Classical Molecular Dynamics

1. Classical Molecular Dynamics CEC, Inha University Chi-Ok Hwang

2. 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

3. Perspectives • Density functional theory - the electronic orbitals are solutions to a Schrödinger 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))

4. Perspectives • Empirical methods - classical molecular dynamics - tight-binding methods; a linear combination of atomic orbitals (LCAO) type • First-principles methods - tight-binding methods - density-functional theory - exact methods; quantum MC

5. Molecular Dynamics: General • Solving classical equations of motion for a system of N molecules interacting via a potential V V ≈ ΣV1(ri) + Σ ΣVeff2(rij) • Lennard-Jones 12-6 potential V IJ(r)= 4ε ((σ/r)12-(σ/r)6)

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

7. 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

8. 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; }

9. Molecular Dynamics: General • Potential truncation • Cell method: linked list and non-overlapping nearby cell sweeping • Thermodynamic quantities - kinetic temperature

10. Molecular Dynamics: General - pressure

11. Molecular Dynamics: General • Mean square displacement: Einstein relation a: step size n: mean number of steps

12. Molecular Dynamics: General • First-passage time probability

13. Molecular Dynamics: General • radial distribution function g(r) • Green-Kubo relation