Radiative Multi-Phonon Emission

D.1 Radiative Multi-Phonon Emission

In a radiative process the energy necessary for a transition from $V1$ to $V2$ is obtained from the binding energy, $EB (q) = V2(q)− V1(q)$ . This energy writes as

1 2 2 1 2 2 EB (q) = E◟2-−◝◜E1◞+ 2M ω 2(q − q2) − 2-M ω1(q − q1) E21 1 ( ω2 ) = E21 + --M ω22 (q22 − 2q1q2 + q21+2q1q2 − q21 − 2qq2 + q2)−-12 (q − q1)2 2 ◟-----◝◜--2--◞ ◟ω◝2◜◞ (q1−q2) R2 1 ( ) = E21 + Sℏ ω2 + -M ω22 (2q1q2 − q21 − 2qq2 + q2)− R2 (q − q1)2 , (D.3) 2

with the relaxation energy $Sℏ ω2 = 12M ω22(q1 − q2)2$ from $V2(q1)$ to $V2(q2)$ . The two points where a radiative transition is possible after the Franck-Condon principle are at $q1$ and $q2$ . Inserting yields

EB (q1) = ϵ12 = E21 + S ℏω2 ( ) 1 2 2 2 1 2 2 EB (q2) = ϵ21 = E21 + S ℏω2 + 2M ω 2 −◟-q1 +-2q◝1◜q2 −-q2◞ − 2M ω1 (q − q1) −(q1− q2)2 ◟--------------◝◜--------------◞ 0 ϵ21 = E21 − S ℏω1. (D.4 )

Analogously to $Sℏω2$ , $Sℏω1$ is defined as the relaxation energy from $V1 (q2)$ to $V1(q1)$ . A full MPE process is schematically depicted in Fig. D.1 for quadratic coupling ( $ω ⁄= ω 1 2$ ). In the case of linear coupling ( $ω = ω 1 2$ ) both relaxation energies coincide.

Figure D.1: Description of a radiative multi-phonon emission process assuming harmonic oscillators with different vibronic frequencies $ω1$ and $ω2$ in a reaction coordinate diagram. The photon energy required to change from $V1$ to $V2$ equals $ϵ12 = E21 + S ℏω2$ . Due to structural relaxation, the photon emitted in the following reverse process is smaller, namely $ϵ = E − S ℏω 21 21 1$ .

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