Session S18: Electronic Structure Methods I

van der Waals density functional with corrected long-range behavior

A new van der Waals density functional, i.e. vdW-DF3, has been developed by maintaining an accurate balance between short- and long-range contributions to the non-local correlation in the original vdW-DF scheme. We stay entirely true to the original design criteria of vdW-DF, adhering to the exact same physical constraints. But, we take advantage of some freedom in defining the plasmon-dispersion model ω_q, which allows us to significantly improve the asymptotic behavior and the resulting calculated C₆ coefficients, overcoming the problem of unreliable C₆ coefficients from previous vdW-DF functionals. Our functional predicts very well the binding energies (<5% error) and separation for molecular dimers of commonly used reference datasets (such as S22, A24, X40, S66 etc.); performance in solids, layered structures, and noble gas dimers is also good.   

A fitted van der Waals density functional with accurate short and long-range interactions

A fitting scheme has been developed to create a new van der Waals density functional, called vdW-DF-fit. The original vdW-DF kernel, as well as the plasmon dispersion model, are numerically modified and fitted to minimize the binding energy and separation errors with respect to experimental/RPA/high-level quantum chemistry results of a list of commonly used reference datasets (S22, A24, X40, S66 etc.). We are also pursuing automated fitting schemes using machine-learning techniques. This is work in progress and we are reporting first, promising results.   

A Generalized Gradient Approximation with Local Parameters.

The dependence of typical generalized gradient approximations (GGA) for exchange-correlation (XC) on fixed, global parameters makes them incapable of providing good accuracy in predicting, simultaneously, molecular and solid properties. This drawback has motivated recent interest in constructing more flexible GGAs incorporating XC parameters that depend on the density, bounding their values to those for the high- and low-density regimes. In this work we explore a scheme to couple X and C functionals through the density-dependence of the gradient expansion coefficient derived beyond the random phase approximation [1,2] employed in different C functionals [3-5]. We discuss the implications of such local dependence of the gradient coefficient and show results of these GGAs for various properties involving test sets of atoms, molecules, and solids.


1. C. D. Hu, D. C. Langreth, Phys. Rev. B 33, 943 (1986).
2. M. Rasolt, D. J. W. Geldart, Phys. Rev. B 34, 1325 (1986).
3. J. P. Perdew, Phys. Rev. B 33, 8822 (1986).
4. J. P. Perdew, A. Ruzsinszky, G. I. Csonka, L. A. Constantin, J. Sun, Phys. Rev. Lett. 103, 026403 (2009).
5. A. Cancio, G. P. Chen, B. T. Krull, K. Burke, J. Chem. Phys. 149, 084116 (2018).   

Neural-Network Implementation of Transferable Kohn-Sham Exchange-Correlation Functionals

The accuracy of Kohn-Sham density functional theory [1] depends crucially on its approximated exchange-correlation functional. Structures of conventional density functionals depend on, however, combination of a number of physical constraints, making thereby the construction technically demanding and systematic improvement of their accuracy difficult. In order to overcome this difficulties, we present a new strategy free from the use of complicated physical constraints: representing the functional structure with Neural-network (NN), which is an extremely flexible function with a large number of parameters.
We demonstrate two types of NN-based functional. (i)Non-local functionals, which is made to reproduce the exact exchange-correlation potential of 1D model systemsconsisting of two electronsapplied with various external potentials [2], (ii)Semi-local functionals, which is made to reproduce densities provided by an accurate quantum chemical calculation of only a few molecules [3]. We show potentials of the NN-based functionals for high accuracy and out-of-training transferability.

[1]W.Kohn and L. J. Sham, Phys Rev. 140, A1133-A1138(1965).
[2]RN, R. Akashi, S. Sasaki, and S. Tsuneyuki, J. Chem. Phys. 148, 241737 (2018).
[3]RN, R. Akashi, O. Sugino, in preparation.   

On the Physical Origin of the 1/4 Exact Exchange in Density Functional Theory

This talk discusses the physical origin of the widely used 1/4 fraction of exact exchange in density functional theory (DFT)’s hybrid functionals. We show that this 1/4 fraction of exact exchange can be attributed to spin-flip exchange interactions between pair states with antiparallel spin, which arise from the coupling between doubly-excited configurations contributing to the ground state energy. To obtain this result, we compare two models of the exchange energy of an electron pair in an unpolarized electron gas. One model treats the spin states of the electron pair as a statistical ensemble, while the other includes their quantum superposition. The quantum mechanical treatment is shown to possess an exchange energy equal to that of the statistical ensemble, plus an additional 1/4 exact exchange energy due to a spin-flip exchange interaction between electron pair states with antiparallel spin. By showing that the statistical ensemble contains 3/4 of the density-based exchange energy, we can attribute, for each pair of occupied orbitals, the 1/4 exact exchange used in hybrid DFT to the spin-flip exchange processes mentioned above. Implications including the tendency of the LDA to overbind and the performance of hybrids for few-electron systems will be discussed.   

The Quasi-2D Electron Gas and Density Functional Theory: Finding a Finite Limit

The uniform electron gas in three and two dimensions is treated exactly by popular Kohn-Sham density functional approximations. However, no general-purpose semi-local functional can find the correct behavior of a 3D electron gas undergoing extreme compression in one dimension. In this talk, I will present our recent work [1] applying the SCAN functional to this perennial problem. While the exact exchange-correlation energy per electron tends to a finite 2D limit, the local density and generalized gradient approximations to it diverge to minus infinity. SCAN tends to a finite limit that is however an order of magnitude too negative. These errors at high compression are in one sense harmless, since the noninteracting kinetic energy, treated exactly in Kohn-Sham density functional theory, overwhelms them. Relevant background in Kohn-Sham density functional theory will be presented, and only passing familiarity is assumed.
[1] A.D. Kaplan, K. Wagle, and J.P. Perdew, Phys. Rev. B. 98, 085147 (2018).   

Response in the local, non-negative kinetic energy density of a perturbed free electron gas: potential functionals and density functionals

We discuss two sets of response functionals for estimating the local, non-negative kinetic energy density of initially free electrons that are perturbed by a static external potential. We begin with first- and second-order functionals of the perturbing potential, which are directly analogous to known response functionals for the electron density and the integrated kinetic energy. We provide reciprocal-space formulations of these functionals that complement previously known real-space expressions. We then present alternate first- and second-order response functionals that operate on the induced electron density and do not depend explicitly on the applied potential. The structure of these latter functionals will help guide the design of the more sophisticated kinetic energy functionals that enable orbital-free density functional theory simulations of materials. We examine the performance of the response functionals, in density-functional form, when applied to electron densities generated from local pseudopotential calculations for Li, Al, and Si solids.   

Large Z scaling of Atomic Pauli potentials

Modeling the Pauli energy, the contribution to kinetic energy of Pauli statistics, without using orbitals is an open problem for electronic structure theory. One aspect of this problem is correctly producing the Pauli potential, which is its response to a change in density. A powerful tool to analyze any density functional quantity is Lieb-Simon scaling, taking its limit in non-relativistic neutral atomic systems as nuclear charge and particle number approach infinity. We calculate the exact orbital-dependent Pauli potential for closed-shell atoms out to element Z=976. Lieb and Simon proposed five distinct regions of behavior in the large Z limit, of which our results show four: a region of constant potential near the nucleus, a core where the potential oscillates about the TF potential, a transition region where the potential deviates unexpectedly from the TF potential, and an evanescent region where it decays exponentially. We compare the exact potential to several models that are variations on the gradient expansion to test their utility as orbital-free approximations. This research may provide insight into semi-classical description of Pauli statistics, and new limiting behaviors to aid the improvement of orbital-free DFT functionals.   

Koopmans compliance: a functional theory for spectral properties

Energy functionals which depend explicitly on each individual orbital density,
rather than the total charge density, appear naturally when applying
self-interaction corrections to density-functional theory.
Rather than a limitation, we argue that this is a powerful feature, and show
how it is possible to interpret the orbital-dependency of densities and
potentials as an effective frequency dependency.
Such a frame is naturally amenable to describe electronic
spectroscopies [1], and is free from e.g. the constraint of having
derivative discontinuities in the exact formulation.
Excellent agreement is achieved for ionization potentials and affinities,
fundamental gaps, and deeper orbital levels (photoemission) for both molecules and solids [2-4].
We believe that a functional theory for the spectral density is therefore
emerging, able to address at the same time total energies and spectral properties.

[1] A. Ferretti, I. Dabo, M. Cococcioni, N. Marzari, Phys. Rev. B 89, 195134 (2014).
[2] N.-L. Nguyen, G. Borghi, A. Ferretti, I. Dabo, N. Marzari, Phys. Rev. Lett. 114, 166405 (2015).
[3] N. Colonna, A. Ferretti, N.-L. Nguyen, N. Marzari, J. Chem. Theory Comput. 14, 2549 (2018).
[4] N.-L. Nguyen, N. Colonna, A. Ferretti, N. Marzari, Phys. Rev. X 8, 021051 (2018).   

The Self-Consistent Field in Kohn-Sham Density Functional Theory: A Review of Methods and Algorithms

Software that computes the electronic structure of a material by searching for an infimum of the Kohn-Sham energy functional - achieving self-consistency - requires the use of an algorithm that iterates an initial estimate of the particle density toward the self-consistent particle density. We present a review of methods and algorithms that solve this problem in the context of plane-wave, pseudopotential density functional theory.

This review involves first identifying features intrinsic to the Kohn-Sham framework that produce sources of numerical difficulty for self-consistency algorithms. A test suite of over fifty Kohn-Sham inputs is then designed in order to exploit these difficulties and target weaknesses in contemporary algorithms. In order to utilise this test suite, a sample of methods from literature are then implemented in the plane-wave, pseudopotential density functional theory code, CASTEP. These methods are then benchmarked on the test suite, and a novel set of analysis tools allow us to determine the effectiveness of each method. We believe this study will not only elucidate the current state of the literature, but also assist future method development by providing a systematic work-flow for presenting new methods and assessing their effectiveness.   

Obtaining Stationary States in Density Functional Theory Using Imaginary Time Propagation

Density functional theory (DFT) is widely successful for determining electronic and structural properties of materials and molecules. In order to perform a DFT calculation, the Kohn-Sham nonlinear equations must be solved self-consistently, a task usually achieved using self-consistent field (SCF) loops. However, for large systems SCF can occasionally struggle to find the ground state. Using time-dependent DFT in imaginary time, a given starting state can be purified into its lowest-energy component with reliable monotonic convergence. With appropriate constraints, it is also possible to obtain excited states with imaginary time propagation (ITP). We compare and analyze the performance of ITP and SCF in a few difficult systems, and present a few excited state calculations performed with the method.
  

Path-integral imaginary-time TDDFT simulation of the hydrogen-bonded systems

We have recently developed a new imaginary-time time-dependent density-functional theory (itTDDFT)-based approach for computing the ground-state electron-ionic structure of atomistic systems[1]. In this method the ionic degrees of freedom are treated in the path-integral formalism, whereas electronic degrees of freedom are treated with itTDDFT. Our method, given an exact functional of the electronic density, is in principle exact in the zero-temperature limit. With approximate functionals it offers a practical path to direct ab initio simulation of molecular and condensed matter systems going beyond the Born-Oppenheimer approximation. In this work we apply our method to study properties of hydrogen-bonded and tautomeric molecules: hydrogen sulfide, formic acid and terephthalic acid dimers. We compare the results of our simulations to known experimental and theoretical results.

[1] Kolesov G, Kaxiras E, Manousakis E. Density functional theory beyond the Born-Oppenheimer approximation: accurate treatment of the ionic zero-point motion. arXiv:1804.10852. Accepted at Phys. Rev. B (2018)   

Benchmarking the structure selection performance of the SCAN functional relative to PBE and PBE+D3

The ability of first-principles computational methods to reproduce ground-state crystal structure is key to their application in the study of structural phase transitions and the discovery of new materials. In assessing the reliability of structure selection made on the basis of density-functional theory, it is critical to understand the impact of various errors on structure stabilization arising from the choice of density functional. Here, we evaluate the SCAN functional performance in structure selection relative to widely-used PBE and PBE+D3 functionals and build on recent reports that the SCAN functional significantly improves crystal structure selection across a wide range of main group compounds [1]. We demonstrate that the origin of this improvement is to a large extent the inclusion of van der Waals interactions.

1. Y. Zhang et al. NPJ Comput. Mater. 4, 9 (2018)   

From one to three, exploring the rungs of Jacob’s ladder in magnetic alloys

Magnetic systems represent an important challenge for electronic structure methods, in particular Density Functional Theory (DFT). We benchmark different exchange correlation (xc) functionals with respect to each other, and with respect to available experimental data, on two families of binary iron alloys. We climb three rungs in Jacob’s ladder of DFT (i) the local density approximation, (ii) the industry standard approximations due to Perdew, Burke and Ernzerhof, and PBEsol, and finally (iii) the meta-GGA functional SCAN. More than 400 structures in ferromagnetic and antiferromagnetic configurations were considered. We compare the Convex Hull, magnetic moment, structure, and formation energy. None of the functionals work in all conditions: the GGAs and mGGA give a fair crystal structure, but SCAN strongly overestimates the formation energy (wrt PBE and experiment). Magnetic moments are better predicted by PBE as well. Our results show that magnetic and strongly correlated materials are a tough litmus test for DFT, and that they represent the next frontier towards a truly universal xc functional.

Romero and Verstraete Eur. Phys. J. B (2018) 91 193