Fig. 3: Calculations on N2, cyclobutadiene, and water dimer.
From: Towards the ground state of molecules via diffusion Monte Carlo on neural networks
a Main panel: calculated energy of N2 at different bond length, plotted as the difference to the experimental data29. For comparison, the green line is the highly accurate r12-MR-ACPF results under a modified basis set based on aug-cc-pV5Z30. Inset: the dissociation curves from experimental data (black line) and FermiNet-DMC (orange squares). The negligible error bars (less than 0.1 mHa) are not plotted. The pink backgrounds highlight the dissociation region where correlations are strong. b Molecular structures of cyclobutadiene’s equilibrium state (bottom) and transition state (top). c Main panel: the ground state energy of cyclobutadiene’s equilibrium state as a function of the VMC training step. FermiNet-DMC energy is calculated using the trial wavefunction at the corresponding training steps. FermiNet-VMC* indicates the result from Spencer et al.21. Inset: the transition barrier of cyclobutadiene calculated with different methods. Pink background indicates the range of experimental estimates between 2.5 and 15.9 mHa, while we show only the part above 9 mHa to highlight differences between QMC results. Gray dashed lines indicate results from five multi-reference coupled cluster (MRCC) methods (top to bottom): MR-DI-EOMCCSD, RMRCCSD(T), Mk-MRCCSD(T), MRCISD+Q and BW-MRCCSD(T)58. d Left: three of the relative energies of the 10 Smith stationary points SPn(n = 1, 2, …, 10)32. For the other results, see Supplementary Fig. 6. The SP1 structure is the global minimum and is taken as the reference. The geometries of SP1 and SP3 are shown as insets while the others are included in Supplementary Fig. 4. Two neural networks have been trained for 105 steps and 3 × 105 steps, which are dubbed as “undertrained” and “well-trained''. All the geometries are optimized by CCSD(T)34. The CCSD(T) energies and the DMC results with the conventional Slater-Jastrow ansatz35 are also plotted for comparison. The error bar of the energy difference is calculated as the square root of the sum of the squares of each energy estimator’s standard error. Right: mean absolute deviation from CCSD(T) results over all the relative energies.
