Classical and quantum molecular dynamics. Simulations of

Report
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 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
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.
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).
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 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!)
Programs for AIMD
CPMD, CP2K, VASP, NWChem, CASTEP,
CP-PAW, fhi98md,…

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