Predictive free energy simulations through hierarchical distillation of quantum Hamiltonians

Significance

Precise computation of the free energy profiles and kinetics of condensed phase chemical reactions involves two seemingly incompatible requirements: extensive statistical sampling and expensive quantum mechanical energies. We overcome this challenge by developing a hierarchical machine learning framework to distill the accuracy of state-of-the-art quantum calculations into a series of coarse-grained quantum models. The distilled quantum models naturally recover the long-range electrostatics and quantum response that are difficult to capture in traditional machine-learning force field approaches. Using this framework, we predict solution-phase weak acid dissociation constants and enzymatic reaction rates from first-principles simulations that are converged to chemical accuracy.

Abstract

Obtaining the free energies of condensed phase chemical reactions remains computationally prohibitive for high-level quantum mechanical methods. We introduce a hierarchical machine learning framework that bridges this gap by distilling knowledge from a small number of high-fidelity quantum calculations into increasingly coarse-grained, machine-learned quantum Hamiltonians. By retaining explicit electronic degrees of freedom, our approach further enables a faithful embedding of quantum and classical degrees of freedom that captures long-range electrostatics and the quantum response to a classical environment to infinite order. As validation, we compute the proton dissociation constants of weak acids and the kinetic rate of an enzymatic reaction entirely from first principles, reproducing experimental measurements within chemical accuracy or their uncertainties. Our work demonstrates a path to condensed phase simulations of reaction free energies at the highest levels of accuracy with converged statistics.

Free energies are the driving force for numerous chemical and biochemical phenomena, but their accurate computation in the condensed phase presents a grand challenge for theory. Bridging the gap between the quantum mechanics of electrons and the macroscopic time and length scales on which real-world (bio-)chemical processes occur requires both high-fidelity electronic structure methods and extensive statistical thermal sampling. Although classical molecular dynamics (MD) simulations using empirical force fields (FFs) can reach long timescales, particularly when combined with specialized hardware (1), these potentials cannot reliably model chemical bond-breaking and forming events. Conversely, high-level quantum chemistry methods that accurately describe electron correlation are limited to small systems, with recent efforts with the gold-standard coupled cluster method and accurate basis sets reaching only picoseconds of dynamics for 19 atoms in the gas phase, even with the use of cost-reduction strategies that utilize the local nature of electron correlation (2).

Machine learning (ML) potentials, which regress quantum ground-state energies from molecular geometries, have emerged as a promising strategy to bridge this gap, but they still face several critical challenges. First, most modern ML potentials rely on equivariant message-passing to achieve accuracy but are highly demanding in terms of computation and memory for large-scale problems (3–8). Second, ML potentials (especially neural-network-based variants) are often data-hungry, requiring large datasets that are prohibitively expensive to generate using high-level quantum chemistry. These two challenges are particularly pronounced in condensed-phase simulation due to the exponentially large configurational and chemical space that must be statistically sampled for computing thermal averages and to be well-represented in the training data. A multiscale approach, similar to the hybrid quantum mechanics/molecular mechanics (QM/MM) method but replacing QM with ML, represents a natural solution but introduces a third challenge: Standard ML potentials lack explicit electronic degrees of freedom, making it difficult to model the crucial response of the ML-described subsystem to the long-range electrostatics generated by the classical environment. To address this, one must either rely on a computationally inexpensive but potentially inaccurate physical model to interact with and respond to the MM potential (9–26) and/or modify the ML architecture to accept the MM information as additional ML input (10–13, 16, 22, 23, 27–31).

Here, we introduce a hierarchical Hamiltonian learning framework that provides a unified solution to the above challenges. Instead of directly regressing the potential energy from atomic coordinates, our approach retains explicit electronic degrees of freedom by systematically coarse-graining and parameterizing different levels of quantum Hamiltonians. This bottom–up strategy begins by distilling the energy information from a small number of high-accuracy wavefunction quantum chemistry calculations into a cheaper, custom-parameterized, density functional theory (DFT). The learned Kohn–Sham functional is then applied in the DFT/MM framework to generate a larger, condensed-phase dataset, which in turn is used to train a final, highly efficient machine-learned semiempirical (SEQM) Hamiltonian (27, 32–40) embedded within a parameterizable classical environment (ML SEQM/MM). This hierarchical structure brings two key advantages: First, it enables data-efficient learning from the ground up, and second, by targeting explicit electronic representations, it provides a physically rigorous, nonperturbative framework for ML/MM coupling that naturally captures long-range electrostatics.

From a technical perspective, our work builds on recent progress made by some of us as well as from the literature. The accurate wavefunction quantum chemistry data utilizes our implementation of differentiable local coupled cluster theory (2, 41) to generate energies and forces at gold-standard accuracy for systems with more than 40 atoms using accurate basis sets. Our DFT/MM simulations use our GPU implementation with multipole acceleration of electrostatics (42) to generate condensed phase data with the proper treatment of long-range electrostatics. Finally, our learning framework utilizes both differentiable Kohn–Sham functionals implemented in this work as well as differential semiempirical quantum models from the literature (43), combined in an ML SEQM/MM setup that incorporates pretrained equivariant models for feature extraction (44).

We demonstrate our approach in the context of two challenging motivating applications. The first, the proton dissociation of weak amino acids [lysine (Lys) and aspartate (Asp)], serves as a model of proton transport in the condensed phase, and features long-range charge separation and significant, nontrivial solvent reorganization. Our framework now enables us to study the free energy profile of proton dissociation with explicitly QM modeled regions with more than two hundred atoms (embedded in the classical environment). We show that this potential of mean force yields the absolute pK_a of the weak acids, independent of any experimental data, to leading accuracy. Our second, the catalysis of the Claisen rearrangement by chorismate mutase (CM), is a prototypical enzyme reaction featuring nontrivial electronic rearrangement in the presence of a complex environment. Our hierarchical setup now allows us to obtain reaction kinetics on converged potential energy surfaces with good control of the statistical thermal sampling, recovering the experimental rate constant to within chemical accuracy. Together, these demonstrate the potential of hierarchical Hamiltonian learning as a path to condensed phase simulations of free energies and kinetics based on the highest accuracy quantum chemistry data in simulations with converged statistics.

Results

We began by calculating the energies and forces for geometries obtained from enhanced sampling simulations of the reactions of interest, using a version of local natural orbital coupled cluster singles and doubles with perturbative triples, LNO-CCSD(T) (2, 45) (see SI Appendix, Data Preparation for details of the enhanced sampling and coupled cluster calculations). We used large computational basis sets (triple-zeta and quadruple-zeta) to extrapolate to the complete basis limit and carefully characterized the convergence of the local truncation error. We find that the local correlation error mainly contributes to a global shift in the energy (SI Appendix, Error Analysis). As this does not affect the (free) energy differences, the more important error to assess is that associated with nonparallelity, or the energy differences between configurations. We estimate this energy difference error from canonical CCSD(T)/CBS to be very small (∼0.2 kcal/mol or less, as detailed in SI Appendix, Error Analysis). Note that directly using such high-level quantum calculations to drive MD simulations is completely impractical for obtaining meaningful statistics and is expensive for generating a large amount of ML training data — even using our efficient implementation of LNO-CCSD(T) (2) on truncated atom clusters (containing 32 atoms for Asp/Lys and 43 atoms for CM; see SI Appendix for more details) a single energy calculation for the largest QM region took approximately 1 d on 1 CPU node. Since we aimed to use only modest levels of computation, we generated only O(10)-O(100) reference data points at this level in this work.

The next step in our approach (Fig. 1) is to distill the CCSD(T) energy surface into a coarse-grained quantum Hamiltonian, for which we adopted a Kohn–Sham Hamiltonian ansatz, specifically the ωB97X-3c density functional (DF) form, extended to contain 17 parameters (SI Appendix, DFT Parameterization and DFT/MM Data Generation). We then reparameterized the functional using the CCSD(T) energies (and no other electronic properties). This training was made feasible by a gradient-based optimization leveraging our GPU implementation of DFT (46, 47), and analytic functional derivatives with respect to the parameters, developed in this work. Choosing a DF imposes a very strong constraint on our model compared to a free-form neural network potential; thus, high data efficiency is expected in this step. Indeed, we found that 10 to 100 CCSD(T) energies were already sufficient to train a robust DF. For example, a DF trained on proton dissociation data from Asp achieved a comparable level of accuracy for Lys (training energy MAE 0.47 kcal/mol, and force MAE 1.1 kcal/mol/Å, compared to the validation errors in Table 1), and a DF trained on small atom clusters (the substrate and R90 side chain) of the CM reaction correctly predicted the energies (within 0.28 kcal/mol MAE) of much larger clusters (the substrate, R90, R7, and E78).

Fig. 1.

Model architecture. Knowledge distillation starts from high-level quantum calculations on small clusters in the gas phase, that is then distilled into a density functional theory. The DFT-based QM/MM generates new data in the condensed phase to train a machine-learned semiempirical Hamiltonian (ML-xTB; framed). In the architecture of ML-xTB, the green blocks represent the inputs with z being the element types and R being the atomic coordinates. The blue blocks represent a neural-network-based featurizer and a neural-network potential as a dispersion correction. The orange blocks represent the tight-binding parameter predictor, MM charge, and radius lookup table, and the ground state solver.

Table 1.

Model accuracy in terms of mean absolute errors in energy E (kcal/mol), QM atom forces F^QM and MM atom forces F^MM (kcal/mol/Å)

	Asp/Lys cluster		Asp/Lys QM/MM			CM cluster	CM QM/MM
	E	F QM	E	F QM	F MM	E	E	F QM	F MM
ωB97X-3c	0.73	1.7				1.9
Reparameterized DFT	0.40	0.95				0.28
Fine-tuned MACE-OFF23			2.4	0.77	25		OOM
DPRc			1.2	1.3	2.8		2.2	1.1	1.6
ML-xTB			1.0	0.9	2.4		0.95	0.74	1.6
GFN1-xTB response			3.5	2.0	3.8		2.1	1.5	1.8

The MM force errors are normalized by the number of QM atoms instead of the number of MM atoms for more informative numbers. Models developed in this work are highlighted in bold in the first column. Errors of DFT are computed from the reference gas-phase LNO-CCSD(T) data. Errors of ML-xTB/MM are computed from the reparameterized DFT QM/MM. The GFN1-xTB response energy is defined as $\mathrm{\Delta }E=E(\text{QM/MM})-E(\text{QM})$ and $\mathrm{\Delta }\mathbf{F}=\mathbf{F}(\text{QM/MM})-\mathbf{F}(\text{QM})$ and the error is obtained by comparing to the reparameterized DFT $\mathrm{\Delta }E$ and $\mathrm{\Delta }\mathbf{F}$. For the trained models, i.e. reparameterized DFT, ML-xTB, DPRc, and MACE-OFF23, the errors are measured on the validation sets. For models not trained or reparameterized (ωB97X-3c and GFN1-xTB) the errors are measured on the full data. See text for definitions of acronyms and models. The dataset sizes and system compositions are summarized in SI Appendix, Dataset Summary.

Using GPU-accelerated QM/MM DFT (42), we were then able to generate significantly more data (∼10 times more), as well as extend the length scale from the above atom clusters to larger QM atom clusters (43 atoms for Asp, 46 atoms for Lys, and 72 atoms for CM) embedded in a full condensed-phase environment: MM water, as well as the protein in the case of CM, described by a standard empirical force field (see SI Appendix for details), including full periodic electrostatics. With this ability to generate highly accurate quantum data in the condensed phase, there remains the challenge of training an ML model given the mixed-resolution nature of the QM/MM data and the large total system size (∼10,000 atoms for Asp/Lys and ∼50,000 atoms for CM). We found that the smallest one of a series of pretrained foundation FF models, MACE-OFF23(S) (44), could barely fit the Asp/Lys system into an Nvidia A100 GPU’s memory (using ∼70 GB of the total 80 GB), while the larger ones could not fit at all. Even the MACE-OFF small model could not fit into memory when used for the entire CM system (denoted OOM in Table 1). As a baseline, we fine-tuned the MACE-OFF23(S) model on the QM/MM Asp and Lys energy and forces, but this did not yield satisfactory accuracy on the MM atoms, shown by the 25 kcal/mol/Å of MM force MAE, and also reflected by the large energy MAE (Table 1).

To address these challenges, we trained an even more coarse-grained semiempirical quantum Hamiltonian from the DFT/MM energies and forces, taking a self-consistent-charge tight-binding Hamiltonian, GFN1-xTB (43, 48), as our Hamiltonian ansatz. Due to the minimal basis and the approximations for electron correlation and electrostatics in GFN1-xTB, it is generally not quantitatively accurate without reparameterization. Crucially, the GFN1-xTB model cannot respond in the same way as DFT to the MM electrostatic potential, with an energy MAE larger than 2 kcal/mol (see GFN1-xTB Response in Table 1). This means that even if we perfectly corrected the GFN1-xTB gas-phase energies and forces using an ML-FF within a Δ-learning framework as recently proposed (15, 18, 26), it would still fail to accurately describe the condensed phase. This deficiency motivates our approach, in which we trained an ML-predicted GFN1-xTB Hamiltonian to correctly respond to the MM long-range electrostatics while adding an ML-potential-based dispersion correction acting solely among QM atoms. A critical component of our approach is that the ground-state potential energy surface where the MD evolves is computed from the self-consistent-field iterations of the ML-xTB Hamiltonian, and thus the response to the MM potential is captured to infinite order, in contrast to finite-order corrections based on atomic charges, polarizabilities (14, 17, 19, 21, 22, 24, 25), and the QM electron density (20). In our architecture, we employed a pretrained equivariant graph neural network, MACE-OFF24(M) (44) as the featurizer and appended individual prediction heads to predict the xTB Hamiltonian parameters, a dispersion energy correction for the QM region. The MM charges and radii entering into the QM/MM electrostatic interaction were also trainable, in a geometry-independent manner (the MM charges and radii from the empirical force field were retained for the pure MM interactions, see SI Appendix, ML-xTB). Importantly, only the QM atoms are visible to the featurizer, and it (and the xTB parameter prediction head) must learn to modulate the response of the xTB Hamiltonian to the external MM potential, without direct knowledge of the MM coordinates. We found that this architecture successfully achieved chemical accuracy in terms of validation MAE (Table 1). As another baseline, we trained a range-corrected deep potential (DPRc) (12, 49), which does not parameterize the xTB Hamiltonian itself, but learns a force-field correction to GFN1-xTB/MM using both the QM atoms and MM atoms as input. We found this yielded larger energy and force errors than our only-QM-visible approach (Table 1), especially in the more complicated CM case.

We next consider the performance of these ML-xTB/MM models in full enhanced sampling MD simulations to compute key observables in our two target applications. For the proton dissociation of Asp and Lys in water, we used the ML-xTB/MM model in conjunction with a large QM region containing the full amino acid and 64 nearby water molecules, in total more than two hundred atoms (Fig. 2). Such a large QM region is crucial to accommodate the solvation of the excess proton in its dissociated limit, ∼5 Å away from the titratable moieties: the Asp carboxylic group and the Lys amine group. In contrast, we cannot observe asymptotically flat behavior of the proton potential of mean force (PMF) in the dissociated limit if we simulate a smaller QM region with 43 waters (SI Appendix, Fig. S2). In the PMF calculations, we employed replica-exchange umbrella sampling (51), biasing the distance of the excess proton [tracked by the center of excess charge (52)] from the titratable groups. To handle the statistical indistinguishability of QM and MM waters, we used the FIRES restraint [see SI Appendix FIRES (53)] which is exact if the QM and MM descriptions give the same potential energy surface. The efficiency of our ML-xTB/MM (400-fold faster than DFT/MM; see Fig. 3) enabled us to run nanosecond-long trajectories where each replica visited every umbrella window at least once. This is needed to sample both the syn and anti conformations of a protonated Asp (Fig. 2, Inset Left column), which are separated by a high free energy barrier in the protonated state (54). The syn↔anti transition is assisted by the exchanges between protonated and deprotonated umbrella windows, the latter of which more easily samples different proton positions relative to the carboxylic oxygen through proton Grotthuss hopping (55) among waters (Fig. 2, Inset Right column). The resulting PMFs show one single well corresponding to the protonated Asp/Lys states, highlighting their weak acid nature. To validate our results, we computed the pK_a values of the two residues by integrating the PMF over the protonated well (see SI Appendix for more details) and found excellent agreement with experimental measurements to within chemical accuracy (Table 2; 1 kcal/mol = 0.73 pH unit at 298.15 K). For context, we note that the best performing absolute pK_a prediction methods require experimental input for related compounds and/or the proton solvation free energy (56–59), and in predictions on drug-like small molecules achieve at best a similar accuracy to our result (59). Perhaps the most comparable theoretical approaches are those that use a free energy perturbation cycle that computes the acid deprotonation free energy (60–62) and obtain the absolute pK_a through an independent estimate of the proton solvation free energy (63), which carries an uncertainty >1 kcal/mol.

Fig. 2.

Potentials of mean force of proton dissociation from Asp and Lys as a function of the center of excess charge distance from their titratable moieties. Typical molecular configurations of protonated Asp (ApsH) and the deprotonated form (Asp⁻) are shown as the Insets. QM atoms are shown as opaque, while the MM atoms are shown as transparent (note only MM atoms near the QM atoms are visible). In AspH, there are syn and anti conformations corresponding to distinct proton positions (circled). In Asp⁻, the hydronium, H3O+ (its oxygen shown by a black sphere) can also take syn- and anti-like positions.

Fig. 3.

Wall time of one MD step on one A100 GPU of an Nvidia DGX100. The MACE-OFF23(S) model ran out of memory for the CM system, and the time was estimated assuming linear scaling in system size. The DFT, GFN1-xTB, and ML-xTB simulations correspond to a QM subsystem embedded in a MM environment, while the MACE-OFF23(S) model simulates the whole system with the ML-FF. The QM regions were one aspartate and 64 water molecules for the Asp in solution, and the substrate, the R90, R7, and E78 residue side chains for CM.

Table 2.

pK_a from ML-xTB/MM MD and experiments

	Theory	Expt.
Asp	$3.8\pm 0.1$ *	3.8†
Lys	$10.5\pm 0.1$ *	11.2†

^*Statistical errors from block averaging.

^†Corrected for nuclear quantum effects and ionic strength from the raw values 3.67 and 10.40 measured by potentiometric titrations (50); see SI Appendix, Experimental pK_a Processing.

For our prototype enzymatic reaction, the CM-catalyzed chorismate-to-prephenate transformation, we ran conformational flooding simulations (64) to compute the rate constant $k_{\text{cat}}$. Unlike the proton dissociation reaction, this reaction is a local chemical transformation, but involves a more complex electronic process (a concerted pericyclic reaction) and takes place in a heterogeneous and complicated environment. Due to the nontrivial electronic structure, standard density functionals (such as a range-separated hybrid DFT) cannot describe the energetics to within chemical accuracy (Table 1). The training of the functional in our framework is thus necessary for a quantitative description of the kinetics, and a proper choice of DF was crucial for even qualitative accuracy, e.g., differentiating the reactive binding pose of the substrate (42). In earlier work, we studied this reaction using GPU accelerated DFT/MM flooding MD and a ωB97X-3c functional, including one reparameterized to accurate reaction barrier energetics (42). The standard ωB97X-3c functional led to an underestimation of the rate constant by three orders of magnitude (42), but after reparameterization, these simulations (for a specific binding mode) achieved good agreement with experiment in the rate constant, but still required a large flooding level $V_f^{\text{max}}=65$ kJ/mol to enhance the barrier crossing sufficiently for practical DFT/MM simulations. Indeed, such a high flooding level is only ∼1 kcal/mol lower than the forward reaction free energy barrier (42), and thus a chemical reaction is typically observed within 20 picoseconds of simulation, but, at this level of flooding, one cannot guarantee that the core assumption of flooding based simulations, namely unperturbed dynamics around the transition state, is satisfied. As a first test, we ran flooding MD with ML-xTB/MM using the same maximum flooding level $V_f^{\text{max}}=65$ kJ/mol as our previous DFT/MM with a similarly revised ωB97X-3c functional, accumulating statistics over 11 independent flooding runs and a total of ∼200 picoseconds accumulated sampling. The resulting $k_{\text{cat}}$ agrees very well with the DFT/MM result (Table 3), while with ML-xTB/MM we achieved the same amount of sampling with a 40-fold speed-up (Fig. 3). Importantly, however, the greater efficiency of the ML-xTB/MM model enabled us to run an order of magnitude longer trajectories (∼2 nanoseconds) with a lower flooding level ($V_f^{\text{max}}=59$ kJ/mol). This meant that we could much better sample the reactant basin–a critical requirement to obtain a reliable rate constant. Using this order of magnitude increase in sampling, we found $k_{\text{cat}}$ to be well converged with respect to $V_f^{\text{max}}$ (Table 3). We can thus conclude that the agreement between the theoretical and experimental rate constant is not accidental (the main remaining unquantified uncertainty is from the size of the QM region). Indeed, using both the converged potential energy surface and converged sampling, the deviation from experiment is within the range of chemical accuracy (1 kcal/mol = 5.4 fold in rates at 300 K) as would be expected from an accurate model.

Table 3.

Catalytic rate constant of chorismate mutase from Bacillus subtilis

	Theory
	DFT	ML-xTB		Expt.
$V_f^{\text{max}}$ (kJ/mol)	65	65	59
Sampling (ns)	0.12	0.17	1.8
kcat (s−1)	1.1±0.2	1.9±0.5	1.5±0.8	16±14

Theoretical errors are statistical errors among 11 flooding runs. The sampling time was the accumulated simulation time over all 11 runs. The DFT results were extracted from previous work (42). The raw experimental value was taken from ref. 65, and corrected for nuclear quantum effects and temperature (42).

Discussion

In summary, we have proposed a hierarchical machine learning strategy that is initiated with a small amount ($O(10-100)$) of high accuracy data and then propagates this information across space and time scales to successfully simulate complex condensed-phase chemical reactions with free energies converged to ∼1 kcal/mol and/or rate constants approaching the experimental uncertainty. Crucially, rather than training ML potentials directly, our approach is based on training a hierarchy of ML quantum Hamiltonians, with a final embedding in empirical force fields. We showed that this approach benefits from physical constraints and an explicit treatment of electronic structure provided by the quantum Hamiltonian (as opposed to free-form ML potentials that lack explicit electrons), to provide a unified solution to challenges associated with data scarcity, long-range electrostatics, and the computational efficiency of learning and inference in large-scale condensed phase problems. Although we only used modest computational resources for simulation and data generation in this work, an interesting future direction is to utilize this same hierarchical framework in conjunction with active learning, for example, to augment both the high-level quantum chemistry wavefunction data and the reparameterized DFT energies and forces on ML-xTB sampled geometries. This would provide an efficient way to test and ensure convergence with respect to training data size. We anticipate this approach will be particularly valuable for even more challenging problems, such as catalysis in metalloenzymes, where the complicated electronic structure invalidates standard parameterized DFT (66), and may require expensive high-level quantum reference calculations beyond coupled-cluster theory. The developments in our work suggest that, even in such complicated chemical reactions, statistical sampling of free energies and kinetics at ambient temperature may soon be conceivable.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Data and figures have been deposited in DOI: 10.6084/m9.figshare.30402142 (67) (Predictive Free Energy Simulations Through Hierarchical Distillation of Quantum Hamiltonians).