where L ω is a lower triangular square matrix with all positive real values along its diagonal. Note that we are not using the Cholesky decomposition to lower the rank of a tensor, as it is usually employed in electronic structure methods, but simply to "evenly distribute" the response matrix to both perturbations. By splitting the response matrix, we can apply L ω to both the electric dipole and magnetic dipole to obtain "hemi-perturbed" densities and contract these to form