In some cases the use of densities may be more convenient, especially in the opposite limit where several singular values are significant, a situation that is discussed in Section 3.4. On the other hand, the NTOs preserve phase information that is lost upon squaring the orbitals, and that information may be useful in some situations, e.g., to distinguish nπ * from ππ * in cases of significant orbital mixing, or to reveal the π → π * in a case with complicated nodal structure, as in the example of Fig. , to which we return below. For a very different example, we turn to x-ray spectroscopy at the titanium K-edge. Calculations on a Ti 16 O 32 H 2 cluster 11 were performed at the PBE0/def2-ma-SVP level where the basis set is an augmented one, 147 intended to describe any nascent band structure. The K-edge consists of transitions from Ti(1s) to valence virtual orbitals at almost 5,000 eV, and to access core-level excitations these calculations invoke the core/valence separation approximation. Here, one omits amplitudes x ia unless ψ i corresponds to a core orbital of interest, meaning Ti(1s) in the present example, while retaining the full virtual space. The principle particle NTO in this example exhibits just one nonzero singular value (λ 2 1 = 1.00) and is depicted in Fig. , where its Ti(3d) character is evident along with some admixture of O(2p). The hole NTO is not shown but corresponds to the Ti(1s) on a nearby atom, meaning that the asymmetry of the cluster has localized this Ti(1s) → Ti(3d) pre-edge feature to one end of the Ti 16 O 32 H 2 cluster.