Fig. 5: Benzene dimer calculated energy and the extrapolation based on VMC-DMC linear relation.

a The energy of the T-shaped benzene dimer with a center-to-center distance of 4.95 Å as well as the CCSD(T) results as baselines. The CCSD(T)/CBS* result is calculated using binding energy and monomer energy. See Supplementary Note 14 for details. b The fitted distribution of binding energy from VMC, DMC, and extrapolation results, where the solid lines inside each violin represent the actual data points. The energy for two separated benzene molecules is calculated using a benzene dimer separated by 10 Å⁴³. The equilibrium configuration is a T-shaped structure with center-to-center distance of 4.95 Å⁴⁰. The experimental range is from Grover et al.⁴⁶. The extrapolation scheme is based on the empirical linear relation between VMC energy (E_VMC) and DMC energy (E_DMC) in the training process, indicated by the blue and yellow points in c. c Linear fitting of E_VMC − E_DMC with respect to E_DMC − E_final on various molecular systems, where E_final represents the DMC energy at final VMC training step. The number within the parenthesis in the legend is the fitted slope for each system. d Linear fitting of E_VMC − E_DMC with respect to E_DMC − E_exact on atoms, where the energy data are the same as Fig. 2e. Note that the data in d is fundamentally different from c in the sense that the linearity in c is about different training steps for the same system while in d the linearity is measured across different systems.