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Ab Initio Molecular DynamicPaul Fleurat-Lessard
AtoSim Master / RFCT
Ab Initio Molecular Dynamic – p.1/67
Summary
I. Introduction
II. Classical Molecular Dynamic
III. Ab Initio Molecular Dynamic (AIMD)
1 - Which one ?
2 - Car-Parinello !
IV. Application : Free Energy Calculations
V. Conclusions
Ab Initio Molecular Dynamic – p.2/67
Introduction
Goal :Compute statistical averages, as in Monte Carlo.Sample phase space⇒ Macroscopic properties
Tool :Time evolution of the system.
Ab Initio Molecular Dynamic – p.3/67
Microscopic becomes Macroscopic
Trajectories sampling phase space
Amacro = 〈A〉 =
∫ ∫A exp(−βE)d−→ri d
−→pi∫ ∫
exp(−βE)d−→ri d−→pi
Ab Initio Molecular Dynamic – p.4/67
Microscopic becomes Macroscopic
Trajectories sampling phase space
Ergodic hypothesis verified :
Amacro = 〈A〉 = limT→∞
AT = limT→∞
1
T
∫ T
0
A(t)dt
Ab Initio Molecular Dynamic – p.4/67
Microscopic becomes Macroscopic
Trajectories sampling phase spaceErgodic hypothesis verified :
Amacro = 〈A〉 = limT→∞
AT = limT→∞
1
T
∫ T
0
A(t)dt
True only if :System at equilibriumSimulation time T much larger than caracteristictimes of the system.
Ab Initio Molecular Dynamic – p.4/67
Short History of MD
Simulations using balls (hard and soft)Computer Simulations
Super-computer MANIAC (Los Alamos in 1952)Monte Carlo : Metropolis, Rosenbluth, Rosenbluth,Teller, and Teller (1953)MD Hard Sphere : Alder and Wainwright (1957)MD with actual movement equations : Rahman(Argonne 1964)
Ab Initio Molecular DynamicEhrenfest 1927 : First approach.C. Leforestier 1978 : Born-OppenheimerCar and Parrinello (Sissa 1985)
Ab Initio Molecular Dynamic – p.5/67
Computer simulations
Four steps :InitializationPropagationEquilibrationAnalysis
Propagation : All schemes based on numerical solution
miri = −∇iV ({−→ri })
Which V ?Empirical : classical force fieldsQuantum : semi-empirical, ab initio and DFT
How to obtain V ?
Ab Initio Molecular Dynamic – p.6/67
Computer simulations
Four steps :InitializationPropagationEquilibrationAnalysis
Propagation : All schemes based on numerical solution
miri = −∇iV ({−→ri })
Which V ?How to obtain V ?
V pre-calculated and tabulated : limited to fewdegrees of freedom, allows quantum nuclei ;V calculated “on the fly” : mostly classical nuclei
Ab Initio Molecular Dynamic – p.6/67
Time scale
V Simulation duration (ps) AtomsForce Fields 100 000 100 000
Semi-empirical 100 1000DFT, HF 10 100post-HF 1 ≈ 10
Ab Initio Molecular Dynamic – p.7/67
Classical Molecular Dynamic
Ab Initio Molecular Dynamic – p.8/67
Equations of motion
Many formalism :Newton
miri = −∇iV ({−→ri })
LagrangeHamilton
Ab Initio Molecular Dynamic – p.9/67
Equations of motion
Many formalism :NewtonLagrange
Lagrangian for a system withn degrees of freedom
L({−→ri } ,{−→
ri
}
) = K({−→ri } ,{−→
ri
}
) − V ({−→ri })
Equation of motion : Euler-Lagrange
d
dt
∂L
∂−→ri
−∂L
∂−→ri
= 0
Impulsion−→pi = ∂L
∂−→ri
, Force−→Fi = ∂L
∂−→ri
Newton is found again :−→pi =
−→Fi
HamiltonAb Initio Molecular Dynamic – p.9/67
Equations of motion
Many formalism :NewtonLagrangeHamilton
Impulsion−→pi used instead of velocity−→ri
Hamiltonian
H({−→ri } , {−→pi }) =3n∑
i=1
qipi − L
= K({−→ri } , {−→pi }) + V ({−→ri })
Ab Initio Molecular Dynamic – p.9/67
Equations of motion
Many formalism :NewtonLagrangeHamilton
Hamiltonian
H({−→ri } , {−→pi }) = K({−→ri } , {−→pi }) + V ({−→ri })
Equations of motion
ri =∂H
∂pipi = −
∂H
∂ri
Ab Initio Molecular Dynamic – p.9/67
Equations of motion
Many formalism :NewtonLagrangeHamilton
Equations of motion
qi =∂H
∂pipi = −
∂H
∂qi
H is constant along the trajectory, interpreted as thetotal energy
dH
dt=∑
i
[∂H
∂riri +
∂H
∂pipi
]
Ab Initio Molecular Dynamic – p.9/67
Equations of motion
Many formalism :NewtonLagrangeHamilton
Equations of motion
qi =∂H
∂pipi = −
∂H
∂qi
H is constant along the trajectory, interpreted as thetotal energy
dH
dt=∑
i
[∂H
∂ri
∂H
∂pi−
∂H
∂pi
∂H
∂qi
]
= 0
Ab Initio Molecular Dynamic – p.9/67
Equations of motion
Many formalism :NewtonLagrangeHamiltonConclusion : all equivalent !
Ab Initio Molecular Dynamic – p.9/67
Working ingredients
Initialization :{−→ri } , {−→pi } at t = 0{−→
r0i
}
: from cristallographic structure, PDB or by
similarity to avoid very repulsive part of thepotential.{−→
p0i
}
: most of the time 0, random or taken from
Maxwell-Bolztman distribution.How to compute V ?Time is discrete, how to choose time step∆t ?
As large as possibleSmaller than characteristic time of the system
Ab Initio Molecular Dynamic – p.10/67
Empirical potentials
Force fields≈ VSEPR extension :Geometries close to a reference :dCC ≈ 1, 54 Å, dCH ≈ 1, 09 Å, α ≈ 109◦4 . . .
Ethane :dCC = 1, 536 Å,
Propane :dCC = 1, 526 Å, α = 112, 4◦,
Butane :d(e)CC = 1, 533 Å, d
(i)CC = 1, 533 Å, α = 112, 8◦.
Energy close the reference energy !
E = E0 + corrections...
Ab Initio Molecular Dynamic – p.11/67
General form of a Force Field
Main contributions :
EMM = Vbonding + Vangle + Vtorsion + Vnb + . . .
Ab Initio Molecular Dynamic – p.12/67
General form of a Force Field
Main contributions :
EMM = Vbonding + Vangle + Vtorsion + Vnb + . . .
Vbonding Bonding energy
Vbonding =1
2
∑
i∈bondings
kr,i(ri − r0i )
2
Ab Initio Molecular Dynamic – p.12/67
General form of a Force Field
Main contributions :
EMM = Vbonding + Vangle + Vtorsion + Vnb + . . .
Vangle Valence angle (bending) modification
Vangle =1
2
∑
i,j∈bondings
kθ,ij(θij − θ0ij)
2
Ab Initio Molecular Dynamic – p.12/67
General form of a Force Field
Main contributions :
EMM = Vbonding + Vangle + Vtorsion + Vnb + . . .
Vtorsion Torsion energy (rocking)
Vtorsion =1
2
∑
i
[V1(1 + cos ϕi)+
V2(1 − cos 2ϕi) + V3(1 + cos 3ϕi)]
Ab Initio Molecular Dynamic – p.12/67
General form of a Force Field
Main contributions :
EMM = Vbonding + Vangle + Vtorsion + Vnb + . . .
Vtorsion Torsion energy
0
0.5
1
1.5
2
2.5
3
-120 0 120
Ene
rgie
ψ (angle HCCH)
Eclipsee
Decalee
E(ψ)
0
1
2
3
4
5
-120 0 120
Ene
rgie
ψ (angle CCCC)
Gauche
Syn
Anti
Butane : V1=1., V2=0., V3=1.
Ab Initio Molecular Dynamic – p.12/67
General form of a Force Field
Main contributions :
EMM = Vbonding + Vangle + Vtorsion + Vnb + . . .
Vnb Non Bonded interactionsVan Der Waals :
VV dW (d) = ǫ∑
i,j∈atomes
(
d0ij
dij
)12
− 2
(
d0ij
dij
)6
Ab Initio Molecular Dynamic – p.12/67
General form of a Force Field
Main contributions :
EMM = Vbonding + Vangle + Vtorsion + Vnb + . . .
Vnb Non Bonded interactionselectrostatic type :
Vel =∑
i,j∈atomes
qiqj
Ddij
H-bonds
Ab Initio Molecular Dynamic – p.12/67
Popular force fields
MM2, MM3 Mainly for organic chemistry.MMFF94 Only for organic chemistre !AMBER, CHARMm Good for biological systemsUFF Universal Force Field. Universaly avoided !ESFF Extensible and Systematic Force Field
Small benchmark :Ab initio MMFF94 MM3 CHARMm AMBER
∆E 0,4 0,4 0,7 0,8 1,1
∆dXH - 0,09 0,5 0,1 0,2
Ab Initio Molecular Dynamic – p.13/67
Propagator
A good propagator must :Be quick and allow for large∆tTime reversibleConserve mechanical energyCompute forces not too frequentlyConserve phase-space volume
Basic idea : Taylor expansion
−→ri (t + ∆t) = −→ri (t) + −→vi (t)∆t +−→ai (t)
2∆t2 + . . .
−→vi (t + ∆t) = −→vi (t) + −→ai (t)∆t +
−→vi (t)
2∆t2 + . . .
Ab Initio Molecular Dynamic – p.14/67
Verlet
General expression :
−→ri (t + ∆t) = −→ri (t) + −→vi (t)∆t +−→ai (t)
2∆t2 + . . .
−→ri (t − ∆t) = −→ri (t) −−→vi (t)∆t +
−→ai (t)
2∆t2 + . . .
thus−→ri (t + ∆t) = 2−→ri (t) −−→ri (t − ∆t) +
−→fi (t)
Mi
∆t2 + O(∆t3)
Store position and forces only
Accuracy in∆t3
Time reversible by construction
Velocities indirectly computed (finite differences) : lessaccurate−→vi = (−→ri (t + ∆t) −−→ri (t − ∆t)) /(2∆t) + O(∆t2)
Higer order term in the propagation formula⇒ numerical noiseAb Initio Molecular Dynamic – p.15/67
Leap-Frog variant
Designed to improve numerical accuracy
−→ri (t + ∆t) = −→ri (t) + −→vi (t +∆t
2)∆t
−→vi (t +∆t
2) = −→vi (t −
∆t
2) + −→ai (t)∆t
−→vi (t) =(−→vi (t + ∆t
2 ) −−→vi (t −∆t2 ))/2 + O(∆t2)
More stable
Ab Initio Molecular Dynamic – p.16/67
Velocity Verlet
Improve velocity accuracy
−→ri (t + ∆t) = −→ri (t) + −→vi (t)∆t +−→ai (t)
2∆t2
−→vi (t + ∆t) = −→vi (t) +−→ai (t) + −→ai (t + ∆t)
2∆t
Velocity accuracy inO(∆t3)Can be seen as a predictor-corrector
Ab Initio Molecular Dynamic – p.17/67
Verlet : Summary
Graphical form :
Velocity verlet often used because :
Simple and efficient, little force computation
Accurate in∆t3
Time reversible, Symplectic (conserve phase space volume)
For realistic time step, Velocity Verlet as good as Gear
predictor-corrector Ab Initio Molecular Dynamic – p.18/67
Constrained Dynamic
Why using constraints : to get larger∆t (frozend(CH)), prevent or force some evolutionPrincipal : Lagrange multiplierConstraint defined byσ ({−→ri }) = 0 , For example :
Distance :σ ({−→ri }) = |−→ri −−→rj | − d0
Difference of two distances :σ ({−→ri }) = |−→ri −
−→rj | − |−→rk −−→rl | − d0
Coordination number :σ ({−→ri }) = ni ({−→ri }) − n0
with ni ({−→ri }) =
∑
j 6=i S(|−→ri −−→rj |) et
S(r) = (1 + exp(κ(r − rc)))−1.
Ab Initio Molecular Dynamic – p.19/67
Constrained Dynamic
Constraint defined byσ ({−→ri }) = 0Extended Lagrangian :
L′({−→ri } , {−→pi }) = L({−→ri } , {−→pi }) −∑
α
λασα ({−→ri })
Equations of motion :
miri = −∂V
∂ri−∑
α
λα∂σα
∂ri
SHAKE : λα tq σα ({−→ri (t + ∆t)}) = 0RATTLE : λα tq σα ({−→ri (t + ∆t)}) = 0
Ab Initio Molecular Dynamic – p.19/67
Which ensembles ?
By default, isolated system⇒ E conserved, i.e. NVEInitialize {−→pi } so thatEtot = Etarget then propagateHow can we obtain other ensembles ?
NVT : E fluctuates
Q(N, V, T ) =1
N !h3N
Z
Πid−→ri d−→pi exp(−βH) = exp (−βF (N, V, T ))
NPT : E and V fluctuate
∆(N, P, T )1
V0N !h3N
Z
Πid−→ri d−→pi exp (−β(H + PV )) = exp (−βG(N, P, T ))
Ab Initio Molecular Dynamic – p.20/67
Canonical ensemble
We know tha⟨∑
kp2
k
2mk
⟩
= 32NkBT leading to
T (t) = 23NkB
∑
kpk(t)2
2mk
At equilibrium, Maxwell-Boltzmann distributionInitialize
Velocities from MB distributionOr
−→p0
i =−→0 , then heating using "inversed
annealing" :−→vi (t) → α−→vi (t), α > 1.Equilibrate : homogenizeT, to obtain〈T (t)〉 = T
Ab Initio Molecular Dynamic – p.21/67
Canonical ensemble
We know tha⟨∑
kp2
k
2mk
⟩
= 32NkBT leading to
T (t) = 23NkB
∑
kpk(t)2
2mk
At equilibrium, Maxwell-Boltzmann distributionInitializeEquilibrate : homogenizeT, to obtain〈T (t)〉 = T
Rescaling : when|T (t) − T | > ∆T , then−→vi (t) →
−→vi (t)√
TT (t) ⇒ Brutal !
Thermostat : coupling with a thermal bath
Ab Initio Molecular Dynamic – p.21/67
Nosé-Hoover
Extended System : a supplementary variable thatmimics the thermal bathEquations
Ab Initio Molecular Dynamic – p.22/67
Nosé-Hoover
Extended SystemEquations
−→ri =
−→pi
mi
−→pi =
−→Fi − ξ−→pi
ξ =1
Q
(∑
k
p2k
2mk
− gkBT
)
=gkB
Q(T − T )
with g number of degrees of freedom,Q mass of thethermostat
Ab Initio Molecular Dynamic – p.22/67
Nosé-Hoover
EquationsQ regulates the speed of exchanges between the bathand the system
Q too small : large not wanted oscillations, slowconvergenceQ too large : slow exchanges.Q → ∞,microcanonical ensemble !Optimum when resonating with the system, ieQ ≈ gkBT
ω2
Ab Initio Molecular Dynamic – p.22/67
Nosé-Hoover Chains
Nosé-Hoover not always ergodic, response time tooslow
Chain of M thermostats :
−→ri =
−→pi
mi
−→pi =
−→Fi − ξ1
−→pi
ξ1 =1
Q1
(∑
k
p2k
2mk
− gkBT
)
− ξ1ξ2
ξn =1
Qn
(Qn−1ξ
2n−1 − kBT
)− ξnξn+1
˙ξM =1
QM
(QM−1ξ
2n−1 − kBT
)
Ab Initio Molecular Dynamic – p.23/67
Test 1D oscillator
H = p2
2m+ mω2x2
2, f(p) =
√β
2πme−βp2/2m, f(x) =
√βmω2
2πe−βω2x2/2
1 thermostat 3 thermostats 4 thermostats
Ab Initio Molecular Dynamic – p.24/67
Andersen or Hybrid Monte Carlo
Mix DM and Monte Carlo to simulate chocs with thethermal bathAndersen
The velocity of a random particle is withdrawn fromthe MD distribution ;Ergodic by construction, converges to the NVTassociated to the MB distribution ;Many particles can be chosen at onceRe-attribution period not so importantRe-attribution does not kick wafefunction too farfrom BO surface
Hybrid Monte Carlo : all velocities are changed atonce
Ab Initio Molecular Dynamic – p.25/67
Langevin
Still stochastic, but white noise added directly in theequation of motion for all particles
q =pt
m
p = −∇V (q) −ξ
mp +
√
2ξ
βdW (t)/dt
with W (t) a gaussian white noisePhysically (mathematically ?) : the most efficient tothermalize, the fastest to convergebut very rare in chemistry, and badly behave in CPMD
Ab Initio Molecular Dynamic – p.26/67
False thermostat : Berendsen
Can be seen as a rescaling, with
α(t + ∆t) =
[
1 +∆t
τ
(T
T (t + ∆t)− 1
)]1/2
=
[
1 +∆t
τT (t + ∆t)(T − T (t + ∆t))
]1/2
T − T (t + ∆t) affectsα and notαVery used in biochemistrybutdo not sample the canonical ensemble !⇒ Use only for equilibration.
Ab Initio Molecular Dynamic – p.27/67
Isobaric-Isotherm ensemble
This is the closest to chemist experimentsVolumebecomes a dynamical variableNo stochastic approachesTwo families :
Isotropic : V change, but not the box shapeHoover, AnderssenAnisotropic :all parameters of the box can changeParrinello-Rahman, Martyna-Tobias-Klein
Ab Initio Molecular Dynamic – p.28/67
Isotropic NPT : Andersen
First approach (1980)Mimic the action of a piston on the box :
Mass QKinetic energyKV = 1
2QV 2
Potential energyVV = PV
Reduced variables :−→s = V −1/3−→r ,−→s = V −1/3−→v
Hamiltonian :HV = K + KV + V + VV
Ab Initio Molecular Dynamic – p.29/67
Isotropic NPT : Andersen
First approach (1980)Mimic the action of a piston on the box
Reduced variables :−→s = V −1/3−→r ,−→s = V −1/3−→v
Hamiltonian :HV = K + KV + V + VV
Equations of motion :
−→s =
−→f
mV 1/3−
2
3
−→s−→V
V
V = (P − P )/Q
with P instantaneous pressureP = 23V (K −W) et
W = −12
∑
i
∑
j>i−→rij
−→fij Internal Virial.
Ab Initio Molecular Dynamic – p.29/67
Isotropic NPT : Andersen
First approach (1980)Mimic the action of a piston on the box
Reduced variables :−→s = V −1/3−→r ,−→s = V −1/3−→v
Hamiltonian :HV = K + KV + V + VV
Equations of motion :
−→s =
−→f
mV 1/3−
2
3
−→s−→V
V
V = (P − P )/Q
In fact, sampling the isobaric-iso-enthalpic ensemble,not so common.
Ab Initio Molecular Dynamic – p.29/67
Isotropic NPT : Hoover
Extended system, similar to the Nosé-Hooverthermostat
−→r = V 1/3−→s
−→si =
−→pi
miV −1/3 −→
pi =−→Fi − (ξ + χ)−→pi
ξ =gkB
Q(T − T )
χ =V
3Vχ =
P(t) − P
τ 2PkBT
with g number of degrees of freedom,Q mass of thethermostat,τP relaxation time of the pressurefluctuations.
Ab Initio Molecular Dynamic – p.30/67
Anisotropic NPT
Parrinello-RahmanIdea : reduced variable−→ri = h
−→si , with
h = [−→a ,−→b ,−→c ] and−→a ,
−→b et−→c cell parameters
We introduceG = hth
P pressure tensor :
Pαβ =1
V
(∑
i
mi ˙riα ˙riβ +∑
i
∑
j
rijαfijβ
)
Hamiltonian
H = K +1
2Q∑
α
∑
β
h2αβ + V + PV
Ab Initio Molecular Dynamic – p.31/67
Anisotropic NPT
Parrinello-RahmanIdea : reduced variable−→ri = h
−→si
We introduceG = hth
Hamiltonian
H = K +1
2Q∑
α
∑
β
h2αβ + V + PV
Equations of motion
m−→s = h−1−→f − mG
−1G−→s
QH = (P − 1P )V(h−1)t
with Q mass of the box Ab Initio Molecular Dynamic – p.31/67
Equilibration
Listening phaseWe check that variables that should be conserved arestable, no drift :
TERMSD : Root Mean square displacementg(r) : pair distribution function
One simple criterion : averages should not depend onthe initial time.
Ab Initio Molecular Dynamic – p.32/67
Analysis
Production phaseCompute the interesting properties :
RMSDStructural parametersg(r)Fluctuations⇒ Cv
Vibrational spectraKinetic constant
Average on a given window, larger than fluctuationtime (see Practical session !)Thermostat, Barostat are perturbing the averages
Ab Initio Molecular Dynamic – p.33/67
Shall we go beyond classical MD
Pros of classical MD :FastGood geometries, vibrations, transport propertiesThermodynamical properties accurate forparameterized systems
ConsParameters!One need to know how atoms are linked :• Chemical reactions cannot be studied by standard
FF• ⇒ ReaxFF, ReBO, LOTF . . .No excited statesNo electronic properties : charge, density, . . .
Ab Initio Molecular Dynamic – p.34/67
Ab Initio Molecular Dynamics
Ab Initio Molecular Dynamic – p.35/67
Which ab initio MD ?
Electron explicitly taken into accountLet us denote by :
Mk, Zk,−→Rk the mass, the charge and the position of
a nucleus−→Rk ou−→vk the velocity of a nucleus−→pk = Mk
−→vk the impulsion of a nucleusme, ri the mass and the position of an electron
Ab Initio Molecular Dynamic – p.36/67
Which ab initio MD ?
Electron explicitly taken into accountWe would like to solve
i~∂φ({−→
Rk
}
, {−→ri } ; t)
∂t= H
({−→Rk
}
, {−→ri })
φ({−→
Rk
}
, {−→ri } ; t)
with
H = −
electrons∑
i
∆i
2︸ ︷︷ ︸
Te
−
elec.∑
i
nuclei∑
k
Zk
rik
+elec.∑
i
elec.∑
j>i
1
rij
+∑
k
∑
l>k
ZkZl
Rkl
︸ ︷︷ ︸
Vne
−
noy.∑
k
∆k
2Mk︸ ︷︷ ︸
TNAb Initio Molecular Dynamic – p.36/67
TDSCF approach
Hypothesis :
φ({−→
Rk
}
, {−→ri } ; t)
≈ Ψ ({−→ri } ; t) χ({−→
Rk
}
; t)
exp
[i
~
∫ t
t0
dt′Ee(t′)
]
with the convenient phase factor
Ee(t′) =
∫∏
i
d−→ri
∏
k
d−→Rk
Ψ∗ ({−→ri } ; t) χ∗
({−→Rk
}
; t)
HeΨ ({−→ri } ; t) χ({−→
Rk
}
; t)
Let us writed−→r =
∏
i d−→ri andd
−→R =
∏
k d−→Rk
Ab Initio Molecular Dynamic – p.37/67
TDSCF approach
Hypothesis :
φ({−→
Rk
}
, {−→ri } ; t)
≈ Ψ ({−→ri } ; t) χ({−→
Rk
}
; t)
exp
[i
~
∫ t
t0
dt′Ee(t′)
]
with
Ee(t′) =
∫
d−→r d−→R
Ψ∗ ({−→ri } ; t) χ∗
({−→Rk
}
; t)
HeΨ ({−→ri } ; t) χ({−→
Rk
}
; t)
Ab Initio Molecular Dynamic – p.37/67
TDSCF approach
Hypothesis :
φ({−→
Rk
}
, {−→ri } ; t)
≈ Ψ ({−→ri } ; t) χ({−→
Rk
}
; t)
exp
[i
~
∫ t
t0
dt′Ee(t′)
]
we get(project on< Ψ|, < χ|, used < H > /dt = 0) :
i~∂Ψ
∂t= −
∑
i
1
2∇2
i Ψ +
{∫
d−→Rχ∗
({−→Rk
}
; t)
Vneχ({−→
Rk
}
; t)}
Ψ
i~∂χ
∂t= −
∑
k
1
2Mk
∇2kχ +
{∫
d−→r Ψ∗ ({−→ri } ; t)HeΨ ({−→ri } ; t)
}
χ
Ab Initio Molecular Dynamic – p.37/67
TDSCF approach
Let us write (A,S : real)χ({−→
Rk
}
; t)
= A({−→
Rk
}
; t)
exp[
iS({−→
Rk
}
; t)
/~
]
we get, after separating real and imaginary part :
i~∂Ψ
∂t= −
∑
i
1
2∇2
i Ψ +
{∫
d−→Rχ∗
({−→Rk
}
; t)
Vneχ({−→
Rk
}
; t)}
Ψ
∂S
∂t= −
∑
k
1
2Mk
(∇kS)2 −
∫
d−→r Ψ∗HeΨ +∑
k
1
2Mk
∇2kA
A
∂A
∂t= −
∑
k
1
Mk
(∇kS) (∇kA) −∑
k
1
2Mk
A(∇2
kS)
Ab Initio Molecular Dynamic – p.37/67
Ehrenfest approach
Classical nuclei :~ = 0
∂S
∂t+∑
k
1
2Mk(∇kS)2 +
∫
d−→r Ψ∗HeΨ = 0
Hamilton-Jacobi equation analogy :
∂S
∂t+∑
k
1
2Mk(pk)
2 + Ve = 0
Using−→pk ≡ ∇kS
MkRk(t) = −∇k
∫
d−→r Ψ∗HeΨAb Initio Molecular Dynamic – p.38/67
Ehrenfest approach
Classical nuclei :
|χ({−→
Rk
}
; t)
|2 =∏
k
δ(−→Rk −
−→Rk(t)
)
lim~→0
∫
d−→Rχ∗
({−→Rk
}
; t)−→Rkχ
({−→Rk
}
; t)
=−→Rk(t)
That is
i~∂Ψ
∂t= −
∑
i
∇2i Ψ + VneΨ = HeΨ
Ab Initio Molecular Dynamic – p.39/67
Ehrenfest summary
Wavefunction explicitly propagated, coupled to thenuclei
MkRk(t) = −∇k
∫
d−→r Ψ∗HeΨ
i~∂Ψ
∂t= HeΨ
No electronic minimization, except fort = 0Transitions between electronic states are explicitlydescribed
∆t imposed by electrons dynamicsVery very smallEhrenfest seldom used
Ab Initio Molecular Dynamic – p.40/67
Born-Oppenheimer
Wavefunction is not propagatedTime independent Schrödinger equation is solved for
each{−→
Rk
}
:
MkRk(t) = −∇k minΨ0
〈Ψ0 |He|Ψ0〉
E0Ψ0 = HeΨ0
Electronic minimization at each stepNo more transitions between electronic states
∆t imposed by the nuclei⇒ relatively large
Ab Initio Molecular Dynamic – p.41/67
Born-Oppenheimer : HF, DFT
Simplifications using independent electron methods :HF ou DFT-KSMolecular Orbitals denoted by{ϕi}Energy minimization with orthogonal MO : :
Le = −〈Ψ0 |He|Ψ0〉 +∑
i,j
Λij (〈ϕi |ϕj〉 − δij)
leads to :Heffϕi =
∑
j
Λijϕj
Ab Initio Molecular Dynamic – p.42/67
Born-Oppenheimer : HF, DFT
Simplifications using independent electron methods :HF ou DFT-KSMolecular Orbitals denoted by{ϕi}Energy minimization with orthogonal MO :leads to :
Heffϕi =∑
j
Λijϕj
Diagonal form :Heffϕi = ǫiϕi
Ab Initio Molecular Dynamic – p.42/67
Born-Oppenheimer : HF, DFT
Simplifications using independent electron methods :HF ou DFT-KSMolecular Orbitals denoted by{ϕi}Energy minimization with orthogonal MO :leads to :
Heffϕi =∑
j
Λijϕj
So that :
MkRk(t) = −∇k minΨ0
〈Ψ0 |He|Ψ0〉
0 = −Heffϕi +∑
j
Λijϕj
Ab Initio Molecular Dynamic – p.42/67
Car-Parrinello (1/2)
Goal : benefit from all advantagesEhrenfest : No electronic minimizationBO : Large time step
Tool : Adiabatic separation between fast electrons andslow nucleiHow : MO described as classical variables
Ab Initio Molecular Dynamic – p.43/67
Car-Parrinello (1/2)
Goal : benefit from all advantagesTool : Adiabatic separation between fast electrons andslow nucleiHow : MO described as classical variables, butdecoupled from the nucleiUsing an extended Lagrangian formulation
Ab Initio Molecular Dynamic – p.43/67
Car-Parrinello (2/2)
Goal : benefit from all advantagesTool : Adiabatic separation between fast electrons andslow nucleiHow : MO described as classical variables, butdecoupled from the nuclei
LCP =∑
k
1
2Mk
−→Rk
2 +∑
i
1
2µi 〈ϕi | ϕi〉
︸ ︷︷ ︸
Kinetic energy
−〈Ψ0 |He|Ψ0〉︸ ︷︷ ︸
Potential energy
+ constraints︸ ︷︷ ︸
orthogonality, geometric...
µ "Fictitious" mass of the electronsAb Initio Molecular Dynamic – p.44/67
Car-Parrinello (2/2)
Goal : benefit from all advantagesTool : Adiabatic separation between fast electrons andslow nucleiConstant of motion
HCP =∑
k
1
2Mk
−→Rk
2 +∑
i
1
2µi 〈ϕi | ϕi〉 + Eel
(
{ϕi} ,{−→
Rk
})
Ab Initio Molecular Dynamic – p.44/67
Car-Parrinello (2/2)
Goal : benefit from all advantagesTool : Adiabatic separation between fast electrons andslow nucleiEquations of motion
MkRk(t) = −∇k 〈Ψ0 |He|Ψ0〉 + ∇k {constraints}
µϕi(t) =δ
δϕ∗i
〈Ψ0 |He|Ψ0〉 +δ
δϕ∗i
{constraints}
∆t approximately 5 to 10 times smaller than BO
Ab Initio Molecular Dynamic – p.44/67
Car-Parrinello : HF,DFT-KS
We use HF or DFT-KS methods
LCP =∑
k
1
2Mk
−→Rk
2 +∑
i
1
2µi 〈ϕi | ϕi〉 −
⟨Ψ0
∣∣Heff |Ψ0
⟩
+∑
i,j
Λij (〈ϕi |ϕj〉 − δij)
Ab Initio Molecular Dynamic – p.45/67
Car-Parrinello : HF,DFT-KS
We use HF or DFT-KS methodsEquations of motion become
MkRk(t) = −∇k
⟨Ψ0
∣∣Heff |Ψ0
⟩
µϕi(t) = −Heffϕi +∑
j
Λijϕj
Very similar to BO :µϕi(t) = 0
Ab Initio Molecular Dynamic – p.45/67
Why does it work ?
Fast electrons, slow nucleiElectronic frequencies forSi8 :
f e(ω) =
∫ ∞
t=0
cos(ωt)∑
i
⟨
Ψ(t)∣∣∣ Ψ(0)
⟩
dt
Triangle : latest nuclear frequencyAb Initio Molecular Dynamic – p.46/67
Why does it work ?
Electronic Oscillations close to the BO surface
Econs =∑
k
1
2Mk
−→Rk
2 +∑
i
1
2µi 〈ϕi | ϕi〉 + 〈Ψ |He|Ψ〉
Ephys =∑
k
1
2Mk
−→Rk
2 + 〈Ψ |He|Ψ〉 = Econs − Te
Model System : Si FCC, 2 atoms/cell
Ab Initio Molecular Dynamic – p.47/67
Why does it work ?
Electronic Oscillations close to the BO surfaceModel System : Si FCC, 2 atoms/cell
Small oscillations, stable in time
Ab Initio Molecular Dynamic – p.47/67
Why does it work ?
Electronic Oscillations close to the BO surfaceModel System : Si FCC, 2 atoms/cell
Small oscillations, stable in time
Ab Initio Molecular Dynamic – p.47/67
Why does it work ?
Forces oscillations very smallModel System : Si FCC, 2 atoms/cell
Small oscillations, stable in timeOscillations averaged to 0
Ab Initio Molecular Dynamic – p.48/67
Why does it work ?
Forces oscillations very smallModel System : Si FCC, 2 atoms/cell
Small oscillations, stable in timeOscillations averaged to 0
Ab Initio Molecular Dynamic – p.48/67
Working Condition CP : adiabaticity
Fast electrons, small nuclei :µ ≪ Mk
ϕi ≈ 0 : wavefunction close to BO, always slightlyabove.Compromise forµ :
∆t large, but we want electron/nuclei separationElectronic frequencyϕi occ. andϕa virtual :ωe
ia =√
2(ǫa − ǫi)/µ
Smallest :ωemin ∝
√
EGAP/µ
Highest :ωemax ∝
√
Ecut/µ
so that∆tmax ∝√
µ/Ecut
Usually :µ = 400-500,∆t = 5-10 au = 0.12-0.24 fs.
Ab Initio Molecular Dynamic – p.49/67
Loosing adiabaticity
If EGAP ≈ 0, thenωemin too close toωnucl
For example for an elongated bond, exSn2
Ab Initio Molecular Dynamic – p.50/67
Loosing adiabaticity
If EGAP ≈ 0, thenωemin too close toωnucl
For example for an elongated bond, exSn2
For metallic systems !Two solutions :
Back to BOElectronic Thermostat
Ab Initio Molecular Dynamic – p.50/67
Comparison BO/CP
BOMD CPMDAlways on BO surface Always slightly above∆t ∼ τnucl ∆t ≪ τnucl
∆tBO ≈ 5 − 10∆tCP
Minimization at each step Only OrthogonalizationProblem when deviatingfrom the BO surface
Stable with respect to devi-ations
Works forEGAP = 0 Electronic thermostatUsed for solid systems Used for liquids
Ab Initio Molecular Dynamic – p.51/67
Comparison BO/CP
Ab Initio Molecular Dynamic – p.52/67
Comparison Ehrenfest/CP
Ehrenfest MD CPMDReal separation (quantum)Fictitious separation (clas-
sical)∆t ∼ τelec ∆t ≫ τelec
∆tCP ≈ 5 − 10∆tEhrenfest
Rigorous Orthonormality Imposed by constraintsDeviations from BO add upStable
Ab Initio Molecular Dynamic – p.53/67
Conclusion and perspectives
AIMD : includes electronic effects, allows forchemical reactions, catalytic processes . . .CPMD : Very used, very fashion !but . . .
Number of atoms still limitedDFT not always sufficient
PerspectivesHybrid methods : QM/MM, QM/QM’Post-HF and TD approachs : excited states, highlycorrelated materials . . .BO ? New algorithms faster, more efficient,becomes competitive with CPMD.
Ab Initio Molecular Dynamic – p.54/67
Application : Free energycalculations
Ab Initio Molecular Dynamic – p.55/67
Free energy profile
Reaction coordinateq =
q({−→
Rk
})
Probability density
P(z) =1
QNV T
∫
d−→Rd
−→P exp(−βH)δ
(
q({−→
Rk
})
− z)
Free energy profile
F (z) = −kBT lnP(z)
Reaction rate :
k =kBT
he−
∆rG‡
RT
Ab Initio Molecular Dynamic – p.56/67
Rare event ?
System position forEa = 2 × kBT
⇒ Diffusive system.
Ab Initio Molecular Dynamic – p.57/67
Rare event ?
System position forEa = 5 × kBT
⇒ Nothing at the TS !
Ab Initio Molecular Dynamic – p.57/67
Rare event ?
System position forEa ≫ kBT
Alternating between the two states.Long residence time, short transition time.kTST ≈ 1s−1
⇒ Slow reaction≡ rare event.
Ab Initio Molecular Dynamic – p.57/67
Why is it so difficult ?
Standard MD is a "real time" method,∆t ≈ 0.1-1 fsChemical reaction time : fast around ns, typical aroundµs-ms, biology can be seconds⇒ Two incompatible time scalesMore, phase space dimension is 6N, impossible tofully sample
Ab Initio Molecular Dynamic – p.58/67
Why is it so difficult ?
⇒ Two incompatible time scalesSolutions
Work at higher T : faster sampling but might be notthe same phenomenaForce the reaction to occur :• Biais potential• Constrained dynamic• Adiabatic dynamic• Metadynamic
In all cases, we calculate∆F , notF
Ab Initio Molecular Dynamic – p.58/67
Bias potential
Many approaches : Umbrella sampling, adaptive biaispotential, accelerated dynamics, flooding potential...Main Idea : we add a potential to "erase" the barrier
V ′
({−→Rk
})
= V({−→
Rk
})
+ ∆V(−→q({−→
Rk
}))
Then get the non-biaised energy :
⟨
O({−→
Rk
})⟩
=
⟨
O({−→
Rk
})
exp (β∆V)⟩′
〈exp (β∆V)〉′
Problem : we do not know the barrier position, heightand shape !In practice : many simulations with a moving modelpotential Ab Initio Molecular Dynamic – p.59/67
Umbrella Sampling
Usual bias :∆V(
q({−→
Rk
}))
∝[
q({−→
Rk
})
− z]2
Simulations for different−→z
Ab Initio Molecular Dynamic – p.60/67
Constrained dynamics
Two families :q evolves continuously fromz0 to zf during thesimulation• Slow growth method : slow evolution, quasi-static• Fast growth method,Jarzinsky: fast evolution,
average taken on many trajectories
〈exp (−βW )〉 = exp (−βF )
Many simulation withq set to different values ofz.
Ab Initio Molecular Dynamic – p.61/67
Thermodynamic integration
Change in free energy calculated by integrating freeenergy derivative :
∆F (z = z0 → z = zf ) =
∫ zf
z0
∂F
∂qdq
Numerical evaluation using discrete valuesA constrained simulation is launched for each value ofq(z).How to obtain∂F
∂q (z) ?
Ab Initio Molecular Dynamic – p.62/67
Free energy derivatives
Two problems :q = z ⇒ q = 0 : no sampling for the impulsionpq.Blue-Moon :
〈O(z)〉 =
⟨Z−1/2O
⟩
cont
〈Z−1/2〉cont
with Z the reduce mass associated toq :
Z =∑
k
1
Mk
∂q
∂Rk
∂q
∂Rk
q is non linear with respect to{−→
Rk
}
, Jacobian matrix
J non unityAb Initio Molecular Dynamic – p.63/67
Free energy derivatives
Two problems :q = z ⇒ q = 0 : no sampling for the impulsionpq.
q is non linear with respect to{−→
Rk
}
, Jacobian matrix
J non unity
∂F
∂q(z) =
⟨∂V
∂q+ kBT
∂ ln |J|
∂q
⟩
q=z
Ab Initio Molecular Dynamic – p.63/67
Free energy derivatives
Two problems :q = z ⇒ q = 0 : no sampling for the impulsionpq.
q is non linear with respect to{−→
Rk
}
, Jacobian matrix
J non unity
∂F
∂q(z) =
⟨Z−1/2 [−λ + kBTG]
⟩
q=z⟨Z−1/2
⟩
q=z
with G = 1Z2
∑
k,l1
MkMl
∂q∂Rk
∂2q∂RkRl
∂q∂Rl
,λ Lagrange multiplier associated to the constraint
σ({−→
Rk
})
= q − z = 0 (SHAKE)
Ab Initio Molecular Dynamic – p.63/67
Adiabatic dynamics
Idea : Decoupling motions alongq from other motionsUse different temperatures
One (chain) thermostat associated toqOne (chain) thermostat associated to the N-1 otherdegrees of freedomTq ≫ Tnuc
Change the mass of atoms concerned byq
Ab Initio Molecular Dynamic – p.64/67
Metadynamic
Idea : Similar to the adiabatic dynamics, usingextended Lagrangian !Reaction coordinates are included as collectivevariables (CV)sα
� Coarse� MD for CV
L = L0 +∑
α
1
2Mαsα
2 −∑
α
1
2kα [Sα(r) − sα]2
Mα large enough⇒ natural adiabatic separation
Ab Initio Molecular Dynamic – p.65/67
Metadynamic
Idea : Similar to the adiabatic dynamics, usingextended Lagrangian !Reaction coordinates are included as collectivevariables (CV)sα
� Coarse� MD for CVon top of this, potential wells are filled with gaussians
∆V (t) =∑
i
W exp
(
−|s − si(t)|
2
2δσ2
)
Important choices for W,σ, time interval betweenadding two gaussians
Ab Initio Molecular Dynamic – p.65/67
Metadynamic
Idea : Similar to the adiabatic dynamics, usingextended Lagrangian !Reaction coordinates are included as collectivevariables (CV)sα
� Coarse� MD for CV
Ab Initio Molecular Dynamic – p.65/67
Metadynamic
Idea : Similar to the adiabatic dynamics, usingextended Lagrangian !Reaction coordinates are included as collectivevariables (CV)sα
� Coarse� MD for CV
Ab Initio Molecular Dynamic – p.65/67
Metadynamic
Idea : Similar to the adiabatic dynamics, usingextended Lagrangian !Reaction coordinates are included as collectivevariables (CV)sα
� Coarse� MD for CV
Ab Initio Molecular Dynamic – p.65/67
Metadynamic
Idea : Similar to the adiabatic dynamics, usingextended Lagrangian !Reaction coordinates are included as collectivevariables (CV)sα
� Coarse� MD for CV
Ab Initio Molecular Dynamic – p.65/67
Metadynamic
Idea : Similar to the adiabatic dynamics, usingextended Lagrangian !Reaction coordinates are included as collectivevariables (CV)sα
� Coarse� MD for CV
Ab Initio Molecular Dynamic – p.65/67
How to choose a good RC ?
In practice, difficult to sam-
ple a coordinate space larger
than 3D
Chemist intuition might
fail ! !
Ab Initio Molecular Dynamic – p.66/67
Conclusion
Already many toolsbut choosing RC is still a hot topic for large systemsClassical nuclei : what about quantum effects ?
Correctionsa posteriori : tunneling, isotopic effectsPath IntegralZPE correction
Ab Initio Molecular Dynamic – p.67/67