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

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

Car-Parrinello Molecular Dynamics Simulations

(CPMD): Basics

Ursula RothlisbergerEPFL Lausanne, Switzerland

Page 2: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland
Page 3: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Literature Car-Parrinello:Literature Car-Parrinello:

• D. Marx and J. HutterD. Marx and J. Hutter Modern Methods and Algorithms Modern Methods and Algorithms of Quantum of Quantum J. Grotendorst (Ed.), NIC J. Grotendorst (Ed.), NIC Forschungszentrum Jülich (2000)Forschungszentrum Jülich (2000) p.301p.301

• D. Sebastiani and U. RothlisbergerD. Sebastiani and U. Rothlisberger Advances in density functional Advances in density functional based modelling techniques: based modelling techniques: Recent extensions of the Recent extensions of the Car-Parrinello approachCar-Parrinello approach in P. Carloni, F. Alber ‘Medicinal in P. Carloni, F. Alber ‘Medicinal Quantum Chemistry’, Wiley-VCH, Quantum Chemistry’, Wiley-VCH, Weinheim (2003)Weinheim (2003)

• R. Car and M. ParrinelloR. Car and M. Parrinello A unified approach for molecularA unified approach for molecular dynamics and density functionaldynamics and density functional

Phys.Rev.Lett. 55, 2471 (1985)Phys.Rev.Lett. 55, 2471 (1985)

• P. Carloni, U. Rothlisberger P. Carloni, U. Rothlisberger andand M.Parrinello M.Parrinello The role and perspective of The role and perspective of abab initio molecular dynamics in initio molecular dynamics in the the study of biological systemsstudy of biological systems Acc. Chem.Res. 35, 455 Acc. Chem.Res. 35, 455 (2002)(2002)

• U RothlisbergerU Rothlisberger 15 years of Car-Parrinello 15 years of Car-Parrinello simulations in Physics, simulations in Physics, Chemistry,Chemistry, and Biologyand Biology Computational Chemistry: Computational Chemistry: Reviews Reviews of Current Trends, J. of Current Trends, J. Leszczynski Leszczynski (Ed.), World Scientific, Vol. 6, (Ed.), World Scientific, Vol. 6, (2001) p.33 (2001) p.33

Page 4: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

When Quantum Chemistry Starts to Move...When Quantum Chemistry Starts to Move...

Traditional QCTraditional QCMethodsMethods

Classical MDClassical MDSimulations

Car-Parrinello Car-Parrinello MDMD

• parameter-free MDparameter-free MD• ab initio force fieldab initio force field• no transferability no transferability problemproblem• chemical reactionschemical reactions

• improved improved optimizationoptimization• finite T effectsfinite T effects• thermodynamic & thermodynamic & dynamic propertiesdynamic properties• solids & liquidssolids & liquids

Page 5: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

When Newton meets Schrödinger...

maF H

Sir Isaac Newton(1642 - 1727)

Erwin Schrödinger(1887 - 1961)

Page 6: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Newt-dinger

maF H

The ideal combination for Ab Initio Molecular Dynamics

Page 7: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Atoms, Molecules and Atoms, Molecules and Chemical BondsChemical Bonds

AtomAtomss N protonsN protons

& neutrons& neutrons

N electronsN electronse-e-

++

Chemical BondsChemical Bonds

Chemical ReactionChemical Reaction

Page 8: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Basic Principles of Quantum Mechanics

Page 9: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Wavefunctions and Probability Distributions

Classical Mechanics: The position and velocity of the particle are precisely defined at any instant in time.

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).

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 volumeelement dV around r.

dVrrP )()( 2 dVrrP )()( 2

The total probability to find the particle anywhere in space integrates to 1.

1)(2 dVrV

1)(2 dVr

V

Page 10: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Classical Mechanics Quantum Classical Mechanics Quantum Mechanics Mechanics

positions and momenta uncertainty positions and momenta uncertainty have sharp defined relationhave sharp defined relationvaluesvalues

t,r*t,r

t,r

hpx

*

v,r

Continous energy spectrum energies are quantizedContinous energy spectrum energies are quantized

nn, E, E, m, m, h, h00

Newton`s Equations Schroedinger EquationNewton`s Equations Schroedinger Equationq

Epotpot

0 q0

n=0n=0

n=1n=1

n=2n=2

n=3n=3

h h

Page 11: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Classical Mechanics: Particle Motion

ro,vo r(t),v(t)

maF maF 21

2)( attvrtr oo 21

2)( attvrtr oo atvtv o

)( atvtv o

)(

Position and velocity of a particle can be calculated exactly at any time t.

rr

r

v

2

2

1mvEkin 2

2

1mvEkin Continuous energy

Page 12: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Goal:Goal:Computational method that provides us Computational method that provides us with a microscopic picture of the structural with a microscopic picture of the structural and dynamic properties of complex and dynamic properties of complex systemssystems

),,...,,,...,,(),,...,,,...,,( 321321321321 trrrrRRRRtrrrrRRRRt

i nNnN

Solution 1: Time-dependent Schrödinger Eq. for a Solution 1: Time-dependent Schrödinger Eq. for a system system of N nuclei and n electronsof N nuclei and n electrons

not possible!not possible!

Page 13: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Approximations:Approximations:1) Born-Oppenheimer Approximation (1927):1) Born-Oppenheimer Approximation (1927): mmelel <<< m <<< mp p electronic and nuclear motion are electronic and nuclear motion are separableseparable

Exceptions: Jahn-Teller instabilities, strong electron-phononExceptions: Jahn-Teller instabilities, strong electron-phonon coupling, molecules in high intensity laser fieldscoupling, molecules in high intensity laser fields nonadiabatic dynamicsnonadiabatic dynamics

),...,,()...,,(),...,,,...,,( 321321321321 nelNnunN rrrrRRRRrrrrRRRR

Product Ansatz for total wavefunction:Product Ansatz for total wavefunction:

),,...,,(),,...,,( 321321 RrrrrERrrrr nelnelel

iI jiiI

I

iiel rijrR

Z

,

2 12/1

Electronic Schrödinger Eq.:Electronic Schrödinger Eq.:

Electronic Hamiltonoperator:Electronic Hamiltonoperator:

Page 14: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Solve electronic Schrödinger Eq. for each set of nuclear Solve electronic Schrödinger Eq. for each set of nuclear coordinatescoordinates

),...,,( 321 NRRRRR

)(RE potential energy surface (PES)potential energy surface (PES)

)....,,()....,,( 321321 NnutotNnunu RRRRERRRRH

JI IJ

JI

II

Inu R

ZZRE

M ,

2 )(2

1

Nuclear SchrödingerEq.Nuclear SchrödingerEq.

Nuclear Hamiltonoperator:Nuclear Hamiltonoperator:

Nuclear Quantum DynamicsNuclear Quantum Dynamics

(review: Makri, Ann. Rev. Phys. 50, 167 (1999)(review: Makri, Ann. Rev. Phys. 50, 167 (1999)

Page 15: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Classical Nuclear Classical Nuclear DynamicsDynamics2) Most atoms are heavy enough so that their 2) Most atoms are heavy enough so that their motion can be motion can be described with described with classical mechanicsclassical mechanics

• ratio of the deBroglie wavelength ratio of the deBroglie wavelength mE

h

2 of an electron and aof an electron and a

proton: proton: 402/1

el

p

p

el

m

m

classical approximation is better: mclassical approximation is better: m, n, n, E, E, T, T

Works surprisingly well in many cases!Works surprisingly well in many cases! what cannot be described:what cannot be described:• zero point energy effectszero point energy effects

• (proton) tunneling(proton) tunneling quantum corrections to classical results quantum corrections to classical results (Wigner&Kirkwood)(Wigner&Kirkwood) classical MD extended to quantum effects on equilibrium classical MD extended to quantum effects on equilibrium properties properties and to some extend also to quantum dynamics and to some extend also to quantum dynamics path path integral MD and centroid dynamicsintegral MD and centroid dynamics

)(RE

Empirical parameterizationEmpirical parameterization→ → force field based MDforce field based MDCalculate Calculate → → Car-Parrinello DynamicsCar-Parrinello Dynamics

HRE )(

Page 16: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

First-Principles Molecular DynamicsHow do we do that?How do we do that?

1) straight-forward:1) straight-forward:

• solve electronic structure problem for a set of solve electronic structure problem for a set of ionic coordinatesionic coordinates

• evaluate forcesevaluate forces

• move atomsmove atoms

Born-Oppenheimer Born-Oppenheimer DynamicsDynamics

Page 17: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Car - Parrinello Molecular Dynamics (1985)

Lagrangian Formulation of Classical Dynamics

)(2

1

)()(

2III

I

II

RVRML

qVqTL

**ii q

L

q

L

dt

d

δ

δ

δ

δ

Euler-Lagrange Equation:

III R

ERM

Page 18: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Car - Parrinello Molecular Dynamics (1985)

Roberto Car

Michele Parrinello

ijji

ijij

Iii iiI IIex

drrr

RERML

,2/1 2

Extended Lagrangian Formulation

Page 19: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Equations of Motion

III R

ERM

jj ijii H

Can be integrated simultaneously (e.g. with Verlet, Velocity-Can be integrated simultaneously (e.g. with Verlet, Velocity-Verlet algorithm etc..)Verlet algorithm etc..)

)()(2

)()(2)( 42

tOtFM

tttRtRttR I

IIII

VerletVerletalgorithmalgorithm

dt ~0.1-0.2 fsdt ~0.1-0.2 fs

Page 20: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Does this fictitious classical dynamics described Does this fictitious classical dynamics described via the extended Lagrangian have anything to do via the extended Lagrangian have anything to do with the real physical dynamics???with the real physical dynamics???

• ifif

total energy of the system becomes the real total energy of the system becomes the real physical total energyphysical total energy

0Ks'M eI

potIpotIe EKEKK

After initial wfct optimization, system is After initial wfct optimization, system is propagated adiabatically and moves withinpropagated adiabatically and moves within finite thickness Kfinite thickness Ke e over the potentialover the potential energy surfaceenergy surface

can be checked via energy conservationcan be checked via energy conservation

Page 21: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

• systems sizes:systems sizes: few hundred to few thousands of atoms few hundred to few thousands of atoms (CP2K)(CP2K)

• Time Steps: ~0.1 fsTime Steps: ~0.1 fs

• Simulation Periods: few tens of ps Simulation Periods: few tens of ps

What’s the price for it ?What’s the price for it ?

Page 22: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

The Quantum ProblemThe Quantum Problem

Stationary Solutions:Stationary Solutions:Time-independent Schrödinger Eq.Time-independent Schrödinger Eq. Eˆ

Variable Separation:Variable Separation:

),,...,,(),,...,,(ˆ321321 RrrrrERrrrr nelnelel

iI jiiI

I

iiel rijrR

Z

,

2 12/1ˆ

Electronic Schrödinger Eq.:Electronic Schrödinger Eq.:

Electronic Hamiltonoperator:Electronic Hamiltonoperator:

)()...()()(),...,,( 321321 nnel rrrrrrrr

Product Ansatz for the wavefunctionProduct Ansatz for the wavefunction::

Effective 1-particle modelEffective 1-particle model

Page 23: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

The Quantum ProblemThe Quantum ProblemSet of N coupled 1-particle equations:Set of N coupled 1-particle equations:

)()(ˆiiii rrh

I jiiI

Iii rijrR

Zh

12/1ˆ 2

Basis Set Expansion:Basis Set Expansion:

m

rGiim

celli

mecV

r 1

llli cr )(

Plane-waves:Plane-waves:

Set of algebraic Eqs. Solved iteratively Set of algebraic Eqs. Solved iteratively (self-consistent field)(self-consistent field)

ca. 10’000-100’000ca. 10’000-100’000

FFTFFT

Choice of QM method: DFTChoice of QM method: DFT

Page 24: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

DENSITY DENSITY FUNCTIONAL FUNCTIONAL

THEORYTHEORY

Page 25: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Walter Kohn and John Pople

Nobelprize in

Chemistry 1998

Page 26: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Literature on DFT:Literature on DFT:

Original Papers:Original Papers:

• P.Hohenberg, W.Kohn, P.Hohenberg, W.Kohn, Phys.Rev.B Phys.Rev.B 1964, 1964, 136, 136, 864-871.864-871.• W.Kohn, L.J.Sham, W.Kohn, L.J.Sham, Phys.Rev.A Phys.Rev.A 1965, 1965, 140, 140, 1133-1138.1133-1138.

Textbooks:Textbooks:• W.Kohn, P.Vashista, W.Kohn, P.Vashista, in Theory of the in Theory of the Inhomogeneous Electron Gas, N.H.March and Inhomogeneous Electron Gas, N.H.March and S.Lundqvist (Eds), Plenum, New YorkS.Lundqvist (Eds), Plenum, New York 1983 1983• R.G.Parr, W.Yang, R.G.Parr, W.Yang, Density Functional Theory of Density Functional Theory of Atoms and Molecules, Oxford University Press, Atoms and Molecules, Oxford University Press, New YorkNew York 1989. 1989. R.M.Dreizler, E.K.U.Gross, R.M.Dreizler, E.K.U.Gross, Density-Functional Density-Functional Theory, Springer, BerlinTheory, Springer, Berlin 1990. 1990.• W.Kohn, W.Kohn, Rev.Mod.Phys. Rev.Mod.Phys. 1999, 1999, 71.71.

Page 27: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Density Functional Theory (DFT)Density Functional Theory (DFT)

Let’s define a new central variable:Let’s define a new central variable:

rx...x,x,x N321

Electron densityElectron density

'21321321* ......,,...,, NNN xdxdxdxxxxxxxxr

Nrdr

Total electron density integrates to the number of electrons:Total electron density integrates to the number of electrons:

Like Hatree-Fock: effective 1-particle HamiltonianLike Hatree-Fock: effective 1-particle Hamiltonian

Page 28: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Theoretical foundations of DFT based on 2 Theoretical foundations of DFT based on 2 theorems:theorems:

Hohenberg and Kohn (1964):Hohenberg and Kohn (1964):(Phys.Rev. 136, 864B)(Phys.Rev. 136, 864B)

• The ground state energy of a system with N The ground state energy of a system with N electrons in an external potential Velectrons in an external potential Vex ex isis a unique a unique functional of the electron density functional of the electron density

VVexex determines the exact determines the exact

vice versa: Vvice versa: Vex ex is determined within an additive is determined within an additive constant by constant by gs expectation value of any observable (i.e. the gs expectation value of any observable (i.e. the H) is a unique functional of the gs densityH) is a unique functional of the gs density

rEE

r

r

r

Page 29: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

•Variational principle: The total energy is Variational principle: The total energy is minimal for the ground state density of minimal for the ground state density of the system the system

r0

rEErE 00min

Kohn and Sham (1965):Kohn and Sham (1965):(Phy. Rev. 1140, 1133A)(Phy. Rev. 1140, 1133A)

The many-electron problem can be mapped exactly The many-electron problem can be mapped exactly onto:onto: •an auxiliary noninteracting reference system with an auxiliary noninteracting reference system with the same density (i.e. the exact gs density)the same density (i.e. the exact gs density) •where each electrons moves in an effective 1-where each electrons moves in an effective 1-particle-potential due to all the other electronsparticle-potential due to all the other electrons

Page 30: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

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

Iionxc'

'

'

ionii

2ii

RErErdrdrr

rr

2

1

rdrrVrE

(1)(1) (2)(2)

(3)(3) (4)(4)

2

ii r2r

Kohn-Sham eqs:Kohn-Sham eqs:

rrVrVrV2

1iiixcHion

2

(5)(5)

Page 31: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Exchange and Exchange and CorrelationCorrelation

Exchange-Correlation HoleExchange-Correlation Hole

Page 32: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

local density approximationlocal density approximation

can be determined exactly:can be determined exactly:Exchange:Exchange:(P.A.M. Dirac, Proc. Cambridge Phil. Soc. 26, 376 (1930), E.P. (P.A.M. Dirac, Proc. Cambridge Phil. Soc. 26, 376 (1930), E.P. Wigner, Trans. Fraraday Soc. 34, 678 (1987))Wigner, Trans. Fraraday Soc. 34, 678 (1987))

3

1

xhomx Cr

3

1

x

3

4

3C

Correlation:Correlation:(D.M. Ceperly, B.J. Alder, Phys. Rev. Lett. 45, 566 (1980), G.Ortiz, P. (D.M. Ceperly, B.J. Alder, Phys. Rev. Lett. 45, 566 (1980), G.Ortiz, P. Ballone, Phys. Rev. B 50, 1391 (1994))Ballone, Phys. Rev. B 50, 1391 (1994))

exact (numerical) results from Quantum Monte exact (numerical) results from Quantum Monte Carlo simulationsCarlo simulations

Universal exchange-correlation functional, Universal exchange-correlation functional, exact form not known!exact form not known!

rr homxcxc

Page 33: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Parametrized analytic forms that interpolate Parametrized analytic forms that interpolate between different density regimes are availablebetween different density regimes are available(e.g. J.P. Perdew, A. Zunger, Phys. Rev. B. 23, 5084 (1981))(e.g. J.P. Perdew, A. Zunger, Phys. Rev. B. 23, 5084 (1981))

- in principle very crude approximation!- in principle very crude approximation!

- E- Excxc of a non uniform system locally ~ uniform of a non uniform system locally ~ uniform electron gaselectron gas resultsresults

- should ‘work’ only for systems with slowly varying - should ‘work’ only for systems with slowly varying densitydensity

but: atoms and molecules are inhomogeneous but: atoms and molecules are inhomogeneous systems!systems!

- works remarkably well in practice:- works remarkably well in practice:Performance of LDA/LSDAPerformance of LDA/LSDA in general good structural properties:in general good structural properties:

bond lenghts up to 1-2%bond lenghts up to 1-2%

bond angles ~ 1-2 degreesbond angles ~ 1-2 degrees

torsional angles ~ a few degreestorsional angles ~ a few degrees

vibrational frequencies vibrational frequencies ~ 10% ( phonon modes up to few %)~ 10% ( phonon modes up to few %)

Page 34: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

cheap and good method for transition cheap and good method for transition metals!: e.g. Crmetals!: e.g. Cr22, Mo, Mo22 in good agreement in good agreement with experiment ( not bound in HF, UHF!)with experiment ( not bound in HF, UHF!)

FF22 r ree within 3% (not bound in HF) within 3% (not bound in HF)

atomization, dissociation energies over atomization, dissociation energies over estimated (mainly due to errors for atoms), estimated (mainly due to errors for atoms), typically by 10-20%typically by 10-20%

hydrogen-bonding overestimatedhydrogen-bonding overestimated

van der Waals-complexes:van der Waals-complexes:strongly overestimated binding (e.g. noble strongly overestimated binding (e.g. noble

gas gas dimers, Mgdimers, Mg22, Be, Be22: factor 2-4: factor 2-4

CrCr22

Re[Å] De (eV)Re[Å] De (eV)HF 1.465 -19.4HF 1.465 -19.4CCSD 1.560 -2.9 CCSD 1.560 -2.9 CCSD(T) 1.621 0.5CCSD(T) 1.621 0.5DFT 1.59 1.5DFT 1.59 1.5exp 1.679 1.4exp 1.679 1.4

(Scuseria 1992)(Scuseria 1992)

Page 35: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Generalized Gradient Approximation Generalized Gradient Approximation (GGA)(GGA)

r,rErdrr

r,rfE

GGAxcxc

xcGGAxc

correction function chosen to fulfill formal correction function chosen to fulfill formal conditions for the properties of the ex-corr holeconditions for the properties of the ex-corr hole

Determination of parameters:Determination of parameters:

• fully non empiricalfully non empirical• fit to exact Ex-Corr energies for atomsfit to exact Ex-Corr energies for atoms• fit to experimental data (empirical)fit to experimental data (empirical)

man different forms (B88, P86, LYP, man different forms (B88, P86, LYP, PW91, PBE, B3LYP etc..)PW91, PBE, B3LYP etc..)

Page 36: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Density-Functional Theory rEE i ii rrr *

rEdrdr

rrrr

drrVrrrE

xc

extii

'

'

'

2*

2/1

2/1

Time-independent electronicSchrödinger Equation:

EH

Page 37: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Practical ImplementationPractical Implementation

• periodic boundary conditionsperiodic boundary conditions• plane wave basis set up to a given kinetic energy plane wave basis set up to a given kinetic energy cutoff Ecutoff Ecutcut

m

rGiim

celli

mecV

r 1

φ

use of FFT techniquesuse of FFT techniques convenient evaluation of different terms in real spaceconvenient evaluation of different terms in real space (E(Eex-corr, ex-corr, EEextext) or in reciprocal space (E) or in reciprocal space (Ekinkin, E, Ehartreehartree))• typical real space grid: ~100typical real space grid: ~10033, ~10000-100000 pws, ~10000-100000 pws• most of the time: FFT most time consuming step (NMlogM)most of the time: FFT most time consuming step (NMlogM)• for large systems: orthogonalization ~Nfor large systems: orthogonalization ~N22

• well parallelizable (over number of electronic states and well parallelizable (over number of electronic states and first index of real space gridfirst index of real space grid

Page 38: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

drrφdrrφ

rc aerc ps2

0

2

0

:

)(

r

rrpseu

allpseu

Pseudo Potentials FrameworkPseudo Potentials Framework

ab initio pseudo

r(a.u.)r(a.u.)

2)rc/r(2l

m

0lll

vps

e)bra()r(V

P)r(V)rc/r(erfr

Z)r(V

• Chemical properties determinedChemical properties determined by valence electronsby valence electrons• perform atomic all electronperform atomic all electron calculationcalculation

r > rr > rcc

smooth fct r < rsmooth fct r < rcc

)()()()(

)()())(2/1(

)()(2

rVrVrVrV

rerrV

rerH

pseuexchartree

pseuallpseu

pseuallpseu

• invert Schrodinger equationinvert Schrodinger equation

rc

Page 39: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

ABINIT www.abinit.org

CASTEP [

i

]

Molecular Simulations Inc.

CPMD

CP2K

[

i

i

]

M. Parrinello, MPI Stuttgart, Germany and IBM Zurich Research Laboratory, Switzerland www.cpmd.orgFree software

Fhi98md [

i

i

i

]

Fritz-Haber Institute Berlin, [email protected]

JEEP François Gygi, Lawrence Livermore National Laboratory, USA

NWCHEM Pacific Northwest National Laboratory, USA

PAW [

i

v

]

P.E. Blöchl, Clausthal University of Technology, Germany

SIESTA [

v

]

P. Ordejon, Institut de Ciencia de Materials de Barcelona, Barcelona, Spain

VASP [

v

i

]

J. Hafner, University of Vienna, Austria

Page 40: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

CPMD (3.9) (CP2K) CPMD (3.9) (CP2K)

Features Features (see also online manual):(see also online manual):• plane wave basis, pseudopotentials, pbc and isolated systemsplane wave basis, pseudopotentials, pbc and isolated systems• LDA, LSD, GGAs (single point hybrid fct calcs possible) LDA, LSD, GGAs (single point hybrid fct calcs possible) • geometry optimizationgeometry optimization• MD (NVE, NVT, NPT, Parrinello-Rahman)MD (NVE, NVT, NPT, Parrinello-Rahman)• path integral MDpath integral MD• different types of constraints and restraintsdifferent types of constraints and restraints• Property calculations: population analysis, multipole moments, Property calculations: population analysis, multipole moments, atomic charges, Wannier fcts, Fukui fcts etc..atomic charges, Wannier fcts, Fukui fcts etc..

www.cpmd.orgwww.cpmd.org

Most Recent Features:Most Recent Features:• QM/MM interfaceQM/MM interface• Response function calculations:Response function calculations: NMR Chemical shifts, electronic spectra, vibrational NMR Chemical shifts, electronic spectra, vibrational spectraspectra• Time Dependent DFT MD in excited statesTime Dependent DFT MD in excited states• History dependent MetadynamicsHistory dependent Metadynamics

Runs on essentially all platforms..Runs on essentially all platforms..

Page 41: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Mixed Quantum-Classical Mixed Quantum-Classical QM/MM- Car-Parrinello Simulations QM/MM- Car-Parrinello Simulations

Classical RegionClassical Region

Interface Interface RegionRegion

QuantumQuantumRegionRegion

• Fully Hamiltonian Fully Hamiltonian QM/MM Car-Parrinello QM/MM Car-Parrinello hybrid codehybrid code QM-Part: CPMD 3.8QM-Part: CPMD 3.8 pbc, PWs, pseudo potentialspbc, PWs, pseudo potentials (n-1) CPUs(n-1) CPUs MM-Part: GROMOS96 + P3M,MM-Part: GROMOS96 + P3M, AMBER (SYBIL, UFF)AMBER (SYBIL, UFF) 1 CPU1 CPU

A. Laio, J. VandeVondele, and U. Rothlisberger, J. Chem. Phys. 116, 6941 (2002);

A. Laio, J. VandeVondele, and U. Rothlisberger, J. Phys. Chem. B (ASAP article)

review in : M. Colombo et al. CHIMIA 56, 11 (2002)

Page 42: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

jijijijiQM

MMQMMMI

IIii

i

rrrdE

EERMrrrdL

,,

*,

/2*

2

1

2

1

QM/MM Car-Parrinello SimulationsQM/MM Car-Parrinello Simulationsmonovalent pseudopotential

QMQM

e-e-

MMMM

i

j

k

l

qo

qp--

++

included in Vext

rErrr

rrdrdrrVrdrrrdRE xcNii

iIiKS

''

1'

2

1

2

1, *

bondednonMM

bondedMMMM EEE

QM/MM Lagrangian

EQM: DFT

EMM: Standard biomolecular Force Field

n

ijklnijkijbb

bondedMM nkkbrkE )cos(1)(

2

1)(

2

10

20

20

op op

po

op

opop

lm lm

mlbondednonMM rrr

qqE

612

04

4

Page 43: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

QM/MM Car-Parrinello in Combination QM/MM Car-Parrinello in Combination with Response Propertieswith Response Properties

• Variational Perturbation Theory:Variational Perturbation Theory: A. Putrino, D. Sebastiani, M. Parrinello, 113, 7103 (2000)A. Putrino, D. Sebastiani, M. Parrinello, 113, 7103 (2000)

• Chemical ShiftsChemical ShiftsD. Sebastiani, M. Parrinello, J. Phys. Chem. A 105, 1951 (2001)D. Sebastiani, M. Parrinello, J. Phys. Chem. A 105, 1951 (2001)

• TDDFT: Spectra and DynamicsTDDFT: Spectra and DynamicsJ. Hutter J.Chem.Phys. 118, 3928 (2003)J. Hutter J.Chem.Phys. 118, 3928 (2003)

• IR and Raman SpectraIR and Raman Spectra• Fukui FunctionsFukui FunctionsR. Vuilleumier, M. Sprik J.Chem.Phys. 115, 3454 (2001)R. Vuilleumier, M. Sprik J.Chem.Phys. 115, 3454 (2001)

Page 44: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

QM/MM Car-Parrinello in Combination QM/MM Car-Parrinello in Combination with Excited State Methodswith Excited State Methods

CP-version:CP-version:I. Frank et al. J. Chem. Phys. 108, I. Frank et al. J. Chem. Phys. 108,

4060 (1998)4060 (1998)

• ROKSROKS

• LR-TDDFT-MDLR-TDDFT-MD

J. Hutter J. Chem.Phys. 118, 3928 (2003)J. Hutter J. Chem.Phys. 118, 3928 (2003)L. Bernasconi et al. J. Chem.Phys. 119, L. Bernasconi et al. J. Chem.Phys. 119, 12417 (2003)12417 (2003)

(Tamm-Dancoff (Tamm-Dancoff Approximation)Approximation)

Landau-ZenerSurface Hopping

• P-TDDFT-MDP-TDDFT-MDI. Tavernelli (to be published)I. Tavernelli (to be published)

Ehrenfest Dynamics

T. Ziegler et al. Theor. Chim. ActaT. Ziegler et al. Theor. Chim. Acta 43, 261 (1977) (sum method)43, 261 (1977) (sum method)

mm11 mm22 tt1,21,2

E(s) = 2E(m) - E(t)E(s) = 2E(m) - E(t)

HOMO-LUMO single excitationsHOMO-LUMO single excitations

Page 45: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Limitations Due to Short Simulation Limitations Due to Short Simulation TimeTime

• MD as sampling MD as sampling tool:tool:

only small portion of phase space is only small portion of phase space is sampled sampled relevant parts might be missed,relevant parts might be missed, especially if there exist largeespecially if there exist large barriers between different barriers between different important regionsimportant regions (e.g. different conformers)(e.g. different conformers)ensemble average have largeensemble average have large statistical errorsstatistical errors (e.g. relative free energies!)(e.g. relative free energies!)

• MD as dynamical tool: Real-time MD as dynamical tool: Real-time simulation of simulation of dynamical processesdynamical processes

many processes lie outside time rangemany processes lie outside time range

)exp( abb

a Fp

p

pA

pB

Page 46: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Techniques from Classical MD:Techniques from Classical MD:

• Sampling at enhanced temperatureSampling at enhanced temperature• Rescaling of atomic mass(es)Rescaling of atomic mass(es)• ConstraintsConstraints (Ryckaert, Ciccotti, Berendsen 1977) (Sprik & Ciccotti 1998) • Umbrella SamplingUmbrella Sampling (Torrie&Valleau 1977)• Quasi-Harmonic AnalysisQuasi-Harmonic Analysis (Karplus, Jushick 1981)• Reaction Path MethodReaction Path Method (Elber & Karplus 1987)• ‘ ‘Hypersurface Deformation’Hypersurface Deformation’ (Scheraga 1988, Wales 1990)• Multiple Time Step MDMultiple Time Step MD (Tuckerman, Berne 1991) (Tuckerman, Parrinello 1994)• Subspace Integration MethodSubspace Integration Method (Rabitz 1993)• Local ElevationLocal Elevation (van Gunsteren 1994)

•Conformational FloodingConformational Flooding (Grubmuller 1995)•Essential DynamicsEssential Dynamics (Amadei&Berendsen 1996)• Path OptimizationPath Optimization (Olender & Elber 1996)• Multidimensional AdaptiveMultidimensional Adaptive Umbrella SamplingUmbrella Sampling (Bartels, Karplus 1997)• HyperdynamicsHyperdynamics (Voter 1997) (Steiner, Genilloud, Wilkins 1998) (Gong & Wilkins 1999)• Transition Path SamplingTransition Path Sampling (Dellago, Bolhuis, Csajka, Chandler 1998)• Adiabatic Bias MDAdiabatic Bias MD (Marchi, Ballone 1999)• Metadynamics(Laio, Iannuzzi, Parrinello PNAS 99, 12562 (2002), PRL 90, 23802 (2003)

Page 47: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Development of Enhanced Sampling MethodsDevelopment of Enhanced Sampling Methods

Two DimensionalTwo DimensionalFree Energy SurfaceFree Energy Surface

with torsionalwith torsionalpotential biaspotential bias

1kcal/mol

• multiple time step multiple time step samplingsampling

Peroxynitrous AcidPeroxynitrous Acid

48ps

J. Chem. Phys. 113 4863 (2000), J. Chem. Phys. 115 7859-7864 (2001), J. Phys. Chem. B 106, 203-208 (2002), J. Am. Chem. Soc. 124, 8163 (2002)

• classical bias potentials and forcesclassical bias potentials and forces• double thermostattingdouble thermostatting• parallel temperingparallel tempering

Configurational SamplingConfigurational Sampling

T = 500K ET = 500K EAA = 30 kcal/mol = 30 kcal/mol

Electronic Bias PotentialsElectronic Bias Potentials• Finite Electronic TemperatureFinite Electronic Temperature• Vibronic CouplingVibronic Coupling• Charge RestraintCharge Restraint

Sampling of Rare Sampling of Rare Reactive Events Reactive Events

Page 48: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

ConstraintsConstraints

• freeze out fast motionsfreeze out fast motions increase integration time step increase integration time step (( linear speed up) linear speed up)• constrain slowest motionconstrain slowest motion guide system ‘manually’ over barrier guide system ‘manually’ over barrier (condition: slowest part of reaction coordinate (condition: slowest part of reaction coordinate is known, all other is known, all other degrees of freedom have time to equilibrate along the path)degrees of freedom have time to equilibrate along the path)

(( free energy differences via thermodynamic integration) free energy differences via thermodynamic integration)

0})N{r(f})N{r(f

VTL

0

ii

iiiig

gfrm

Lagrangian:Lagrangian:Equations of motion:Equations of motion:

'ddF

'd)(F)(FF2

1

1212

integral replaced by aintegral replaced by a discrete set of points discrete set of points (R)= (R)= ’’

''ddF

for a simple distance constraint for a simple distance constraint (R)= lR(R)= lRII-R-RJJl:l:

Page 49: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Umbrella Sampling: Bias PotentialsUmbrella Sampling: Bias Potentials(Torrie&Valleau 1977)(Torrie&Valleau 1977)

'H')q,p(H)q,p('H'

'H')q,p(H)q,p('H'

He

e)q,p(f)q,p(f

H)q,p(H

H)q,p(H

He

e)q,p(f)q,p(f

‘‘Ideal’ Bias:Ideal’ Bias:

• high overlap with originalhigh overlap with original ensembleensemble• close match PES or freeclose match PES or free energy surfaceenergy surface• low dimensionalitylow dimensionality• computationally inexpensivecomputationally inexpensive

‘‘Golden Rules’Golden Rules’

(Grubmuller 1995, Voter 1997, Karplus 1997, Wilkins 1998…)

Page 50: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Sampling Error in ab initio MD:Sampling Error in ab initio MD:

mol

kcal5.4F

00991.0p

0135.0p

996.0p

MAX

)360240(

)240120(

)1200(

Methyl Group RotationMethyl Group Rotation in Ethane Cin Ethane C22HH66

(500K, 7.25 ps)(500K, 7.25 ps)

Probability Distribution Probability Distribution HCCHHCCH

EEAA = 2.8 kcal/mol = 2.8 kcal/mol

Page 51: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Atomic Bias PotentialsAtomic Bias Potentials

(500K, 7.25 ps)(500K, 7.25 ps)

Before correctionBefore correction

Torsional Bias 0.0017auTorsional Bias 0.0017au

After correctionAfter correction

mol

kcal02.0F

331.0p

331.0p

338.0p

MAX

)360240(

)240120(

)1200(

Methyl Group RotationMethyl Group Rotation in Ethane Cin Ethane C22HH66

))3cos(1(V2/1V 0bias

Page 52: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Bias Potentials from Classical Force FieldBias Potentials from Classical Force Field

Peroxynitrous AcidPeroxynitrous Acid

ONOOHONOOH

-100 0 100 200 300 400-100

0

100

200

300

400

J. VandeVondele, U.R. J. Chem. Phys. 113 4863 (2000)

Trajectory in biased spaceTrajectory in biased space (48 ps) (48 ps)

Free Energy SurfaceFree Energy Surface

Page 53: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

CAFES:• Partitioning into reactive system / environment

Canonical, Adiabatic Free Energy Sampling

ER mm Slow dynamics of the reactive

subsystem

ETRTRCAFES

RREAL )x(ρ)x(ρ

adiabatic decoupling

different temperatures TR/TE (2 thermostats)

Sampling efficiency at TR can be estimatedEa= 20 kcal/mol, TE=300K, TR=1200K -> 1013

R

E

mmt

J. VandeVondele, U.R. J. Phys. Chem. B 106, 203 (2002)

Page 54: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Nucleophilic substitution with anchimeric assistance

• QMMM SPC/CPMD• CAFES 100 / 2000K /

300K

• ~22 kcal/mol• shows that the reaction coordinate is not simple

Page 55: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Transition State Path Sampling

Given: - initial state A - final state B - one path connecting the two

1

0100

1

010

)()()()(})({

: haspath reactiveA

)()(})({

: haspath generalA

L

LBAAB

L

xxpxxhxhxf

xxpxxf

generate the ensemble of ‘reactive paths’ calculate transition rates

Page 56: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Dispersion Interactions in DFT

QM

MM

Page 57: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

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…)

Page 58: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Optimized Effective Atom Centered Potentials

xchartreeextKStot VVVV ˆˆˆˆ NLV

Expansion in linear combination of atom-centered (nonlocal) potentials

III

effNL RrRrVrrV ',',ˆ

Analytic pseudopotentials by Goedecker et al.

)2/)exp(

'ˆ'ˆ',

2exp

2

',)'(',ˆ

22)1(2

*3

1,

6

4

4

3

2

21

2

2

lhl

lh

lmljlhjhj

lh

l

lmlm

nll

loclocloc

locloc

ionloc

nlloceff

rrrrp

rYrphrprYrrV

r

rC

r

rC

r

rCC

r

r

r

rerf

r

ZrV

rrVrrrVrrV

Page 59: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Optimization Penalty Functional

jj

i

d

rdn

rnrrd

d

d

n(r)rrdrn

)(

)()(

)(,

3

3

F

wP

FwP

Linear density response calculated via first order perturbation theory with perturbation Hamiltonian

j

ieff

j

VH

)1(ˆ

For : nlV 2

2

intint ionsN

I

refII

refrefrefrefdisp RFwRERERP

Page 60: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

BLYP

MP2

OECP

Is this potential transferable???

1 = -0.003522 = 3.280

Page 61: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

BLYP

MP2

OECP

BLYP

MP2

OECP

Page 62: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

BLYP

OECP

exp

z = 3.35AE=35 meV/atom

z = 3.3A

E=32 meV/atom

Page 63: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

BLYP

MP2Klopper et al. J.Chem.Phys. 101, 9747 (1994)

OECP

Reference system: Ar2

1 additional f-channel:1 = -0.002062 = 2.902

BLYP

MP2

OECP

Page 64: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

What about the intramolecular geometry??

Bond lengths in benzene: << 0.01 A

What about the electronic properties??

BLYP OECP MP2Dipole moment: benzene-Ar 0.047 0.035 0.037 Quadrupole moment: benzene -5.35 -5.50 -6.46Polarizability: argon 12.30 12.31 11.15 xx- yy benzene 39.18 38.45 35.07 zz benzene-Ar 55.0 58.1 59.2

Page 65: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland
Page 66: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland
Page 67: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

ω

αΦ

α

Φ

ωΦ

α

Increasing kinetic energyback to reactant geometry

relaxation to product geometry

minimum on S1

start on S1

Formaldimine. Excited state dynamics after excitation S0→S1

The region of conical intersection, CI, is reached only in case of non-thermostatted trajectories.

Page 68: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Landau-Zener SH

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.

( ) ( *) ( *) , ,m m mU q U q F q q m D A

0qq

en

erg

y

UD UA

qq*

*

)(*)(

qq

mm q

qUqF

))((*)( 0 couplDvibDA VtHqUTH

AAtvFDDtvFtH AD **)(0

tvq *

DAVADVV ADDA

2, DUDPD

The asymptotic value for the survival probability of the electron for remaining at the donor

The donor survival probability is

ePDAD

DA

FF

V

v

2

*

12

Page 69: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Units: atomic unitsUnits: atomic units used throughout used throughout

Page 70: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Transition Rate Constants

)kk(

ek )0(h

)t(h)0(h)t(k

ABBArxn1

/tBA

A

BA rxn

• Can be calculated with trajectories starting at the TS• Is difficult if a RC/TS cannot be defined.

Reactive Flux Correlation Function

Page 71: Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics Ursula Rothlisberger EPFL Lausanne, Switzerland

Rate constants in the TPE

)()(

)()(

)(

)()(

)()()()(

000

000

tCth

thtC

kdt

tdC

xhxdx

xhxhxdxtC

ABB

ABB

BA

A

tBA

• 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.