*2.2. Effective Single-Particle Energies*

Once the cross-shell monopole matrix elements are determined the above-described way, one can obtain proton and neutron shell evolutions. The shell evolution is characterized by the effective-single-particle energy (ESPE), which includes the effects of valence nucleons on the single-particle energy. While the ESPEs can be defined for any wave function (see Ref. [1]), they are often estimated by filling configurations, so that one can directly connect monopole matrix elements to shell evolution. To simplify the discussion, a case of mass-independent two-body interactions is considered here. The ESPE of the neutron orbital, *jn*, changes by filling protons in the orbital *jp* as

$$\varepsilon\_{\nu j\_n}(\pi j\_p : \text{filled}) = \varepsilon\_{\nu j\_n}(\pi j\_p : \text{empty}) + (2j\_p + 1) V\_{pn}^{\text{m}}(j\_n, j\_p) \,. \tag{5}$$

When one defines the change of the ESPE of *νjn* with filling *πjp* as

$$
\Delta\_{\pi j\_{\mathcal{P}}} \varepsilon\_{\nu j\_{\mathcal{u}}} \equiv \varepsilon\_{\nu j\_{\mathcal{u}}} (\pi j\_{\mathcal{P}} : \text{filled}) - \varepsilon\_{\nu j\_{\mathcal{u}}} (\pi j\_{\mathcal{P}} : \text{empty}), \tag{6}
$$

the evolution of the *shell gap* between *νjn* and *νj <sup>n</sup>* with filling *πjp* is expressed as

$$
\Delta\_{\pi j\_p} (\varepsilon\_{\nu j\_n} - \varepsilon\_{\nu j\_n'}) = (2j\_p + 1) \{ V\_{pn}^{\mathbf{m}} (j\_{n'} j\_p) - V\_{pn}^{\mathbf{m}} (j\_{n'}' j\_p) \}.\tag{7}
$$

Figure 1 provides a schematic illustration of what is represented in Equation (7). One of the most important properties of Δ*πjp* (*ενjn* − *εν<sup>j</sup> <sup>n</sup>* ) is that this quantity does not depend on the choice of the core to define the ESPE. For example, the evolution of the *N* = 34 shell gap can be probed not only by the systems with the *N* = 34 core but also by those with the *N* = 28 core or the *N* = 20 core. This means that one can investigate a specific shell evolution for very neutron-rich isotones by using that of less neutron-rich ones, which will be utilized in some cases considered in Section 3.

**Figure 1.** Schematic illustration of what is investigated in this paper. The blue and green lines are, respectively, the effective single-particle energies (ESPEs) of the neutron orbital, *jn* and *j <sup>n</sup>*, that change with the proton orbital, *jp*, filled. The evolution of the shell gap, denoted as Δ*πjp* (*ενjn* − *εν<sup>j</sup> <sup>n</sup>* ), is the main focus of this paper.

When one uses a mass-dependent interaction, Equation (7) is not exact but it is still useful for estimating shell evolution from monopole matrix elements.
