5 Table II: Absolute squares | A$_{↑}$ | 2 and | A$_{↓}$ | $^{2}$, where A$_{↑}$ and A$_{↓}$ represent the up- and down-spin parts of the overlap integral, respectively, between | f 0 〉 and s σ | g 〉 in the σ -spin channel, that is, 〈 f 0 | s σ | g 〉 = A$_{↑}$A$_{↓}$ . up-spin core-hole dn-spin core-hole | A$_{↑}$ | 2 | A$_{↓}$ | 2 | A$_{↑}$ | 2 | A$_{↓}$ | 2 Ni 0 . 940 0 . 234 0 . 770 0 . 572 Co 0 . 983 0 . 049 0 . 980 0 . 601 Fe 0 . 967 0 . 127 0 . 969 0 . 909 the 3 d band. On the other hand, the d -DOS’s of the up- spin states are slightly pulled down for Ni, and pushed upward to the higher energy region for Co and Fe. The d -DOS’s at the site without a core-hole are essentially the same as those in the ground state with no core-hole Since up-spin electrons are prevented from coming close to the core-hole site due to the exchange interac- tion with the up-spin core hole, the screening is almost done by down-spin electrons. This tendency is clear in Ni; Δ n$_{d}$$_{↓}$ = 0 . 87 , while Δ n$_{d}$$_{↑}$ = 0 . 07 , as shown in Table I. For Co and Fe, the core-hole potential is overscreened by down-spin electrons; Δ n$_{d}$$_{↓}$ = 1 . 55 (Co), Δ n$_{d}$$_{↓}$ = 2 . 38 (Fe). This overscreening is compensated by up-spin electrons; Δ n$_{d}$$_{↑}$ = − 0 . 52 (Co), Δ n$_{d}$$_{↑}$ = − 1 . 44 (Fe). As a result, the screening electron numbers become almost unity: 0 . 94 , 1 . 03 , and 0 . 94 for Ni, Co, and Fe, respectively, indi- cating that the screening is nearly completed by the d - symmetric states. The magnetic moments at the core- hole site are − 0 . 2 µ$_{B}$ , − 0 . 5 µ$_{B}$ , and − 1 . 8 µ$_{B}$ for Ni, Co, and Fe, respectively, which are opposite to those without the core-hole, 0 . 6 µ$_{B}$ , 1 . 6 µ$_{B}$ , 2 . 1 µ$_{B}$ , for Ni, Co, and Fe, respectively. We note that the magnitude of the overscreening by the down-spin electrons becomes weaker Fe, Co, and Ni in that order. In Ni, the number of the down-spin electrons screening the core-hole is just about unity. Consequently, the up-spin 3 d states become to be less pushed upward to the high energy region in that order. In Ni, the effects of the core-hole potential and the Coulomb repulsion from the screening down electrons subtly cancel out each other and the d -DOS of up-spin states is hardly modified. It is inferred from these changes in the d -DOS’s that one-electron wave functions are largely modified by the core-hole potential particularly for down-spin states. It is necessary to use both the occupied and unoccupied states of the ground state in order to expand those modified one-electron wave functions for the down-spin electrons, since the down-spin 3 d bands are partially occupied in the ground state. For this reason, the absolute square | A$_{↓}$ | 2 becomes rather small, as shown in table II. Here A$_{↑}$ ( A$_{↓}$ ) represents the the up(down)-spin part of the overlap integral between the lowest-energy final state | f$_{0}$ 〉 (no e-h pair) and s$_{↑}$ | g 〉 , and thereby 〈 f$_{0}$ | s$_{↑}$ | g 〉 = A$_{↑}$A$_{↓}$ . The one-electron wave functions for up-spin electrons are also modified from those in the ground state. In spite of such modification, | A$_{↑}$ | $^{2}$’s are nearly unity as shown in First we consider the situation that a up-spin 3 s elec- tron is removed in a unit cell. As shown in Figs. 2 (b) and 3(b), the d -DOS at the core-hole site are strongly modified by the core-hole potential for Ni and Co. The situation is similar to Fe, as shown in Fig. 3 in Ref. 19. The d -DOS’s of the down-spin states are strongly pulled down, forming quasi-bound states around the bottom of 1. Up-spin channel Figure 3: Calculated d -DOS at the core-hole site in the super- cell system in ferromagnetic cobalt; (a) d -DOS with no core- hole, (b) d -DOS when the 3 s up-spin electron is removed, (c) d -DOS when the 3 s down-spin electron is removed. -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 Co -8 -6 -4 -2 0 2 4 Energy (eV) 4 2 0 2 4 DOS (states/eV) c site 3d up spin dn spin (b) -8 -6 -4 -2 0 2 4 Energy (eV) 4 2 0 2 4 DOS (states/eV) c site 3d up spin dn spin (c)