Advanced methods of molecular dynamics

1. Advanced methods of molecular dynamics • Monte Carlo methods • Free energy calculations • Ab initio molecular dynamics • Quantum molecular dynamics • 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 HΨe= EΨe 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 HΨe= EΨe, i.e., the ground state energy iteratively. Then solving the classical (Newton) equations of motion for the nuclei: MI2RI/t2 = -I<ΨeIHeIΨe> 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: MI2RI/t2 = -I<Ψ0IHeIΨ0> mi2φi/t2 = -/φi<Ψ0IHeIΨ0> mi is the fictitious mass of the orbital φi (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 HΨe= EΨe: • 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,…