Lets assume we have $N$ Atoms and we treat them within the Born-Oppenheimer Approximation. We can calculate the adiabatic electronic groundstate potential. Lets assume we observe two local minima, denoted by $\alpha$ and $\beta$, on this potential energy surface. Lets denote these minima nuclear coordinates as $R_\alpha$ and $R_{\beta} $. At each minimum we can do an harmonic approximation of the potential, centered on the corresponding minimum, \begin{eqnarray} V_\alpha(R) = V(R_\alpha)+(R-R_\alpha)^T V'_{\alpha} + 1/2(R-R_\alpha)^TV''_{\alpha}(R-R_\alpha) \tag{1}\\ V_\beta(R) = V(R_\beta)+(R-R_\alpha)^T V'_{\beta} + 1/2(R-R_\beta)^TV''_{\beta}(R-R_\beta) \tag{2}\\ \end{eqnarray}
where the gradent is $V'_\alpha=\nabla_RV(R)|_{R=R_\alpha}$ and the Hessian $V''_{\alpha,nm} = \frac{\partial^2 V}{\partial R_n \partial R_m}|_{R=R_\alpha}$.
Since we are in a local minimum, we know that the gradients vanish,$V'=0$. For simplicities sake, we also assume that both potentials evaluated at the minimum strcucture are zero, $V(R_{\alpha/\beta})=0$.
The associated harmonic vibrational Hamiltonian in atomic units then takes on the form $$\tag{3} H_\alpha(R) = -\frac{1}{2}\nabla^T_RM^{-1}\nabla_R + 1/2(R-R_\alpha)^TV''_\alpha(R-R_\alpha) $$ where $M$ is a diagonal matrix with the nuclear masses.
Each harmonic potential allows the definition of vibrational normal modes and vibrational normal coordinates. The transformation to normal coordinates is defined as follows.
First we define the mass weighted displacement coordinates $X_\alpha = M^{1/2}(R-R_\alpha)$, in these coordinates we have \begin{eqnarray} H_\alpha(X_\alpha)= -\frac{1}{2}\nabla^T_{X_\alpha}\nabla_{X_\alpha} +1/2 X^T_\alpha M^{-1/2}V''_\alpha M^{-1/2} X_\alpha \tag{4}\\ H_\alpha(X_\alpha)= -\frac{1}{2}\nabla^T_{X_\alpha}\nabla_{X_\alpha} +1/2 X^T_\alpha W_\alpha X_\alpha \tag{5}\\ \end{eqnarray} with the mass weighted Hesse matrix $W=M^{-1/2}V''M^{-1/2}$.
The last step to obtain normal coordinates is the transformation to coordinates that diagonalize the mass weighted Hesse matrix. Let $$ WK = K\Lambda \tag{6} $$ where $K$ is the matrix with eigenvectors of $W$ as columns and $\Lambda$ a diagonal matrix with corresponding eigenvalues. The required transformation is then $$ X= KQ\tag{7} $$ which leads to $$ H_\alpha(Q_\alpha)= -\frac{1}{2}\nabla^T_{Q_\alpha}\nabla_{Q_\alpha} +1/2 Q^T_\alpha \Lambda_\alpha Q_\alpha \tag{8} $$
Summarized we have two different sets of transformations, one for each minimum, \begin{eqnarray} Q_\alpha = K^{-1}_\alpha M^{1/2}(R-R_\alpha)\tag{9}\\ Q_\beta = K^{-1}_\beta M^{1/2}(R-R_\beta) \tag{10} \end{eqnarray}
After this rather lengthy setup comes my question. How can I transform from one set of normal coordinates $Q_\alpha$ to the other $Q_\beta$ and how do I interpret this. Does that mean that I can express any molecules vibrational coordinates in the vibrational coordinates of another molecule(made from the same number and type of atoms)?
Intuitively I see problems when the geometries are translated and rotated with respect to each other assuming a global cartesian coordinate system for both molecules. There should be no simple internal vibrational coordinate that could connect two such geometries but on the other hand, internal vibrational coordinates should form a complete coordinate system and allow me to express any structure with the same amount of atoms in it. I think I am missing something very obvious to reconcile this "contradiction", which is why I need some input.
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