I have a diabatic potential energy matrix, $V(r,Z)$, (real symmetric) for a 2-level system with two nuclear coordinates, $(r,Z)$: $$ V(r,Z)= \begin{pmatrix} V_{11}(r,Z) & V_{12}(r,Z)\\ V_{12}(r,Z) & V_{22}(r,Z) \end{pmatrix} $$ To run the dynamics I would like to write a subroutine that computes the eigenvalue/vector derivatives (with respect to r and Z). A literature search lead me to these formulas by J.R. Magnus, $$ d\lambda\;=\;(u_0^T\otimes u_0^T)\text{vec}\;dX\\ du\;=\;[u_0^T\otimes(\lambda_0I-X_0)^+]\text{vec}\;dX\\ $$ where $X_0$ is the real symmetric matrix, $u_0$ its eigenvectors, $\lambda_0$ the simple eigenvalues, $I$ the identity matrix, $\Lambda^{+}$ is the Moore-Penrose psuedoinverse of $\Lambda$, and $dX$, $du$, $d\lambda$ are differentials.
Am I on the right track in using these formulas? The eigenvalues and eigenvectors of $V(r,Z)$ are easily obtained, and I assume $dX$ in my case translates to the gradient of $V(r,Z)$, but the elements of the $\nabla V(r,Z)$ matrix are 2-vectors, and so I'm unsure how to represent $\text{vec}\;dX$,or make any more sense of this for that matter. Any insight would be greatly appreciated!
First, you need to find the differential of $V.\,\,\,$ This will be a matrix whose components are $$\eqalign{ dV_{ij} &= \frac{\partial V_{ij}}{\partial r}\,dr + \frac{\partial V_{ij}}{\partial z}\,dz \cr }$$ or more succinctly $$\eqalign{ dV &= V_r\,dr + V_z\,dz \cr }$$ The $(V_r,V_z)$ terms are $2\times 2$ matrices, while $(dr,dz)$ are scalars.
Substitute the differential directly into Magnus's formulae. Since we're now dealing with scalar variables, there's no need to vectorize the matrix terms. $$\eqalign{ d\lambda &= u^T\,dV\,u \cr &= (u^T\,V_r\,u)\,dr + (u^T\,V_z\,u)\,dz \cr \frac{\partial \lambda}{\partial r} &= u^T\,V_r\,u \cr \frac{\partial \lambda}{\partial z} &= u^T\,V_z\,u \cr \cr du &= (\lambda I-V)^+\,dV\,u \cr &= (\lambda I-V)^+(V_r\,u)\,dr + (\lambda I-V)^+(V_z\,u)\,dz \cr \frac{\partial u}{\partial r} &= (\lambda I-V)^+ V_r\,u \cr \frac{\partial u}{\partial z} &= (\lambda I-V)^+ V_z\,u \cr\cr }$$