Chapter 5
Fast Nonadiabatic Dynamics
Abstract In this chapter we present the fast dynamics of a molecular system in
regions (avoided crossings, conical intersections) where the Born–Oppenheimer
approximation breaks down because the electronic and the nuclear motion are
strongly coupled. When a nuclear wavepacket reaches a region where the PESs are
close to each other, the nonadiabatic transitions are far from being negligible, and
time-dependent perturbation theory cannot be applied. We will show that, in spite of
the strong interplay between electronic and nuclear motion, interesting information
can be obtained from an approximated mixed quantum/classical model, which leads
to the celebrated Landau–Zener formula. Moreover, the main features of conical
intersections will be described in some detail.
Keywords Nonadiabatic dynamics · Avoided crossings · Landau-Zener
Conical intersections · Berry’s phase · Surface hopping
5.1 Noncrossing Rule and Avoided Crossings
In Sects. 3.10 and 3.11 we argued that large energy gaps imply slow nonadiabatic
transitions. Here we want to explore what happens when the PESs get close to
each other. Given a pair of electronic adiabatic states |ϕ 1 and |ϕ 2 , our first aim is to
understand under which circumstances their respective energies U 1 and U 2 can reach
a degeneracy point Q x where U 1 (Q x ) = U 2 (Q x ). More complicated intersections
involving three or more states are not considered in this chapter (however, in some
cases |ϕ 1 and |ϕ 2 have to be doubly degenerate; see Sect. 5.4.5). We assume the
other states to be far in energy, so that their coupling with |ϕ 1 and |ϕ 2 is small and
we can limit ourselves to consider the subspace spanned by these two states. Let |η 1 ,
|η 2 be an orthogonal basis of that subspace. According to Appendix D we have
|ϕ 1 = cos θ |η 1 + sin θ |η 2
|ϕ 2 = − sin θ |η 1 + cos θ |η 2
(5.1)
which can be written in matrix form as |ϕ = |η C, where
© Springer International Publishing AG, part of Springer Nature 2018
M. Persico and G. Granucci, Photochemistry, Theoretical Chemistry
and Computational Modelling, https://doi.org/10.1007/978-3-319-89972-5_5
141
Fast Nonadiabatic Dynamics
Abstract In this chapter we present the fast dynamics of a molecular system in
regions (avoided crossings, conical intersections) where the Born–Oppenheimer
approximation breaks down because the electronic and the nuclear motion are
strongly coupled. When a nuclear wavepacket reaches a region where the PESs are
close to each other, the nonadiabatic transitions are far from being negligible, and
time-dependent perturbation theory cannot be applied. We will show that, in spite of
the strong interplay between electronic and nuclear motion, interesting information
can be obtained from an approximated mixed quantum/classical model, which leads
to the celebrated Landau–Zener formula. Moreover, the main features of conical
intersections will be described in some detail.
Keywords Nonadiabatic dynamics · Avoided crossings · Landau-Zener
Conical intersections · Berry’s phase · Surface hopping
5.1 Noncrossing Rule and Avoided Crossings
In Sects. 3.10 and 3.11 we argued that large energy gaps imply slow nonadiabatic
transitions. Here we want to explore what happens when the PESs get close to
each other. Given a pair of electronic adiabatic states |ϕ 1 and |ϕ 2 , our first aim is to
understand under which circumstances their respective energies U 1 and U 2 can reach
a degeneracy point Q x where U 1 (Q x ) = U 2 (Q x ). More complicated intersections
involving three or more states are not considered in this chapter (however, in some
cases |ϕ 1 and |ϕ 2 have to be doubly degenerate; see Sect. 5.4.5). We assume the
other states to be far in energy, so that their coupling with |ϕ 1 and |ϕ 2 is small and
we can limit ourselves to consider the subspace spanned by these two states. Let |η 1 ,
|η 2 be an orthogonal basis of that subspace. According to Appendix D we have
|ϕ 1 = cos θ |η 1 + sin θ |η 2
|ϕ 2 = − sin θ |η 1 + cos θ |η 2
(5.1)
which can be written in matrix form as |ϕ = |η C, where
© Springer International Publishing AG, part of Springer Nature 2018
M. Persico and G. Granucci, Photochemistry, Theoretical Chemistry
and Computational Modelling, https://doi.org/10.1007/978-3-319-89972-5_5
141