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at the core-hole site. 

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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. 

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. 

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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 

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