dispersion interactions in solids using the exchange-hole
TRANSCRIPT
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 XDM Damping Dimers Implementation Applications Conclusions
Dispersion interactions in solids
Molecular crystal packing, aggregation and growth.Crystal structure prediction. Polymorph stability.Energetics of surface adsorption.Crystal Engineering.Supramolecular chemistry.
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 2 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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)
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 3 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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
Dispersion interactions XDM Damping Dimers Implementation 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-D.I Variable C6: Tkatchenko-Scheffler, Sato-Nakai, XDM.
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 5 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 6 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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).
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 7 / 29
Dispersion interactions XDM Damping Dimers Implementation 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 in solids ACS Meeting (March 2012) 8 / 29
Dispersion interactions XDM Damping Dimers Implementation 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 .Johnson et al. JCP 127 (2007) 154108
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 9 / 29
Dispersion interactions XDM Damping Dimers Implementation 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 in solids ACS Meeting (March 2012) 10 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 11 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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.
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 12 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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.
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 14 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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.
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 15 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 16 / 29
Dispersion interactions XDM Damping Dimers Implementation 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 in solids ACS Meeting (March 2012) 17 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 18 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 19 / 29
Dispersion interactions XDM Damping Dimers Implementation 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 in solids ACS Meeting (March 2012) 20 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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.
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 21 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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.
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 22 / 29
Dispersion interactions XDM Damping Dimers Implementation 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 in solids ACS Meeting (March 2012) 23 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 24 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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.
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 25 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
Prediction of crystal structures
Compare to experimental geometries: vibrations.
Thermal pressure
pth = −∂Fvib
∂V
Equilibrium condition
∂E∂V
= pth = −psta
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 26 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 27 / 29
Dispersion interactions XDM Damping Dimers Implementation 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
B86B-XDM PBE-D PBE-TS1.32 1.55 1.14
PBE-XDM PW86-vdwDF2 revPBE-vdwDF11.90 1.56 3.81
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 28 / 29
Dispersion interactions XDM Damping Dimers Implementation Applications Conclusions
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,...
A. Otero & E. Johnson (UC Merced) XDM in solids ACS Meeting (March 2012) 29 / 29