dispersion interactions in solids using the exchange-hole

Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions

Dispersion interactions in solids using theexchange-hole dipole moment model

Alberto Otero-de-la-Roza and Erin R. Johnson

School of Natural Sciences, University of California, Merced, 5200North Lake Road, Merced, California 95343, USA

A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 1 / 29


Dispersion interactions in solids

Molecular crystal packing, aggregation and growth.Crystal structure prediction. Polymorph stability.Energetics of surface adsorption.Crystal Engineering.Supramolecular chemistry.



Dispersion interactions, origin

Long-range non-local correlation effect.Interaction of induced instantaneous dipoles.Electrostatic interaction excited state fragments.Not captured by semilocal functionals.

A B

VB(rA) VA(rB)



Dispersion interactions, methods

CCSD(T): very good accuracy but N7 scaling.MP2: available in condensed phases but systematic overbinding.DFT: beyond semilocal.

Benzene dimer

(Johnson et al. CPL 394 (2004) 334)A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 4 / 29


Review of dispersion methods in solids

ACFDT/RPA.Non-local correlation functionals (vdw-DFx).Dispersion corrected potentials (DCP).meta-GGAs fit to binding energies.Post-SCF pairwise energy.

I Popular choice: simple and efficient.I Fixed C6: DFT-D.I Variable C6: Tkatchenko-Scheffler, Sato-Nakai, XDM.



Dispersion via pairwise interaction energies

Edisp = −12

∑ij

C6f6(Rij)

R6ij

+

[C8f8(Rij)

R8ij

+C10f6(Rij)

R10ij

+ . . .

]

Popular methods:DFT-Dx:

I x = 2 – fixed C6 coefficients (J. Comp. Chem. 27 (2006) 1787).I x = 3 – not implementable in solids (JCP 132 (2010) 154104).

Tkatchenko-Scheffler (PRL 102 (2009) 073005)

C6AB =2C6AAC6BB

α0B

α0AC6AA +

α0A

α0BC6BB

; C6AA =

(Veff

A

V freeA

)2

Cfree6AA



XDM: second order perturbation theory

Edisp = −12

∑ij

C6f6(Rij)

R6ij

+

[C8f8(Rij)

R8ij

+C10f6(Rij)

R10ij

+ . . .

]

comes from second order PT:

E(2) =〈V̂int〉∆E

A B

VB(rA) VA(rB)

where:Interaction between neutral fragments (classical electrostaticinteractions already captured at semilocal level).Asymptotic expression.Incorrect in some cases (conductors).



The exchange-hole model

The exchange hole:

hxσ(1, 2) = −|ρ1σ(1, 2)|2

ρ1σ(1)

Probability of exclusion of same-spin electron.On-top depth condition: hxσ(1, 1) = −ρ1σ(1)

Normalization:∫

hxσ(1, 2)d2 = −1 for all 1.ρ1σ(1, 2) =

∑σi ψ∗i (1)ψi(2)



The exchange-hole model

nucleus

referenceelectron

holecenter

dipole

Model for dispersion: interaction of electron-hole dipoles.Dipole: dxσ(r) =

∫r′hxσ(r, r′)dr′ − r

Assumption: dipole oriented to nearest nucleus.〈M2

l 〉i =∑

σ

∫ωi(r)ρσ(r)[rl

i − (ri − dXσ)l]2dr .Johnson et al. JCP 127 (2007) 154108



The Becke-Roussel model of exchange-hole

Becke-Roussel model of hx.(PRA 39 (1989) 3761)

Parameters (A,a,b) obtained:NormalizationValue at reference point.Curvature at reference point(reqs. kinetic energy density).

referencepoint

holecenter

b

Ae-ar

Advantages:1 Semilocal model of the dipole (dx = b).2 XDM dispersion model −→ meta-GGA.3 Better performance than exact hole (HF) version in molecules.



The XDM equations: interaction coefficients

Multipole moments

〈M2l 〉i =

∑σ

∫ωi(r)ρσ(r)[rl

i − (ri − dXσ)l]2dr

use Hirshfeld partition:

ωi(r) =ρat

i (r)∑j ρ

atj (r)

Non-empirical dispersion coefficients:

C6,ij =αiαj〈M2

1〉〈M21〉j

〈M21〉αj + 〈M2

1〉jαiC8,ij =

32αiαj

(〈M2

1〉i〈M22〉j + 〈M2

2〉i〈M21〉j)

〈M21〉iαj + 〈M2

1〉jαi

C10,ij = 2αiαj

(〈M2

1〉i〈M23〉j + 〈M2

3〉i〈M21〉j)

〈M21〉iαj + 〈M2

1〉jαi+

215

αiαj〈M22〉i〈M2

2〉j〈M2

1〉iαj + 〈M21〉jαi



The damping function

Edisp = −12

∑ij

C6f6(Rij)

R6ij

Damping function. Some options:f6 = 1

1+6(R/(sR0))−γ

f6 = 11+e−γ(R/sR0−1)

Becke-Johnson expression:

Edisp = −12

∑L

∑′

ij

C6,ij

R6vdw,ij + R6

ijL+

C8,ij

R8vdw,ij + R8

ijL+

C10,ij

R10vdw,ij + R10

ijL

Rvdw,ij = a1Rc,ij + a2

where a1 and a2 are parameters.Correct finite R −→ 0 limit.Grimme (J. Comp. Chem. 32 (2011) 1456–1465): the choice isnot very relevant if correctly parametrized.



The damping function: effect of the exchangefunctional

Ne dimer

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

Inte

ract

ion

ener

gy (

mH

)

dNe−Ne (Å)

HFLDAPBE

revPBEB86b

PW86B97

B97D

Best exchange functionals: B86b, PW86. Also, PBE, revPBE, PBEsol.A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 13 / 29


The damping function: parametrization

Molecular calculations:Kannemann and Becke,JCTC 6 (2010) 1081.Crystal binding energies,geometries not available.Supercell molecularcalculations.Training set: 65 dimers.



The damping function: parametrization

0 1 2 3 4 5a2 (Å)

0.0

0.2

0.4

0.6

0.8

1.0

a 1

10

20

30

40

50

60

70

Coefficients calculated once. a1 and a2 fitted to the training set.



The optimized parameters

Statistics for supercell (PS/PW) and gaussian calculations.Training set

B86b PW86 PBE revPBE PBEsola1 0.136 0.881 (0.82) 0.000 0.316 0.000

a2(Å) 3.178 0.937 (1.16) 3.879 2.065 4.185N 65 65 49 49 49

MAE (kcal/mol) 0.41 0.47 (0.33) 0.61 0.65 0.88MAPE 15.5 16.9 16.9 16.6 20.8

S22MAE 0.62 0.74 (0.31) 0.75 0.57 1.05

MAE (refit) 0.42 0.43 0.55 0.59 0.98S66

MAE 0.28



Binding energies of dimers (supercell)

-80

-60

-40

-20

0

20

40

60

80

He-H

eH

e-N

eH

e-A

rH

e-K

rN

e-N

eN

e-A

rN

e-K

rA

r-Ar

Ar-K

rK

r-Kr

He-N

2 L

He-N

2 T

He-F

Cl

FC

l-He

CH

4-C

2H

4C

F4-C

F4

SiH

4-C

H4

CO

2-C

O2

OC

S-O

CS

C10H

8-C

10H

8 P

C10H

8-C

10H

8 P

CC

10H

8-C

10H

8 T

C10H

8-C

10H

8 T

CC

H4-N

H3

SiH

4-H

FC

H4-H

FC

2H

4-H

FC

H3F

-CH

3F

H2C

O-H

2C

OC

H3C

N-C

H3C

NH

CN

-HF

NH

3-N

H3

H2O

-H2O

H2C

O2-H

2C

O2

Form

am

ide-F

orm

am

ide

Ura

cil-U

racil H

BP

yrid

oxin

e-A

min

opyrid

ine

Adenin

e-T

hym

ine W

CC

1C

H4-C

H4

C2H

4-C

2H

4C

6H

6-C

H4

C6H

6-C

6H

6 P

DP

yra

zin

e-P

yra

zin

eU

racil-U

racil s

tack

Indole

-C6H

6 s

tack

Adenin

e-T

hym

ine s

tack

C2H

4-C

2H

2C

6H

6-H

2O

C6H

6-N

H3

C6H

6-H

CN

C6H

6-C

6H

6 T

Indole

-C6H

6 T

Phenol-P

henol

HF

-HF

NH

3-H

2O

H2S

-H2S

HC

l-HC

lH

2S

-HC

lC

H3C

l-HC

lH

CN

-CH

3S

HC

H3S

H-H

Cl

CH

4-N

EC

6H

6-N

EC

2H

2-C

2H

2C

6H

6-C

6H

6 s

tack

Re

lative

err

or

(%)

B86bB86b (opt)

PW86PBE

revPBEPBEsol

Noble gases not included in the fit.Universal trend for all functional/parameter combination.



Implementation in solids

Implemented in Quantum ESPRESSO.

Uniform 3D grid:

I dxσ, valence τ , ρ.I ωi, all-electron ρ, ρat.

Core reconstruction.

Insensitive to grid density (CO2)ngrid = 64 80 120

C6 (C-C) 22.300 22.425 22.426C6 (O-O) 11.580 11.627 11.627Edisp (Ry) -0.062965 -0.063374 -0.063374



Implementation in solids

Computational cost.

I Comparable to DFT-D.I Computation of Edisp fast compared to

EDFT.

Optimization: atomic forces and stresses.

I Coefficients from the self-consistentdensity.

I Calculated at the first step.I Easy expressions for atomic forces and

stress.

Fdisp,i = −∂E∂xi

; σdisp,ξη = − 1V∂E∂εξη

ξ, η = x, y, z



Graphite

−90−85−80−75−70−65−60−55−50−45−40−35−30−25−20−15

5.5 6 6.5 7 7.5 8 8.5 9

Eex

(m

eV/a

tom

)

c (Å)

B86bPW86

PBErevPBEPBEsol

Expt.Expt.

ACFDT−RPALDA

GGA+vdwQMC

VDW−DF



Prediction of sublimation enthalpies

Benchmark:No reference wave-function data.Experimental sublimation enthalpies not directly comparable.

31 crystals, small systems, low polymorphism.Well known sublimation enthalpies at or below room temperature.Different intermolecular interactions.



Prediction of sublimation enthalpies

Methods compared:

1 XDM: B86b-XDM, PBE-XDM.2 DFT-D2: PBE-D.3 Tkatchenko-Scheffler: PBE-TS.4 Langreth et al.: revPBE-vdw1, PW86-vdw2.

Ecrys and Emol converged separately (≈ 0.1 mRy).PW86 numerical noise. Memory limitations.Reduction of ∆Hsub with Cp.



∆Hsub: zero-point and thermal correction

∆Hsub(V,T) = Emolel + Etrans + Erot + Emol

vib + pV

−(Ecrys

el + Ecrysvib

)Ecrys

el −→ DFT+dispersionEmol

el −→ DFT+dispersion, supercellEtrans + Erot + pV −→ 4RT (7/2RT)Rigid molecule approximation Emol

vib = Ecrysvib for intramolecular

Intermolecular Ecrysvib −→ Dulong-Petit 6RT (5RT)

Zero-point vibrational contributions neglected



Sublimation enthalpy: results.

B86b-XDM PBE-D PBE-TS PBE-XDMRMS (kJ/mol) 9.22 10.03 23.91 9.12MAE (kJ/mol) 7.44 8.14 20.24 7.39

MAPE 10.11 11.85 27.23 9.81

-80

-60

-40

-20

0

20

40

60

80

13-d

initro

benzene

14-c

yclo

hexanedio

ne

14-d

ichlo

robenzene

aceta

mid

e

acetic

acid

adam

anta

ne

am

monia

benzene

bip

henyle

ne

cate

chol

co2

cyanam

ide

cyto

sin

e

dib

enzoth

iophene

eth

ylc

arb

am

ate

form

am

ide

hexaflu

oro

benzene

imid

azole

naphth

ale

ne

oxalic

acid

alp

ha

oxalic

acid

beta

phenol

pyra

zin

e

pyra

zole

thym

ine

trans-a

zobenzene

trans-s

tilbene

triazin

e

trioxane

ura

cil

ure

aR

ela

tive e

rror

(%)

B86b-XDMPBE-D

PBE-TSPBE-XDM



Surface adsorption: is variable C6 important?Solids (QE)

adamantane 17.64graphite 17.84

biphenylene 18.19formamide 18.87anthracene 19.08naphthalene 19.21

benzene 19.55C6F6 19.96

cyanamide 21.29CO2 22.29

DFT-D2 30.36

Molecules (gaussian)

Ti 1232TiBr4 502TiCl4 469

Ti(Cp)2Cl2 387TiF4 312TiO2 279Ti+2 46

DFT-D2 187.33

Zn 312ZnO 208

Zn(OH)2 158Zn(CH3)2 147

Zn(porphine) 146Zn(CN)2 133

Zn+2 24

DFT-D2 187.33

PBE-D most popular for surface adsorption.PBE-D: same C6 to all first-row transition metals.C6 variable in metallic atoms.DFT-D C6 large compared to XDM.



Prediction of crystal structures

Compare to experimental geometries: vibrations.

Thermal pressure

pth = −∂Fvib

∂V

Equilibrium condition

∂E∂V

= pth = −psta



Calculation of pth

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 200 400 600 800 1000 1200 1400

DO

S (

1/c

m-1

)

Frequency (cm-1

)

0.04

0.042

0.044

0.046

0.048

0.05

0.052

0.054

0.056

0.058

600 700 800 900 1000 1100 1200 1300 1400 1500 1600F

vib

(H

y)

V (bohr3)

Quartic fitParabolic fit 3 pts



Geometries, results

0

1

2

3

4

5

6

7

8

9

10

14-c

yclo

hexanedio

ne

acetic

acid

adam

anta

ne

am

monia

benzene

co2

cyanam

ide

eth

ylc

arb

am

ate

form

am

ide

imid

azole

naphth

ale

ne

oxalic

acid

alp

ha

oxalic

acid

beta

pyra

zin

e

ure

aR

ela

tive

err

or

(%)

B86b-XDMPBE-XDM

PBE-DPBE-TS

revPBE-vdwdf1pw86-vdwdf2

B86B-XDM PBE-D PBE-TS1.32 1.55 1.14

PBE-XDM PW86-vdwDF2 revPBE-vdwDF11.90 1.56 3.81



Conclusions

1 Implemented XDM dispersion model in solids(pseudopotentials/planewaves, QE).

2 Interaction coefficients from the density.3 Cost comparable to DFT-D, variable C6.4 Crystal binding energies improved compared to other methods.5 Accurate crystal structures.6 Next: surface adsorptions,...


dispersion interactions in solids using the exchange-hole

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