Symmetry Breaking with the SCAN Density Functional Describes Strong Correlation in the Singlet C arbon Dimer John P. Perdewab Shah Tanvir ur Rahman Chowdhuryc Chandra Shahia Aaron D. Kaplana

2025-05-02 0 0 434.24KB 10 页 10玖币

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Symmetry Breaking with the SCAN Density Functional Describes

Strong Correlation in the Singlet Carbon Dimer

John P. Perdewa,b,*, Shah Tanvir ur Rahman Chowdhuryc, Chandra Shahia, Aaron D. Kaplana,

Duo Songd, and Eric J. Bylaskad

aDepartment of Physics, Temple University, Philadelphia, PA 19122

bDepartment of Chemistry, Temple University, Philadelphia, PA 19122

cThayer School of Engineering, Dartmouth College, Hanover, NH 03755

dPacific Northwest National Laboratory, Richland, WA 99353

Abstract: The SCAN (strongly constrained and appropriately normed) meta-generalized gradient

approximation (meta-GGA), which satisfies all 17 exact constraints that a meta-GGA can satisfy,

accurately describes equilibrium bonds that are normally correlated. With symmetry breaking, it also

accurately describes some sd equilibrium bonds that are strongly correlated. While sp equilibrium bonds

are nearly always normally correlated, the C2 singlet ground state is known to be a rare case of strong

correlation in an sp equilibrium bond. Earlier work that calculated atomization energies of the molecular

sequence B2, C2, O2, and F2 in the local spin density approximation (LSDA), the Perdew-Burke-Ernzerhof

(PBE) GGA, and the SCAN meta-GGA, without symmetry breaking in the molecule, found that only

SCAN was accurate enough to reveal an anomalous under-binding for C2. This work shows that spin

symmetry breaking in singlet C2, the appearance of net up- and down-spin densities on opposite sides (not

ends) of the bond, corrects that under-binding, with a small SCAN atomization-energy error more like that

of the other three molecules, suggesting that symmetry-breaking with an advanced density functional might

reliably describe strong correlation. This article also discusses some general aspects of symmetry breaking,

and the insights into strong correlation that symmetry-breaking can bring.

*Corresponding author: perdew@temple.edu, 1-215-204-1407

Introduction

The spin-density functional theory of Kohn and Sham1,2 is an exact in principle self-consistent-

field formalism for the ground-state energy and spin densities of a system of interacting non-relativistic

electrons in the presence of an external possibly-spin-dependent potential  In practice, only the

exchange-correlation energy as a functional of the spin densities must be approximated. The

computationally-efficient approximations on the first three rungs of Jacob’s Ladder3 are single integrals

of the form



 

,   . (1)

Here     



 is the total electron density, and

Symmetry Breaking with the SCAN Density Functional Describes Strong Correlation in the Singlet Carbon Dimer

Page 2 of 10

 





 (2)

is the positive orbital kinetic energy density. Rung 1 of the ladder (e.g., LSDA1,2.4) uses only the local

spin densities, rung 2 (e.g., PBE5 GGA) adds the gradients of the spin densities, and rung 3 (e.g., SCAN6

or r2SCAN7 meta-GGA) adds the kinetic energy densities.

There is a nearly straight line8 from the exact density functional theory to its first-principles

approximations (e.g., LSDA, PBE, or SCAN), which proceeds from exact (if impractical) expressions for

 to the derivation of mathematical properties of the exact functional that can be satisfied by an

approximation on a given rung, and finally to imposing those constraints on the approximate functional.

Appropriate norms, or systems in which a given rung can be accurate for the exchange energy alone and

the correlation energy alone, can be safely fitted. Appropriate norms for all three rungs include electron

gases of uniform spin densities. Appropriate norms for rung 3 include closed-subshell atoms. LSDA

inherits 8 exact constraints from its uniform gas norm, PBE satisfies 11 exact constraints appropriate to

the GGA level, and SCAN satisfies all 17 known exact constraints that a meta-GGA can satisfy. The

energies of molecules and solids, often fitted by empirical functionals, are not appropriate norms, because

of an understood but imperfect and uncontrolled cancellation between errors in the approximate exchange

and correlation energies. While empirical functionals interpolate between bonds, first-principles

approximations predict9 them, typically working better than empirical functionals for artificial molecules

that are unlike those commonly fitted.

SCAN with a long-range van der Waals correction was more accurate than other tested

functionals from the first three rungs, and more accurate than some hybrid functionals employing a

fraction of exact exchange, in a 2017 test10 on the GMTKN55 suite of 55 molecular test sets. SCAN is

also accurate for insulating solids. For some strongly-correlated sd-bound solids, such as the cuprates11

and manganese dioxides12, symmetry-broken SCAN is accurate without a “Hubbard-like:”+U correction

(sometimes interpreted as a self-interaction correction13). In other strongly-correlated transition-metal

oxides, symmetry-broken SCAN requires a +U correction, but one significantly smaller than PBE

requires14. Range-separated GGA hybrid functionals with short-range exact exchange can describe15 many

strongly-correlated solids better than LSDA or GGA and without a material-dependent parameter like U.

Such functionals16 and other nonlocal functionals17 can also describe polaronic symmetry breaking.

For equilibrium bonds, a self-interaction correction seems to be much more needed for sd

than for sp bonds. The Perdew-Zunger self-interaction correction18 is first-principles, but is not reliably

accurate due to the lobedness of its localized one-electron densities19. Perhaps in the future an improved

self-interaction correction to a SCAN-like functional will lead to a reliable and widely-useful description

of strong correlation via symmetry breaking. For now, any strongly-correlated equilibrium sp bond is of

special interest as a test of the ability of symmetry breaking to describe strong correlation. Such bonds are

rare, but the singlet ground state of the molecule C2 is known to be strongly correlated. This molecule has

an avoided crossing of two energy surfaces near its equilibrium bond length, and has been studied carefully

with the full configuration interaction quantum Monte Carlo (FCIQMC) correlated-wavefunction

approach.20

Symmetry breaking (in both wavefunction and density functional theories) and strong

correlation have attracted the interest of many, notably Gustavo E. Scuseria and Richard L. Martin15,22,23.

Strong correlation arises when degenerate or nearly degenerate Slater determinants are strongly mixed by

the electron-electron pair interaction. This leads to correlation or exchange-correlation energies more

negative than those that a standard approximate density functional can produce. A standard approximate

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SymmetryBreakingwiththeSCANDensityFunctionalDescribesStrongCorrelationintheSingletCarbonDimerJohnP.Perdewa,b,*,ShahTanvirurRahmanChowdhuryc,ChandraShahia,AaronD.Kaplana,DuoSongd,andEricJ.BylaskadaDepartmentofPhysics,TempleUniversity,Philadelphia,PA19122bDepartmentofChemistry,TempleUniversity,Philade...

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Symmetry Breaking with the SCAN Density Functional Describes Strong Correlation in the Singlet C arbon Dimer John P. Perdewab Shah Tanvir ur Rahman Chowdhuryc Chandra Shahia Aaron D. Kaplana

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