A Link for the corresponding potential documentation might also help.
As far as I know, there is no ready-to-use pair style for this. Your quote also does not say anything about the definition of the f(B_i+B_j) function.
Since this is obviously a pairwise additive potential, it could be implemented with pair style table. As of LAMMPS version 22 December 2022 we bundle a small Python script module that can be used to conveniently generate such tables.
If you are an adventurous person, then you could download the latest LAMMPS development snapshot from github and use the LEPTON package, which comes with pair style lepton that allows to enter pairwise potentials with an expression string. This is quite a bit slower than programming in C++ but significantly faster (~5x) than using Python and can save you the burden of C++ programming. So it should be good enough for testing. If you want better performance, you can always use the pair_write command to generate tables for pair style table. Since the expression is evaluated analytically (including its first derivative for the force), it can be used to figure out the optimal settings for tabulation to maintain the desired accuracy (which is always a challenge with tabulation).
In any case, it is usually better to exclude the coulomb term in U(r_{ij}) in the table or Lepton expression and instead use pair style hybrid/overlay and import the Coulomb interaction from pair style coul/cut or (even better) coul/long (in combination with Ewald or PPPM).
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