Density Functional Theory Problem Research Articles

Theoretical exploration of band gap error dependency on band gap size in density functional theory Calculations: CdTe and GeTe as representative cases of two band structure semiconductor types

This study addresses the prevalent band gap error problem in density functional theory (DFT) with local density or generalized gradient approximation (LDA/GGA), widely considered as the “standard model” for investigating the band structure of semiconductors. Despite previous DFT efforts focusing on improving the exchange correlation energy to mitigate the underestimation of the band gap, the correlation between the band gap error and the intrinsic electronic structure of semiconductors has been overlooked. From two principal observations: (i) traditional semiconductors typically have a p-s type band structure, and (ii) a smaller band gap is linked to less underestimation, we derive two crucial questions: (i) How is band gap error manifested in s-p type semiconductors? (ii) Does the error vanish as the band gap approaches zero, akin to metals? Through the examination of CdTe and GeTe, representing p–s and s–p type semiconductors with strain-engineered band gaps, our calculations indicate that for CdTe, the band gap is underestimated, whereas for GeTe, it is overestimated. Significantly, these errors persist even as the band gap of semiconductors tends to zero, unlike in metals. This persistent discrepancy stems from the discontinuity in wavefunctions between the semiconductors' VBM and CBM.

Seven useful questions in density functional theory

We explore a variety of unsolved problems in density functional theory, where mathematicians might prove useful. We give the background and context of the different problems, and why progress toward resolving them would help those doing computations using density functional theory. Subjects covered include the magnitude of the kinetic energy in Hartree–Fock calculations, the shape of adiabatic connection curves, using the constrained search with input densities, densities of states, the semiclassical expansion of energies, the tightness of Lieb–Oxford bounds, and how we decide the accuracy of an approximate density.

Some problems in density functional theory

Though calculations based on density functional theory (DFT) are used remarkably widely in chemistry, physics, materials science, and biomolecular research and though the modern form of DFT has been studied for almost 60 years, some mathematical problems remain. For context, we provide an outline of the basic structure of DFT, then pose several questions regarding both its time-independent and time-dependent forms. Progress on any of these would aid in development of better approximate functionals and in interpretation.

Building Kohn–Sham Potentials for Ground and Excited States

We analyze the inverse problem of Density Functional Theory using a regularized variational method. First, we show that given k and a target density $$\rho $$ , there exist potentials having kth bound mixed states which densities are arbitrarily close to $$\rho $$ . The state can be chosen pure in dimension $$d=1$$ and without interactions, and we provide numerical and theoretical evidence consistently leading us to conjecture that the same pure representability result holds for $$d=2$$ , but that the set of pure-state v-representable densities is not dense for $$d=3$$ . Finally, we present an inversion algorithm taking into account degeneracies, removing the generic blocking behavior of standard ones.

Toward a Resolution of the Static Correlation Problem in Density Functional Theory from Semidefinite Programming

Kohn-Sham density functional theory (DFT) has long struggled with the accurate description of strongly correlated and open shell systems, and improvements have been minor even in the newest hybrid functionals. In this Letter we treat the static correlation in DFT when frontier orbitals are degenerate by the means of using a semidefinite programming (SDP) approach to minimize the system energy as a function of the N-representable, non-idempotent 1-electron reduced density matrix. While showing greatly improved singlet-triplet gaps for local density approximation and generalized gradient approximation (GGA) functionals, the SDP procedure reveals flaws in modern meta and hybrid GGA functionals, which show no major improvements when provided with an accurate electron density.

(Invited) First-Principles Understanding of Deep-Level Defects and Impurities in GaN and Nitride Alloys

The presence of native defects or unwanted impurities can strongly affect semiconductor device performance. Such defects can give rise to deep levels in the band gap, acting as sources of non-radiative recombination, carrier trapping, and sub-band-gap luminescence. Although predicting the properties of deep defects in semiconductors has long been hampered by the band-gap problem of density functional theory (DFT), methodological developments now allow for quantitative predictions. Here we discuss how first-principles calculations have shed light on the role of deep-level defects and impurities in gallium nitride and related alloys.We begin by discussing computational methodologies for calculating the properties of native defects and discuss how to overcome the band-gap problem of DFT. In particular, we examine to what extent calculations performed with semilocal functionals (such as the generalized gradient approximation), combined with correction schemes, can produce accurate results. The properties of vacancy, interstitial, and antisite defects in GaN are discussed, as well as their interactions with impurities such as hydrogen and oxygen. We also discuss first-principles results in the context of experimental observations, and examine how native defects and their complexes impact the performance of nitride devices [1].We also explore how the properties of defects are affected by presence of alloying elements such as indium [2]. With explicit alloy calculations we monitor how defect levels depend on indium content and the distribution of In atoms. The relative shifts of the charge‐state transition levels of different defects are explained by the atomic character of the defect state. Such shifts depend on whether the defect state is composed of valence‐band or conduction‐band states of the host material, or whether it acts as an atomic‐like impurity. Various possible atomic configurations of In and Ga cations in InGaN alloys of given compositions lead to a distribution of charge‐state transition levels. Defects on the nitrogen site lead to a larger spread of defect levels, while defects on the cation site lead to a smaller spread.Finally, we also investigate the properties of heavily C-doped GaN using both optical experiments and hybrid DFT calculations. Previous calculations had established that carbon acceptors (CN) give rise to a yellow luminescence (YL) band near 2.2 eV [3], along with a blue luminescence band near 2.9 eV. Photoluminescence measurements showed these optical transitions shifting as a function of carbon concentration, suggesting a change in the behavior of carbon. With hybrid DFT we calculated the electrical and optical behavior of carbon species containing multiple carbon impurities, which may arise in heavily doped material. We compared the behavior of these complexes to the isolated centers, and find that the CGa-CN complex is a good candidate to explain the shift in the YL peak. Furthermore, we determined the local vibrational modes of carbon impurity centers, and compared these results to recent experiments.[1] J. L. Lyons and C. G. Van de Walle, npj Computational Materials 3, 12 (2017).[2] D. Wickramaratne et al., phys. Stat. sol. (b) 2020 (https://doi.org/10.1002/pssb.201900534).[3] J. L. Lyons, A. Janotti, and C. G. Van de Walle, Phys. Rev. B 89, 035204 (2014).

Universal nature of different methods of obtaining the exact Kohn–Sham exchange-correlation potential for a given density

Finding the external potential or the Kohn–Sham (KS) potential for a given ground state density is a fundamental inverse problem in density functional theory. Furthermore, it is important in generating the exact exchange-correlation potential for a density which can then serve as a benchmark for testing the accuracy of an exchange-correlation energy functional. Over the years different methods have been proposed to do the density-to-potential inversion and all of these appear to be disjoint. In this paper we show that these methods are connected to and can be derived from one general algorithm that utilizes two forms of the equation for the density; these are the KS equation in terms of orbitals and the Euler equation in terms of the density. We obtain the condition for the convergence of the general method and show that all the methods considered by us satisfy this condition. We demonstrate the method and its flexibility by obtaining the KS potential for some spherical systems in a variety of ways.

Representability problem of density functional theory for superconductors

Aiming at a unified treatment of correlation and inhomogeneity effects in superconductors, Oliveira, Gross and Kohn proposed in 1988 a density functional theory for the superconducting state. This theory relies on the existence of a Kohn-Sham scheme, i.e., an auxiliary noninteracting system with the same electron and anomalous densities of the original superconducting system. However, the question of noninteracting $v$-representability has never been properly addressed and the existence of the Kohn-Sham system has always been assumed without proof. Here, we show that indeed such a noninteracting system does not exist in at zero temperature. In spite of this result, we also show that the theory is still able to yield good results, although in the limit of weakly correlated systems only.

New Optimal Control Problems in Density Functional Theory Motivated by Photovoltaics

We present and study novel optimal control problems motivated by the search for photovoltaic materials with high power-conversion efficiency. The material must perform the first step: convert light...

Unified formulation of fundamental and optical gap problems in density-functional theory for ensembles

Solving the fundamental and optical gap problems, which yield information about charged and neutral excitations in electronic systems, is one of the biggest challenges in density-functional theory. Despite their intrinsic differences, we show that the two problems can be made formally identical by introducing a universal and canonical ensemble weight-dependent exchange-correlation (xc) density functional. The weight dependence of the xc energy turns out to be the key ingredient for describing the infamous derivative discontinuity and represents an alternative path for its approximation.

Communication: Correct charge transfer in CT complexes from the Becke'05 density functional.

It has been known for over twenty years that density functionals of the generalized-gradient approximation (GGA) type and exact-exchange-GGA hybrids with low exact-exchange mixing fraction yield enormous errors in the properties of charge-transfer (CT) complexes. Manifestations of this error have also plagued computations of CT excitation energies. GGAs transfer far too much charge in CT complexes. This error has therefore come to be called "delocalization" error. It remains, to this day, a vexing unsolved problem in density-functional theory (DFT). Here we report that a 100% exact-exchange-based density functional known as Becke'05 or "B05" [A. D. Becke, J. Chem. Phys. 119, 2972 (2003); 122, 064101 (2005)] predicts excellent charge transfers in classic CT complexes involving the electron donors NH3, C2H4, HCN, and C2H2 and electron acceptors F2 and Cl2. Our approach is variational, as in our recent "B05min" dipole moments paper [Dale et al., J. Chem. Phys. 147, 154103 (2017)]. Therefore B05 is not only an accurate DFT for thermochemistry but is promising as a solution to the delocalization problem as well.

Structural and electronic properties of silicon carbide polytypes as predicted by exact exchange calculations

A full-potential band structure calculation within the exact exchange-optimized effective potential (EXX-OEP) approach was performed for the 3C, 6H, 15R, 4H, and 2H polytypes of SiC. The calculated lattice constants and energy band gaps were found to be in a good agreement with experimental values; therefore, the EXX-OEP method, although computationally expensive, is a viable method for solving the long-standing band gap problem of density functional theory (DFT). Throughout the paper, the results obtained using EXX-OEP are compared with those obtained using regular DFT calculations.

Numerical methods for the inverse problem of density functional theory

AbstractThe inverse problem of Kohn–Sham density functional theory (DFT) is often solved in an effort to benchmark and design approximate exchange‐correlation potentials. The forward and inverse problems of DFT rely on the same equations but the numerical methods for solving each problem are substantially different. We examine both problems in this tutorial with a special emphasis on the algorithms and error analysis needed for solving the inverse problem. Two inversion methods based on partial differential equation constrained optimization and constrained variational ideas are introduced. We compare and contrast several different inversion methods applied to one‐dimensional finite and periodic model systems.

On the v-Representabilty Problem in Density Functional Theory: Application to Non-Interacting Systems

Based on a computational procedure for determining the functional derivative with respect to the density of any antisymmetric N-particle wave function for a non-interacting system that leads to the density, we devise a test as to whether or not a wave function known to lead to a given density corresponds to a solution of a Schrödinger equation for some potential. We examine explicitly the case of non-interacting systems described by Slater determinants. Numerical examples for the cases of a one-dimensional square-well potential with infinite walls and the harmonic oscillator potential illustrate the formalism.

Variational fractional-spin density-functional theory for diradicals

Accurate computation of singlet-triplet energy gaps of diradicals remains a challenging problem in density-functional theory (DFT). In this work, we propose a variational extension of our previous work [D. H. Ess, E. R. Johnson, X. Q. Hu, and W. T. Yang, J. Phys. Chem. A 115, 76 (2011)], which applied fractional-spin density-functional theory (FS-DFT) to diradicals. The original FS-DFT approach assumed equal spin-orbital occupancies of 0.5 α-spin and 0.5 β-spin for the two degenerate, or nearly degenerate, frontier orbitals. In contrast, the variational approach (VFS-DFT) optimizes the total energy of a singlet diradical with respect to the frontier-orbital occupation numbers, based on a full configuration-interaction picture. It is found that the optimal occupation numbers are exactly 0.5 α-spin and 0.5 β-spin for diradicals such as O(2), where the frontier orbitals belong to the same multidimensional irreducible representation, and VFS-DFT reduces to FS-DFT for these cases. However, for diradicals where the frontier orbitals do not belong to the same irreducible representation, the optimal occupation numbers can vary between 0 and 1. Furthermore, analysis of CH(2) by VFS-DFT and FS-DFT captures the (1)A(1) and (1)B(1) states, respectively. Finally, because of the static correlation error in commonly used density functional approximations, both VFS-DFT and FS-DFT calculations significantly overestimate the singlet-triplet energy gaps for disjoint diradicals, such as cyclobutadiene, in which the frontier orbitals are confined to separate atomic centers.

Improved Prediction of Properties of π-Conjugated Oligomers with Range-Separated Hybrid Density Functionals.

Range-separated hybrid functionals along with global hybrids and pure density functionals have been employed to calculate geometries, ionization energies (IP)s, electron affinities (EA)s, and excitation energies of neutral and oxidized polyenes, thiophene, and furan oligomers. Long-range correction with 100% HF exchange solves the problem of density functional theory with incorrect chain length dependence of IPs and energy gaps. There is a possibility of overcorrection, if the short-range part of the functional with no or low HF exchange is too small. The wB97XD functional with 22% of HF exchange in the short-range and a range-separation parameter of 0.2 seems to be just right for conjugated systems at all chain lengths. The wB97XD functional additionally produces negative orbital energies in very good agreement with IPs and EAs. With correct orbital energies, band gaps correspond to transport gaps (Et) and not to optical gaps (Eg). Et is much larger than Eg in the gas phase, but the difference is significantly smaller in the solid state. The accuracy of the negative orbital energies is good down to about 30 eV so that valence and innervalence PE spectra can be modeled. wB97XD is therefore suitable for calculating band structures of conjugated polymers employing orbital energies.

Robust acceleration of self consistent field calculations for density functional theory

We show that the type 2 Broyden secant method is a robust general purpose mixer for self consistent field problems in density functional theory. The Broyden method gives reliable convergence for a large class of problems and parameter choices. We directly mix the approximation of the electronic density to provide a basis independent mixing scheme. In particular, we show that a single set of parameters can be chosen that give good results for a large range of problems. We also introduce a spin transformation to simplify treatment of spin polarized problems. The spin transformation allows us to treat these systems with the same formalism as regular fixed point iterations.

Twelve outstanding problems in ground-state density functional theory: A bouquet of puzzles

Kohn–Sham density functional theory and its many applications constitute perhaps the most populated area of both physics and chemistry. The theory permits calculation of the ground-state density and energy of a many-electron system (e.g., an atom, molecule, or solid) by the selfconsistent solution of a one-electron Schroedinger equation. The theory is exact in principle for these quantities, and in practice the required approximations can be usefully accurate. Here, with few equations and without any claim of optimal selection, we present 12 problems in density functional theory that remain to be solved, with an emphasis on formal problems and on the construction of approximate density functionals. The solutions to some of these problems could go a long way toward deepening understanding of the exact theory and toward improving the accuracy of electronic structure calculations.

Electronic and Magnetic Structure of Transition-Metal Carbodiimides by Means of GGA+U Theory

The electronic structures and magnetic properties of MNCN (M = Fe, Co, and Ni) have been investigated by density-functional theory including explicit electronic correlation through an ad hoc Coulomb potential (GGA+U). The results evidence CoNCN and NiNCN as type-II anti-ferromagnetic semiconductors (that is, intralayer ferromagnetic and interlayer anti-ferromagnetic), in accordance with experimental observations. Just like the prototype MnNCN, the MNCN phases, with M = Ni and Co, thus resemble the corresponding MO monoxides with respect to their magnetic and transport properties. By contrast, FeNCN remains (semi)metallic even upon applying a strong Coulomb correlation potential. This, most probably, is in contradiction with its observed optical transparency and expected insulating behavior and points toward a serious density-functional theory problem.

Accelerated Block Preconditioned Gradient method for large scale wave functions calculations in Density Functional Theory

An Accelerated Block Preconditioned Gradient (ABPG) method is proposed to solve electronic structure problems in Density Functional Theory. This iterative algorithm is designed to solve directly the non-linear Kohn–Sham equations for accurate discretization schemes involving a large number of degrees of freedom. It makes use of an acceleration scheme similar to what is known as RMM-DIIS in the electronic structure community. The method is illustrated with examples of convergence for large scale applications using a finite difference discretization and multigrid preconditioning.