Ab initio molecular dynamics and nuclear quantum effects
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1 Ab initio molecular dynamics and nuclear quantum effects Luca M. Ghiringhelli Fritz Haber Institute Hands on workshop density functional theory and beyond: First principles simulations of molecules and materials Berlin, Germany, July 13 July 23, 2015
2 Why statistical sampling Thermodynamic ensemble properties Static ensemble properties:
3 Why statistical sampling Thermodynamic ensemble properties Static ensemble properties: Dynamic (ensemble) properties:
4 Why statistical sampling Thermodynamic ensemble properties Static ensemble properties: Dynamic (ensemble) properties:
5 Why statistical sampling Ergodic hypothesis: ensemble average equal to time average
6 Why statistical sampling Ergodic hypothesis: ensemble average equal to time average
7 Road map Justification of ergodicity Molecular dynamics: basic cycle The Verlet integrator Why numerically integrated trajectories are not accurate (Lyapunov instability) Why this is not really a problem (Shadow trajectories) (Born Oppenhemier) ab initio MD Extended Lagrangians: Car Parrinello MD Canonical ensemble: thermostats and more thermostats Application of vanilla ab initio MD: Infrared spectroscopy Quantum nuclei: (Born Oppenhemier) path integral MD.
8 Justification of ergodicity r
9 Justification of ergodicity r
10 Justification of ergodicity r
11 Justification of ergodicity r
12 Molecular dynamic: the basic loop 1. Assign initial R (positions) and p (momenta) 2. Evolve (numerically) Newton's equations of motion for a discrete time increment (requires evaluation of the forces)
13 Molecular dynamic: the basic loop 1. Assign initial R (positions) and p (momenta) 2. Evolve (numerically) Newton's equations of motion for a discrete time increment (requires evaluation of the forces)
14 Molecular dynamic: the basic loop 1. Assign initial R (positions) and p (momenta) 2. Evolve (numerically) Newton's equations of motion for a discrete time increment (requires evaluation of the forces) 3. Assign new positions and momenta
15 Molecular dynamic: the basic loop 1. Assign initial R (positions) and p (momenta) 2. Evolve (numerically) Newton's equations of motion for a discrete time increment (requires evaluation of the forces) 3. Assign new positions and momenta
16 Molecular dynamic: thermodynamic ensembles Microcanonical (NVE) ensemble: Number of particles, Volume, and total Energy are conserved Natural ensemble to simulate MD (follows directly from Hamilton's equations of motion Canonical (NVT) ensemble: Number of particles, Volume, and Temperature are conserved System in contact with a heat bath NPT, NPH ensembles (H: enthalpy) for studying phase transitions Grand canonical (μvt) (μ: chemical potential) for adsorption/desorption Computer experiment : equilibrate the system and measure
17 Numerical integration This is an N body problem, which can only be solved numerically (except in very special cases) at least in principle. One (always) starts from a Taylor expansion: Naïve implementation: truncation of Taylor expansion Wrong! The naive forward Euler algorithm is not time reversible does not conserve volume in phase space suffers from energy drift Better approach: Verlet algorithm
18 Verlet algorithm compute position in next and previous time steps Or Verlet algorithm: is time reversible does conserve volume in phase space, i.e., it is symplectic (conservation of action element ) does not suffer from energy drift...but is it a good algorithm? i.e. does it predict the time evolution of the system correctly???
19 Chaos Molecular chaos Dynamics of well behaved classical many body system is chaotic. Consequence: Trajectories that differ very slightly in their initial conditions diverge exponentially ( Lyapunov instability )
20 Chaos Why should anyone believe in Molecular Dynamics simulations???
21 Chaos Why should anyone believe in Molecular Dynamics simulations??? Answers: 1. Good MD algorithms (e.g. Verlet) can also be considered as good (NVE!) Monte Carlo algorithm they therefore yield reliable STATIC properties ( Hybrid Monte Carlo ) 2. What is the point of simulating dynamics, if we cannot trust the resulting time evolution??? 3. All is well (probably), because of... The Shadow Theorem.
22 Shadow trajectory Shadow theorem (hypothesis) For any realistic many body system, the shadow theorem is merely a hypothesis. It basically states that good algorithms generate numerical trajectories that are close to a REAL trajectory of the many body system. Question: Does the Verlet algorithm indeed generate shadow trajectories? In practice, it follows an Hamiltonian, depending on the timestep, which is close to the real Hamiltonian, in the sense that for, converges to Take a different look at the problem. Do not discretize NEWTON's equation of motion...but discretize the ACTION
23 Shadow trajectory Lagrangian Classical mechanics Newton: Lagrange (variational formulation of classical mechanics): Consider a system that is at a point r0 at time 0 and at point rt at time t, then the system follows a trajectory r(t) such that: is an extremum.
24 Shadow trajectory Discretized action For a one dimensional system this becomes
25 Shadow trajectory Minimize the action Now do the standard thing: Find the extremum for small variations in the path, i.e. for small variations in all xi. This will generate a discretized trajectory that starts at time t0 at X0, and ends at time t at Xt.
26 Shadow trajectory which is the Verlet algorithm! The Verlet algorithm generates a trajectory that satisfies the boundary conditions of a REAL trajectory both at the beginning and at the endpoint. Hence, if we are interested in statistical information about the dynamics (e.g. time correlation functions, transport coefficients, power spectra...)...then a good MD algorithm (e.g. Verlet) is fine.
27 First principle (ab initio) Molecular dynamics Same as before but with forces from an ab initio potential V (Hellmann Feynman theorem, Pulay terms, etc.) Possible flavors: Born Oppenhemier MD (electrons always in the ground state) Car Parrinello MD Beyond Born Oppenhemier (Ehrenfest, Surface hopping) Reachable time scale (Born Oppenhemier): tens of picoseconds to few nanoseconds
28 (Ab initio) MD in practice, tuning (some) parameters 1. Read initial R(t0) and p(t0) 2. Converge electronic structure va a self consistent cycle 3. Calculate forces. 4. Integrate the equations of motion to evolve R(t) and p(t) 5. Determine R(t+Δt) and p(t+δt) and GOTO 2.
29 (Ab initio) MD in practice, tuning (some) parameters 1. Read initial R(t0) and p(t0) 2. Converge electronic structure va a self consistent cycle 3. Calculate forces. 4. Integrate the equations of motion to evolve R(t) and p(t) 5. Determine R(t+Δt) and p(t+δt) and GOTO 2.
30 (Ab initio) MD in practice, tuning (some) parameters
31 (Ab initio) MD in practice, tuning (some) parameters
32 (Ab initio) MD in practice, tuning (some) parameters 1. Read initial R(t0) and p(t0) 2. Converge electronic structure va a self consistent cycle 3. Calculate forces. 4. Integrate the equations of motion to evolve R(t) and p(t) 5. Determine R(t+Δt) and p(t+δt) and GOTO 2.
33 (Ab initio) MD in practice, tuning (some) parameters
34 Avoiding scf: the Car Parrinello MD Extended Lagrangian: add (fictitious) degrees of freedom for the electrons (KS orbitals) in the Lagrangian and solve couples equations of motion
35 Avoiding scf: the Car Parrinello MD Extended Lagrangian: add (fictitious) degrees of freedom for the electrons (KS orbitals) in the Lagrangian and solve couples equations of motion Adiabatic separation: electron mass needs to be very small small time step (1/50 fs) Electrons follow nuclei: No self consistency needed (at each step)
36 Sampling the canonical ensemble: thermostats
37 Sampling the canonical ensemble: thermostats
38 Stochastic thermostat: Andersen Stochastic thermostat: the Andersen thermostat Every particle has a fixed probability to collide with the Andersen demon After collision the particle is give a new Velocity The probabilities to collide are uncorrelated (Poisson distribution) Downside: momentum not conserved. Fixed by Lowe Andersen (2006)
39 Deterministic Stochastic thermostat: thermostat: Andersen Nosé(-Hoover) Deterministic thermostat: the Nosé Hoover
40 Deterministic Stochastic thermostat: thermostat: Andersen Nosé(-Hoover) Deterministic thermostat: the Nosé Hoover
41 Deterministic Stochastic thermostat: thermostat: Andersen Nosé(-Hoover) Deterministic thermostat: the Nosé Hoover
42 Deterministic Stochastic velocity thermostat: thermostat: rescaling Andersen Nosé(-Hoover) thermostat Stochastic velocity rescaling thermostat
43 Deterministic Stochastic velocity thermostat: thermostat: rescaling Andersen Nosé(-Hoover) thermostat Stochastic velocity rescaling thermostat
44 Advanced:dynamics Deterministic Stochastic velocity thermostat: Langevin thermostat: rescaling dynamics Andersen Nosé(-Hoover) thermostat Langevin
45 Advanced:dynamics Deterministic Stochastic velocity thermostat: Langevin thermostat: rescaling dynamics Andersen Nosé(-Hoover) thermostat Langevin
46 Advanced:dynamics Deterministic Stochastic velocity thermostat: Langevin thermostat: rescaling dynamics Andersen Nosé(-Hoover) thermostat Langevin
47 Advanced:dynamics Deterministic Stochastic velocity thermostat: Langevin thermostat: rescaling dynamics Andersen Nosé(-Hoover) thermostat Langevin
48 Advanced: thermostat: Deterministic Stochastic velocity Langevin thermostat: rescaling dynamics Andersen Nosé(-Hoover) thermostat Generalised Langevin dynamics: colored noise
49 Advanced: thermostat: Deterministic Stochastic velocity Langevin thermostat: rescaling dynamics Andersen Nosé(-Hoover) thermostat Generalised Langevin dynamics: colored noise
50 Advanced: thermostat: Deterministic Stochastic velocity Langevin thermostat: rescaling dynamics Andersen Nosé(-Hoover) thermostat Generalised Langevin dynamics: colored noise
51 Advanced: thermostat: Deterministic Stochastic velocity Langevin thermostat: rescaling dynamics Andersen Nosé(-Hoover) thermostat Generalised Langevin dynamics: colored noise
52 Advanced: thermostat: Deterministic Stochastic velocity Langevin thermostat: rescaling dynamics Andersen Nosé(-Hoover) thermostat Applications of plain MD Average structure / static properties at finite temperature Vibrations (IR, Raman spectroscopy) NMR spectroscopy Transport properties For realistic processes, phase transitions: enhanced sampling
53 IR spectroscopy: application to small gold clusters FHI-aims code: all-electron basis sets, scalar relativity PBE functional + van der Waals interactions via a C6[n]/r6 term derived from the self-consistent electronic density n tight settings, tier 2 basis set IR spectra from Fourier transform of dipole timeautocorrelation function, statistics from BO NVT-MD
54 IR spectroscopy: application to small gold clusters LMG et al. NJP (2013) FHI-aims code: all-electron basis sets, scalar relativity PBE functional + van der Waals interactions via a C6[n]/r6 term derived from the self-consistent electronic density n tight settings, tier 2 basis set IR spectra from Fourier transform of dipole timeautocorrelation function, statistics from BO NVT-MD
55 IR spectroscopy: application to small gold clusters LMG et al. NJP (2013) Harmonic IR spectra Eb = ev `` Is this the correct structure? Exp. T=100 K Eb = ev Harmonic IR spectra Eb = ev Vibrational frequencies [cm 1]
56 IR spectroscopy: application to small gold clusters LMG et al. NJP (2013) Harmonic IR spectrum IR spectrum at 100K MPD FIR spectrum
57 IR spectroscopy: application to small gold clusters LMG et al. NJP (2013)
58 IR spectroscopy: application to small gold clusters Harmonic IR spectra Theor. bare Au7 IR T=100 K Vibrational frequencies [cm 1]
59 IR spectroscopy: application to small gold clusters Vibrational frequencies [cm 1]
60 Advanced. All quantum: Path integrals Path Au7, integrals fluxionality Sum over paths: Suppose only two paths: interference
61 Advanced. All quantum: Path integrals Path Au7, integrals fluxionality Introduce of a large number of intermediate gratings, each containing many slits. Electrons may pass through any sequence of slits before reaching the detector Take the limit in which infinitely many gratings empty space
62 Do nuclei really behave classically?
63 Do nuclei really behave classically?
64 Do nuclei really behave classically?
65 Path integral molecular dynamics the adiabatic separation of electrons and nuclei where the electrons are kept in their ground state without any coupling to electronically excited states (Born Oppenheimer or clamped nuclei approximation), using a particular approximate electronic structure theory in order to calculate the interactions, approximating the continuous path integral for the nuclei by a finite discretization (Trotter factorization) and neglecting the indistinguishability of identical nuclei (Boltzmann statistics)
66 Path integral molecular dynamics Time dependent Schrödinger equation Probability All paths contribute: An interesting isomorphism (the famous imaginary time propagation)
67 Path integral molecular dynamics
68 Example of PIMD: proton transfer in malonaldehyde Tuckerman and Marx, Phys. Rev. Lett. (2001)
69 Proton in Water Hydroxyl (Ice) Overlayers on Metal Surfaces T = 160 K Li et al., Phys. Rev. Lett. (2010)
70 Proton in Water Hydroxyl (Ice) Overlayers on Metal Surfaces Li et al., Phys. Rev. Lett. (2010)
71 What about dynamical observables? Path Integral MD uses ring polymer trajectories to estimate static averages of the form: However, many important quantities are given by dynamic averages: PIMD does NOT give access to real time propagation (momenta are fictitious)
72 What about dynamical observables? PIMD does NOT give access to real time propagation. However, by normal mode transformation of free ring polymer modes: Ring polymer molecular dynamics (RPMD) [1] Masses of the beads are the real masses Problem: beads frequencies resonate with physical frequencies Centroid MD (CMD) [2] Centroid moves in the effective potential generated by the internal modes of the ring Beads have small masses: need for a special thermostat Problem: curvature problem
73 References
74 Summary Ensemble and time averages: Ergodicity Justification of ergodicity Molecular dynamics: basic cycle The Verlet integrator Why numerically integrated trajectories are not accurate (Lyapunov instability) Why this is not really a problem (Shadow trajectories) (Born Oppenhemier) ab initio MD Extended Lagrangians: Car Parrinello MD Canonical ensemble: thermostats and more thermostats Application of vanilla ab initio MD: Infrared spectroscopy Quantum nuclei: (Born Oppenhemier) path integral MD. Thanks to Mariana Rossi and Christian Carbogno, who originally conceived some of the slides shown here
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