A quantum Monte Carlo calculation of the ground state energy of the hydrogen molecule

We have calculated the ground state energy of the hydrogen molecule using the quantum Monte Carlo (QMC) method of solving the Schrodinger equation, without the use of the Born-Oppenheimer or any other adiabatic approximations. The wave function sampling was carried out in the full 12-dimensional configuration space of the four particles (two electrons and two protons). Two different methods were employed: the diffusion quantum Monte Carlo (DQMC) method and the Green’s function quantum Monte Carlo (GFQMC) method. This computation is very demanding because the configurations must be evolved on the time scale of the electronic motion, whereas the finite nuclear mass effects are resolved accurately only after equilibration on the much slower time scale of the nuclear motion. Thus, a very large number of iterations is required. The calculations were performed on the CM-2 Connection Machine computer, a massively parallel supercomputer. The enormous speedup afforded by the massive parallelism allowed us to complete the computation in a reasonable amount of time. The total energy from the DQMC calculations is 1.163 97 + 0.000 05 a.u. A more accurate result was obtained from the GFQMC calculations of 1.164 024 + 0.000 009 a.u. Expressed as a dissociation energy, the GFQMC result is 36 117.9 f 2.0 cm ‘, including the corrections for relativistic and radiative effects. This result is in close agreement with accurate nonadiabatic-relativistic dissociation energies from variational calculations (corrected for radiative effects) in the range of 36 117.9-36 118.1 cm’ and with the best experimentally determined dissociation energy of McCormack and Eyler 36 118.1 + 0.2 cm ‘.