Classical Density Functional Theory of Solvation in Molecular Solvents
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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).

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Rosa ramirez universit d evry shuangliang zhao ens paris

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)


Rosa ramirez universit d evry shuangliang zhao ens paris

For a given molecule in a given solvent, can we predict efficiently

and with « chemical accuracy:

  • The solvation free energy

  • The microscopic solvation profile

Solvation: Some issues

  • A few applications:

  • Differential solvation (liquid-liquid extraction)

  • Solubility prediction

  • Reactivity

  • Biomolecular solvation, ….

Explicit solvent/FEP


Rosa ramirez universit d evry shuangliang zhao ens paris

Solvation: Implicit solvent methods

Dielectric continuum approximation (Poisson-Boltzmann)

electrostatics

+ non-polar

Solvent Accessible Surface Area (SASA)

Biomolecular modelling: PB-SA method

Quantum chemistry: PCM method


Rosa ramirez universit d evry shuangliang zhao ens paris

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)

  • Molecular picture (G. Patey, P. Fries, …)

Site-site OZ + closure

Molecular OZ + closure

  • Classical Density Functional Theory

This work: Can we use classical DFT to define an improved and

well-founded implicit solvation approach?


Rosa ramirez universit d evry shuangliang zhao ens paris

Fpol

entropy

Fext

Fexc

Solvent-solvent

P(r)

DFT formulation of electrostatics


Rosa ramirez universit d evry shuangliang zhao ens paris

Plane wave expansion

Soft « pseudo-potentials »

On-the-fly minimization with extended

Lagrangian

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


Rosa ramirez universit d evry shuangliang zhao ens paris

Dielectric Continuum Molecular Dynamics

a-helix

horse-shoe


Rosa ramirez universit d evry shuangliang zhao ens paris

Dielectric Continuum Molecular Dynamics

Energy conservation

Adiabaticity


Rosa ramirez universit d evry shuangliang zhao ens paris

Beyond continuum electrostatics: Classical DFT of solvation

In the grand canonical ensemble, the grand

potential can be written as a functional

of r(r,W):

Intrinsic to a given solvent

Functional minimization:

D. Mermin(« Thermal properties of the

inhomogeneous electron gas », Phys. Rev., 137 (1965))

Thermodynamic equilibrium


Rosa ramirez universit d evry shuangliang zhao ens paris

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 ?

But what is the functional ??


Rosa ramirez universit d evry shuangliang zhao ens paris

The exact functional


Rosa ramirez universit d evry shuangliang zhao ens paris

g(r)

h(r)

The homogeneous reference fluid approximation

Neglect the dependence of c(2)(x1,x2,[ra]) on the parameter a, i.e use

direct correlation function of the homogeneous system

c(x1,x2) connected to the pair correlation function h(x1,x2) through

theOrnstein-Zernike relation


Rosa ramirez universit d evry shuangliang zhao ens paris

g(r)

h(r)

The homogeneous reference fluid approximation

Neglect the dependence of c(2)(x1,x2,[ra]) on the parameter a, i.e use

direct correlation function of the homogeneous system

c(x1,x2) connected to the pair correlation function h(x1,x2) through

theOrnstein-Zernike relation


Rosa ramirez universit d evry shuangliang zhao ens paris

The picture

Functional minimization


Rosa ramirez universit d evry shuangliang zhao ens paris

Rotational invariants expansion


Rosa ramirez universit d evry shuangliang zhao ens paris

The case of dipolar solvents

The Stockmayer solvent


Rosa ramirez universit d evry shuangliang zhao ens paris

A generic functional for dipolar solvents

Particle density

Polarization density

R. Ramirez et al, Phys. Rev E, 66, 2002

J. Phys. Chem. B 114, 2005


Rosa ramirez universit d evry shuangliang zhao ens paris

A generic functional for dipolar solvents


Rosa ramirez universit d evry shuangliang zhao ens paris

A generic functional for dipolar solvents


Rosa ramirez universit d evry shuangliang zhao ens paris

A generic functional for dipolar solvents


Rosa ramirez universit d evry shuangliang zhao ens paris

A generic functional for dipolar solvents

Connection to electrostatics:R. Ramirez et al, JPC B 114, 2005


Rosa ramirez universit d evry shuangliang zhao ens paris

The picture

Functional minimization


Rosa ramirez universit d evry shuangliang zhao ens paris

Step 1: Extracting the c-functions from MD simulations

Pure Stockmayer solvent, 3000 particles, few ns

s = 3 A, n0 = 0.03 atoms/A3

m0 = 1.85 D, e = 80

h-functions

c-functions

O-Z


Rosa ramirez universit d evry shuangliang zhao ens paris

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)

Minimisation step


Rosa ramirez universit d evry shuangliang zhao ens paris

N-methylacetamide: Particle and polarization densities

trans

cis


Rosa ramirez universit d evry shuangliang zhao ens paris

C

N

N-methylacetamide: Radial distribution functions

O

H

CH3


Rosa ramirez universit d evry shuangliang zhao ens paris

N-methylacetamide: Isomerization free-energy

cis

trans

Umbrella sampling

DFT


Rosa ramirez universit d evry shuangliang zhao ens paris

Begin with a linear model of

Acetonitrile (Edwards et al)

DFT: General formulation

(with Shuangliang Zhao)

To represent:

One needs higher spherical invariants expansions or angular grids


Rosa ramirez universit d evry shuangliang zhao ens paris

Step 1: Inversion of Ornstein-Zernike equation


Rosa ramirez universit d evry shuangliang zhao ens paris

Vexc(r1,W1)

Step 2: Minimization of the discretized functional


Rosa ramirez universit d evry shuangliang zhao ens paris

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)

  • Minimization in direct space by quasi-Newton (BFGS-L)

  • (8x106 variables !!)

  • 2 xNW = 64 FFTs per minimization step

~20 s per minimization step on a

single processor


Rosa ramirez universit d evry shuangliang zhao ens paris

Solvation in acetonitrile: Results

Solvent structure

Na

Na+

MD

MD

DFT

DFT


Rosa ramirez universit d evry shuangliang zhao ens paris

Solvation in acetonitrile: Results

MD (~20 hours)

DFT (10 mn)


Rosa ramirez universit d evry shuangliang zhao ens paris

Solvation in acetonitrile: Results

Halides solvation free energy

Parameters for ion/TIP3P interactions


Rosa ramirez universit d evry shuangliang zhao ens paris

Z

Y

X

Solvation in SPC/E water

Solute-Oxygen radial distribution functions

MD

DFT

Three angles:


Rosa ramirez universit d evry shuangliang zhao ens paris

Solvation in SPC/E water

N

C

CH3


Rosa ramirez universit d evry shuangliang zhao ens paris

Solvation in SPC/E water

Cl-q


Rosa ramirez universit d evry shuangliang zhao ens paris

Solvation in SPC/E water

Water in water

HNC

PL-HNC

HNC+B

gOO(r)


Rosa ramirez universit d evry shuangliang zhao ens paris

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

  • 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.


Rosa ramirez universit d evry shuangliang zhao ens paris

V(r)

DCMD: « Soft pseudo-potentials »

V(r) = c(r)-1= 4p /(e(r)-1)

c=0

V(r)

r

r


Rosa ramirez universit d evry shuangliang zhao ens paris

Dielectric Continuum Molecular Dynamics

Hexadecapeptide P2

Ca2+

La3+


Rosa ramirez universit d evry shuangliang zhao ens paris

DCMD: Computation times

linear in N !

Each time step correspond to a solvent free energy, thus

an average over many solvent microscopic configurations


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