148
5 Fast Nonadiabatic Dynamics
5.3 Landau–Zener Rule
As discussed in Sect. 5.1, in avoided crossing regions the nonadiabatic coupling is
large and the energy difference is small; therefore nonradiative transitions between
electronic states are very likely. This problem was first tackled by Landau and Zener
in 1932, using a classical approximation for the nuclear motion [6, 7]. In particular,
it is assumed that the classical description is acceptable for the nuclei, at least as a
first approximation, so that the nuclear motion is given by a classical trajectory Q(t).
Then, we consider a time-dependent Schrödinger equation only for the electrons
i
dΨ el (r, t; Q(t))
dt
= ˆ
H el (Q(t))Ψ el (r, t; Q(t))
(5.21)
where the parametric dependence of the electronic wavefunction Ψ el and Hamiltonian
ˆ
H el on the nuclear trajectory Q(t) has been made explicit. Note in particular that
ˆ
H el , as well as its eigenfunctions and eigenvalues, depend on time through Q(t).
Therefore, the time evolution operator (2.26) cannot be applied. The Ψ el can be
expanded in terms of the adiabatic basis
Ψ el (t) =
l
a l (t)e
−iγ l (t)
ϕ l
γ l (t) =
1
t
0
U l (Q(t
))dt
(5.22)
where e
−iγ l (t) is the so-called dynamical phase, as opposed to the “static” phase that
would be present for a time-independent Hamiltonian (see, for example, Eq. (3.3)).
Inserting the above expansion in Eq. (5.21) we obtain
l
˙
a l ϕ l e
−iγ l = −
l
a l e
−iγ l ˙
ϕ l .
(5.23)
Multiplying both sides by ϕ
∗
k and integrating on the electronic coordinates we get
the differential equation for the adiabatic coefficients a k
˙
a k = −
l
a l (t)e
−i(γ l (t)−γ k (t))
ϕ k | ˙
ϕ l
(5.24)
and, according to the chain derivative rule, we have
ϕ k | ˙
ϕ l =
α
g
(α)
kl
˙
Q α
(5.25)
We see that the variation of the coefficients a k (and thus the probability to make
a transition from an adiabatic state to another) depends on the scalar product
between the nuclear velocity and the nonadiabatic coupling vectors, which is analogous to the coupling terms g
(α)
kl
∂
∂ R α
of Eq. (2.62). Note however that in this mixed
5 Fast Nonadiabatic Dynamics
5.3 Landau–Zener Rule
As discussed in Sect. 5.1, in avoided crossing regions the nonadiabatic coupling is
large and the energy difference is small; therefore nonradiative transitions between
electronic states are very likely. This problem was first tackled by Landau and Zener
in 1932, using a classical approximation for the nuclear motion [6, 7]. In particular,
it is assumed that the classical description is acceptable for the nuclei, at least as a
first approximation, so that the nuclear motion is given by a classical trajectory Q(t).
Then, we consider a time-dependent Schrödinger equation only for the electrons
i
dΨ el (r, t; Q(t))
dt
= ˆ
H el (Q(t))Ψ el (r, t; Q(t))
(5.21)
where the parametric dependence of the electronic wavefunction Ψ el and Hamiltonian
ˆ
H el on the nuclear trajectory Q(t) has been made explicit. Note in particular that
ˆ
H el , as well as its eigenfunctions and eigenvalues, depend on time through Q(t).
Therefore, the time evolution operator (2.26) cannot be applied. The Ψ el can be
expanded in terms of the adiabatic basis
Ψ el (t) =
l
a l (t)e
−iγ l (t)
ϕ l
γ l (t) =
1
t
0
U l (Q(t
))dt
(5.22)
where e
−iγ l (t) is the so-called dynamical phase, as opposed to the “static” phase that
would be present for a time-independent Hamiltonian (see, for example, Eq. (3.3)).
Inserting the above expansion in Eq. (5.21) we obtain
l
˙
a l ϕ l e
−iγ l = −
l
a l e
−iγ l ˙
ϕ l .
(5.23)
Multiplying both sides by ϕ
∗
k and integrating on the electronic coordinates we get
the differential equation for the adiabatic coefficients a k
˙
a k = −
l
a l (t)e
−i(γ l (t)−γ k (t))
ϕ k | ˙
ϕ l
(5.24)
and, according to the chain derivative rule, we have
ϕ k | ˙
ϕ l =
α
g
(α)
kl
˙
Q α
(5.25)
We see that the variation of the coefficients a k (and thus the probability to make
a transition from an adiabatic state to another) depends on the scalar product
between the nuclear velocity and the nonadiabatic coupling vectors, which is analogous to the coupling terms g
(α)
kl
∂
∂ R α
of Eq. (2.62). Note however that in this mixed