Molecular Mechanics / ReaxFF

Molecular Dynamics simulations Lecture 09: Molecular Mechanics / ReaxFF Dr. Olli Pakarinen University of Helsinki Fall 2012 Lecture notes based on notes by Dr. Jani Kotakoski, 2010 CONTENTS Molecular mechanics (MM) Force fields for molecular interactions Force field approach in MD GROMACS ReaxFF

Molecular Mechanics Molecular mechanics (MM) is a method to describe complex molecular systems Structurally there are two differences between MD and MM: The constituent particles can be molecules or parts of molecules rather than single atoms Interactions are typically described via force fields instead of potentials For example, methyl ( CH 3 ) and methylene ( CH 2 ) groups are often modeled as single entities Also, large proteins are sometimes modeled as a set of attached beads Bonds are presented as springs no bond breakings occur However, the prototypical application is simple energy minimization, not time evolution different optimization methods However, many variations exist The example below used Folding@Home for about a million CPU days of simulation time [http://folding.stanford.edu] [Christopher D. Snow, Houbi Nguyen, Vijay S. Pande and Martin Gruebele, Nature 420, 102 (2002)]

Force Fields Molecular interactions become easily complicated because bond energies depend on the overall structure instead of only the local neighborhood U = U bond + U angle + U torsion + U oop + U cross + U nonbond (1) Ubond is the energy related to bond length Uangle is associated with changes in the bond angles Utorsion term including variations coming from the rotation of two parts of the molecule with respect to each other Uoop out-of-plane -term, which affects the planarity of the molecule Unonbond ionic or the van der Waals terms which are not associated with covalent bonding MM2: [J. Am. Chem. Soc. 99, 8127 (1977)], AMBER: [J. Comp. Chem. 24, 1999 (2003)], CHARMm: [J. Comp. Chem. 4, 187 (2004)] From http://www.ncsc.org/, original site removed Some terms included in a force field parametrization

BOND TERM The term describing covalent bonding depends on the bond length variation from an ideal value b 0 It is thus a simple pair potential, which can be of Morse or Lennard-Jones type At least until recently these potentials have been too expensive for molecular interactions In fact, the simple harmonic form is often used (AMBER, CHARMm): U bond = 1 2 k b(b b 0 ) 2 (2) bonds Since bond breaking is not allowed, the harmonic approximation is more or less ok for most cases in solids this corresponds to the harmonic regime (Hooke s law) However, it fails for large inter-atomic distances Another often used form is the cubic approximation (MM2) U bond = bonds 1 2 k b(b b 0 ){1 2(b b 0 )} (3) Morse Harmonic Cubic Morse potential is the most realistic of these However, as mentioned, it is sometimes too expensive Bond energy Bond length b 0 Cubic is asymmetric, and in that sense better than harmonic Unfortunately it becomes unphysical at long b

ANGLE TERM Also the angle dependency is typically (AMBER, CHARMm) modeled with a harmonic function U angle = bonds 1 2 k θ(θ θ 0 ) 2 (4) A better description is offered by a more expensive sixth-order function (MM2) U angle = k θ (θ θ 0 ) 2 {1 7 10 8 (θ θ 9 ) 4 } (5) Bond angle bending force constants (k θ ) are proportionately smaller than the bond distortion force constants (k b ) because much lower energies are required for distorting angles TORSIONAL INTERACTIONS Torsional angle rotations are the most important intramolecular terms in a force field τ They differ from bond stretching and bending interactions in two important ways: Internal rotation barriers are low, and thus changes in dihedral angles can be large Torsional potential U torsion is periodic through a 360 rotation

The U torsion depends heavily on the atoms forming the molecule Therefore, the functional form must be chosen so that it can model a wide variety of different potentials Commonly, a Fourier series is used for this purpose Different implementations include: AMBER: U torsion = 1 2 k τ{1 + cos(nτ δ)} CHARMm: U torsion = k τ k τ cos(nτ) MM2: U torsion = 1 2 {k τ 1 (1 + cos τ) + k τ2 (1 cos 2τ) + k τ3 (1 + cos 3τ) These functions include different numbers of minima The last one is a sum of functions with 1, 2 and 3 minima OUT-OF-PLANE BENDING INTERACTIONS Out-of-plane interactions are closely related to the torsional terms For four atoms (A,B,C,D) which for three angles (A-B-C, A-B-D, C-B-D), there can be an energy cost for moving one of the atoms out of the plane spanned by the three other atoms This can be accounted for in one of two main ways: Harmonic potential Improper torsion term

NON-BONDING INTERACTION Non-bonded interactions act both between atoms in the same molecule and those in other molecules These are often divided to Electrostatic interactions Van der Waals interactions The electrostatic part arises from charge differences within a molecule This can be modeled by distributing point charges in the molecule Of course, the simple approach is to implement an interaction of the form (CHARMm & AMBER) U elect = 1 4πε 0 q 1 q 2 r ij (6) Also a dipole-dipole interaction can be implemented to act between polar groups A functional form is (MM2): U e = µ iµ j ε 0 rij 3 {cos χ 3 cos α i cos α j } (7) The µ i,j are dipole moments, χ is the angle between them. The α i,j are the angles between the moments The van der Waals interaction is understood to include all non-bonding interactions

Most typically, this is included via a LJ potential (something similar to this is used in CHARMm and AMBER) [ (σij ) 12 ( ) ] 6 σij U LJ = 4ε ij r ij r ij (8) A better description of the repulsive part would be an exponential function This leads to the Hill/Buckingham potential (MM2) U Hill vdw = A ij exp( B ij r ij ) c ij /r 6 ij (9) As mentioned during the pair potential lecture, this leads to a maximum at finite r ij and thus allows for a Buckingham explosion at close distances Also the Morse potential is occasionally used Comparison of the behavior of LJ and Buckingham potentials Note that this is a schematic, albeit with correct functional forms Buckingham U LJ r

ADDITIONAL TERMS The above-listed terms are common for all force fields However, there are typically many additional terms They are often used as coupling terms between the different listed terms, e.g., to join the the bond length and bond angle Also, it s possible to include terms describing electronegativity or conjugative effects Another missing feature is hydrogen bonding, which has been implemented in different ways CHOOSING A FORCE FIELD Obviously, the force field has to be chosen based on an analysis on the accuracy vs. efficiency of the method Some have been implemented for simulating bulk phases (e.g., AMBER) These force fields have generally a simple form with harmonic terms Other force fields, such as the MM-group FFs (MM4: [J. Comput. Chem. 14, 642 (1996)], [J. Comput. Chem. 14, 669 (1996)], [J. Comput. Chem. 14, 695 (1996)], [J. Comput. Chem. 14, 747 (1996)]), are designed to determine structures and vibrational frequencies

In this case, more accurate high order polynomials are used for bond stretches and bends and a Hill/Buckingham potential is used for the van der Waals interaction The different force fields are classified as class 1, 2 or 3 force fields, where the class 1 FFs are of the simplest form Another choise to be made is whether all the atoms are to be explicitly described (all-atom force field) The other option is to disregard the unimportant atoms, usually hydrogen (united-atom force field) The neglected atoms are accounted for by increasing the size of the atoms they are bonded to (by increasing the van der Waals radius) This obviously helps greatly in reducing the required number of computations Also entire groups of atoms can be replaced by a single site However, one needs to remember that neglecting the hydrogen atoms will yield a poorer description of electrostatic effects (e.g., charge separation) A hybrid method has also been proposed [J. Mol. Struct. (Theochem) 464, 39 (1999)], in which an all-atom description is used for hydrogen atoms bonded to phenyl rings and a united-atom approach for hydrogen atoms in aliphatic groups

PARAMETRIZATION OF A FORCE FIELD For a two-atomic system (here C-H), the following parameters need to be parametrized: 2 parameters for the two different bond types (H-H neglected) 2 for each three different angle types (C-C-C, C-C-H, H-C-H) 3 for each dihedral type (C-C-C-C, H-C-C-C, H-C-C-H) 2 van der Waals parameters for the three different atom combinations (C-C, C-H, H-H) up to 5 different charges So, the simplest possible two-atomic system of any interest includes at least 25 parameters For example the MM2 force field (for 30 atoms) has 3722 parameters ON CURRENT FORCE FIELDS The source: [http://cmt.dur.ac.uk/sjc/thesis_dlc/node68.html] is from 2003 The force fields mentioned above are of an intuitive simple form More accurate description have been developed more recently The new ones are fitted largely to experimental data (i.e., heats of formation and vibrational frequencies) A complicated functional form is required for the accurate description Methods based on the MM2 aim at making accurate prediction of molecular structures and properties Two more recent ones are: MM3: [J. Am. Chem. Soc. 111, 8551 (1989)], [J. Am. Chem. Soc. 111, 8566 (1989)], [J. Am. Chem. Soc. page 8576 (1989)] MM4: [J. Comput. Chem. 14, 642 (1996)], [J. Comput. Chem. 14, 669 (1996)], [J. Comput. Chem. 14, 695 (1996)], [J. Comput. Chem. 14, 747 (1996)]

The other group of force fields aims at modelling large molecules (polymers/proteins) Typical force fields in this category are AMBER [J. Am. Chem. Soc. 106 (1984)], [J. Am. Chem. Soc. 117, 5179 (1995)] CHARMm [J. Comput. Chem. 4, 1234 (1983)] OPLS [J. Am. Chem. Soc. 118, 11225 (1996)] These have a simpler functional form, typically only containing the terms covered during this lecture Often they also utilize the united-atom approach For references to more advanced methods, see the mentioned online material GROMACS GROMACS is a GPL d MD code which is able to use various force fields (Groningen Machine for Chemical Simulations) It is available in many Linux distributions as a standard package Website: http://www.gromacs.org/, Manual: http://manual.gromacs.org/current/ This software is primarily designed for biochemical molecules like proteins, lipids and nucleic acids that have many complicated bonded interactions, with solvent or implicit solvent GROMACS at least claims to be extremely fast at calculating the nonbonded interactions which typically dominate the simulation time The code is parallelized with MPI and can thus be run using several PC computers or several cores of a single PC

The actual manual is 370 pages long ([http://www.gromacs.org/api/deki/files/126/gromacs_manual-4.5.pdf]) Manual gives a very good introduction to the method (MD/MM) It even highlights the approximations done in the simulations: The simulations are classical, Electrons are in the ground state, Force fields are approximate, The force field is pair-additive, Long-range interactions are cut off, Boundary conditions are unnatural This manual is instructive to any MD user, even if not using GROMACS The biggest problem in using GROMACS is that no bond breaking is allowed all force fields are non-reactive An example from YouTube: [http://www.youtube.com/watch?v=y79xl0lfyi4] ReaxFF ReaxFF is an approach to combine force fields with a bond order approach The name comes from Reactive Force Field This method has been developed by van Duin, Goddard and others to allow chemical reactions within a force field approach ReaxFF aims at being as general as possible, and has been parametrized and tested for hydrocarbon reactions, transition-metal-catalyzed nanotube formation and high-energy materials ReaxFF manual is available at Adri van Duin s website at (updated in 2003) http://www.wag.caltech.edu/home/duin ReaxFF is trying to provide a transferable potential which would be applicable to a wide range of chemical environments

To ensure transferability, the following guidelines have been adopted No discontinuities in energy or forces, even during reactions (Same condition as for any other inter-atomic potential!) Each element is described by just one force field atom type No pre-definition of reactive sites (although it is possible to drive reactions with restraints) ReaxFF software has following features implemented NVT and NVE dynamics (Berendsen method) Steepest descent and conjugate gradient minimization methods Optimization of simulation cell parameters Simulations using a crystal unit cells keeping track of the system between periodic images of atoms EEM charge derivation method allowing calculation of geometry dependent charge distributions From the ReaxFF manual, the following parameter sets are available (website update date: July 13, 2003): Name System Quality CH Hydrocarbons *** CH All-carbon (fullerenes, nanotubes) ** RDX Nitramines *** CONSH CONSH-systems, proteins ** SiO Si/SiO 2 clusters and condensed phase *** AlO Al/AlO *** SiN SiN clusters and condensed phase * ZrO Zr and ZrO 2 condensed phase * MoO Mo and Mo-oxides ** Pt Pt bulk metal; Pt-C, Pt-H and Pt-O systems * Quality: * means that parameterization has only been performed for against a fairly limited training set, ** indicates that the parameters have been tested against a reasonably good training set but that further modifications and improvements are expected, *** indicates that these parameters have reached the application-stage.

Originally, the method was described in a publication in 2001: [J. Phys. Chem. A 2001, 105, 9396-9409] From a lecture by the main author found online; Current status of ReaxFF (2008): Code is (apparently) used by 70 research groups around the world List of implementations:

The ReaxFF approach is attractive, but due to the huge number of involved parameters, it gets difficult to make sure that the fit works in all circumstances An example from YouTube (reactions): [http://www.youtube.com/watch?v=6voau_ejw_k] According to (now removed) YouTube videos, at least someone has had problems with the Si fit: badly unstable at wrong lattice constant. Also, due to a large number of different terms, ReaxFF is much more demanding than the bond order potentials Another example from YouTube: [http://www.youtube.com/watch?v=5rn_gvlq1dy] EXAMPLE OF A REAXFF FIT From Development of a ReaxFF description for gold by Tommi Järvi and co-workers [Eur. Phys. J. B 66, 75 79 (2008)] Gold is not an obvious target for a ReaxFF potential The main motivation of the fit was to allow atomistic simulations of a thiol-gold system A thiol is a organosulfur compound that contains a carbon-bonded sulfhydryl (-C-SH or R-SH) group which are used in many areas of nanotechnology, typically on gold surfaces Of the possible interactions, only three terms are included U system = U bond + U over + U vdw (10) These are supposed to be enough to describe a metal U over is implemented to correct over-coordinations, other terms are self-explanatory

APPLICATION FOR THIOL-GOLD DESCRIPTION From Binding of deposited gold clusters to thiol self-assembled monolayers on Au(111) surfaces by Leila Costelle and co-workers [Appl. Phys. Lett. 98(4) (2011)] For this simplified ReaxFF potential, we get the following equations with fitted parameters

Properties of different structures as given by the potential Particular goals for the potential were reasonable descriptions of surface and dimer properties For the surface energetics, the potential gives γ (100) : 1.21 J/m 2 γ (111) : 1.07 J/m 2 Experiments give γ: 1.54 J/m 2 Adatom/vacancy energetics turn out to be reasonable (LDA/PBE DFT values in parentheses): E adat(100) : 0.59 (0.30/0.14) ev, E vac(100) : 0.59 (0.40/0.31) ev, E adat(111),fcc : 0.843 (0.95/0.61) ev, E adat(100),hcp : 0.843 (1.00/0.64) ev, E adat(100) : 0.87 (0.88/0.61) ev

Of all the potentials the ReaxFF parametrization was compared to, it gave clearly the best description of the dimer dissociation energetics This is easy to understand since the other potentials were not fitted to this Taking into account to goal of the potential, the fit seems to perform well SUMMARY Molecular mechanics is a wide term partially overlapping with molecular dynamics biggest differences are that the former is typically more concerned with molecules than individual atoms, and focuses on finding minimum energy configurations or other properties by several types of optimization methods, not necessarily following the time evolution Description of molecules often requires implementing constraints For example, some atoms may be described as a single large particle (united-atom approach) Terms included in force fields include: bond term, angle term, torsion term, out-of-plane term and non-bonding interactions Additionally, cross-terms are often implemented GROMACS is a GPL d software developed for force field MD ReaxFF builds upon the force field approach, but implements continuous functions to allow bond breaking It has a large set of parameters which allows for better fits but may yield problems in cases not included in the fitting procedure