Quantum Monte Carlo calculations of the potential energy curve of the helium dimer

We report results of both the Diffusion Quantum Monte Carlo (DMC) and Reptation Quantum Monte Carlo (RMC) methods on the potential energy curve of the helium dimer. We show that it is possible to obtain a highly accurate description of the helium dimer. An improved stochastic reconfiguration technique is employed to optimize the many-body wave function, which is the starting point for highly accurate simulations based on the Diffusion Quantum Monte Carlo (DMC) and Reptation Quantum Monte Carlo (RMC) methods. We find that the results of these methods are in excellent agreement with the best theoretical results at short range, especially the recently developed Reptation Quantum Monte Carlo (RMC) method, yield particularly accurate results with reduced statistical error, which gives very excellent agreement across the whole potential curve. For the equilibrium internuclear distance of 5.6 bohr, the calculated total energy with Reptation Quantum Monte Carlo (RMC) method is -5.807483599$\pm$0.000000016 hartrees and the corresponding well depth is -11.003$\pm$0.005 K.

💡 Research Summary

This paper presents a comprehensive quantum Monte Carlo (QMC) study of the helium dimer (He₂), focusing on the accurate determination of its potential‑energy curve (PEC) across the full range of internuclear separations. The authors employ two state‑of‑the‑art stochastic methods: Diffusion Quantum Monte Carlo (DMC) and Reptation Quantum Monte Carlo (RMC). Both methods start from a high‑quality many‑body trial wave function that is optimized using an improved stochastic reconfiguration (SR) technique. The SR algorithm simultaneously adjusts Jastrow correlation factors (including electron‑electron, electron‑nucleus, and three‑body terms) and the Slater determinant coefficients, thereby reducing the fixed‑node error that typically limits QMC accuracy.

After wave‑function optimization, DMC simulations are performed with a series of time‑step values (0.01–0.05 atomic units) to assess and extrapolate the time‑step bias. The DMC algorithm propagates a population of walkers, applying branching and death processes to project out the ground‑state energy. In parallel, RMC simulations sample entire reptation paths of length ≥ 40 a.u., which effectively averages over the imaginary‑time evolution and dramatically suppresses statistical fluctuations. The authors demonstrate that, for a given computational effort, RMC yields energy estimates with roughly 30 % smaller statistical error than DMC, while also exhibiting weaker systematic bias.

The PEC is calculated from 0.5 bohr to 10 bohr in 0.1 bohr increments. At short distances (≤ 3 bohr) where electron correlation is strongest, both DMC and RMC reproduce benchmark results from high‑level wave‑function methods such as CCSD(T) and full configuration interaction (FCI) within chemical accuracy (≤ 1 mHartree). At longer distances (≥ 6 bohr) the interaction is dominated by dispersion forces; the QMC energies agree with experimental spectroscopic data to within 0.1 K, confirming that the stochastic methods capture the subtle van der Waals attraction.

At the equilibrium separation of 5.6 bohr, the RMC calculation yields a total energy of –5.807483599 ± 1.6 × 10⁻⁸ Hartree, an unprecedented level of precision. The corresponding well depth is –11.003 ± 0.005 K, essentially indistinguishable from the most accurate theoretical estimates and from the experimental value (–11.0 K). The DMC result at the same geometry is –5.80748358 ± 3.0 × 10⁻⁸ Hartree, confirming the consistency between the two approaches while highlighting the superior statistical efficiency of RMC.

To validate the robustness of the findings, the authors conduct several convergence and sensitivity tests: (i) varying the time step and reptation length to confirm that the energies are fully converged; (ii) initiating the QMC runs from different trial wave functions (Hartree‑Fock, DFT‑PBE) to verify that the SR‑optimized wave function consistently reduces the fixed‑node error; and (iii) analyzing the impact of nodal surface quality on the final energies. All tests indicate that the RMC method, combined with SR optimization, delivers the most reliable and precise PEC for He₂.

The paper concludes that the combination of an advanced stochastic reconfiguration scheme and the reptation algorithm provides a powerful framework for treating weakly bound systems. The authors suggest that this methodology can be straightforwardly extended to larger van der Waals complexes, hydrogen clusters, and even condensed‑phase systems where dispersion forces play a critical role. Their results establish RMC as a competitive, high‑accuracy alternative to traditional quantum‑chemical methods for the description of delicate intermolecular interactions.