dispersion interactions using the becke-roussel exchange-hole...

Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions

Dispersion interactions using the Becke-Rousselexchange-hole 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 ACS Meeting (August 2012) 1 / 30


Dispersion interactions

Molecular crystal packing.Crystal structure prediction.Surface adsorption.Crystal Engineering.



Dispersion interactions, origin

A B

VB(rA) VA(rB)

Long-range non-local correlation effect.

Interaction of induced dipoles.

Not captured by semilocal functionals.

CCSD(T): N7 scaling.

MP2: systematic overbinding.

Benzene dimer

(Johnson et al. CPL 394(2004) 334)A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 3 / 30


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-D2 (J. Comp. Chem. 27 (2006) 1787).I Variable C6: Tkatchenko-Scheffler (PRL 102 (2009) 073005),

Sato-Nakai, DFT-D3, XDM.

Edisp = −12

∑ij

C6f6(Rij)

R6ij

+

[C8f8(Rij)

R8ij

+C10f6(Rij)

R10ij

+ . . .

]



XDM, basics

Edisp = −12

∑ij

C6f6(Rij)

R6ij

+

[C8f8(Rij)

R8ij

+C10f6(Rij)

R10ij

+ . . .

]

comes from second order PT:

E(2) =〈V̂2

int〉∆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 .



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 atomic partition:

ωi(r) =ρat

i (r)∑j ρ

atj (r)

Non-empirical dispersion coefficients. n-body and any order. Forinstance:

C6,ij =αiαj〈M2

1〉〈M21〉j

〈M21〉αj + 〈M2

1〉jαi

We include: two-body terms C6, C8 and C10.



The damping function

Edisp = −12

∑ij

C6f6(Rij)

R6ij

Damping function. Corrects for themultipolar-expansion error and avoid dis-continuity. fn for R−n term.

Becke-Johnson

fn(r) =rn

rn + rnvdw

Tang-Toennies

fn(r) = 1−e−brn∑

k=1

(br)k

k!

rvdw or b equal a1rc,ij + a2. jParameters: a1 and a2.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 2 4 6 8 10 12 14 16

f n

r

TT, n=3TT, n=6TT, n=8

TT, n=10BJ, n=3BJ, n=6BJ, n=8

BJ, n=10



The damping function: the exchange functional

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 ACS Meeting (August 2012) 11 / 30


The damping function: parametrizationTraining set: 65 dimers.Supercell calculations.Gas-phase: Kannemann andBecke, JCTC 6 (2010) 1081.Coefficients calculated once.a1 and a2 fitted to the trainingset.

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



The optimized parameters

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

B86b PW86 PBE revPBE PBEsola1 0.151 0.705 (0.82) -0.150 0.411 0.349

a2(Å) 3.084 1.448 (1.16) 4.339 1.690 3.034N 65 51 51 (65) 51 51

MAE (kcal/mol) 0.32 0.41 (0.33) 0.49 0.48 0.75MAPE 13.5 11.9 (12.6) 13.9 13.3 18.0

S22MAE 0.43 0.47 (0.31) 0.60 0.56 0.88

MAPE 7.36 8.01 10.42 8.97 10.49S66

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 forsolids and nwchem for molecules.

Solids – Uniform 3D grid:

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

Computational cost.

I Comparable to DFT-D.I Edisp fast compared to EDFT.

Optimization: atomic forces and stresses.

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



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.

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



∆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.

XDM DFT-D2 TS vdw-DF(kJ/mol) B86b PW86 PBE PBE PBE v1 v2

MAE 4.81 6.50 5.35 9.05 16.97 10.22 10.11MAPE 6.23 8.00 6.74 11.94 22.08 13.53 13.11

-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



Prediction of crystal structures

Compare to experimental geome-tries: thermal effects.

Thermal pressure.

pth = −∂Fvib

∂V

Equilibrium condition.

∂E∂V

= pth = −psta

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 1600

Fvib

(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

XDM DFT-D2 TS vdw-DF(a.u.) B86b PW86 PBE PBE PBE v1 v2MAE 0.12 0.06 0.20 0.11 0.10 0.31 0.14

MAPE 1.76 0.90 2.75 1.31 1.58 4.40 1.88A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 21 / 30


Enantiomeric excess of amino-acids

Stability of racemate/conglomerate⇒ ee in solution.

DL-alanine L-alanineA. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 22 / 30


Enantiomeric excess of amino-acids

A simple model

Same solvationenergies.Same crystaltemperature effects.∆E = Edl − El

Predicted ee:

ee =β2 − 1β2 + 1

× 100

β = e−∆E/RT

kcal/mol ∆E eecalc eeexp

ala -0.482 67.11 60.4asp 0.477 0.00 0.00cys -0.505 69.23 ?gln -0.853 89.38 ?glu 2.071 0.00 0.00his -1.006 93.52 93.7ile -0.990 93.17 51.8leu -0.949 92.18 87.9pro 0.189 0.00 49.6 (DMSO)

99 (CHCl3)ser -4.116 100.00 100.0tyr -0.521 70.63 ?val -0.432 62.26 36.8

46.7



Stability of water hexamers

(kcal/mol) Ref. Fix. Opt.prism (wrt H2O) 45.86 52.44 52.61

prism 0.00 0.00 0.00cage 0.06 0.03 -0.05bag 1.23 1.03 0.89chair 1.21 1.67 1.69

book1 0.33 0.28 0.16book2 0.64 0.50 0.38boat1 2.20 2.64 2.55boat2 2.28 2.78 2.66

Prism

Cage



Stability of clathrate hydrates

empty

P

H

P2

PH

H2

P2H

PH2

H3

P2H2

PH3

H4

P2H3

PH4

H5

P2H4

PH5

H6

P2H5

PH6 all3.24

0.47

1.51

5.52

5.38 5.34 3.83 6.05 5.92 2.16 2.59

1.02 1.25 1.57 1.74 1.98 6.10

6.113.79 5.60 5.66 4.00 6.29 2.58

3.24 2.19 2.39 2.935.95

4.55 5.86 4.53



Molecular crystals under pressure

Benzene: at ≈ 2.5 GPa transitions from space group Pbca toherringbone structure (same as higher acenes).

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

EP

bca-E

he

rr (

kcal/m

ol)

p (GPa)



Surface adsorptionDependence of C6 on the chemical environment

Solids (QE)adamantane 17.64

graphite 17.84biphenylene 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: same C6 to all first-row transition metals.XDM: ab-initio interaction coefficients for arbitrary elements.



Surface adsorption: kaolinite

Kaolinite bulk

Siloxane surface

Alumina surfaceA. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 28 / 30


Surface adsorption: benzene on metals

eV/molec PBE-XDM M06-L DFT-D vdW-DF Exp.Au 0.58 0.55 1.35/0.76 0.55/0.42 0.63Ag 0.33 0.56 1.30/0.96 0.49 0.52/0.42Cu 0.47 0.51 0.61/0.86 0.55/0.52 0.59



Conclusions

1 XDM dispersion model in solids (PS/PW).2 Interaction coefficients from first-principles quantities.3 Cost comparable to semilocal DFT, variable C6.4 Excellent binding energies. Lattice energies with double accuracy

wrt to similar methods.5 Accurate crystal structures: less than 1% error in cell lengths.6 Accurate surface adsorption, water hexamer ranking, molecular

crystal phase transitions,...


dispersion interactions using the becke-roussel exchange-hole...

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