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- Density functional theory of atoms and molecules
- Density Functional Theory of Atoms and Molecules
- Density-Functional Theory of Atoms and Molecules
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Density functional theory of atoms and molecules
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Their accuracy heavily relies on the exchange-correlation energy functional, which encapsulates electron—electron interaction beyond the classical model. As its universal form remains undiscovered, approximated functionals constructed with heuristic approaches are used for practical studies.
However, there are problems in their accuracy and transferability, while any systematic approach to improve them is yet obscure. In this study, we demonstrate that the functional can be systematically constructed using accurate density distributions and energies in reference molecules via machine learning. Surprisingly, a trial functional machine learned from only a few molecules is already applicable to hundreds of molecules comprising various first- and second-row elements with the same accuracy as the standard functionals.
This is achieved by relating density and energy using a flexible feed-forward neural network, which allows us to take a functional derivative via the back-propagation algorithm. In addition, simply by introducing a nonlocal density descriptor, the nonlocal effect is included to improve accuracy, which has hitherto been impractical. Our approach thus will help enrich the DFT framework by utilizing the rapidly advancing machine-learning technique.
Machine learning ML is a method to numerically implement any mapping, relationship, or function that is difficult to formulate theoretically, only from a sampled dataset. In the past decade, it has rapidly been proven to be effective for many practical problems. In studies on materials, the ML scheme is often applied to predict material properties from basic information, such as atomic configurations, by bypassing the heavy calculation required by electronic structure theory, as is done in the material informatics or the construction of atomic forcefields 1 , 2.
However, the trained ML model thus obtained often fails to be applicable for materials whose structures or component elements are not included in the training dataset. Meanwhile, ML schemes treating electron density are shown to have large transferability even with a limited training dataset 3 , 4 , 5. This transferability originates from the fact that the spatial distribution of the density has more information about the intrinsic physical principles than the scalar quantities such as energy.
Thus, various physical or chemical properties are expected to be predicted more accurately by considering electron density than by directly predicting them from atomic positions.
Furthermore, ML has also been applied to more fundamental physical concepts: density functional theory DFT. In this theory, the solution of the KS equation. However, its explicit form remains undiscovered. Some modern functionals were criticized for being biased toward the energy accuracy than density accuracy 12 , despite the fact that both are important.
The functionals have been formulated so that the physical conditions, such as asymptotic behaviors and scaling properties, are satisfied, but their detailed forms rely on human heuristics, especially in the intermediate regime where the asymptotes do not apply.
On the other hand, there are many accurate densities and energies available, thanks to the theoretical and experimental development, which should help us to augment the functionals toward the ideal unbiased and transferable form. In this paper, we demonstrate the development of V xc utilizing such accurate reference data with the ML. The pioneering studies on ML application of the density functionals have been conducted by Burke and coworkers 13 , 14 , 15 , where the universal Hohenberg—Kohn functional F HK [ n ] as a sum of the kinetic energy T [ n ] and the interaction energy functionals V ee [ n ] was constructed for orbital-free DFT, whose framework avoids the heavy calculation to solve the KS equation.
Our approach contrasts to theirs, as we target V xc and adopt the KS framework. Therein, the neural network NN form was adopted because of its ability to represent any well-behaved functions with arbitrary accuracy 17 , We have found that, when applied to V ion not referenced in the training, the explicit treatment of the kinetic energy suppresses the effect from spurious oscillation in the predicted V xc , and it reduces the error of finally obtained n r.
This result suggests that the machine-learning approach to V xc with the KS equation is a promising route. The challenge is then to make the ML of V xc feasible for real materials.
Our strategy is to restrict the functional form to the semi- local one, as adopted in most existing functionals for KS-DFT. The xc energy density is constructed using the fully-connected NN, which takes the spatially local descriptors. This semi- local NN form has practical advantages compared with the fully nonlocal form, which is adopted in the previous studies.
As we demonstrate later, those features enable us to construct functionals whose performance is comparable to standard functionals by training with data of only a few of molecules. Gunnarsson et al. Construction in such a nonlocal form has been uncommon, except for the van der Waals systems, because of the absence of appropriate physical conditions.
To test the performance of this approach, we constructed a functional using a few molecules to train the NN. We selected three molecules according to the following criteria: i the structures of the molecules should be distinct from each other and have low symmetry. Note that the NO radical is spin-polarized. The functionals are trained to reproduce the atomization energy AE and the density distribution DD of them. We generated the training data using accurate quantum chemical calculations, i.
See Table 1. The larger error implies the difficulty of reproducing TE within the semi- local approximations, whereas the relative energy such as the AE can be predicted more accurately due to the error cancellation. It is also worth emphasizing that DD contains abundant information of the electronic structure all over the three-dimensional space, which is expected to contribute to determining a large number of NN parameters. We selected the above conditions simply for demonstration, though how the accuracy depends on the choice of the training dataset remains a target for future studies.
At each step, the KS equation was self-consistently solved for the three molecules to obtain their densities and total energies. Subsequently, errors from the reference values of the AE and DD were evaluated for the update of parameters. This procedure was repeated until the error was minimized. Using the trained NN-based functionals, we calculated the AE, DD, barrier heights BH of chemical reactions, ionization potentials IP , and TE of the hundreds of molecular benchmark systems 32 , 33 , 34 , which comprise first- to third-row elements Table 1.
The performances are compared with those calculated with the existing analytic functionals. For the wide range of unreferenced molecular systems and unreferenced quantities BH, IP, and TE , the NN-based functionals exhibit comparable or superior performance to existing functionals in every approximation level.
In particular, the nonlocal NRA-type functional is comparable to the hybrid functionals, which partly contain nonlocal effects. It is also noteworthy that the NN-based functionals are comparable to ML, B3LYP, or M06, which were implemented with the parameter fitting referring to more than systems.
This remarkable transferability with the small training dataset is nontrivial in the context of conventional ML methods predicting material properties.
It reflects the advantage of our method when using electron density, which is common to any material, as the input for ML mapping. Even for unreferenced molecules, this NN-based functional would work if its local DD is similar to the one included in the reference molecules. Actually, the NN-based functional shows comparable accuracies for the AE of hydrocarbons AE HC 28 to other molecules, even though no carbon element is included in the reference molecules.
Furthermore, some hydrocarbons such as benzene and butadiene have delocalized electrons owing to their conjugated structures. This means that, as the descriptor of DD increases, the NN gains the ability to distinguish whether the electrons are localized or delocalized. The accuracy for TE and DD should be related because the Hohenberg—Kohn theorem proves their one-to-one correspondence.
This improvement for those basic quantities would increase the accuracy of all other properties. Thus, training using density is effective not only for determining a large number of NN parameters, but also for improving their accuracy. Tozer et al. These results suggest that systematic improvement of the functional is realized by adding further descriptors to g , and by training with DDs.
The panels represent accuracy for atomization energy AE , reaction barrier height BH76 , and total energy TE against accuracy for density distribution DD , corresponding to Table 1. The closed and open markers represent the accuracy of existing and the NN-based functionals, respectively.
Figure 3 also represents the improvement along with the approximation level for each benchmark molecule. This implies that their electron DD cannot be trained sufficiently with the current reference molecules. Actually, they have tetrahedrally coordinated structures, which do not appear in the reference molecules. Large parts of their DD are considered to not appear in the reference molecules, leading to inaccurate prediction of the functional value.
Each point represents the atomization energy from DFT calculation using NN-based functionals against the experimental value They agree very well, even though the NN-based functional is trained only for the molecules in equilibrium structures. This transferability for unreferenced structures is nontrivial in typical ML applications that predict the material properties directly from atomic configurations with skipping basic physical theories.
This indicates the advantage of explicitly solving the KS equation, where the kinetic energy operator mitigates nonphysical noises of ML xc potential that may appear when the ML prediction is used for unreferenced inputs, thereby enhancing the transferability of the functional out of the training dataset, as shown in ref.
For C 2 H 2 , the two C—H bonds are dissociated symmetrically along the original bond direction. The horizontal axis shows the bond length, and the vertical axis shows energy relative to the atomized limit E atomized. The densities of the single CH radical and N atom are plotted as blue lines, and the densities of the bonded molecules are plotted as red lines. They are plotted along the 1D coordinate x penetrating the centers of those molecules.
Panels c and d of Fig. The difference in density between binding and unbinding structures is well reproduced with the NN-based functional. This transferability is also nontrivial compared with conventional ML methods to predict density from nuclear coordinates, as they usually have to account for the change in environment around each nucleus in their ML models. On the other hand, as our ML method is incorporated into the KS equation, bonding can be easily reproduced, similarly to ordinary DFT studies.
In Fig. The vertical coordinates are defined using Eq. As meta-GGA-type functionals have four variables, the panels show the dependency on one of them while the others are fixed. The fixed parameters in panels a — d are set to reproduce the UEG limit, whereas the generalized gradient s is set to 0. This divergence does not affect the calculation of molecules because s is greater than 0. However, this divergence is recently found to cause ill-convergence when implemented in periodic boundary codes; therefore, it should be suppressed in future work.
Except for this divergence, the NN-meta-GGA functional behaves similarly to the other analytical functionals. This result suggests that some physical conditions can be automatically reproduced through this ML approach; this property would contribute to the development of DFT with unconventional nonlocal descriptors such as those in our NRA, for which the exact asymptotic behavior is not straightforwardly derived.
Our results suggest that improvements can be made by following a simple strategy: preparing a maximally flexible NN-based functional form and then training it with the electron DDs and energy-related properties of appropriate reference materials.
The NN-based functionals trained using only a few reference molecules exhibit comparable or superior performance to the representative standard. We have revealed that employing the semi- local form and including DDs in the training dataset contribute to this transferability, as well as the determination of a large number of NN parameters. Furthermore, this approach enables the systematic construction of a functional with minimum assumptions, as demonstrated by the NRA functional with a nonlocal variable R , which is difficult to construct using conventional methods because of the lack of physical conditions.
On the other hand, our approach of introducing nonlocality retains the classical framework of solving the KS equation with the explicit functional of density, which makes the calculation more feasible for larger systems. For those problems with complicated nonlocality, our approach seems effective as it can systematically construct a maximally flexible functional form.
Density Functional Theory of Atoms and Molecules
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Density-Functional Theory of Atoms and Molecules
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Horizons of Quantum Chemistry pp Cite as. Current studies in density functional theory and density matrix functional theory are reviewed, with special attention to the possible applications within chemistry.
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