# Property of nonadiabatic vector coupling matrix

I just tried to derive the "dressed" kinetic energy operator (for the Hamiltonian $\mathbf{H} = \frac{1}{2M}\left(\mathbf{P} -\mathrm{i}\hbar \mathbf{F} \right)^2 +\mathbf{V}$) in the adiabatic basis [1] and came across the following relationship

$$\nabla \cdot \mathbf{F} = (\nabla \cdot \mathbf{F}) + \mathbf{F} \cdot \nabla$$

for the vector coupling matrix $\mathbf{F}$. I could not figure out what the difference between $\nabla \cdot \mathbf{F}$ and $(\nabla \cdot \mathbf{F})$ is. Hence, my question is what this notation means.

[1] cp. eg. Domcke, Wolfgang, David Yarkony, and Horst Köppel. Conical intersections electronic structure, dynamics & spectroscopy. River Edge, N.J. London: World Scientific, 2004, page 8 (https://books.google.ch/books?id=Oo9pDQAAQBAJ&lpg=PA8&ots=ULT0gLSCoe&dq=nonadiabatic%20vector%20coupling%20gradient%20dressed%20operator&pg=PA8#v=onepage&q=nonadiabatic%20vector%20coupling&f=false)

The difference in notation tells you if you should use chain rule when you apply another operator or wavefunction. For example, $$(\nabla F)\Psi$$ is just $$(\nabla F)\Psi$$, whereas, $$\nabla F \Psi = (\nabla F) \Psi + F\nabla\Psi$$. In order to understand more deeply why this is true, you should read over the definition of the momentum operator in Sakurai's Modern Quantum Physics textbook. After reading the relevant chapter, you will see that this follows from the definition of $$\nabla$$, which is kind of a tricky operator that needs a wavefunction to be properly defined. I always learned in undergrad that you should apply a dummy wavefunction to the right side of $$\nabla$$ to see what it does, then remove it afterward for the sake of notation.