dispersion interactions from the exchange-hole dipole...

XDM Solids Sublimation enthalpies ee in solution Clathrates Adsorption Summary

Dispersion Interactions from the Exchange-HoleDipole Moment. II. Applications

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 (April 2013) 1 / 20


XDM, basics

Edisp = −12

∑ij

C6f6(Rij)

R6ij

+

[C8f8(Rij)

R8ij

+C10f6(Rij)

R10ij

+ . . .

]

comes from perturbation theory:

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.



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.



Implementation in solids

PS/PW (Quantum ESPRESSO)

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



Damping function parametrization

Edisp = −12

∑ij

C6f6(Rij)

R6ij

+

[C8f8(Rij)

R8ij

+C10f6(Rij)

R10ij

+ . . .

]

fn(Rij) =Rn

ij

Rnij + (a1Rij,c + a2)n

Kannemann-Becke 65-set.

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

Supercell calculations.



Damping parameters and statistics of the fit

Statistics for supercell (PS/PW) and Gaussian calculations.Training set (KB49)

B86bPBE PW86PBE BLYPa1 0.684 0.407 0.934 0.267 0.774

a2(Å) 1.368 2.415 0.965 2.227 0.839MAE (kcal/mol) 0.41 0.46 0.42 0.41 0.31

MAPE 11.3 13.8 11.8 14.3 9.8S22

MAE (kcal/mol) 0.43 0.46 0.35 0.32 0.22MAPE 7.00 8.12 5.92 8.24 4.85



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

Exf

olia

tion

ener

gy (

meV

/ato

m)

Inter−layer distance (Å)

B86bPW86

PBErevPBEPBEsol

ACFDT−RPALDA

GGA+vdwQMC

VDW−DFExpt.Expt.



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 neglectedApproximations tested for CO2 crystal. Average experimentalaccuracy ≈ 1 kcal/mol.



Sublimation enthalpies

XDM DFT-D2 TS09 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

Vibrational Helmholtz free energy:

Fvib(V,T) =

3n∑j=1

[ωj

2+ kBT ln

(1− e−ωj/kBT

)]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

)

Relax the crystal under negative pressure pth



Crystal structures

-60

-40

-20

0

20

40

60

Rela

tive e

rror

(%)

B86b-XDMPBE-D

PBE-TSPBE-XDM

vdwDF1vdwDF2

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 1.88



Enantiomeric excess of amino-acids



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 clathrate hydrates

2900

2920

2940

2960

2980

3000

3020

3040

3060

3080

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2O

H s

tre

tch

ing

fre

qu

en

cy (

cm

-1)

p (GPa)

ch4co2

empty



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



Summary

1 XDM implemented for solids.2 Very accurate lattice energies and geometries.3 Work in progress: ee in solution, clathrates, phase transitions,

adsorption,...

Electrides and alkalides using the exchange-hole dipole momentmodel

Stephen G. DaleThursday, 9:00 AM, Quantum Chemistry Session

Morial Convention Center, Room: 354


dispersion interactions from the exchange-hole dipole...

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