Stepping-Stone CBH: Benchmark and Application of a Multilayered Isodesmic-Based Correction Scheme.

We present a generalization of the connectivity-based hierarchy (CBH) of isodesmic-based correction schemes to a multilayered fragmentation platform for overall cost reduction while retaining high accuracy. The newly developed multilayered CBH approach, called stepping-stone CBH (SSCBH), is benchmarked on a diverse set of 959 medium-sized organic molecules. Applying SSCBH corrections to the PBEh-D3 density functional resulted in an average error of 0.76 kcal/mol for the full test set compared to accurate CCSD(T)-quality enthalpies and an even lower error of 0.44 kcal/mol on a subset containing only acyclic molecules. These results rival the traditional CBH-3 approach at a greatly reduced cost, allowing larger fragment corrections to be made at the MP2 level of theory rather than with G4. Our SSCBH approach will enable more widespread applications of CBH methods to a broader range of organic and biomolecular systems.

Approaching Coupled Cluster Accuracy with Density Functional Theory Using the Generalized Connectivity-Based Hierarchy.

This Perspective reviews connectivity-based hierarchy (CBH), a systematic hierarchy of error-cancellation schemes developed in our group with the goal of achieving chemical accuracy using inexpensive computational techniques ("coupled cluster accuracy with DFT"). The hierarchy is a generalization of Pople's isodesmic bond separation scheme that is based only on the structure and connectivity and is applicable to any organic and biomolecule consisting of covalent bonds. It is formulated as a series of rungs involving increasing levels of error cancellation on progressively larger fragments of the parent molecule. The method and our implementation are discussed briefly. Examples are given for the applications of CBH involving (1) energies of complex organic rearrangement reactions, (2) bond energies of biofuel molecules, (3) redox potentials in solution, (4) pKa predictions in the aqueous medium, and (5) theoretical thermochemistry combining CBH with machine learning. They clearly show that near-chemical accuracy (1-2 kcal/mol) is achieved for a variety of applications with DFT methods irrespective of the underlying density functional used. They demonstrate conclusively that seemingly disparate results, often seen with different density functionals in many chemical applications, are due to an accumulation of systematic errors in the smaller local molecular fragments that can be easily corrected with higher-level calculations on those small units. This enables the method to achieve the accuracy of the high level of theory (e.g., coupled cluster) while the cost remains that of DFT. The advantages and limitations of the method are discussed along with areas of ongoing developments.

Interpretable Graph-Network-Based Machine Learning Models via Molecular Fragmentation.

Chemists have long benefitted from the ability to understand and interpret the predictions of computational models. With the current shift to more complex deep learning models, in many situations that utility is lost. In this work, we expand on our previously work on computational thermochemistry and propose an interpretable graph network, FragGraph(nodes), that provides decomposed predictions into fragment-wise contributions. We demonstrate the usefulness of our model in predicting a correction to density functional theory (DFT)-calculated atomization energies using Δ-learning. Our model predicts G4(MP2)-quality thermochemistry with an accuracy of <1 kJ mol-1 for the GDB9 dataset. Besides the high accuracy of our predictions, we observe trends in the fragment corrections which quantitatively describe the deficiencies of B3LYP. Node-wise predictions significantly outperform our previous model predictions from a global state vector. This effect is most pronounced as we explore the generality by predicting on more diverse test sets indicating node-wise predictions are less sensitive to extending machine learning models to larger molecules.

Quantitative Prediction of Vertical Ionization Potentials from DFT via a Graph-Network-Based Delta Machine Learning Model Incorporating Electronic Descriptors.

While accurate wave function theories like CCSD(T) are capable of modeling molecular chemical processes, the associated steep computational scaling renders them intractable for treating large systems or extensive databases. In contrast, density functional theory (DFT) is much more computationally feasible yet often fails to quantitatively describe electronic changes in chemical processes. Herein, we report an efficient delta machine learning (ΔML) model that builds on the Connectivity-Based Hierarchy (CBH) scheme─an error correction approach based on systematic molecular fragmentation protocols─and achieves coupled cluster accuracy on vertical ionization potentials by correcting for deficiencies in DFT. The present study integrates concepts from molecular fragmentation, systematic error cancellation, and machine learning. First, we show that by using an electron population difference map, ionization sites within a molecule may be readily identified, and CBH correction schemes for ionization processes may be automated. As a central feature of our work, we employ a graph-based QM/ML model, which embeds atom-centered features describing CBH fragments into a computational graph to further increase accuracy for the prediction of vertical ionization potentials. In addition, we show that the incorporation of electronic descriptors from DFT, namely electron population difference features, improves model performance well beyond chemical accuracy (1 kcal/mol) to approach benchmark accuracy. While the raw DFT results are strongly dependent on the underlying functional used, for our best models, the performance is robust and much less dependent on the functional.

A Fragmentation-Based Graph Embedding Framework for QM/ML.

We introduce a new fragmentation-based molecular representation framework "FragGraph" for QM/ML methods involving embedding fragment-wise fingerprints onto molecular graphs. Our model is specifically designed for delta machine learning (Δ-ML) with the central goal of correcting the deficiencies of approximate methods such as DFT to achieve high accuracy. Our framework is based on a judicious combination of ideas from fragmentation, error cancellation, and a state-of-the-art deep learning architecture. Broadly, we develop a general graph-network framework for molecular machine learning by incorporating the inherent advantages prebuilt into error cancellation methods such as the generalized Connectivity-Based Hierarchy. More specifically, we develop a QM/ML representation through a fragmentation-based attributed graph representation encoded with fragment-wise molecular fingerprints. The utility of our representation is demonstrated through a graph network fingerprint encoder in which a global fingerprint is generated through message passing of local neighborhoods of fragment-wise fingerprints, effectively augmenting standard fingerprints to also include the inbuilt molecular graph structure. On the 130k-GDB9 dataset, our method predicts an out-of-sample mean absolute error significantly lower than 1 kJ/mol compared to target G4(MP2) calculated energies, rivaling current deep learning methods with reduced computational scaling.

Effective Molecular Descriptors for Chemical Accuracy at DFT Cost: Fragmentation, Error-Cancellation, and Machine Learning.

Recent advances in theoretical thermochemistry have allowed the study of small organic and bio-organic molecules with high accuracy. However, applications to larger molecules are still impeded by the steep scaling problem of highly accurate quantum mechanical (QM) methods, forcing the use of approximate, more cost-effective methods at a greatly reduced accuracy. One of the most successful strategies to mitigate this error is the use of systematic error-cancellation schemes, in which highly accurate QM calculations can be performed on small portions of the molecule to construct corrections to an approximate method. Herein, we build on ideas from fragmentation and error-cancellation to introduce a new family of molecular descriptors for machine learning modeled after the Connectivity-Based Hierarchy (CBH) of generalized isodesmic reaction schemes. The best performing descriptor ML(CBH-2) is constructed from fragments preserving only the immediate connectivity of all heavy (non-H) atoms of a molecule along with overlapping regions of fragments in accordance with the inclusion-exclusion principle. Our proposed approach offers a simple, chemically intuitive grouping of atoms, tuned with an optimal amount of error-cancellation, and outperforms previous structure-based descriptors using a much smaller input vector length. For a wide variety of density functionals, DFT+ΔML(CBH-2) models, trained on a set of small- to medium-sized organic HCNOSCl-containing molecules, achieved an out-of-sample MAE within 0.5 kcal/mol and 2σ (95%) confidence interval of <1.5 kcal/mol compared to accurate G4 reference values at DFT cost.

Accurate Thermochemistry for Organic Cations via Error Cancellation using Connectivity-Based Hierarchy.

Connectivity-Based Hierarchy (CBH) is an effective error-cancellation scheme for the determination of chemically accurate thermochemical properties of a variety of organic and biomolecules. Neutral molecules and open-shell radicals have already been treated successfully by this approach utilizing inexpensive computational methods such as density functional theory. Herein, we present an extension of the method to a new class of molecules, specifically, organic cations. Because of the presence of structural rearrangements involving hydrogen migrations as well as unusual structures such as bridged cations, the application of the standard CBH protocol to a test set of 25 cations leads to significant errors due to ineffective bond-type matching. We propose an adjusted protocol to overcome such limitations to achieve highly effective error cancellation. The modified CBH methods, in conjunction with a wide range of density functionals, reproduce G4 energies for the test set of organic cations accurately within 1-2 kcal/mol at a reduced computational cost.