Session T24: Electronic Structure Methods IV

Reduced Density-Matrix Functional Theory: correlation and spectroscopy

In this work we explore the performance of approximations to electron correlation in reduced density-matrix functional theory (RDMFT) and of approximations to the observables calculated within this theory\footnote{S. Di Sabatino, J.A. Berger, L. Reining, and P. Romaniello, submitted}. Particular focus is put on the spectral function, which determines, for example, photoemission spectra, and which cannot be obtained in a straightforward way from the density matrix, and on the regime of strong electron correlation, which is difficult to treat by standard methods. Using the simple Hubbard model as test case shines light on the content, successes and limits of current RDMFT approaches.   

A problem with the stress theorem commonly used in DFT codes

The change in energy when an affine transformation (strain) is applied to a lattice of ions can be obtained by evaluating the algebraic derivative of the DFT energy (in practice a local or other approximation) of an electron density that has been similarly strained [1]. Because the DFT energy is stationary in the density, it is only required that the strained density reduces to the exact density at zero strain; it does. The algebraic derivatives of the Hartree and exchange energies are straightforward. The derivative with respect to strain of the non-interacting kinetic energy depends on two assumptions: 1) the modulus squared of the strained orbitals equals the strained electron density, and 2) the strained orbitals minimize the non-interacting kinetic energy. The first assumption is correct. I find that the second assumption applies only in special cases. The limitations and possible modifications of the stress theorem are discussed.\\[0pt] [1] Nielsen, O. H. \& Martin, R. M. 1983 First-Principles Calculation of Stress. Physical Review Letters 50, 697.\\[0pt] [2] D. M. Nicholson, Madhusudan Ojha, and T. Egami, Journal of Physics Condensed Matter 10/2013; 25(43):435505.   

Predictive DFT$+$U Methods for Small Molecule Binding in MOF-74

In order to use density functional theory (DFT) to reliably treat small molecule binding at open metal sites in metal-organic frameworks (MOFs), electron correlation effects associated with the localized d-states present at the metal centers must be accounted for. Incorporation of a Hubbard U-like term can be an approximate but computationally efficient means, yielding excellent agreement with experiment provided an appropriate value for the parameter U is chosen. To predict adsorption energetics for as-yet unsynthesized MOFs, we would need to select U using a systematic, physically motivated approach rather than the ad hoc methods typically employed. Here, we use an \textit{ab initio}linear response approach to calculate U. We show that U values determined with this method reproduce previous results for the binding of carbon dioxide in Co-MOF-74 and Cu-MOF-74, and we discuss the method's application to other 3d metals in theMOF-74 framework; our preliminary results suggest that a wide range of U's above a critical value will produce accurate binding energies. Finally, we present U values calculated for Co2$+$ ions in other systems, probing the environment dependence of this parameter. This work supported by DOE, and computational resources provided by NERSC.   

Density versus spin-density functional in DFT+U and DFT+DMFT

The construction of multi-variable effective action theories such as DFT+U and DFT+DMFT requires the choice of a local subspace of correlated orbitals and an additional variable being either the charge density or spin density. This talk examines the differences between using charge-only and spin-dependent exchange-correlation functionals with the aim of providing guidance for constructing more sophisticated beyond-density functional theories. The widely used spin-dependent approximations to the exchange-correlation functional are found to lead to a large and in some cases unphysical effective exchange coupling within the correlated subspace. Additionally, the differences between Wannier and Projector based definitions of the correlated orbitals are examined, and only small differences are found provided that the orbitals are orthonormal and strongly localized. These results are documented in the context of the rare earth nickelates.   

The Hubbard Dimer: A Complete DFT Solution to a Many-Body Problem

In this work we explain the relationship between density functional theory and strongly correlated models using the simplest possible example, the two-site asymmetric Hubbard model. We discuss the connection between the lattice and real-space and how this is a simple model for stretched H$_2$. We can solve this elementary example analytically, and with that we can illuminate the underlying logic and aims of DFT. While the many-body solution is analytic, the density functional is given only implicitly. We overcome this difficulty by creating a highly accurate parameterization of the exact function. We use this parameterization to perform benchmark calculations of correlation kinetic energy, the adiabatic connection, etc. We also test Hartree-Fock and the Bethe Ansatz Local Density Approximation. We also discuss and illustrate the derivative discontinuity in the exchange-correlation energy and the infamous gap problem in DFT.   

Development of a DFT$+$DMFT method using multi-orbital impurity solver

The density functional theory (DFT), often performed with the local density approximation or the generalized gradient approximation, is very successful for ab initio calculations of various materials. However, it has limited accuracy for strongly correlated materials. The dynamical mean field theory (DMFT), which maps a correlated lattice system to an interacting impurity site in a non-interacting bath, may describe local correlation effects. Combination of above methods, DFT$+$DMFT, can be an adequate approach for investigation of strongly correlated materials. We have implemented a DFT$+$DMFT method based on the ab-initio pseudopotential method of the SIESTA code, where electronic wavefunctions are expanded with pseudo-atomic orbitals. An exact diagonalization method is used in our DFT$+$DMFT method to obtain the local Green function of the impurity site with multiple orbitals. We apply our DFT$+$DMFT method to the electronic structure of LaFeAsO and compare the results with those from DFT and experiments. This work is supported by the NRF of Korea (Grant No.2011-0018306). Computational resources have been provided by KISTI Supercomputing Center (Project No. KSC-2013-C3-062).   

Nonorthogonal generalized hybrid Wannier functions for linear scaling DFT simulations of surfaces and interfaces

Semiconductor-based thin-films have applications in microelectronics, from transistors to nano-capacitors. Many of their properties depend on phenomena at multiple length scales, but their complexity makes it difficult to obtain a detailed understanding of their behavior from experiment alone. First-principles simulations based on density-functional theory (DFT) are invaluable for providing insight into materials' properties including for the study of thin films. In particular, hybrid Wannier functions (WFs), fully extended in the surface plane, but localized along the direction normal to the surface, have been successfully used to explore the properties of systems layered along a given direction. The large length scales associated with structures and processes in more realistic surfaces, however, are beyond the scope of such calculations, because they rely on first performing a traditional cubic-scaling DFT calculation. To overcome this limitation we extend the concept of hybrid WFs to nonorthogonal orbitals that are directly optimized in situ in the electronic structure calculation. We show that this method, implemented in the ONETEP linear scaling DFT code, enables the study of large-scale surfaces and interfaces with plane-wave accuracy but at reduced computational expense.   

Performance and Accuracy of Recursive Subspace Bisection for Hybrid DFT Calculations

The high cost of computing the Hartree-Fock exchange has resulted in limited use of Hybrid Functionals in DFT calculations. Approximations based on transformation to localized orbitals provide one way to reduce this cost. One such method is the recursive subspace bisection method (RSB)[1]. Such localization methods involve truncation of localized orbitals, which introduces an additional approximation. We take advantage of our ability to systematically reduce the error in RSB calculations through a single parameter to study this approximation. We present the errors in ground state energy, forces, and relative energy differences between configurations for a variety of systems, including tungsten oxide, a silicon-water interface, and liquid water including the calculation of empty states. \\{} [1] F.Gygi, Phys. Rev. Lett. 102, 166406 (2009). \\{} [2] Qbox code, http://eslab.ucdavis.edu/software/qbox/ \\{}   

Speeding up DFT: A faster method for integrating band energy in SCF cycles

Typically in SCF cycles, a ``rectangle rule'' is used on uniformly spaced points (Monk Pack meshes)$^{1}$ to integrate the band energy. The use of rectangles is motivated by their fast convergence when used on the fully occupied bands of semiconductors. Unfortunately integration with rectangles is extremely inefficient for metals. This motivates the use of gauss quadrature (or other higher order methods) for integrating the band energy. As we show, however, even in the case of semiconductors where the rectangle convergence is extremely efficient, higher order methods are still {\em more} efficient. The savings in semiconductors alone are sufficient to motivate the implementation of a higher order method in current DFT codes. Even though higher order quadrature methods were discussed immediately following the original Monkhorst and Pack$^{1}$ paper, we revisit the issue in light of modern DFT calculations. [1] H.J. Monkhorst and J.D. Pack, Phys. Rev. B 13, 5188 (1976).   

Efficient parameter-free calculation of absorption spectra for insulators, semiconductors and metals from time-dependent current DFT

In this work we show that with a simple dynamical kernel we can obtain good absorption spectra from time-dependent current-density functional theory (TDCDFT) for insulators, semiconductors and metals. Our approach is fully parameter free since no artificial broadening parameter is used to match calculated and measured spectra. The cost of a calculation is equal to an RPA calculation. Moreover, our TDCDFT approach scales better with system size than standard TDDFT implementations.   

Gauge invariant calculation of magnetic properties from time-dependent current DFT

We present a method to calculate magnetic properties from the current density that does not depend on the gauge choice for the vector potential when a finite basis set is used [1]. To obtain this we put paramagnetic and diamagnetic contributions to the current on equal footing by making use of a sum rule [1]. Our method is general. Here we use it to calculate static and dynamical magnetizabilities of molecules within Time-Dependent Current-Density-Functional Theory. \\[4pt] [1] N. Raimbault, P.L. de Boeij, P. Romaniello, and J.A. Berger, submitted   

Accurate Energies and Orbital Description in Semi-Local Kohn-Sham DFT

We present our progress on a scheme in semi-local Kohn-Sham density-functional theory (KS-DFT) for improving the orbital description while still retaining the level of accuracy of the usual semi-local exchange-correlation (xc) functionals. DFT is a widely used tool for first-principles calculations of properties of materials. A given task normally requires a balance of accuracy and computational cost, which is well achieved with semi-local DFT. However, commonly used semi-local xc functionals have important shortcomings which often can be attributed to features of the corresponding xc potential. One shortcoming is an overly delocalized representation of localized orbitals. Recently a semi-local GGA-type xc functional was constructed to address these issues [1], however, it has the trade-off of lower accuracy of the total energy. We discuss the source of this error in terms of a surplus energy contribution in the functional that needs to be accounted for, and offer a remedy for this issue which formally stays within KS-DFT, and, which does not harshly increase the computational effort. The end result is a scheme that combines accurate total energies (e.g., relaxed geometries) with an improved orbital description (e.g., improved band structure). [1] PRL 111, 036402 (2013)   

Understanding Density Functional Theory (DFT) and Completing it in Practice

A brief review of the seminal article by Hohenberg and Kohn leads to two conditions that have to be met by electronic structure calculations in order for their results to represent the physics content of DFT. One of these conditions is often the verifiable attainment of the absolute minima of the occupied energies. \textit{Using the second Hohenberg Kohn theorem, we show that results of calculations that do not meet this condition, when it applies, do not necessarily represent DFT findings}. We illustrate this fact with over 100 calculated band gaps that are much smaller than corresponding, measured ones; in contrast, we list calculations that strictly adhered to the aforementioned conditions and whose results are in excellent agreement with experiment. \textit{We describe two crucial steps in the latter calculations that add to or complete DFT in practice.} Some implications of our findings for academia, industry, and program package developers will be discussed. Acknowledgments: This work was funded in part by the National Science Foundation (NSF) and the Louisiana Board of Regents, through LASiGMA [Award Nos. EPS- 1003897, NSF (2010-15)-RII-SUBR] and NSF HRD-1002541, the US Department of Energy -- National, Nuclear Security Administration (NNSA) (Award Nos. DE-NA0001861 and DE- NA0002630), LaSPACE, and LONI-SUBR.   

Elimination of the fractional dissociation problem in Kohn-Sham DFT using the ensemble-generalization approach

Many approximations within density-functional theory (DFT) spuriously predict that a many-electron system can dissociate into fractionally charged fragments. Here, we revisit the case of infinitely separated diatomic molecules, known to exhibit this problem when studied within standard approximations, including the local spin-density approximation (LSDA). We apply the recently suggested ensemble-generalization to LSDA (eLSDA) [1,2] and find that fractional dissociation is eliminated in all systems examined. The eLSDA Kohn-Sham potential develops a spatial step, associated with the emergence of the derivative discontinuity in the exchange-correlation energy functional. This step, predicted in the past for the exact Kohn-Sham potential and observed in some of its more advanced approximate forms, is a desired feature that prevents any fractional charge transfer between the system's fragments. Our findings show that, if appropriately generalized for fractional electron densities, even the most simple approximate functionals correctly predict integer dissociation [3]. [1] E. Kraisler, L. Kronik, Phys. Rev. Lett. 110, 126403 (2013) [2] E. Kraisler, L. Kronik, J. Chem. Phys. 140, 18A540 (2014) [3] E. Kraisler, L. Kronik, submitted.   

Representability of Bloch states on Projector-augmented-wave (PAW) basis sets

Design of small, yet `complete', localized basis sets is necessary for an efficient dual representation of Bloch states on both plane-wave and localized basis\footnote{Agapito, Ferretti, Calzolari, Curtarolo and Buongiorno Nardelli, PRB {\bf 88}, 165127 (2013).}. Such simultaneous dual representation permits the development of faster more accurate (beyond DFT) electronic-structure methods for atomistic materials (e.g. the ACBN0 method\footnote{Agapito, Curtarolo and Buongiorno Nardelli, \texttt{arXiv:1406.3259 [cond-mat.str-el]}}.) by benefiting from algorithms (real and reciprocal space) and hardware acceleration (e.g. GPUs) used in the quantum-chemistry and solid-state communities. Finding a `complete' atomic-orbital basis (partial waves) is also a requirement in the generation of robust and transferable PAW pseudopotentials. We have employed the atomic-orbital basis from available PAW data sets, which extends through most of the periodic table, and tested the representability of Bloch states on such basis. Our results show that PAW data sets allow systematic and accurate representability of the PAW Bloch states, better than with traditional quantum-chemistry double-zeta- and double-zeta-polarized-quality basis sets.