*3.1. IMME Coefficients for Masses and Excitation Spectra of Proton-Rich Nuclei*

It was recognized long ago that the quadratic IMME, Equation (8), been successful throughout the nuclear chart, can provide a powerful method to determine masses, called the method of Coulomb displacement energies [28,79,89–92]. Namely, the mass excess of a proton-rich nucleus (with *MT* = −*T*) on the basis of an experimental mass excess of its neutron-rich mirror (with *MT* = *T*) and the theoretical *b* coefficient as

$$\mathcal{M}(\eta, T, M\_T = -T) = \mathcal{M}(\eta, T, M\_T = T) - 2\,\text{b}(\eta, T)\,\text{T} \,. \tag{17}$$

If theoretically Coulomb displacement energies are calculated, then they can be used straight instead of 2*bT* in Equation (18), as is done in Ref. [28,92]. Since the IMME is also applicable to describe excited multiplets, the method can be used to predict the positions of excited states in proton-rich nuclei.

Even more precise determination of the energy-level position is possible in triplets if two of three members of an isobaric multiplet are known experimentally:

$$\mathcal{M}(\eta, T, M\_T = -1) = 2\mathcal{M}(\eta, T, M\_T = 0) - \mathcal{M}(\eta, T, M\_T = 1) + 2c(\eta, T) \,. \tag{18}$$

Since the rms (root-mean-square) deviation for *c* coefficients is typically smaller than that for *b* coefficients, one would expect to have a smaller theoretical uncertainty value. These methods can be advantageous for determination of the level in proton-rich nuclei of astrophysical interest (e.g., [93]), as pointed out in Section 5.

The methods described above rely on the quadratic IMME given by Equation (8). Indeed, for isobaric mutliplets with *T* > 1, which involve more than three members, deviations from the quadratic law can be expected. An extended IMME equation would include terms proportional to *M*<sup>3</sup> *<sup>T</sup>* and *<sup>M</sup>*<sup>4</sup> *<sup>T</sup>*, i.e.,

$$\mathcal{M}(\eta, T, M\_T) = a(\eta, T) + b(\eta, T)M\_T + c(\eta, T)M\_T^2 + d(\eta, T)M\_T^3 + e(\eta, T)M\_T^4,\tag{19}$$

which can be tested on quartets and quintets. Up till now, very few cases of non-zero *d* or *e* coefficients have been reported [5,6,94]; see also Refs. [95,96] and references therein. Typical values reach tens of keV.

Theoretically [94,97], deviations from a quadratic IMME are possible due to the presence of charge-dependent three-nucleon forces and/or due to isospin-mixing with nearby states. It is worth noting that the diagonalization of an INC shell-model Hamiltonian can generate an extended IMME, and several calculations have been reported [96,98]. To understand the challenge of getting reliable estimations of cubic and quartic terms on purely theoretical grounds, it is sufficient to notice that the rms errors of *b* and *c* coefficients are of the same order of magnitude or even larger than possible non-zero values of *d* and *e* coefficients. To avoid these ambiguities, a dedicated analysis constraining theory by available experimental information on *A* = 32 quintet have recently been performed [99]. Further efforts towards required precision will be crucial to advance our understanding of the origin of the IMME beyond its quadratic form.
