Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

Chapter 8
Molecules
8.1 Born-Oppenheimer approximation
Let us consider a system of interacting nuclei and electrons. In general, the
Hamiltonian of the system will depend upon all nuclear coordinates, R µ , and
all electronic coordinates, r i . For a system of n electrons under the field of N
nuclei with charge Z µ , in principle one has to solve the following Schrödinger
equation:
(T I + V II + V eI + T e + V ee ) Ψ(R µ , r i ) = EΨ(R µ , r i ) (8.1)
where T I is the kinetic energy of nuclei, V II is the Coulomb repulsion between
nuclei, V eI is the Coulomb attraction between nuclei and electrons, T e is the
kinetic energy of electrons, V ee is the Coulomb repulsion between electrons:
T I = − ∑
µ=1,N
¯h 2
2M µ
∇ 2 µ,
V ee = q2 e
2
T e = − ∑
∑
i≠j
i=1,n
¯h 2
2m ∇2 i ,
1
|r i − r j | , V eI = −q 2 e
V II = q2 e
2
∑
µ≠ν
∑ ∑
µ=1,N i=1,n
Z µ Z ν
|R µ − R ν | ,
Z µ
|R µ − r i | . (8.2)
This looks like an impressive problem. It is however possible to exploit the mass
difference between electrons and nuclei to separate the global problem into an
electronic problem for fixed nuclei and a nuclear problem under an effective
potential generated by electrons. Such separation is known as adiabatic or Born-
Oppenheimer approximation. The crucial point is that the electronic motion
is much faster than the nuclear motion: while forces on nuclei and electrons
have the same order of magnitude, an electron is at least ∼ 2000 times lighter
than any nucleus. We can thus assume that at any time the electrons ”follow”
the nuclear motion, while the nuclei at any time ”feel” an affective potential
generated by electrons. Formally, we assume a wave function of the form
Ψ(R µ , r i ) = Φ(R µ )ψ (l)
R (r i) (8.3)
where the electronic wave function ψ (l)
R (r i) solves the following Schrödinger
equation:
(T e + V ee + V eI ) ψ (l)
R (r i) = E (l)
R ψ(l) R (r i). (8.4)
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