Enhanced Free Energy Based Structure Prediction in Materials Science

Transcription

1 Enhanced Free Energy Based Structure Prediction in Materials Science Mark E. Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Sciences New York University, 100 Washington Square East, NY NYU-ECNU Center for Computational Chemistry at NYU Shanghai , China 纽约大学 - 华东师范大学计算化学中心在上海纽约大学 NEW YORK UNIVERSITY MRSEC

2 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)

3 Targets of the sixth blind structure prediction test

4 Accuracy of intermolecular interactions Standard force fields qiq j aij b ij U( R) U 12 6 bonded ( R) ij rij rij rij Kohn-Sham density functional theory (plus dispersion): 1 1 n( r) n( r) U d d E n E n U 2 ( R) min i i xc[ ] ext[, ] NN ( ) { } 2 i 2 r r R R rr' Empirical potentials fit to SAPT(DFT) calculations ( n) ( l) l ijr q ij (1) iq j C ( n) ij U( R) 1 Aij rij e f1( ij, rij ) fn( ij, rij ) Ubonded ( ) n R ij l rij n6,8,10 rij m n r r fn(, r) 1e m! Machine-learning ( mathematical ) potentials 2 n( r) ( r) i i i j ij m0

5 Structure prediction protocol for Target XXII Optimize monomer geometry using DFT [PBE0 + D3, 6-311G**(d,p) Generate Unit Cells (UPACK) Run MD at exptl. P,T 25,000 generated structures 4,042 unique after clustering 107 within 10 kj/mol of lowest energy Lowest Energy Structures C2/c Pbca Pbca P P2 1 /n P2 1 /n

6 Summary of 50 predicted structures based on MD averaged energies

7 I: P2 1 /n II: Pbca, E = 3.46 kj/mol III: P2 1 /c, E = 5.37 kj/mol IV: P2 1 /n, E = 5.99 kj/mol V: P2 1 /c, E = 6.31 kj/mol

8 Prediction of the experimental structure Final ranking based on POLY1 SAPT(DFT) potential At 300 K: RMSD 20 from experimental structure = 0.18 Å. Rank = 4 on our list At 150 K: RMSD 20 from experimental structure = 0.15 Å. Rank = 4 on our list Spc grp a b c β Expt. P2 1 /n Calc. P2 1 /n Attempted predictions = 21, Times predicted = 12, Times ranked 5 or lower = 7

9 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 HIV-1 protease

10 Pharmaceutical Polymorphism C 13 H 22 N 4 O 3 S Ranitidine hydrochloride (Zantac) Acetylsalicylic acid Bond et al. Amer. Pharma. Rev. (2007)

11 Rough energy landscapes in proteins and crystals Dunitz and Scheraga PNAS 101, (2004)

12 Sampling via collective variables in molecular simulation Peptides: φ ψ Crystals: Cell shape: Packing: Q 4 l 2 l f rb Ylm b b (2l1) N ml b1 N b ( ) (, ) h a b c a b c a b c x x x y y y z z z Orientation:

13 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, (2008) For prediction of crystal structures: Yu and MET Phys. Rev. Lett. 107, (2011); Yu et al. J. Chem. Phys. 150, (2014) Suppose n collective variables characterize a free energy landscape of interest Canonical probability distribution and free energy surface: q q (,..., ) q( ) 1,..., n r1 rn r n H (, ) ( 1,..., n) pr d p d r ( ( r) ) 1 P s s e q s A( s,..., s ) k T ln P( s,..., s ) 1 n B 1 n Write δ-functions as product of Gaussians: 1/2 2 { } q ( r) s lim exp q ( r) s 2 2 Dynamically sample Gaussian centers s 1,,s n by introducing a kinetic energy for them And extending the phase space, which gives the following Hamiltonian: n 2 p n s H ( p, r, s, ps) H( p, r) q( r) s 2m

14 Adiabatic free energy (temperature-accelerated molecular) dynamics (AFED/TAMD) Introduce high temperature T T for extended variables and high masses m m s i Adiabatically decoupled equations of motion: H q ( ) miri s q r heat bath( T) r r i m s s q r T s ( ) heat bath( ) Under adiabatic conditions, we generate a distribution { } ({ }) lim kts ln Padb ( s1,..., sn, Ts, T) { } { } i P ( s,..., s, T, T) ({ }) adb 1 / lim P ( s,..., s, T, T) P( s,..., s, T) ({ }) adb 1 n s 1 n T lim kts ln P( s1,..., sn, T) A( s1,..., sn, T) Ts TT s n s

15 Other developments: 1. Combining with metadynamics used as a bias: M. Chen. M. Cuendet, MET J. Chem. Phys. 137, (2012) 2. Optimal free energy reconstruction protocols: M. Cuendet and MET J. Chem. Theor. Comput. 10, 2975 (2014) Alanine Decamer (gas phase) Force field: CHARMM22 20 CVs: All (φ,ψ) pairs CV Temperature: T s = 900 K Physical Temp: T = 300 K CV mass: m (ψ,φ) = 600m H Harmonic coupling: kcal/mol/rad 2

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17 Constant pressure version with cell matrix as CVs GROMOS FF, T 100 K, T s 32,000 40,000 K N 216 (333), P 2 GPa, Total run time = 500 ps, 5 ns

18 X-ray powder diffraction patterns Yonetani et al. (2001). II(98) II(01) 98 01

19 Free energies of benzene polymorphs Six-dimensional free energy surface analyzed using clustering based on a Gaussian mixture model. (see, also, J. Yang et al. Science 345, 640 (2014)) P2 1 /c Cmca P C2/c P2 1 /c II(98) II(01) MURI/ARO

20 Polymorphs of resorcinol Resorcinol (C 6 H 6 O 2 ) 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 Pna2 11. High pressure phase (δ phase) exists and arises from an α δ at pressures above 3 GPa 2. 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 scattering 2. Another ambient polymorph (γ phase) in the R b conformation recently found 3. Its exact structure is unknown 4. Recent new experimental activity in study of resorcinol. 5 1 Robertson and Ubbelohde Proc. Royal Soc. London (1938) 2 Rao et al. Phys. Rev. B (2002). 3 Kahr et al. Cryst. Growth Design (2011). 4 Zhu et al. Angew. Chem. Intl. Ed. (submitted) 5 J. Phys. Chem. B (2015); J. Mol. Struct. (2015).

21 polymorph

22 P21

23 transition at 3 GPa

24 Preliminary free energy estimates for resorcinol Supercell length A form δ form a/a b/a c/a a/a b/a c/a Cell length Cell angle Space GP Pna2 1 Pna2 1

25 Energetic Co-crystals: DADP.TxTNB DADP TCTNB Polarizable system without pre-determined force field parameters = opportunity for ab initio enhanced sampling! Experimental Structures α β ( chloro- ) ( iodo- ) ( bromo- ) 1 3 Exp. Structures (Maztger Group): J. Am. Chem. Soc. 2015, 137, Angew. Chem. Int. Ed. 2013, 52,

26 Energetic Co-crystals: DADP.TxTNB α ( chloro- ) ( bromo- ) β ( iodo- ) For optimized unit cells within PBE+D3, DZVP, 450 Ry in CP2K Cl α < Cl β Br α > Br β ΔE 1.6 kj/mol ΔE 4.9 kj/mol I α >> I β ΔE 15.8 kj/mol 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 CP2K 2.5: PBE+D3/DZVP-MOLOPT-(SR)-GTH

27 Ab initio molecular dynamics M R I I I min E[{ }, R] { } Start with nuclei Add electrons Compute i, n, i n F Add electrons Propagate nuclei a short time Δt with F e.g. Verlet: 2 t RI ( t) 2 RI (0) RI ( t) FI (0) M I

28 11.0 picosecond of ab initio crystal-afed with CP2K for TCTNB/DADP co-crystal

29 Experimental structure: α New Polymorph Alternate polymorph: β a b c a b c a b c g/cm g/cm g/cm

30 Crystal structures of Xenon at 2700 K and 25 GPa 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) Using Q 4 and Q 6 only as collective variables T s = 1.5 x 10 5 K m s = 10 8 amu Using Q 4, Q 6, and the cell matrix h d as collective variables T s = T h = 10 5 K, m s = 10 8 amu MURI/ARO

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32 The mechanism of equilibrium melting of a solid A. Samanta, T.-Q. Yu, W. E, MET Science (November, 2014) Classical nucleation theory: 4 G r r r ( ) 4 s, < 0 r* r 2 s Gr ( ) s 2

33 Crystal d-afed simulations of the melting of Cu A. Samanta, T.-Q. Yu, W. E, MET Science (November, 2014) Collective variables: h d, Q 4, Q 6, T = T m, T s =T h = 10 7 K Potential model: Embedded Atom Model (EAM) of Mishin et al. Phys. Rev. B 63, (2001).

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

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36 From a minimum, we ask for the nearest accessible saddle points. These can be found using gentlest ascent dynamics (GAD) on the FES. Activation: Navigating on a high-dimensional surface A. Samanta, M. Chen, T. Q. Yu, W. E, MET J. Chem. Phys. 140, (2014). M. Chen, T. Q. Yu, MET PNAS 112, 3235 (2015) Combine with relaxation to next minimum: s F Termed STochastic Activation-Relaxation Technique or START

37 Graph representation of free energy surfaces 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. Example: Alanine tripeptide

38 Full network diagram for met-enkephalin using 10 Ramachandran angles as CVs: 1081 minima, 1431 saddles Graph nodes sorted based on Sketch Map analysis 2 F( Rij ) f ( rij ) i j 2 M. Ceriotti, G. A. Tribello, M. Parrinello Proc. Natl. Acad. Sci. 108, (2011)

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

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41 M. Chen, T. Q. Yu, MET (in prep) Spc grp a b c β Expt. P2 1 /c Calc. P2 1 /c Experimental structure first found in 1 ns using orientational CVs. RMSD 4 of carbons = 0.16 Å, full RMSD 4 = 0.44 Å using a simple, universal force field

42 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 6 th 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.

43 Acknowledgments External Krzysztof Szalewicz (U. Del.) Michael Metz (U. Del.) Andrew Rohl (Curtin U.) Bart Kahr (NYU) Qiang Zhu (Stony Brook) Postdocs Elia Schneider Linas Vilciauskas Leslie Vogt Students Ming Chen Tang-Qing Yu Manav Kumar Hongxing Song