Thus, to calculate all the derivatives of the Slater determinant, we only need the derivatives of the single particle wave functions (\( \vec\nabla_i \phi_j(\mathbf{r}_i) \) and \( \nabla^2_i \phi_j(\mathbf{r}_i) \)) and the elements of the corresponding inverse Slater matrix (\( \hat{D}^{-1}(\mathbf{r}_i) \)). A calculation of a single derivative is by the above result an \( O(N) \) operation. Since there are \( d\cdot N \) derivatives, the time scaling of the total evaluation becomes \( O(d\cdot N^2) \). With an \( O(N^2) \) updating algorithm for the inverse matrix, the total scaling is no worse, which is far better than the brute force approach yielding \( O(d\cdot N^4) \).

Important note: In most cases you end with closed form expressions for the single-particle wave functions. It is then useful to calculate the various derivatives and make separate functions for them.