# Polyatomic molecule vibrational spectra - Molecular dipole expansion in Taylor series

To give some context: I am studying the molecular spectra in the context of an introductory course to atomic and molecular physics. My professor is using the transition rate expression from an initial state $$|a\rangle$$ to a final state $$|b\rangle$$ in the dipole approximation: $$$$W_{a\rightarrow b} = \frac{4\pi^{2}}{e\hbar} \, \alpha \, I(\omega_{ba}) \, |\vec{\varepsilon}\cdot\vec{D}_{ba}|^{2}$$$$ to derive the selection rules for the vibrational (and rotational) spectra. $$\vec{D}_{ba}=\langle b|\vec{D}|a\rangle$$ is the matrix element of the molecular dipole between the two states, and for a generic molecule composed of $$N$$ nuclei of positions and atomic numbers $$\vec{R}_{I}$$, $$Z_{I}$$ and $$n$$ electrons of positions $$\vec{r}_{i}$$ the dipole is given by: $$$$\vec{D} = e \left( \sum_{I=1}^{N} Z_{I} \vec{R}_{I} - \sum_{i=1}^{n} \vec{r}_{i} \right)$$$$ For a diatomic molecule with relative distance between the nuclei 1 and 2 given by $$R=|\vec{R}_{1}-\vec{R}_{2}|\,$$, if $$\vec{D}$$ depends on $$R$$ one can expand it in series around the distance of equilibrium between the two nuclei $$R_{\mathrm{eq}}$$ ["physics of Atoms and Molecules", B.H. Bransden & C.J. Joachain]: $$$$\vec{D}(R) = \vec{D}(R_{\mathrm{eq}}) + (R-R_{\mathrm{eq}}) \, \frac{\mathrm{d}\vec{D}(R)}{\mathrm{d}R} \, \bigg|_{R=R_{\mathrm{eq}}} + \cdots$$$$ Which is logic enough, since the derivation of the vibrational and rotational eigenvalues has been made only for a diatomic molecule on the assumption that the nuclear motion is centred around $$R_{\mathrm{eq}}\,$$.
The problem arises when my professor wrote that expansion for a polyatomic molecule like this: $$$$D_{\beta}(\vec{R}) \simeq D_{\beta}(\vec{R}_{\mathrm{eq}}) + \sum_{I=1}^{N} \ \sum_{\alpha=x,y,z} (R_{I,\alpha}-R_{I,\alpha}^{\mathrm{eq}}) \, \frac{\partial D_{\beta}(\vec{R})}{\partial R_{I,\alpha}} \bigg|_{\vec{R}=\vec{R}_{\mathrm{eq}}}$$$$ What does it mean? Has anyone ever seen this expression? And if so, could you suggest me where to search?
It seems like he is expanding every coordinate of the dipole $$\vec{D}$$ around an equilibrium point $$\vec{R}_{\mathrm{eq}}$$ , but then there are the terms $$R_{I,\alpha}^{\mathrm{eq}}$$ which suggests that every nucleus $$I$$ has its own equilibrium position. So the vectors $$\vec{R}$$ and $$\vec{R}_{\mathrm{eq}}$$ are vectors of vectors, with every component corresponding respectively to a position and an equilibrium point of one of the nuclei...right? But does this make sense?
Has anyone ever seen this method to deal with the spectra of polyatomic molecules, and if so, could you suggest me where to search?
Every previous calculation has been made with diatomic molecules, so this jump to the polyatomic case it's very confusing.

Its essentially just a convention for summing over indices of all relevant quantities. The Dipole vector has three dimensions and in a Cartesian coordinate system it is conventional to call these $$x,y,z$$, and the index $$\beta$$ is a place holder for them, $$\beta \in {x,y,z}$$.
Each nucleus also "lives" in a three dimensional Cartesian space, and in the mathematical treatment we use the direct sum of these spaces, i.e. for each nucleus we have a set of $$[x,y,z]$$ coordinates, which we label additionally with an index for the corresponding nucleus, for example $$\vec R_1 = (R_{1x},R_{1y},R_{1z})$$ for the first nucleus. The complete vector that describes the structure of the molecule is then $$\vec R_1\oplus \vec R_2 \oplus \dots \vec R_N :=\left ( \begin{matrix}R_{1x}\\ R_{1y}\\ R_{1z}\\ \vdots \\ R_{Iz} \end{matrix}\right )=\vec R$$ In your example that index is $$I\in {[1\dots N]}$$. So summing over all nuclei coordinates requires summation over a combination of $$[x,y,z]$$ and $$[1\dots N]$$, which is done in your example using $$\alpha$$ and $$I$$.
Its also important to understand that each component of the dipole moment is a function of all these coordinates, i.e. $$\vec D = \left(\begin{matrix}D_x \\ D_y \\ D_z \end{matrix}\right)$$ where $$D_x=D_x(\vec R) =D_x(R_{1x},R_{1y}, R_{1z}, \dots R_{Iz} )$$, likewise $$D_y =D_y(R_{1x},R_{1y}, R_{1z}, \dots R_{Iz} )$$, and $$D_z =D_z(R_{1x},R_{1y}, R_{1z}, \dots R_{Iz} )$$. Each component is function of all nuclei coordinates and each function is expanded around the equilibrium coordinates, $$\vec D^{Taylor} \approx \left(\begin{matrix}D_x(\vec R_{eq}) + \vec \nabla_RD_x|_{R_{eq} }\cdot (R-R_{eq}) \\D_y(\vec R_{eq}) + \vec \nabla_RD_y|_{R_{eq} }\cdot (R-R_{eq}) \\ D_z(\vec R_{eq}) + \vec \nabla_RD_z|_{R_{eq} }\cdot (R-R_{eq}) \end{matrix}\right)\\$$ The last equation of yours is index notation for this expression.