If you are sure the analytic form of the potential you want is not in
LAMMPS, then you can add a new potential - e.g. pair_lj_delta - by
following the procedure outlined in Section 8 of the documentation.
Basically, create a new pair_lj_delta.cpp and *.h using an existing file
as a starting point. Write the code for the formula, coeffs you want.
Add 2 lines to style_user.h and you're done.