Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

The index R is a reminder that both the wave function and the energy depend
upon the nuclear coordinates, via V eI ; the index l classifies electronic states.
We now insert the wave function, Eq.(8.3), into Eq.(8.2) and notice that T e
does not act on nuclear variables. We will get the following equation:
(
T I + V II + E (l)
R
)
Ψ(R µ )ψ (l)
R (r i) = EΨ(R µ )ψ (l)
R (r i). (8.5)
If we now neglect the dependency upon R of the electronic wave functions in
the kinetic term:
(
T I Φ(R µ )ψ (l) )
R (r i) ≃ ψ (l)
R (r i) (T I Φ(R µ )) . (8.6)
we obtain a Schrödinger equation for nuclear coordinates only:
(
T I + V II + E (l) )
R
Ψ(R µ ) = EΨ(R µ ), (8.7)
where electrons have ”disappeared” into the eigenvalue E (l)
R . The term V II +E (l)
R
plays the role of effective interaction potential between nuclei. Of course such
potential, as well as eigenfunctions and eigenvalues of the nuclear problem,
depends upon the particular electronic state.
The Born-Oppenheimer approximation is very well verified, except the special
cases of non-adiabatic phenomena (that are very important, though). The
main neglected term in Eq.8.6 has the form
∑
µ
¯h 2
(
(∇ µ Φ(R µ )) ∇ µ ψ (l) )
R
M (r i)
µ
and may if needed be added as a perturbation.
8.2 Potential Energy Surface
(8.8)
The Born-Oppenheimer approximation allows us to separately solve a Schrödinger
equation for electrons, Eq.(8.4), as a function of atomic positions, and
a problem for nuclei only, Eq.(8.7). The latter is in fact a Schrödinger equation
in which the nuclei interact via an effective interatomic potential, V (R µ ) ≡
V II +E (l) , a function of the atomic positions R µ and of the electronic state. The
interatomic potential V (R µ ) is also known as potential energy surface (“potential”
and “potential energy” are in this context synonyms), or PES. It is clear
that the nuclear motion is completely determined by the PES (assuming that
the electron states does not change with time) since forces acting on nuclei are
nothing but the gradient of the PES:
F µ = −∇ µ V (R µ ), (8.9)
while equilibrium positions for nuclei, labeled with R (0)
µ , are characterized by
zero gradient of the PES (and thus of any force on nuclei):
F µ = −∇ µ V (R (0)
µ ) = 0. (8.10)
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