9 For Cu, | A$_{↑}$ | 2 and | A$_{↓}$ | 2 are close to unity, although the one-electron wave functions are strongly modified. Since the 3 d bands are fully occupied in the ground state, one-electron wave functions constituting | f$_{0}$ 〉 are nearly expressed by a unitary transform of those con- stituting | g 〉 , except for the shrink of the radial part of the atomic wavefunctions at the core-hole site. There- fore, the squares of the overlap determinant | A$_{↑}$ | 2 and | A$_{↓}$ | 2 are nearly unity. Final states | f$_{1}$ 〉 (one e-h pair), f$_{2}$ 〉 (two e-h pairs), and so on, could have merely very small overlaps with s$_{↓}$ | g 〉 , since the one-electron state on which the excited electron sits is nearly orthogonal to the one-electron states constituting the ground state. Thus we have a simple single peak structure coming from | f$_{0}$ 〉 without any noticeable intensities on the higher binding energy side, as shown in Fig. 10. This result is consis- tent with the experimental observation. 7 Note that the change of the wave functions is not directly related to the spectral shape. The effect due to forming strong bound states is not seen. Table IV: Absolute squares | A$_{↑}$ | 2 and | A$_{↓}$ | $^{2}$, where A$_{↑}$ and A$_{↓}$ represent the up- and down-spin parts of the overlap integral between | f 0 〉 and s$_{↓}$ | g 〉 , that is, 〈 f 0 | s$_{↓}$ | g 〉 = A$_{↑}$A$_{↓}$ . dn-spin core-hole | A$_{↑}$ | 2 | A$_{↓}$ | 2 Cu 0 . 910 0 . 935 V 0 . 571 0 . 938 Ru 0 . 681 0 . 949 -15 -10 -5 0 5 10 15 Binding Energy (eV) 0.0 0.1 0.2 0.3 Intensity (arb.units) total |f$_{0}$> Cu 3s XPS Figure 10: 3 s XPS spectra in nonmagnetic copper as a func- tion of binding energy in the down-spin channel. For V, | A$_{↓}$ | 2 is close to unity. This suggests that one- electron wave functions for down-spin electrons are little modified from those in the ground state. On the other hand, | A$_{↑}$ | 2 is rather smaller than unity. This suggests that up-spin one-electron wave functions constituting | f$_{0}$ 〉 include the amplitudes of the unoccupied one-electron states in the ground state. In such a situation, the final states | f$_{1}$ 〉 ’s containing one e-h pair with up-spin could have finite overlaps with the ground state. Figure 11 We calculate the up- and down-spin parts of overlap integrals between the lowest-energy final state | f$_{0}$ 〉 and the ground state | g 〉 , which values are listed in Table IV. Table III: Screening electron number with respect to the d symmetry within the muffin-tin sphere at the 3 s ( 4 s ) down- spin core-hole site. The radii of the muffin-tin spheres are 2 . 0 , 2 . 1 , and 2 . 3 Bohr for Cu, V, and Ru, respectively. 3s hole spin Δ n $^{d}$↑ Δ n $^{d}$↓ Δ n $^{d}$↑ + Δ n $^{d}$↓ Cu dn 0 . 31 0 . 24 0 . 55 V dn 1 . 44 − 0 . 44 1 . 00 Ru dn 0 . 77 0 . 24 1 . 01 Figure 9: (a) DOS calculated in a system of super-cell with no core-hole in nonmagnetic ruthenium. The solid line represents the DOS projected onto the d -symmetry within the muffin-tin sphere. (b) DOS at the site of 4 s down-spin core-hole. -8 -6 -4 -2 0 2 4 Energy (eV) 2 0 2 DOS (states/eV) c site 4d up spin dn spin (a) fcc Ru -8 -6 -4 -2 0 2 4 Energy (eV) 2 0 2 DOS (states/eV) c site 4d up spin dn spin (b) at the core-hole site.