Molecular-orbital-based Machine Learning for Open-shell and Multi-reference Systems with Kernel Addition Gaussian Process Regression

TL;DR

This work introduces kernel addition Gaussian process regression (KA-GPR) within molecular-orbital-based ML (MOB-ML) to directly learn total correlation energies for general electronic structure theories, including open-shell and multi-reference systems. By leveraging Nesbet's theorem and ROHF-based MOB features, KA-GPR unifies predictions for closed- and open-shell cases and demonstrates state-of-the-art accuracy across criegee, H10, small radicals, water dissociation, QM9/QM7b-T/GDB-13-T benchmarks, and the QMSpin carbene dataset. Key findings show chemical accuracy achievable with modest training data, robust PES predictions, and strong transferability within similar molecular sizes, albeit with some loss in absolute-energy transfer when crossing dataset spaces. Overall, KA-GPR advances MOB-ML toward practical, high-accuracy predictions of electronic energies beyond CCSD(T) for diverse chemical systems, with significant implications for studying challenging open-shell reactions and radical chemistry.

Abstract

We introduce a novel machine learning strategy, kernel addition Gaussian process regression (KA-GPR), in molecular-orbital-based machine learning (MOB-ML) to learn the total correlation energies of general electronic structure theories for closed- and open-shell systems by introducing a machine learning strategy. The learning efficiency of MOB-ML (KA-GPR) is the same as the original MOB-ML method for the smallest criegee molecule, which is a closed-shell molecule with multi-reference characters. In addition, the prediction accuracies of different small free radicals could reach the chemical accuracy of 1 kcal/mol by training on one example structure. Accurate potential energy surfaces for the H10 chain (closed-shell) and water OH bond dissociation (open-shell) could also be generated by MOB-ML (KA-GPR). To explore the breadth of chemical systems that KA-GPR can describe, we further apply MOB-ML to accurately predict the large benchmark datasets for closed- (QM9, QM7b-T, GDB-13-T) and open-shell (QMSpin) molecules.

Figures (7)

• Figure 1: Mean absolute errors (MAEs) of predicted correlation energies for 100 test criegee geometries using MOB-ML. For the reference theory of CCSD(T)/cc-pVTZ (blue), we display the results from learning the pair energies by GPR and total correlation energies by KA-GPR, respectively. For the reference theory of MRCI+Q/cc-pVTZ (orange), only the results learnt by KA-GPR using total correlation energy labels are reported. All the features are generated from the computation results of RHF/cc-pVTZ. All the MAEs are plotted as functions of the number of training geometries on a logarithm scale ("learning curves"). The Lewis structures of the criegee molecule are also shown. All the prediction MAEs are also listed in the Supporting Information Table S1.
• Figure 2: Mean absolute errors (MAEs) of predicted correlation energies for 9 test closed-shell H10 configurations using MOB-ML. An example structure of H10 are also shown, where $a$ is the bond length of each H2 and $x$ is the distance between two H2. All the prediction MAEs are also listed in the Supporting Information Table S2.
• Figure 3: Learning curves of MOB-ML for test QM9 molecules at the MP2/cc-pVTZ level of theory. The models are generated with different numbers of randomly selected QM9 molecules. The learning curves of OrbNet-Equi orbnetequi and QML Karandashev2022 using with a reference theory of B3LYP/6-31G(2df,p) ramakrishnan2014quantum are also displayed for comparison. The results of MOB-ML (GMM/GPR) obtained from Ref. sun2022molecular are also shown to compare with the ones from MOB-ML (KA-GPR). The corresponding the level of theory used for feature generation of each ML approach is also listed in the legend. The MAEs of MOB-ML (KA-GPR) are also listed in Table S3 in the Supporting Information.
• Figure 4: Learning curves for MP2/cc-pVTZ energy predictions from MOB-ML trained on QM7b-T and applied to (a) QM7b-T and (b) GDB-13-T. The literature results of learning pair energies by unsupervised clustering via Gaussian mixture and local regression with AltBBMM regressor in MOB-ML, i.e., MOB-ML (GMM/GPR, pair-wise), and QML (MO, $\Delta$-learning) using features generated from HF/cc-pVTZ computations are plotted. In panel (b), the prediction accuracies of the relative conformer energies (dash lines) computed by subtracting the corresponding true and predicted optimized structure energies of each GDB-13-T molecule are also shown. The corresponding computational cost of the feature generation in each ML approach is also listed in the legend, and the shaded areas correspond to the chemical accuracy region of 1 kcal/mol MAE.
• Figure 5: Mean absolute errors (MAEs) of predicted correlation energies for different test small radical structures using MOB-ML. The corresponding test set contains 120 test structures for each small radical molecule. Panel (a) and (c) display the results of KA-GPR at the level of LUCCSD/cc-pVTZ, and panel (b) shows the results at the level of MRCI+Q/cc-pVTZ. For the reference theory of LUCCSD/cc-pVTZ, we display the results from learning the pair energies by GPR and total correlation energies by KA-GPR, respectively. All the results are plotted on a logarithm scale.