# What is the “direction” of the transition dipole moment? (Understanding Eq. 9.29, Charge and Energy Transfer 3rd Ed, May & Kuhn)

For a real vector $\mathbf{r}$, the direction is given by: $\hat{\mathbf{n}}=\mathbf{r}/\left|\mathbf{r}\right|$.

The transition dipole moment is a complex vector. How do you define its direction?

The point of this question is that I am trying to understand the meaning of equation 9.29 of the book Charge and Energy Transfer 3rd Ed by May & Kuhn, which reads:

$$J_{mn}=\frac{\left|\mathbf{d}_{m}\right|\left|\mathbf{d}_{n}^{*}\right|}{R_{mn}^3}\left[\mathbf{n}_{m}\cdot\mathbf{n}_{n}-3\left(\mathbf{e}_{mn}\cdot\mathbf{n}_{m}\right)\left(\mathbf{e}_{mn}\cdot\mathbf{n}_{n}\right)\right]$$

In the next paragraph I attempt to explain the notation in this equation. I'll point out the parts that confuse me.

Here we are considering two molecules labeled $m$ and $n$, and we only consider two electronic levels in each, the ground state $g$ and one excited state $e$. $\mathbf{d}_{m}$ is the transition dipole moment for the transition $g\rightarrow e$ of the $m$th molecule, and similarly for $\mathbf{d}_{n}$. $\mathbf{n}_{m}$ is a unit vector pointing in the direction of $\mathbf{d}_{n}$ (I don't understand this), and similarly for $\mathbf{n}_{n}$. Finally, $R_{mn}$ is the distance between the centers of mass of the molecules. The quantity we are calculating here, $J_{mn}$, is the excitonic coupling between the two molecules. This can be seen as the rate at which an exciton at a molecule $n$ will be transfered to a molecule $m$ initially in its ground state. Here we calculate $J_{mn}$ using the dipole-dipole interaction approximation, valid when the molecules are sufficiently far apart.

Note also the factor $\left|\mathbf{d}_{n}^{*}\right|$ in the equation. I don't understand this bit of notation either. I am not sure how to interpret the complex conjugation sign ($*$) as it appears inside absolute-value brackets ($|\square |$).

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