PE expression for vibrational spectra of molecules using Newton's second law
It's probably simplest to start with the familiar example of the 1D harmonic oscillator. Here
$$F=-\frac{dV}{dx}=-\frac{d\left(\frac{1}{2}kx^2\right)}{dx}=-kx\tag{1}$$
which is often called Hooke's law, $F=-kx$, where x is the displacement from the equilibrium position, and the potential is assumed quadratic (parabolic) in the position coordinate. The derivation above requires the substitution $V(x) = \frac12 kx^2$, which can be obtained by trimming a Taylor expansion of the potential about the energy minimum (assumed = 0) at the quadratic term, and the force constant is defined as
$$k=\left(\frac{d^2V}{dx^2} \right)_e\tag{2}$$
Also,
$$F=ma=m\frac{d^2x}{dt^2}\tag{3}$$
If we transform to mass-weighted coordinates (or equivalently set m=1), we therefore can write that
$$\frac{d^2x}{dt^2} = -\frac{dV}{dx}=-\left(\frac{d^2V}{dx^2} \right)_e x\tag{4}$$
The derivation in your book does essentially the same thing. Note that since the potential is assumed quadratic and the other listed conditions are imposed, the potential can be written as
$$V = \frac12 \sum_{i = 1}^{3N}\sum_{j = 1}^{3N} \left( \dfrac{\partial^2{V}}{\partial{q_i}\partial{q_j}} \right)_e q_i q_j \tag{5}$$
The double summation can be collapsed into a single one when you take the derivative with respect to a spatial coordinate, say $q_k$:
$$\begin{align} \dfrac{d^2 q_k}{d t^2}  &amp;=-\left( \dfrac{\partial{V}}{\partial{q_k}} \right)  \\&amp;=-\left(\dfrac{\partial{}}{\partial{q_k}} \left[  \frac12 q_k^2  \left( \dfrac{\partial^2{V}}{\partial{q_k^2}} \right)_e   + \frac12 q_k\sum_{j \neq k }^{3N} \left( \dfrac{\partial^2{V}}{\partial{q_k}\partial{q_j}} \right)_e  q_j  + \frac12 q_k \sum_{i \neq k }^{3N} \left( \dfrac{\partial^2{V}}{\partial{q_i}\partial{q_k}} \right)_e q_i  + \frac12 \sum_{i \neq k }^{3N}\sum_{j \neq k}^{3N} \left( \dfrac{\partial^2{V}}{\partial{q_i}\partial{q_j}} \right)_e q_i q_j  \right] \right)  \\ &amp;= -\left( \left[ \left( \dfrac{\partial^2{V}}{\partial{q_k^2}} \right)_e  q_k  + \frac12 \sum_{j \neq k}^{3N} \left( \dfrac{\partial^2{V}}{\partial{q_k}\partial{q_j}} \right)_e  q_j  + \frac12 \sum_{i \neq k }^{3N} \left( \dfrac{\partial^2{V}}{\partial{q_i}\partial{q_k}} \right)_e q_i    \right] \right)    \\ &amp;= - \sum_{j = 1}^{3N} \left( \dfrac{\partial^2{V}}{\partial{q_k} \partial{q_j}} \right)_e q_j \end{align}$$