Introduction to Molecular Dynamics

Dr. Kasra Momeniwww.KNanoSys.com

PI: K. Momeni

Outline

• Overview of the MD

• Classical Dynamics

• 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

PI: K. Momeni

Overview of the MD

• The first step to MD simulations is to define a model

• The proposed model encompasses two parts1. Model of inter-molecular interactions2. 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

PI: K. Momeni

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 𝒓𝑁 ; N=#atoms

• Defining 𝒓𝑁, determines the system configuration

• Properties that only depend on 𝒓𝑁 are called configurational properties

• The intermolecular force (⟷) applied on a molecule (no intermolecular dissipative force):

𝑭𝑖 = −𝜕𝑈 𝒓𝑁

𝜕𝒓𝑖

AdHiMad Lab 4

𝒓𝑖

PI: K. Momeni

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

• …

AdHiMad Lab 5

https://upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Limiteperiodicite.svg/512px-Limiteperiodicite.svg.pngB. K. Truong Quoc Vo, Scientific Reports 6, 280 (2016)

PI: K. Momeni

Overview of the MD

• Different forms of MD:1. Equilibrium

2. Nonequilibrium

• Consider the simplest system – i.e. isolated system with fixed volume • V, N, E are fixed

AdHiMad Lab 6

𝑭𝑖 = 𝑚 ሷ𝒓𝑖 = −𝜕𝑈 𝒓𝑁

𝜕𝒓𝑖

𝒑𝑖 = න𝑚 ሷ𝒓𝑖𝑑𝑡; 𝒙 = ඵ𝑚 ሷ𝒓𝑖𝑑𝑡𝑑𝑡

PI: K. Momeni

Overview of the MD• Time average of property A

• <A> must be independent of t0

• The above equation is valid for calculating • Thermodynamic properties (static)

• Dynamic properties

AdHiMad Lab 7

𝐴 = lim𝑡→∞

1

𝑡න𝑡0

𝑡0+𝑡

𝐴 𝜏 𝑑𝜏

Dynamic Molecular Modeling

Model Development MD Simulations

Molecular Interactions

Boundary Conditions

Equations of Motion

Generating Trajectories

Analyzing Trajectories

Adapted from J.M. Haile, MD Simulations, John Wiley and Sons (1992)

PI: K. Momeni

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

• Limited to short-lived phenomena

AdHiMad Lab 8

lim𝑟→𝑅/2

𝑭𝑖 ≈ 0

R

R

r

PI: K. Momeni

Classical Dynamics

• Newton’s second law of motion

• For a system of N atoms, there are 3N second-order ODEs

• For 𝑭𝑖 =0, 2nd law reduces to 1st law: ሶ𝒓𝑖 = 𝑐𝑡𝑒

• How you can prove 3rd law from the 2nd law?Ftotal = 0;

Ftotal = F12+F21=0

F12=-F21

AdHiMad Lab 9

𝑭𝑖 = 𝑚 ሷ𝒓𝑖

PI: K. Momeni

Classical Dynamics

𝒓𝑖=𝒓𝑖(t); 𝑭𝑖 = 𝑭𝑖(𝑡);

• Functional form of Newton’s second law is time-independent

• There must be a function that remain constant as time passes• Called Hamiltonian

• Special case: For isolated system, total energy is conserved

𝐻 𝒓𝑁, 𝒑𝑁 = 𝐾. 𝐸.+𝑃. 𝐸.=1

2𝑚 ∙ 𝒗𝑖

2 + 𝑈 𝒓𝑁 =1

2𝑚𝒑𝑖

2 + 𝑈 𝒓𝑁

AdHiMad Lab 10

𝑭𝑖 = 𝑚 ሷ𝒓𝑖

https://en.wikipedia.org/wiki/William_Rowan_Hamilton

𝐻 𝒓𝑁, 𝒑𝑁 = 𝑐𝑡𝑒; 𝒑𝑁=m ሶ𝒓𝑖

PI: K. Momeni

Classical Dynamics• Hamilton’s Equations of motion

𝐻 𝒓𝑁, 𝒑𝑁 =1

2𝑚𝒑𝑖

2 + 𝑈 𝒓𝑁

𝑑𝐻

𝑑𝑡=

𝜕𝐻

𝜕𝒑𝑖

𝜕𝒑𝑖𝜕𝑡

+𝜕𝐻

𝜕𝒓𝑖

𝜕𝒓𝑖𝜕𝑡

• For an isolated system𝑑𝐻

𝑑𝑡=1

𝑚𝒑𝑖 ∙ ሶ𝒑𝑖 +

𝜕𝑈

𝜕𝒓𝑖∙ ሶ𝒓𝑖 =0

• Equations of motion for a conservative system 𝜕𝐻

𝜕𝒑𝑖=𝒑𝑖𝑚= ሶ𝒓𝑖

thus𝑑𝐻

𝑑𝑡= ሶ𝒓𝑖 ∙ ሶ𝒑𝑖 +

𝜕𝐻

𝜕𝒓𝑖∙ ሶ𝒓𝑖 = ሶ𝒑𝑖 +

𝜕𝐻

𝜕𝒓𝑖∙ ሶ𝒓𝑖 = 0 ⇒

𝜕𝐻

𝜕𝒓𝑖= − ሶ𝒑𝑖

• For a system of N spherical particles• Newton’s view: 3N 2nd order ODEs• Hamilton’s view: 6N 1st order PDEs

AdHiMad Lab 11

PI: K. Momeni

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:

AdHiMad Lab 12

1

3

6

𝑁

2=

𝑁!

2! 𝑁 − 2 !

PI: K. Momeni

B&T: Pairwise Interacting Objects

• Calculating the potential energy for a system of N atoms

• Example: N=4

AdHiMad Lab 13

𝑈 =1

2

𝑖

𝑗≠𝑖

𝑈𝑖𝑗 =

𝑖

𝑗>𝑖

𝑈𝑖𝑗

PI: K. Momeni

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

PI: K. Momeni

B&T: Interatomic Potentials

• Coulomb–Three-body potential

AdHiMad Lab 15

• Atomistic Model

• Buckingham-style potential

• Linear piezoelectricity

6exp

i j ij

ij

ij ij

q q r CE r A

r r

Elct. + Repul. + Attr.

K. Momeni, H. Attariani, and R. A. LeSar, Phys. Chem. Chem. Phys. 18,

19873 (2016).

PI: K. Momeni

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

AdHiMad Lab 16

W. D. Callister, Materials Science and Engineering (Wiley, 1999).

PI: K. Momeni AdHiMad Lab 17

Questions