Kohn-Sham Energy Research Articles

Exploration of near the origin and the asymptotic behaviors of the Kohn-Sham kinetic energy density for two-dimensional quantum dot systems with parabolic confinement.

The behaviors of the positive definite Kohn-Sham kinetic energy density near the origin and at the asymptotic region play a major role in designing meta-generalized gradient approximations (meta-GGAs) for exchange in low-dimensional quantum systems. It is shown that near the origin of the parabolic quantum dot, the Kohn-Sham kinetic energy differs from its von Weizsäcker counterpart due to the p orbital contributions, whereas in the asymptotic region, the difference between the above two kinetic energy densities goes as ∼ρ(r)r2. All these behaviors have been explored using the two-dimensional isotropic quantum harmonic oscillator as a test case. Several meta-GGA ingredients are then studied by making use of the above findings. Also, the asymptotic conditions for the exchange energy density and the potential at the meta-GGA level are proposed using the corresponding behaviors of the two kinetic energy densities.

An Empirical, yet Practical Way To Predict the Band Gap in Solids by Using Density Functional Band Structure Calculations

Band structure calculations based on density functional theory (DFT) with local or gradient-corrected exchange-correlation potentials are known to severely underestimate the band gap of semiconducting and insulating materials. Alternative approaches have been proposed: from semiempirical setups, such as the so-called DFT+U, to hybrid density functionals using a fraction of nonlocal Fock exchange, to modifications of semilocal density functionals. However, the resulting methods appear to be material dependent and lack theoretical rigor. The rigorous many-body perturbation theory based on GW methods provides accurate results but at a very high computational cost. Hereby, we show that a linear correlation between the electronic band gaps obtained from standard DFT and GW approaches exists for most materials and argue that (1) this is a strong indication that the problem of predicting band gaps from standard DFT calculation arises from the assignment of a physical meaning to the Kohn–Sham energy levels rather...

Redox Levels through Constant Fermi-Level ab Initio Molecular Dynamics.

We develop a method to determine redox levels of half reactions through the use of ab initio molecular dynamics evolving at constant Fermi energy. This scheme models the effect of an electrode by controlling the charge transfer between the single-particle energy levels of the system and an electron reservoir set at a given potential during the dynamics. Like the thermodynamic integration method, our scheme does not require a priori knowledge of the products of the reaction, which can simply be obtained by driving the reaction through the variation of the Fermi level. The simulations are performed subject to periodic boundary conditions in the absence of any counterelectrode. We extract the redox level from the evolution of the Kohn-Sham energies upon charging (or discharging) the system. Using Janak's theorem and assuming a quadratic evolution of the Kohn-Sham energy of the highest occupied state upon charging, we demonstrate that our scheme for the determination of redox levels is equivalent to the thermodynamic integration method. The approach is illustrated for the redox couples Fe2+/Fe3+, HO2•/HO2-, and MnO4-/MnO4-2 in aqueous solution and yields redox potentials differing by less than 0.1 eV from respective ones achieved with the thermodynamic integration method.

Frequency-dependent dielectric function of semiconductors with application to physisorption

The dielectric function is one of the most important quantities that describes the electrical and optical properties of solids. Accurate modeling of the frequency-dependent dielectric function has great significance in the study of the long-range van der Waals (vdW) interaction for solids and adsorption. In this work, we calculate the frequency-dependent dielectric functions of semiconductors and insulators using the $GW$ method with and without exciton effects, as well as efficient semilocal density functional theory (DFT), and compare these calculations with a model frequency-dependent dielectric function. We find that for semiconductors with moderate band gaps, the model dielectric functions, $GW$ values, and DFT calculations all agree well with each other. However, for insulators with strong exciton effects, the model dielectric functions have a better agreement with accurate $GW$ values than the DFT calculations, particularly in high-frequency region. To understand this, we repeat the DFT calculations with scissors correction, by shifting DFT Kohn-Sham energy gap to match the experimental band gap. We find that scissors correction only moderately improves the DFT dielectric function in low-frequency region. Based on the dielectric functions calculated with different methods, we make a comparative study by applying these dielectric functions to calculate the vdW coefficients ($C_3$ and $C_5$) for adsorption of rare-gas atoms on a variety of surfaces. We find that the vdW coefficients obtained with the nearly-free electron gas-based model dielectric function agree quite well with those obtained from the $GW$ dielectric function, in particular for adsorption on semiconductors, leading to an overall error of less than 7% for $C_3$ and 5% for $C_5$. This demonstrates the reliability of the model dielectric function for the study of physisorption.

Atmospheric implication of the hydrogen bonding interaction in hydrated clusters of HONO and dimethylamine in the nighttime.

In this study, the stability of clusters formed by the trans- and cis-isomers of nitrous acid (HONO) with dimethylamine (DMA) and water has been characterized by density functional theory. The large red shifts of the OH-stretching transitions of both HONO isomers in the clusters indicate the formation of strong hydrogen bonds. At standard temperature and pressure, H2O (acceptor) binds to HONO (donor) with binding energies of -25.0 to -24.6 kJ mol-1, less stable than those of DMA (acceptor) with HONO (donor) (-50.5 to -45.3 kJ mol-1). Our findings indicate that hydration enhances proton transfer from HONO to DMA, and consequently increases the interaction strength (binding energies = -67.8 to -78.6 kJ mol-1). The topological and generalized Kohn-Sham energy decomposition confirms strong hydrogen bond interactions. The clustering of HONO with DMA in the atmosphere is negligible as compared to the important H2SO4-DMA clusters.

Dissection of H-bonding interactions in a glycolic acid-water dimer.

The binding strength and collective effects of multiple H-bonds in the glycolic acid-water dimer were studied in comparison to the aromatic analog, 9-hydroxy-9-fluorene carboxylic acid (9HFCA). Quantitative analysis by the generalized Kohn-Sham energy decomposition analysis shows that the energy difference in each specific physical interaction, from a glycolic acid-water dimer to a 9HFCA-water dimer, is small and amounts to less than 5% of the binding energy of the 9HFCA-water dimer. Extensive comparison of further, similar H-bonded complexes with widely varying binding strengths reinforces their excellent analogy in that the fluorene group acts as a non-interfering spectator for intermolecular H-bonding interactions. With reference to the spectroscopic measurement on the 9HFCA-water dimer (8.51 ± 0.09 kcal mol-1), the binding energy of the glycolic acid-water dimer is estimated to be 8.51 ± 0.31 kcal mol-1, a much better accuracy than previous reports. Furthermore, correlating the infrared spectra of 9HFCA H-bonded complexes provides a circumstantial probing of the existence and consequences of cooperative and anti-cooperative behaviors in the glycolic acid-water dimer. Our studies point to the interesting H-bonding phenomena in the glycolic acid-water dimer, which may inspire challenging experiments in future.

GW-BSE approach on S1 vertical transition energy of large charge transfer compounds: A performance assessment.

In this work, we apply many-body perturbation theory (MBPT) on large critical charge transfer (CT) complexes to assess its performance on the S1 excitation energy. Since the S1 energy of CT compounds is heavily dependent on the Hartree-Fock (HF) exchange fraction in the reference density functional, MBPT opens a new way for reliable prediction of CT S1 energy without explicit knowledge of suitable amount of HF-exchange, in contrary to the time-dependent density functional theory (TD-DFT), where depending on various functionals, large errors can arise. Thus, simply by starting from a (semi-)local reference functional and performing update of Kohn-Sham (KS) energies in the Green's function G while keeping dynamical screened interaction (W(ω)) frozen to the mean-field level, we obtain impressingly highly accurate S1 energy at slightly higher computational cost in comparison to TD-DFT. However, this energy-only updating mechanism in G fails to work if the initial guess contains a fraction or 100% HF-exchange, and hence considerably inaccurate S1 energy is predicted. Furthermore, eigenvalue updating both in G and W(ω) overshoots the S1 energy due to enhanced underscreening of W(ω), independent of the (hybrid-)DFT starting orbitals. A full energy-update on top of HF orbitals even further overestimates the S1 energy. An additional update of KS wave functions within the Quasi-Particle Self-Consistent GW (QSGW) deteriorates results, in stark contrast to the good results obtained from QSGW for periodic systems. For the sake of transferability, we further present data of small critical non-charge transfer systems, confirming the outcomes of the CT-systems.

Communication: Physical origins of ionization potential shifts in mixed carboxylic acids and water complexes.

The ionization potential (IP) of the aromatic alpha hydroxy carboxylic acid, 9-hydroxy-9-fluorene carboxylic acid (9HFCA), is shifted by complexation with hydrogen bonding ligands such as water and formic acid. Generalized Kohn-Sham energy decomposition analysis decomposes the intermolecular binding energies into a frozen energy term, polarization, correlation, and/or dispersion energy terms, as well as terms of geometric relaxation and zero point energy. We observe that in each dimer the attractive polarization always increases upon ionization, enhancing binding in the cation and shifting the IP toward the red. For 9HFCA-H2O, a substantial decrease of the repulsive frozen energy in cation further shifts the IP toward red. For 9HFCA-HCOOH, the increase of the frozen energy actually occurs in the cation and shifts the IP toward blue. Consistent with the experimental measurements, our analysis provides new, non-intuitive perspectives on multiple hydrogen bonds interactions in carboxylic acids and water complexes.

Van der Waals molecular interactions in the organic functionalization of graphane, silicane, and germanane with alkene and alkyne molecules: a DFT-D2 study.

Density functional theory with the addition of a semi-empirical dispersion potential was applied to the conventional Kohn-Sham energy to study the adsorption of alkene and alkyne molecules on hydrogen-terminated two-dimensional group IV systems (graphane, silicane, and germanane) by means of a radical-initiated reaction. In particular, we investigated the interactions of acetylene, ethylene, and styrene with those surfaces. Although we had studied these systems previously, we included van der Waals interactions in all of the cases examined in the present work. These forces, which are noncovalent interactions, can heavily influence different processes in molecular chemistry, such as the adsorption of organic molecules on semiconductor surfaces. This unified approach allowed us to perform a comparative study of the relative reactivities of the various organic molecule/surface systems. The results showed that the degree of covalency of the surface, the lattice size, and the partial charge distribution (caused by differences in electronegativity) are all key elements that determine the reactivity between the molecules and the surfaces tested in this work. The covalent nature of graphane gives rise to energetically favorable intermediate states, while the opposite polarities of the charge distributions of silicane and germanane with the organic molecules favor subsequent steps of the radical-initiated reaction. Finally, the lattice size is a factor that has important consequences due to steric effects present in the systems and the possibility of chain reaction continuation. The results obtained in this work show that careful selection of the substrate is very important. Calculated energy barriers, heats of adsorption, and optimized atomic structures show that the silicane system offers the best reactivity in organic functionalization.

On the errors of local density (LDA) and generalized gradient (GGA) approximations to the Kohn-Sham potential and orbital energies.

In spite of the high quality of exchange-correlation energies Exc obtained with the generalized gradient approximations (GGAs) of density functional theory, their xc potentials vxc are strongly deficient, yielding upshifts of ca. 5 eV in the orbital energy spectrum (in the order of 50% of high-lying valence orbital energies). The GGAs share this deficiency with the local density approximation (LDA). We argue that this error is not caused by the incorrect long-range asymptotics of vxc or by self-interaction error. It arises from incorrect density dependencies of LDA and GGA exchange functionals leading to incorrect (too repulsive) functional derivatives (i.e., response parts of the potentials). The vxc potential is partitioned into the potential of the xc hole vxchole (twice the xc energy density ϵxc), which determines Exc, and the response potential vresp, which does not contribute to Exc explicitly. The substantial upshift of LDA/GGA orbital energies is due to a too repulsive LDA exchange response potential vxresp (LDA) in the bulk region. Retaining the LDA exchange hole potential plus the B88 gradient correction to it but replacing the response parts of these potentials by the model orbital-dependent response potential vxresp (GLLB) of Gritsenko et al. [Phys. Rev. A 51, 1944 (1995)], which has the proper step-wise form, improves the orbital energies by more than an order of magnitude. Examples are given for the prototype molecules: dihydrogen, dinitrogen, carbon monoxide, ethylene, formaldehyde, and formic acid.

Visualization and analysis of the Kohn-Sham kinetic energy density and its orbital-free description in molecules.

We visualize the Kohn-Sham kinetic energy density (KED) and the ingredients--the electron density, its gradient, and Laplacian--used to construct orbital-free models of it, for the AE6 test set of molecules. These are compared to related quantities used in metaGGA's, to characterize two important limits--the gradient expansion and the localized-electron limit typified by the covalent bond. We find the second-order gradient expansion of the KED to be a surprisingly successful predictor of the exact KED, particularly at low densities where this approximation fails for exchange. This contradicts the conjointness conjecture that the optimal enhancement factors for orbital-free kinetic and exchange energy functionals are closely similar in form. In addition we find significant problems with a recent metaGGA-level orbital-free KED, especially for regions of strong electron localization. We define an orbital-free description of electron localization and a revised metaGGA that improves upon atomization energies significantly.

First principles many-body calculations of electronic structure and optical properties of SiC nanoribbons

A first principles many-body approach is employed to calculate the band structure and optical response of nanometer-sized ribbons of SiC. Many-body effects are incorporated using the GW approximation, and excitonic effects are included using the Bethe–Salpeter equation. Both unpassivated and hydrogen-passivated armchair SiC nanoribbons are studied. As a consequence of low dimensionality, large quasiparticle corrections are seen to the Kohn–Sham energy gaps. In both cases quasiparticle band gaps are increased by up to 2 eV, as compared to their Kohn–Sham energy values. Inclusion of electron–hole interactions modifies the absorption spectra significantly, giving rise to strongly bound excitonic peaks in these systems. The results suggest that hydrogen passivated armchair SiC nanoribbons have the potential to be used in optoelectronic devices operating in the UV-Vis region of the spectrum. We also compute the formation energies of these nanoribbons as a function of their widths, and conclude that hydrogen-saturated ribbons will be much more stable as compared to bare ones.

Kinetic Energy of Hydrocarbons as a Function of Electron Density and Convolutional Neural Networks.

We demonstrate a convolutional neural network trained to reproduce the Kohn-Sham kinetic energy of hydrocarbons from an input electron density. The output of the network is used as a nonlocal correction to conventional local and semilocal kinetic functionals. We show that this approximation qualitatively reproduces Kohn-Sham potential energy surfaces when used with conventional exchange correlation functionals. The density which minimizes the total energy given by the functional is examined in detail. We identify several avenues to improve on this exploratory work, by reducing numerical noise and changing the structure of our functional. Finally we examine the features in the density learned by the neural network to anticipate the prospects of generalizing these models.

The nature of hydrogen-bonding interaction in the prototypic hybrid halide perovskite, tetragonal CH3NH3PbI3.

In spite of the key role of hydrogen bonding in the structural stabilization of the prototypic hybrid halide perovskite, CH3NH3PbI3 (MAPbI3), little progress has been made in our in-depth understanding of the hydrogen-bonding interaction between the MA+-ion and the iodide ions in the PbI6-octahedron network. Herein, we show that there exist two distinct types of the hydrogen-bonding interaction, naming α- and β-modes, in the tetragonal MAPbI3 on the basis of symmetry argument and density-functional theory calculations. The computed Kohn-Sham (K-S) energy difference between these two interaction modes is 45.14 meV per MA-site with the α-interaction mode being responsible for the stable hydrogen-bonding network. The computed bandgap (Eg) is also affected by the hydrogen-bonding mode, with Eg of the α-interaction mode (1.73 eV) being significantly narrower than that of the β-interaction mode (2.03 eV). We have further estimated the individual bonding strength for the ten relevant hydrogen bonds having a bond critical point.

Source of Cooperativity in Halogen‐Bonded Haloamine Tetramers

Inspired by the isostructural motif in α-bromoacetophenone oxime crystals, we investigated halogen-halogen bonding in haloamine quartets. Our Kohn-Sham molecular orbital and energy decomposition analysis reveal a synergy that can be traced to a charge-transfer interaction in the halogen-bonded tetramers. The halogen lone-pair orbital on one monomer donates electrons into the unoccupied σ*N-X orbital on the perpendicular N-X bond of the neighboring monomer. This interaction has local σ symmetry. Interestingly, we discovered a second, somewhat weaker donor-acceptor interaction of local π symmetry, which partially counteracts the aforementioned regular σ-symmetric halogen-bonding orbital interaction. The halogen-halogen interaction in haloamines is the first known example of a halogen bond in which back donation takes place. We also find that this cooperativity in halogen bonds results from the reduction of the donor-acceptor orbital-energy gap that occurs every time a monomer is added to the aggregate.

Gentlest ascent dynamics for calculating first excited state and exploring energy landscape of Kohn-Sham density functionals.

We develop the gentlest ascent dynamics for Kohn-Sham density functional theory to search for the index-1 saddle points on the energy landscape of the Kohn-Sham density functionals. These stationary solutions correspond to excited states in the ground state functionals. As shown by various examples, the first excited states of many chemical systems are given by these index-1 saddle points. Our novel approach provides an alternative, more robust way to obtain these excited states, compared with the widely used ΔSCF approach. The method can be easily generalized to target higher index saddle points. Our results also reveal the physical interest and relevance of studying the Kohn-Sham energy landscape.

Application of DFT for the modeling of the valence region photoelectron spectra of boron and d‐element complexes and macromolecules

Using density functional theory (DFT) in conjunction with ultraviolet (UPS) and X‐ray photoelectron spectroscopy (XPS), we investigated a number of complexes and macromolecules. We have shown on a large set of UPS, XPS, and DFT data that the calculated Kohn–Sham energies of organic and metalorganic complexes can be used as approximate ionization energies (IEs). It is possible to evaluate IEs with an accuracy of 0.1 eV with the density functional approximation (DFA) defect approach. This method has been successfully tested on a large number of boron β‐diketonates and d‐metal chelate and sandwich complexes. We interpreted the bands in the valence region of the XP spectra of macromolecular organosilicon compounds in the solid state by taking into account the density of states and the ionization cross‐sections. According to DFT calculation results, the one‐electron states in the valence region of the model compounds correlate with the positions of the spectral band maxima. © 2015 Wiley Periodicals, Inc.

Prediction of core level binding energies in density functional theory: Rigorous definition of initial and final state contributions and implications on the physical meaning of Kohn-Sham energies.

A systematic study of the N(1s) core level binding energies (BE's) in a broad series of molecules is presented employing Hartree-Fock (HF) and the B3LYP, PBE0, and LC-BPBE density functional theory (DFT) based methods with a near HF basis set. The results show that all these methods give reasonably accurate BE's with B3LYP being slightly better than HF but with both PBE0 and LCBPBE being poorer than HF. A rigorous and general decomposition of core level binding energy values into initial and final state contributions to the BE's is proposed that can be used within either HF or DFT methods. The results show that Koopmans' theorem does not hold for the Kohn-Sham eigenvalues. Consequently, Kohn-Sham orbital energies of core orbitals do not provide estimates of the initial state contribution to core level BE's; hence, they cannot be used to decompose initial and final state contributions to BE's. However, when the initial state contribution to DFT BE's is properly defined, the decompositions of initial and final state contributions given by DFT, with several different functionals, are very similar to those obtained with HF. Furthermore, it is shown that the differences of Kohn-Sham orbital energies taken with respect to a common reference do follow the trend of the properly calculated initial state contributions. These conclusions are especially important for condensed phase systems where our results validate the use of band structure calculations to determine initial state contributions to BE shifts.

Subsystem density functional theory with meta-generalized gradient approximation exchange-correlation functionals.

We analyze the methodology and the performance of subsystem density functional theory (DFT) with meta-generalized gradient approximation (meta-GGA) exchange-correlation functionals for non-bonded molecular systems. Meta-GGA functionals depend on the Kohn-Sham kinetic energy density (KED), which is not known as an explicit functional of the density. Therefore, they cannot be directly applied in subsystem DFT calculations. We propose a Laplacian-level approximation to the KED which overcomes this limitation and provides a simple and accurate way to apply meta-GGA exchange-correlation functionals in subsystem DFT calculations. The so obtained density and energy errors, with respect to the corresponding supermolecular calculations, are comparable with conventional approaches, depending almost exclusively on the approximations in the non-additive kinetic embedding term. An embedding energy error decomposition explains the accuracy of our method.

On the relation between adiabatic time dependent density functional theory (TDDFT) and the ΔSCF-DFT method. Introducing a numerically stable ΔSCF-DFT scheme for local functionals based on constricted variational DFT

In ΔSCF density functional theory studies of a i → a transition one performs separate fully self-consistent field calculations on the ground state configuration (i)n (n = 1,2) and the excited state configuration (i)n − 1a. The excitation energy for the transition i → a is subsequently determined as the Kohn-Sham energy difference ΔEi → a = E[in − 1a] − E[in] between the ground state (i)n and the excited state configuration (i)n − 1a. The ΔSCF scheme has been applied extensively and works well for lower energy excitations provided that they can be represented by a single orbital replacement or transition i → a. However, for excitations of higher energy ΔSCF tends to become numerically unstable with a variational collapse to transitions of lower energy. We demonstrate here a numerically stable ΔSCF scheme for local functionals that is guaranteed not to collapse on excited configurations of lower energy as well as the ground state. The new scheme is based on constricted variational density functional theory in which the canonical ground state orbitals are allowed to relax (R-CV(∞)-DFT). Since it is restricted to a single orbital replacement i → a it is termed SOR-R-CV(∞)-DFT.