Mark E. Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Sciences

New York University, 100 Washington Square East, NY 10003 NYU-ECNU Center for Computational Chemistry at NYU Shanghai 200062, China

纽约大学 - 华东师范大学计算化学中心在上海纽约大学

Enhanced Free Energy Based Structure Prediction

in Materials Science

NEW YORK UNIVERSITY MRSEC

Crystallization and the Pharmaceutical Industry: Materials Science?

Pharmaceutical Materials Science:

An Active New Frontier in Materials

Research

J. Elliott and B. Hancock

Guest Editors

MRS Bulletin 31, Issue 11 (2006)

Pharmaceutical Industry

• $500 billion global market

• $250 billion US market

• Majority of pharmaceuticals are

small molecule crystals

“One of the continuing scandals in the physical sciences is that it remains impossible to

predict the structure of even the simplest crystalline solids from their chemical composition.”

-- John Maddox, Nature (1998)

Targets of the sixth blind structure prediction test

Accuracy of intermolecular interactions

Empirical potentials fit to SAPT(DFT) calculations

( )

( ) (1) ( )

1 bonded

6,8,10

( ) 1 ( , ) ( , ) ( )ij ij

n

r i j ijl l n

ij ij ij ij n ij ij nij l nij ij

q q CU A r e f r f r U

r r

R R

0

( , ) 1!

mn

r

n

m

rf r e

m

Machine-learning (‘mathematical’) potentials

Standard force fields

bonded12 6( ) ( )

i j ij ij

ij ij ij ij

q q a bU U

r r r

R R

Kohn-Sham density functional theory (plus dispersion):

2

{ }

1 1 ( ) ( )( ) [ ] [ , ] ( )

2 2 'min i i xc ext NN

i

n nU d d E n E n U

r r

R r r R Rr r

2( ) ( ) i i j ij

i

n r r

Structure prediction protocol for Target XXII

P212121

Lowest Energy Structures

P21/n P21/n

Pbca C2/c Pbca

Optimize monomer

geometry using DFT

[PBE0 + D3, 6-311G**(d,p)

Generate Unit Cells

(UPACK)

25,000 generated structures

4,042 unique after clustering

107 within 10 kJ/mol of lowest energy

Run MD at exptl. P,T

Summary of 50 predicted structures based on MD averaged energies

I: P21/n II: Pbca, E = 3.46 kJ/mol III: P21/c, E = 5.37 kJ/mol

IV: P21/n, E = 5.99 kJ/mol V: P21/c, E = 6.31 kJ/mol

Spc grp a b c β

Expt. P21/n 11.95 6.70 12.60 108.6

Calc. P21/n 12.02 6.72 12.48 108.7

Prediction of the experimental structure

At 150 K: RMSD20 from experimental structure = 0.15 Å. Rank = 4 on our list

Final ranking based on POLY1 SAPT(DFT) potential

At 300 K: RMSD20 from experimental structure = 0.18 Å. Rank = 4 on our list

Attempted predictions = 21, Times predicted = 12, Times ranked 5 or lower = 7

Pharmaceutical polymorphism

HIV-1 protease

Polymorphism refers to the ability

of a compound to form multiple

crystal structures.

Ritonavir: HIV protease inhibitor

• NIH: $3,500,000 investment

• Abbott Labs: $200,000,000 investment

• Originally dispensed in 1996 as ordinary

capsules, no refrigeration required

• Converted to lower energy (hitherto unknown)

polymorph (form I to form II) on the shelf

• Form II: Poor solubility, lower bioavailability

• Required recall (1998) and reformulation as

gel caps (2002).

Pharmaceutical Polymorphism

Ranitidine hydrochloride (Zantac) C13H22N4O3S

Bond et al. Amer. Pharma. Rev. (2007) Acetylsalicylic acid

Rough energy landscapes in proteins and crystals

Dunitz and Scheraga

PNAS 101, 14309 (2004)

Sampling via collective variables in molecular simulation

φ ψ

Peptides:

Crystals:

Packing:

Orientation:

Cell shape:

x x x

y y y

z z z

a b c

a b c

a b c

h

2

1

4( ) ( , )

(2 1)

bNl

l b lm b b

m l b

Q f r Yl N

( , )

1

1

( ,..., ) d d ( ( ) )n

H

nP s s e q s

p r

p r r

Adiabatic free energy (temperature-accelerated molecular) dynamics (AFED/TAMD)

L. Rosso, P. Minary, Z. Zhu, MET J. Chem. Phys. 116, 4389 (2002)

Margliano and Vanden-Eijnden, Chem. Phys. Lett. 426, 168 (2006); J. B. Abrams and MET, J. Phys. Chem. B 112, 14752 (2008)

For prediction of crystal structures: Yu and MET Phys. Rev. Lett. 107, 015701 (2011); Yu et al. J. Chem. Phys. 150, 214109 (2014)

1/2

2

{ }( ) lim exp ( )

2 2q s q s

r r

Write δ-functions as product of Gaussians:

Canonical probability distribution and free energy surface:

2

2

1 1

( , , , ) ( , ) ( )2 2

n ns

s

ps p H q s

m

p r p r rH

1( ,..., ) ( ) 1,...,Nq q q n r r r

Suppose n collective variables characterize a free energy landscape of interest

1 1( ,..., ) ln ( ,..., )n B nA s s k T P s s

Dynamically sample Gaussian centers s1,…,sn by introducing a kinetic energy for them

And extending the phase space, which gives the following Hamiltonian:

Adiabatically decoupled equations of motion:

( ) heat bath( )i i

i i

qHm s q T

r rr r

( ) heat bath( )sm s s q T r

Introduce high temperature for extended variables and high masses s iT T m m

Under adiabatic conditions, we generate a distribution ({ })

adb 1( ,..., , , )n sP s s T T

({ })

adb 1{ }

1 1{ }

lim ln ( ,..., , , )

lim ln ( ,..., , ) ( ,..., , )

s n s

s n n

s

kT P s s T T

TkT P s s T A s s T

T

/({ })

adb 1 1{ }lim ( ,..., , , ) ( ,..., , ) sT T

n s nP s s T T P s s T

Adiabatic free energy (temperature-accelerated molecular) dynamics (AFED/TAMD)

Alanine Decamer (gas phase)

Force field: CHARMM22

20 CVs: All (φ,ψ) pairs

CV Temperature: Ts = 900 K

Physical Temp: T = 300 K

CV mass: m(ψ,φ) = 600mH

Harmonic coupling: 300.5 kcal/mol/rad2

Other developments: 1. Combining with metadynamics used as a bias:

2. Optimal free energy reconstruction protocols:

M. Chen. M. Cuendet, MET J. Chem. Phys. 137, 024102 (2012)

M. Cuendet and MET J. Chem. Theor. Comput. 10, 2975 (2014)

GROMOS FF, 100 K, 32,000 40,000 KsT T

216 (3 3 3), 2 GPa, Total run time = 500 ps, 5 nsN P

Constant pressure version

with cell matrix as CVs

II(01)

X-ray powder diffraction patterns

Yonetani et al. (2001).

II(98)

98

01

MURI/ARO

Free energies of benzene polymorphs

Six-dimensional free energy surface analyzed using

clustering based on a Gaussian mixture model.

II(98) II(01)

Cmca

P41212

P21/c

C2/c

P21/c

(see, also, J. Yang et al.

Science 345, 640 (2014))

Polymorphs of resorcinol

• Resorcinol (C6H6O2) is a simple benzene derivative.

• At ambient conditions, forms two stable forms (α and β) depending on the relative orientations of the OH bonds, both having space group Pna21

1.

• High pressure phase (δ phase) exists and arises from an α → δ at pressures above 3 GPa2. Structure unknown.

• Another high pressure phase (ε phase) can be reached from β phase above 5.6 GPa. This phase is also thought to be disordered based on Raman scattering2.

• Another ambient polymorph (γ phase) in the Rb conformation recently found3. Its exact structure is unknown4.

• Recent new experimental activity in study of resorcinol.5

1Robertson and Ubbelohde Proc. Royal Soc. London 167 122 (1938) 2 Rao et al. Phys. Rev. B 65 054108 (2002). 3 Kahr et al. Cryst. Growth Design 11 2070 (2011). 4 Zhu et al. Angew. Chem. Intl. Ed. (submitted) 5 J. Phys. Chem. B (2015); J. Mol. Struct. (2015).

polymorph

12P

transition

at 3 GPa

Preliminary free energy estimates for resorcinol

form δ form

a/A b/A c/A a/A b/A c/A

Cell length 10.53 9.53 5.60 8.20 11.65 4.88

Cell angle 90 90 90 90 90 90

Space GP Pna21 Pna21

Supercell length A

Energetic Co-crystals: DADP.TxTNB

DADP

TCTNB

β

( iodo- )

α

( chloro- )

( bromo- )

Experimental Structures

Polarizable system without pre-determined force field parameters

= opportunity for ab initio enhanced sampling!

Exp. Structures (Maztger Group):

J. Am. Chem. Soc. 2015, 137, 5074−5079

Angew. Chem. Int. Ed. 2013, 52, 6468-6471

1 3

CP2K 2.5: PBE+D3/DZVP-MOLOPT-(SR)-GTH

For optimized unit cells

within PBE+D3, DZVP, 450 Ry in CP2K

Cl α < Cl β ΔE ≈ 1.6 kJ/mol

Br α > Br β ΔE ≈ 4.9 kJ/mol

I α >> I β ΔE ≈ 15.8 kJ/mol

Energetic Co-crystals: DADP.TxTNB

β

( iodo- )

α

( chloro- )

( bromo- )

• Observed DADP.TBTNB co-crystal is a

kinetic product – aggregation promotes

co-crystal growth?

• Can we obtain the actual free energy

difference between the two forms for

each co-crystal?

• Implementation of crystal-AFED in CP2K

Start with nuclei Compute ,i n ,i n F

Propagate nuclei a

short time Δt with F

Add electrons

Add electrons

Ab initio molecular dynamics

2

( ) 2 (0) ( ) (0)I I I I

I

tt t

M

R R R F

e.g. Verlet:

{ }min [{ }, ]I I IM E

R R

11.0 picosecond of ab initio crystal-AFED with CP2K for TCTNB/DADP co-crystal

Experimental structure: α New Polymorph Alternate polymorph: β

3

14.344

15.272

9.006

91.3

1.549 g/cm

a

b

c

3

25.797

9.450

12.638

99.6

1.542 g/cm

a

b

c

3

16.724

9.450

12.638

69.7

1.631 g/cm

a

b

c

MURI/ARO

Crystal structures of Xenon at 2700 K and 25 GPa

Using Q4 and Q6 only

as collective variables

Ts = 1.5 x 105 K

ms = 108 amu

Using Q4, Q6, and the

cell matrix hd as

collective variables

Ts = Th = 105 K,

ms = 108 amu

T. Q. Yu, P. –Y. Chen, A. Samanta, E. Vanden-Eijnden, MET J. Chem. Phys. (2014)

Experimental evidence mixed fcc-hcp structures

in pressure-induced Martensitic transitions

from fcc to hcp as intermediates:

Jephcoat et al. Phys. Rev. Lett. 59, 2670 (1987)

Cynn et al. Phys. Rev. Lett. 86, 4552 (2001)

The mechanism of equilibrium melting of a solid

A. Samanta, T.-Q. Yu, W. E, MET Science (November, 2014)

Classical nucleation theory:

3 24( ) 4 , < 0

3sG r r r

3

2

2

16 ( )

3

s

s

r

G r

r*

Crystal d-AFED simulations of the melting of Cu

Collective variables: hd, Q4, Q6, T = Tm, Ts=Th = 107 K

Potential model: Embedded Atom Model (EAM) of Mishin et al.

Phys. Rev. B 63, 224106 (2001).

A. Samanta, T.-Q. Yu, W. E, MET Science (November, 2014)

Free energy surfaces

One-dimensional profiles constructed using string method [Maragliano et al. JCP (2006)]

Navigating on a high-dimensional surface

From a minimum, we ask for the

nearest accessible saddle points.

These can be found using gentlest

ascent dynamics (GAD) on the FES.

A. Samanta, M. Chen, T. –Q. Yu, W. E, MET J. Chem. Phys. 140, 164109 (2014).

M. Chen, T. –Q. Yu, MET PNAS 112, 3235 (2015)

s F

Activation:

Combine with relaxation to next minimum:

Termed “STochastic Activation-Relaxation Technique” or START

Example: Alanine tripeptide

Landmarks determined using density-based clustering methods.

Minima and saddles are assigned as vertices of different colors.

An edge connects a saddle vertex to a minimum vertex if a relaxation initiated at

the saddle ends up at the corresponding minimum.

Graph representation of free energy surfaces

Full network diagram for

met-enkephalin using

10 Ramachandran angles

as CVs:

1081 minima, 1431 saddles

Graph nodes sorted based on

Sketch Map analysis

M. Ceriotti, G. A. Tribello, M. Parrinello Proc. Natl. Acad. Sci. 108, 13023 (2011)

22 ( ) ( )ij ij

i j

F R f r

If we want to compute free energies, use the minima obtained in a START run to

indicate where a set of bins should be placed and use AFED sampling to obtain the

populations in those bins.

Can be adapted for crystal structure prediction! [M. Chen, T. –Q. Yu, MET (in prep)]

Can handle explicit solvent!

Spc grp a b c β

Expt. P21/c 9.45 6.04 8.56 103.4

Calc. P21/c 9.95 5.90 8.79 106.6

Experimental structure first found in 1 ns using orientational CVs. RMSD4 of carbons = 0.16 Å, full RMSD4 = 0.44 Å using a simple, universal force field

M. Chen, T. –Q. Yu,

MET (in prep)

Conclusions and Perspectives

1. Protocol for crystal structure prediction including specialized force field

generation [SAPT(DFT)] combined with molecular dynamics refinement.

2. Enhanced sampling approaches further allow for polymorph prediction and

ranking based on free energy, which is the proper figure of merit.

3. Surface navigation combined with enhanced sampling looks to be a promising

approach going forward, however, it is important to have a robust set of

internal collective variables.

4. Considerable challenges remain including:

i. Highly flexible molecules.

ii. High-temperature and/or high-pressure polymorphs.

iii. Stacking faults, screw axes, mixed phases,….

iv. Isotopic polymorphism (requires, e.g., path integrals).

5. From Nature press release based on CCDC 6th blind test, “It is fair to claim that,

to a large extent, this blind test has shown that the problem of organic structure

prediction has been solved.” – M. Neumann. This claim might be exaggerated,

and the field of crystal structure prediction far from dead.

Acknowledgments

Postdocs • Elia Schneider • Linas Vilciauskas • Leslie Vogt

External

• Krzysztof Szalewicz (U. Del.) • Michael Metz (U. Del.) • Andrew Rohl (Curtin U.) • Bart Kahr (NYU) • Qiang Zhu (Stony Brook)

Students

• Ming Chen • Tang-Qing Yu • Manav Kumar • Hongxing Song