Nodal error behind discrepancies between coupled cluster and diffusion Monte Carlo in hydrogen-bonded systems

TL;DR

The study interrogates the origin of disagreements between coupled-cluster (CC) theory and diffusion Monte Carlo (DMC) in hydrogen-bonded noncovalent systems, using the AcOH dimer and water–peptide as benchmarks. It systematically probes CC by exploring CBS extrapolation, core treatments, local CC, and higher excitations, and interrogates DMC via time-step, localization, and nodal quality, including backflow. The key finding is that fixed-node error in Slater-Jastrow DMC dominates the discrepancy, and applying backflow nodes brings DMC energies into near agreement with CC results, supporting CC as the benchmark for these systems. This suggests that improving DMC nodal surfaces, rather than CC methodology, will be crucial for resolving residual differences in hydrogen-bonded interactions and guides future work toward more accurate nodal representations.

Abstract

The small magnitude and long-range character of non-covalent interactions pose a significant challenge for computational quantum chemical and electronic-structure methods alike. State-of-the-art coupled cluster (CC) theory and benchmark-grade diffusion Monte Carlo (DMC) are ideally positioned to tackle these problems, but concerning differences between both methods have been reported in numerous studies of the interaction energy of non-covalently bound dimers. Given that the basic theoretical frameworks underpinning both methods are exact in principle, the error must arise from one or several of the approximations required to make the calculations computationally tractable. Here, we carry out a rigorous and systematic examination of the effect of each of these approximations using the acetic acid dimer and water-peptide systems as convenient testing grounds. Thanks to the use of stringently optimized backflow wave functions we are able to find that the significant discrepancies are dominated by the fixed-node error incurred by the Slater-Jastrow DMC result, while errors in the CC calculations do not significantly alter the result. This finding, likely applicable to other hydrogen-bonded systems, helps establish that CC should be regarded as the benchmark for these systems, and can potentially guide the search for pragmatic solutions to the fixed-node problem in the future.

Figures (2)

• Figure 1: Interaction energy of (a) the AcOH dimer and (b) the water-peptide dimer obtained using CC theory (blue) and DMC (red). The interaction energies that allow for the most direct comparison between each methodology (bold) with a 'two-sigma' confidence interval are shown as a shaded area. For the CC calculations, 0.5CP-corrected values are reported with non-statistical error bars extending from the CP-uncorrected to the CP-corrected result; note that reference CC values are plotted as single points. The error bars on the DMC values are purely statistical in nature, representing 95% (two-sigma) confidence intervals, and include optimization uncertainty where pertinent, see text. The insets show the structures considered with C atoms in grey, O atoms in red, H atoms in pink and N atoms in blue.
• Figure 2: (a) Scatter plot of the SJB-VMC energy as a function of SJB-DMC energy of the ECP AcOH monomer for 20 independent random instances of wave function optimization at each of four optimization sample sizes $n_\text{opt}$, and for the final optimization performed at $n_\text{opt}=10^7$; note the lack of visible correlation between VMC and DMC energies and the evident mismatch of the location of the VMC and DMC energy minima, schematically represented in the inset. (b) Resulting uncertainty (1-sigma confidence interval) on the SJB-VMC and SJB-DMC energy arising from the stochastic nature of optimization as a function of $n_{\rm opt}$, computed as the standard deviation of the corresponding energies in Fig. \ref{['fig:opt_figure']}(a), in log-log scale. The target optimization uncertainty and chosen sample size are shown as dotted lines, and solid lines are linear fits in log-log scale to the data points using a fixed slope of $-1/2$. Shaded areas and errorbars represent 95% (two-sigma) confidence intervals. Also shown is the (average) standard error on the mean value of the $n_\text{opt}$ local energies used in optimization, which is an order of magnitude greater than the optimization uncertainty on the backflow VMC energy: correlated-sampling optimizers can determine the location of the energy minimum to much better accuracy than the statistical resolution of the local energy sample mean would suggest.