Nonadiabatic transitions

analytical expression derived in reference [43] and shown in Equation 2.24.

D

Note that the matrix element above contains two Wigner 3-j symbols. A nice introduction to these symbols can be found in reference [44] where the relation to Clebsch-Gordan coefficients is explained as well. Furthermore, quantities and quantum numbers for the upper and lower state are marked by⁰and⁰⁰. µ⁽_s^1,σ) is the dipole moment in the laboratory frame. It is connected to the components along the Cartesian coordinates X, Y, Z via Equations 2.25 and 2.26.

µ⁽_s^1,±1) =(∓µ_X +iµ_Y)(2)^−0.5 (2.25)

µ⁽_s^1,0) = µ_Z (2.26)

Analogously, the dipole moment in the molecular frame is defined as

µ⁽_m^1,±1) = (∓µ_b+iµ_c)(2)^−0.5 (2.27)

µ⁽_m^1,0) = µ_a. (2.28)

2.3. Nonadiabatic transitions

In the following sections, a basic introduction to the theoretical treatment of nonadia-batic transitions is given following the remarks in reference [45] and [10]. First, the Born-Oppenheimer separation is introduced and limits of the Born-Oppenheimer ap-proximation[46] are discussed. Then the difference between the adiabatic and the dia-batic representation is explained. Finally, the transition probability in the Landau-Zener-Stueckelberg approximation is given.

2.3.1. Born-Oppenheimer separation

The description of molecular structure and dynamics is given by the Schrödinger equation containing the Hamiltonian operator H, the wave functionΨand the eigenvaluesE.

2. Scientific context

(H−E)Ψ(r,R)=0 (2.29)

Randr denote the nuclear and electronic coordinates, respectively. The Hamiltonian operator can be separated into two parts:

H =T_R+H₀ (2.30)

with

T_R =

N

Õ

α

~²

2M_α∇²_M. (2.31)

Here, the operator T_R describes the kinetic energy of the nuclei and Mα is the mass of the nucleus α. H₀ is the electronic Hamiltonian describing the kinetic energy of the electron, the electron-electron, electron-nuclei, and nuclei-nuclei interactions for fixed nuclear positions. The basis set of electronic wave functionsφ_k(r;R)is chosen such that the wave functions depend only parametrically on the nuclear coordinates. The complete system can be described by the overall wave functionΨ(r,R)given in Equation 2.32.

Ψ(r,R)=Õ

k

φ_k(r;R)χ_k(R) (2.32) In this equation, χ_k(R)is a nuclear wave function that is defined for a particular electronic statek. Substituting Equation 2.30 and 2.32 into 2.29 yields a set of coupled equations.

(T_R+T_{k k}⁰⁰ +U_{k k} −E)χ_k(R)= Õ

k⁰,k

(T_{k k}⁰ 0 +T_{k k}⁰⁰0 +U_{k k}⁰)χ_k(R) (2.33) Here,Tdenotes matrix elements arising from the nuclear kinetic energy operator acting on the electronic wave functions, whereasUspecifies elements arising from the electronic Hamiltonian operator. The indexk klabels a diagonal element, whereask k⁰indicates that two electronic states are coupled via these matrix elements. Note that there are diagonal elementsT_{k k}⁰⁰ arising from the nuclear kinetic energy operator acting on electronic wave functions, which can be thought of as nonadiabatic corrections to the potential energy surface. However, these are usually small enough to be neglected. The off-diagonal elementsT_{k k}⁰ 0 andT_{k k}⁰⁰0 are given here explicitly because these elements are neglected in the Born-Oppenheimer approximation and give rise to so-called nonadiabatic transitions between adiabatic potential energy surfaces.

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2.3. Nonadiabatic transitions

T_{k k}⁰ 0 =

N

Õ

α

~²

2M_αd^(M)_{k k}0 ∇_M (2.34)

d^(M)_{k k}0 = φ_k

∇_Mφ⁰_k

(2.35) T_{k k}⁰⁰0 =

ÕN

α

~²

2MαD⁽_{k k}^M0⁾ (2.36)

D^(M)_{k k}0 = φ_k

∇²_Mφ⁰_k

(2.37) Here, the contribution of T_{k k}⁰ 0 is in general larger than the one of T_{k k}⁰⁰0. Since the nonadiabatic coupling T_{k k}⁰ 0 includes ∇_M, the operator for the translational energy of the nuclei, Equation 2.34 already shows that the nonadiabatic coupling will be velocity dependent. Often, the Massey criterion[47]can be used as a rough guide to decide whether a process can be described in the framework of the Born-Oppenheimer approximation:

~vd_{k k}⁰

E_k −E_k⁰ 1. (2.38)

In this equation,vis the classical velocity along the coordinate of interest andE_k−E_k⁰ is the energetic separation between two electronic states k and k⁰. If Criterion 2.38 holds, the Born-Oppenheimer approximation is expected to be a good description.

2.3.2. Adiabatic and diabatic representations

Equation 2.33 shows that for an arbitrary selection of an electronic basis there are non-zero off-diagonal elements for the electronic Hamiltonian operator. However, if the electronic wave functions are chosen as the eigenfunctions ofH₀, then all off-diagonal elementsU_{k k}⁰ become zero. This representation is called the adiabatic basis. If, in addition, T_{k k}⁰ 0 and T_{k k}⁰⁰0 are neglected, the right hand side of Equation 2.33 becomes zero.

(T_R+U_{k k} −E)χ_k(R)= 0 (2.39)

Hence, following the Born-Oppenheimer approximation and using an adiabatic basis, the nuclear motion is described by the Schrödinger Equation 2.39. This description is only valid as long as the motion along the nuclear coordinates is slow and the electronic character does not change rapidly with R. If this is not the case, the nonadiabatic couplingsT_{k k}⁰ 0

andT_{k k}⁰⁰0 have to be taken into account and nonadiabatic transitions between the adiabatic potential energy surfaces occur. Most often this is the case in a region where two adiabatic

2. Scientific context

Figure 2.4.: An avoided crossing in the diabatic and adiabatic picture. The diabatic potential energy curves are shown as solid lines. A transition between these curves is most likely induced by an off-diagonal element of the electronic HamiltonianU_{k k}⁰. The adiabatic curves are shown as dashed lines. A so called nonadiabatic transition takes place between two adiabatic states and is mediated by an off-diagonal elementT_{k k}⁰arising from the nuclear kinetic energy operator acting on electronic wave functions.

curves approach each other closely, for example when the electronic character changes from neutral to anionic along the nuclear coordinate. In the case of very fast motion through this region an alternative diabatic representation can be useful for the description of the process. In the diabatic representation, the eletronic basis functions can be chosen as the approximate functions for the respective electronic configuration, for instance neutral and ionic states. However, there is no general definition of the diabatic representation.

The functions are often chosen such thatT_{k k}⁰ 0 andT_{k k}⁰⁰0 are minimized. Now, transitions between the diabats are caused by the nonzero off-diagonal elements of the electronic HamiltonianU_{k k}⁰. Figure 2.4 depicts an avoided crossing and gives an overview of the terminology used in the two different representations. As shown in the figure, diabats cross each other whereas a so called avoided crossing occurs in the adiabatic picture.

Note that a nonadiabatic transition describing the jump between two different adiabatic potential energy surfaces corresponds to the evolution of the system on a single diabatic surface.

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