Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

3.2 Scattering of H atoms from rare gases
The total cross section σ tot (E) for the scattering
of H atoms by rare gas atoms was measured by
Toennies et al., J. Chem. Phys. 71, 614 (1979).
At the right, the cross section for the H-Kr system
as a function of the energy of the center of
mass. One can notice “spikes” in the cross section,
known as “resonances”. One can connect
the different resonances to specific values of the
angular momentum l.
The H-Kr interaction potential can be modelled quite accurately as a Lennard-
Jones (LJ) potential:
[ (σ ) 12 ( ) ]
σ
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V (r) = ɛ − 2
(3.9)
r r
where ɛ = 5.9meV, σ = 3.57Å. The LJ potential is much used in molecular and
solid-state physics to model interatomic interaction forces. The attractive r −6
term describes weak van der Waals (or “dispersive”, in chemical parlance) forces
due to (dipole-induced dipole) interactions. The repulsive r −12 term models the
repulsion between closed shells. While usually dominated by direct electrostatic
interactions, the ubiquitous van der Waals forces are the dominant term for the
interactions between closed-shell atoms and molecules. These play an important
role in molecular crystals and in macro-molecules. The LJ potential is the first
realistic interatomic potential for which a molecular-dynamics simulation was
performed (Rahman, 1965, liquid Ar).
It is straightforward to find that the LJ potential as written in (3.9) has a
minimum V min = −ɛ for r = σ, is zero for r = σ/2 1/6 = 0.89σ and becomes
strongly positive (i.e. repulsive) at small r.
3.2.1 Derivation of Van der Waals interaction
The Van der Waals attractive interaction can be described in semiclassical terms
as a dipole-induced dipole interaction, where the dipole is produced by a charge
fluctuation. A more quantitative and satisfying description requires a quantummechanical
approach. Let us consider the simplest case: two nuclei, located
in R A and R B , and two electrons described by coordinates r 1 and r 2 . The
Hamiltonian for the system can be written as
H = − ¯h2
2m ∇2 1 −
q 2 e
|r 1 − R A | − ¯h2
2m ∇2 2 −
q 2 e
|r 2 − R B |
(3.10)
−
q 2 e
|r 1 − R B | − q 2 e
|r 2 − R A | + q 2 e
|r 1 − r 2 | + q 2 e
|R A − R B | ,
where ∇ i indicates derivation with respect to variable r i , i = 1, 2. Even this
“simple” hamiltonian is a really complex object, whose general solution will be
the subject of several chapters of these notes. We shall however concentrate on
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