Energy of the nearest neighbor RVB state by systematic loop expansion

We present an approximation scheme for the calculation of the norm and energy of the nearest-neighbor-RVB state for the Heisenberg antiferromagnet on the 2D square lattice. The approximation leads to a systematic expansion of norm and energy, with the `expansion parameter' being the maximum length of loops taken into account in the calculation of energy and norm. The expansion converges well, the best estimate for the ground state energy/site is $-0.434473J$.