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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

24. DENSITY
FUNCTIONAL
THEORY

25. Walter Kohn and John Pople
Nobelprize in
Chemistry 1998

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

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

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

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

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

31. Exchange and Correlation
Exchange-Correlation Hole

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

56. Dispersion Interactions in DFTQM MM

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

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

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

60. Is this potential transferable???
BLYP
OECP
MP2

61. BLYP
BLYP
OECP
OECP
MP2
MP2

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

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

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

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

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

69. Units: atomic units used throughout

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

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