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

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.

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

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.

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).

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.

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.

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.

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.

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!)

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