Ab initio Molecular Dynamics Born Oppenheimer and beyond

Transcription

1 Ab initio Molecular Dynamics Born Oppenheimer and beyond

2 Reminder, reliability of MD MD trajectories are chaotic (exponential divergence with respect to initial conditions), BUT... With a good integrator (e.g., Verlet or velocity Verlet) the discrete trajectory, with the same initial and final point as the exact analytical one, can be made infinitely close to the latter (shadow theorem/hypothesis). So: the dynamical information are reliable. What about the ensemble averages? The ergodic theorem/hypothesis tells us that time and ensemble averages are equivalent (at the limit of infinite sampling time). Not true for all systems, though (indeed called non ergodic).

3 Ab initio MD, general formalism Schrödringer equation for a system of N nuclei and n electrons: acts on nuclei acts on electrons

4 Ab initio MD, general formalism Suppose the solutions of the time independent electronic Schrödinger equation are known : correction to the adiabatic eigenvalue of

5 Ab initio MD, Born Oppenheimer approximation If all are negligble: Adiabatic approximation (Born Oppenheimer): Valid in many cases, but, notably, not for electron transfer, photoisomerisation,...

6 Ab initio MD, semi classical approach classical trajectory adiabatic eigenfunctions : probability of finding the system in adiabatic state i at time t

7 Ab initio MD, validity of the BO approximation characteristic length time scale of electronic motion velocity of the nuclei Massay parameter

8 Ab initio MD, semi classical BO approximation The system, when prepared in one adiabatic state i stays there forever Nuclei move according to Newton's equations:

9 Non adiabatic: mean field (Ehrenfest) dynamics The atoms evolve on an effective potential representing an average over the adiabatic states weighted by their state populations Forces: Hellmann Feynman theorem: non adiabatic coupling vectors

10 Ehrenfest dynamics A system that was initially prepared in a pure adiabatic state will be in a mixed state when leaving the region of strong nonadiabatic coupling. The pure adiabatic character of the wavefunction cannot be recovered even in the asymptotic regions of configuration space. The total wavefunction may contain significant contributions from adiabatic states that are energetically inaccessible.

11 Ehrnefest dynamics, violation of microscopic reversibility

12 Surface hopping (fewest switches) Let there be N trajectories: At a later time: Supposing:

13 Surface hopping (fewest switches) Transition selected by using random numbers

14 Surface hopping (fewest switches) Transition from i to k invoked if : Sum of the transition probabilities of the first k states Uniform random number Between 0 and 1

15 Photoisomerization of formaldimine

16 Photoisomerization of formaldimine R P R R

17 Car Parrinello MD, a classical system Suppose a fluid of polarizable molecules E.g., dipolar molecules: permanent + induced dipole moment dipole dipole tensor Induced dipoles follow nuclei adiabatically and is always at its minimum: Iterative solution of (3N) linear equations?

18 Car Parrinello MD, a classical system Extended Lagrangian: Equation of motion for dipoles: Role of M? Adiabadicity: two temperatures?

19 Car Parrinello MD, the idea Starting from minimized Kohn Sham orbitals, define orbital velocities and kinetic energy: units? Lagrange multipliers Equations of motion: Functional derivative. In practice wrt the coefficients of the chosen basis set

20 Car Parrinello MD, the algorithm Initial conditions: Update velocities (Δt/2) Update positions (Δt) Constraint? Initial forces:

21 Car Parrinello MD, the algorithm Iterative solution: Starting from:

22 Car Parrinello MD, the algorithm (summary) Velocities: Positions: Forces update: 5 6

23 Molecular dynamics: summary of extended systems and applications of Born Oppenheimer MD

24 Reminder, reliability of MD MD trajectories are chaotic (exponential divergence with respect to initial conditions), BUT... With a good integrator (e.g., Verlet or velocity Verlet) the discrete trajectory, with the same initial and final point as the exact analytical one, can be made infinitely close to the latter (shadow theorem/hypothesis). So: the dynamical information are reliable. What about the ensemble averages? The ergodic theorem/hypothesis tells us that time and ensemble averages are equivalent (at the limit of infinite sampling time). Not true for all systems, though (indeed called non ergodic).

25 Summary of extended systems: NVT (Nosé) Lagrangian: Conjugated momenta: Hamiltonian Eq. of motion:

26 Summary of extended systems: NPH (Andersen)

27 Summary of extended systems: NPH (Parrinello Rahman)

28 Summary of extended systems: on the fly optimization Car Parrinello MD Classical example: polarizable fluid

29 Application of plain MD: nature of liquid C Galli et al. Science C atoms 300 fs

30 Application of plain MD: nature of liquid C Galli et al. Phys. Rev. Lett K

31 Application of plain MD: dissociative adsorption Phys. Rev. Lett. 1993

32 Collision Final (400fs) Trajectory 1 Trajectory 2 Trajectory 3 Trajectory 4 Trajectory 5 Kinetic energy: 1 ev

33 Application of plain MD: solvation of DMSO in water Kirchner and Hutter, J. Chem. Phys water molecules K gas phase DMSO Water DMSO