As a starting point we may consider that each time a new position is suggested in the Metropolis algorithm, a row of the current Slater matrix experiences some kind of perturbation. Hence, the Slater matrix with its orbitals evaluated at the new position equals the old Slater matrix plus a perturbation matrix,

$$ \begin{equation*} \tag{12} d_{jk}(\mathbf{x^{new}}) = d_{jk}(\mathbf{x^{old}}) + \Delta_{jk}, \end{equation*} $$

where

$$ \begin{equation*} \tag{13} \Delta_{jk} = \delta_{ik}[\phi_j(\mathbf{x_{i}^{new}}) - \phi_j(\mathbf{x_{i}^{old}})] = \delta_{ik}(\Delta\phi)_j . \end{equation*} $$