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 XDM Damping Dimers Impl. Applications Conclusions
Dispersion interactions
Molecular crystal packing.Crystal structure prediction.Surface adsorption.Crystal Engineering.
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 2 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
+ . . .
]
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 4 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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).
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 5 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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)
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 6 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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 .
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 7 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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.
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 8 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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.
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 9 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 10 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 12 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 13 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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.
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 14 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 15 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 16 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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.
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 17 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
∆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
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 18 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 19 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 20 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 23 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 24 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 25 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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)
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 26 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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.
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 27 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
Surface adsorption: kaolinite
Kaolinite bulk
Siloxane surface
Alumina surfaceA. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 28 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 29 / 30
Dispersion interactions XDM Damping Dimers Impl. Applications Conclusions
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,...
A. Otero & E. Johnson (UC Merced) XDM ACS Meeting (August 2012) 30 / 30