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The large-Z asymptotic expansion of atomic exchange energies has been useful in determining exact conditions for corrections to the local density approximation in density functional theory. We find that the necessary correction is fit well with a leading ZlnZ term, and find its coefficient numerically. The gradient expansion approximation also displays such a term, but with a substantially smaller coefficient. Analytic results in the limit of vanishing interaction with hydrogenic orbitals (a Bohr atom) are given, leading to the conjecture that the true coefficients for all atoms are precisely 2.7 times larger than their gradient expansion counterpart. Combined with the hydrogen atom result, this yields an analytic expression for the exchange-energy correction which is accurate to ∼5% for all Z.
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Deriving approximate functionals with asymptotics
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.
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- Award ID(s):
- 1856165
- PAR ID:
- 10220560
- Date Published:
- Journal Name:
- Faraday Discussions
- Volume:
- 224
- ISSN:
- 1359-6640
- Page Range / eLocation ID:
- 98 to 125
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1039/D0FD00057D
