36
2 Molecular States
of the molecule. To separate the rotational motion one has first to define a vector
ω representing the angular velocity of rotation of the whole molecule, which plays
the same role as ˙
R C M for translations, and then to set up a reference frame rotating
at angular velocity ω. If the system is formed by only two particles, the vector ω is
easily defined, because in the reference frame centered in the center of mass, the two
particles necessarily rotate at the same angular velocity ω = J/μr
2 , where μ is the
reduced mass and r is the interparticle distance. Of course, in general we may always
define ω in such a way that J = Iω, where I is the inertia tensor. However, I is not
known a priori: its form depends on the positions of all the particles constituting the
molecular system. Therefore, apart from the simple case of a two-particle system, the
rotational motion cannot be separated exactly. An approximate separation is however
possible: we shall come back on this point in Sect. 2.5.
2.3 The Born–Oppenheimer Approximation and Its
Breakdown: The Nonadiabatic Couplings
In a molecule, the interaction between electrons and nuclei is never negligible. In
fact, at least in bound states it is the Hamiltonian term with the largest mean value
in module. As a consequence, the complete separation of nuclear and electronic
variables is not possible, not even approximately. However, the electron mass is at
least three orders of magnitude smaller than the nuclear masses, so that the timescales
for the respective motions are different (femtoseconds for nuclei and attoseconds for
electrons). This entails a different kind of separation of variables. We start observing
that a given eigenfunction ψ(r, R) of the molecular Hamiltonian can always be
written in this way
ψ(r, R) = ϕ(r; R)χ (R)
(2.53)
where r and R collect the electronic and nuclear Cartesian coordinates, respectively.
For the sake of simplicity the dependence on spin, which does not play a relevant
role in this context, is omitted. We also assume that ψ is normalized. In the above
equation ϕ and χ are, respectively, electronic and nuclear wavefunctions. In fact,
we want |χ(R)|
2 to be the probability density to find the nuclei in R, which can be
obtained by integrating |ψ(r, R)|
2 with respect to the electronic coordinates:
|χ(R)|
2
=
+∞
−∞
|ϕ(r; R)χ (R)|
2 dr
3
1 . . . dr
3
N e
.
(2.54)
This identity requires ϕ(r; R) to be normalized for any fixed set of nuclear coordinates:
ϕ |ϕ r = 1
∀ R .
(2.55)
2 Molecular States
of the molecule. To separate the rotational motion one has first to define a vector
ω representing the angular velocity of rotation of the whole molecule, which plays
the same role as ˙
R C M for translations, and then to set up a reference frame rotating
at angular velocity ω. If the system is formed by only two particles, the vector ω is
easily defined, because in the reference frame centered in the center of mass, the two
particles necessarily rotate at the same angular velocity ω = J/μr
2 , where μ is the
reduced mass and r is the interparticle distance. Of course, in general we may always
define ω in such a way that J = Iω, where I is the inertia tensor. However, I is not
known a priori: its form depends on the positions of all the particles constituting the
molecular system. Therefore, apart from the simple case of a two-particle system, the
rotational motion cannot be separated exactly. An approximate separation is however
possible: we shall come back on this point in Sect. 2.5.
2.3 The Born–Oppenheimer Approximation and Its
Breakdown: The Nonadiabatic Couplings
In a molecule, the interaction between electrons and nuclei is never negligible. In
fact, at least in bound states it is the Hamiltonian term with the largest mean value
in module. As a consequence, the complete separation of nuclear and electronic
variables is not possible, not even approximately. However, the electron mass is at
least three orders of magnitude smaller than the nuclear masses, so that the timescales
for the respective motions are different (femtoseconds for nuclei and attoseconds for
electrons). This entails a different kind of separation of variables. We start observing
that a given eigenfunction ψ(r, R) of the molecular Hamiltonian can always be
written in this way
ψ(r, R) = ϕ(r; R)χ (R)
(2.53)
where r and R collect the electronic and nuclear Cartesian coordinates, respectively.
For the sake of simplicity the dependence on spin, which does not play a relevant
role in this context, is omitted. We also assume that ψ is normalized. In the above
equation ϕ and χ are, respectively, electronic and nuclear wavefunctions. In fact,
we want |χ(R)|
2 to be the probability density to find the nuclei in R, which can be
obtained by integrating |ψ(r, R)|
2 with respect to the electronic coordinates:
|χ(R)|
2
=
+∞
−∞
|ϕ(r; R)χ (R)|
2 dr
3
1 . . . dr
3
N e
.
(2.54)
This identity requires ϕ(r; R) to be normalized for any fixed set of nuclear coordinates:
ϕ |ϕ r = 1
∀ R .
(2.55)