Completing density functional theory by machine learning hidden messages from molecules

Modern computational approaches to electronic structures rely heavily on Kohn-Sham density functional theory (DFT). The accuracy of DFT hinges on the exchange-correlation (xc) energy functional, which describes electron-electron interactions beyond the classical model. However, the universal form of this functional remains elusive, necessitating the use of approximated functionals constructed heuristically. These approximations suffer from limitations in accuracy and transferability, lacking a systematic improvement approach. Machine learning (ML), capable of numerically implementing mappings from sampled datasets, offers a potential solution. While ML is often used to predict material properties directly from atomic configurations, bypassing computationally expensive electronic structure calculations, this approach often lacks transferability to materials outside the training dataset. In contrast, ML methods using electron density exhibit superior transferability due to the density's rich information on intrinsic physical principles. This paper explores applying ML to construct the xc functional within the DFT framework, targeting the xc potential (Vxc) and utilizing the Kohn-Sham (KS) equation. Previous pioneering work by Burke and coworkers focused on the universal Hohenberg-Kohn functional for orbital-free DFT. This research differs by targeting Vxc within the KS framework and leveraging a neural network (NN) to map density to Vxc, addressing the challenge of making this ML approach feasible for real materials. The strategy employed involves restricting the functional form to the (semi-)local form commonly used in existing functionals, assuming an xc-energy functional (Exc) that is subsequently differentiated to obtain the xc potential. The xc-energy density is formulated using a feed-forward NN, enabling functional derivative evaluation via back-propagation. This NN-based approach offers advantages over fully nonlocal forms in terms of transferability and data requirements.

Machine learning has rapidly gained traction in materials science, particularly for predicting material properties. Methods focusing on predicting properties directly from atomic configurations, such as those used in material informatics and atomic force field constructions, are computationally efficient but often suffer from limited transferability beyond the training dataset. Conversely, machine learning approaches employing electron density as input show enhanced transferability, stemming from the density's intrinsic connection to fundamental physical principles. Previous work has explored applying machine learning to density functional theory, notably by Burke and coworkers, who focused on constructing the universal Hohenberg-Kohn functional for orbital-free DFT. This study distinguishes itself by targeting the exchange-correlation potential within the Kohn-Sham DFT framework, using a neural network to model the mapping between density and the exchange-correlation potential.

The research employed a feed-forward neural network (NN) with multiple layers to represent the exchange-correlation energy density (Exc). The input vector (g) for the NN included local density descriptors, such as those used in LSDA (Local Spin Density Approximation), GGA (Generalized Gradient Approximation), and meta-GGA (meta-Generalized Gradient Approximation). A novel approach, termed “near region approximation” (NRA), was also introduced, incorporating a nonlocal descriptor (R) representing a weighted average of the density around a point. This nonlocal descriptor was inspired by Gunnarsson et al.'s work, demonstrating the efficiency of averaged density in describing Exc. The functional derivative, δExc/δn(r), needed to obtain the xc potential, was calculated efficiently using the back-propagation algorithm. The NN was trained using accurate data from quantum chemical calculations (Gaussian-2 for atomization energies and coupled-cluster method with single and double excitations (CCSD) for density distributions) for three reference molecules (H2O, NH3, and NO) chosen for structural diversity and inclusion of spin-polarization. The atomization energy (AE) was chosen for training instead of the total energy to leverage error cancellation. The NN parameters were optimized using a Metropolis-type Monte Carlo method, iteratively solving the Kohn-Sham equation for the reference molecules and updating the NN parameters to minimize the error between predicted and reference AE and density distributions. The performance of the trained functionals was evaluated on hundreds of molecular benchmark systems containing first- to third-row elements, and the results were compared to those obtained with various existing analytical functionals.

The machine-learned functionals demonstrated comparable or superior performance to existing functionals across various approximation levels (LSDA, GGA, meta-GGA, and NRA) for a wide range of unreferenced molecular systems and properties (atomization energies, barrier heights, ionization potentials, and total energies). Notably, the nonlocal NRA functional performed comparably to hybrid functionals, which incorporate nonlocal effects. The functionals showed remarkable transferability, achieving high accuracy even for systems and properties not included in the training dataset. This transferability was particularly evident for hydrocarbons, even though the training set didn't include carbon-containing molecules. The accuracy of the learned functionals improved systematically as the level of approximation increased (from LSDA to NRA), highlighting the effectiveness of adding more descriptors to the NN input and incorporating density distributions in the training dataset. The improvement in accuracy for total energy and ionization potential corroborated the importance of training with density distributions, as their accuracy is directly linked to density accuracy. The NN-LSDA functional significantly outperformed the standard SVWN functional, suggesting that the ML approach effectively overcomes the non-uniqueness issue in local approximations. Analysis of the bond dissociation of C2H2 and N2 showed that the NN-based functionals accurately reproduced the dissociation curves and density transformations, even for bond lengths outside the training set range, underscoring the strength of incorporating the ML functional within the Kohn-Sham equation. Examination of enhancement factors relative to LSDA highlighted the improvement of the NN-based functionals with increasing levels of approximation, indicating systematic improvement via descriptor addition and density-based training.

The findings demonstrate the potential of machine learning to systematically improve density functional theory. The remarkable transferability and accuracy of the trained functionals, even with a limited training dataset, highlight the advantages of using electron density as the primary input for the machine learning model. The systematic improvement observed with increasing levels of approximation suggests a pathway for further refinement and enhancement of DFT functionals. The success of the nonlocal NRA functional underscores the ability of this approach to incorporate nonlocal effects, a challenge for traditional DFT functional development. The results suggest that the ML approach overcomes limitations of conventional DFT approximations, particularly the non-uniqueness of the exchange-correlation potential in local approximations. The work's significance lies in providing a new framework for constructing more accurate and transferable DFT functionals, potentially addressing challenges posed by systems with complex nonlocal interactions.

This study presents a novel machine learning-based approach to construct exchange-correlation functionals for Kohn-Sham density functional theory, achieving accuracy comparable to standard functionals with a significantly smaller training dataset. The approach utilizes a neural network to model the relationship between density and energy, enabling systematic improvement by adding descriptors and leveraging density-based training. Future work could extend this method to systems challenging for existing functionals, such as those with dispersion interactions, self-interaction error, or strong correlation effects.

The current study focused on a limited set of reference molecules for training. While the results demonstrated impressive transferability, the performance might be sensitive to the choice of training molecules, especially for systems with significantly different electronic structures. Further investigation is required to explore the robustness and generalization capabilities with more diverse and extensive training datasets. The computational cost of the nonlocal NRA functional scales quadratically with system size, which might become limiting for very large systems.