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I am still a bit confused\. Many papers used Bjerrum length=3\.0 Sigma
when doing coarse-grained simulations, does this mean that the dielectric
constant should be set as 1/3 when the unit is lj?
This is not so simple. As you must know the Bjerrum length is simply
defined as the distance between two elementary charges where the
thermal energy compensate the Coulomb interaction energy. In lammps
the Coulomb interaction energy is defined as
E = C q_i q_j / (dielectric r), with q_i = q_j = 1
where the C constant depends on the units
E is equal to the thermal energy kB T when r = lB = C / (dielectric kB T).
So you should set dielectric to C/(lB kB T). Therefore it will depend
on the temperature of your system and the units. In LJ units C = 1 so
you end up with dielectric = 1/(lB* T*).