2 with energy ϵ and spin σ could be proportional to tra in a series of ferromagnetic metals Ni, Co, and Fe. The usefulness of our ab initio method is demonstrated. We have already reported the spectra in ferromagnetic Fe in Ref. 19. In these metals, spectral shapes have char- acteristic dependence on elements and spin channels; the spectra have satellite or shoulder in the majority spin channel, while the spectra show single peak structures in the minority spin channel. Here we define the majority (minority) spin channel by the process that the 3 s -core electron is photo-excited to the vacuum state with the same spin as the majority (minority) spin in the con- duction band states. As far as we know, such spectra have been analyzed only by using a single band Hub- bard model, and has been related to the 3 s - 3 d exchange interaction. 21 However, the model is, we think, too sim- ple to compare the calculated results quantitatively with the experimental data and to draw definite conclusion. Applying the ab initio method, we calculate the spec- tra in good agreement with the experimental observa- tions. The screening effects are quite different between the spin channels due to the exchange interaction be- tween the 3 d electrons and the core hole. The 3 d states are modified by the core-hole potential at the core-hole site, and sometimes quasi-bound states are created near the bottom of the 3 d band. The e-h pair excitations from such quasi-bound states to the empty states cor- respond to the satellite or shoulder intensities. We find that the presence of the quasi-bound states is not suffi- cient and the 3 d bands have to be partially occupied in the ground state, in order that the satellite or shoulder structure appears. These considerations well explain the characteristics of the XPS spectra. Furthermore, to clearly show that the presence of satel- lite or shoulder has no direct relation to magnetic states, we calculate the 3 s XPS spectra in nonmagnetic metals V and Cu, and the 4 s XPS spectra in nonmagnetic metal Ru. We obtain shoulders in the high binding-energy re- gion in V and Ru. On the other hand, we have no such structure in Cu, although the localized bound states are clearly created below the bottom of the 3 d band. We could explain these behaviors in the same way as in the ferromagnetic metals. The present paper is organized as follows. In Sec. II, we formulate the XPS spectra with the ab initio method. In Sec. III, we present the calculated XPS spectra and discuss the behavior. The last section is devoted to the concluding remarks. In order to simulate the photo-excited states, we con- sider a periodic array of super-cells with one core-hole per cell, and calculate the one-electron states by means of the band structure calculation based on the full po- tential linear augmented plane wave (FLAPW) method. We use the 3 × 3 × 3 bcc super-cell for Fe and V as shown in Fig. 1 in Ref. 19, and the 3 × 3 × 3 fcc super-cell for Co, Ni, Cu and Ru as shown in Fig. 1, where the core-hole sites form a bcc lattice and an fcc lattice, re- spectively. The larger the unit cell size is, the better results are expected to come out. The 3 s - or 4 s -core states in transition metals are treated as localized states within a muffin-tin sphere, so that we could specify the core-hole site. To ensure the charge neutrality, we as- sume n$_{e}$ + 1 band electrons per unit cell instead of n$_{e}$ band electrons, where n$_{e}$ is the number of band electrons per cell in the ground state. One additional electron per unit cell would not cause large errors in evaluating one- electron states in the limit of large unit-cell size. The self-consistent potential is obtained as the potential for the fully relaxed (screened) state. We write the resulting one-electron state with energy eigenvalue ϵ$_{σn}$ ( k ) as ψ$_{σn}$$_{k}$ ( r ) = 1 √$_{N}$$_{c}$ ∑ j φ$_{σn}$$_{k}$ ( r − R $_{j}$) exp( i k · R $_{j}$) , (2) B. Construction of final and initial states where w represents the transition matrix element from the core state localized at a particular site to the state of photo-electron, and is assumed to be independent of energy ϵ and spin σ . The s$_{σ}$ is the annihilation opera- tor of a relevant core electron, which is assumed to have only spin σ as the internal degrees of freedom. The kets | g 〉 and | f 〉 represent the ground state with energy E$_{g}$ and the final state with energy E$_{f}$ , respectively. We de- fine | f 〉 by excluding the photo-electron. In the following calculation, we replace the δ -function by the Lorentzian function with the full width of half maximum (FWHM) 2Γ$_{s}$ with Γ$_{s}$ = 1 . 0 eV in order to take account of the life-time broadening of the core level. I XPS σ ( ω$_{q}$ − ϵ ) = 2 π | w | 2 ∑ f |〈 f | s$_{σ}$ | g 〉| $^{2}$δ ( ω$_{q}$ + E$_{g}$ − ϵ − E$_{f}$ ) , (1) II. PROCEDURE OF CALCULATION A. Formula for XPS spectra with φ$_{σn}$$_{k}$ ( r ) = u$_{σn}$$_{k}$ ( r ) e i k · $^{r}$, where u$_{σn}$$_{k}$ ( r ) has the pe- riod of the super-cell, and j runs over N$_{c}$ super-cells. Needless to say, wave vector k ’s have N$_{c}$ discrete val- ues in the irreducible Brillouin zone. We use these one- electron states as substitutes of the states under a single core-hole. We distribute n$_{e}$ band electrons per super- cell on these states to construct the excited states. In addition, we carry out the band calculation in the ab- sence of the core-hole with assuming n$_{e}$ band electrons We consider the situation that a core electron is excited to a high energy state with energy ϵ by absorbing an x-ray photon with energy ω$_{q}$ and that the interaction between the escaping photo-electron and the other electrons could be neglected. The probability of finding a photo-electron