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 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) Biomolecular modelling: PB-SA method Solvent Accessible Surface Area (SASA) electrostatics+ non-polar Quantum chemistry: PCM method

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) Site-site OZ + closure Molecular OZ + closure

F pol entropy F exc Solvent-solvent F ext P(r)P(r) DFT formulation of electrostatics

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 On-the-fly minimization with extended Lagrangian Plane wave expansion Soft « pseudo-potentials »

Dielectric Continuum Molecular Dynamics  -helix horse-shoe

Dielectric Continuum Molecular Dynamics Energy conservation Adiabaticity

Beyond continuum electrostatics: Classical DFT of solvation In the grand canonical ensemble, the grand potential can be written as a functional of  (r  Functional minimization: Thermodynamic equilibrium D. Mermin (« Thermal properties of the inhomogeneous electron gas », Phys. Rev., 137 (1965)) Intrinsic to a given solvent

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 ??

The exact functional

The homogeneous reference fluid approximation Neglect the dependence of c (2) (x 1,x 2,[   ]) on the parameter , i.e use direct correlation function of the homogeneous system c(x 1,x 2 ) connected to the pair correlation function h(x 1,x 2 ) through the Ornstein-Zernike relation g(r) h(r)

The homogeneous reference fluid approximation Neglect the dependence of c (2) (x 1,x 2,[   ]) on the parameter , i.e use direct correlation function of the homogeneous system c(x 1,x 2 ) connected to the pair correlation function h(x 1,x 2 ) through the Ornstein-Zernike relation g(r) h(r)

The picture Functional minimization

Rotational invariants expansion

The case of dipolar solvents The Stockmayer solvent

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

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

The picture Functional minimization

O-Z h-functions c-functions Step 1: Extracting the c-functions from MD simulations Pure Stockmayer solvent, 3000 particles, few ns  = 3 A, n 0 = 0.03 atoms/A 3  0 = 1.85 D,  = 80

Step 2: Functional minimisation around a solvated molecule Minimization with respect to Discretization on a cubic grid (typically 64 3 ) Conjugate gradients technique Non-local interactions evaluated in Fourier space (8 FFts per minimization step) Minimisation step

N-methylacetamide: Particle and polarization densities transcis

N-methylacetamide: Radial distribution functions H CH 3 O N C

N-methylacetamide: Isomerization free-energy cis trans Umbrella sampling DFT

DFT: General formulation One needs higher spherical invariants expansions or angular grids To represent: Begin with a linear model of Acetonitrile (Edwards et al) (with Shuangliang Zhao)

Step 1: Inversion of Ornstein-Zernike equation

Step 2: Minimization of the discretized functional V exc (r 1,  1 )

Step 2: Minimization of the discretized functional Discretization of on a cubic grid for positions and Gauss-Legendre grid for orientations (typically 64 3 x 32) Minimization in direct space by quasi-Newton (BFGS-L) (8x10 6 variables !!) 2 x N  = 64 FFTs per minimization step ~20 s per minimization step on a single processor

MD DFT Solvent structure Na + Na Solvation in acetonitrile: Results 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 MD DFT Z X Y Three angles:

CH 3 C N Solvation in SPC/E water

Cl -q Solvation in SPC/E water

Water in water HNCPL-HNC HNC+B g OO (r)

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 L. Gendre at al, Chem. Phys. Lett. S. Zhao et al, In prep.

DCMD: « Soft pseudo-potentials » V(r) r V(r) =  (r) -1 = 4  /(  (r)-1) V(r) r  =0

Dielectric Continuum Molecular Dynamics Hexadecapeptide P2 La 3+ Ca 2+

DCMD: Computation times Each time step correspond to a solvent free energy, thus an average over many solvent microscopic configurations linear in N !