# How is the probability of intersystem crossing influenced by small differences in energy levels?

Say I have some dye molecule in some low level triplet state, e.g. $T_1$, and it's decaying slowly to the ground state via phosphorescence. However, there are also events where the dye in the $T_1$ state can undergo intersystem crossing to transition to the first singlet excited state, $S_1$, which then decays rapidly to the ground state releasing a photon in the process.

Predicting the probability of intersystem crossing is difficult, and usually requires experimental support. So I'm not asking here about how to ab initio calculate this probability. What I'm interested in is how the probability changes as a function of what we'll call $\Delta G$, which is the difference in energy between the $T_1$ and $S_1$ state.

Can we understand the influence of the energy gap on the probability of $T_1 \to S_1$ intersystem crossing using a simple Boltzmann approximation of the form: $P(T_1 \to S_1) \approx P_c \times e^{(\frac{\Delta G}{k_B*T})}$, where $k_B$ is Boltzmann's constant, and $P_c$ is the probability of intersystem crossing assuming that $T_1$ and $S_1$ have identical energies within some small error $\epsilon$?

• For small delta-G's the exponential will become linear. – Kevin Kostlan Dec 19 '13 at 15:58
• @KevinKostlan Do you mean this in the sense that it will appear to scale linearly, or that there will be a different phenomena at the limit of small energy gaps? – RGrey Dec 19 '13 at 21:27