3 Δ E = E$_{f}$$_{ν}$ − E$_{g}$ = E$_{f}$$_{ν}$ E$_{f}$$_{0}$ + E$_{f}$$_{0}$ − E$_{g}$, (5) III. RESULTS AND DISCUSSION A. Ferromagnetic Transition Metals C. Overlap integrals We assume that the transition matrix elements be- tween the core state and the photo-excited states are constant. The remaining matrix elements connecting the ground and final states are expressed by Figure 1: Sketch of a super-cell containing core-holes in fcc structure. Core-hole sites indicated by solid circles are as- sumed forming a 3 × 3 × 3 fcc lattice. The super-cell for the bcc structure is shown in fig. 1 in Ref. 19. with 〈 f$_{ν}$ | s$_{σ}$ | g 〉 = ∣ ∣ S$_{1}$$_{,}$$_{1}$ S$_{2}$$_{,}$$_{1}$ ..... S$_{N}$$_{e}$$_{,}$$_{1}$ S$_{1}$$_{,}$$_{2}$ S$_{2}$$_{,}$$_{2}$ ..... S$_{N}$$_{e}$$_{,}$$_{2}$ .... .... .... .... S$_{1}$$_{,N}$$_{e}$ S$_{2}$$_{,N}$$_{e}$ ...... S$_{N}$$_{e}$$_{,N}$$_{e}$ ∣ ∣ ∣$^{,}$ (3) S$_{i,i}$ ′ = ∫ φ ∗ $_{i}$( r ) φ (0) i ′ ( r )d $^{3}$r, (4) In this subsections, we refer to majority(minority) spin as up(down)-spin. The Ni metal takes an fcc structure. For simplicity, the Co metal is assumed to take an fcc structure, although it actually takes an hcp structure. where the integral is carried out within a unit cell. Sub- scripts i = ( σ, n, k ) and i ′ = ( σ $^{′}$, n$^{′}$, k $^{′}$) are running over Before closing this section, we briefly mention the XPS intensity at the energy of threshold. The final state | f$_{0}$ 〉 with the lowest energy (no e-h pair) has a finite overlap with the ground state | g 〉 , giving rise to intensities at the threshold. In principle, such overlap converges at zero with N$_{e}$ → ∞ , according to the Anderson orthogonal- ity theorem. 1 In such infinite systems, energy levels be- come continuous near the Fermi level and thereby infinite numbers of e-h pairs could be created with infinitesimal excitation energies, leading to the so called Fermi edge singularity in the XPS spectra. The finite contribution obtained above arises from the discreteness of energy lev- els and could be interpreted as the integrated intensity of singular spectra near the threshold, in consistent with the model calculations for other systems. 6,24 In the actual calculation, instead of N$_{c}$ k -points, we pick up only the Γ point as the sample states for cal- culating XPS spectra. For Ni, we pick up the X point (and the equivalent Y and Z points) in addition to the Γ point, since the 3 d band states at the Γ point are fully occupied by both up-spin and down-spin electrons even though the 3 × 3 × 3 fcc super-cell is used. where ϵ$_{i}$ ’s are the Kohn-Sham eigenvalues, and ϵ$_{j}$ − ϵ$_{i}$ stands for the energy of e-h pair of an electron at level j and a hole at level i . Although the Kohn-Sham eigenval- ues may not be proper quasi-particle energies, they prac- tically give a good approximation to quasi-particle ener- gies, except for the fundamental energy gap. 22,23 Substi- tuting Eqs. (3) and (5) into Eq. (1), we obtain the XPS spectra. where E$_{f}$$_{0}$ − E$_{g}$ includes the energy of core hole, and is treated as an adjustable parameter in the present study such that the threshold of XPS spectra coincides with the experimental value. The excitation energy E$_{f}$$_{ν}$ − E$_{f}$$_{0}$ with ν e-h pairs is given by occupied conduction states in the presence of core hole and in the ground state, respectively. We eliminate the overlaps between the wave functions for the core levels. The corresponding energies difference is given by per super-cell. The wave function and energy eigenvalue are denoted as ψ (0) σn $_{k}$( r ) and ϵ (0) $_{σn}$( k ) , respectively. All the lowest N$_{e}$ = N$_{c}$ × n$_{e}$ levels are occupied in the ground state. The final states | f 〉 ’s are constructed by using the one- electron states calculated in the presence of the core-hole in accordance with the following procedure. Defining n (0) σ ( k ) by the number of levels with spin σ and wave vector k below the Fermi level in the ground state, we distribute n (0) σ ( k ) electrons with spin σ and wave vec- tor k in the states given in the presence of core hole. The final state | f$_{0}$ 〉 containing no e-h pair is constructed by distributing electrons from the lowest energy level up to the n (0) σ ( k ) ’th level with spin σ for each wave vec- tor k . The final states | f$_{ν}$ 〉 ’s containing ν e-h pairs are constructed by annihilating ν electrons in the occupied conduction states and creating ν electrons in the unoc- cupied conduction states from | f$_{0}$ 〉 . E$_{f}$$_{ν}$ − E$_{f}$$_{0}$ = ∑ ( i,j ) ( ϵ$_{j}$ − ϵ$_{i}$ ) , (6) Core Hole Site