The diffusion equation for function $\Psi=e^{-ir\vec{k}}$ is: $$ \nabla \cdot (D\nabla\psi)= (-{\rm i})\nabla \cdot (D\psi \vec{k})=(-{\rm i})\sum_i \frac{\partial}{\partial x_i}(D\psi \vec{k})_i=(-{\rm i})\sum_{i,j} \frac{\partial}{\partial x_i}D_{ij}\psi k_j$$
$$=-\sum_{i,j} D_{ij}\psi k_ik_j=-(k^TDk)\psi$$
I would like to calculate it analogously and get the result in the same form for more general case: $ \Psi=e^{i \vec{\phi}} $
My try is:
$$ \nabla \cdot (D\nabla e^{i\vec{\phi}}) = i\nabla \cdot (D\frac{\partial \phi}{\partial r}e^{i\vec{\phi}}) = i\sum_i \frac{\partial}{\partial x_i}(D\frac{\partial \phi}{\partial r}e^{i\vec{\phi}})_i = $$
$$ = -(\frac{\partial \phi}{\partial r})^{T}D\frac{\partial \phi}{\partial r}e^{i\vec{\phi}} + iD\frac{\partial^2 \phi}{\partial r^2}e^{i\vec{\phi}} $$
I've been learning the vector calculus for a short time and I am still a little bit confused. Is it correct?