8 Figure 8 shows the d -DOS for V. The d -DOS at the core-hole site is shifted downward to the deeper energy region for up-spin states, while the d -DOS is shifted up- ward for down-spin states. No bound state is formed. As listed in Table III, the screening electron numbers are given by Δ n$_{d}$$_{↑}$ = 1 . 21 , Δ n$_{d}$$_{↓}$ = − 0 . 21 , and the total num- ber by Δ n$_{d}$$_{↑}$ + Δ n$_{d}$$_{↓}$ = 1 . 00 . The core-hole potential is overscreened by up-spin electrons, and the overscreening is compensated by down-spin electrons. Note that the screening is complete within the 3 d electrons at the core- hole site. The screening is effective because the 3 d bands are partially occupied. below the 3 d bands in both up- and down-spin states. As listed in Table III, the screening electron numbers are given by Δ n$_{d}$$_{↑}$ = 0 . 31 , Δ n$_{d}$$_{↓}$ = 0 . 24 , and the total screen- ing electron number is given by Δ n$_{d}$$_{↑}$ +Δ n$_{d}$$_{↓}$ = 0 . 55 . This indicates that the core-hole potential is not sufficiently screened by the 3 d electrons at the core-hole site. Here we note that the screening mechanism seems to be somewhat different from the other metals which have partially oc- cupied d -bands and do not show the bound states. Since the 3 d band states are almost fully occupied in the ground state, the change of the charge density is hardly achieved by a unitary transform of the 3 d band states. Since the bound states are split off from the band bottom edge, the radial part of the local atomic wavefunctions constituting the bound states at the core-hole site is slightly shrunk compared to that in the ground state with no core-hole, leading to the small change of the charge density. This shrink of radial part of the local atomic wavefunctions cannot be described by a unitary transform of the 3 d band states in the ground state. The states with much higher energy in the ground state necessarily constitute the bound states to some extent. -4 -2 0 2 4 6 8 Energy (eV) 4 2 0 2 4 DOS (states/eV) c site 3d up spin dn spin (a) bcc V -4 -2 0 2 4 6 8 Energy (eV) 4 2 0 2 4 DOS (states/eV) c site 3d up spin dn spin (b) -10 -8 -6 -4 -2 0 2 4 Energy (eV) 4 2 0 2 4 DOS (states/eV) c site 3d up spin dn spin (a) fcc Cu -10 -8 -6 -4 -2 0 2 4 Energy (eV) 4 2 0 2 4 DOS (states/eV) total/27 c site 3d up spin dn spin (b) Figure 8: (a) DOS calculated in a system of super-cell with no core-hole in nonmagnetic vanadium. The solid line represents the DOS projected onto the d -symmetry within the muffin-tin sphere. (b) DOS at the site of 3 s down-spin core-hole. The solid line represents the DOS projected onto the d - symmetry within the muffin-tin sphere. Figure 7: (a) DOS calculated in a system of super-cell with no core-hole in nonmagnetic copper. The solid line represents the DOS projected onto the d -symmetry within the muffin-tin sphere. (b) DOS at the site of 3 s down-spin core-hole. The thin line represents the total DOS divided by the number of atoms in a unit cell. Figure 9 shows the d -DOS for Ru. The d -DOS’s at the core-hole site are shifted downward to the deeper energy region with both spin states, but the change is the smallest in the three cases without any bound states. This is probably related to the fact that the 4 d electrons are more itinerant than the 3 d electrons. As listed in Table III, the screening electron numbers are Δ n$_{d}$$_{↑}$ = 0 . 77 , Δ n$_{d}$$_{↓}$ = 0 . 24 , and Δ n$_{d}$$_{↑}$ + Δ n$_{d}$$_{↓}$ = 1 . 01 , indicating that the screening is nearly completed by the 4 d electrons