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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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)


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


Solvation: Implicit solvent methods efficiently

Dielectric continuum approximation (Poisson-Boltzmann)

electrostatics

+ non-polar

Solvent Accessible Surface Area (SASA)

Biomolecular modelling: PB-SA method

Quantum chemistry: PCM method


Improved implicit solvent models efficiently

(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?


F efficientlypol

entropy

Fext

Fexc

Solvent-solvent

P(r)

DFT formulation of electrostatics


Plane wave expansion efficiently

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


Dielectric Continuum Molecular Dynamics efficiently

a-helix

horse-shoe


Dielectric Continuum Molecular Dynamics efficiently

Energy conservation

Adiabaticity


Beyond continuum electrostatics: Classical DFT of solvation efficiently

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


In analogy to electronic DFT, how to use efficientlyclassical DFT as a « theoretical chemist »

tool to compute the solvation properties of molecules, in particular their solvation

free-energy ?

But what is the functional ??



g(r) efficiently

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


g(r) efficiently

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


The picture efficiently

Functional minimization



The case of dipolar solvents efficiently

The Stockmayer solvent


A generic functional for dipolar solvents efficiently

Particle density

Polarization density

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

J. Phys. Chem. B 114, 2005





A generic functional for dipolar solvents efficiently

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


The picture efficiently

Functional minimization


Step 1: Extracting the c-functions from MD simulations efficiently

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


Step 2: Functional minimisation around a solvated molecule efficiently

  • 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



C efficiently

N

N-methylacetamide: Radial distribution functions

O

H

CH3


N-methylacetamide: Isomerization free-energy efficiently

cis

trans

Umbrella sampling

DFT


Begin with a linear model of efficiently

Acetonitrile (Edwards et al)

DFT: General formulation

(with Shuangliang Zhao)

To represent:

One needs higher spherical invariants expansions or angular grids



V efficientlyexc(r1,W1)

Step 2: Minimization of the discretized functional


Step 2: Minimization of the discretized functional efficiently

  • 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


Solvation in acetonitrile: Results efficiently

Solvent structure

Na

Na+

MD

MD

DFT

DFT


Solvation in acetonitrile: Results efficiently

MD (~20 hours)

DFT (10 mn)


Solvation in acetonitrile: Results efficiently

Halides solvation free energy

Parameters for ion/TIP3P interactions


Z efficiently

Y

X

Solvation in SPC/E water

Solute-Oxygen radial distribution functions

MD

DFT

Three angles:


Solvation in SPC/E water efficiently

N

C

CH3


Solvation in SPC/E water efficiently

Cl-q


Solvation in SPC/E water efficiently

Water in water

HNC

PL-HNC

HNC+B

gOO(r)


Conclusion DFT efficiently

  • 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.


V(r) efficiently

DCMD: « Soft pseudo-potentials »

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

c=0

V(r)

r

r


Dielectric Continuum Molecular Dynamics efficiently

Hexadecapeptide P2

Ca2+

La3+


DCMD: Computation times efficiently

linear in N !

Each time step correspond to a solvent free energy, thus

an average over many solvent microscopic configurations


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