Chemical bonds between atoms are stabilized by the exchange-correlation (xc) energy, a quantum-mechanical effect in which “social distancing” by electrons lowers their electrostatic repulsion energy. Kohn-Sham density functional theory (DFT) ( 1 ) states that the electron density determines this xc energy, but the density functional must be approximated. This is usually done by satisfying exact constraints of the exact functional (making the approximation predictive), by fitting to data (making it interpolative), or both. Two exact constraints—the ensemble-based piecewise linear variation of the total energy with respect to fractional electron number ( 2 ) and fractional electron z -component of spin ( 3 )—require hard-to-control nonlocality. On page 1385 of this issue, Kirkpatrick et al. ( 4 ) have taken a big step toward more accurate predictions for chemistry through the machine learning of molecular data plus the fractional charge and spin constraints, expressed as data that a machine can learn.
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The Predictive Power of Exact Constraints and Appropriate Norms in Density Functional Theory
Ground-state Kohn-Sham density functional theory provides, in principle, the exact ground-state energy and electronic spin densities of real interacting electrons in a static external potential. In practice, the exact density functional for the exchange-correlation (xc) energy must be approximated in a computationally efficient way. About 20 mathematical properties of the exact xc functional are known. In this work, we review and discuss these known constraints on the xc energy and hole. By analyzing a sequence of increasingly sophisticated density functional approximations (DFAs), we argue that ( a) the satisfaction of more exact constraints and appropriate norms makes a functional more predictive over the immense space of many-electron systems and ( b) fitting to bonded systems yields an interpolative DFA that may not extrapolate well to systems unlike those in the fitting set. We discuss both how the class of well-described systems has grown along with constraint satisfaction and the possibilities for future functional development.
more » « less- Award ID(s):
- 2344734
- PAR ID:
- 10501614
- Publisher / Repository:
- NSF-PAR
- Date Published:
- Journal Name:
- Annual Review of Physical Chemistry
- Volume:
- 74
- Issue:
- 1
- ISSN:
- 0066-426X
- Page Range / eLocation ID:
- 193 to 218
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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In density-functional theory, the exchange–correlation (XC) energy can be defined exactly through the coupling-constant (λ) averaged XC hole n̄xc(r,r′), representing the probability depletion of finding an electron at r′ due to an electron at r. Accurate knowledge of n̄xc(r,r′) has been crucial for developing XC energy density-functional approximations and understanding their performance for molecules and materials. However, there are very few systems for which accurate XC holes have been calculated since this requires evaluating the one- and two-particle reduced density matrices for a reference wave function over a range of λ while the electron density remains fixed at the physical (λ = 1) density. Although the coupled-cluster singles and doubles (CCSD) method can yield exact results for a two-electron system in the complete basis set limit, it cannot capture the electron–electron cusp using finite basis sets. Focusing on Hooke’s atom as a two-electron model system for which certain analytic solutions are known, we examine the effect of this cusp error on the XC hole calculated using CCSD. The Lieb functional is calculated at a range of coupling constants to determine the λ-integrated XC hole. Our results indicate that, for Hooke’s atoms, the error introduced by the description of the electron–electron cusp using Gaussian basis sets at the CCSD level is negligible compared to the basis set incompleteness error. The system-, angle-, and coupling-constant-averaged XC holes are also calculated and provide a benchmark against which the Perdew–Burke–Ernzerhof and local density approximation XC hole models are assessed.
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1146/annurev-physchem-062422-013259
