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We present calculations of shell-model effective interactions for bothdegenerate and non-degenerate model spaces using the Krenciglowa-Kuo (KK) andthe extended Krenciglowa-Kuo iteration method recently developed by Okamoto,Suzuki {\it et al.} (EKKO). The starting point is the low-momentumnucleon-nucleon interaction $V_{low-k}$ obtained from the N$^3$LO chiraltwo-nucleon interaction. The model spaces spanned by the $sd$ and $sdpf$ shellsare both considered. With a solvable model, we show that both the KK and EKKOmethods are convenient for deriving the effective interactions fornon-degenerate model spaces. The EKKO method is especially desirable in thissituation since the vertex function $\hat Z$-box employed therein is wellbehaved while the corresponding vertex function $\hat Q$-box employed in theLee-Suzuki (LS) and KK methods may have singularities. The convergedshell-model effective interactions given by the EKKO and KK methods areequivalent, although the former method is considerably more efficient. Thedegenerate $sd$-shell effective interactions given by the LS method arepractically identical to those from the EKKO and KK methods. Results of the$sd$ one-shell and $sdpf$ two-shell calculations for $^{18}$O, $^{18}$F,$^{19}$O and $^{19}$F using the EKKO effective interactions are compared, andthe importance of the shell-model three-nucleon forces is discussed.

Non-degenerate shell-model effective interactions from the Okamoto-Suzuki and Krenciglowa-Kuo iteration methods