any displacement from the equilibrium point produces forces opposed to the
displacement, i.e. the second derivative is positive everywhere) or unstable (a
maximum or a saddle point: for at least some directions of displacement from
equilibrium, there are forces in the direction of the displacement, i.e. there
are negative second derivatives). Among the various minima, there will be a
global minimum, the lowest-energy one, corresponding to the ground state of
the nuclear system, for a given electronic state. If the electronic state is also
the ground state, this will be the ground state of the atomic system. All other
minima are local minima, that is, metastable states that the nuclear system can
leave by overcoming a potential barrier.
8.3 Diatomic molecules
Let us consider now the simple case of diatomic molecules, and in particular,
the molecule of H 2 . There are 6 nuclear coordinates, 3 for the center of mass
and 3 relative to it, but just one, the distance R between the nuclei, determines
the effective interatomic potential V (R): the PES is in fact invariant, both
translationally and for rotations around the axis of the molecule. Given a
distance R, we may solve Eq.(8.4) for electrons, find the l-th electronic energy
level E (l) (R) and the corresponding interatomic potential V (R) = E II (R) +
E (l) (R). Note that the nuclear repulsion energy E II (R) is simply given by
E II (R) = q2 eZ 1 Z 2
R
(8.11)
where Z 1 and Z 2 are nuclear charges for the two nuclei.
Let us consider the electronic ground state only for H 2 molecule. At small
R, repulsion between nuclei is dominant, V (R) becomes positive, diverging like
q 2 e/R for R → 0. At large R, the ground state becomes that of two neutral
H atoms, thus V (R) ≃ 2Ry. At intermediate R, the curve has a minimum at
about R 0 = 0.74Å, with V (R 0 ) ≃ V (∞) − 4.5eV. Such value – the difference
between the potential energy of atoms at large distances and at the minimum,
is known as cohesive or binding energy. The form of the interatomic potential is
reminiscent of that of model potentials like Morse (in fact such potentials were
proposed to model binding).
What is the electronic ground state for R ∼ R 0 ? We can get an idea by
using the method of molecular orbitals: an approximate solution in which singleelectron
states are formed as linear combinations of atomic states centered
about the two nuclei. Combinations with the same phase are called ligand, as
they tend to accumulate charge in the region of space between the two nuclei.
Combinations with opposite phase are called antiligand, since they tend to
deplete charge between the nuclei. Starting from molecular orbitals, one can
build the wave function as a Slater determinant. Two ligand states of opposite
spin, built as superpositions of 1s states centered on the two nuclei (σ orbitals),
yield a good approximation to the ground state. By construction, the total spin
of the ground state is zero.
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