[Paper Review] Atomic correlation energies and the generalized gradient approximation

Kieron Burke, Antonio C. Cancio+2 more|arXiv (Cornell University)|Sep 16, 2014

Advanced Chemical Physics Studies3 citations

TL;DR

This paper proposes a non-empirical generalized gradient approximation (acGGA) for atomic correlation and exchange energies by deriving asymptotic behavior in the high-Z limit, where correlation energy scales as $-AZ\ln Z + BZ$. The method corrects the asymptotic coefficient $B$ using quantum chemical reference data and real-space GGA construction, reducing atomic correlation errors by a factor of 2 compared to PBE and by a factor of 4 for total XC energy when combined with improved exchange.

ABSTRACT

Careful extrapolation of atomic correlation energies suggests that $E_c$ tends to $-AZ\log{Z} + BZ$ as $Z$ tends to infinity, where $Z$ is the atomic number, $A$ is known, and $B$ is about 38 milliHartrees. The coefficients roughly agree with those of the high-density limit of the real-space construction of the generalized gradient approximation. An asymptotic coefficient, missed by previous derivations, is included in a revised approximation. The exchange is also corrected, reducing atomic errors considerably.

Motivation & Objective

• To determine the correct asymptotic behavior of atomic correlation energy in the $Z \to \infty$ limit, specifically the coefficient $B$ in $E_C \to -AZ\ln Z + BZ$.
• To identify and correct a missing asymptotic coefficient in existing generalized gradient approximations (GGA) that affects accuracy in the high-density limit.
• To develop a non-empirical, systematically derived GGA functional (acGGA) that satisfies exact asymptotic conditions for both correlation and exchange.
• To reduce errors in atomic correlation and exchange energies by leveraging quantum chemical reference data and semiclassical asymptotic analysis.
• To validate the new functional across the periodic table using RPA-calibrated correlation energies and assess its performance relative to established functionals like PBE and LYP.

Proposed method

• Extrapolate high-accuracy quantum chemical (QC) correlation energies for spherical atoms to $Z \to \infty$ to extract the asymptotic coefficient $B$.
• Use the real-space construction of the GGA to model the gradient correction to correlation energy, matching the derived asymptotic form.
• Derive a corrected exchange functional by adjusting the PBE exchange coefficient $\mu$ by +13% to better match the high-Z limit of exact exchange.
• Combine the improved correlation and exchange functionals into a new non-empirical GGA, acGGA, using a consistent asymptotic framework.
• Validate the functional using RPA calculations for non-spherical atoms up to Xe, correcting them to match QC reference data for correlation energy.
• Apply Lieb-Simon scaling to confirm the generalizability of the asymptotic behavior to molecules and solids.

Experimental results

Research questions

• RQ1 What is the correct asymptotic coefficient $B$ in the large-$Z$ expansion of atomic correlation energy, $E_C \to -AZ\ln Z + BZ$?
• RQ2 Why do standard GGAs like LYP and PBE fail or underperform in the high-$Z$ limit, and what asymptotic condition is missing in their construction?
• RQ3 Can a non-empirical GGA be systematically derived from the high-density limit of the uniform and slowly-varying electron gas to improve accuracy?
• RQ4 How does correcting the asymptotic coefficient $B$ and the exchange parameter $\mu$ affect the total atomic correlation and exchange energy errors?
• RQ5 To what extent do the errors in acGGA for correlation and exchange cancel in the total XC energy, and how does this compare to PBE?

Key findings

• The coefficient $B$ in the large-$Z$ correlation energy expansion is found to be approximately 38 milliHartrees, with a precise value extracted from extrapolated quantum chemical data.
• The real-space GGA construction correctly reproduces the asymptotic behavior, validating its use in non-empirical functional development.
• The acGGA functional reduces the mean absolute error (MAE) in atomic correlation energy by a factor of 2 compared to PBE.
• When combined with a corrected exchange functional, the total XC energy error of acGGA is reduced by a factor of 4 compared to PBE.
• The largest errors in acGGA occur at elements where a new angular momentum shell (2p, 3d, 4f) is first filled, due to limitations in the semiclassical expansion at such points.
• The acGGA functional is non-empirical and fully derived from asymptotic conditions, with explicit formulas provided in the supplementary information.