Analytical Expression for the Gradient

The derivative of $I_{ij}^{c}$ with respect to a coordinate $\mu$ (Eq. 12 of Yip and Case 1989) can be written as

$${\bf\nabla}_{\mu} I_{ij}^{c} = {\bf\nabla}_{\mu}[\exp(- {\bf... ...({\bf\nabla}_{\mu}{\bf R}) {\bf L} {\bf F}^{(ij)} {\bf L}^{T}]$$ (39.7)

where ${\bf F}^{(ij)}$ is defined as

$${\bf F}_{ru}^{(ij)} \equiv \left\{ \begin{array}{ll} - {\... ...}^{T} \exp(-\lambda_{r} \tau) &\mbox{else} \end{array}\right.$$ (39.8)

L and ${\bf\Lambda}$ are the matrix of eigenvectors and eigenvalues of R, respectively. $\lambda_{r}$ is the rth eigenvalue of the relaxation matrix.