  Similar to the modular vector fields in Poisson geometry, modular derivations
are defined for smooth Poisson algebras with trivial canonical bundle. By
twisting Poisson module with the modular derivation, the Poisson cochain
complex with values in any Poisson module is proved to be isomorphic to the
Poisson chain complex with values in the corresponding twisted Poisson module.
Then a version of twisted Poincar\'{e} duality is proved between the Poisson
homologies and cohomologies. Furthermore, a notion of pseudo-unimodular Poisson
structure is defined. It is proved that the Poisson cohomology as a
Gerstenhaber algebra admits a Batalin-Vilkovisky operator inherited from some
one of its Poisson cochain complex if and only if the Poisson structure is
pseudo-unimodular. This generalizes the geometric version due to P. Xu. The
modular derivation and Batalin-Vilkovisky operator are also described by using
the dual basis of the K\"{a}hler differential module.
