I'm trying to understand the derivation provided in the section 2.4 of this paper. I have modified the notation and cut out the unimportant parts of the equations for clarity purposes, but for reference I will be dealing with equation 29, 30, 31 and 33.

In the relevant part, they have the action of a system originally expressed in terms of clasical nuclei coordinates and quantum electronic wavefunctions and they make a change of variables to consider now the same clasical coordinates ($\mathbf{R_N}$) but with the coeficients in the expansion of a basis ($c_{i,j}$). So far so good.

Then, having the following (part of) the action:

\begin{equation} A_q = \int_{t_1}^{t_2} F_q(t) dt \end{equation} where \begin{equation} F_q = \sum_{j} \sum_{\alpha \beta} c_{\alpha,j}^{\ast} \left[ A_{\alpha,\beta} \, c_{\beta,j} + B_{\alpha,\beta} \, \dot{c}_{\beta,j} \right] \end{equation} and the matrices $A$ and $B$ are a function of the $\mathbf{R_N}$ (but not of $\mathbf{\dot{R}_N}$) and the $^{\ast}$ is just conjugating the complex number. Now the problem is that when they derive that part of the action, they do it differently for the coeficients and for the spatial coordinates: \begin{align} \frac{\delta A_q}{\delta c_{\alpha,j}^{\ast}} = \frac{\partial F_q}{\partial c_{\alpha,j}^{\ast}} \qquad \qquad \frac{\delta A_q}{\delta \mathbf{R_N}} = \frac{\partial F_q}{\partial \mathbf{R_N}} + \frac{d}{dt}\left( \frac{\partial F_q}{\partial \mathbf{\dot{R}_N}} \right) \end{align}

Moreover, the final expresion does not derive the non conjugate coeficients. It's like it is taking both to be independent (but I understand they are not) and only obtaining the equations for one of them.

\begin{equation} \frac{\partial F_q}{\partial c_{\alpha,j}^{\ast}} = \sum_{\beta} A_{\alpha,\beta} \, c_{\beta,j} + B_{\alpha,\beta} \, \dot{c}_{\beta,j} \end{equation}

Why are these general coordinates being treated differently?


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