Ground-state Electron Density Research Articles

Electronic structure prediction of multi-million atom systems through uncertainty quantification enabled transfer learning

The ground state electron density — obtainable using Kohn-Sham Density Functional Theory (KS-DFT) simulations — contains a wealth of material information, making its prediction via machine learning (ML) models attractive. However, the computational expense of KS-DFT scales cubically with system size which tends to stymie training data generation, making it difficult to develop quantifiably accurate ML models that are applicable across many scales and system configurations. Here, we address this fundamental challenge by employing transfer learning to leverage the multi-scale nature of the training data, while comprehensively sampling system configurations using thermalization. Our ML models are less reliant on heuristics, and being based on Bayesian neural networks, enable uncertainty quantification. We show that our models incur significantly lower data generation costs while allowing confident — and when verifiable, accurate — predictions for a wide variety of bulk systems well beyond training, including systems with defects, different alloy compositions, and at multi-million-atom scales. Moreover, such predictions can be carried out using only modest computational resources.

Cavity Quantum Electrodynamics Enables para- and ortho-Selective Electrophilic Bromination of Nitrobenzene.

Coupling molecules to a quantized radiation field inside an optical cavity has shown great promise to modify chemical reactivity. In this work, we show that the ground-state selectivity of the electrophilic bromination of nitrobenzene can be fundamentally changed by strongly coupling the reaction to the cavity, generating ortho- or para-substituted products instead of the meta product. Importantly, these are products that are not obtained from the same reaction outside the cavity. A recently developed ab initio approach was used to theoretically compute the relative energies of the cationic Wheland intermediates, which indicate the kinetically preferred bromination site for all products. Performing an analysis of the ground-state electron density for the Wheland intermediates inside and outside the cavity, we demonstrate how strong coupling induces reorganization of the molecular charge distribution, which in turn leads to different bromination sites directly dependent on the cavity conditions. Overall, the results presented here can be used to understand cavity induced changes to ground-state chemical reactivity from a mechanistic perspective as well as to directly connect frontier theoretical simulations to state-of-the-art, but realistic, experimental cavity conditions.

Earth Mover's Charge Transfer Distance: A General and Robust Approach for Describing Excited State Locality.

A novel approach for assessing the extent of electron displacement in optical transitions is proposed by implementing the Earth Mover's Distance (EMD) method, which quantifies the spatial dissimilarity between ground and excited state electron density distributions. In contrast to previous descriptors, this index provides a representative and intuitively understandable distance under a robust and computationally efficient scheme for all possible forms of locality, even in the most difficult to dissect topological cases. The theoretical differences among the existing indices and our method are first illustrated with the help of a simplified model system, followed by a benchmarking of several partial atomic charge models using experimentally relevant push-pull compounds with diverse symmetries. These same molecules are finally employed to further demonstrate the principal advantages of the EMD index and its capabilities in rationalizing charge transfer phenomena.

Assessing the source of error in the Thomas-Fermi-von Weizsäcker density functional.

We investigate the source of error in the Thomas-Fermi-von Weizsäcker (TFW) density functional relative to Kohn-Sham density functional theory (DFT). In particular, through numerical studies on a range of materials, for a variety of crystal structures subject to strain and atomic displacements, we find that while the ground state electron density in TFW orbital-free DFT is close to the Kohn-Sham density, the corresponding energy deviates significantly from the Kohn-Sham value. We show that these differences are a consequence of the poor representation of the linear response within the TFW approximation for the electronic kinetic energy, confirming conjectures in the literature. In so doing, we find that the energy computed from a non-self-consistent Kohn-Sham calculation using the TFW electronic ground state density is in very good agreement with that obtained from the fully self-consistent Kohn-Sham solution.

First-Principles Investigation of Electronic and Related Properties of Cubic Magnesium Silicide (Mg2Si)

We present results from ab initio, self-consistent calculations of electronic, transport, and bulk properties of cubic magnesium silicide (Mg2Si). We employed a local density approximation (LDA) potential to perform the computation, following the Bagayoko, Zhao, and Williams (BZW) method, as improved by Ekuma and Franklin (BZW-EF). The BZW-EF method guarantees the attainment of the ground state as well as the avoidance of over-complete basis sets. The ground state electronic energies, total and partial densities of states, effective masses, and the bulk modulus are investigated. As per the calculated band structures, cubic Mg2Si has an indirect band gap of 0.896 eV, from Γ to X, for the room temperature experimental lattice constant of 6.338 Å. This is in reasonable agreement with the experimental value of 0.8 eV, unlike previous ab initio DFT results of 0.5 eV or less. The predicted zero temperature band gap of 0.965 eV, from Γ to X, is obtained for the computationally determined equilibrium lattice constant of 6.218 Å. The calculated value of the bulk modulus of Mg2Si is 58.58 GPa, in excellent agreement with the experimental value of 57.03 ± 2 GPa.

SnS2-covalent organic framework Z-scheme van der Waals heterojunction for enhanced photocatalytic reduction of uranium (VI) in rare earth tailings wastewater

Uranium removal by photocatalytic reduction is one of the most promising methods to reduce radioactive contamination in wastewater. Herein, a Z-scheme van der Waals heterojunction photocatalyst (SnS2COF) was synthesized in situ by combining covalent organic frameworks (COF) with semiconductor (SnS2) for U (VI) reduction in rare earth tailings wastewater. The synthesis method of van der Waals heterojunction is simple and solves the problem of no hanging bond in composite components. In this heterojunction, large areas of van der Waals interaction form high-speed electron transport channels. In addition, it is deduced that SnS2COF fits the Z-scheme heterojunction electron transport mode through the theoretical calculation of the ground state and excited state electron density difference and the related band structure. Under the photoexcitation, the direction of electron flow is reversed, which further promotes the separation of the photogenerated electron (e−)-hole (h+) under the action of the built-in electric field, maintains the high reducibility of the conduction band, and avoids the photocorrosion of SnS2. Compared with inorganic-inorganic heterojunction, SnS2COF has a wider light absorption range, more active sites, and higher e−-h+ separation and transfer efficiency. Therefore, it had a higher U (VI) reduction removal capacity, up to 1123.3 mg g−1, far surpassing the SnS2 and COF counterparts under ultraviolet/visible light. And the U (VI) removal rate reached 98.5 % in rare earth tailings wastewater. The design concept of organic–inorganic heterojunction materials provides an alternative strategy for improving the photocatalytic performance.

The Predictive Power of Exact Constraints and Appropriate Norms in Density Functional Theory.

Ground-state Kohn-Sham density functional theory provides, in principle, the exact ground-state energy and electronic spin densities of real interacting electrons in a static external potential. In practice, the exact density functional for the exchange-correlation (xc) energy must be approximated in a computationally efficient way. About 20 mathematical properties of the exact xc functional are known. In this work, we review and discuss these known constraints on the xc energy and hole. By analyzing a sequence of increasingly sophisticated density functional approximations (DFAs), we argue that (a) the satisfaction of more exact constraints and appropriate norms makes a functional more predictive over the immense space of many-electron systems and (b) fitting to bonded systems yields an interpolative DFA that may not extrapolate well to systems unlike those in the fitting set. We discuss both how the class of well-described systems has grown along with constraint satisfaction and the possibilities for future functional development.

Composite basis set of plane wave and Gaussian function or spline function

By combining plane waves with Gaussian or spline functions, a new composite basis set is constructed in this work. As a non local basis vector, the plane wave basis group needs a large number of plane waves to expand all parts of the physical space, including the intermediate regions that are not important to our problems. Our basis set uses the local characteristics of Gaussian function or spline function at the same time, and controls the energy interval by selecting different plane wave vectors, in order to realize the partition solution of Hamiltonian matrix. Orthogonal normalization of composite basis sets is performed by using Gram-Schmidt’s orthogonalization method or Löwdin’s orthogonalization method. Considering the completeness of plane wave vector, a certain value of positive and negative should be selected at the same time. Here, by changing the absolute value of wave vector, we can select the eigenvalue interval to be solved. The plane wave with a specific wave vector value is equivalent to a trial solution in the region with gentle potential energy. The algorithm automatically combines local Gaussian or spline functions to match the difference in wave vector value between the trial solution and the strict solution. By selecting the absolute value of the wave vector in the plane wave function, the calculation of large Hamiltonian matrices turns into the calculation of multiple small matrices, together with reducing the basis numbers in the region where the electron potential changes smoothly, therefore, we can significantly reduce the computational time. As an example, we apply this basis set to a one-dimensional finite depth potential well. It can be found that our method significantly reduce the number of basis vectors used to expand the wave function while maintaining a suitable degree of computational accuracy. We also study the influence of different parameters on calculation accuracy. Finally, the above calculation method can be directly applied to the density functional theory (DFT) calculation of plasmons in silver nanoplates or other metal nanostructures. Given a reasonable tentative initial state, the ground state electron density distribution of the system can be solved by self-consistent solution through using DFT theory, and then the electromagnetic field distribution and optical properties of the system can be solved by using time-dependent density functional theory.

A conceptual density functional theory approach to substituent effects in fluorescence processes: The case of naphthalimide derivatives

AbstractThe theoretical framework of conceptual density functional theory is applied to the study of molecular excitations by considering that the differences between the ground and excited state electronic densities may be treated as if they were the consequence of intramolecular charge transfer processes. Through this approach, it is found that the hardness is directly related with the absorption and emission energies, and that the dual descriptor provides a description of the regions of the molecule where charge is added or removed, that in its condensed form leads to an estimation of the effective charge involved in the excitation. All this chemically significative information is then used to analyze the effects of the substituent in the fluorescence of naphthalimide derivatives.

Cluster Amplitudes and Their Interplay with Self-Consistency in Density Functional Methods.

Density functional theory (DFT) provides convenient electronic structure methods for the study of molecular systems and materials. Regular Kohn-Sham DFT calculations rely on unitary transformations to determine the ground-state electronic density, ground state energy, and related properties. However, for dissociation of molecular systems into open-shell fragments, due to the self-interaction error present in a large number of density functional approximations, the self-consistent procedure based on the this type of transformation gives rise to the well-known charge delocalization problem. To avoid this issue, we showed previously that the cluster operator of coupled-cluster theory can be utilized within the context of DFT to solve in an alternative and approximate fashion the ground-state self-consistent problem. This work further examines the application of the singles cluster operator to molecular ground state calculations. Two approximations are derived and explored: i) A linearized scheme of the quadratic equation used to determine the cluster amplitudes. ii) The effect of carrying the calculations in a non-self-consistent field fashion. These approaches are found to be capable of improving the energy and density of the system and are quite stable in either case. The theoretical framework discussed in this work could be used to describe, with an added flexibility, quantum systems that display challenging features and require expanded theoretical methods.

Machine learning the Hohenberg-Kohn map for molecular excited states

The Hohenberg-Kohn theorem of density-functional theory establishes the existence of a bijection between the ground-state electron density and the external potential of a many-body system. This guarantees a one-to-one map from the electron density to all observables of interest including electronic excited-state energies. Time-Dependent Density-Functional Theory (TDDFT) provides one framework to resolve this map; however, the approximations inherent in practical TDDFT calculations, together with their computational expense, motivate finding a cheaper, more direct map for electronic excitations. Here, we show that determining density and energy functionals via machine learning allows the equations of TDDFT to be bypassed. The framework we introduce is used to perform the first excited-state molecular dynamics simulations with a machine-learned functional on malonaldehyde and correctly capture the kinetics of its excited-state intramolecular proton transfer, allowing insight into how mechanical constraints can be used to control the proton transfer reaction in this molecule. This development opens the door to using machine-learned functionals for highly efficient excited-state dynamics simulations.

Hydrates of active pharmaceutical ingredients: A 35Cl and 2H solid-state NMR and DFT study

This study uses 35Cl and 2H solid-state NMR (SSNMR) spectroscopy and dispersion-corrected plane-wave density functional theory (DFT) calculations to characterize the molecular-level structures and dynamics of hydrates of active pharmaceutical ingredients (APIs). We use 35Cl SSNMR to measure the EFG tensors of the chloride ions to characterize hydrated forms of hydrochloride salts of APIs, along with two corresponding anhydrous forms. DFT calculations are used to refine the crystal structures of the APIs and determine relationships between the 35Cl EFG tensors and the spatial arrangements of proximate hydrogen bonds, which are particularly influenced by interactions with water molecules. We find that the relationship between 35Cl EFG tensors and local hydrogen bonding geometries is complex, but meaningful structure/property relationships can be garnered through use of DFT calculations. Specifically, for every case in which such a comparison could be made, we find that the hydrate has a smaller magnitude of CQ than the corresponding anhydrous form, indicating a chloride ion environment with a ground-state electron density of higher spherical symmetry in the former. Finally, variable-temperature 35Cl and 2H SSNMR experiments on a deuterium-exchanged sample of the API cimetidine hydrochloride monohydrate are used to monitor temperature-dependent influences on the spectra that may arise from motional influences on the 35Cl and 2H EFG tensors. From the 2H SSNMR spectra, we determine that the motions of water molecules are characterized by jump-like motions about their C2 rotational axes that occur on timescales that are unlikely to influence the 35Cl central-transition (+1/2 ↔︎ −1/2) powder patterns (this is confirmed by 35Cl SSNMR). Together, these methods show great promise for the future study of APIs in their bulk and dosage forms, especially variable hydrates in which crystallographic water content varies with external conditions such as humidity.

Moment functional based spectral density functional theory

We describe a density-functional method which aims at computing the ground state electron density and the spectral function at the same time. One basic ingredient of our method is the construction of the spectral function from the first four spectral moment matrices. The second basic ingredient is the construction of the spectral moment matrices from density functionals. We call our method moment-functional based spectral density-functional theory (MFbSDFT), because it is based on density-functionals for the spectral moments and because it allows us to compute the spectral function. If it is implemented in second variation our method consumes only a fraction more computer time than a standard DFT calculation with the PBE functional. We show that MFbSDFT captures correlation effects such as the valence-band satellite in Ni and the formation of lower and upper Hubbard bands in SrVO$_3$. For the purpose of constructing the spectral function from the first four $N\times N$ spectral moment matrices we describe an efficient algorithm based on the diagonalization of one hermitean $2N\times 2N$ matrix.

Efficient time-dependent orbital-free density functional theory: Semilocal adiabatic response

Orbital-free density functional theory and its time-dependent extension are efficient ab initio methods for calculating the electronic structure and dynamics of large systems. Through the calculation of the optical spectra of selected clusters, we reach three important conclusions: (1) The quality of the spectra is strongly affected by the quality of the corresponding ground-state electron density; (2) the adiabatic part of the electronic response to external perturbations can be safely evaluated at the semilocal level; and (3) the nonadiabatic, current-dependent part of the time-dependent Pauli potential is key to recover correct spectral envelopes.

Quantifying energetic information in density functional theory.

Energy and information are two fundamental concepts in physics and chemistry. In density functional theory (DFT), all information pertaining to stability, reactivity, and other properties is encompassed in the ground state electron density. The basic theorems of DFT govern that energy is a universal functional of the density, and thus, it can be regarded as a special kind of information. In this work, we quantify the energetic information in terms of Shannon entropy and Fisher information for energetic distributions of atoms and molecules. Two identities are unveiled for an energetic density, its gradient, and Laplacian to rigorously satisfy. A new partition scheme to decompose atoms in molecules has been proposed using the energetic distribution. We also show that our approach can simultaneously quantify both two-body and many-body interactions. This new framework should provide new analytical tools for us to appreciate electronic properties of molecular systems, including stability and reactivity. More importantly, this work establishes the missing link in DFT between energy and information, the two most fundamental quantities in quantum theory.

Reconstruction of Exchange-Correlation Potentials from Their Matrix Representations.

Within a basis set of one-electron functions that form linearly independent products (LIPs), it is always possible to construct a unique local (multiplicative) real-space potential that is precisely equivalent to an arbitrary given operator. Although standard basis sets of quantum chemistry rarely form LIPs in a numerical sense, occupied and low-lying virtual canonical Kohn-Sham orbitals often do so, at least for small atoms and molecules. Using these principles, we construct atomic and molecular exchange-correlation potentials from their matrix representations in LIP basis sets of occupied canonical Kohn-Sham orbitals. The reconstructions are found to imitate the original potentials in a consistent but exaggerated way. Since the original and reconstructed potentials produce the same ground-state electron density and energy within the associated LIP basis set, the procedure may be regarded as a rigorous solution to the Kohn-Sham inversion problem within the subspace spanned by the occupied Kohn-Sham orbitals.

Orbital-free methods for plasmonics: Linear response.

Plasmonic systems, such as metal nanoparticles, are widely used in different areas of application, going from biology to photovoltaics. The modeling of the optical response of such systems is of fundamental importance to analyze their behavior and to design new systems with required properties. When the characteristic sizes/distances reach a few nanometers, nonlocal and spill-out effects become relevant and conventional classical electrodynamics models are no more appropriate. Methods based on the Time-Dependent Density Functional Theory (TD-DFT) represent the current reference for the description of quantum effects. However, TD-DFT is based on knowledge of all occupied orbitals, whose calculation is computationally prohibitive to model large plasmonic systems of interest for applications. On the other hand, methods based on the orbital-free (OF) formulation of TD-DFT can scale linearly with the system size. In this Review, OF methods ranging from semiclassical models to the Quantum Hydrodynamic Theory will be derived from the linear response TD-DFT, so that the key approximations and properties of each method can be clearly highlighted. The accuracy of the various approximations will then be validated for the linear optical properties of jellium nanoparticles, the most relevant model system in plasmonics. OF methods can describe the collective excitations in plasmonic systems with great accuracy and without system-tuned parameters. The accuracy of these methods depends only on the accuracy of the (universal) kinetic energy functional of the ground-state electronic density. Current approximations and future development directions will also be indicated.

Noninteracting v-Representable Subspaces of Orbitals in the Kohn-Sham Method.

The notion of noninteracting v-representability is extended from electron densities to finite-dimensional linear subspaces of orbitals. Unlike electron densities, orbital subspaces can be tested for noninteracting v-representability using a transparent necessary condition: the subspace must be invariant under the action of some one-electron Kohn-Sham Hamiltonian. This condition allows one to determine in principle, and sometimes in practice, whether a given one-electron basis set can represent an N-electron density within the Kohn-Sham method and to find the corresponding Kohn-Sham effective potential v if it exists. If the occupied Kohn-Sham orbitals form linearly independent products, then their subspace is determined by the corresponding ground-state electron density. This means that the Kohn-Sham effective potential corresponding to certain finite-basis-set electron densities can be deduced from the basis set itself.

Machine learning based prediction of the electronic structure of quasi-one-dimensional materials under strain

We present a machine learning based model that can predict the electronic structure of quasi-one-dimensional materials while they are subjected to deformation modes such as torsion and extension/compression. The technique described here applies to important classes of materials such as nanotubes, nanoribbons, nanowires, miscellaneous chiral structures and nano-assemblies, for all of which, tuning the interplay of mechanical deformations and electronic fields is an active area of investigation in the literature. Our model incorporates global structural symmetries and atomic relaxation effects, benefits from the use of helical coordinates to specify the electronic fields, and makes use of a specialized data generation process that solves the symmetry-adapted equations of Kohn-Sham Density Functional Theory in these coordinates. Using armchair single wall carbon nanotubes as a prototypical example, we demonstrate the use of the model to predict the fields associated with the ground state electron density and the nuclear pseudocharges, when three parameters - namely, the radius of the nanotube, its axial stretch, and the twist per unit length - are specified as inputs. Other electronic properties of interest, including the ground state electronic free energy, can then be evaluated with low-overhead post-processing, typically to chemical accuracy. We also show how the nuclear coordinates can be reliably determined from the pseudocharge field using a clustering based technique. Remarkably, only about 120 data points are found to be enough to predict the three dimensional electronic fields accurately, which we ascribe to the symmetry in the problem setup, the use of low-discrepancy sequences for sampling, and presence of intrinsic low-dimensional features in the electronic fields. We comment on the interpretability of our machine learning model and discuss its possible future applications.

Comment on “Electron Density Analysis for the H2+System Confined by Hard Walls: The Chemical Bond under Extreme Conditions” [Ann. Phys. (Berlin) 2019,531, 1800476

AbstractThe numerical results of Hernandez‐Esparza et al. demonstrate that the maximum electron density of the H2+molecule confined by hard walls is sometimes displaced away from the nuclei of the molecule. It is here shown analytically that hard confinement can significantly change the ground state electron density. For the free system, the electronic energyE(without the internuclear repulsion term) is strongly negative for all internuclear distancesR, while for a system under tight confinement, a critical point of QTAIM type (3,−3) occurs whenE> 0.