Unless I misunderstood the problem, this tweaking of the LJ parameters will be a tractable optimization problem once we know these two constraints:
- How much more favorable we want the new V to be?
- Over which additional tail range we want the favorable part to persist?
- In these examples, the new V is still favorable at r=8:
- Is the favorable difference sufficient, say at r=5, and do we have a handle on which extra energy is needed?
Fig.1 - New V(n.m) without additional width on sigma: