Internal coordinates transformations

Since the Hessian and gradients are computed in CCs, whereas the FF is usually expressed through RICs, some coordinate transformation is required. For infinitesimal displacements with respect to a given geometrical conformation, the RICs are related to the nuclear CCs $x$ through a non invertible transformation

$$\delta q=B\ \delta x$$ (25)

where $\delta q$ and $\delta x$ are colum vectors. The Wilson rectangular $B$ matrix

$$B_{\mu i}= \left(\frac{\partial q_\mu}{\partial x_i}\right)$$ (26)

is related to the geometry the displacements are referred to, and can be accurately computed both in analytical [36] and numerical ways.

The normal coordinates are computed from the Hessian matrix in CCs

$$H_{ij}= \left(\frac{\partial^2E}{\partial x_i\partial x_j}\right) = E_{ij}^{\prime\prime}$$ (27)

obtained by a QM calculation at a given geometry. $H$ is transformed to the mass weighted CCs form and diagonalized by a unitary matrix $C$

$$M^{-1/2}\ H\ M^{-1/2}C = C\Lambda$$ (28)

The matrix $M$ is diagonal and for each CC contains the mass $m$ of the related atom. The columns of the $C$ matrix are the linear combinations of the mass weighted CCs that correspond to the NCs displacements

$$\delta Q_K = \sum_{i=1}^{3N} \sqrt{m_i}C_{iK} \delta x_{i}$$ (29)

or in matrix form

$$\delta Q = \widetilde{C\ }M^{1/2} \delta x$$ (30)

where $\delta Q$ and $\delta x$ are column vectors. In the case the geometry corresponds to an absolute or local energy minimum, $3N-6$ eigenvalues $\Lambda_{K}$ are positive and refer to vibrations, whereas the 3 translational and 3 rotational modes are identified by zero eigenvalues. In other cases negative eigenvalues can occur and these do not correspond to vibrational modes. If all the NCs are retained, the transformation of equation (29) is fully invertible

$$\delta x=M^{-1/2}C\delta Q$$ (31)

The relation between the RICs and the NCs can be easily obtained exploiting the completeness of the CCs basis set. Using equations (25) and (31)

$$\delta q = B M^{-1/2}\ C \delta Q = T \delta Q$$ (32)

where the $T$ matrix is defined as

$$T_{\mu K}= \left( \frac{\partial q_{\mu}}{\partial Q_{K}}\right)$$ (33)

Thus the RICs may be expressed in terms of the NCs and the inclusion or not of the rotational and translational modes is uninfluential since they leave the RICs unchanged.