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

2. Introduction: Force Fields Power an glory of empirical force fields: Fitted to experiment, simple, and cheap. Can be refined by including additional terms (polarization, cross intramolecular terms, …). Misery of empirical force fields: You or others do the fitting/fidling – results can become GIGA (Garbage-In-Garbage-Out). Difficult to improve in a systematic way. No bond making/breaking – no chemistry! Alternative: Potentials and forces from quantum chemistry.

3. Ab initio Potentials Instead of selecting a model potential selecting a particular approximation to HΨ e = EΨ e Price of dramatically increased computational costs: much smaller systems and timescales. Constructing the whole potential energy surface in advance: exponential dimensionality bottleneck, possibly only for very small systems (<5 atoms) Alternative: on-the-fly potentials constructed along the molecular dynamics trajectory

4. Dynamical Schemes I: Born-Oppenheimer Dynamics Finding the lowest solution of HΨ e = EΨ e, i.e., the ground state energy iteratively. Then solving the classical (Newton) equations of motion for the nuclei: M I  2 R I /  t 2 = -  I In principle posible also for excited states but that almost always involves mixing of states: Ehrenfest dynamics or surface hopping.

5. Dynamical Schemes II: Car-Parrinello Dynamics Real dynamics for nuclei + fictitious dynamics of electrons. Takes advantage of the adiabatic separation between slow nuclei and fast electrons: M I  2 R I /  t 2 = -  I m i  2 φ i /  t 2 = -  /  φ i m i is the fictitious mass of the orbital φ i (typically hundreds times the mass of electron in order to increase the time step).

6. Dynamical Schemes III: Comparison Car-Parrinello – for right choice of parameters usually close to Born-Oppenheimer dynamics. Methods of choice in the orignal 1985 paper due to relatively low computational costs. Born-Oppenheimer dynamics – rigorously adiabatic potential but more costly iterative solution. Today becoming more and more the method of choice.

7. Electronic Structure Methods Different approaches tested: Hartree-Fock, Semiempirical Methods, Generalized Valence Bond, Complete Active Space SCF, Configuration Interaction, and … (overwhelmingly) Density Functional Theory. Why DFT? Best price/performance ratio. Better scaling with systém size than HF and generally more accurate. Originally LDA, today mostly GGA (BLYP, PBE, …) functionals.

8. Basis Sets Plane waves: Traditional solution suitable for periodic systems. Independent of atomic positions & systematically extendable (increasing energy cutoff). Need for pseudopotentials for core electrons. Gaussians: Relatively new, suitable for molecular (chemical) problems. Gaussians for Kohn-Sham orbitals can be combined with plane wavesfor the density. Wavelets: Localized functions in the coordinate space.

9. Boundary Conditions Periodic: 3D periodic boundary conditions mimic condensed phase systems. Natural with plane waves. 2D periodic boundary conditions for slab systems. Non-periodic: Cluster boundary conditions for isolated molecules or clusters. Requires large boxes unless localized basis functions (wavelets) are used to replace plane waves.

10. Problems with DFT Only aproximate solution of HΨ e = EΨ e : -inaccurate physical properties (e.g., too low density and diffusion constant of water), -self-interaction error leads to artificially favoring of delocalized states. Problematic particularly for radicals and reaction intermediates. - inadequate description of dispersion interactions. Fixtures: - runs at elevated temperatures, -empirical correction schemes for self-interaction, -empirical dispersion terms, Possible use of hybrid functionals (costly!)

11. Programs for AIMD CPMD, CP2K, VASP, NWChem, CASTEP, CP-PAW, fhi98md,…