Is there a relationship between the transition moment and position? Suppose that there exists two states which are not eigenstates of the position operator in a superposition:
$$
\Psi(t) = a \phi_1(r, t) + b \phi_2(r, t) . 
$$
The expectation value of the position operator is then given as follows:
$$
\langle \hat{r} \rangle = | a |^2 \langle \phi_1 | \hat{r} | \phi_1 \rangle + | b |^2 \langle \phi_2 | \hat{r} | \phi_2 \rangle + a^* b \langle \phi_1 | \hat{r} | \phi_2 \rangle + b^* a \langle \phi_2 | \hat{r} | \phi_1 \rangle. 
$$
The meaning of this is transparent and, if these states have different energies, the expectation value of position will vary in time according to the above equation.
Integrals of this form are oftentimes seen in quantum mechanics, where one is calculating transition rates between two different states through the dipole interaction (proportional to the position operator). Specifically, the transition between two different states, $d_{nm}$, is the off-diagonal component of the position operator:
$$
d_{nm} \propto \langle \phi_n | \hat{r} | \phi_m \rangle . 
$$
Given the similar form in this and the position expectation value of the above equation, is there an interpretation of transition moments related to position?
The motivation for this question is there are transition moments which may change sign or vanish depending on the orbitals involved (e.g. Cooper minima). Is there a position-related interpretation for situations like this?
 A: 
is there an interpretation of transition moments related to position?

AFAIK only in the trivial sense you point out - expected average position (or center of mass of electronic charge) for given psi function can be expressed using similar integrals for pairs of Hamiltonian eigenfunctions. But note $d_{mn}$'s (fixed numbers) are only part of the expression, they are not sufficient, since the time-dependent expansion coefficients are important too.
Also, average position is often not considered interesting in light-molecule interaction because the material system is considered to be not moving or moving negligibly from the assumed position. But this would change if effect of radiation on translatory motion of the system is studied, or if there are multiple such systems with non-trivial motion.
It is known that if there is to be a non-zero energy transfer in dipole interaction (linear in electric field) between light and a pair of stationary states $m,n$ of an atom/molecule, the corresponding non-zero dipole moment element $d_{mn}$ has to be non-zero (this depends on the Hamiltonian) and also the assumed atom/molecule psi function has to have overlap with both corresponding basis functions $\phi_m,\phi_n$. The expected center of charge for this psi function will then manifest partial oscillation at the corresponding frequency $\frac{E_n-E_m}{h}$.

The motivation for this question is there are transition moments which may change sign or vanish depending on the orbitals involved (e.g. Cooper minima).

If transition moment element of an atom/molecule changes sign, something in the Hamiltonian has to change. What is it, can you give an example?
