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Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics

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1 Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics
Ursula Rothlisberger EPFL Lausanne, Switzerland

2

3 Literature Car-Parrinello:
R. Car and M. Parrinello A unified approach for molecular dynamics and density functional Phys.Rev.Lett. 55, 2471 (1985) P. Carloni, U. Rothlisberger and M.Parrinello The role and perspective of ab initio molecular dynamics in the study of biological systems Acc. Chem.Res. 35, 455 (2002) U Rothlisberger 15 years of Car-Parrinello simulations in Physics, Chemistry, and Biology Computational Chemistry: Reviews of Current Trends, J. Leszczynski (Ed.), World Scientific, Vol. 6, (2001) p.33 D. Marx and J. Hutter Modern Methods and Algorithms of Quantum J. Grotendorst (Ed.), NIC Forschungszentrum Jülich (2000) p.301 D. Sebastiani and U. Rothlisberger Advances in density functional based modelling techniques: Recent extensions of the Car-Parrinello approach in P. Carloni, F. Alber ‘Medicinal Quantum Chemistry’, Wiley-VCH, Weinheim (2003)

4 When Quantum Chemistry Starts to Move...
Traditional QC Methods Classical MD Simulations Car-Parrinello MD improved optimization finite T effects thermodynamic & dynamic properties solids & liquids parameter-free MD ab initio force field no transferability problem chemical reactions

5 When Newton meets Schrödinger...
Sir Isaac Newton Erwin Schrödinger ( ) ( )

6 The ideal combination for Ab Initio Molecular Dynamics
Newt-dinger The ideal combination for Ab Initio Molecular Dynamics

7 Atoms, Molecules and Chemical Bonds
N protons & neutrons + N electrons e- Chemical Reaction Chemical Bonds

8 Basic Principles of Quantum Mechanics
- give example for size ration (in number and in a real object comparison)

9 Wavefunctions and Probability Distributions
Classical Mechanics: The position and velocity of the particle are precisely defined at any instant in time. Quantum Mechanics: The particle is better described via its wave character, with a wave function (r,t). The square of wave function is a measure for the probability P(r) to find the particle in an infinitesimal volume element dV around r. - give example for size ration (in number and in a real object comparison) The total probability to find the particle anywhere in space integrates to 1.

10 Epot q q Classical Mechanics Quantum Mechanics
positions and momenta uncertainty have sharp defined relation values Continous energy spectrum energies are quantized Epot q n=0 n=1 n=2 n=3 h w q Newton`s Equations Schroedinger Equation n, E, m, h0

11 Classical Mechanics: Particle Motion
ro,vo  r(t),v(t) - give example for size ration (in number and in a real object comparison) Position and velocity of a particle can be calculated exactly at any time t. Continuous energy

12 Goal: Computational method that provides us with a microscopic picture of the structural and dynamic properties of complex systems Solution 1: Time-dependent Schrödinger Eq. for a system of N nuclei and n electrons  not possible!

13 Electronic Schrödinger Eq.: Electronic Hamiltonoperator:
Approximations: 1) Born-Oppenheimer Approximation (1927): mel <<< mp  electronic and nuclear motion are separable Exceptions: Jahn-Teller instabilities, strong electron-phonon coupling, molecules in high intensity laser fields  nonadiabatic dynamics Product Ansatz for total wavefunction: Electronic Schrödinger Eq.: Electronic Hamiltonoperator:

14 potential energy surface (PES)
Solve electronic Schrödinger Eq. for each set of nuclear coordinates potential energy surface (PES) Nuclear SchrödingerEq. Nuclear Hamiltonoperator: Nuclear Quantum Dynamics (review: Makri, Ann. Rev. Phys. 50, 167 (1999)

15  classical approximation is better: m, n, E, T
Empirical parameterization → force field based MD Calculate → Car-Parrinello Dynamics Classical Nuclear Dynamics 2) Most atoms are heavy enough so that their motion can be described with classical mechanics ratio of the deBroglie wavelength of an electron and a proton:  classical approximation is better: m, n, E, T  Works surprisingly well in many cases!  what cannot be described: zero point energy effects (proton) tunneling  quantum corrections to classical results (Wigner&Kirkwood)  classical MD extended to quantum effects on equilibrium properties and to some extend also to quantum dynamics  path integral MD and centroid dynamics

16 First-Principles Molecular Dynamics
How do we do that? 1) straight-forward: solve electronic structure problem for a set of ionic coordinates evaluate forces move atoms Born-Oppenheimer Dynamics

17 Car - Parrinello Molecular Dynamics (1985)
Lagrangian Formulation of Classical Dynamics Euler-Lagrange Equation:

18 Car - Parrinello Molecular Dynamics (1985)
Extended Lagrangian Formulation Roberto Car Michele Parrinello

19 Equations of Motion Can be integrated simultaneously (e.g. with Verlet, Velocity-Verlet algorithm etc..) Verlet algorithm dt ~ fs

20 Does this fictitious classical dynamics described via the extended Lagrangian have anything to do with the real physical dynamics??? if  total energy of the system becomes the real physical total energy  can be checked via energy conservation After initial wfct optimization, system is propagated adiabatically and moves within finite thickness Ke over the potential energy surface

21 What’s the price for it ? systems sizes:
few hundred to few thousands of atoms (CP2K) Time Steps: ~0.1 fs Simulation Periods: few tens of ps

22 The Quantum Problem Stationary Solutions:
Time-independent Schrödinger Eq. Variable Separation: Electronic Schrödinger Eq.: Electronic Hamiltonoperator: Product Ansatz for the wavefunction: Effective 1-particle model

23 The Quantum Problem Set of N coupled 1-particle equations:
Basis Set Expansion:  Set of algebraic Eqs. Solved iteratively (self-consistent field) ca. 10’ ’000 Plane-waves:  FFT Choice of QM method: DFT

24 DENSITY FUNCTIONAL THEORY

25 Walter Kohn and John Pople
Nobelprize in Chemistry 1998

26 Literature on DFT: Original Papers: Textbooks:
P.Hohenberg, W.Kohn, Phys.Rev.B 1964, 136, W.Kohn, L.J.Sham, Phys.Rev.A 1965, 140, Textbooks: W.Kohn, P.Vashista, in Theory of the Inhomogeneous Electron Gas, N.H.March and S.Lundqvist (Eds), Plenum, New York 1983 R.G.Parr, W.Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, New York 1989. R.M.Dreizler, E.K.U.Gross, Density-Functional Theory, Springer, Berlin 1990. W.Kohn, Rev.Mod.Phys. 1999, 71.

27 Density Functional Theory (DFT)
Like Hatree-Fock: effective 1-particle Hamiltonian Let’s define a new central variable: Electron density Total electron density integrates to the number of electrons:

28 Theoretical foundations of DFT based on 2 theorems:
Hohenberg and Kohn (1964): (Phys.Rev. 136, 864B) The ground state energy of a system with N electrons in an external potential Vex is a unique functional of the electron density  Vex determines the exact  vice versa: Vex is determined within an additive constant by  gs expectation value of any observable (i.e. the H) is a unique functional of the gs density

29 Variational principle: The total energy is minimal for the ground state density of the system
Kohn and Sham (1965): (Phy. Rev. 1140, 1133A) The many-electron problem can be mapped exactly onto: an auxiliary noninteracting reference system with the same density (i.e. the exact gs density) where each electrons moves in an effective 1-particle-potential due to all the other electrons

30 Kohn-Sham eqs: (1) (2) (4) (5) (3)
(1) Kinetic energy of the non interacting system (2) External potential due to ionic cores (3) Hartree-term ~ classical Coulomb energy (4) exchange-correlation energy functional (5) Core -core interaction Kohn-Sham eqs:

31 Exchange and Correlation
Exchange-Correlation Hole

32 Universal exchange-correlation functional, exact form not known!
 local density approximation can be determined exactly: Exchange: (P.A.M. Dirac, Proc. Cambridge Phil. Soc. 26, 376 (1930), E.P. Wigner, Trans. Fraraday Soc. 34, 678 (1987)) Correlation: (D.M. Ceperly, B.J. Alder, Phys. Rev. Lett. 45, 566 (1980), G.Ortiz, P. Ballone, Phys. Rev. B 50, 1391 (1994)) exact (numerical) results from Quantum Monte Carlo simulations

33 - in principle very crude approximation!
Parametrized analytic forms that interpolate between different density regimes are available (e.g. J.P. Perdew, A. Zunger, Phys. Rev. B. 23, 5084 (1981)) - in principle very crude approximation! - Exc of a non uniform system locally ~ uniform electron gas results - should ‘work’ only for systems with slowly varying density but: atoms and molecules are inhomogeneous systems! - works remarkably well in practice: Performance of LDA/LSDA  in general good structural properties:  bond lenghts up to 1-2%  bond angles ~ 1-2 degrees  torsional angles ~ a few degrees  vibrational frequencies ~ 10% ( phonon modes up to few %)

34  cheap and good method for transition
metals!: e.g. Cr2, Mo2 in good agreement with experiment ( not bound in HF, UHF!)  F2 re within 3% (not bound in HF)  atomization, dissociation energies over estimated (mainly due to errors for atoms), typically by 10-20%  hydrogen-bonding overestimated  van der Waals-complexes: strongly overestimated binding (e.g. noble gas dimers, Mg2, Be2: factor 2-4 Re[Å] De (eV) HF CCSD CCSD(T) DFT exp Cr2 (Scuseria 1992)

35 Generalized Gradient Approximation (GGA)
correction function chosen to fulfill formal conditions for the properties of the ex-corr hole Determination of parameters: fully non empirical fit to exact Ex-Corr energies for atoms fit to experimental data (empirical)  man different forms (B88, P86, LYP, PW91, PBE, B3LYP etc..)

36 Time-independent electronic Schrödinger Equation:
Density-Functional Theory

37 Practical Implementation
periodic boundary conditions plane wave basis set up to a given kinetic energy cutoff Ecut  use of FFT techniques convenient evaluation of different terms in real space (Eex-corr, Eext) or in reciprocal space (Ekin, Ehartree) typical real space grid: ~1003, ~ pws most of the time: FFT most time consuming step (NMlogM) for large systems: orthogonalization ~N2 well parallelizable (over number of electronic states and first index of real space grid

38 Pseudo Potentials Framework
Chemical properties determined by valence electrons perform atomic all electron calculation ab initio pseudo r > rc smooth fct r < rc rc invert Schrodinger equation r(a.u.)

39 ABINIT CASTEP [i] Molecular Simulations Inc. CPMD CP2K [ii] M. Parrinello, MPI Stuttgart, Germany and IBM Zurich Research Laboratory, Switzerland Free software Fhi98md [iii] Fritz-Haber Institute Berlin, Germany JEEP François Gygi, Lawrence Livermore National Laboratory, USA NWCHEM Pacific Northwest National Laboratory, USA PAW [iv] P.E. Blöchl, Clausthal University of Technology, Germany SIESTA [v] P. Ordejon, Institut de Ciencia de Materials de Barcelona, Barcelona, Spain VASP [vi] J. Hafner, University of Vienna, Austria

40 Features (see also online manual):
CPMD (3.9) (CP2K) Features (see also online manual): plane wave basis, pseudopotentials, pbc and isolated systems LDA, LSD, GGAs (single point hybrid fct calcs possible) geometry optimization MD (NVE, NVT, NPT, Parrinello-Rahman) path integral MD different types of constraints and restraints Property calculations: population analysis, multipole moments, atomic charges, Wannier fcts, Fukui fcts etc.. Runs on essentially all platforms.. Most Recent Features: QM/MM interface Response function calculations: NMR Chemical shifts, electronic spectra, vibrational spectra Time Dependent DFT MD in excited states History dependent Metadynamics

41 Mixed Quantum-Classical QM/MM- Car-Parrinello Simulations
Classical Region Interface Region Quantum Region Fully Hamiltonian QM/MM Car-Parrinello hybrid code QM-Part: CPMD 3.8 pbc, PWs, pseudo potentials (n-1) CPUs MM-Part: GROMOS96 + P3M, AMBER (SYBIL, UFF) 1 CPU A. Laio, J. VandeVondele, and U. Rothlisberger, J. Chem. Phys. 116, (2002); A. Laio, J. VandeVondele, and U. Rothlisberger, J. Phys. Chem. B (ASAP article) review in : M. Colombo et al. CHIMIA 56, 11 (2002)

42 QM/MM Car-Parrinello Simulations
monovalent pseudo potential QM/MM Lagrangian QM MM j l k i e- qo qp - + included in Vext EQM: DFT EMM: Standard biomolecular Force Field

43 QM/MM Car-Parrinello in Combination with Response Properties
Variational Perturbation Theory: A. Putrino, D. Sebastiani, M. Parrinello, 113, 7103 (2000) IR and Raman Spectra Fukui Functions R. Vuilleumier, M. Sprik J.Chem.Phys. 115, 3454 (2001) Chemical Shifts D. Sebastiani, M. Parrinello, J. Phys. Chem. A 105, 1951 (2001) TDDFT: Spectra and Dynamics J. Hutter J.Chem.Phys. 118, 3928 (2003)

44 QM/MM Car-Parrinello in Combination with Excited State Methods
ROKS m1 m2 t1,2 HOMO-LUMO single excitations T. Ziegler et al. Theor. Chim. Acta 43, 261 (1977) (sum method) CP-version: I. Frank et al. J. Chem. Phys. 108, 4060 (1998) E(s) = 2E(m) - E(t) LR-TDDFT-MD (Tamm-Dancoff Approximation) J. Hutter J. Chem.Phys. 118, 3928 (2003) L. Bernasconi et al. J. Chem.Phys. 119, (2003) P-TDDFT-MD I. Tavernelli (to be published) Landau-Zener Surface Hopping Ehrenfest Dynamics

45 Limitations Due to Short Simulation Time
MD as dynamical tool: Real-time simulation of dynamical processes  many processes lie outside time range MD as sampling tool: only small portion of phase space is sampled relevant parts might be missed, especially if there exist large barriers between different important regions (e.g. different conformers) ensemble average have large statistical errors (e.g. relative free energies!) pA pB

46 Techniques from Classical MD:
Sampling at enhanced temperature Rescaling of atomic mass(es) Constraints (Ryckaert, Ciccotti, Berendsen 1977) (Sprik & Ciccotti 1998) Umbrella Sampling (Torrie&Valleau 1977) Quasi-Harmonic Analysis (Karplus, Jushick 1981) Reaction Path Method (Elber & Karplus 1987) ‘Hypersurface Deformation’ (Scheraga 1988, Wales 1990) Multiple Time Step MD (Tuckerman, Berne 1991) (Tuckerman, Parrinello 1994) Subspace Integration Method (Rabitz 1993) Local Elevation (van Gunsteren 1994) Conformational Flooding (Grubmuller 1995) Essential Dynamics (Amadei&Berendsen 1996) Path Optimization (Olender & Elber 1996) Multidimensional Adaptive Umbrella Sampling (Bartels, Karplus 1997) Hyperdynamics (Voter 1997) (Steiner, Genilloud, Wilkins 1998) (Gong & Wilkins 1999) Transition Path Sampling (Dellago, Bolhuis, Csajka, Chandler 1998) Adiabatic Bias MD (Marchi, Ballone 1999) Metadynamics (Laio, Iannuzzi, Parrinello PNAS 99, 12562 (2002), PRL 90, (2003)

47 Development of Enhanced Sampling Methods
Configurational Sampling Sampling of Rare Reactive Events multiple time step sampling classical bias potentials and forces double thermostatting parallel tempering Electronic Bias Potentials Finite Electronic Temperature Vibronic Coupling Charge Restraint Two Dimensional Free Energy Surface with torsional potential bias T = 500K EA = 30 kcal/mol Peroxynitrous Acid 48ps  1kcal/mol J. Chem. Phys (2000), J. Chem. Phys (2001), J. Phys. Chem. B 106, (2002), J. Am. Chem. Soc. 124, 8163 (2002)

48 Constraints ( linear speed up)
Lagrangian: Equations of motion: freeze out fast motions  increase integration time step ( linear speed up) constrain slowest motion  guide system ‘manually’ over barrier (condition: slowest part of reaction coordinate  is known, all other degrees of freedom have time to equilibrate along the path) ( free energy differences via thermodynamic integration)  integral replaced by a discrete set of points (R)= ’ for a simple distance constraint (R)= lRI-RJl:

49 Umbrella Sampling: Bias Potentials (Torrie&Valleau 1977)
(Grubmuller 1995, Voter 1997, Karplus 1997, Wilkins 1998…) high overlap with original ensemble close match PES or free energy surface low dimensionality computationally inexpensive ‘Ideal’ Bias: ‘Golden Rules’

50 Sampling Error in ab initio MD:
Methyl Group Rotation in Ethane C2H6 (500K, 7.25 ps) Probability Distribution  HCCH EA = 2.8 kcal/mol

51 Atomic Bias Potentials
Methyl Group Rotation in Ethane C2H6 (500K, 7.25 ps) Before correction After correction Torsional Bias au

52 Bias Potentials from Classical Force Field
Peroxynitrous Acid Trajectory in biased space (48 ps) ONOOH Free Energy Surface J. VandeVondele, U.R. J. Chem. Phys (2000)

53 CAFES: Canonical, Adiabatic Free Energy Sampling
Partitioning into reactive system / environment adiabatic decoupling Slow dynamics of the reactive subsystem different temperatures TR/TE (2 thermostats) Sampling efficiency at TR can be estimated Ea= 20 kcal/mol, TE=300K, TR=1200K -> 1013 J. VandeVondele, U.R. J. Phys. Chem. B 106, 203 (2002)

54 Nucleophilic substitution with anchimeric assistance
QMMM SPC/CPMD CAFES 100 / 2000K / 300K ~22 kcal/mol shows that the reaction coordinate is not simple

55 Transition State Path Sampling
Given: - initial state A - final state B - one path connecting the two  generate the ensemble of ‘reactive paths’  calculate transition rates

56 Dispersion Interactions in DFT
QM MM

57 Suggested Remedies add -C6/r6 -term (with damping function)
(LeSar 1984 ,Sprik 1996, Scoles 2001, Parrinello 2003, Wang, York2004…) specially designed (local) functionals (PW91, PBE, mPBE, X3LYP, …) density partitioning schemes (Wesolowski 2003…) nonlocal correlation functionals for special cases (Langreth,Lundvist 2000, 2003…) perturbation calculation of dispersion forces (Kohn 1998, Szalewicz 2003…)

58 Optimized Effective Atom Centered Potentials
Expansion in linear combination of atom-centered (nonlocal) potentials Analytic pseudopotentials by Goedecker et al.

59 Optimization Penalty Functional
Linear density response calculated via first order perturbation theory with perturbation Hamiltonian For :

60 Is this potential transferable???
BLYP OECP MP2

61 BLYP BLYP OECP OECP MP2 MP2

62 BLYP OECP z = 3.3A E=32 meV/atom exp z = 3.35A E=35 meV/atom

63 1 additional f-channel:
BLYP MP2 OECP Reference system: Ar2 1 additional f-channel: s1 = s2 = 2.902 BLYP OECP MP2 Klopper et al. J.Chem.Phys. 101, 9747 (1994)

64 What about the electronic properties??
What about the intramolecular geometry?? Bond lengths in benzene: D << 0.01 A What about the electronic properties?? BLYP OECP MP2 Dipole moment: benzene-Ar Quadrupole moment: benzene Polarizability: argon axx- ayy benzene azz benzene-Ar

65

66

67 Formaldimine. Excited state dynamics after excitation S0→S1
The region of conical intersection, CI, is reached only in case of non-thermostatted trajectories. relaxation to product geometry α Φ Φ start on S1 back to reactant geometry Increasing kinetic energy α Φ ω ω minimum on S1 α

68 Landau-Zener SH UD UA q* q
Classical treatment for the derivation of an analytical formula for the transfer rate which is valid for any value of the coupling matrix element spanning the range between adiabatic and nonadiabatic ET. energy UD UA q q* The asymptotic value for the survival probability of the electron for remaining at the donor The donor survival probability is

69 Units: atomic units used throughout

70 Transition Rate Constants
Reactive Flux Correlation Function Can be calculated with trajectories starting at the TS Is difficult if a RC/TS cannot be defined.

71 Rate constants in the TPE
C(t) = the fraction of trajectories of length t, starting in A, that arrives in B Can be calculated with a reversible work calculation. Contrary to direct MD, the computational efficiency does not depend on the height of the barrier.


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