1. 1. MIT 10.637 Lecture 2 Molecular mechanics Heather J Kulik hjkulik@mit.edu Tues 09-09-14
2. 2. MIT 10.637 Lecture 2 Overview Bonding components of force fields Non-bonding components of force fields Extensions to force fields Amber force field example Methods for parameterizing force fields
3. 3. MIT 10.637 Lecture 2 Choosing a computational model Empirical models functional form with parameters from experimental or other calculated data: Pair potentials Many body potentials Semi-empirical models model Hamiltonians: Tight binding MNDO, AM1 Quantum mechanical models approximations to the Schrdinger equation: Hartree-Fock Density functional theory Post-Hartree-Fock (Configuration interaction, MP2) Moreefficient Moretransferable Best tool for the job? Depends on the job!
4. 4. MIT 10.637 Lecture 2 Empirical model potentials Species are repulsive for small distances Attractive for longer distances Only need to calculate potentials for atoms within a certain distance. V(DR) DR
5. 5. MIT 10.637 Lecture 2 Limitations of pair potentials Counts only bonds, not organization: Nbonds = 3 Nbonds = 3 =
6. 6. MIT 10.637 Lecture 2 Limitations of pair potentials Counts only bond length, no orientation or angular effects (e.g. ethylene): qq Pair potentials: C-H, C-C bonds. No treatment: H-C-H angle, H-C-C-H dihedral.
7. 7. MIT 10.637 Lecture 2 Limitations of pair potentials Preference for high number/high density of bonds formed to lower total energy: Bond energy for blue atom on left is four times the right. In real systems: more bonds means lower energy/bond.
8. 8. MIT 10.637 Lecture 2 Force field energy terms bend non-bonded Coupling terms
9. 9. MIT 10.637 Lecture 2 Transferability of force fields Force field parameters are designed to be transferrable: e.g. most C-H bond lengths are 1.06 1.10 in almost all molecules, stretching frequencies are 2900-3300 cm-1. Separating van der Waals and electrostatic terms out makes force field parameters more transferable across atom types.
10. 10. MIT 10.637 Lecture 2 Stretch energy Taylor expansion around equilibrium R0: Set to zero Expanded around minimum so = 0 Typically only use first term in most force fields. If expand to cubic term then limit at long bond lengths is negative infinity (bad!) usually terminate with quartic positive infinity energy at long bond length.
11. 11. MIT 10.637 Lecture 2 Morse Potential 0.0 1.0 2.0 3.0 4.0 r () 0 1 2 3 4 5 V(r)(eV) H2 De re H2 parameters: De=4.75 eV b=1.93 1/ re=0.741 adapted from McQuarrie and Simon V(r)= De(1-e-b(r-re) )2 Most accurate but too slow/poor restoring force at large R so seldom used in force fields.
12. 12. MIT 10.637 Lecture 2 Stretch energy MM3 and other force fields include up to quartic term, bio force fields use only quadratic: Need three parameters can be difficult to get from spectroscopy, but can relate to Morse potential. V(r)= De(1-e-b(r-re) )2 Can use expt. k and De to get k(3) and k(4).
13. 13. MIT 10.637 Lecture 2 Example: CH4 stretch adapted from Jensen Energy window for force fields
14. 14. MIT 10.637 Lecture 2 Angular terms Taylor expansion around natural or equilibrium bond angle: General organic force field MM3 continues to sextic terms Biomolecular force fields are typically truncated at quadratic terms: q
15. 15. MIT 10.637 Lecture 2 Adding angular terms H2O Bending adapted from Jensen Harmonic works well at +/- 20o around equilibrium Cubic truncation of Taylor expansion can be forced to have zero derivative at 180 degrees. Energy window for force fields
16. 16. MIT 10.637 Lecture 2 Out of plane bending d c Energy penalty associated with bending an atom out of the plane called Eoop or Einv based on angle (c) or distance out of the plane (d). Also called improper torsions. e.g. case where equilibrium angle is planar (0), like benzene: Could use this for inversion barriers but usually not needed because there is a barrier from increasing bond angles during the inversion. Many ways to define this can also define as a torsion (next slides).
17. 17. MIT 10.637 Lecture 2 Torsional terms Torsional angle is angle between plane formed by ijk and jkl, that is looking along bond j-k angle formed between ij and kl bonds. Torsion barriers have contributions from non-bonded terms in addition to torsional energy. Coupling between parameters! Torsional energy must be periodic in angle bond is rotated 360o then energy must be the same. Energetic cost for distorting by rotation around bond is usually low so large deviations may occur Taylor expansion wont work.
18. 18. MIT 10.637 Lecture 2 Torsional terms Best fit for terms is Fourier series: Example of typical torsional term: Rewritten so it is non-negative.
19. 19. MIT 10.637 Lecture 2 Adding torsional terms Ethane eclipsed Ethane staggered Torsional potential is steric, non-bonded electrostatics. Periodicity needs to be enforced, e.g. ethane. 0 60 120 180 240 300 360 Dihedral (degrees) Energy
20. 20. MIT 10.637 Lecture 2 Components of torsional terms Example: V1=0.5, V2=-0.2, V3=0.5 Like butane anti gauche gauche
21. 21. MIT 10.637 Lecture 2 Choosing torsional terms Force fields for large systems only have one term 3w for single bonds, 2w for double bonds. Generally n=3,6,9 for single bonds, n=2,4 for double bonds. May need more terms for complicated systems, e.g. octahedral metal-ligand complexes. Systems with bulky substituents could have as many as 6 minima. Can also have a phase term to shift the torsional term, but most commonly phase shift is 0 or 180.
22. 22. MIT 10.637 Lecture 2 Interpreting torsional terms n = 1 corresponds to dipole interactions n = 2 corresponds to hyperconjugation n= 3 corresponds to steric 1-4 interactions
23. 23. MIT 10.637 Lecture 2 van der Waals interactions Also called London or dispersion forces non-classical and arises from correlation of electronic wavefunctions. Is an induced dipole-dipole interaction that lowers energy at intermediate distances. Other terms dipole-quadrupole, quadrupole-quadrupole and so on but typically neglected. Not important for non-polar molecules. Since it is long-range, can be dominant cost of force-field calculation. When atoms get too close, densities overlap and exchange repulsion makes interaction repulsive.
24. 24. MIT 10.637 Lecture 2 Forms of the van der Waals energy Correct form including exchange repulsion: Form typically used for saving computational cost is Lennard-Jones: No mathematical justification for R12, just computationally efficient.
25. 25. MIT 10.637 Lecture 2 Lennard-Jones Potential 2.0 4.0 6.0 8.0 10.0 r () -150 -100 -50 0 50 100 150 200 E/kB(K) r0 VLJ Ar parameters: R0=3.345 /KB=125.7 K J. A. White, JCP (1999). Argon Simplest example of balance between repulsive and attractive energies
26. 26. MIT 10.637 Lecture 2 Choosing parameters Van der Waals distance and softness parameters depend on atoms A and B that are interacting. Usually take sum of van der Waals radii for minimum distance and geometric mean for softness: Usually parameterized on experimental data which may include manybody effects Hydrogen is not spherical so some force fields scale the vdw distance. Electronegative atoms give smaller effective vdW for hydrogen. Force field may have different types of hydrogen. Lone pairs can make atom larger in some directions. Some force fields give lone pairs pseudoatom types. Most biological force fields just use simple spherical parameters in LJ.
27. 27. MIT 10.637 Lecture 2 Hydrogen bonding Hydrogen bonding can be treated as a special form. Pairs of atoms that can form hydrogen bonds and vdW replaced by special hydrogen bonding parameters. Sometimes multiply by a directional term (1- cos(wXHY)).
28. 28. MIT 10.637 Lecture 2 Electrostatic energy Coulomb interactions between atoms A and B with partial charges: e is Effective dielectric constant, typically 1 in vacuum. However, may be chosen arbitrarily to be larger (e.g. distance dependent) to kill off electrostatic contributions faster and make them easier to calculate (avoids square root), also mimics screening of interactions by the presence of solvent.
29. 29. MIT 10.637 Lecture 2 Electrostatic energy Alternative approach to electrostatics: Polarized bonds are dipoles and we compute the electrostatic interaction between the dipoles A B c aA aB This is used in the MM2 and MM3 force fields gives same result but harder to parameterize.
30. 30. MIT 10.637 Lecture 2 Where do q come from? Atomic/partial charges: these are not physical observable quantities but are used in computational chemistry. Restrained Electrostatic Potential (RESP) charges. Potential is fitted just outside of the vdW radius: Buried atoms are restrained to have low charge with hyperbolic constraint around 0. Minimize error function:
31. 31. MIT 10.637 Lecture 2 Challenges with electrostatics Electrostatics are computed for nonbonded atoms and die off slowly ~R-1 so it is difficult to treat with a small cutoff. Electrostatic focus on intermolecular interactions, but intramolecular energies also matter and arent fitted. Just partial charges gives crude representation need non- centered partial charges or dipole/quadrupole/etc. Dependence on geometry is neglected. Point charge models do not always accurately reproduce electrostatic potentials, charges cant change with geometry, only pairwise interactions are considered but theres 10-20% change in presence of a third body polarization effect.
32. 32. MIT 10.637 Lecture 2 Improving with the fluctuating charge model Expand energy as a function of the number of electrons N: Take expansion point as one with no net atomic charges and terminate at 2nd order.
33. 33. MIT 10.637 Lecture 2 Fluctuating charge model Sum over sites (atomic centers): External potential term Energy must be stationary with respect to charges: Solve these equations iteratively: potential depends on charges
34. 34. MIT 10.637 Lecture 2 Introducing polarizability Fluctuating charge model couples electrostatic energy to geometry but it does not e.g. account for polarization of planar molecules where field is perpendicular to the plane. Need explicit polarization: dipole polarizability tensor Electric field Each atom contributes to field, dipoles must be solved selfconsistently. Can go beyond induced dipole to quadrupole, octopole
35. 35. MIT 10.637 Lecture 2 Challenges with polarizability To incorporate electric multipoles, fluctuating charges, and atomic polarizabilities: many more fitting parameters needed must rely on electronic structure calculation. Need to deconvolute electronic structure calculation to obtain separated parameters. e.g. decompose dipole into permanent and induced contribution. Gas phase polarization to condensed phase may lead to errors. Computational cost is increased dramatically.
36. 36. MIT 10.637 Lecture 2 Cross terms This is important to reproduce vibrational spectra. Stretch of one bond affects stretch of other bond, for example: Class of force fields that includes cross terms are more sophisticated, require more computational time. bend Or stretch/bend combinations:
37. 37. MIT 10.637 Lecture 2 Classes of force fields Class I: harmonic functions for stretch and bends, no cross terms, L-J for vdW. Also called harmonic or diagonal. Common for large systems like proteins or DNA. Class II: cross terms, cubic or quartic expansions of stretch and bend, exponential- type potentials for vdW. For small to medium molecules. Class III: Hyperconjugation and polarization also included.
38. 38. MIT 10.637 Lecture 2 Computational cost n Natoms Estr Ebend Etors Evdw 10 32 31 (5%) 30(10%) 81(14% ) 405(70%) 20 62 61 (3%) 60(6%) 171(8% ) 1,710(83%) 50 152 151 (1%) 300(3%) 441(4% ) 11,025(93%) 100 302 301 (1%) 600 (1%) 891(2% ) 44,550(96%) N (N-1) 2(N-2) 3(N-5) 1/2N(N-1)-3N+5 e.g. CH3(CH2)n-2CH3 The stretch, bend and torsion terms grow linearly with system size but the non- bonded contributions grow with the square of the system size. Individual non-bonded terms fall off quickly with system size but number of contributions increase. Can use a cutoff of 10 Angstroms but a cutoff of 20 Angstroms cutoff is really needed to converge vdW. Electrostatics might need 30 A. Also need to keep a non-bonded/neighbor list. Fast multipole or Ewald for faster electrostatic energies.
39. 39. MIT 10.637 Lecture 2 Accuracy of force fields Relative energies of conformers are harder to get right than geometries. Conformer energies depend strongly on various non-bonded and torsional terms. All force fields will have some extra artificial minima and will miss some real minima, e.g. cyclododecane MM2 has 122 conformations while MM3 has only 98. Can validate with the heats of formation compared to experiment (just absolute energy isnt a meaningful quantity). Compound Type Avg. Error DHf (kcal/mol) Hydrocarbons 0.42 Aliphatic amines 0.46 Ethers and alcohols 0.50 Carbonyls 0.81 Silanes 1.08 Aromatic amines 2.90 Works best for simple, well parameterized compounds. Error on hydrocarbons is within experimental uncertainty.
40. 40. MIT 10.637 Lecture 2 Force fields for metal compounds It is really hard to come up with good force field parameters for metals. Bonding around metals is more variable and ligands can be exchanged. Energetic cost for geometrical distortion (stretch/bend) smaller around a metal than around carbon. May want to define pseudoatom metal-binding to a center of a bond. Electronic structure calculations must be fit to but those are also difficult for metal complexes.
41. 41. MIT 10.637 Lecture 2 Reactive force fields Standard force fields work only around energy minima. Extended Valence Bond (EVB) reactant and products are diabatic surfaces. Solve for the adiabatic surface. ReaxFF: force field parameters are dependent on geometry. Elaborate interpolation methods that allow the force constant to go to zero as the bond length increases.
42. 42. MIT 10.637 Lecture 2 Follow-up reading Force fields for biological systems: J. W. Ponder and D. A. Case, Force fields for protein simulations Adv. Prot. Chem. (2003). A. D. Mackerell Empirical force fields for biological macromolecules: Overview and issues J. Comput. Chem. (2004). Non-biological force fields: H. Balamane, T. Halicioglu, W. A. Tiller Comparative study of silicon empirical interatomic potentials Phys. Rev. B. (1992). M. Finnis, Interatomic Forces in Condensed Matter, Oxford University Press (2003). M. J. Buehler, A. C. T. van Duin, W. A. Goddard, III Multiparadigm modeling of dynamical crack propagation in silicon using a reactive force field Phys. Rev. Lett. (2006). Force field optimization water examples: L.P. Wang et al Systematic improvement of a classical molecular model of water J. Phys. Chem. B. (2013). L.P. Wang, T.J. Martinez, and V.S. Pande Building Force Fields: An automatic, systematic and reproducible approach J. Phys. Chem. Lett (2014).
43. 43. MIT 10.637 Lecture 2 Survey! svy.mk/1rV7epN
44. 44. MIT 10.637 Lecture 2 AMBER Assisted Model Building with Energy Refinement: AMBER is a force field and a software package. http://www.ambermd.org 1) AMBER Force field: ffXX (year) peptides and nucleic acids, some ions (Mg2+). 2) GAFF (Generalized Amber Force Field): generalized scheme for force field for any organic molecule based upon topology. 3) Parameter sets available on the web: http://www.pharmacy.manchester.ac.uk/bryce/amber/
45. 45. MIT 10.637 Lecture 2 The AMBER force field E = kb (l -l0 )2 + ka (q -q0 )2 + Vn 2 [1+cos(nf -f0 )]+ torsions angles bonds (1) (2) (3) ei, j r0ij rij 12 -2 r0ij rij 6 i=j+1 N j=1 N-1 + qiqj 4pe0riji=j+1 N j=1 N-1 + Cij rij 12 - Dij rij 10 i=j+1 N j=1 N-1 (4) (5) (6) 1, 2, 3: Harmonic oscillator-like bonding, angular, torsional terms van der Waals electrostatic hydrogen bonding
46. 46. MIT 10.637 Lecture 2 Things that are missing More advanced force fields: Cross terms: e.g. bend affects stretch in a water molecule. Polarizability from point charges or multipoles: e.g. AMOEBA. Bond breaking and formation: e.g. ReaxFF. Not all advanced methods are feasible for very large simulations. bend
47. 47. MIT 10.637 Lecture 2 Generalized Amber FF Same functional form as AMBER but more general (http://ambermd.org/antechamber/gaff.html):
48. 48. MIT 10.637 Lecture 2 GAFF parameters Bond constants, force constants, torsional dependence: based on assigned bonding topology (you or a code decides). Across large test set: MP2/6-31g*, MP4/6- 31g* and Cambridge Structural Database training set. Charges for electrostatics from semi- empirical methods or HF/6-31g*: RESP. Can generate custom charges for each molecule.
49. 49. MIT 10.637 Lecture 2 RESP charges
50. 50. MIT 10.637 Lecture 2 RESP charges
51. 51. MIT 10.637 Lecture 2 REDS example Catechol Neutral, default spin, default methods no need for system.config/project.config Otherwise useful info in project.config: MOLECULE1-TOTCHARGE = Neutralize terminating fragments (for amino acids): MOLECULE1-INTRA-MCC1 = 0.0 | 1 2 3 4 5 6 | Remove system.config: CHR_TYP = RESP-X1 Corresponds to using DFT instead of HF for charges.
52. 52. MIT 10.637 Lecture 2 REDS example RED server optimizes structure, assesses bond topology and assigns force constants from GAFF parameters (default is AMBER ff10):
53. 53. MIT 10.637 Lecture 2 REDS example
54. 54. MIT 10.637 Lecture 2 REDS example
55. 55. MIT 10.637 Lecture 2 REDS example Charges for each atom from HF/6-31G* GAMESS calculation are stored in the mol2 file and reported in the charge.txt file:
56. 56. MIT 10.637 Lecture 2 REDS example Charges for each atom from HF/6-31G* GAMESS calculation are stored in the mol2 file and reported in the charge.txt file:
57. 57. MIT 10.637 Lecture 2 REDS example Note bond connectivity information also in mol2 format:
58. 58. MIT 10.637 Lecture 2 Parameterizing force fields MM2 has 71 atom types and assume 30 atom types can form bonds with each other. 2x71 = 142 vDW parameters 1/2x30x30x2 = 900 stretch parameters (k and R0) 1/2x30x30x30x2 = 27000 bend terms (k and q0) 1/2x30x30x30x30x3 = 1,215,000 torsional terms (V1, V2, V3) Cross terms can add 1 million parameters In practice many fewer parameters are used: Term Theoretical Actual vdw 142 142 str 900 290 bend 27000 824 tors 1215000 2466 Only ~0.2% of torsions but these encompass most interesting compounds. Roughly 20% of 1.5 Million known compounds can be modeled. We can also guess torsions from chemically similar ones: e.g. guess H-X-Y-O from H-X-Y-C.
59. 59. MIT 10.637 Lecture 2 Parameterizing force fields Need set of reference data, e.g. from experiments or electronic structure calculations. Then assign values to parameters to match the reference data. Can weight reference data, e.g. bond distances, angles, relative energies, frequencies, etc. Then minimize error function: Can be done sequentially (one class of compounds at a time) or combined. Can vary one parameter or many parameters.
60. 60. MIT 10.637 Lecture 2 Water: an example Lets compare functional forms and features of force fields using water as an example. Recall, most water force fields have incomplete physics: Fixed point charges (no electronic polarization), Classical mechanics (no isotope effects), Fixed bond topology (no chemistry) However, much can be recovered through parameterization: Increase the partial charges to recover polarization effects Tune vdW parameters to recover the experimental density Some force fields are more accurate than quantum chemistry Generally, we characterize our water model by: (1) number of interaction sites, (2) flexibility of bonds, and (3) whether polarization is included.
61. 61. MIT 10.637 Lecture 2 Water: simple models H O H 3-site models: Typically, the O-H bond length and H-O-H angle are kept rigid and fixed to experimental values: r(OH) = 0.9572 HOH = 104.52o Interaction between molecules a & b: electrostatics + vdW Three sites means that atom centered sites are points of interaction. Widely used TIP3P model: q(O) = -0.834 q(H) = +0.417 (q(O)=-2xq(H)) A = 582000 kcal 12/mol B = 595.0 kcal 6/mol
62. 62. MIT 10.637 Lecture 2 Water: simple models H O H 3-site models: Slightly more accurate 3-site models include SPC/E and Flexible SPC: SPC/E: Slightly different geometry r(OH) = 1.0 HOH = 109.47o SPC/E adds an average polarization correction its a constant, averaged correction to the energy of 1.25 kcal/mol per molecule: Difference in dipole between SPC/E and isolated water Isotropic polarizability Other SPC/E parameters: q(O) = -0.8476 q(H) = +0.4238 (q(O)=-2xq(H)) A = 629400 kcal 12/mol B = 625.5 kcal 6/mol
63. 63. MIT 10.637 Lecture 2 Water: simple models H O H 3-site models: SPC: Similar to SPC/E slightly different charges. Flexible SPC (SPC/Fw): Reparameterized three-site model O-H stretching and angular term added. Can reproduce properties of liquid water more accurately. 4-site models: H O H M Four-site models add one dummy atom near the oxygen with a negative charge instead of on oxygen: improves the electrostatic distribution of the water molecule. TIP4P (1983) is a popular 4-site water model. Same geometry as TIP3P. Other parameters: r(OM) = 0.15 q(M) = -1.04 q(H) = +0.52 A = 600000 kcal 12/mol B = 610.0 kcal 6/mol 5- and 6-site water models: charges on lone pair sites + dummy atom.
64. 64. MIT 10.637 Lecture 2 Water: polarizable models Polarizable water models are an example of more sophisticated water models. e.g. AMOEBA force field: cross term electrostatic terms: from permanent and induced dipoles
65. 65. MIT 10.637 Lecture 2 Water: polarizable models Point charge, dipole, quadrupole similar to other electrostatic terms we already covered. Direct polarization: the polarizable dipoles are induced by electric fields from the permanent multipoles. Full self-consistent polarizability: electric fields from the other induced dipoles is included. atomic polarizability Permanent multipole components on site j Interaction matrix element Extra term: Electric fields from the induced dipoles
66. 66. MIT 10.637 Lecture 2 Construct an objective function measuring the disagreement between the reference data and corresponding simulation result. An optimization algorithm searches for parameters that minimize the objective function. kk kk k 2 22 min ResultSimulation DataReference c c opt SR S R Grid Scan Newton-Raphson Simulated Annealing Force field optimization
67. 67. MIT 10.637 Lecture 2 Iterative force field optimization Start by generating parameters and running MD here: Calculate properties and their derivatives with FF parameters Differences and also how to improve them?
68. 68. MIT 10.637 Lecture 2 Force-matching
69. 69. MIT 10.637 Lecture 2 Water: improving TIP4P
70. 70. MIT 10.637 Lecture 2 Water: modified FF properties
71. 71. MIT 10.637 Lecture 2 Water: modified FF properties
72. 72. MIT 10.637 Lecture 2 Fit against experimental and theoretical data 1) Energies and forces for 12,000 geometries from QM theory 2) Gas-phase cluster binding energies from QM theory 3) Experimental monomer geometry, vibrational modes, and multipole moments 4) Experimental density and heat of vaporization curves Property AMOEBA This work Experiment Density (kg m-3) 1000 1 999 1 997 DHvap (kJ mol-1) 43.8 0.1 43.8 0.1 44.0 Dielectric constant 81 10 81 5 78.4 Diffusion constant (10-5 cm2 s-1) 2.0 0.1 2.3 0.1 2.3 Density maximum (C) 15 - 25 0 - 10 4 iAMOEBA example