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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)
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
Solvation: Implicit solvent methods
Dielectric continuum approximation (Poisson-Boltzmann)
  80
   r  r   40 r 
 r 
electrostatics
  i
F
+ non-polar
1
dr  0 r  r    A

2
Solvent Accessible Surface Area (SASA)
Biomolecular modelling: PB-SA method
Quantum chemistry: PCM method
Improved implicit solvent models
(based on « modern » liquid state theory)
• Integral equations
• Interaction site picture (RISM) (D. Chandler, P. Rossky, M. Pettit,
F. Hirata, A. Kovalenko)
hij r , cij (r)
Site-site OZ + closure
• Molecular picture (G. Patey, P. Fries, …)
Molecular OZ + closure
hr12 , Ω1 , Ω2 , cr12 , Ω1 , Ω2 
• Classical Density Functional Theory
This work: Can we use classical DFT to define an improved and
well-founded implicit solvation approach?
DFT formulation of electrostatics
ri
P(r)
Uelec  V0 (ri )  F[P(r);ri ]
F P(r)
1
4
1
2


d
r
P
(
r
)

dr
P
(
r
)

E
r

drdr ' P(r )  T(r  r ' )  P(r ' )
0



2
 (r)  1
2
Fpol
entropy
Fext
Fexc
Solvent-solvent
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
Plane wave expansion
Soft « pseudo-potentials »
On-the-fly minimization with extended
Lagrangian
P(r)   P(k ) exp(ik  r)
k
1
1 1 1
     H (r)
 r   s   i  s 
d 2 P(k )
F
MP

2
dt
P(k )
d 2ri F V0
mi 2 

dt
ri ri
Dielectric Continuum Molecular Dynamics
a-helix
horse-shoe
Dielectric Continuum Molecular Dynamics
Energy conservation
Adiabaticity
Beyond continuum electrostatics: Classical DFT of solvation
 r, Ω  position/orientation solventdensity
In the grand canonical ensemble, the grand
potential can be written as a functional
of (r,W:
   T S    Fexc     drdΩ Vext r, Ω  r, Ω    c N
Intrinsic to a given solvent
 
Functional minimization:
0

D. Mermin (« Thermal properties of the
inhomogeneous electron gas », Phys. Rev., 137 (1965))
 0 r, Ω
 0 r, Ω
Thermodynamic equilibrium
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 solvation
free-energy ?
c ,  0
c ,Vext (r, Ω)
 0 
 r, Ω
F       0 
Fmin  Solvationfree energy
But what is the functional ??
The exact functional
F  x  Fid    Fexc    Fext  
x  (r, Ω)


  x1  
   x1    0 
Fid    k B T  dx1   x1  ln
 0 


 (r, Ω)
Fext     dx1 Vext x1   x1 
Fexc    k B T  dx1 dx 2  x1  C x1 , x 2   x1 ,
C x1 , x 2    da a  1 c ( 2) x1 , x 2 ; a 
1
0
 x   x  0
a x   0  a  x
The homogeneous reference fluid approximation
Neglect the dependence of c(2)(x1,x2,[a]) on the parameter a, i.e use
direct correlation function of the homogeneous system
c ( 2) x1 , x 2 ; a   c ( 2) x1 , x 2 ; 0   cx1 , x 2 
c(x1,x2) connected to the pair correlation function h(x1,x2) through
the Ornstein-Zernike relation
hx1 , x 2   cx1 , x 2  
 0  dx 3 cx1 , x 3  hx 3 , x 2 
g(r)
h(r)
hx1 , x 2   g x1 , x 2   1
The homogeneous reference fluid approximation
Neglect the dependence of c(2)(x1,x2,[a]) on the parameter a, i.e use
direct correlation function of the homogeneous system
c ( 2) x1 , x 2 ; a   c ( 2) x1 , x 2 ; 0   cx1 , x 2 
c(x1,x2) connected to the pair correlation function h(x1,x2) through
the Ornstein-Zernike relation
hr12 , Ω1 , Ω 2   cr12 , Ω1 , Ω 2  
 0  dr3dΩ3 c(r13 , Ω1 , Ω3 ) hr32 , Ω3 , Ω 2 
g(r)
h(r)
hx1 , x 2   g x1 , x 2   1
The picture
h( r12 ,Ω1 ,Ω2 )
c( r12 ,Ω1 ,Ω2 )
Functional minimization
Rotational invariants expansion
Ω2
Ω1
r12
lmn
ˆ
r12  lmn
h(r12 , Ω1 , Ω2 )   h
 (r12 , Ω1 , Ω2 )
lmn
ˆ
r12  lmn
c(r12 , Ω1 , Ω2 )   c
 (r12 , Ω1 , Ω2 )
The case of dipolar solvents
Ω2
Ω1
r12
The Stockmayer solvent
000  1, 110  Ω1  Ω2 , 112  3(Ω1  r12 )(Ω1  r12 )  Ω1  Ω2
c(r12 , Ω1, Ω2 )  c000 (r12 ) 000  c110 (r12 ) 110  c112 (r12 ) 112
A generic functional for dipolar solvents
 r, Ω  position/orientation solventdensity
Particle density
nr    dΩ  r , Ω 
F  r, Ω
Polarization density
Pr    0  dΩ Ω  r, Ω 
F nr , Pr 
R. Ramirez et al, Phys. Rev E, 66, 2002
J. Phys. Chem. B 114, 2005
A generic functional for dipolar solvents
F n, P  Fid n, P  Fext n, P  Fexc n, P
 n(r ) 
Fid n, P  k BT  dr1 n(r1 ) ln 1   n(r1 )  n0
 n0 
 L1 P(r) /  0 n(r) 
1
P(r) /  0 n(r)
 k BT  dr ln 

P
(
r
)
L

1
 sinh L P(r) /  0 n(r) 


P(r)  P(r)
L1 ( x)  Inversede la fonctionde LangevinL(x)
A generic functional for dipolar solvents
F n, P  Fid n, P  Fext n, P  Fexc n, P
 n(r ) 
Fid n, P  k BT  dr1 n(r1 ) ln 1   n(r1 )  n0
 n0 
P(r) 2
 k BT  dr
2a d n(r)
ad 
2
3k BT
 local orientational polarizability
A generic functional for dipolar solvents
F n, P  Fid n, P  Fext n, P  Fexc n, P
Fext n, P    dr VLJ (r ) n(r ) 
 dr E
q
(r )  P(r )
A generic functional for dipolar solvents
F n, P  Fid n, P  Fext n, P  Fexc n, P
Fexc n, P   
k BT
000
d
r
d
r

n
(
r
)
c
(r12 ) n(r2 )
1
2
1

2
k BT

2

110
d
r
d
r
c
 1 2 (r12 ) P(r1 )  P(r2 )
k BT
112
d
r
d
r
c
(r12 ) 3P(r1 )  r12 P(r2 )  r12   P(r1 )  P(r2 )
2

1
2
Connection to electrostatics: R. Ramirez et al, JPC B 114, 2005
The picture
h 000 (r12 )
h110 ( r12 )
h112 ( r12 )
c 000 (r12 )
c110 (r12 )
c112 (r12 )
Functional minimization
Step 1: Extracting the c-functions from MD simulations
Pure Stockmayer solvent, 3000 particles, few ns
s = 3 A, n0 = 0.03 atoms/A3
0 = 1.85 D,  = 80
h-functions
c-functions
O-Z
Step 2: Functional minimisation around a solvated molecule
• Minimization with respect to n(r) and P(r)
• Discretization on a cubic grid (typically 643)
• Conjugate gradients technique
• Non-local interactions evaluated in Fourier space (8 FFts
per minimization step)
Minimisation step
N-methylacetamide: Particle and polarization densities
trans
cis
N-methylacetamide: Radial distribution functions
O
C
N
H
CH3
N-methylacetamide: Isomerization free-energy
trans
cis
Umbrella sampling
DFT
DFT: General formulation
(with Shuangliang Zhao)
To represent:  r, Ω and cr12 , Ω1 , Ω2 
One needs higher spherical invariants expansions or angular grids
Begin with a linear model of
Acetonitrile (Edwards et al)
N  4
N  8
NW  N  N  32
Step 1: Inversion of Ornstein-Zernike equation
hk , Ω1 , Ω 2   ck , Ω1 , Ω 2    0  dΩ3 ck , Ω1 , Ω3  hk , Ω3 , Ω 2 
 C(k)  H(k)(I  0 WH(k))1
Step 2: Minimization of the discretized functional
F  x  Fid    Fexc    Fext  


  r, Ω  
   r, Ω    0 
Fid    k BT  drdΩ   r, Ω  ln
 0 


Fext     drdΩ Vext r, Ω   r, Ω 
1
Fexc     dr1dΩ1  r1 , Ω1   dr2 dΩ 2 c(r1  r2 , Ω1 , Ω 2 )  r2 , Ω 2 
2
Vexc(r1, W1)
Step 2: Minimization of the discretized functional
• Discretization of  r, Ω  r, Ω on a cubic grid for positions and
Gauss-Legendre grid for orientations (typically 643 x 32)
2
• Minimization in direct space by quasi-Newton (BFGS-L)
(8x106 variables !!)
• 2 x NW = 64 FFTs per minimization step
~20 s per minimization step on a
single processor
Solvation in acetonitrile: Results
Solvent structure
Na+
Na
MD
DFT
MD
DFT
Solvation in acetonitrile: Results
MD (~20 hours)
DFT (10 mn)
Solvation in acetonitrile: Results
Halides solvation free energy
Parameters for ion/TIP3P interactions
Solvation in SPC/E water
Solute-Oxygen radial distribution functions
Z

MD
DFT

Y

X
Three angles:
 ,  ,
Solvation in SPC/E water
N
C
CH3
Solvation in SPC/E water
Cl-q
Solvation in SPC/E water
Water in water
gOO(r)
HNC
PL-HNC
HNC+B
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 Pr  or nr , Pr 
•
Direct correlation functions can be computed from MD simulations
•
For general solvents, one can use angular grids instead of rotational
invariants expansion
•
BEYOND:
-- Ionic solutions
-- Solvent mixtures
-- Biomolecule solvation
R. Ramirez et al, Phys. Rev E, 66, 2002
J. Phys. Chem. B 114, 2005
Chem. Phys. 2005
L. Gendre at al, Chem. Phys. Lett.
S. Zhao et al, In prep.
DCMD: « Soft pseudo-potentials »
V(r) = c(r)-1= 4 /((r)-1)
c=0
1
1 1 1
     H (r)
 r   s   i  s 
V(r)
V(r)
r
r
Dielectric Continuum Molecular Dynamics
Hexadecapeptide P2
La3+
Ca2+
DCMD: Computation times
System Nb of
atoms
CPU
total
CPU
forces
CPU
TIP3P
Dipep
-tide
Octa
22
3.2
0.1
2.45
83
3.3
0.3
2.45
BPTI
892
5.7
2.7
2.72
linear in N !
Each time step correspond to a solvent free energy, thus
an average over many solvent microscopic configurations