This equation scales as \( O(N^2) \). The evaluation of the determinant of an \( N \times N \) matrix by standard Gaussian elimination requires \( \mathbf{O}(N^3) \) calculations. As there are \( Nd \) independent coordinates we need to evaluate \( Nd \) Slater determinants for the gradient (quantum force) and \( Nd \) for the Laplacian (kinetic energy). With the updating algorithm we need only to invert the Slater determinant matrix once. This can be done by standard LU decomposition methods.