Crossing of Two Electronic States

Abstract

In this chapter, we discuss crossing between two or more Born–Oppenheimer states. We begin with wave packet motion which allows to introduce the classical limit for nuclear motion. The matrix elements of the nonadiabatic coupling can become very large or even divergent, whenever two electronic states come close. The “adiabatic to diabatic” transformation eliminates at least the singular parts of the derivative coupling. We derive the so-called diabatic Schrödinger equation and discuss the simplest case of a crossing between two states. For a Hamiltonian depending on only one nuclear coordinate, the transformation to a diabatic basis is possible and yields a diabatic coupling which is given by half the splitting of the adiabatic states. The semiclassical approximation makes use of narrow localized wavepackets and describes nuclear motion as a classical trajectory defined as the time-dependent average position. The famous Landau Zener model uses a linear approximation of the trajectory in the vicinity of the crossing point and obtains an explicit solution for the transition probability. If more coordinates are involved, conical intersections appear which are very important for ultrafast transitions. We discuss the linear vibronic coupling model for the dynamics in the vicinity of a conical intersection.

Problems

21.1

Crude Adiabatic Model

Consider the crossing of two electronic states along a coordinate Q. As basis functions we use two coordinate independent electronic wavefunctions which diagonalize the Born–Oppenheimer Hamiltonian at the crossing point $Q_{0}$

$$ (T_{el}+V(Q_{0}))\varphi ^{1,2}= E^{1,2}\varphi ^{1,2}. $$

Use the following ansatz functions

$$ \varPsi _{1}(r, Q)=(\cos \zeta (Q)\varphi ^{1}(r)-\sin \zeta (Q)\varphi ^{2}(r))\chi ^{1}(Q) $$

$$ \varPsi _{2}(r, Q)=(\sin \zeta (Q)\varphi ^{1}(r)+\cos \zeta (Q)\varphi ^{2}(r))\chi ^{2}(Q) $$

which can be written in more compact form

$$ (\varPsi _{1},\varPsi _{2})=(\varphi ^{1},\varphi ^{2})\left( \begin{array}{cc} c &{} s\\ -s &{} c \end{array}\right) \left( \begin{array}{c} \chi _{1}\\ \chi _{2} \end{array}\right) . $$

The Hamiltonian is partitioned as

$$ H=T_{N}+T_{el}+V(r, Q_{0})+\varDelta V(r, Q). $$

Calculate the matrix elements of the Hamiltonian

$$\begin{aligned}&\left( \begin{array}{cc} H_{11} &{} H_{12}\\ H_{21} &{} H_{22} \end{array}\right) =\nonumber \\&=\left( \begin{array}{cc} c &{} -s\\ s &{} c \end{array}\right) \left( \begin{array}{c} \varphi _{1}^{\dagger }\\ \varphi _{2}^{\dagger } \end{array}\right) \left( -\frac{\hbar ^{2}}{2m}\frac{\partial ^{2}}{\partial Q^{2}}+T_{el}+V(Q_{0})+\varDelta V\right) \left( \varphi ^{1},\varphi ^{2}\right) \left( \begin{array}{cc} c &{} s\\ -s &{} c \end{array}\right) \end{aligned}$$

(21.139)

where $\chi (Q)$ and $\zeta (Q)$ depend on the coordinate Q whereas the basis functions $\varphi ^{1,2}$ do not. Chose $\zeta (Q)$ such that $T_{el}+V$ is diagonalized. Evaluate the nonadiabatic interaction terms at the crossing point $Q_{0}$.