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relation effect, especially in the case with quarter-filling as shown in Fig.3 (b). On the other hand, when g ̸ = 0, the degeneracy is resolved, and then, the intra-orbital spin-singlet correlation 〈 ˆ n$_{filσ}$ ˆ n$_{fil}$$_{¯}$ σ 〉 is enhanced, while the inter-orbital spin-triplet correlation 〈 ˆ n$_{filσ}$ ˆ n$_{fi}$ ¯ lσ 〉 and the inter-orbital spin-singlet correlation ˆ n$_{filσ}$ ˆ n$_{fi}$ ¯ l σ are suppressed due to the effect of g . As the results, 〈 τ 2 z 〉 given in eq.(5) is largely enhanced for half-filling with large values of the double-occupancy probabilities, while it is slightly enhanced for quarter-filling with small values of those as shown in Fig.2. 

## 3.4 Local charge fluctuation 

The local charge fluctuation of f -electrons, i. e., the valence fluctuation is written by 

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and is plotted as a function of g for several values of U at half- and quarter-filling in Fig.4. When g = 0, the valence fluctuation decreases with increasing U to- gether with decreasing double-occupancy probabilities for both half- and quarter-filling due to the electron cor- relation effect. When g ̸ = 0, the valence fluctuation is suppressed or almost constant due to the effect of g in contrast to the case with the single-orbital periodic An- derson model where the valence fluctuation is coupled with local phonons and is enhanced due to the effect of g . 30–32) We note that the sign of the final term (the inter- orbital correlation) in eq.(6) is opposite to that in eq.(5), and then, the changes in the intra- and inter- orbital cor- relations cancel with each other (see Fig.3) resulting in the decrease in the valence fluctuation with g in contrast to the increase in the local orbital fluctuation. 

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## 3.5 Local moment 

The local moment, 〈 S $^{2}$〉 = 〈 ˆ S 2 i 〉 = 〈 ˆ S 2 ix + ˆ S 2 iy + ˆ S 2 iz 〉 , written by 

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is plotted as a function of g for several values of U at half- and quarter-filling in Fig.5. When g = 0, 〈 S $^{2}$〉 is enhanced due to the effect of U . When g increases, 〈 S decreases with decreasing (increasing) spin-triplet (singlet) corre- lation as shown in Fig.3. The decrease in 〈 S $^{2}$〉 with g for half-filling is larger than that for quarter-filling, as the double-occupancy probabilities for half-filling are larger than those for quarter-filling (see Fig.3). 

