  In this work, a long-range resonating valence bond state is proposed as a
variational wave function for the ground state of the $S=1/2$ antiferromagnetic
Heisenberg model on the honeycomb lattice. Employing Variational Monte Carlo
(VMC) method, we show that the ground state energy obtained from such RVB wave
function, lies well below the energy of the N\'eel state and compares very well
to the energies evaluated from spin-wave theory and series expansion method. We
also obtain the spin-spin correlation function along zig-zag and armchair
directions and find that the two correlations are different, which indicates
the anisotropic nature of the system. We compare our results with the square
lattice and we show that although the quantum fluctuations on honeycomb lattice
are much stronger, but do not completely destroy the magnetic order.
