not too dependent on the specifics of the model.

In more detail, the($x,y$) Hamiltonian is:

$$H=\sum T_i + \sum_{i<j}V(ij)$$

where

$$V=V_c+xV_{so}+yV_t$$

with $c \equiv$central, $s.o. \equiv$spin-orbit, and $t
\equiv$tensor.

For ($x,y$)=(1,1) the matrix elements of this interaction are close to those of realistic G matrices such as Bonn A. We can study the effects the spin-orbit and tensor interactions by varying $x$ and $y$.

Note that we do not add any single-particle energies to the above Hamiltonian. Rather, we let the single-particle energies be implicitly generated by $H$. Hence, if we set $x$=0 i.e. turn off the spin-orbit interaction, we will also be turning off the one-body spin-orbit splitting coming from this interaction.

As a counterpoint, we repeat all the calculations with the $Q \cdot Q$ Hamiltonian

$$H_{Q}=\sum_i \frac{p_i^2}{2m} + \frac{1}{2}m\omega^2r_i^2
-\chi\sum_{i<j}Q \cdot Q$$

Note that we have added the term $\frac{1}{2}m\omega^2r^2$ which is not present for the ($x,y$) interaction. The reason for this is that $Q \cdot Q$ cannot generate any single-particle potential energy splitting whereas the ($x,y$) interaction can.

Whereas the ($x,y$) interaction like all realistic interactions is of short range, the $Q \cdot Q$ interaction is long range. Yet, as we shall see some of the results 