  We develop a variational Monte Carlo (VMC) method for electron-phonon coupled
systems. The VMC method has been extensively used for investigating strongly
correlated electrons over the last decades. However, its applications to
electron-phonon coupled systems have been severely restricted because of its
large Hilbert space. Here, we propose a variational wave function with a large
number of variational parameters which is suitable and tractable for systems
with electron-phonon coupling. In the proposed wave function, we implement an
unexplored electron-phonon correlation factor which takes into account the
effect of the entanglement between electrons and phonons. The method is applied
to systems with diagonal electron-phonon interactions, i.e. interactions
between charge densities and lattice displacements (phonons). As benchmarks, we
compare VMC results with previous results obtained by the exact
diagonalization, the Green function Monte Carlo and the density matrix
renormalization group for the Holstein and Holstein-Hubbard model. From these
benchmarks, we show that the present method offers an efficient way to treat
strongly coupled electron-phonon systems.
