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## Classical Molecular Dynamics

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**Classical Molecular Dynamics**CEC, Inha University Chi-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 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))**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**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)**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)**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**Molecular Dynamics: General**- pressure**Molecular Dynamics: General**• Mean square displacement: Einstein relation a: step size n: mean number of steps**Molecular Dynamics: General**• First-passage time probability**Molecular Dynamics: General**• radial distribution function g(r) • Green-Kubo relation