The discontinuous nature of the exchange-correlation functional -- critical for strongly correlated systems

Standard approximations for the exchange-correlation functional have been found to give big errors for the linearity condition of fractional charges, leading to delocalization error, and the constancy condition of fractional spins, leading to static correlation error. These two conditions are now unified for states with both fractional charge and fractional spin: the exact energy functional is a plane, linear along the fractional charge coordinate and constant along the fractional spin coordinate with a line of discontinuity at the integer. This sheds light on the nature of the derivative discontinuity and calls for explicitly discontinuous functionals of the density or orbitals that go beyond currently used smooth approximations. This is key for the application of DFT to strongly correlated systems.Comment: 5 pages 2 figure

Fractional spins and static correlation error in density functional theory

Electronic states with fractional spins arise in systems with large static correlation (strongly correlated systems). Such fractional-spin states are shown to be ensembles of degenerate ground states with normal spins. It is proven here that the energy of the exact functional for fractional-spin states is a constant, equal to the energy of the comprising degenerate pure spin states. Dramatic deviations from this exact constancy condition exist with all approximate functionals, leading to large static correlation errors for strongly correlated systems, such as chemical bond dissociation and band structure of Mott insulators. This is demonstrated with numerical calculations for several molecular systems. Approximating the constancy behavior for fractional spins should be a major aim in functional constructions and should open the frontier for DFT to describe strongly correlated systems. The key results are also shown to apply in reduced density-matrix functional theory.Comment: 6 pages, 4 figure

Localization and delocalization errors in density functional theory and implications for band-gap prediction

The band-gap problem and other systematic failures of approximate functionals are explained from an analysis of total energy for fractional charges. The deviation from the correct intrinsic linear behavior in finite systems leads to delocalization and localization errors in large or bulk systems. Functionals whose energy is convex for fractional charges such as LDA display an incorrect apparent linearity in the bulk limit, due to the delocalization error. Concave functionals also have an incorrect apparent linearity in the bulk calculation, due to the localization error and imposed symmetry. This resolves an important paradox and opens the possibility to obtain accurate band-gaps from DFT.Comment: 4 pages 4 figure

The derivative discontinuity of the exchange-correlation functional.

The derivative discontinuity is a key concept in electronic structure theory in general and density functional theory in particular. The electronic energy of a quantum system exhibits derivative discontinuities with respect to different degrees of freedom that are a consequence of the integer nature of electrons. The classical understanding refers to the derivative discontinuity of the total energy as a function of the total number of electrons (N), but it can also manifest at constant N. Examples are shown in models including several hydrogen systems with varying numbers of electrons or nuclear charge (Z), as well as the 1-dimensional Hubbard model (1DHM). Two sides of the problem are investigated: first, the failure of currently used approximate exchange-correlation functionals in DFT and, second, the importance of the derivative discontinuity in the exact electronic structure of molecules, as revealed by full configuration interaction (FCI). Currently, all approximate functionals, including hybrids, miss the derivative discontinuity, leading to basic errors that can be seen in many ways: from the complete failure to give the total energy of H2 and H2(+), to the missing gap in Mott insulators such as stretched H2 and the thermodynamic limit of the 1DHM, or a qualitatively incorrect density in the HZ molecule with two electrons and incorrect electron transfer processes. Description of the exact particle behaviour of electrons is emphasised, which is key to many important physical processes in real systems, especially those involving electron transfer, and offers a challenge for the development of new exchange-correlation functionals.We gratefully acknowledge funding from the Royal Society (AJC) and Ramon y Cajal (PMS). PMS also acknowledges grant FIS2012-37549 from the Spanish Ministry of Science.This is the final published version, which can also be viewed on the publisher's website here: http://pubs.rsc.org/en/content/articlelanding/2014/cp/c4cp01170h#!divAbstrac

Dramatic changes in electronic structure revealed by fractionally charged nuclei.

Discontinuous changes in the electronic structure upon infinitesimal changes to the Hamiltonian are demonstrated. These are revealed in one and two electron molecular systems by full configuration interaction (FCI) calculations when the realm of the nuclear charge is extended to be fractional. FCI electron densities in these systems show dramatic changes in real space and illustrate the transfer, hopping, and removal of electrons. This is due to the particle nature of electrons seen in stretched systems and is a manifestation of an energy derivative discontinuity at constant number of electrons. Dramatic errors of density functional theory densities are seen in real space as this physics is missing from currently used approximations. The movements of electrons in these simple systems encapsulate those in real physical processes, from chemical reactions to electron transport and pose a great challenge for the development of new electronic structure methods.We thank the Royal Society and Ramón y Cajal for funding. P.M.S. also acknowledges Grant No. FIS2009-12721 from the Spanish Ministry of Science and Innovation.This article appeared in the Journal of Chemical Physics 140 and may be found at http://scitation.aip.org/content/aip/journal/jcp/140/4/10.1063/1.4858461.Copyright 2014 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics

Landscape of an exact energy functional

One of the great challenges of electronic structure theory is the quest for the exact functional of density functional theory. Its existence is proven, but it is a complicated multivariable functional that is almost impossible to conceptualize. In this paper the asymmetric two-site Hubbard model is studied, which has a two-dimensional universe of density matrices. The exact functional becomes a simple function of two variables whose threedimensional energy landscape can be visualized and explored. A walk on this unique landscape, tilted to an angle defined by the one-electron Hamiltonian, gives a valley whose minimum is the exact total energy. This is contrasted with the landscape of some approximate functionals, explaining their failure for electron transfer in the strongly correlated limit. We show concrete examples of pure-state density matrices that are not v representable due to the underlying nonconvex nature of the energy landscape. The exact functional is calculated for all numbers of electrons, including fractional, allowing the derivative discontinuity to be visualized and understood. The fundamental gap for all possible systems is obtained solely from the derivatives of the exact functional.We gratefully acknowledge funding from Ramon y Cajal (P.M.-S.) and the Royal Society (A.J.C.). P.M.S. also acknowledges funding from MINECO Grant No. FIS2015-64886-C5-5-P

Fractional charge perspective on the band-gap in density-functional theory

The calculation of the band-gap by density-functional theory (DFT) methods is examined by considering the behavior of the energy as a function of number of electrons. It is found that the incorrect band-gap prediction with most approximate functionals originates mainly from errors in describing systems with fractional charges. Formulas for the energy derivatives with respect to number of electrons are derived which clarify the role of optimized effective potentials in prediction of the band-gap. Calculations with a recent functional that has much improved behavior for fractional charges give a good prediction of the energy gap and also $\epsilon_{{\rm homo}}\simeq-I$ for finite systems. Our results indicate it is possible, within DFT, to have a functional whose eigenvalues or derivatives accurately predict the band-gap

Failure of the random-phase-approximation correlation energy

The random phase approximation (RPA) is thought to be a successful method; however, basic errors have been found that have massive implications in the simplest molecular systems. The observed successes and failures are rationalized by examining its performance against exact conditions on the energy for fractional charges and fractional spins. Extremely simple tests reveal that the RPA method satisfies the constancy condition for fractional spins that leads to correct dissociation of closed-shell molecules and no static correlation error (such as in H2 dissociation) but massively fails for dissociation of odd electron systems, with an enormous delocalization error (such as H2 + dissociation). Other methods related to the RPA, including the Hartree-Fock response (RPAE) or range-separated RPA, can reduce this delocalization error but only at the cost of increasing the static correlation error. None of the RPA methods have the discontinuous nature required to satisfy both exact conditions and the full unified condition (e.g., dissociation of H2 + and H2 at the same time), emphasizing the need to go beyond differentiable energy functionals of the orbitals and eigenvaluesSupport from Ramón y Cajal (PMS), the Royal Society (A.J.C.), and the Office of Naval Research (N00014-09-0576) and National Science Foundation (CHE-2609-11119) (W.Y.) is greatly appreciated. P.M.S. also acknowledges grant FIS2009- 12721 from the Spanish Ministry of Science and Innovatio