Introduction to Molecular Dynamics Dr. Kasra Momeni www.knanosys.com
Overview of the MD Classical Dynamics Outline Basics and Terminology Pairwise interacting objects Interatomic potentials (short-range vs long range forces) Unit cell (cubic/non-cubic) and simulation cell Periodic boundary conditions Cutoffs and their effects Increasing computational efficiency (e.g. Neighboring lists) AdHiMad Lab 2
Overview of the MD The first step to MD simulations is to define a model The proposed model encompasses two parts 1. Model of inter-molecular interactions 2. Model of the interactions with surrounding environment The intermolecular interactions are assumed to be independent from interactions with environment The intermolecular interactions can be described by either an force law or by a potential energy function. Thus fixing the intermolecular interactions fixes the symmetry of the molecule, nature of interactions, and geometry of the molecule The force law (potential energy function) can be defined analytically or numerically AdHiMad Lab 3
Overview of the MD Only spherically symmetric atoms will be considered The intermolecular potential function is only a function of relative position of molecules U = U r N ; N=#atoms Defining r N, determines the system configuration Properties that only depend on r N are called configurational properties The intermolecular force ( ) applied on a molecule (no intermolecular dissipative force): F i = U rn r i r i AdHiMad Lab 4
Overview of the MD To complete the atomistic model we need to define the interactions between the molecules ( ) and the surrounding environment i.e. define Boundary Conditions Bulk material Nonuniform regions Shear https://upload.wikimedia.org/wikipedia/commons/thumb/2/2e/limite periodicite.svg/512px-limiteperiodicite.svg.png B. K. Truong Quoc Vo, Scientific Reports 6, 280 (2016) AdHiMad Lab 5
Different forms of MD: 1. Equilibrium 2. Nonequilibrium Overview of the MD Consider the simplest system i.e. isolated system with fixed volume V, N, E are fixed F i = m ሷ r i = p i = න m ሷ r i dt; U rn r i x = ඵ m ሷ r i dtdt AdHiMad Lab 6
Overview of the MD Time average of property A t 1 0 +t A = lim t t න A τ dτ t 0 Dynamic Molecular Modeling <A> must be independent of t 0 The above equation is valid for calculating Thermodynamic properties (static) Dynamic properties Molecular Interactions Model Development Boundary Conditions Equations of Motion MD Simulations Adapted from J.M. Haile, MD Simulations, John Wiley and Sons (1992) Generating Trajectories Analyzing Trajectories AdHiMad Lab 7
Overview of the MD The MD simulations are computationally expensive, and are limited to a few thousands of atoms for a few nanoseconds Limited to short-range forces lim r R/2 Fi 0 Limited to short-lived phenomena R r R AdHiMad Lab 8
Newton s second law of motion Classical Dynamics F i = m ሷ r i For a system of N atoms, there are 3N second-order ODEs For F i =0, 2 nd law reduces to 1 st law: r i ሶ = cte How you can prove 3 rd law from the 2 nd law? F total = 0; F total = F 12 +F 21 =0 F 12 =-F 21 AdHiMad Lab 9
r i =r i (t); Classical Dynamics F i = F i (t); Functional form of Newton s second law is time-independent There must be a function that remain constant as time passes Called Hamiltonian F i = m ሷ r i H r N, p N = cte; p N =m ሶ Special case: For isolated system, total energy is conserved H r N, p N = K. E. +P. E. = 1 2 m v i 2 + U r N = 1 2m p i 2 + U r N r i https://en.wikipedia.org/wiki/william_rowan_hamilton AdHiMad Lab 10
Classical Dynamics Hamilton s Equations of motion thus For an isolated system H r N, p N = 1 2m p i 2 + U r N dh dt = H p i p i t + H r i r i t dh dt = 1 m p i Equations of motion for a conservative system H p i = p i m = dh dt = r i ሶ pሶ i + H r i For a system of N spherical particles Newton s view: 3N 2 nd order ODEs Hamilton s view: 6N 1 st order PDEs r i ሶ = pሶ i + U r i ሶ r i r i ሶ = 0 pሶ i + H rሶ r i = 0 H = pሶ i r i i AdHiMad Lab 11
B&T: Pairwise Interacting Objects There are often times that you need to calculate summation of terms over pairs of objects Potential energy of a system of atoms interacting via a pairwise potential i.e. the total intermolecular potential energy is the sum of mutual interactions How many ways exist that a system of N atoms can interact with each other? 2 atoms: 3 atoms: 4 atoms: N atoms: 1 3 6 N 2 = N! 2! N 2! AdHiMad Lab 12
B&T: Pairwise Interacting Objects Calculating the potential energy for a system of N atoms U = 1 2 i U ij = U ij j i i j>i Example: N=4 AdHiMad Lab 13
B&T: Interatomic Potentials The interatomic potential may have different terms and format depending on the material of study Electronic interactions van der Waals interactions Covalent bonds The simplest and most common interatomic potential is Lennard-Johns Sum of repulsive and attractive terms AdHiMad Lab 14
B&T: Interatomic Potentials Atomistic Model Buckingham-style potential q q r C rij rij i j exp ij ij Elct. + ARepul. + Attr. 6 E r Linear piezoelectricity Coulomb Three-body potential K. Momeni, H. Attariani, and R. A. LeSar, Phys. Chem. Chem. Phys. 18, 19873 (2016). AdHiMad Lab 15
B&T: Unit-cell Crystalline materials can be grouped into 7 Crystal systems 14 Crystal lattices A simulation cell May comprised of multiple unit cells Is not necessarily cubic It needs to fill the space A cell can be defined using three linearly independent vectors (a.b 0) i.e basis vectors W. D. Callister, Materials Science and Engineering (Wiley, 1999). AdHiMad Lab 16
Questions AdHiMad Lab 17