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Ab InitioAb Initio Molecular Dynamics with a Molecular Dynamics with a
Continuum Solvation ModelContinuum Solvation Model
Ab InitioAb Initio Molecular Dynamics with a Molecular Dynamics with a
Continuum Solvation ModelContinuum Solvation Model
Hans Martin Senn, Tom Ziegler
University of Calgary
CSC 2002 Conference, Vancouver, BC, 2 June 2002
OutlineOutline
Background Car–Parrinello ab initio molecular dynamics The Projector-Augmented Wave (PAW) method Continuum solvation / The Conductor-like Screening Model (COSMO)
Continuum solvation within ab initio MD Surface with smooth analytic derivatives Surface charges as dynamic variables
Tests and first applications Hydration energies Conformers of glycine
CO insertion into Ir–CH3
Ab initio molecular dynamics
Ab initio molecular dynamicsForces on nuclei derived from
instantaneous electronic structure
Ab initio molecular dynamicsForces on nuclei derived from
instantaneous electronic structure
Ab InitioAb Initio Molecular Dynamics Molecular DynamicsR. Car, M. Parrinello, Phys. Rev. Lett. 1985, 55,
2471Nuclei
Classical mechanics
ElectronsDensity-functional
theory
€
MJ ˙ ̇ R J = −∂E {RI},{c i}[ ]
∂RJ
mc ˙ ̇ c j = −dE {RI},{c i}[ ]
d c j+ c i L iji∑
Car–Parrinello scheme for ab initio MD Fictitious dynamics of the wavefunctions Simultaneous treatment of atomic and electronic structure
Friction dynamics for minimizations
€
+ =T −V = 12
M J˙ R J
2J∑
nuclear kin. energy1 2 4 3 4
+ mc˙ c i
˙ c ii∑fictitious kin. energyof the wavefunctions
1 2 4 4 3 4 4 − E {RI },{c i}[ ]
DFT total energy1 2 4 4 3 4 4
+ c i c j −dij( )L jii, j∑orthonormality constraint
1 2 4 4 4 4 3 4 4 4 4
Basis set: Augmented plane waves Spatial decomposition of the wavefunction according to its shape
Smooth plane-wave expansion far from nuclei
Rapidly oscillating augmentation functions around the nuclei
Integrations for total energy and matrix elements decompose accordingly Smooth parts in Fourier representation
One-centre parts on radial grids in spherical harmonics expansions
All-electron method Frozen-core approximation
c = ˜ c + f k ˜ p k ˜ c −
k∑ ˜ f k ˜ p k ˜ c k∑
The Projector-Augmented Wave (PAW) MethodThe Projector-Augmented Wave (PAW) MethodP.E. Blöchl, Phys. Rev. B 1994, 50,
17953
€
˜ c (r) = c (G)eiG rG∑ Plane-wave expansion
f k (r) = x k Yl ,m Numerical, atomic-orbital-like functions
˜ f k (r) Smoothened atomic orbitals
˜ p k (r) Projector functions
Continuum Solvation ModelsContinuum Solvation Models
Implicit electrostatic solvation “Solvent” = homogeneous, isotropic dielectric medium
Fully characterized by relative permittivity (r)
Electrostatic interactions
1. Solute polarizes the dielectric ( reaction field) Epol > 0
2. Solute charge density interacts with polarized medium Eint < 0
3. Solute electronic and atomic structure adapt (“back-polarization”) ∆Een > 0
4. Repeat to self-consistency!
€
Gdiel = Eint + Epol = 12 Eint
€
Gdiel = Eint + Epol = 12 Eint
Electrostatic free energy For a linear dielectric:
The Solvation EnergyThe Solvation Energy
Non-electrostatic contributions Cavity formation Dispersion and exchange-repulsion Approximately proportional to cavity surface area
€
Gsol = Eensol +Gdiel + Gnp
ΔsolvG = Gsol − Eeng = Gdiel +Gnp + DEen
€
Gsol = Eensol +Gdiel + Gnp
ΔsolvG = Gsol − Eeng = Gdiel +Gnp + DEen
€
Gnp = b +gA
€
Gnp = b +gA
Free energy of solvation
Neglect changes in thermal motion upon solvation
The Conductor-Like Screening Model (COSMO)The Conductor-Like Screening Model (COSMO)
Main approximations in COSMO
1. Dielectric with infinite permittivity (r ∞)
No volume polarization; polarization expressed as surface charge density only
Vanishing potential at the cavity boundary
€
Gdiel∞ s[ ] = d3r d2s
r (r)s (s)r − sS∫V∫ +
12
d2s d2 ′ s s (s)s ( ′ s )
s − ′ s S∫S∫
€
Gdiel∞ s[ ] = d3r d2s
r (r)s (s)r − sS∫V∫ +
12
d2s d2 ′ s s (s)s ( ′ s )
s − ′ s S∫S∫
€
f ∞(s) =dGdiel
∞
dss
= d3rr (r)r − sV∫ + d2 ′ s
s ( ′ s )s − ′ s S∫ = 0
€
f ∞(s) =dGdiel
∞
dss
= d3rr (r)r − sV∫ + d2 ′ s
s ( ′ s )s − ′ s S∫ = 0
€
GdielCOSMO = fGdiel
∞ s[ ]
s (s) = fs (s)
€
GdielCOSMO = fGdiel
∞ s[ ]
s (s) = fs (s)
€
GdielCOSMO {qi}( ) = qi d3r
r (r)r − siV∫
i∑ +
12 f
qiq j
si − s ji, j( j≠i )
∑ +1.07 4p
2 fqi
2
aii∑
€
GdielCOSMO {qi}( ) = qi d3r
r (r)r − siV∫
i∑ +
12 f
qiq j
si − s ji, j( j≠i )
∑ +1.07 4p
2 fqi
2
aii∑
2. Recover true dielectric behaviour by scaling Derived for rigid multipoles in spherical cavity
f = f(r) = (r – 1) / (r + x) [x = 0, 1/2 for monopole, dipole]
3. Discretize cavity surface and surface charge Surface segments carrying point charges qi
Energy expression is variational in qi
Surface Charges as Dynamic VariablesSurface Charges as Dynamic Variables Extended Lagrangean including solvation
Equations of motion for charges
€
+ =+ CP + mq ˙ q i2
i∑ − Gdiel
COSMO {RJ},{q j}( ) −Gnp {RJ}( )
€
+ =+ CP + mq ˙ q i2
i∑ − Gdiel
COSMO {RJ},{q j}( ) −Gnp {RJ}( )
€
mq˙ ̇ q i =∂ +∂qi
−mq ˙ q zq
€
mq˙ ̇ q i =∂ +∂qi
−mq ˙ q zq
Additional forces acting on the nuclei Surface segments and charges implicitly depend on atomic positions via the surface
construction
Forces obtained as analytic derivatives
Energy-conserving dynamics requires smooth derivatives Segments/charges must not vanish or be created during the simulation
Each segment/charge must be unambiguously assigned to an atom
Conventional schemes for building the cavity are not compatible with this!
Surface ConstructionSurface Construction
A surface construction compatible with AIMD
1. Surface = union of atomic spheres
2. Each sphere is triangulated
3. All segments and charges are retained
Each segment/charge moves rigidly along with its atom
€
GdielCOSMO = qiQ i d3r
r (r)r − siV∫
i∑ +
12 f
qiQ iq jQ j
si − sji, j( j≠i)
∑ +1.07 p
fqi
2Q i2
aii∑
4. A switching function effectively removes charges lying within another
sphere from the energy expression
i = 1 if the charge is exposed on the molecular surface
i = 0 if the charge is buried
Smooth transition between “on” and “off”
Total switching function
Charge is smoothly switched off if it lies within any one of the other spheres
Switching FunctionsSwitching Functions
Building the switching function Rectangular pulse
Centred at d0 = Rsolv – c
Vanishes smoothly at d0
€
h(d, d0) =1 − exp − d −d0( )
2n ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥, n∈N
€
h(d,d0) =1 − exp − d −d0( )
2n ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥, n∈N
n = 1n = 10n = 24
d0 Rsolv
€
uJ,i =h RJ − si ,dJ
0( ) for RJ − si > dJ
0
0 for RJ − si ≤ dJ0
⎧ ⎨ ⎪
⎩ ⎪
€
uJ,i =h RJ − si ,dJ
0( ) for RJ − si > dJ
0
0 for RJ − si ≤ dJ0
⎧ ⎨ ⎪
⎩ ⎪
d
€
Qi = uJ ,iJ
(J ≠I(i))
∏
€
Qi = uJ ,iJ
(J ≠I(i))
∏
Atomic switching function
RAsolv
RBsolv
Rounding the Edges…Rounding the Edges…
Behaviour of hidden charges Magnitude of switched-off charges physically undetermined
Problematic if charge becomes again exposed
Restoring potential Added to the Lagrangean
Artificial self-interaction, acting only on hidden charges
€
Erest = k(1−Qi )qi2
i∑
€
Erest = k(1−Qi )qi2
i∑
pot.
|si – sj|
Rijc
Modified Coulomb potential at short range Charges can collide at seams
Modified Coulomb potential True 1/r behaviour replaced by linear continuation
for |si – sj| < Rijc
Rijc derived from average of self-interaction potentials
Representation of the Solute DensityRepresentation of the Solute Density
Decoupling of periodic images Plane-wave-based methods always create periodic images
Artificial in molecular calculations
Electrostatic decoupling scheme in PAW Model density
Superposition of atom-centred Gaussians
Coefficients determined by fit to true density
Reproduces long-range behaviour
Surface charges couple to model density Not instrumental Computationally convenient
3–5 Gaussians per atom
Fitting procedure relays forces acting on the wave functions due to the charges
€
ˆ r (r) = QI ,mgI ,m (r)I ,m∑
€
ˆ r (r) = QI ,mgI ,m (r)I ,m∑
€
Es–m =qiQ iQI ,m
RI − sierf RI − si rI ,m
c[ ]
i,I ,m∑
€
Es–m =qiQ iQI ,m
RI − sierf RI − si rI ,m
c[ ]
i,I ,m∑
Parameterization Free energies of hydration
Parameters: Solvation radii RIsolv, non-polar parameters ,
Fit set: Neutral organic molecules containing C, H, O, N
Mean unsigned error (MUE) over whole set: 4.1 kJ/mol On par with similar DFT/COSMO results
5.2 kJ/mol4.2 kJ/molMUE: 3.2 kJ/mol
Parameterization and ValidationParameterization and Validation
CH4
H
H
OH
OH
OH
O
O
H
O
O
OH
O
O
OH
OH
OH O
H
NH3
H2N
NH N
N
NH2
NO2
NH
O
NH2
ONH
O
NH
O
Conformers of GlycineConformers of Glycine
H-bonding patterns
Relative stabilities (kJ/mol)
Hydration energies (kJ/mol)
A
BC Z
Gas phase Aqueous phaseMethod C A B Method Z A BPAW 0 4.1 9.6 PAW/COSMO 0 49 55Other pure DFT 0 1.1…4.6 6.8 Other pure DFT 0 36…39Hybrid DFT 0 –1.6…–1.1 4.7…5.6 Hybrid DFT 0 13Expt. 0 –5.9 Expt. 30…32
Method ∆hydGA(g) → A(a )q A(g) → Z(a )q
PAW/COSMO –57 –105Ot her pureDFT –54 –91Hybri d DFT –49…–42 –61…–57Expt. –80
Zwitterion in solution Most stable conformer
Not present in the gas phase
Energy ConservationEnergy Conservation
Conservation of total energy Glycine without and with continuum solvation
∆t = 7 a.u., T ≈ 300 K (no thermostats, no friction)
Drift in total energy: solvation off +2.78 5 Eh/ps per atom
solvation on 1.43 5 Eh/ps per atom
Energy is conserved
A First ApplicationA First Application
Carbonylation of methanol “Monsanto” / ”Cativa” acetic-acid process
M = Rh, Ir
Via [MI3(CH3)(CO)2]– [MI3(COCH3)(CO)]–
AIMD simulation of CO insertion step Solvent: iodomethane; T = 300 K
Slow-growth thermodynamic integration along d(CMe–CCO)
∆A‡ = 111 (128) kJ/mol∆E‡ = 126 (155) kJ/mol; ∆S‡ = 60 (91) J/(K mol)
Dissociation of I– trans to incipient acyl! Only in solution at finite temperature
Enthalpy of Ir–I bond traded for entropy of free (solvated) I–
MeOHCO
MeCOOHcat. [MI2(CO)2]
–, HI, H2OMeOH
COMeCOOH
cat. [MI2(CO)2]–, HI, H2O
RC
SummarySummary
COSMO continuum solvation in AIMD
Surface charges as dynamic variables
Surface construction
Switching function smoothly disables unexposed charges
Smooth analytic derivatives wrt. atomic positions
Energy-conserving
Solvation energies
On par with other DFT/COSMO implementations
Modelling of finite-temperature and solvation effects
AcknowledgmentsAcknowledgments
Peter E. Blöchl,Institute of Theoretical Physics, Clausthal University of Technology, Germany
Peter M. Margl,Corporate R&D Computing, Modeling and Information Sciences, Dow
Rochus Schmid,Institute of Inorganic Chemistry, Munich University of Technology, Germany
$ Swiss National Science Foundation
$ NSERC
$ Petroleum Research Fund