Classical Molecular Dynamics

CEC, Inha UniversityChi-Ok Hwang

Perspectives

• Many-electron problem; many electrons moving in a potential field

- considering the nuclei as being fixed; Born-Oppenheimer approximation

- Hartree-Fock method; a variational method, a kind of mean-field approach in statistical mechanics

Perspectives

• Density functional theory - the electronic orbitals are solutions to a Sc

hrödinger equation which depends on the electron density rather than on the indivisual electron orbitals

- the dependence of the one-particle Hamiltonian on this density is in principle nonlocal (cf. local density approximation (LDA))

Perspectives

• Empirical methods - classical molecular dynamics - tight-binding methods; a linear combination of a

tomic orbitals (LCAO) type• First-principles methods - tight-binding methods - density-functional theory - exact methods; quantum MC

Molecular Dynamics: General

• Solving classical equations of motion for a system of N molecules interacting via a potential V

V ≈ Σ V1(ri) + Σ Σ Veff2(rij)

• Lennard-Jones 12-6 potential V IJ(r)= 4ε ((σ/r)12-(σ/r)6)

Molecular Dynamics: General

• Algorithms 1: Verlet algorithm r(t+δt)=r(t) + δtv(t)+1/2 (δt)2 a(t) (1) r(t-δt)=r(t) - δtv(t)+1/2 (δt)2 a(t) (2) from the above two equations, we get r(t+δt)= 2r(t) - r(t-δt) + (δt)2 a(t) v(t) = (r(t+δt) - r(t-δt))/(2δt)

Molecular Dynamics: General

• Algorithms 2: Leap-Frog algorithm r(t+δt)=r(t) + δt v(t+δt/2) v(t+δt/2) = v(t-δt/2) + δt a(t); update first• Algorithms 3: Velocity Verlet algorithm r(t+δt)= r(t) + δt v(t) + (δt)2/2 a(t) v(t+δt) = v(t) + δt (a(t) + a(t+δt))/2

Molecular Dynamics: General

• Periodic boundary conditions 1) for( i=1;i <= Cell_N_x; i++){ Cell_P[i] = i+1; Cell_M[i] = i-1; } Cell_P[Cell_N_x] = 1; Cell_M[1] = Cell_N_x; 2) while( (*xnew) < 0 ){ *xnew = *xnew + Sx; }

Molecular Dynamics: General

• Potential truncation• Cell method: linked list and non-

overlapping nearby cell sweeping• Thermodynamic quantities - kinetic temperature

N

iii tvm

dNN

tK

dtkT

tKtkTd

N

1

2 )(1)(2

)(

)()(2

Molecular Dynamics: General

- pressure

jiijij

jiijij

ideal

tFtrdNkTNkT

PV

tFtrdV

tkTV

NtP

tkTV

NP

))()((1

1

)()(1

)()(

)(

Molecular Dynamics: General

• Mean square displacement: Einstein relation

22

2

)(

6)(

natR

DttR

a: step size

n: mean number of steps

Molecular Dynamics: General

• First-passage time probability

D

Rdtt

Ptt

r

tDmrtP

m

m

6

)exp()1(21);(

2

0

2

22

1

Molecular Dynamics: General

• radial distribution function g(r)

• Green-Kubo relation

i ij

ijrrN

Vrg )()(

2

0

)()0(3

1tvvdtD