1. WHEN AND WHY DO SEMILOCAL DENSITY FUNCTIONALS WORK?
JOHN P. PERDEW
DEPARTMENTS OF PHYSICS AND CHEMISTRY
TEMPLE UNIVERSITY
PHILADELPHIA, PA
WINTERSCHOOL ON COMPUTATIONAL CHEMISTRY 2016
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2. KOHN-SHAM DENSITY FUNCTIONAL THEORY HAS GROWN UP IN 50 YEARS TO BECOME THE MOST WIDELY-USED METHOD OF ELECTRONIC STRUCTURE CALCULATION IN PHYSICS AND CHEMISTRY, AND THE LYNCH-PIN OF THE COMPUTATIONAL MATERIALS-DESIGN INITIATIVE. IT REDUCES THE PROBLEM OF FINDING GROUND-STATE ENERGIES AND ELECTRON DENSITIES TO A SELFCONSISTENT ONE-ELECTRON PROBLEM, IN A WAY THAT IS IN PRINCIPLE EXACT. IN PRACTICE, THE DENSITY FUNCTIONAL FOR THE EXCHANGE-CORRELATION ENERGY MUST BE APPROXIMATED.
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3. JACOB`S LADDER OF DENSITY FUNCTIONAL APPROXIMATIONS
exact x info.
unoccupied KS orbitals
HERE
POSITIVE KE DENSITY.
BY ADDING INGREDIENTS, WE CAN SATISFY ADDITIONAL EXACT CONSTRAINTS AND ACHIEVE GREATER ACCURACY, FOR SOME INCREASE IN
COMPUTATIONAL COST.
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4. THIS IS A FIVE-RUNG LADDER OF APPROXIMATIONS, IN WHICH EACH RUNG ADDS A NEW INGREDIENT TO THE ENERGY DENSITY. THE FIRST THREE RUNGS ARE COMPUTATIONALLY SEMILOCAL AND THUS EFFICIENT, AND CAN BE NON-EMPIRICAL. THEY ARE NEEDED FOR: LARGE SYSTEMS, LONG-TIME MOLECULAR DYNAMICS SIMULATIONS, HIGH-THROUGHPUT MATERIALS SEARCHES. LOCAL SPIN DENSITY APPROXIMATION (LSDA) GENERALIZED GRADIENT APPROXIMATION (GGA): (E.G., PBE) META-GGA (MGGA): (E.G., SCAN) THE TWO HIGHEST RUNGS ARE FULLY NONLOCAL. THE FOURTH RUNG OF HYBRID FUNCTIONALS WITH EXACT EXCHANGE CAN BE MUCH MORE EXPENSIVE IN PLANE-WAVE CODES. THE FIFTH RUNG OF RPA-LIKE FUNCTIONALS IS MUCH MORE EXPENSIVE THAN THE FOURTH.
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5. THE EXACT EXCHANGE-CORRELATION (XC) ENERGY IS NEGATIVE
THE EXACT EXCHANGE-CORRELATION (XC) ENERGY IS NEGATIVE. IT IS THE (NONLOCAL OR DOUBLE-INTEGRAL) ELECTROSTATIC INTERACTION BETWEEN THE ELECTRON DENSITY AT POSITION A AND THE DENSITY AT POSITION B OF THE XC HOLE AROUND AN ELECTRON AT POSITION A. (LANGRETH AND PERDEW 1975, GUNNARSSON AND LUNDQVIST 1976) THE XC HOLE IS THE SUM OF AN X HOLE AND A COUPLING-CONSTANT- AVERAGED C HOLE. HOLE SUM RULES: THE X HOLE DENSITY IS NEGATIVE AND INTEGRATES TO -1, BECAUSE THE ELECTRON AT POSITION A IS MISSING FROM THE REST OF THE DENSITY. THE C HOLE DENSITY INTEGRATES TO 0.
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6. THE ELECTRONS ARE LIKE A SWARM OF BEES
THE ELECTRONS ARE LIKE A SWARM OF BEES. THEY AVOID BUMPING INTO ONE ANOTHER, AN EFFECT THAT LOWERS THE ELECTROSTATIC ENERGY. EACH ELECTRON IS SURROUNDED BY A HOLE OR PERSONAL SPACE INTO WHICH ANOTHER ELECTRON RARELY INTRUDES. AS TWO ATOMS APPROACH ONE ANOTHER, THERE IS MORE OPPORTUNITY FOR ELECTRONS TO AVOID ONE ANOTHER. THIS IS THE ORIGIN OF MOST OF THE BONDING IN REAL MOLECULES AND SOLIDS. THE XC ENERGY IS “NATURE’S GLUE”, BINDING ONE ATOM TO ANOTHER.
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7. A SEMILOCAL DENSITY FUNCTIONAL FOR X CAN ONLY BE ACCURATE WHEN THE EXACT X HOLE REMAINS CLOSE TO ITS ELECTRON. THIS HAPPENS IN A UNIFORM OR SLOWLY-VARING ELECTRON DENSITY, AND ALSO IN AN ATOM. THESE SYSTEMS ARE “APPROPRIATE NORMS” TO WHICH SEMILOCAL FUNCTIONALS FOR X CAN BE FITTED. MOLECULES AND REAL SOLIDS ARE NOT APPROPRIATE NORMS, AND SEMILOCAL FUNCTIONALS FOR X ARE NOT ACCURATE FOR THEM. BUT SEMILOCAL FUNCTIONALS FOR XC CAN STILL WORK DUE TO AN UNDERSTOOD BUT NOT FULLY RELIABLE ERROR CANCELLATION BETWEEN X AND C: THE EXACT XC HOLE IS DEEPER AND MORE LOCALIZED AROUND ITS ELECTRON THAN THE EXACT X HOLE IS.
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8. SEMILOCAL FUNCTIONALS MUST FAIL WHEN THE EXACT XC HOLE IS NOT EVERYWHERE CLOSE TO ITS ELECTRON. THE CLASSIC EXAMPLE IS STRETCHED H2+, A ONE-ELECTRON SYSTEM IN WHICH THE EXACT XC HOLE EQUALS THE EXACT X HOLE AND IS EQUALLY SHARED OVER TWO DISTANT NUCLEAR CENTERS. ONLY A FULLY NONLOCAL FUNCTIONAL (SUCH AS A SELF-INTERACTION-CORRECTED SEMILOCAL FUNCTIONAL) CAN BE ACCURATE IN SUCH A CASE.
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9. THE LOCAL DENSITY APPROXIMATION (EXACT BY CONSTRUCTION FOR AN ELECTRON GAS OF UNIFORM DENSITY) WAS MORE ACCURATE THAN EXPECTED BECAUSE IT SATISFIED THE HOLE SUM RULES. THE HOLE SUM RULES WERE EARLY EXACT CONSTRAINTS USED TO CONSTRUCT EARLY GENERALIZED GRADIENT APPROXIMATIONS (GGA’S). THE LATER DISCOVERY OF EXACT CONSTRAINTS ON THE XC ENERGY FUNCTIONAL ITSELF (LEVY, LIEB, …) HAS LED TO BETTER GGA’S AND META-GGA’S. ONLY THE BEST AND MOST FULLY-CONSTRAINED META-GGA’s CAN BE EXPECTED TO BE ACCURATE WHEN THE EXACT XC HOLE IS WELL LOCALIZED AROUND ITS ELECTRON.
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10. INTRODUCTION TO META-GGA
EFFICIENT SEMILOCAL APPROXIMATIONS TO 𝐸 𝑥𝑐 :
THE ROLE OF 𝜏 (SPIN SUPPRESSED)
LIMITING FORMS:
SINGLE-ORBITAL SYSTEMS:
𝜏 𝑤 = 1 8 |𝛻𝑛| 2 /𝑛
UNIFORM ELECTRON GAS (UEG):
𝜏 𝑢𝑛𝑖𝑓 = (3 𝜋 2 ) 2/3 𝑛 5/3
LSDA: 𝑛 𝜎 = 𝑖 𝑜𝑐𝑐𝑢𝑝 | 𝜓 𝑖,𝜎 | 2
GGA: ADDS 𝛻𝑛 ↑ AND 𝛻𝑛 ↓
MGGA: ADDS 𝜏 𝜎 = 𝑖 𝑜𝑐𝑐𝑢𝑝 |𝛻 𝜓 𝑖,𝜎 | 2
𝜓 𝑖,𝜎 : KOHN−SHAM ORBITALS
RIGHT DIMENSIONLESS INGREDIENTS FOR MGGA: 𝛼= 𝜏− 𝜏 𝑤 𝜏 𝑢𝑛𝑖𝑓
REGION 𝛼
SINGLE ORBITAL (COVALENT)
SLOWLY-VARYING DENSITY (METALLIC) ~1
OVERLAP OF CLOSED SHELLS (NONCOVALENT)
>>1
Now, let’s show an example that already reveal the advantages of MGGAs. The right upper panel shows the adsorption of CO molecules on the Pt (111) surface at the top site, meaning that the carbon atom of the molecule sits on top of a Pt atom of the surface. Here we calculate two properties, 1) the energy to desorb the molecule from the surface; 2) the energy needed to cut the bulk Pt into two semi-infinite surfaces. We found that modifying GGA forms can not simultaneously improve both descriptions, as shown in the right lower panel. The GGAs move along the line but never move towards the Expt points. However, the revTPSS metaGGA obviously improves the descriptions simultaneously, and move towards the expt data. Of course, the high-level RPA gives very accurate descriptions for these two energies. But it is computationally too expensive. About 100 times more expensive.
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J. SUN et al, PRL, 111, (2013)

11. LSDA AND GGA ARE SEMILOCAL FUNCTIONALS OF THE ELECTRON DENSITY
LSDA AND GGA ARE SEMILOCAL FUNCTIONALS OF THE ELECTRON DENSITY. META-GGA IS A FULLY NONLOCAL FUNCTIONAL OF THE DENSITY, BECAUSE THE OCCUPIED KOHN-SHAM ORBITALS ARE SUCH FUNCTIONALS. BUT META-GGA IS STILL COMPUTATIONALLY SEMILOCAL, BECAUSE IT YIELDS THE EXCHANGE-CORRELATION ENERGY AS A SINGLE INTEGRAL OVER 3D SPACE OF A FUNCTION OF INGREDIENTS AT EACH POINT WHICH ARE READILY AVAILABLE IN ANY KOHN-SHAM CALCULATION.
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12. SCAN META-GGA: AN ACCURATE, EFFICIENT, AND SOUNDLY-BASED SEMILOCAL DENSITY FUNCTIONAL FOR MOLECULES AND MATERIALS?
SUPPORTED BY NSF-DMR, AND BY DOE-BES THROUGH THE ENERGY FRONTIER RESEARCH CENTER CCDM.
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13. STRONGLY CONSTRAINED AND APPROPRIATELY NORMED (SCAN) META-GENERALIZED GRADIENT APPROXIMATION FOR EXCHANGE AND CORRELATION Jianwei Sun, Adrienn Ruzsinszky, & John P. Perdew Physical Review Letters 115, (2015) Jianwei Sun, et al., submitted
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14. BASIC IDEAS AND GOALS OF SCAN
MODEL 𝜀 𝑥/𝑐 0 FOR SINGLE-ORBITAL REGIONS (α=0) AND 𝜀 𝑥/𝑐 1 FOR SLOWLY- VARYING REGIONS (α≈1), RESPECTIVELY, AND USE 𝑓 𝑥/𝑐 α TO INTERPOLATE BETWEEN THEM AND EXTRAPOLATE TO α→∞.
SATISFY AS MANY (17) EXACT CONSTRAINTS AS POSSIBLE (INCLUDING ALL CONSTRAINTS SATISFIED BY PBE)
NORM OR FIT APPROPRIATELY
A SEMILOCAL FUNCTIONAL SHOULD BE ACCURATE FOR SYSTEMS IN WHICH THE EXACT XC HOLE IS WELL-LOCALIZED AROUND ITS ELECTRON. THIS IS THE CASE FOR BONDS NEAR EQUILIBRIUM IN WEAKLY-CORRELATED SYSTEMS.
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15. APPROPRIATE NORMS: GUIDING SCAN FROM 𝛼=0 TO 𝛼=1, THEN TO 𝛼→∞
APPROPRIATE NORMS: SYSTEMS WITH LOCALIZED XC HOLES (SUFFICIENT IF X HOLES ARE LOCALIZED)
UNIFORM ELECTRON GAS AND H ATOM
RARE-GAS ATOMS
THREE COEFFICIENTS OF LARGE-Z LIMIT OF THE EXCHANGE AND CORRELATION ENERGIES OF NEUTRAL ATOMS
lim 𝑍→∞ 𝐸 𝑥 𝑍 = 𝐸 𝑥 𝐿𝐷𝐴 + 𝛾 𝑥1 𝑍+ 𝛾 𝑥2 𝑍 2/3
lim 𝑍→∞ 𝐸 𝑐 𝑍 = 𝐸 𝑐 𝐿𝐷𝐴 + 𝛾 𝑐1 𝑍 (KIERON BURKE)
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16. EARLY RESULTS FOR MOLECULES AND SOLIDS
MAE: MEAN ABSOLUTE ERROR
G3 (kcal/mol)
BH76 (kcal/mol)
S22 (kcal/mol)
LC20 (Å) ME
MAE
GGA
PBE
22.2
9.2
2.8
0.059
MGGA
SCAN -
5.7
7.7
0.9
0.016
LC20: LATTICE CONSTANTS OF 20 SOLIDS INCLUDING METALS, SEMICONDUCTORS, AND INSULATORS
The reason why SCAN seems to do better jobs for solids than molecules probably is that solids have more localized exchange-correlation holes than molecules do. This is also partly the reason why LSDA can give reasonable accurate predictions for solids and is still popular in solid state physics.
G3: ATOMIZATION ENERGIES OF 223 MOLECULES
BH76: 76 REACTION BARRIERS
S22: 22 MOLECULAR COMPLEXES BOUND BY WEAK BONDS
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17. THE NONEMPIRICAL SCAN META-GGA IS MORE ACCURATE THAN THE NONEMPIRICAL PBE GGA, ESPECIALLY FOR WEAK BONDS, PRESUMABLY BECAUSE SCAN SATISFIES MORE EXACT CONSTRAINTS. (SOFT MATTER AS WELL AS HARD!) NOTE THAT SCAN IS NOT FITTED TO ANY BONDED SYSTEM. THUS ITS RESULTS FOR BONDS ARE GENUINE PREDICTIONS, NOT FITS.
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18. FOR MANY PROBLEMS (E.G., THE CRITICAL PRESSURES OF STRUCTURAL PHASE TRANSITIONS, OR INTERSTITIAL FORMATION ENERGIES), SCAN IS AS ACCURATE AS THE MORE COMPUTATIONALLY EXPENSIVE HYBRID FUNCTIONALS THAT MIX SEMILOCAL AND EXACT EXCHANGE. FOR vdW BONDS, SCAN IS MUCH MORE ACCURATE THAN STANDARD HYBRID FUNCTIONALS.
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19. ALTHOUGH SCAN SLIGHTLY OVERBINDS ONE WATER MOLECULE TO ANOTHER, IT STILL PREDICTS THE CORRECT ENERGETIC ORDER OF THE FOUR WATER HEXAMERS OF LOWEST ENERGY. MOREOVER, FOR A FIXED NUMBER OF HYDROGEN BONDS, IT CORRECTLY PREDICTS THE ENERGY DIFFERENCES PER H2O UNIT BETWEEN THESE CLUSTERS, AND BETWEEN THE DIFFERENT PHASES OF ICE. THIS SHOWS THAT SCAN CAPTURES MOST OF THE VAN DER WAALS INTERACTION IN WATER.
3 TESTED FERROELECTRICS/MULTIFERROICS: SCAN MUCH BETTER THAN PBE
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20. STABILITIES OF CRYSTAL STRUCTURES
Gerd Ceder and collaborators found the wrong (alpha) ground-state crystal structure for MnO2, using PBE and PBE+U. Haowei Peng tried SCAN, and found the right (beta) ground-state crystal structure for this material, in agreement with experiment. In fact, SCAN seems to be accurate for all tested properties of MnO2. SCAN BAND GAPS IN A GKS SCHEME ARE SIMILAR TO OR BETTER THAN PBE GAPS, AND (FOR STRONGLY-CORRELATED SYSTEMS) OFTEN CLOSE TO PBE+U GAPS
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21. NONLOCAL CORRECTIONS TO SCAN
LONG-RANGE VAN DER WAALS INTERACTION
(SCAN+rVV10, DEVELOPED BY HAOWEI PENG).
IMPORTANT MAINLY FOR THE EXFOLIATION ENERGIES OF
LAYERED MATERIALS.
(2) FOR SELF-INTERACTION CORRECTION
(UNDER DEVELOPMENT).
NEEDED FOR STRETCHED BONDS OR STRONG CORRELATION.
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22. SUMMARY OF SCAN
A META-GGA, STRONGLY CONSTRAINED AND APPROPRIATELY NORMED.
COMPUTATIONALLY EFFICIENT AND NONEMPIRICAL.
SYSTEMATIC IMPROVEMENT OVER THE PBE GGA FOR A WIDE RANGE OF PROPERTIES FROM THE SCAN META-GGA.
OFTEN PERFORMS LIKE A HYBRID FUNCTIONAL, AT LOWER COST AND WITH INTERMEDIATE-RANGE VAN DER WAALS INTERACTION.
WE EXPECT TO FIND MAJOR IMPROVEMENTS IN THE FORMATION ENERGIES AND THUS IN THE RELATIVE STABILITIES OF MOLECULES AND SOLIDS (EVEN WITHOUT NONLOCAL CORRECTIONS).
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23. SUMMARY OF THE LECTURE
SEMILOCAL DENSITY FUNCTIONALS ARE COMPUTATIONALLY EFFICIENT. THEY CAN ALSO BE ACCURATE, BUT ONLY TO THE EXTENT THAT THE EXACT XC HOLE IS WELL LOCALIZED AROUND ITS ELECTRON. THE MOST ACCURATE SEMILOCAL FUNCTIONALS ARE META-GGA’s. THOSE LIKE SCAN THAT SATISFY MANY EXACT CONSTRAINTS ARE WIDELY RELIABLE. THE MOST IMPORTANT EXACT CONSTRAINT THAT CAN ONLY BE SATISFIED BY A FULLY NONLOCAL (EXPENSIVE) FUNCTIONAL IS PROBABLY EXACTNESS FOR ALL ONE-ELECTRON DENSITIES (FOR WHICH THE EXACT XC ENERGY SHOULD PERFECTLY CANCEL THE HARTREE ELECTROSTATIC ENERGY).
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