Lets say i have an initial state $|i\rangle$ and a final state $|f\rangle$. A transition from $|i\rangle \rightarrow |f\rangle$ is coupled by an operator $\hat O$. Is the relevant coupling matrix element for the transition starting in state $|i\rangle$ going to $|f\rangle$ the matrix element $$ \langle i|\hat O |f\rangle =O_{if} $$ or is the relevant matrix element $$\langle f | \hat O | i \rangle = O_{fi} $$ ?
I thought that the relevant matrix element is $$ O_{fi} $$
but a paper discussing nonadiabatic coupling is confusing me, since the operator always appears with indices switched in comparison to what i would expect.
For Context:
My source of confusion is this paper, https://doi.org/10.1016/j.chemphys.2008.01.044, which is about Nonadiabatic Surface Hopping. The section "2.2 Velocity adjustment" equation (21) uses the nonadiabatic coupling element of first order $\vec d_{ij}$ to rescale velocities after a hop between two adiabatic electronic states. The hop i.e. coupling is from state $i$ to state $j$, which is why was expecting the coupling element $ \vec d_{ji}$. The indices on this coupling element are reversed in comparison to what i would have expected.
The nonadiabatic coupling element of first order is defined as $$ \vec d_{ij}(R) = \langle \phi_i(r,R)| \nabla_R \phi_j(r,R)\rangle_r $$ where $\phi_{i/j}(r,R)$ are adiabatic electronic states, i.e. eigenstates to the electronic Hamilton operator within the Born Oppenheimer Approximation.