dispersion interactions in density-functional...

Introduction XDM Molecules Solids Applications Summary

Dispersion Interactions inDensity-Functional Theory

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

Chemistry and Chemical Biology, University of California, Merced

E. R. Johnson (UC Merced) Dispersion from XDM DFT2013 (Sept 2013) 1 / 33


Dispersion interactions

Biological molecules.Surface adsorption.Molecular crystal packing.Crystal structure prediction.



Dispersion interactions

Long-range non-local correlation effect, not captured by semi-localfunctionals. Johnson et al. CPL 394 (2004) 334.



Review of dispersion methods

Fluctuation-dissipation theorem/RPA.

Non-local correlation functionals (vdw-DFx).

Dispersion correcting potentials (DCP).

Meta-GGAs fit to binding energies.

Post-SCF pairwise energy, fixed C6: DFT-D2.

Variable C6: Tkatchenko-Scheffler, DFT-D3, XDM.



The XDM method

Dispersion arises from interaction of induced dipoles.

The source of the instantaneous dipole moments is taken to be thedipole moment of the exchange hole.

Becke and Johnson JCP 127 (2007) 154108.



The exchange hole

The exchange hole measures the depletion in probability of findinganother same-spin electron in the vicinity of a reference electron.

nucleus

referenceelectron

holecenter

dipole

An electron plus its exchange hole has zero total charge, but anon-zero dipole moment in general.



The exchange-hole model

The magnitude dX of the exchange-hole dipole moment is obtainedusing the Becke-Roussel exchange-hole model; PRA 39 (1989) 3761.

referencepoint

holecenter

b

Ae-ar

Parameters (A,a,b) obtained from normalization, density, andcurvature at reference point.Advantages: semi-local (meta-GGA) model of the dipole: dx = b.



The XDM method

The dispersion energy comes from second-order perturbation theory

E(2) =〈V2

int〉∆E

Vint(rA, rB) = multipole moments of electron + hole at rA

interacting withmultipole moments of electron + hole at rB

∆E is the average excitation energy, obtained from second-orderpertubation theory applied to polarizability.



The XDM equations

The XDM dispersion energy is:

Edisp = −12

∑ij

C6f6(Rij)

R6ij

+C8f8(Rij)

R8ij

+C10f10(Rij)

R10ij

+ . . .

The dispersion coefficients are non-empirical.

C6,ij =αiαj〈M2

1〉〈M21〉j

〈M21〉αj + 〈M2

1〉jαi

Atomic multipole moment integrals use Hirshfeld atomic partitioning.

〈M2l 〉i =

∑σ

∫ωi(r)ρσ(r)[rl

i − (ri − dXσ)l]2dr



Damping function

Corrects for the multipolar-expansion error and avoids discontinuities.

fn(R) =Rn

Rn + Rnvdw

Rvdw = a1Rc,ij + a2

a1 and a2 are parameters fit for use with a particular XC functional.

Rc,ij are proportional to atomic volumes.



Parametrization set

49 gas-phase dimers from Kannemann and Becke; JCTC 6 (2010) 1081.

dispersionπ-stackingdipole - induced dipolemixeddipole - dipolehydrogen-bonding



Implementation for molecules

Pair XDM with GGA, hybrid, and range-separated functionals.

XDM calculations used Gaussian 09 and the postg program.

From the wfn file, postg gives:

XDM dispersion energyforces for geometry optimization (fixed coefficients)second derivatives for frequencies

Download postg from http://gatsby.ucmerced.edu



Binding energies

XDM with aug-cc-pVTZ; mean absolute errors in kcal/mol.

Pure functionals:

Quantity BLYP PW86 PBEMAE 0.31 0.40 0.50

MA%E 9.8 11.8 14.3

Hybrid and range-separated functionals:

Quantity B3LYP BH&HLYP PBE0 CAM-B3LYP LC-ωPBEMAE 0.28 0.37 0.41 0.39 0.28

MA%E 6.7 7.8 10.2 8.3 7.8



Role of exchange

The exact exchange potential decays as −1/r far from a molecule.

In terms of the exchange hole, hX remains on the molecule as thereference point moves away from it.

The −1/r asymptotic dependence was used to design the B88exchange functional.

Functionals based on B88 or range-separated hybrids with the fullexact-exchange limit (LC-ωPBE) give more accurate intermolecularexchange contributions.



Thermochemistry

Mean absolute errors in XDM thermochemical benchmarks.

(kcal/mol) BLYP B3LYP LC-ωPBEG3/99 atomization energies

Base functional 11.4 7.8 5.1With XDM 6.5 4.0 5.5

Main group bond energiesBase functional 7.6 5.7 4.4

With XDM 5.1 4.1 3.6Charge-transfer complexes

Base functional 1.4 4.5 1.2With XDM 2.8 3.0 0.7

JCP 138 (2013) 204109



Implementation for solids

Quantum ESPRESSO, planewaves/psuedopotentialsdxσ, valence τ , ρ.ωi, all-electron ρ, ρat.Edisp fast compared to EDFT.

Errors in calculated XDM binding energies

(kcal/mol) PBE PW86 B86b BLYPMAE 0.52 0.46 0.41 0.43

MA%E 14.9 13.8 11.3 14.4

JCP 136 (2012) 174109



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

PBErevPBE

Expt.Expt.

ACFDT−RPALDA

GGA+vdwQMC

VDW−DF



Sublimation enthalpies

∆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 molecules: Emol

vib = Ecrysvib for intramolecular modes

Dulong-Petit limit: Ecrysvib −→ 6RT (5RT) for intramolecular modes

Zero-point vibrational contributions neglected



Sublimation enthalpies

Mean absolute errors in sublimation enthalpies, in kJ/mol

Quantity XDM DFT-D2 TS vdw-DFB86b PW86 PBE PBE PBE v1 v2

MAE 4.81 6.50 5.35 9.05 16.97 10.22 1 0.11MA%E 6.23 8.00 6.74 11.94 22.08 13.53 1 3.11

21 molecular crystals, low polymorphism, varied interaction types.

Well known sublimation enthalpies at or below room temperature.



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

Mean absolute errors in cell lengths, in atomic units.

XDM DFT-D2 TS vdw-DFQuantity B86b PW86 PBE PBE PBE v1 v2

MAE 0.12 0.06 0.20 0.11 0.10 0.31 0.14MA%E 1.76 0.90 2.75 1.31 1.58 4.40 1.88

JCP 137 (2012) 054103



Chiral crystals



Enantiomeric excess

Enantiomers have the samesolvation energies.

Assume same temperatureeffects.

∆E = Edl − El

Predicted ee:

ee =β2 − 1β2 + 1

× 100

β = e−∆E/RT

Enantiomeric excess values (%):

Amino acid DFT Expt.Serine 100.0 100.0

Histidine 93.5 93.7Leucine 92.2 87.9Alanine 67.1 60.4Cysteine 69.2 58.4Tyrosine 70.6 51.7Valine 62.3 44.1Proline 0.0 39.7

Aspartic acid 0.0 0.0Glutamic acid 0.0 0.0



Enantiomeric excess

0

0.2

0.4

0.6

0.8

1

−2 −1 0 1 2 3 4

Ena

ntio

mer

ic E

xces

s

∆E (kcal/mol)



Electrides

An electride is an ionic substancein which a localized electron actsas an anion.

Existing electrides require a cagelike structure to stabilise the cation:crown ethers and cryptands.

High magnetic susceptibilities,variable conductivities, very strongreducing agents.



NCI analysis

Reduced density gradient:

s =1

2(3π2)1/3

|∇ρ|ρ4/3

Maps intermolecular contacts(i.e. Pauli repulsion zones).Simpler to calculate thanBader’s topologyColor-mapping and regionextent: interaction strength.



NCI analysis

Implemented in nciplot for molecules and critic2 for solids.

JACS 132 (2010) 6498 and PCCP 14 (2012) 12165



Electrides

Use the NCI index to visualize the electrons - non-nuclear attractorswill have low density and reduced gradient.



Electrides

J. L. Dye used van der Waals radii to generate approximate channelsand vacancies of electrons - JACS 1996, 118 (1996) 7329.



Graphite surfaces

2

FIG. 1. Illustration of the molecular dynamics simulation ofa 5.6 nm diameter AFM tip sliding over graphite with a stepedge. Details are given in the text.

trolled the tip shape by removing atoms from the tipapex. In the following, the hemisphere without any trun-cation models a sharp tip and truncated hemispheresmodel blunt tips. We modeled two blunt tips, one with25% of the hemisphere removed and one with 50% of thehemisphere removed. A constant external normal loadwas maintained on the rigid body at the top of the tip.The rigid body was connected by a harmonic spring tothe support (virtual atom) that moved at 4 m/s in thex-direction. Even though there is large difference be-tween the sliding speed in this simulation and that ac-cessible to typical AFM experiments, previous researchhas shown that the energetics predicted by a high speedMD simulations can be reliably used in interpreting AFMdata13. The spring had stiffness of 8 N/m in the x- andy-directions, but did not resist motion in the z-direction(normal to the graphite surface). A Langevin thermostatwas applied to the free atoms in the system to maintaina temperature of 300K. The root-mean-square roughnessof the graphite terrace was calculated to be 0.01-0.03 nm,which is reasonable compared to the range of values mea-sured experimentally (0.002-0.017 nm) for graphite ona boron nitride substrate14. The inter-atomic interac-tions within the tip and substrate were described viathe Adaptive Intermolecular Reactive Empirical BondOrder (AIREBO) potential15, and the long range in-teractions between tip and substrate were modeled us-ing the Lennard-Jones (LJ) potential (energy minimum0.016 eV, zero-crossing distance 0.28 nm). The simu-lations were performed using the LAMMPS simulationsoftware16.

DFT is widely used to accurately model the energies ofsurfaces, molecules, and solid-state systems. In this work,we use DFT to study the energy profile of a tip slidingon a graphite step with different trajectories. Two DFTcomputational models were tested for the graphite step:a periodic supercell in the pseudopotentials/plane-wavesapproach (using the Quantum ESPRESSO program17)and a finite representation of the step with local orbitals(using Gaussian 0918). In both cases, capping hydrogenswere added at a 1.07A distance from the carbons. Thetip was represented by a single methane molecule slidingon the step model.

Regarding the periodic calculations, we found that 8

FIG. 2. The two-layer finite DFT model for the step edge isshown, together with the methane molecule representing thetip.

graphite two-dimensional cells (containing two carbonatoms) were enough to represent a step. However, in or-der to avoid spurious self-interactions between adjacentmethane molecules, at least 4 cells were required in thecomplementary direction, exceeding our computationalresources at present. Because of this limitation, we de-cided to use a finite model. Several finite graphite stepswere tested and we decided to use a model containing twostacked graphite flakes as shown in Figure 2. The bottomlayer is a 5 × 4 graphite flake and the top layer contains2 × 4 cells. The interlayer distance is 3.42A. In all cal-culations, the coordinates of the atoms in the step arefixed whereas the coordinates of the hydrogens in the tip(and possibly the z coordinate of the carbon) are varied,depending on the type of geometry optimization.

The calculations were carried out using the LC-ωPBE19,20 functional (with range-separation parameterω = 0.4) as implemented in Gaussian0918 and the 6-31G∗ basis-set. Dispersion effects are expected to be es-sential in the accurate description of the energy surfaceso we coupled the functional above with our XDM dis-persion functional21–23 with coefficients a1 = 1.0225 anda2 = 0.8613A, as described in ref. 24. The dispersion en-ergy was calculated using the converged wave functionsand the postg code24,25.

Figure 3 shows the MD-predicted lateral force experi-enced by the sharp (a) and blunt (b and c) tips as theymove down the step. A negative lateral force correspondsto an assisting force that helps the tip move forward anda positive lateral force corresponds to a resistive forcethat impedes the tip’s motion. We can compare theseplots in terms of three characteristic features: the firstlateral force peak, the minimum lateral force, and thesecond lateral force peak, as shown in Figure 3 (a). Thefirst lateral force peak occurs before the tip moves downthe step and is the feature identified in experiments asthe lateral force that resists the motion of tip4–9. Theminimum lateral force is an assistive force observed inmeasurements taken using a sharp tip only6. The secondlateral force peak occurs after the tip has moved downthe step; this feature has not been reported experimen-

2

FIG. 1. Illustration of the molecular dynamics simulation ofa 5.6 nm diameter AFM tip sliding over graphite with a stepedge. Details are given in the text.

trolled the tip shape by removing atoms from the tipapex. In the following, the hemisphere without any trun-cation models a sharp tip and truncated hemispheresmodel blunt tips. We modeled two blunt tips, one with25% of the hemisphere removed and one with 50% of thehemisphere removed. A constant external normal loadwas maintained on the rigid body at the top of the tip.The rigid body was connected by a harmonic spring tothe support (virtual atom) that moved at 4 m/s in thex-direction. Even though there is large difference be-tween the sliding speed in this simulation and that ac-cessible to typical AFM experiments, previous researchhas shown that the energetics predicted by a high speedMD simulations can be reliably used in interpreting AFMdata13. The spring had stiffness of 8 N/m in the x- andy-directions, but did not resist motion in the z-direction(normal to the graphite surface). A Langevin thermostatwas applied to the free atoms in the system to maintaina temperature of 300K. The root-mean-square roughnessof the graphite terrace was calculated to be 0.01-0.03 nm,which is reasonable compared to the range of values mea-sured experimentally (0.002-0.017 nm) for graphite ona boron nitride substrate14. The inter-atomic interac-tions within the tip and substrate were described viathe Adaptive Intermolecular Reactive Empirical BondOrder (AIREBO) potential15, and the long range in-teractions between tip and substrate were modeled us-ing the Lennard-Jones (LJ) potential (energy minimum0.016 eV, zero-crossing distance 0.28 nm). The simu-lations were performed using the LAMMPS simulationsoftware16.

DFT is widely used to accurately model the energies ofsurfaces, molecules, and solid-state systems. In this work,we use DFT to study the energy profile of a tip slidingon a graphite step with different trajectories. Two DFTcomputational models were tested for the graphite step:a periodic supercell in the pseudopotentials/plane-wavesapproach (using the Quantum ESPRESSO program17)and a finite representation of the step with local orbitals(using Gaussian 0918). In both cases, capping hydrogenswere added at a 1.07A distance from the carbons. Thetip was represented by a single methane molecule slidingon the step model.

Regarding the periodic calculations, we found that 8

FIG. 2. The two-layer finite DFT model for the step edge isshown, together with the methane molecule representing thetip.

graphite two-dimensional cells (containing two carbonatoms) were enough to represent a step. However, in or-der to avoid spurious self-interactions between adjacentmethane molecules, at least 4 cells were required in thecomplementary direction, exceeding our computationalresources at present. Because of this limitation, we de-cided to use a finite model. Several finite graphite stepswere tested and we decided to use a model containing twostacked graphite flakes as shown in Figure 2. The bottomlayer is a 5 × 4 graphite flake and the top layer contains2 × 4 cells. The interlayer distance is 3.42A. In all cal-culations, the coordinates of the atoms in the step arefixed whereas the coordinates of the hydrogens in the tip(and possibly the z coordinate of the carbon) are varied,depending on the type of geometry optimization.

The calculations were carried out using the LC-ωPBE19,20 functional (with range-separation parameterω = 0.4) as implemented in Gaussian0918 and the 6-31G∗ basis-set. Dispersion effects are expected to be es-sential in the accurate description of the energy surfaceso we coupled the functional above with our XDM dis-persion functional21–23 with coefficients a1 = 1.0225 anda2 = 0.8613A, as described in ref. 24. The dispersion en-ergy was calculated using the converged wave functionsand the postg code24,25.

Figure 3 shows the MD-predicted lateral force experi-enced by the sharp (a) and blunt (b and c) tips as theymove down the step. A negative lateral force correspondsto an assisting force that helps the tip move forward anda positive lateral force corresponds to a resistive forcethat impedes the tip’s motion. We can compare theseplots in terms of three characteristic features: the firstlateral force peak, the minimum lateral force, and thesecond lateral force peak, as shown in Figure 3 (a). Thefirst lateral force peak occurs before the tip moves downthe step and is the feature identified in experiments asthe lateral force that resists the motion of tip4–9. Theminimum lateral force is an assistive force observed inmeasurements taken using a sharp tip only6. The secondlateral force peak occurs after the tip has moved downthe step; this feature has not been reported experimen-

3

FIG. 3. Lateral force for (a) sharp, (b) 25% truncated and(c) 50% truncated tips scanning across the graphite step edge.The tip diameter is 5.6 nm and the normal load is 0 nN inthese three plots. (d) Variation of specific features of thelateral force traces with amount of tip truncation.

tally but, as will be discussed later, is reasonable in termsof the energetic landscape near the step.

Our simulation results show that the first lateral forcepeak and the minimum lateral force valley exist regard-less of the tip shape, but their relative magnitudes changesignificantly. We evaluate the effect of truncation by com-paring these two features for our three model tips: per-fect hemisphere (i.e. 0% truncation), 25% truncation,and 50% truncation. As shown in Figure 3 (d), the mag-nitude of the first lateral force peak increases (becomesmore positive) while the magnitude of the minumum lat-eral force decreases (becomes less negative) with increas-ing degree of truncation. This explains the experimentalobservation that an assistive force is only observable (i.e.large enough to be measured) for sharp tips and that theresistive force is dominant for blunt tips6. The magnitudeof the relatively small second lateral force peak changeslittle with truncation.

We explored the effect of tip truncation for differentnormal loads (from 2 to 15 nN) and tip radii (from 3.2 to20 nm) in our MD simulations. The results are consistentwith the preceding discussion and the results in Figure 3,although the actual friction forces naturally depend onthe tip geometry and load.

We explore now the connection between tip bluntnessand friction. One of the benefits of MD simulation isthat we have atomic-scale information about the con-tact throughout the sliding process. We tracked the po-sitions of the bottommost atoms of the three different tipsduring scanning; the results are shown in Figure 4 (a).The vertical position of the bottom of each tip does notchange during the first 2 nm of sliding, indicating that thetip is moving along the upper terrace. The sharper tipsare slightly closer to the graphite (0.24 nm with sharp

FIG. 4. (a) The vertical positions of tips of different bluntnessscanning forward across the step edge. Potential energy ofthe (a) sharp, (b) 25% truncated and (b) 50% truncated tipsscanning forward across the step edge.

tip, 0.26 nm with 25% truncated tip and 0.265 nm with50% truncated tip) potentially because they experiencea greater pressure for the same load. Then, at a slid-ing distance that is dependent on the tip shape, thereis a sharp drop in the vertical position of the tip indi-cating that it has moved down the step. The distancefrom the step at which this drop occurs is significant andwill be called the step-down distance. After the drop,the vertical position of each tip is again roughly constantindicating they are moving along the lower terrace. Itis important to note that the change in vertical positionas the tip moves from the upper to the lower terrace isapproximately equal to the graphite interlayer distanceof 0.335 nm. However, the key observation from Figure 4(a) is that the step-down distance moves further from thestep as the bluntness of the tip is increased. Specifically,the step-down distance is 0.65 nm for the sharp tip, 1.05nm for the 25% truncated tip and 2.25 nm for the 50%truncated tip.

Previous experimental studies interpreted their re-sults using a one-dimensional empirical potential energymodel5,6. These models predict there is an energy barrier(the Schwoebel barrier) at the step which can be used toexplain the extra friction force. If the energy barrier islarger, a larger friction force is necessary to overcome it.We repeat this type of analysis using our MD simulation.The potential energy profiles predicted by our model foreach of the three tips are shown in Figure 4 (b), (c) and(d). The energy is expressed relative to that of the tip onthe lower terrace. We observed that the distance at whichthe energy is at a maximum is the same as the tip step-down distance. In addition, the maximum energy (theenergy barrier) increases with tip bluntness. Thus, ourfindings corroborate the previous suggestion that frictionat a step is due to an energy barrier, and the magnitude



Graphite surfaces

Appl. Phys. Lett. In press.



XDM summary

XDM dispersion coefficients are obtained from first-principles.

Computational cost is comparable to semi-local DFT.

Excellent binding energies.

Improved thermochemistry.

Applications for molecules and the solid state.



Acknolwdgements

Group members:

Alberto Otero-de-la-RozaMatthew ChristianStephen DaleJoseph DizonJoel Mallory


dispersion interactions in density-functional...

Documents