Properties of Molecular Crystals from Density-Functional ...

XDM Benchmarks Chirality Polymorphism End

Properties of Molecular Crystalsfrom Density-Functional Theory

Erin R. Johnson

E. R. Johnson (Dalhousie) Molecular Crystals from DFT Waterloo SCP (Nov 2015) 1 / 30


Dispersion interactions

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



Dispersion interactions

Long-range non-local correlation not captured by semi-local functionals



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.

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) =〈V̂2

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

Rc,ij are proportional to atomic volumes.

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



Parametrization set

49 gas-phase dimers

dispersionπ-stackingdipole - induced dipolemixeddipole - dipolehydrogen-bonding



Implementation

Calculation of Edisp is fast compared to EDFT.

For molecules: pair XDM with GGA, hybrid, and range-separatedfunctionals using Gaussian 09 and the postg program.

Download postg from http://schooner.chem.dal.ca

For solids: plane waves/pseudopotentials GGA calculations using theQuantum ESPRESSO program.

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



Molecular benchmarks

XDM has been benchmarked for:

Dispersion coefficients:C6, C8, C10, and 3-body C9

Binding energies ofsmall-molecule dimersSupermolecular complexesThermochemistryKinetics

JCP 138 (2013) 204109; JCTC 11 (2015) 4033; JPCA 119 (2015) 5883



Graphite exfoliation

−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 of molecular crystals

∆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 intermolecular modes

Zero-point vibrational contributions neglected

JCP 137 (2012) 054103



Molecular crystals

21 molecular crystals, low polymorphism, varied interaction types.

Well known sublimation enthalpies at or below room temperature.

Errors for XDM-corrected functionals:

Quantity Sublimation enthalpies Cell lengths(kcal/mol) (au)

B86b PW86 PBE B86b PW86 PBEMAE 1.15 1.55 1.28 0.12 0.06 0.20

MA%E 6.2 8.0 6.7 1.8 0.9 2.8

Without XDM, PBE gives a MAE of 8.6 kcal/mol and a MAPE of 47.2%for sublimation enthalpies.



Chiral crystals



Theoretical model

Enantiomers have the samesolvation energies.

Assume the same temperatureeffects so that

∆E = Edl − El

The predicted e.e. is:

ee =[L]− [D]

[L] + [D]=β2 − 1β2 + 1

β = e−∆E/RT

Using experimental heats of formation for the L and DL crystals givesthe following e.e. values (%):

Amino acid Model Expt.Leucine 85.3 87.9Alanine 74.6 60.4



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 44.4

Aspartic acid 0.0 0.7Glutamic acid 0.0 0.7

Mean Absolute Error – 10.5

Angew. Chem. Int. Ed. 118 (2014) 17577.



More e.e. values (%)

Molecule DFT Expt.

51.3 62

35.5 34

>99.9 >99.9



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)E. R. Johnson (Dalhousie) Molecular Crystals from DFT Waterloo SCP (Nov 2015) 20 / 30


Isoleucine – an Allo diastereomer

DFT gives an e.e. 93.2%compared with 51.7% fromexperiment.

Chiral HPLC-MS chro-matography reveals a minoramount of allo-isoleuceinethat alters the equilibrium byforming a co-crystal.



Selective crystallization

Contact with an enantiopure seed drives crystallization from asupersaturated solution without nucleation of the other enantiomer.

Flask A Flask B

Pump D

Pump C1

2

=OH

NH2

OH

O

=OH

NH2

OH

O

[ ] [ ]=

[ ] [ ]>

A)

Flask A Flask B Flask A Flask B

time initial time finalB)

By predicting the e.e. at the chiral eutectic, DFT-XDM can guide thechoice of compounds for selective crystallization.



Polymorphism

Polymorphs have different:

packing arrangementselectronic energiessublimation energiesmelting pointssolubilitiesbioavailability

First-principles crystal-structure prediction(CSP) is a grand challenge in chemisty.



CSP blind tests

The Cambridge CrystallographicData Centre (CCDC) announcesa set of 4-5 compounds.

Groups have a year to predictthe crystal structure.

Predictions are compared toexperimental x-ray structures.

There have been 5 blind tests, for a total of 21 compounds.



Polymorph ranking

Can DFT-XDM be used forenergy ranking of polymorphs?

Use the CSP blind tests as ourbenchmark set.

Consider “Class 1” and “Class 2”compounds - small rigid moleculeswith < 36 atoms.



Polymorph results

Molecule XDM Base Molecule XDM Base

3 7 3 7

3 7 3 3

3 7 3 3



Polymorph results

The base functional tends to favour less dense crystals.

Experimental/DFT-XDM Base functional



An organic salt

Do not expect good performance because GGAsfavour fractional charge-transfer (0.82 e−).

Experimental structure GGA-favoured structure



Summary

XDM dispersion coefficients are obtained from first-principles.

Computational cost is comparable to semi-local DFT.

DFT-XDM yields accurate sublimation energies and cell lengthsfor molecular crystals.

DFT-XDM can predict relative solubilities of chiral crystals.

Preliminary data show promise for polymorph ranking and CSP.



Acknowledgements

Current group members:

Matthew ChristianStephen DaleLuc LeBlanc

Post-docs and collaborators:

Alberto Otero-de-la-RozaSarah WhittletonJason Hein


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