Car-Parrinello Method and Applications Moumita Saharay Jawaharlal Nehru Center for Advanced Scientific Research, Chemistry and Physics of Materials Unit, Bangalore.

Outline Difference between MD and ab initio MD Why to use ab initio MD ? Born-Oppenheimer Molecular Dynamics Car-Parrinello Molecular Dynamics Applications of CPMD Disadvantages of CPMD Other methods Conclusions

DFT, MD, and CPMD Properties of liquids/fluids depend a lot on configurational entropy MD with improved empirical potentials DFT calculation of a frozen liquid configuration Configurational Entropy part of the free energy will be missing in that case. Ab initio MD offers a path that mixes the goodness of both MD and of DFT AIMD is expensive.

Molecular simulations  Classical MD Hardwired potential No electronic degrees of freedom No chemical reaction Accessible length scale ~100 Å Accessible time scale ~ 10 ns  Ab initio MD On-the-fly potential Electronic degrees of freedom Formation and breaking of bonds Accessible length scale ~ 20 Å Accessible time scale ~ 10 ps

Livermore’s Nova LaserSandia National Laboratories Z accelerators A short intense shock caused the hydrogen to form a hot plasma and become a conducting metal The experiments found different compressibilities which could affect the equation of state of hydrogen and its isotope Quantum simulations could give the proper reasons for different results Conditions of the Nova and Z flyer were different : Time scales of the pulse were different

Why ab Initio MD ? Chemical processes Poorly known inter atomic interactions e.g. at high Pressure and/or Temperature Properties depending explicitly on electronic states ; IR spectra, Raman scattering, and NMR chemical shift Bonding properties of complex systems

Born-Oppenheimer approximation Electronic motion and nuclear motion can be separated due to huge difference in mass Different time scale for electronic and ionic motion Fast electrons have enough time to readjust and follow the slow ions

Born-Oppenheimer MD Electron quantum adiabatic evolution and classical ionic dynamics Effective Hamiltonian : H o I → Ionic k.e. and ion-ion interaction 2 nd term → Free energy of an inhomogeneous electron gas in the presence of fixed ions at positions (R I ) Electronic ground state – electron density ρ(r) – F({R I }) min Born-Oppenheimer Potential Energy Surface

Minimization to BO potential surface E{ρ(r)} ρ(r)ρ 0 (r)

Born-Oppenheimer MD Forces on the ions due to electrons in ground state Ionic Potential Energy Ψ i (r) one particle electron wave function 1 st → Electronic k.e. ; 2 nd → Electrostatic Hartree term 3 rd → integral of LDA exchange and correlation energy density ε xc 4 th → Electron-Ion pseudopotential interaction ; 5 th → Ion-Ion interaction

Born-Oppenheimer MD Electronic density; f i → occupation number E eI → Electron-Ion coupling term includes local and nonlocal components Kohn-Sham Hamiltonian operator Time evolution of electronic variables Time dependence of H ks ← slow ionic evolution given by Newton’s equations U ks = minimum of E ks w.r.t. ψ i -

Merits and Demerits of BOMD Advantages Disadvantages True electronic Adiabatic Evolution on the BO surface Need to solve the self- consistent electronic-structure problem at each time step Minimization algorithms require ~ 10 iterations to converge to the BO forces Poorly converged electronic minimization → damping of the ionic motion Computationally demanding procedure

Car-Parrinello MD CP Lagrangian Ψ i → classical fields μ → mass like parameter [1 Hartree x 1 atu 2 ] 4 th → orthonormality of the wavefunctions Constraints on the KS orbitals are holonomic No dissipation

Choice of μ Folkmar Bornemann and Christof Schutte demonstrate If the gap between occupied and unoccupied states = 0 (Insulators and semiconductors) (Metals) Fictitious kinetic energy of the electrons grow without control Use electronic thermostat μ must be small → small integration time step μ ~ 400 au, time step ~ 0.096x s

CP Equations of motion Equations of motion from L cp : Ionic time evolution Electronic time evolution Constraint equation Boundary conditions

Hellmann-Feynman Theorem If Ψ is an exact eigenfunction of a Hamiltonian H, and E is the corresponding energy eigenvalue : λ is any parameter occurring in H For an approximate wavefunction Ψ For an exact Ψ

Force on Ions G I → constraint force + G I When, ψi is an eigenfunction Force on the ions due to electronic configuration, when electronic wavefunction is an eigen function is zero

Constants of motion Vibrational density of states of electronic degrees of freedom Comparison with the highest frequency phonon mode of nuclear subsystem

Constants of motion

Merits and Demerits of CPMD  Advantages Fast dynamics compared to BOMD No need to perform the quenching of electronic wave function at each time step  Disadvantages Dynamics is different from the adiabatic evolution on BO surface Forces on ions are different from the BO forces Ground state Ψ i ≡ Ψ ks i → good agreement with the BOMD

Velocity Verlet algorithm for CPMD.

References  R. Car and M. Parrinello; Phys. Rev. Lett. 55 (22), 2471 (1985)  D. Marx, J. Hutter;  F. Buda et. al; Phys. Rev. A 44 (10), 6334 (1991)  D.K. Remler, P.A. Madden; Mol. Phys. 70 (6), 921 (1990)  B.M. Deb; Rev. Mod. Phys. 45 (1), 22 (1973)  M. Parrinello; Comp. Chemistry 22, (2000)  M.C. Payne et. al; Rev. Mod. Phys. 64 (4), 1045 (1992)

CPMD CPMD code is available at Code developers : Michele Parrinello, Jurg Hutter, D. Marx, P. Focher, M. Tuckerman, W. Andreoni, A. Curioni, E. Fois, U. Roethlisberger, P. Giannozzi, T. Deutsch, A. Alavi, D. Sebastiani, A. Laio, J. VandeVondele, A. Seitsonen, S. Billeter and others PWscf (Plane Wave Self Consistent field) PINY-MD

Applications

Autoionization in Liquid Water Chandler, Parrinello et. al Science 2001, 291, 2121 pH determination of water by CPMD Intact water molecules dissociate → OH - + H 3 O + Rare event ~ 10 hours >>>> fs Transition state separation between the charges ~ 6Å Proposed theory → Autoionization occurs due to specific solvent structure and hydrogen bond pattern at transition state Diffusion of ions from this transition state Role of solvent structure in autoionization Diffusion of ions Microsecond motion of a system as it crosses transition state can not be resolved experimentally pH = - log [H + ]

Nature of proton transfer in water Grotthuss’s idea : Proton has very high mobility in liquid water which is due to the rearrangement of bonds through a long chain of water molecule; effective motion of proton than the real movement + +

Charge separation Chandler, Parrinello et. al Science 2001, 291, Dissociation: Fluctuation in solvent electric field ; cleavage of OH bond 2 H 3 O+ moves by proton transfer within 30 fs 34 Conduction of proton through H-bond network 60 fs 5 Crucial fluctuations carries system to transition state ; breaking of H-bond : 30 fs 6 NO fast ion recombination

Order parameter for autoionization Fluctuations that control routes for proton : No. of hydrogen bond connecting the ions : ℓ ℓ = 2 ; recombination occurs within 100 fs reactant ℓ = 0 ; product ℓ ≥ 3 Critical ion separation is 6 Å At ℓ = 2, sometimes reactant basin ; Thus ℓ is not the only order parameter Potential of proton in H-bonded wire → fluctuation q → configuration description ; q = 1 neutral ; q = 0 charge separated ΔE = E[r(1) – r(0)] → solvent preference for separated ions over neutral molecules

Potential of protons in hydrogen bonded wires connecting H 3 O+ and OH- ions Chandler, Parrinello et. al Science 2001, 291, 2121 Neutral state, bond destabilizing electric field has not appeared Electric field starts to appear ; metastable state w.r.t. proton motion ; 2kcal/mol higher than neutral state Field fluctuations increase ; stable charge separated state ; 20kcal/mol more stable

Nature of the hydrated excess proton in liquid water Two proposed theories : 1. Formation of H 9 O 4 + (by Eigen) 2. Formation of H 5 O 2 + (by Zundel) Charge migration happens in a few picoseconds Tuckermann, Parrinello et. al J. Chem. Phys. 1997, 275, H9O4+H9O4+ H5O2+H5O2+ + Hydrogen bonds in solvation shells of the ions break and reform and the local environment reorders Ab initio calculations show that transport of H+ and OH- are significantly different

Proton transport Tuckermann, Parrinello et. al Nature 1997, 275, 817 Proton diffusion does not occur via hydrodynamic Stokes diffusion of a rigid complex Continual interconversion between the covalent and hydrogen bonds

Proton transport δ = RO aH - RO bH + OaOa ObOb H For small δ ; equal sharing of excess proton → Zundel’s H 5 O 2 + For large δ ; threefold coordinated H3O+ → Eigen’s H 9 O 4 + Tuckermann, Parrinello et. al Nature 1997, 275, 817 ΔF(ν) = -k B T ln [ ∫ dR OO P(R OO,ν) ] Free energy : H 5 O 2 + : at δ = 0 ± 0.05Å, Roo ~ Å ΔF < 0.15 kcal/mol, thermal energy = 0.59 kcal/mol Numerous unclassified situations exists in between these two limiting structures

Breaking bonds by mechanical stress Frank et. al J. Am. Chem. Soc. 2002, 124, 3402 Reactions induced by mechanical stress in PEG 1. Formation of ions corresponds to heterolytic bond cleavage 2. Motion of electrons during the reaction Polymer is expanded with AFM tip Unconstrained reactions can not be observed by classical MD Quantum chemical approaches are more powerful in describing the general chemical reactivity of complex systems

H H C2 C C C O1 O2 H H HH HH H H O H H Solvent Small piece of PEG in water Breaking bonds by mechanical stress Method ΔE (C-O) kcal/mol ΔE (C-C) kcal/mol BLYP Exp Radicaloid bond breaking After equilibration, distance between O1 and O2 was increased continuously by au/time Reaction started at 250 K ; C2O1 ~ 3.2 Å

Snapshots of the reaction mechanisms O O H O H H O H H O H + - O H O H O H O H H O H O O H O H H O H H O H O O H O H H O H H O H O O H O H H O H H O H + - O O H O H O H H O H H 250 K 320 K Frank et. al J. Am. Chem. Soc. 2002, 124, 3402

Hydrogen bond driven chemical reaction Parrinello et. al J. Am. Chem. Soc. 2004, 126, 6280 Beckmann rearrangement of Cyclohexanone Oxime into ε-Caprolactam in SCW SCW accelerates and make selective synthetic organic reactions System description : CPMD simulation, BLYP exchange correlation MT norm conserving pseudo potential Plane wave cut-off 70 Ry, Nose-Hoover thermostat T = 673K, 300K 64 H2O + 1 solute, 18 ps analysis + 11 ps equil. Disrupted hydrogen bond network of SCW alters the solvation of O and N

Proton attack on the Cyclohexanone Oxime Parrinello et. al J. Am. Chem. Soc. 2004, 126, 6280

Problems  Computationally costly  Can not simulate slow chemical processes that take place beyond time scales of 10 ps  Inaccuracy in the assumption of exchange and correlation potential Limitation in the number of atoms and time scale of simulation Inaccurate van der Waals forces, height of the transition energy barrier  BOMD not applicable for photochemistry; transition between different electronic energy levels

Other methods  QM/MM – quantum mechanics / molecular mechanics Classical MD AIMD e.g. catalytic part in enzyme  Path-sampling approach combined with ab-initio MD for slow chemical processes  Metadynamics, for slow processes

Conclusions CPMD : nuclear and electronic degrees of freedom Interaction potential is evaluated on-the-fly Bond formation and breaking is accessible in CPMD : direct access to the chemistry of materials Transferability over different phases of matter CPMD is computationally expensive

Acknowledgement Prof. S. Balasubramanian Dr. M. Krishnan, Bhargava, Sheeba, Saswati

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