Car-Parrinello Method and Car-Parrinello Method and ApplicationsApplications

Moumita SaharayJawaharlal Nehru Center for Advanced Scientific Research,

Chemistry and Physics of Materials Unit,

Bangalore.

Outline

Difference between MD and ab initio MDWhy to use ab initio MD ?Born-Oppenheimer Molecular DynamicsCar-Parrinello Molecular DynamicsApplications of CPMDDisadvantages of CPMDOther methodsConclusions

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 MDOn-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 Laser Sandia 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 dynamicsEffective Hamiltonian :

HoI → Ionic k.e. and ion-ion interaction

2nd term → Free energy of an inhomogeneous electron gas in the presence of fixed ions at positions (RI)

Electronic ground state – electron density ρ(r) – F({RI}) min

Born-Oppenheimer Potential Energy Surface

Minimization to BO potential surfaceE

{ρ(r

)}

ρ(r)ρ 0(r)

Born-Oppenheimer MDForces on the ions due to electrons in ground state

Ionic Potential Energy

Ψi (r) one particle electron wave function1st → Electronic k.e. ; 2nd → Electrostatic Hartree term

3rd → integral of LDA exchange and correlation energy density εxc

4th → Electron-Ion pseudopotential interaction ; 5th → Ion-Ion interaction

Born-Oppenheimer MD

Electronic density ; fi → occupation number

EeI → Electron-Ion coupling term includes local and nonlocal components

Kohn-Sham Hamiltonian operator

Time evolution of electronic variables

Time dependence of Hks ← slow ionic evolution given by Newton’s equations

Uks = minimum of Eks w.r.t. ψi-

Merits and Demerits of BOMDAdvantages 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 MDCP Lagrangian

Ψi → classical fields μ → mass like parameter [1 Hartree x 1 atu 2 ]

4th → orthonormality of the wavefunctions

Constraints on the KS orbitals are holonomic No dissipation

Choice of μFolkmar Bornemann and Christof Schutte demonstrate

If the gap between occupied and unoccupied states = 0

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 10-15 s

CP Equations of motionEquations of motion from Lcp :

Ionic time evolution

Electronic time evolution

Constraint equation

Boundary conditions

Hellmann-Feynman TheoremIf 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

I

I

REF

Force on Ions

GI → constraint force

+ GI

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

DisadvantagesDynamics is different from the adiabatic evolution on BO surface

Forces on ions are different from the BO forces

Ground state

Ψi ≡ Ψksi → 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; http://www.fz-juelich.de/nic-series/ 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)

CPMDCPMD code is available at http://www.cpmd.org

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) http://www.pwscf.org

PINY-MD http://homepages.nyu.edu/~mt33/PINY_MD/PINY.html

Applications

Autoionization in Liquid Water

Chandler, Parrinello et. al Science 2001, 291, 2121

pH determination of water by CPMDIntact water molecules dissociate → OH- + H3O+ 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 waterGrotthuss’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, 2121

1

Dissociation: Fluctuation in solvent electric field ; cleavage of OH bond

2

H3O+ moves by proton transfer within 30 fs

3

4

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 autoionizationFluctuations that control routes for proton :

No. of hydrogen bond connecting the ions : ℓ

ℓ = 2 ; recombination occurs within 100 fsreactant 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 H3O+ 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 H9O4+ (by Eigen)

2. Formation of H5O2+ (by Zundel)

Charge migration happens in a few picoseconds

Tuckermann, Parrinello et. al J. Chem. Phys. 1997, 275, 817

+ +

H9O4+

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 transportδ = ROaH - RObH

+Oa

ObH

For small δ ; equal sharing of excess proton → Zundel’s H5O2+

For large δ ; threefold coordinated H3O+ → Eigen’s H9O4+

Tuckermann, Parrinello et. al Nature 1997, 275, 817

ΔF(ν) = -kBT ln [ ∫ dROO P(ROO,ν) ]

Free energy :

H5O2+ : at δ = 0 ± 0.05Å, Roo ~ 2.46-2.48 ÅΔF < 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 PEG1. 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 H H

H H 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

83.979.1

85.0 83.0

Radicaloid bond breaking

After equilibration, distance between O1 and O2 was increased continuously by 0.0001 au/time

Reaction started at 250 K ; C2O1 ~ 3.2 Å

Snapshots of the reaction mechanisms

O

O

H

O

H

HO

H

H OH

+

-

OH

OH

O

H

O H

H

OH

O

O

H

O

H HO

HH

O

H

O

O

H

O

H H

O

H HO

H

OO

H

O

H

HO

H

H OH

+

-

O

OH

O

H

O H

H

OH

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 freedomInteraction potential is evaluated on-the-flyBond formation and breaking is accessible in CPMD : direct access to the chemistry of materialsTransferability over different phases of matterCPMD is computationally expensive

Acknowledgement

Prof. S. Balasubramanian

Dr. M. Krishnan, Bhargava, Sheeba, Saswati

THANK YOU