Rosa Ramirez ( Université d’Evry ) Shuangliang Zhao ( ENS Paris)

1. Classical Density Functional Theory of Solvation in Molecular Solvents Daniel Borgis Département de Chimie Ecole Normale Supérieure de Paris daniel.borgis@ens.fr • Rosa Ramirez (Université d’Evry) • Shuangliang Zhao (ENS Paris)

2. 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

3. 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

4. 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?

5. Fpol entropy Fext Fexc Solvent-solvent P(r) DFT formulation of electrostatics

6. 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

7. Dielectric Continuum Molecular Dynamics a-helix horse-shoe

8. Dielectric Continuum Molecular Dynamics Energy conservation Adiabaticity

9. 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

10. 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 ??

11. The exact functional

12. 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

13. 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

14. The picture Functional minimization

15. Rotational invariants expansion

16. The case of dipolar solvents The Stockmayer solvent

17. 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

18. A generic functional for dipolar solvents

19. A generic functional for dipolar solvents

20. A generic functional for dipolar solvents

21. A generic functional for dipolar solvents Connection to electrostatics:R. Ramirez et al, JPC B 114, 2005

22. The picture Functional minimization

23. 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

24. 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

25. N-methylacetamide: Particle and polarization densities trans cis

26. C N N-methylacetamide: Radial distribution functions O H CH3

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

28. 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

29. Step 1: Inversion of Ornstein-Zernike equation

30. Vexc(r1,W1) Step 2: Minimization of the discretized functional

31. 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

32. Solvation in acetonitrile: Results Solvent structure Na Na+ MD MD DFT DFT

33. Solvation in acetonitrile: Results MD (~20 hours) DFT (10 mn)

34. Solvation in acetonitrile: Results Halides solvation free energy Parameters for ion/TIP3P interactions

35. Z Y X Solvation in SPC/E water Solute-Oxygen radial distribution functions MD DFT Three angles:

36. Solvation in SPC/E water N C CH3

37. Solvation in SPC/E water Cl-q

38. Solvation in SPC/E water Water in water HNC PL-HNC HNC+B gOO(r)

39. 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.

40. V(r) DCMD: « Soft pseudo-potentials » V(r) = c(r)-1= 4p /(e(r)-1) c=0 V(r) r r

41. Dielectric Continuum Molecular Dynamics Hexadecapeptide P2 Ca2+ La3+

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