rosa ramirez ( université d’evry ) shuangliang zhao ( ens paris)
DESCRIPTION
Classical Density Functional Theory of Solvation in Molecular Solvents. Daniel Borgis Département de Chimie Ecole Normale Supérieure de Paris [email protected]. Rosa Ramirez ( Université d’Evry ) Shuangliang Zhao ( ENS Paris). - PowerPoint PPT PresentationTRANSCRIPT
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• Rosa Ramirez (Université d’Evry)• Shuangliang Zhao (ENS Paris)
Classical Density Functional Theory of Solvation in Molecular Solvents
Daniel Borgis Département de Chimie
Ecole Normale Supérieure de [email protected]
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Solvation: Some issues
For a given molecule in a given solvent, can we predict efficiently and with « chemical accuracy:
• The solvation free energy• The microscopic solvation profile
A few applications:• Differential solvation (liquid-liquid extraction)• Solubility prediction• Reactivity• Biomolecular solvation, ….
Explicit solvent/FEP
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Solvation: Implicit solvent methods
Dielectric continuum approximation (Poisson-Boltzmann)
rrr 04 r
i
80
Biomolecular modelling: PB-SA method
AdF rrr 021
Solvent Accessible Surface Area (SASA)
electrostatics + non-polar
Quantum chemistry: PCM method
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Improved implicit solvent models
• Integral equations
• Interaction site picture (RISM) (D. Chandler, P. Rossky, M. Pettit, F. Hirata, A. Kovalenko)
• Molecular picture (G. Patey, P. Fries, …)
• Classical Density Functional Theory
This work: Can we use classical DFT to define an improved and well-founded implicit solvation approach?
(based on « modern » liquid state theory)
)(, rcrh ijijSite-site OZ + closure
Molecular OZ + closure 21122112 ,,,,, ΩΩrΩΩr ch
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)'()'()('21)()(
1)(4
21)( 0
2 rPrrTrPrrrErPrPr
rrP
dddrdF
Fpol
entropy
Fexc
Solvent-solvent
Fext
P(r)
ir
]);([)(0 iielec FVU rrPr
DFT formulation of electrostatics
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Dielectric Continuum Molecular Dynamics
M. Marchi, DB, et al., J. Chem Phys. (2001), Comp. Phys. Comm. (2003)
Use analogy with electronic DFT calculations and CPMD method
k
rkkPrP )exp()()( i
ii
ii
P
VFdtdm
Fdt
dM
rrr
kPkP
02
2
2
2
)()(
On-the-fly minimization with extended Lagrangian
Plane wave expansion
Soft « pseudo-potentials »
)(1111 rr
Hsis
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Dielectric Continuum Molecular Dynamics
-helix horse-shoe
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Dielectric Continuum Molecular Dynamics
Energy conservation Adiabaticity
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Beyond continuum electrostatics: Classical DFT of solvation
densitysolventnorientatioposition/, Ωr
In the grand canonical ensemble, the grandpotential can be written as a functional of (r
NVddFST cextexc ΩrΩrΩr ,,
0
Ωr,0
Ωr,0Functional minimization:
Thermodynamic equilibriumD. Mermin (« Thermal properties of the inhomogeneous electron gas », Phys. Rev., 137 (1965))
Intrinsic to a given solvent
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In analogy to electronic DFT, how to use classical DFT as a « theoretical chemist »tool to compute the solvation properties of molecules, in particular their solvationfree-energy ?
0 Ωr,
0 F
energy freeSolvation min F
0, c ),(, Ωrextc V
But what is the functional ??
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The exact functional
extexcid FFFF x
01
0
111 ln
xxxx
dTkF Bid
111 xxx extext VdF
,, 121121 xxxxxx CddTkF Bexc 0 xx
;,1, 21)2(1
021 xxxx cdC xx 0
),( Ωrx
),( Ωr
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The homogeneous reference fluid approximation
Neglect the dependence of c(2)(x1,x2,[]) on the parameter , i.e use direct correlation function of the homogeneous system
21021)2(
21)2( ,;,;, xxxxxx ccc
c(x1,x2) connected to the pair correlation function h(x1,x2) through the Ornstein-Zernike relation
2331302121 ,,,, xxxxxxxxx hcdch
1,, 2121 xxxx ghg(r)
h(r)
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The homogeneous reference fluid approximation
Neglect the dependence of c(2)(x1,x2,[]) on the parameter , i.e use direct correlation function of the homogeneous system
21021)2(
21)2( ,;,;, xxxxxx ccc
c(x1,x2) connected to the pair correlation function h(x1,x2) through the Ornstein-Zernike relation
2332311333021122112 ,,),,(,,,, ΩΩrΩΩrΩrΩΩrΩΩr hcddch
1,, 2121 xxxx ghg(r)
h(r)
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),,(h 2112 ΩΩr
),,(c 2112 ΩΩr
The picture
Functional minimization
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Rotational invariants expansion
),,ˆ(),,( 2112122112 ΩΩrΩΩr lmnlmn rhh
),,ˆ(),,( 2112122112 ΩΩrΩΩr lmnlmn rcc
1Ω
2Ω
12r
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21121121112
21110000 ))((3,,1 ΩΩrΩrΩΩΩ
The case of dipolar solvents
The Stockmayer solvent
1Ω
2Ω
12r
11212
11211012
11000012
0002112 )()()(),,( rcrcrcc ΩΩr
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Particle density Polarization density
ΩrΩr , dn ΩrΩΩrP ,0 d
Ωr,F rPr ,nF
densitysolventnorientatioposition/, Ωr
R. Ramirez et al, Phys. Rev E, 66, 2002 J. Phys. Chem. B 114, 2005
A generic functional for dipolar solvents
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A generic functional for dipolar solvents
PPPP ,,,, nFnFnFnF excextid
010
111 )(
)(ln)(, nn
nn
ndTknF Bid
r
rrrP
)(/)(L)(
)(/)(Lsinh)(/)(L
ln 01
01
01
rrrrr
rrr nPP
nPnP
dTkB
)()( rPr P
L(x)LangevindefonctionladeInverse)(L 1 x
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A generic functional for dipolar solvents
PPPP ,,,, nFnFnFnF excextid
010
111 )(
)(ln)(, nn
nn
ndTknF Bid
r
rrrP
)(2)( 2
rrPrn
dTkd
B
litypolarizabinalorientatiolocal3
2
TkB
d
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A generic functional for dipolar solvents
PPPP ,,,, nFnFnFnF excextid
)()()()(, rPrrrrrP qLJext EdnVdnF
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A generic functional for dipolar solvents
PPPP ,,,, nFnFnFnF excextid
)()()(2
, 212000
121 rrrrP nrcnddTknF Bexc
)()()()(3)(2 2112212112
11221
rPrPrrPrrPrr rcddTkB
)()()(2 2112
11021 rPrPrr rcddTkB
Connection to electrostatics: R. Ramirez et al, JPC B 114, 2005
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)(
)(
)(
12112
12110
12000
rh
rh
rh
The picture
Functional minimization
)(
)(
)(
12112
12110
12000
rc
rc
rc
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O-Z
h-functions c-functions
Step 1: Extracting the c-functions from MD simulations
Pure Stockmayer solvent, 3000 particles, few ns
= 3 A, n0 = 0.03 atoms/A3 0 = 1.85 D, = 80
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Step 2: Functional minimisation around a solvated molecule
• Minimization with respect to • Discretization on a cubic grid (typically 643)• Conjugate gradients technique• Non-local interactions evaluated in Fourier space (8 FFts per minimization step)
)(and)( rPrn
Minimisation step
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N-methylacetamide: Particle and polarization densities
trans cis
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N-methylacetamide: Radial distribution functions
H
CH3
O
N
C
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N-methylacetamide: Isomerization free-energy
cis trans
Umbrella sampling
DFT
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DFT: General formulation
One needs higher spherical invariants expansions or angular grids 2112 ,,cand, ΩΩrΩrTo represent:
4N 8N
32 NNN
Begin with a linear model ofAcetonitrile (Edwards et al)
(with Shuangliang Zhao)
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Step 1: Inversion of Ornstein-Zernike equation
2331302121 ,,,,,,,, ΩΩkΩΩkΩΩΩkΩΩk hcdch
10 ))()(()( kHWIkHkC
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Step 2: Minimization of the discretized functional
extexcid FFFF x
0
0
ln
Ωr,Ωr,Ωr,ΩrddTkF Bid
Ωr,Ωr,Ωr extext VddF
222121221111 ),,(21 Ω,rΩΩrrΩrΩ,rΩr cddddFexc
Vexc(r1,1)
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Step 2: Minimization of the discretized functional
• Discretization of on a cubic grid for positions and Gauss-Legendre grid for orientations (typically 643 x 32)
2,, ΩrΩr
• Minimization in direct space by quasi-Newton (BFGS-L) (8x106 variables !!)
• 2 x N = 64 FFTs per minimization step
~20 s per minimization step on a single processor
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MDDFT
Solvent structure
Na+Na
Solvation in acetonitrile: Results
MDDFT
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Solvation in acetonitrile: Results
MD (~20 hours)
DFT (10 mn)
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Solvation in acetonitrile: Results
Halides solvation free energy
Parameters for ion/TIP3P interactions
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Solvation in SPC/E water
Solute-Oxygen radial distribution functions
MDDFT
Z
X
Y
Three angles: ,,
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CH3
C
N
Solvation in SPC/E water
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Cl-q
Solvation in SPC/E water
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Solvation in SPC/E water
Water in water
HNC PL-HNC HNC+B
g OO(r
)
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Conclusion DFT
• One can compute solvation free energies and microscopic solvation profiles using « classical » DFT
• Solute dynamics can be described using CPMD-like techniques
• For dipolar solvents, we presented a generic functional of or
• Direct correlation functions can be computed from MD simulations • For general solvents, one can use angular grids instead of rotational invariants expansion
rP
• BEYOND: -- Ionic solutions -- Solvent mixtures -- Biomolecule solvation
rPr ,n
R. Ramirez et al, Phys. Rev E, 66, 2002 J. Phys. Chem. B 114, 2005 Chem. Phys. 2005L. Gendre at al, Chem. Phys. Lett.S. Zhao et al, In prep.
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DCMD: « Soft pseudo-potentials »
V(r)
r
V(r) = (r)-1= 4/((r)-1)
V(r)
r
)(1111 rr
Hsis
=0
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Dielectric Continuum Molecular Dynamics
Hexadecapeptide P2
La3+ Ca2+
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DCMD: Computation times
System Nb of atoms
CPU total
CPU forces
CPU TIP3P
Dipep-tide
22 3.2 0.1 2.45
Octa 83 3.3 0.3 2.45
BPTI 892 5.7 2.7 2.72
Each time step correspond to a solvent free energy, thus an average over many solvent microscopic configurations
linear in N !