Ensemble density functional theory (EDFT) is a generalization of ground-state DFT, which is based on an exact formal theory of finite collections of a system’s ground and excited states. EDFT in various forms has been shown to improve the accuracy of calculated energy level differences in isolated model systems, atoms, and molecules, but it is not yet clear how EDFT could be used to calculate band gaps for periodic systems. We extend the application of EDFT toward periodic systems by estimating the thermodynamic limit with increasingly large finite one-dimensional ‘particle in a box’ systems, which approach the uniform electron gas (UEG). Using ensemble-generalized Hartree and local spin density approximation exchange-correlation functionals, we find that corrections go to zero in the infinite limit, as expected for a metallic system. However, there is a correction to the effective mass, with results comparable to other calculations on 1D, 2D, and 3D UEGs, which indicates promise for non-trivial results from EDFT on periodic systems.
Lieb's most useful contribution to density functional theory?
The importance of the Lieb-Simon proof of the relative exactness of Thomas-Fermi theory in the large-Z limit to modern density functional theory (DFT) is explored. The principle, that there is a specific semiclassical limit in which functionals become local, implies that there exist well-defined leading functional corrections to local approximations that become relatively exact for the error in local approximations in this limit. It is argued that this principle might be used to greatly improve the accuracy of the thousand or so DFT calculations that are now published each week. A key question is how to find the leading corrections to any local density approximation as this limit is approached. These corrections have been explicitly derived in ridiculously simple model systems to ridiculously high order, yielding ridiculously accurate energies. Much analytic work is needed to use this principle to improve realistic calculations of molecules and solids.
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- Award ID(s):
- 1856165
- PAR ID:
- 10332733
- Date Published:
- Journal Name:
- ArXivorg
- ISSN:
- 2331-8422
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Approaching periodic systems in ensemble density functional theory via finite one-dimensional modelsmore » « less
Abstract -
We argue that the success of DFT can be understood in terms of a semiclassical expansion around a very specific limit. This limit was identified long ago by Lieb and Simon for the total electronic energy of a system. This is a universal limit of all electronic structure: atoms, molecules, and solids. For the total energy, Thomas-Fermi theory becomes relatively exact in the limit. The limit can also be studied for much simpler model systems, including non-interacting fermions in a one-dimensional well, where the WKB approximation applies for individual eigenvalues and eigenfunctions. Summation techniques lead to energies and densities that are functionals of the potential. We consider several examples in one dimension (fermions in a box, in a harmonic well, in a linear half-well, and in the Pöschl-Teller well. The effects of higher dimension are also illustrated with the three-dimensional harmonic well and the Bohr atom, non-interacting fermions in a Coulomb well. Modern density functional calculations use the Kohn-Sham scheme almost exclusively. The same semiclassical limit can be studied for the Kohn-Sham kinetic energy, for the exchange energy, and for the correlation energy. For all three, the local density approximation appears to become relatively exact in this limit. Recent work, both analytic and numerical, explores how this limit is approached, in an effort to deduce the leading corrections to the local approximation. A simple scheme, using the Euler-Maclaurin summation formula, is the result of many different attempts at this problem. In very simple cases, the correction formulas are much more accurate than standard density functionals. Several functionals are already in widespread use in both chemistry and materials that incorporate these limits, and prospects for the future are discussed.more » « less
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Modern density functional approximations achieve moderate accuracy at low computational cost for many electronic structure calculations. Some background is given relating the gradient expansion of density functional theory to the WKB expansion in one dimension, and modern approaches to asymptotic expansions. A mathematical framework for analyzing asymptotic behavior for the sums of energies unites both corrections to the gradient expansion of DFT and hyperasymptotics of sums. Simple examples are given for the model problem of orbital-free DFT in one dimension. In some cases, errors can be made as small as 10 −32 Hartree suggesting that, if these new ingredients can be applied, they might produce approximate functionals that are much more accurate than those in current use. A variation of the Euler–Maclaurin formula generalizes previous results.more » « less
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Abstract Time-dependent density functional theory continues to draw a large number of users in a wide range of fields exploring myriad applications involving electronic spectra and dynamics. Although in principle exact, the predictivity of the calculations is limited by the available approximations for the exchange-correlation functional. In particular, it is known that the exact exchange-correlation functional has memory-dependence, but in practise adiabatic approximations are used which ignore this. Here we review the development of non-adiabatic functional approximations, their impact on calculations, and challenges in developing practical and accurate memory-dependent functionals for general purposes.
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null (Ed.)A major challenge in density functional theory (DFT) is the development of density functional approximations (DFAs) to overcome errors in existing DFAs, leading to more complex functionals. For such functionals, we consider roles of the noninteracting reference systems. The electron density of the Kohn–Sham (KS) reference with a local potential has been traditionally defined as being equal to the electron density of the physical system. This key idea has been applied in two ways: the inverse calculation of such a local KS potential for the reference from a given density and the direct calculation of density and energy based on given DFAs. By construction, the inverse calculation can yield a KS reference with the density equal to the input density of the physical system. In application of DFT, however, it is the direct calculation of density and energy from a DFA that plays a central role. For direct calculations, we find that the self-consistent density of the KS reference defined by the optimized effective potential (OEP), is not the density of the physical system, when the DFA is dependent on the external potential. This inequality holds also for the density of generalized KS (GKS) or generalized OEP reference, which allows a nonlocal potential, when the DFA is dependent on the external potential. Instead, the density of the physical system, consistent with a given DFA, is given by the linear response of the total energy with respect to the variation of the external potential. This is a paradigm shift in DFT on the use of noninteracting references: the noninteracting KS or GKS references represent the explicit computational variables for energy minimization, but not the density of the physical system for external potential-dependent DFAs. We develop the expressions for the electron density so defined through the linear response for general DFAs, demonstrate the results for orbital functionals and for many-body perturbation theory within the second-order and the random-phase approximation, and explore the connections to developments in DFT.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4171/90-1/7
