*3.2. Single-Particle Contributions: Vll and Vls*

It was recognised by Zuker et al. [16] that the single-particle splitting between neutron and proton orbitals, induced by the Coulomb interaction, should be accounted for. In the shell model, this can be achieved through introducing shifts between the neutron and proton single particle levels before diagonalisation. The required Coulomb shifts (*Vll*) can be determined through the formalism derived by Duflo and Zuker [48]. Since MED are normalised to the ground state, this term will only become significant where configurations change along the yrast line, and where there are different orbital occupancies between protons and neutrons. This term was eventually neglected by Zuker et al. [16], since they showed that other monopole effects (see Section 3.3) dominate in the specific region being tested. However, this will not always be the case, and the term is routinely included in MED calculations.

A second single particle term, not originally included by Zuker et al. [16], is the electromagnetic spin-orbit effect (*Vls*). This is a purely electromagnetic effect, affecting both proton and neutron levels, associated with the spin magnetic moment of the nucleon interacting with the Coulomb field of the nucleus. The formalism was introduced by Nolen and Schiffer [49] in their description of Coulomb displacement energies. This effect, which has opposite signs for protons and neutrons, can be significant for MED, especially where occupancy of orbitals with *j* = *l* + *s* and *j* = *l* − *s* (where *j*, *l* and *s* are the total, orbital and spin angular momentum quantum numbers) are both changing (e.g., [50]).

In calculations of MED, both these effects are routinely included. Generally, however, these contributions are expected to cancel in TED calculations, due to the double-difference method of determining TED.
