Theory

Overview

Optical absorption, emission, and nonradiative decay processes in molecules are of both fundamental and applied interests. In organic light-emitting materials, the emission color is determined by both the electronic transition energy from the excited state to the ground state and the vibrationally resolved emission spectrum. The light-emitting efficiency is determined by a competition between radiative and nonradiative decay rates.

Harmonic Oscillator Model

We apply Born-Oppenheimer approximation and suppose a transition (absorption, emission, or nonradiative decay) happens between initial vibronic state \(\left|\Psi_{av}\right\rangle = \left|\Phi_{a}\Theta_{av}\right\rangle\) and final vibronic state \(\left|\Psi_{bu}\right\rangle = \left|\Phi_{b}\Theta_{bu}\right\rangle\), where \(\left|\Phi_{a}\right\rangle\) and \(\left|\Phi_{b}\right\rangle\) are adiabatic electronic states, \(\left|\Theta_{av}\right\rangle = \left|\chi_{av_1} \chi_{av_2} \cdots \chi_{av_N} \right\rangle\) and \(\left|\Theta_{bu}\right\rangle = \left|\chi_{bu_1} \chi_{bu_2} \cdots \chi_{bu_N} \right\rangle\) are vibrational states. \(\left|\chi_{av_k} \right\rangle\) and \(\left|\chi_{bu_l} \right\rangle\) are eigenstates of one-dimensional harmonic oscillator Hamiltonian:

\[\hat{H}_{ak} = \frac{1}{2} \left( \hat{P}_{ak}^2 + \omega_{ak}^2 \hat{Q}_{ak}^2 \right)\]
\[\hat{H}_{bl} = \frac{1}{2} \left( \hat{P}_{bl}^2 + \omega_{bl}^2 \hat{Q}_{bl}^2 \right)\]

The normal mode coordinates \(Q_{ak}\) and \(Q_{bl}\) can be represented by mass-weighted Cartesian coordinates

\[Q_{ak} = \sum_{\sigma=1}^{n} \sum_{j=x,y,z} L_{a\sigma j, k} \left( q_{a\sigma j} - q_{a\sigma j}^0 \right)\]
\[Q_{bl} = \sum_{\sigma=1}^{n} \sum_{j=x,y,z} L_{b\sigma j, l} \left( q_{b\sigma j} - q_{b\sigma j}^0 \right)\]

The ground state molecular geometry and excited state molecular geometry may be put in one frame of reference, and the normal mode coordinates \(Q_{ak}\) and \(Q_{bl}\) can be related by a Duschinsky rotation matrix \(\mathbf{S}_{a \leftarrow b}\) and a coordinate displacement vector \(\mathbf{D}_{a \leftarrow b}\).

\[Q_{ak} = \sum_{l=1}^N \mbox{S}_{a \leftarrow b, kl} Q_{bl} + {D}_{a \leftarrow b, k}\]

where

\[\mathbf{S}_{a \leftarrow b} = \mathbf{L}_{a}^T \mathbf{L}_{b}\]
\[\mathbf{D}_{a \leftarrow b} = \mathbf{L}_{a}^T \left( \mathbf{q}_b^0 - \mathbf{q}_a^0 \right)\]

\(\mathbf{S}_{a \leftarrow b}\) is a unitary matrix, the Duschinsky rotation matrix, whose elements represent the mixing of normal modes in the initial and final electronic states. \(\mathbf{D}_{a \leftarrow b}\) is a displacement vector connecting the minima of the parabolas of the two electronic states. The Duschinsky rotation matrix and mode displacement vector can be abbreviated as \(\mathbf{S}\) and \(\mathbf{D}\).

Optical Absorption and Emission Spectra

The absorption spectrum is given as the absorption cross section \(\sigma_{\mathrm{abs}(\omega)}\) with dimensions of \(\mathrm{cm}^2\). This cross section is defined as the rate of photon energy absorption per molecule and per unit radiant energy flux, which is equivalent to the ratio of the power absorbed by the molecule to the incident power per unit area.

\[\sigma_{\mathrm{abs}}(\omega) = \frac{4\pi \omega}{3c} \sum_{v,u} P_{a v} (T) \left| \left\langle \Theta_{bu} \left| \vec{\mu}_{ba} \right| \Theta_{av} \right\rangle \right|^2 \delta \left( \hbar \omega - E_{ba} - E_{bu} + E_{av} \right)\]

where \(P_{av}\) is Boltzmann distribution function for the initial vibronic manifold.

The emission spectrum is given as the differential spontaneous photon emission rate \(\sigma_{\mathrm{em}(\omega)}\). This is a dimensionless quantity defined as the rate of spontaneous photon emission per molecule and per unit frequency between \(\omega\) and \(\omega\) + d\(\omega\). The explicit expression for \(\sigma_{\mathrm{em}(\omega)}\) is given by the following formula,

\[\sigma_{\mathrm{emi}}(\omega) = \frac{4 \omega^3}{3c^3} \sum_{v,u} P_{a v} (T) \left| \left\langle \Theta_{bu} \left| \vec{\mu}_{ba} \right| \Theta_{av} \right\rangle \right|^2 \delta \left( E_{ab} + E_{av} - E_{bu} - \hbar \omega \right)\]

In general, \(\mathbf{\mu}_{ba}\) is not a constant vector, which is depend on the nuclear coordinate. Thus it can be expanded as

\[\vec{\mu} = \vec{\mu}_0 + \sum_k \vec{\mu}_k Q_k + \sum_{kl} \vec{\mu}_{kl} Q_k Q_l + \cdots\]

For strongly dipole-allowed transitions, the spectrum is usually dominated by the zero-order term of the expansion (Franck-Condon approximation), while for weakly dipole-allowed or -forbidden transitions, one must at least consider the first order term (Herzberg-Teller approximation).

\[\vec{\mu} = \vec{\mu}_0 + \sum_k \vec{\mu}_k Q_k\]

Applying the Fourier transformation to the delta functions in the above equations,

\[\delta(\omega) = \frac{1}{2\pi} \int_{-\infty}^\infty e^{i \omega t} dt\]

The spectrum formula can be represented as three terms:

\[ \sigma_{\mathrm{emi}} (\omega) = \sigma_{\mathrm{emi}}^{\mathrm{FC}} (\omega) + \sigma_{\mathrm{emi}}^{\mathrm{FC/HT}} (\omega) + \sigma_{\mathrm{emi}}^{\mathrm{HT}} (\omega)\]
\[\sigma_{\mathrm{emi}}^{\mathrm{FC}} (\omega) = \frac{2\omega^3}{3\pi \hbar c^3} \left| \vec{\mu}_0 \right|^2 \int_{-\infty}^{\infty} e^{-i(\omega - \omega_{ab})t} \rho_{\mathrm{emi},0}^{\mathrm{FC}} (t,T)\]
\[\sigma_{\mathrm{emi}}^{\mathrm{FC/HT}} (\omega) = \frac{2\omega^3}{3\pi \hbar c^3} \sum_k \vec{\mu}_0 \cdot \vec{\mu}_k \int_{-\infty}^{\infty} e^{-i(\omega - \omega_{ab})t} \rho_{\mathrm{emi},k}^{\mathrm{FC/HT}} (t,T)\]
\[\sigma_{\mathrm{emi}}^{\mathrm{HT}} (\omega) = \frac{2\omega^3}{3\pi \hbar c^3} \sum_{kl} \vec{\mu}_k \cdot \vec{\mu}_l \int_{-\infty}^{\infty} e^{-i(\omega - \omega_{ab})t} \rho_{\mathrm{emi},kl}^{\mathrm{HT}} (t,T)\]

where the three kinds of thermal vibration correlation functions can be defind as

\[\rho_{\mathrm{emi},0}^{\mathrm{FC}} (t,T) = Z_{iv}^{-1} \mathrm{Tr} \left[ e^{-i\tau_b \hat{H}_b } e^{-i\tau_a \hat{H}_a } \right]\]
\[\rho_{\mathrm{emi},k}^{\mathrm{FC/HT}} (t,T) = Z_{iv}^{-1} \mathrm{Tr} \left[ Q_{bk} e^{-i\tau_b \hat{H}_b } e^{-i\tau_a \hat{H}_a } \right]\]
\[\rho_{\mathrm{emi},kl}^{\mathrm{HT}} (t,T) = Z_{iv}^{-1} \mathrm{Tr} \left[ Q_{bk} e^{-i\tau_b \hat{H}_b } Q_{bl} e^{-i\tau_a \hat{H}_a } \right]\]

The analytical forms of the spectra correlation functions can be find in the reference [1].

Internal Conversion Rate

According to Fermi’s golden rule, the internal conversion (IC) rate can be presented as

\[ k_{\mathrm{ic}} = \frac{2\pi}{\hbar} \left| -\hbar^2 \sum_{l} \left\langle \Phi_b \Theta_{bu} \left| \frac{\partial \Phi_a}{\partial Q_{bl}} \frac{\partial \Theta_{av}}{\partial Q_{bl}} \right. \right\rangle \right|^2 \delta\left( E_{ba} + E_{bu} - E_{av} \right)\]

Apply Condon approximation, Fourier transformation, the IC rate can be represented as

\[k_{\mathrm{ic}} = \sum_{kl} k_{\mathrm{ic},kl}\]
\[k_{\mathrm{ic}} = \frac{1}{\hbar^2} R_{kl} \int_{-\infty}^{\infty} dt e^{i\omega_{ab}t} \rho_{\mathrm{ic},kl} (t,T)\]

where

\[R_{kl} = \left\langle \Phi_b \left| \hat{P}_{bk} \right| \Phi_a \right\rangle \left\langle \Phi_a \left| \hat{P}_{bk} \right| \Phi_b \right\rangle\]

and

\[\rho_{\mathrm{ic},kl} (t,T) = Z_{av}^{-1} \mathrm{Tr} \left[ P_{bk} e^{-i\tau_b \hat{H}_b } P_{bl} e^{-i\tau_a \hat{H}_a } \right]\]

The analytical forms of the IC correlation functions can be find in the reference [1,2].

Intersystem crossing (ISC)

Based on the time-dependent second-order perturbation theory and the Born-Oppenheimer adiabatic approximation, the thermal average rate constant from the initial electronic state \(a\) with the vibrational quantum numbers \(v\) to the final electronic state \(b\) with the vibrational quantum numbers \(u\) reads:

\[k_{b \rightarrow a} = \frac{2\pi}{\hbar} \sum_{vu} P_{av} \left| H'_{bu,av} + \sum_{cw} \frac{ H'_{bu,cw} H'_{cw,av} }{ E_{bv} - E_{cw} } \right|^2 \delta(E_{av} - E_{bu})\]

For the radiationless transitions, the interaction Hamiltonians can be considered as

\[H' \Psi_{av} = -\hbar^2 \frac{\partial \Phi_a}{\partial Q_{bl}} \frac{\partial \Theta_{av}}{\partial Q_{bl}} + \hat{H}^{\mathrm{SO}} \Phi_a \Theta_{av}\]

The ISC rate can be derived as

\[k_{\mathrm{isc}} = k_{\mathrm{isc}}^{(0)} + k_{\mathrm{isc}}^{(1)} + k_{\mathrm{isc}}^{(2)}\]

where

\[k_{\mathrm{isc}}^{(0)} = \frac{1}{\hbar^2} R_{ba}^{\mathrm{isc}} \int_{-\infty}^{\infty} dt e^{i\omega_{ab}t} \rho_{\mathrm{isc}}^{(0)}(t,T)\]
\[k_{\mathrm{isc}}^{(1)} = Re \left[ \frac{1}{\hbar^2} \sum_{k} 2 R_{ba,k}^{\mathrm{isc}} \int_{-\infty}^{\infty} dt e^{i\omega_{ab}t} \rho_{\mathrm{isc},k}^{(1)}(t,T) \right]\]
\[k_{\mathrm{isc}}^{(2)} = \frac{1}{\hbar^2} \sum_{kl} R_{ba,kl}^{\mathrm{isc}} \int_{-\infty}^{\infty} dt e^{i\omega_{ab}t} \rho_{\mathrm{isc},kl}^{(2)}(t,T)\]

where the correlation function is the same as that defined in IC process.

The analytical forms of the ISC correlation functions and electron coupling terms \(R_{ba}^{\mathrm{isc}}\), \(R_{ba,k}^{\mathrm{isc}}\) and \(R_{ba,kl}^{\mathrm{isc}}\) can be find in the reference [3].

[1]. Niu, Y.; Peng, Q.; Deng, C.; Gao, X.; Shuai, Z., The Journal of Physical Chemistry A, 2010, 114, 7817-7831. [2]. Niu, Y.; Peng, Q.; Shuai, Z., Science in China Series B: Chemistry, 2008, 51, 1153-1158. [3]. Peng, Q.; Niu, Y.; Shi, Q.; Gao, X.; Shuai, Z., Journal of Chemical Theory and Computation, 2013, 9, 1132-1143.