Scientists Reveal a Linear Relation between Radiative and Nonradiative Decay Rates of Molecules

Date：17-05-2023 Print

Efficient fluorescent molecules have extensive applications in fields such as molecular biological imaging, organic light-emitting diodes and laser applications. The fluorescence efficiency of molecules depends on two competing relaxation processes of the excited state, namely radiative and nonradiative transitions. Radiative transitions involve the emission of a photon with energy equal to or less than the excitation energy, whereas nonradiative channels transfer excitation energy to the vibrational levels without any photon emission. The ratio of the radiative decay rate (*k _{R}*) to the nonradiative decay rate (

On April 28th, 2023, the Journal of Physical Chemistry Letters published an online report on the research work conducted by the WENG Yuxiang research group in Soft Matter Physics Laboratory (SM06) at the Institute of Physics, Chinese Academy of Sciences/Beijing National Laboratory for Condensed Matter Physics. The letter is titled "Theoretical and Experimental Investigation of the Electronic Propensity Rule: A Linear Relation between Radiative and Nonradiative Decay Rates of Molecules" (Figure 1). The doctoral students ZHANG Ying and LIU Heyuan are the first and second authors, respectively.

This work summarized the theoretical derivation of *k _{R}* and

\(J\left(k_R\right) \propto\left\langle\phi_f\left|e \boldsymbol{r}_e\right| \phi_i\right\rangle_e ; J\left(k_{N R}\right) \propto\left\langle\phi_f\left|\frac{\partial}{\partial R_{a n}}\right| \phi_i\right\rangle_e\).

Formally, the two are not directly related. SCHUURMANS *et al*. introduced the velocity expression of the electronic dipole operator, given by CHANDRASEKHAR X.:

\(\left\langle\phi_f|\boldsymbol{\mu}| \phi_i\right\rangle_e=\frac{e \hbar^2}{m_e \mit\Delta E}\left\langle\phi_f\left| \boldsymbol{\Sigma}_e \boldsymbol{\nabla}_{r_e}\right| \phi_i\right\rangle_e\).

By specifying the symmetry of the molecular internal interaction potential on the electronic and nuclear coordinates \(\boldsymbol{\Sigma} _e \boldsymbol{\nabla}_{r_e} V(\boldsymbol{r}, \boldsymbol{R})+\boldsymbol{\Sigma} _n \boldsymbol{\nabla}_{R_n} V(\boldsymbol{r}, \boldsymbol{R})=0\), and performing the commutativity of quantum mechanical operators, the equivalent relationship between the electronic coordinate gradient operator and the nuclear coordinate gradient operator acting between the initial and final electronic states was obtained, i.e., \(\left\langle\phi_f \mid \Sigma_e \nabla_{r_e} \phi_i\right\rangle_e=-\left\langle\phi_f \mid \Sigma_n \nabla_{R_n} \phi_i\right\rangle_e\). This established the inherent connection between the electronic coupling terms for radiative and nonradiative transitions, allowing for the ratio of *k _{R}* to

To verify the relationship between *k _{R}* and

This work provides a theoretical understanding of the linear relation between radiative and nonradiative decay rates of molecules under external electric field perturbation in weak coupling limit, and verifies this relationship through experimental data from dextran-dye molecules and LH2. This work provides further insights into the relationship between radiative and nonradiative processes and has some theoretical guidance for the efficiency of molecular luminescence, among other applications.

The study was supported by the Ministry of Science and Technology, the National Natural Science Foundation of China, the Strategic Priority Research Program of the Chinese Academy of Sciences, and the Chinese Academy of Sciences Frontier Science Key Programs.

Figure 1. Schematic representation of the radiative and nonradiative decay processes in weak coupling limit. (a) Radiative process releasing a photon without phonons. (b) Nonradiative processes for energy relaxation by promoting modes and accepting modes. (Image by Institute of Physics)

Figure 2. Plot of *k _{NR}* against

**Contact**:

Institute of Physics

WENG Yuxiang

Email: yxweng@iphy.ac.cn

**Key words**:

Fluorescence efficiency; radiative decay rate; nonradiative decay rate; electronic propensity rule

**Abstract**:

This study provides a theoretical derivation of a linear relation between radiative and nonradiative decay rates of molecules under external electric field perturbation in weak coupling limit, and verifies this relationship through experimental data from dextran-dye molecules and the photosynthetic bacterial light-harvesting antenna protein complexes.

The Institute of Physics, Chinese Academy of Sciences P.O.Box 603,Beijing 100190,China

Tel:86-10-82649361 Fax:86-10-82649531