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)
• D. Marx and J. Hutter
Modern Methods and Algorithms
of Quantum
• 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
J. Grotendorst (Ed.), NIC
Forschungszentrum Jülich (2000) 15 years of Car-Parrinello
simulations in Physics, Chemistry,
p.301
and Biology
• 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)
Computational Chemistry: Reviews
of Current Trends, J. Leszczynski
(Ed.), World Scientific, Vol. 6,
(2001) p.33
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
When Newton meets Schrödinger...
Sir Isaac Newton
(1642 - 1727)
Erwin Schrödinger
(1887 - 1961)
F  ma
Hˆ   
Newt-dinger
F  ma
Hˆ   
The ideal combination for
Ab Initio Molecular Dynamics
Atoms, Molecules and
Chemical Bonds
Atoms
+
e-
N protons
& neutrons
N electrons
Chemical Bonds
Chemical 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.
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.

2 
P ( r )   ( r )dV
2 
  ( r )dV  1
V
The total probability to find the particle
anywhere in space integrates to 1.
Classical Mechanics
positions and momenta
have sharp defined  
values
r,v
Continous energy spectrum
Epot
Quantum Mechanics
uncertainty xp  h

relation
 r , t 

* 
 r , t  *  r , t 
energies are quantized
n=3
n=2
n=1
n=0
q
0
Newton`s Equations
hw
q
0
Schroedinger Equation
n, E, m, h0
Classical Mechanics: Particle Motion

r
F  ma
ro,vo  r(t),v(t)



 
r ( t )  ro  v o t  12 at2 v ( t )  v o  at

Position r and velocity vof a particle
can be calculated exactly at any time t.
1
E kin  mv 2
2
Continuous energy
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
       
       

 i  ( R1, R2 , R3...RN , r1, r2 , r3 ,...rn , t )   ( R1, R2 , R3...RN , r1, r2 , r3 ,...rn , t )
t
 not possible!
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:
       
   
   
(R1, R2 , R3...RN , r1, r2 , r3,...rn )  nu (R1, R2 , R3...RN )el (r1, r2 , r3,...rn )
Electronic Schrödinger Eq.:
   
   
el el (r1, r2 , r3,...rn , R)  Eel (r1, r2 , r3,...rn , R)
Electronic Hamiltonoperator:
ZI
1
2
 el  1 / 2 i  
 
i
I ,i RI ri i  j rij
Solve electronic Schrödinger Eq. for each set of nuclear coordinates
  

R  (R1, R2 , R3,...RN )
E (R )
potential energy surface (PES)
Nuclear SchrödingerEq.
  

  

Hnunu (R1, R2 , R3....RN )  Etotnu (R1, R2 , R3....RN )
Nuclear Hamiltonoperator:
ZI ZJ
1
2
 nu  
 I  E ( R)  
I 2M I
I , J RIJ
Nuclear Quantum Dynamics
(review: Makri, Ann. Rev. Phys. 50, 167 (1999)
E (R )
Empirical parameterization
→ force field based MD
Calculate E ( R)   H 
→ 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  
proton:
el  m p
 
 p  mel
1/ 2




h
of an electron and a
2mE
 40
 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
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
Car - Parrinello Molecular Dynamics (1985)
Lagrangian Formulation of Classical Dynamics
L  T (qI )  V (qI )
1
2

L   M I RI  V (RI )
I 2
Euler-Lagrange Equation:
d  δL  δL

dt  δqi *  δqi *
E


M I RI  
RI
Car - Parrinello Molecular Dynamics (1985)
Extended Lagrangian Formulation
2

L   I 1 / 2 M I RI  i  i i  E i , RI 
ex
   ij
ij
  r  r dr  
Roberto
Car
i
j
ij
Michele
Parrinello
Equations of Motion
E


M I RI  
RI
i   Hi   j ij j
Can be integrated simultaneously (e.g. with Verlet, Velocity-Verlet
algorithm etc..)



t 2 
RI ( t  t )  2 RI ( t )  RI ( t  t ) 
FI ( t )  O(t 4 )
2M I
Verlet
algorithm
dt ~0.1-0.2 fs
Does this fictitious classical dynamics described via the
extended Lagrangian have anything to do with the real
physical dynamics???
• if
  MI ' s  K e  0
 total energy of the system becomes the real physical
total energy
K e  K I  Epot  K I  Epot
 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
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
The Quantum Problem
Stationary Solutions:
Time-independent Schrödinger Eq.
ˆ   E

Variable Separation:
Electronic Schrödinger Eq.:
   
   
ˆ
 el el (r1 , r2 , r3 ,...rn , R)  Eel (r1 , r2 , r3 ,...rn , R)
ZI
1
2
ˆ



1
/
2



 i


Electronic Hamiltonoperator:
el
i
I , i RI ri i  j rij
Product Ansatz for the wavefunction:
   




el (r1 , r2 , r3 ,...rn )  (r1 )(r2 )(r3 )...(rn )
Effective 1-particle model
The Quantum Problem
Set of N coupled 1-particle equations:


ˆ
hi (ri )   i (ri )
ˆh  1 / 2 2   Z I   1
i
i
I RI ri i  j rij
Basis Set Expansion:

(ri )   c l  l
 Set of algebraic Eqs. Solved iteratively
(self-consistent field)
l
Plane-waves:

 i r  
1
Vcell
Choice of QM method: DFT
 c im e
m


iG m  r
ca. 10’000-100’000
 FFT
DENSITY
FUNCTIONAL
THEORY
Walter Kohn and John Pople
Nobelprize in
Chemistry 1998
Literature on DFT:
Original Papers:
• P.Hohenberg, W.Kohn, Phys.Rev.B 1964, 136, 864-871.
• W.Kohn, L.J.Sham, Phys.Rev.A 1965, 140, 1133-1138.
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.
Density Functional Theory (DFT)
Like Hatree-Fock: effective 1-particle Hamiltonian
Let’s define a new central variable:

  

 x 1 , x 2 , x 3 ... x N   r 
Electron density


   
 

*   
 r      x1 , x2 , x3 ... x N   x1 , x2 , x3 ... x N dx1dx2 ...dx N '
Total electron density integrates to the number of electrons:
 
 r dr  N
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  r


E  Er 

 Vex determines the exact r 
 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  r

•Variational principle: The total energy is minimal for the

ground state density  0 r  of the system


Er min  E 0  E 0 r 
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-particlepotential due to all the other electrons

  
E i       i   i r   Vion r r d r
2
i
(1)
(2)
 
 '


1 r  r   '
    ' d r d r  E xc r   E ion R I
2
r r
 
(4)
(3)
(5)
(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

2
r   2  i r 
i
Kohn-Sham eqs:




 1 2
 2   Vion r   VH r   Vxc r   i   i  i r 


Exchange and Correlation
Exchange-Correlation Hole
Universal exchange-correlation functional,
exact form not known!
 local density approximation


hom
 xc r    xc r 
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))
1
hom
3
x
x
1
3


r   C 
33
Cx   
4
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
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 %)
 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
1.465
-19.4
Cr2
CCSD
1.560
-2.9
CCSD(T) 1.621
0.5
(Scuseria 1992)
DFT
1.59
1.5
exp
1.679
1.4
Generalized Gradient Approximation (GGA)




E
 fxc r , r 
   GGA 

   xc r r dr  E xc r , r 
GGA
xc


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..)
Time-independent electronic
Schrödinger Equation:
H  E
Density-Functional Theory
E  E r 
 r   
*

i i
r  i r 
E  1/ 2  i* r   2i r     r  Vext r  dr
 1/ 2
 r  r ' 
r  r'
drdr'  Exc  r 
Practical Implementation
• periodic boundary conditions
• plane wave basis set up to a given kinetic energy cutoff
Ecut
 

φ i r  
1
 cime
Vcell m
iG m  r
 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, ~10000-100000 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
Pseudo Potentials Framework
• Chemical properties determined
by valence electrons
• perform atomic all electron
calculation
ab initio pseudo
 pseu r    all ( r )
 pseu r  :
rc
ps
 φ
0
2
r  rdrc 
rc
ae
 φ
0
r 
r > rc
smooth fct
r < rc
2
dr
• invert Schrodinger equation
r(a.u.)
m


Zv
Vps (r )  
erf (r / rc )   Vl (r )Pl
r
l0
2
Vl (r )  (a  br )e
 (r / rc)2
H pseu ( r )  e all pseu ( r )
( 1 / 2 2  V ( r )) pseu ( r )  e all pseu ( r )
V ( r )  V hartree ( r )  V exc ( r )  V pseu ( r )
ABINIT
CASTEP
www.abinit.org
[
i
Molecular Simulations Inc.
]
CPMD
[
i
i
CP2K
]
Fhi98md
[
i
i
i
M. Parrinello, MPI Stuttgart, Germany and IBM Zurich
Research Laboratory, Switzerland www.cpmd.org
Free software
Fritz-Haber Institute Berlin, Germany
fhim@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
VASP
[
v
]
P. Ordejon, Institut de Ciencia de Materials de Barcelona,
Barcelona, Spain
[
v
J. Hafner, University of Vienna, Austria
CPMD (3.9)
(CP2K)
www.cpmd.org
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
Mixed Quantum-Classical
QM/MM- Car-Parrinello Simulations
• 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
Interface Region
Quantum
Region
Classical Region
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)
QM/MM Car-Parrinello Simulations
monovalent
pseudo
potential
QM/MM Lagrangian
QM
 2
 *  1
1
L     dr  i r  i r    M I RI  E MM  EQM / MM
2 i
2I
 * 
 EQM    i , j  dr  i r  j r    i , j
i
i, j
e-

MM
j

l
k
qp -
+
EQM: DFT
qo

 *


  1    1


1
E KS  i , RI     dr  i r  i r    drVN r  r    dr dr '  r     r '  E xc  r 
included
2i
2
r  r'


EMM: Standard biomolecular Force Field
bonded
non bonded
E MM  E MM
 E MM
in Vext

1
1
bonded
EMM
  kb (rij  b0 ) 2   k ( ijk   0 ) 2    k n 1  cos(n ijkl  0 )
b2
 2
n
   12   6 
op 
 op 
qq
nonbonded

EMM
  l m   4 op  

 rop  
lm 4 0 rlm op
  rop 

 


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)
QM/MM Car-Parrinello in Combination
with Excited State Methods
• ROKS
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)
• LR-TDDFT-MD
(Tamm-Dancoff Approximation)
J. Hutter J. Chem.Phys. 118, 3928 (2003)
L. Bernasconi et al. J. Chem.Phys. 119, 12417
(2003)
• P-TDDFT-MD
I. Tavernelli (to be published)
Landau-Zener
Surface Hopping
Ehrenfest Dynamics
m1 m2
t1,2
E(s) = 2E(m) - E(t)
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,
pa
 exp( Fab )
especially if there exist large
pb
barriers between different
important regions
(e.g. different conformers)
ensemble average have large
statistical errors
(e.g. relative free energies!)
pA
pB
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, 23802 (2003)
Development of Enhanced Sampling Methods
Configurational Sampling
• multiple time step sampling
• classical bias potentials and forces
• double thermostatting
• parallel tempering
Two Dimensional
Free Energy Surface
with torsional
potential bias
Sampling of Rare
Reactive Events
Electronic Bias Potentials
• Finite Electronic Temperature
• Vibronic Coupling
• Charge Restraint
T = 500K EA = 30 kcal/mol
Peroxynitrous Acid
48ps
 1kcal/mol
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)
Constraints
Lagrangian:
L  T  V     

   f (r {N})  f0 (r {N})  0
Equations of motion:
miri  fi  gi
gi     i 

• 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)
2
dF
F12  F( 2 )  F(1)   d'
1 d'
 integral replaced by a
discrete set of points (R)= ’
for a simple distance constraint (R)= lRI-RJl:
dF
   '
d'
Umbrella Sampling: Bias Potentials
(Torrie&Valleau 1977)
 f (p, q) H 
(Grubmuller 1995,
Voter 1997, Karplus 1997,
Wilkins 1998…)
‘Ideal’ Bias:
‘Golden Rules’
 f (p, q) H 
 f (p, q)e H(p,q) H
 e H(p,q) H
 f (p, q)e'H'(p,q)H(p,q) 'H'
 e'H'(p,q)H(p,q) 'H'
• high overlap with original
ensemble
• close match PES or free
energy surface
• low dimensionality
• computationally inexpensive
Sampling Error in ab initio MD:
Methyl Group Rotation
in Ethane C2H6
(500K, 7.25 ps)
Probability Distribution  HCCH
EA = 2.8 kcal/mol


 0.0135 

 0.00991
p (0120 )  0.996
p (120 240 )
p (240 360 )
 FMAX  4.5

kcal
mol
Atomic Bias Potentials
Methyl Group Rotation
in Ethane C2H6
(500K, 7.25 ps)
Before correction
After correction
Torsional Bias 0.0017au
p (0120 )  0.338

kcal
p (120 240 )  0.331  FMAX  0.02
mol

p (240 360 )  0.331
Vbias  1/ 2V0 (1  cos(3))


Bias Potentials from Classical Force Field
400
300
200
100
0
Peroxynitrous Acid
ONOOH
-100
-100
0
100
200
300
400
Trajectory in biased space
(48 ps)
Free Energy Surface
J. VandeVondele, U.R. J. Chem. Phys. 113 4863 (2000)
CAFES: Canonical, Adiabatic Free Energy Sampling
• Partitioning into reactive system / environment
adiabatic decoupling
E
m
t 
m
m  m
R
E
Slow dynamics of the reactive
subsystem
R
different temperatures TR/TE (2 thermostats)
ρ REAL (x
R

)  ρ CAFES (x
R
)

TR TE
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)
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
 generate the ensemble of
‘reactive paths’
 calculate transition rates
A general path has :
L 1
f ({x})   ( x0 ) p( x  x 1 )
 0
A reactive path has :
L 1
f AB ({x})  hA ( x0 )hB ( xL )  ( x0 ) p( x  x 1 )
 0
Dispersion
Interactions in DFT
MM
QM
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
VˆtotKS  Vˆext  Vˆhartree  Vˆxc  Vˆ NL
Expansion in linear combination of
atom-centered (nonlocal) potentials
Vˆ NL r , r'  V eff  r  RI , r' RI

I
Vˆ eff r , r'  V loc r  ( r  r' )  V nl r , r'
V
Analytic pseudopotentials by
Goedecker et al.
loc
 r 
 r2 
 Z ion
r  
erf 
  e xp  2  
r
 2rloc 
 rloc 2 
2
4
6








r
r
r
C  C 
  C3 
  C4 
 
1
2

 rloc 
 rloc 
 rloc  

Vl
nl
l
3
m l
j , h1
r , r'  Ylm rˆ    plh r hlhj plj r'Ylm* rˆ'
plh r r l  2( h1) e xp( r 2 ) / 2rl2 )
Optimization Penalty Functional
P nr ,  i    d 3rw ( r )F n(r)
dP
F dn( r )
  d 3rw ( r )
d j
n( r ) d j
V
Hˆ (j 1) 
Linear density response calculated via first order
perturbation theory with perturbation Hamiltonian
For Vˆ nl :
 
 
2
    w F R 
ref
P disp Rref  Eint
Rref  Eint Rref
2
N ions
ref
I
I
I
eff
 i 
 j
1 = -0.00352
2 = 3.280
BLYP
OECP
MP2
Is this potential
transferable???
BLYP
BLYP
OECP
OECP
MP2
MP2
BLYP
z = 3.3A
OECP
E=32 meV/atom
exp
z = 3.35A
E=35 meV/atom
Reference system: Ar2
1 additional f-channel:
1 = -0.00206
2 = 2.902
BLYP
BLYP
OECP
OECP
MP2
Klopper et al. J.Chem.Phys. 101, 9747 (1994)
MP2
What about the intramolecular geometry??
Bond lengths in benzene:
 << 0.01 A
What about the electronic properties??
Dipole moment:
benzene-Ar
Quadrupole moment:
benzene
Polarizability:
argon
xx- yy benzene
zz
benzene-Ar
BLYP
OECP
MP2
0.047
0.035
0.037
-5.35
-5.50
-6.46
12.30
39.18
55.0
12.31
38.45
58.1
11.15
35.07
59.2
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
ω
ω
Φ
minimum on S1
α
Landau-Zener SH
energy
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.
Um (q)  Um (q*)  Fm (q*) q , m  D, A
UA
UD
Fm (q*)  
q*
q
0
q
q
U m (q )
 q q  q*
q  v *t
H DA  Tvib  U D (q*)  ( H0 (t )  Vcoupl )
H0 (t )  FD v *t D D  FA v *t A A
V  VDA D A  VAD A D
The asymptotic value for the survival probability of the electron for remaining at the donor
PD  D U , D
2
The donor survival probability is
2
PD  e
2 1 VDA

 v * FD  FA
Units: atomic units
used throughout
Transition Rate Constants
Reactive Flux Correlation Function
 (t ) 
 hA (0)h
B
k( t ) 
 k A Be  t / rxn
 hA (0) 
 1rxn  (k A B  kB A )
• Can be calculated with
trajectories starting at the
TS
• Is difficult if a RC/TS
cannot be defined.
Rate constants in the TPE
dx  ( x )h ( x )h ( x )

C (t ) 
 dx  ( x )h ( x )
0
0
0
A
0
B
0
A
0
t
dC (t )
 k A B
dt
 hB (t )  AB
C (t ) 
C (t )
 hB (t )  AB
Contrary to direct MD,
the computational
efficiency does not
depend on the height of
the barrier.
• C(t) = the fraction of
trajectories of length t,
starting in A, that arrives in B
• Can be calculated with a
reversible work calculation.