  The variational cluster approximation (VCA) proposed by M. Potthoff {\it et
al.} [Phys. Rev. Lett. {\bf 91}, 206402 (2003)] is extended to electron or spin
systems with nonlocal interactions. By introducing more than one source field
in the action and employing the Legendre transformation, we derive a
generalized self-energy functional with stationary properties. Applying this
functional to a proper reference system, we construct the extended VCA (EVCA).
In the limit of continuous degrees of freedom for the reference system, EVCA
can recover the cluster extension of the extended dynamical mean-field theory
(EDMFT). For a system with correlated hopping, the EVCA recovers the cluster
extension of the dynamical mean-field theory for correlated hopping. Using a
discrete reference system composed of decoupled three-site single impurities,
we test the theory for the extended Hubbard model. Quantitatively good results
as compared with EDMFT are obtained. We also propose VCA (EVCA) based on
clusters with periodic boundary conditions. It has the (extended) dynamical
cluster approximation as the continuous limit. A number of related issues are
discussed.
