What is molecular dynamics (MD) simulation and how does it work?

What is molecular dynamics (MD) simulation and how does it work? A lecture for CHM425/525 Fall 2011

The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact solution of these laws leads to equations much too complicated to be soluble. Paul Dirac on Quantum Mechanics (1929). Yesterday s Calculation Today s Calculation 2

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What can we do with MD simulations? Predict static and dynamical properties of materials, including chemical and biological systems. Compute certain properties of the system where experiment cannot be done. Allows us to understand our system at the most fundamental level (atomistic level) 4

Outline of the lecture: Fundamental physics law governing our system Ab initio molecular dynamics (AIMD) (Classical) molecular dynamics (MD) Technical details and examples 5

Fundamental physics law: Quantum Mechanics Time-independent Schrodinger equation: Hˆ ( r,r) E ( r,r) r = {r 1, r 2, r 3,..} (electron coordinates) R = {R 1, R 2, R 3,..}(nuclei coordinates) The solution of this equation will be a set of wavefunctions ({ n }) and energies ({E n }) for all the allowed states. Time evolution of a given system: ( r,r;t ) c ( r,r) e n n n ie n t 6

Born-Oppenheimer Approximation Mass disparity: M H 2000m e (r,r) (r ;R) (R) Electrons in a fixed nuclei geometry R Tˆ Vˆ ( rˆ) Vˆ ( rˆ, R) ( x, R) ( R) ( x, R) e ee en electron kinetic energy electron-electron interaction energy electron-nuclei interaction energy Nuclei on each electronic energy surface Tˆ ( Rˆ ) Vˆ ( Rˆ ) ( R) E ( R) n nn nucleri kinetic energy nuclei-nuclei interaction energy 7

Why solving nuclear Schrodinger equation is so hard? Examples of exact quantum calculations. H + H 2 H 2 + H (collinear) D. Truhlar and A. Kupperman (1970) Dimensionality issue: Adding one more atom to the system increase the dimensionality by THREE. For example, a nuclear wavefunction for a system with 10 atom is a function with 30 variables. 8

Demote to a classical Hamiltonian: Hˆ T ˆ ( R ˆ ) V ( R ˆ ) n 0 Molecular dynamics simulation nn N 2 PI H ( P, R) 0( R) Vnn( R) 2M Nuclei motions now given by Hamilton s equations: I 1 I R I H P I P I H R I V R I Ab initio MD: Compute the potential energy and force by solving electronic Schrodinger equation for a given nuclei configuration (almost always using DFT) Classical MD: Compute the potential energy and force by using a model (called force field ). 9

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Ab initio molecular dynamics (AIMD) simulation Electrons Nuclei Start with nuclei Add electrons Compute i, n, i n F e.g. Verlet: Add electrons 2 t RI ( t) 2 RI (0) RI ( t) FI (0) M I Propagate nuclei a short time Δt with F t 0. 1fs 11

How can we solve electronic Schrodinger equation? Traditional approach: Hartree-Fock (cheap, too crude), MP2 (accurate, expensive), Coupled-Cluster (accurate, really expensive) Density Functional Theory (DFT): Good balance between the cost and the accuracy. Hohenberg-Kohn Theorem (1964): External potential, and hence the total energy is a unique functional of electron density (always a three dimensional object). Minimize total energy, E[n(r)], with respect to electron density, n(r). 12

Hydrated proton Ab initio MD in action T. C. Berkelbach, H.-S. Lee and M. E. Tuckerman Phys. Rev. Lett. 103, 238302 (2009) 13

Organic reaction Ab initio MD in action Self-assembled molecular wire - styrene forms lines on Si (100) - precursor to molecular electronics 14

Butadiene on silicon surface 15

Product Distribution from 40 butadiene trajectories P. Minary and M. Tuckereman JACS (2005); STM Teague and Boland JPCB 107, 3820 (2003) Product Theory (%) Exp. (%) Product Theory (%) Exp. (%) A 15 11 3 D(+E) 30 22 15 B 30 31 6 H 10 21 5 C 15 16 7 A: DA [4+2] B: X-dimer intra-row C: X-dimer inter-row D(+E): [2+2] Intermediate Fluxional species 16

Force Field: Classical Molecular dynamics Define empirical potential energy function, V(R), to model molecular interaction These model potential should be differentiable in order to compute the forces acting on each atom Implementation A theoretical functional form (with adjustable parameter) is assumed for a given type of interaction: e.g. Stretching, bending, Each functional form has a set of adjustable parameters. These parameters are determined so as to reproduce the experimental data The same atoms in different chemical environment are treated differently (different atom type): e.g. The carbons in C=O and C-C are of different types. 17

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Protein Folding 23

Residue Trp6 is caged Trp-cage Folding Folds in about 4 s AMBER99 Force Field Continuum Solvent GBSA Trp-cage Blue: MD simulation Grey: NMR structure C. Simmerling, B. Strockbine, A. Roitberg, JACS 124, 11258, (2002) 24

90 Å MD simulation of peptide/popc system 128 POPC lipid 1-2 peptides Gromacs FF Choline (+) Phosphate (-) glycerol Lipid tail 65 Å 25

Antimicrobial peptide ( -lysin) in action Hydrophobic residues 80 ns 100 ns 118 ns 135 ns 26

Ab initio vs. Classical Molecular dynamics simulation Classical MD Ab initio MD 1. Fixed number of atom type No need to define atom type 2. No bond breaking/forming No problem with bond breaking and (can t be far away from equilibrium forming bond length 3. No electronic polarization (fixed partial charge) 4. Can handle large systems (10 4 atoms is routine and up to 10 6 atoms is possible) 5. Typical time scale is ns and simulation up to s becomes more tractable. Electronic polarization is fully incorporated Limited to small systems (few hundred atoms) Very short simulation length (typically, in the order of tens of ps) 6. No nuclear quantum effect No nuclear quantum effect (Path Integral AIMD is needed) 27