1. Advanced methods of molecular dynamics 1.Monte Carlo methods 2.Free energy calculations 3.Ab initio molecular dynamics 4.Quantum molecular dynamics II 5.Trajectory analysis

2. Time-dependent Schrodinger equation: numerically exact solution 1. Discretization of the wavefunction: - expansion into a basis (orthogonal polynomials) - expansion on a grid (equidistant or non-equidistant), n points, k max =  h/  vibrations: harmonic oscillator functions – Hermite polynomials rotations: free rotor functons- spherical harmonics 

3. ih  (t)/  t =H  (t)  (t+  t) = U  (t) = e -iH  t/ћ  (t) i) 2 nd order differences (SOD) – Taylor expansion U U = 1 - iH  t/ћ +...  (t+  t) =  (t) - i  t H  (t)/ ћ 2. Rozvoj evolučního operátoru U: Numerically unstable - symmetrization (epansion +/-  t )  (t+  t) =  (t-  t) - 2i  t H  (t)/ ћ conditionally stable: pro  t<h/E max error OF the SOD method: O(  t 3 ) SOD: Local (small  t), simple, but not very accurate method, applicable also to time-dep. Hamiltonians.

4. ii) Split Hamiltonian method (S-O): U  e -iT  t/ћ. e -iV  t/ћ...operators T & V non-commuting! U  e -iT  t/(2h). e -iV  t/(2ћ). e -iV  t/(2ћ). e -iT  t/(2ћ) or U  e -iV  t/(2h). e -iT  t/(2ћ). e -iT  t/(2ћ). e -iV  t/(2ћ) (equiv.) Error of S-O method: O(  t 3 ), but smaller prefactor than SO S-O: Local (small  t), more accurate than SO, applicable also to time-dependent Hamiltonians.

5. iii) Chebyshev method: U =  a n P n (-iHt/ћ)...expansion into orthogonal polynomials P n = cos [n arccos(x)]...n th Chebyshev polynomial a n = 2 J n [(E max -E min )t/(2ћ)]...Bessel functions error of the Chebyshev method ~ e -N Chebyshev: Global (large step), accurate (exponential convergence), applicable only to time- independent Hamiltonians. iii) Other global methods: Lanczos method orthogonal polynomyials by repeated action od Hamiltonian on the wave function tt’- method auxilliary time variable t’, also for H(t)

6. 3. Operation of Hamilton H on wave function  : H = T + V...operators of kinetic and potential energies In coordinate representation:  =  (x,t), V = V(x) V(x)  (x,t)...local multiplication point-by-point on the grid. T=(-iћ/2m) ..nonlocal Laplace diferential operator   (x i,t) = (  (x i+1,t) +  (x i-1,t) - 2  (x i,t))/  x 2 semilocal approximation – violates uncertainity relations. Fourier transform  (x,t)  (k,t), multiplication by k 2 inverse Fourier transform. Fast – using FFT.

7. Propagation of a wavepacket in 2D Disociation of C-H bonds in acethylene using a train of intense ultrashort IR pulses: intenzivních ultrakrátkých infračervených pulzů: J. T. Muckerman, Brookhaven National Laboratory