Session T23: Density Functional Theory II

Particle Number and Probability Density Functional Theory, and A-representability

In Hohenberg-Kohn (HK) density functional theory (DFT), the functional $F_{HK}[\rho]$ of the density $\rho({\bf{r}})$ representing the expectation of the electron-interaction and kinetic energy operators is universal. Knowledge of $F_{HK} [\rho]$ by itself is insufficient to obtain the energy: the electron number $N$ is primary. By emphasizing this primacy of $N$, we rewrite the energy $E$ as a nonuniversal functional of $N$ and probability density $p({\bf{r}}): E = E[N, p]$, with $p ({\bf{r}})$ satisfying the constraints of normalization to unity and positivity. A particle number $N$ and probability density $p ({\bf{r}})$ functional theory is constructed, and examples of exact functionals provided. The concept of $A$-representability is introduced as the set of functions $\psi_{p}$ that lead to quantum mechanical $p({\bf{r}})$ as the expectation of the probability density operator. We show via the Harriman and Gilbert constructions that the $A$- and $N$-representable probability density $p({\bf{r}})$ sets are equivalent, with the latter defined as $p({\bf{r}}) = \rho({\bf{r}})/N$.   

Improved Constraint-based GGA Functionals in Extended Systems and Molecules

Despite wide-spread interest in explicitly orbitally-dependent exchange-correlation functionals, there is aimed at the basic vision of Density Functional Theory, orbital-free implementation. Here we report on further development of our non-empirical X functionals. We give results for the VMT functional (J. Chem. Phys.\ \textbf{130} 244103 (2009)) combined with the PBE C functional on simple solids and ultra-thin films. We also give results for molecules for a more sophisticated family of functionals, VPTmn, which satisfy more constraints than VMT. VMT11 with PBE C tested on a widely used 20 molecule set shows essentially no change in atomization energies compared to VMT, an illustration that enforcing more constraints does not necessarily improve outcomes. We also consider VMT11 in extended systems and in combination with both the LYP and rev-TCA correlation functionals on molecules.   

The role of vdW interactions for the cohesive properties of the coinage metals

Density-functional theory has been successfully applied over many years to calculate binding energies and bond lengths for a wide range of systems. More recently, it has been extended to include long-range correlation interactions and hence could be used for purely van der Waals (vdW) bound systems such as noble gas solids or organic crystals. On the other hand, early calculations for noble metals have revealed a vdW contribution to the cohesive energy [Rehr, Zaremba, Kohn, PRB 12, 2062 (1975)]. We investigate this issue by employing the vdW-DF approach [Dion et al., PRL 92, 296401 (2004)], where the exchange interactions are treated within revPBE, and the non-local correlations are based on an approximation to the adiabatic connection formula. We find that the latter give a substantial contribution to the cohesive energies but, overall, vdW-DF underestimates their magnitude while overestimating the lattice parameters. We attribute this shortcoming to the local part of the correlation energy.   

Efficient band gap estimation for solids from DFT

The ab initio prediction of band gaps for solids is important for fundamental and practical reasons. Unfortunately the most popular ab initio computation tool for solids, Density Functional Theory (DFT) in the Kohn-Sham implementation with local or semilocal exchange correlation functionals (LDA/GGA), suffers from the famous ``band gap problem.'' In Kohn-Sham DFT with LDA/GGA, the energy gaps between occupied and unoccupied single-particle states (Kohn-Sham gaps) are typically far below the experimentally-measured band gaps. In this talk we propose an efficient method for the estimation of fundamental band gaps in solids using DFT with local and semi-local functionals. We demonstrate significant improvements compared to Kohn-Sham band gaps on over 100 compounds with experimental band gaps between 0.5 and 4 eV (mean absolute error reduced from 0.84 to 0.25 eV, standard deviation from 0.5 to 0.2 eV). Our proposed method has an accuracy similar to the screened hybrid functionals and modified Becke-Johnson potentials, while requiring computational costs similar to typical DFT LDA/GGA calculations.   

A Scalable Implementation of Van der Waals Density Functionals

Recently developed Van der Waals density functionals[1] offer the promise to account for weak intermolecular interactions that are not described accurately by local exchange-correlation density functionals. In spite of recent progress [2], the computational cost of such calculations remains high. We present a scalable parallel implementation of the functional proposed by Dion et al.[1]. The method is implemented in the Qbox first-principles simulation code (http://eslab.ucdavis.edu/software/qbox). Application to large molecular systems will be presented. \\[4pt] [1] M. Dion et al. Phys. Rev. Lett. 92, 246401 (2004).\\[0pt] [2] G. Roman-Perez and J. M. Soler, Phys. Rev. Lett. 103, 096102 (2009).   

Total energy performance of non-Koopmans functionals

We have recently introduced a non-Koopmans correction to local and semilocal exchange-correlation functionals, able to identify orbital energies with opposite removal energies in the frozen orbital approximation. This approach directly improves on some of the key failures of common exchange-correlation functionals, often related to self-interaction (e.g. the dissociation energies of simple molecules or the polarizability of linear chains), and provides a link between the Hartree-Fock and the density-functional theory approaches to the electronic-structure problem. We evaluate here the performance of this corrected functional to total energy properties; these include geometries, dissociation energies, and linear and non linear polarizabilities of linear molecules and chains. Our results indicate that the method is very effective in correcting for the self-interaction problem of local and semi-local functionals, with large improvements, among others, for the predictions of polarizabilities and hyperpolarizabilities.   

Universal extension to the Becke-Johnson exchange potential

The Becke-Johnson exchange potential [J. Chem. Phys. {\bf 124}, 221101 (2006)] has been successfully used in electronic structure calculations within density-functional theory. However, in its original form the potential may dramatically fail in systems with non-Coulombic external potentials, or in the presence of external electric or magnetic fields. Here we present a system-independent extension to the Becke-Johnson approximation by (i) enforcing its gauge-invariance and (ii) making it exact for any single-electron system. The resulting approximation is then better designed to deal with current-carrying states, and recovers the correct asymptotic behavior for systems with arbitrary number of electrons. Our approximation is shown to give very good results for atoms, atomic chains, and molecules with and without external electric or magnetic fields.   

Self-interaction-free nonlocal correlation energy functional associated with a Jastrow function

We propose a self-interaction-free nonlocal correlation energy functional based on the transcorrelated method [1]. An effective Hamiltonian, ${\cal H}_{\rm eff}=\frac{1}{F} {\cal H} F$, is derived from a similarity transformation with respect to a `Jastrow' correlation factor, $F$. The total energy is given by the expectation value of ${\cal H}_{\rm eff}$ with respect to a single Slater determinant. If a two-body Jastrow function is adopted, the resulting method resembles a Kohn-Sham density functional theory in which the correlation energy functional consists of two- and three-body interactions [2]. To simplify our calculations, we exclude the three-body terms and instead multiply the two-body term by an adjustable parameter that ensures convergence of the correlation energy to the exact limit for the homogeneous electron gas. The computational cost of the proposed method is comparable to the Hartree-Fock method. Moreover, the present correlation functional does not include self-interaction terms. The performance of this functional for various atoms and molecules will be presented. [1]S. F. Boys and N. C. Handy, Proc. Roy. Soc. A, {\bf 309}, 209; {\bf 310}, 43; {\bf 310}, 63; {\bf 311}, 309 (1969). [2] N. Umezawa and T. Chikyow, Phys. Rev. A {\bf 73}, 062116 (2006).   

New expressions for the correlation energy functional within the GW-RPA approximation

Approaches such as the Luttinger-Ward formalism allow one, in principle, to compute both total energies and quasiparticle excitations (i.e., electron and hole energies and wave functions) simultaneously from first principles by working with total energy functionals of the one-particle Green's function. We briefly review our recent results on exact and approximate expressions for the GW-RPA correlation energy. We then provide some numerical results on atoms assessing the quality and superior convergence properties of these new expressions. We end by discussing our finding that unconstrained optimization of the total energy functional over non-interacting Green's functions or equivalently over non-local exchange correlation potentials leads to unphysical results: This shows that when using such approaches, one must place constraints on the Green's function or exchange-correlation potential.   

Extended LDA+U+V approach for covalent systems

A novel DFT+U energy functional (named DFT+U+V) is introduced based on a corrective Hubbard Hamiltonian that includes both on-site (U) and inter-site (V) electron-electron interactions. The competition between these interactions avoids the over-stabilization of occupied atomic orbitals, usually observed within the ``on-site-only" DFT+U approach, and allows to describe systems in which hybridization plays an important role (as, e.g. semiconductors doped with magnetic impurities, or high T$_c$ superconductors) and whose behavior is intermediate between that of Mott and band insulators. In addition, the inter-site interaction parameter V can be straightforwardly obtained (at no additional cost) from the same linear-response approach used to calculate U [1]. The flexibility and reliability of the novel functional are demonstrated by its application to prototypical covalent (Si) and ionic (GaAs) semiconductors and to charge-transfer insulators (NiO). \noindent{[1] M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71, 035105 (2005).}   

Locating Low-Energy Solutions within DFT+U

The widely employed DFT+U formalism is known to give rise to many self-consistent yet energetically distinct solutions in correlated systems, which can be highly problematic for reliably predicting the thermodynamic and physical properties of such materials. While this phenomenon has been previously demonstrated for 5f compounds, we demonstrate it for 3d systems as well. We characterize the scope of these multiple energetic minima, indicating both in which systems they should occur, and the magnitude of the potential errors they introduce. We then propose an efficient method for locating the lowest-energy solution. Finally, we suggest that our method could also be extended to hybrid functionals.   

Defect levels of the O vacancy in ZnO in DFT, hybrid-DFT, and GW

The band gap problem of the LDA and GGA approximations to density-functional theory (DFT) introduce a significant uncertainty in the prediction of the charge transition levels of electrically active defects or impurities. For the case of the O vacancy in ZnO, we here compare the predictions of three methods with an increasing level of computational effort, i.e., (i) GGA+U plus a rigid shift of the conduction band minimum, (ii) hybrid-DFT using the HSE functional, and (iii) GW calculations of the quasi-particle energies of the defect states. In addition to finite-size corrections of DFT (or hybrid-DFT) supercell total energies, which are applied to all methods, we demonstrate here also the need of corrections for the GW quasi-particle energies of charged defect states. Applying the GW quasi-particle energy corrections to the self-consistent GGA+U and HSE calculations, we then obtain the 2+/0 donor level between 1.4 eV (GGA+U) and 1.7 eV (HSE) above the valence band maximum. Without the GW corrections, the transition levels lie at 1.0 eV and 2.3 eV, respectively in GGA+U and HSE, where in the HSE calculation the fraction of the Fock exchange was adjusted so to match the experimental band gap. This work was supported through the Center for Inverse Design, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences.   

Periodic image errors in the electronic hyperpolarizabilities

Second-order nonlinear optical properties of large organic nonlinear optical (NLO) molecules have attracted increasing attention, and static and dynamical hyperpolarizabilities are now important target of the ab-initio theoretical studies. In the DFT and TDDFT, the Poisson equation is commonly solved with the periodic boundary condition (PBC) to reduce the computational cost, but it produces serious the periodic image error due to large dipole moment of the NLO molecules. Here, we show that a Green's function technique, i.e., effective screening medium (ESM) method of Otani and Sugino, provides an efficient way of overcoming this problem. The ESM method allows to apply an external electric field to the molecule by changing the boundary condition of the Poisson equation. Moreover, the method requires the computational cell just enough to contain the electron density, although PBC calculation requires the dipole-interaction tail to sufficiently small at the cell boundary. Other correction methods such as cut-off or multipole method also requires larger, say, doubled cell. The ESM method has a further merit of using various boundary conditions to model surrounding vacuum, metallic as well as dielectric medium. This suggests a future possibility of incorporating the solvent effects into the theoretical study. The ESM method thus would stimulate further studies of a large photonic molecule.   

Hybrid Density Functional Calculations of Redox Potentials of Transition Metal Compounds

Prior works have shown that density functional theory (DFT) with the DFT+U method resolves the underestimation of redox potentials calculated by conventional functionals for a number of transition metal compounds relevant for battery applications, including the olivine Li$_x$MPO$_4$ (M = Fe, Mn, Co, Ni), layered Li$_x$MO$_2$ (M = Co, Ni) and spinel-like Li$_x$Mn$_2$O$_4$. We show that the redox potentials of these compounds are also well reproduced by the hybrid density functional by Heyd-Scuseria-Ernzerhof (HSE06). Hybrid functionals combine a conventional DFT functional with a part of Hartree-Fock (HF) exchange. While the HF part increases the computational expense by at least one order of magnitude, it provides, in contrast to DFT+U, a correction for the self-interaction error that does not rely on special treatment of the occupancies of the orbital states of ions or species-specific parameters. We compare the accuracy of regular DFT, DFT+U and HSE06 for the redox potentials, lattice constants, and other properties. Examples of electron delocalization problems connected to the self-interaction error in the systems are discussed, and shown to be resolved both by the hybrid functional and DFT+U methods. Comments are made on the possibility to approach the delocalization problem with a semi-local functional.   

Water splitting for energy storage: first principles insights into the reaction mechanism of Ru-based homogeneous catalysts

Water oxidation has been recognized as the key bottleneck toward the development of artificial photosynthesis, where the goal is to use solar energy to produce chemicals for energy conversion and storage. This reaction requires the loss of four electrons and four protons and has a standard reduction potential of 1.23 V (pH=0, NHE). The recent development of an all-inorganic tetra-ruthenium polyoxometallate homogeneous catalyst [1,2] that oxidizes water at a low overpotential ($\sim$ 0.2 V) is a breakthrough in this field, since it combines the stability of inorganic compounds and the high activity of homogeneous catalysts. Here we report the results of a first-principles DFT study of the properties of this catalyst and the mechanism it promotes. We focus on the analysis of the thermodynamics of the water oxidation cycle, considering the relative stability of different candidate intermediates as a function of the external bias that drives the reaction. The comparison with available cyclic voltammetry allows to shed some light on the possible oxidation states of the Ru centers involved in the catalytic mechanism. [1] A. Sartorel et al. J. Am. Chem. Soc. 130, 5006 (2008) [2] Y. Geletti et al. Angew. Chem. Int. Ed. 47, 3896 (2008)   

Continuum mechanics for quantum many-body systems

Continuum mechanics is a theory of the dynamics of classical liquids and solids in which the state of the body is described by a small set of collective, such as density and current. A similar description is possible for quantum many-body systems, and indeed its existence is guaranteed by the basic theorems of time-dependent current density functional theory. In this paper we show how the exact Heisenberg equation of motion for the current density of a many-body system can be closed by expressing the quantum stress tensor as a functional of the current density. Several approximation schemes for this functional are discussed. The simplest scheme allows us to bypass the solution of the time-dependent Schr\"odinger equation, resulting in an equation of motion for the current that requires only ground-state properties as an input. We illustrate the formalism by applying it to the calculation of excitation energies in simple one- and two-electron systems.