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Car-Parrinello Molecular Dynamics Simulations (CPMD): Basics. Ursula Rothlisberger EPFL Lausanne, Switzerland. Literature Car-Parrinello:. R. Car and M. Parrinello A unified approach for molecular dynamics and density functional Phys.Rev.Lett. 55, 2471 (1985). - PowerPoint PPT Presentation
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Car-Parrinello Molecular Dynamics Simulations
(CPMD): Basics
Ursula RothlisbergerEPFL 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
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
When Newton meets Schrödinger...
maF H
Sir Isaac Newton(1642 - 1727)
Erwin Schrödinger(1887 - 1961)
Newt-dinger
maF H
The ideal combination for Ab Initio Molecular Dynamics
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
Basic Principles of Quantum Mechanics
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
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
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
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!
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:
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)
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 )(
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
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
Car - Parrinello Molecular Dynamics (1985)
Roberto Car
Michele Parrinello
ijji
ijij
Iii iiI IIex
drrr
RERML
,2/1 2
Extended Lagrangian Formulation
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
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
• 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 ?
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
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
DENSITY DENSITY FUNCTIONAL FUNCTIONAL
THEORYTHEORY
Walter Kohn and John Pople
Nobelprize in
Chemistry 1998
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.
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
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
•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
(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)
Exchange and Exchange and CorrelationCorrelation
Exchange-Correlation HoleExchange-Correlation Hole
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
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 %)
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)
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..)
Density-Functional Theory rEE i ii rrr *
rEdrdr
rrrr
drrVrrrE
xc
extii
'
'
'
2*
2/1
2/1
Time-independent electronicSchrödinger Equation:
EH
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
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
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, Germanyfhim@fhi-berlin.mpg.de
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
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..
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)
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
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)
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
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
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)
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
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:
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…)
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
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
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
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)
Nucleophilic substitution with anchimeric assistance
• QMMM SPC/CPMD• CAFES 100 / 2000K /
300K
• ~22 kcal/mol• shows that the reaction coordinate is not simple
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
Dispersion Interactions in DFT
QM
MM
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…)
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
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
BLYP
MP2
OECP
Is this potential transferable???
1 = -0.003522 = 3.280
BLYP
MP2
OECP
BLYP
MP2
OECP
BLYP
OECP
exp
z = 3.35AE=35 meV/atom
z = 3.3A
E=32 meV/atom
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
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
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.
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*
*
)(*)(
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
Units: atomic unitsUnits: atomic units used throughout used throughout
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
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.
Recommended