SCAN META-GGA: THE PREDICTIVE POWER OF 17 EXACT CONSTRAINTS

JOHN P. PERDEW

DEPARMENTS OF PHYSICS AND CHEMISTRY

TEMPLE UNIVERSITY, PHILADELPHIA, PA 19122 USA

ES17, Princeton U., JUNE 27, 2017

SUPPORTED BY NSF-DMR, AND BY DOE-BES THROUGH THE

ENERGY FRONTIER RESEARCH CENTER CCDM.

THE RIGHT ANSWER FOR THE RIGHT REASON AT THE RIGHT PRICE.

NOT AN IMPOSSIBLE DREAM!

HOW CAN WE APPROXIMATE THE DENSITY FUNCTIONAL FOR THE EXCHANGE-CORRELATION ENERGY?

IT IS OFTEN BUT INCORRECTLY SAID THAT “THERE IS NO SYSTEMATIC WAY TO CONSTRUCT A DENSITY-FUNCTIONAL APPROXIMATION”.

THERE IS A SYSTEMATIC OR “PHYSICS” WAY THAT PRODUCES EXTRAPOLATIVE OR PREDICTIVE FUNCTIONALS.

THERE IS ALSO A “PHYTTING” WAY THAT PRODUCES MAINLY INTERPOLATIVE FUNCTIONALS.

FIVE STEPS TO A PHYSICS-BASED DFT APPROXIMATION

(I) PROVE THE EXISTENCE OF THE EXACT DENSITY FUNCTIONAL, AND DERIVE EXACT FORMAL EXPRESSIONS FOR IT.

(II) DISCOVER MATHEMATICAL PROPERTIES OF THE EXACT FUNCTIONAL (EXACT CONSTRAINTS), INCLUDING LIMITS, SCALING RELATIONS, EQUALITIES, AND BOUNDS.

(III) DEVELOP APPROXIMATE BUT COMPUTATIONALLY TRACTABLE FORMS FOR THE APPROXIMATIONS AT VARIOUS LEVELS OF FLEXIBILITY: ADD INGREDIENTS (LSDA, GGA, META-GGA, HYBRID, RPA-LIKE)

(IV) IMPOSE THE EXACT CONSTRAINTS FROM STEP (II) ON A GIVEN FORM, AS APPROPRIATE.

(V) IF A FORM RETAINS SOME FLEXIBILITY, FIT TO APPROPRIATE NORMS, NON-BONDED SYSTEMS FOR WHICH THE FORM CAN BE EXPECTED TO BE ACCURATE.

ALL EXACT CONSTRAINTS ARE CREATED EQUAL, AND ENDOWED BY THEIR CREATOR WITH THE RIGHT TO BE SATISFIED.

STRONGLY CONSTRAINED AND APPROPRIATELY NORMED (SCAN) META-GENERALIZED

GRADIENT APPROXIMATION FOR EXCHANGE AND CORRELATION

CONSTRUCTED VIA STEPS (I)-(V). SATISFIES ALL 17 KNOWN EXACT CONSTRAINTS THAT A META-GGA CAN SATISFY. ITS ACCURACY FOR DIVERSELY-BONDED SYSTEMS CONFIRMS

STEPS (I)-(V).

J. SUN, A. RUZSINSZKY, AND J.P. PERDEW, PRL 2015J. SUN ET AL., NATURE CHEMISTRY 2016

THE 17 EXACT CONSTRAINTS SATISFIED BY SCAN.

AN ASTERISK INDICATES THE 11 EXACT CONSTRAINTS SATISFIED BY PBE.

FOR EXCHANGE:

*1. NEGATIVITY

*2. SPIN SCALING EQUALITY

*3. UNIFORM DENSITY SCALING

4. FOURTH-ORDER GRADIENT EXPANSION

5. FINITE NON-UNIFORM SCALING LIMIT

6. TIGHT LOWER BOUND FOR TWO-ELECTRON DENSITIES

FOR CORRELATION:

*7. NON-POSITIVITY

*8. SECOND-ORDER GRADIENT EXPANSION

*9. UNIFORM DENSITY SCALING TO THE HIGH-DENSITY LIMIT

*10. UNIFORM DENSITY SCALING TO THE LOW-DENSITY LIMIT

11. ZERO CORRELATION ENERGY FOR ONE-ELECTRON DENSITIES

12. FINITE NON-UNIFORM SCALING LIMIT

FOR BOTH:

*13. SIZE EXTENSIVITY

*14. GENERAL LIEB-OXFORD LOWER BOUND

*15. WEAK SPIN-POLARIZATION DEPENDENCE IN LOW-DENSITY LIMIT

*16. STATIC LINEAR RESPONSE OF THE UNIFORM GAS

17. LIEB-OXFORD LOWER BOUND FOR TWO-ELECTRON DENSITIES

INTRODUCTION TO META-GGA

THE ROLE OF 𝜏 (SPIN SUPPRESSED)

• LIMITING FORMS:

• SINGLE-ORBITAL SYSTEMS:

𝜏𝑤 =1

8|𝛻𝑛|2/𝑛

• UNIFORM ELECTRON GAS (UEG):

𝜏𝑢𝑛𝑖𝑓 =3

10(3𝜋2)2/3𝑛5/3

REGION 𝛼

SINGLE ORBITAL (COVALENT) 0

SLOWLY-VARYING DENSITY (METALLIC)

~1

OVERLAP OF CLOSED SHELLS (NONCOVALENT)

>>1

RIGHT DIMENSIONLESS INGREDIENT FOR MGGA: 𝛼 =𝜏−𝜏𝑤

𝜏𝑢𝑛𝑖𝑓

),,,,,(],[ 3

nnnnrndnnE xcxcEFFICIENT SEMILOCAL APPROXIMATIONS TO 𝐸𝑥𝑐:

• LSDA: 𝑛𝜎 = σ𝑖𝑜𝑐𝑐𝑢𝑝

|𝜓𝑖,𝜎|2

• GGA: ADDS 𝛻𝑛↑ AND 𝛻𝑛↓• MGGA: ADDS 𝜏𝜎 = σ𝑖

𝑜𝑐𝑐𝑢𝑝 1

2|𝛻𝜓𝑖,𝜎|

2

• 𝜓𝑖,𝜎 ∶ KOHN − SHAM ORBITALS

G3 (kcal/mol) BH76 (kcal/mol)

S22 (kcal/mol) LC20 (Å)

ME MAE ME MAE ME MAE ME MAE

GGA

PBE 22.2 9.2 2.8 0.059

MGGA SCAN - 5.7 7.7 - 0.9 0.016

-

EARLY RESULTS FOR MOLECULES AND SOLIDS

MAE: MEAN ABSOLUTE ERROR

G3: ATOMIZATION ENERGIES OF 223 MOLECULES

BH76: 76 REACTION BARRIERS

S22: 22 MOLECULAR COMPLEXES BOUND BY WEAK BONDS

LC20: LATTICE CONSTANTS OF 20 SOLIDS INCLUDING METALS, SEMICONDUCTORS, AND INSULATORS

THE NONEMPIRICAL SCAN META-GGA IS MORE ACCURATE THAN THE NONEMPIRICAL PBE GGA, ESPECIALLY FOR WEAK BONDS AND INTERMEDIATE-RANGE VAN DER WAALS ATTRACTION. (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.

Phase transition pressure and defect formation energies of Si: The situation before SCAN

STRUCTURAL PHASE TRANSITIONINTERSTITIAL DEFECT FORMATION ENERGY

REF: E.R. Batista, et al, PRB, 74, 121102 (2006)

PERFORMANCE OF SCAN FOR SOLID Si

Phase transition Interstitial defect formation (eV)

𝛿𝐸(meV) 𝑝𝑡(Gpa) X H T

PBE 290 8.4 3.61 3.65 3.78

SCAN 411 14.3 4.29 4.36 4.59

Ref 480±50 11-15 4.5-4.8

ENERGETIC ORDERING OF WATER HEXAMERS(BINDING ENERGY IN meV/H2O)

method Prism Cage Book Cyclic

DMC 331.9 329.5 327.8 320.8

CCSD(T) 347.6 345.5 338.9 332.5

PBE 336.1 339.4 345.6 344.1

SCAN 376.8 375.6 370.0 359.7

PBE+vd

W(TS) 369.6 372.6 370.6 360.7

Ref: B. Santra, et al, JCP, 129, 194111 (2008)

~7 ICE PHASES: CORRECT ENERGY DIFFERENCES

LIQUID WATER BY AB INITIO MOLECULAR DYNAMICS:

PBE OVERSTRUCTURES WATER, AND MAKES ICE SINK IN WATER. SCAN CORRECTS THESE ERRORS. WATER IS ESSENTIAL FOR CHEMISTRY AND BIOLOGY.

TESTED FERROELECTRICS/MULTIFERROICS: SCAN MUCH BETTER THAN PBE FOR STRUCTURAL DISTORTIONS, POLARIZATIONS, AND TRANSITION TEMPERATURES.

STABILITIES OF CRYSTAL STRUCTURES

WORK WITH GERD CEDER SHOWS THAT SCAN HALVES THE ERRORS OF PBE FOR FORMATION ENERGIES OF 200 BINARY SOLIDS AND FOR PREDICTION OF THE GROUND-STATE CRYSTAL STRUCTURES.

SCAN BAND GAPS OF SOLIDS IN A GKS SCHEME ARE LESS UNDERESTIMATED THAN PBE GAPS, AND (FOR STRONGLY-CORRELATED SYSTEMS) OFTEN CLOSE TO PBE+U GAPS.

BUT HYBRID GAPS IN GKS ARE EVEN BETTER. ALL FOR THE RIGHT REASON. (PERDEW, YANG, BURKE ET AL., PNAS, 2017)

NONLOCAL CORRECTIONS TO SCAN

(1) LONG-RANGE VAN DER WAALS INTERACTION(SCAN+rVV10). IMPORTANT FOR THE EXFOLIATION ENERGIES OFLAYERED MATERIALS,

AND FOR THE SURFACE ENERGIES AND WORK FUNCTIONS OF METALS.

H. PENG, Z. YANG, J. SUN, & J.P. PERDEW, PHYS. REV. X (2016)

(2) SELF-INTERACTION CORRECTION, PIECEWISE LINEARITY OF E(N), FLAT PLANE CONDITION (UNDER DEVELOPMENT).

NEEDED FOR STRETCHED BONDS OR STRONG CORRELATION.

SUMMARY OF SCAN

• A META-GGA, STRONGLY CONSTRAINED AND APPROPRIATELY NORMED.

• COMPUTATIONALLY EFFICIENT AND NONEMPIRICAL. AVAILABLE IN VASP, ADF, QCHEM, QUANTUM ESPRESSO,…

• SYSTEMATIC IMPROVEMENT OVER THE PBE GGA FOR A WIDE RANGE OF PROPERTIES.

• OFTEN PERFORMS LIKE A HYBRID FUNCTIONAL, AT LOWER COST AND WITH INTERMEDIATE-RANGE VAN DER WAALS INTERACTION.

LONG-RANGE vdW INTERACTION FROM SCAN+rVV10. SELF-INTERACTION CORRECTIONS TO SCAN (SATISFYING THE LAST KNOWN EXACT CONSTRAINTS) ARE UNDER DEVELOPMENT.

EXCHANGE

EXACT CONSTRAINTS

𝐸𝑥𝑆𝐶𝐴𝑁 𝑛 = න𝑑3𝑟𝑛𝜀𝑥

𝑢𝑛𝑖𝑓(𝑛)𝐹𝑥

𝑆𝐶𝐴𝑁(𝑠, 𝛼)

𝐹𝑥𝑆𝐶𝐴𝑁 𝑠, 𝛼 = {ℎ𝑥

1 𝑠, 𝛼 + 𝑓𝑥 𝛼 [ℎ𝑥0 − ℎ𝑥

1 𝑠, 𝛼 ]}𝑔𝑥 𝑠

UNDETERMINED PARAMETERS: 𝑘1, 𝑐1x, 𝑐2x, 𝑑x

1. TIGHT LOWER BOUND FOR EXCHANGE 𝐹𝑥𝑆𝐶𝐴𝑁 ≤ 1.174 → ℎ𝑥

0=1.174

3. NON-UNIFORM SCALING LIMIT: lim𝑠→∞

𝐹𝑥𝑆𝐶𝐴𝑁 𝑠, 𝛼 ∝ 𝑠−

1

2 → 𝑔𝑥 𝑠 = 1 − 𝐸𝑥𝑝[−𝑎1𝑠−1/2]

2. 4TH-ORDER GRADIENT EXPANSION OF SLOWLY VARYING DENSITY LIMIT (𝑠 →0 and 𝛼 →1):

𝐹𝑥GE4 𝑠, 𝛼 = 1 +

10

81𝑠2 −

1606

18225𝑠4 +

511

13500𝑠2 1 − 𝛼 +

5913

405000(1 − 𝛼)2

RESUMMATION OF 4TH-ORDER GRADIENT EXPANSION (RGE):

ℎ𝑥1 𝑠, 𝛼 = 1 + 𝑘1 −

𝑘1

1 +𝜇𝐴𝐾𝑠

2(1 +𝑏4 𝑠2/𝜇𝐴𝐾𝐸𝑥𝑝(−|𝑏4|𝑠

2/𝜇𝐴𝐾)) + {𝑏1𝑠2 + 𝑏2 1 − 𝛼 𝐸𝑥𝑝 −𝑏3 1 − 𝛼

2 }2

𝑘1

𝑎1=4.9479: Ex(H)

INTERPOLATION AND EXTRAPOLATION: 𝑓𝑥 𝛼 = 𝐸𝑥𝑝 −𝑐1x𝛼

1−𝛼𝜃 1 − 𝛼 − 𝑑x𝐸𝑥𝑝

𝑐2x

1−𝛼𝜃 𝛼 − 1

𝜇𝐴𝐾 = 10/81, 𝑏1 =0.156632, 𝑏2 = 0.12083, 𝑏3 = 0.5, 𝑏4 =𝜇𝐴𝐾2

𝑘1− 0.112654

NO CORRECTION TO RGE IN ANY POWER OF 𝛻𝑛 AT THE SLOWLY VARYING DENSITY LIMIT

𝜃(𝑥): STEP FUNCTION

3/12 )3)(4/3()( nnunifx 3/43/12 )()3(2

||

n

ns

CORRELATION

EXACT CONSTRAINTS

UNDETERMINED PARAMETERS: 𝑐1c, 𝑐2c, 𝑑𝑐

1. ONE ELECTRON SELF-CORRELATION FREE

3. 2ND-ORDER GRADIENT EXPANSION OF SLOWLY VARYING DENSITY LIMIT → A SIMPLIFIED PBE CORRELATION FOR 𝜀𝑐1

INTERPOLATION AND EXTRAPOLATION: 𝑓𝑐 𝛼 = 𝐸𝑥𝑝 −𝑐1c𝛼

1−𝛼𝜃 1 − 𝛼 − 𝑑c𝐸𝑥𝑝

𝑐2c

1−𝛼𝜃 𝛼 − 1

𝐸𝑐𝑆𝐶𝐴𝑁 𝑛 = න𝑑3𝑟𝑛𝜀𝑐

𝑆𝐶𝐴𝑁 )𝜀𝑐𝑆𝐶𝐴𝑁 = 𝜀𝑐

1 + 𝑓𝑐 𝛼 (𝜀𝑐0 − 𝜀𝑐

1

2. LOWER BOUND ON 𝐸𝑥𝑐 OF 2-ELECTRON DENSITIES: 𝐸𝑥𝑐 𝑛 ≥ 1.67𝐸𝑥𝐿𝐷𝐴

NONEMPIRICAL 𝜀𝑐0

4. NON-POSITIVITY → 𝜀𝑐0 ≥ 𝜀𝑐

1