Chemistry 6440 / 7440

1. Chemistry 6440 / 7440 Models for Solvation

2. Resources • Foresman and Frisch, Exploring Chemistry with Electronic Structure Methods, Chapter 10 • Jensen, Chapter 16.3 • Cramer, Chapter 11 • Young, Chapter 24 • Tomasi & Mennucci, ECC pg 2547

3. Explicit Solvent Models • Includes individual solvent molecules • Calculate the free energy of solvation by simulating solute-solvent interactions • Monte Carlo (MC) calculations, molecular dynamics (MD) simulations • Very lengthy calculations • Requires an empirical interaction potential between the solvent and solute, and between the solvent molecules

4. Monte Carlo Simulations • Box containing a solute and solvent molecules (periodic boundary conditions) • Random moves of molecules • If energy goes down, accept the move • If energy goes up, accept according to Boltzmann probability • MC calculations can be used to compute free energy differences, radial distribution functions, etc. • Cannot be used to compute time dependent properties such as diffusion coefficients, viscosity, etc.

5. Molecular Dynamics Simulations • Use classical equations to simulate the motion of the molecules for a suitably long time (100’s ps to ns) • Requires energies and gradients of the potential • In addition to free energies, can be used to compute time dependent properties transport properties, correlation functions, etc.

6. “It cannot be overemphasized that solvation changes the solute electronic structure. Dipole moments in solution are larger than the corresponding dipole moments in the gas phase. Indeed, any property that depends on the electronic structure will tend to have a different expectation value in solution than in the gas phase.” -Cramer1 “A continuum model in computational molecular sciences can be defined as a model in which a number of the degrees of freedom of the constituent particles are described in a continuous way, usually by means of a distribution function.” -Tomasi, Mennucci, and Cammi2

7.  = 78.39 Explicit vs. Implicit Solvation

8. Implicit Solvent Model • Solvent is treated as a polarizable continuum with a dielectric constant, , instead of explicit solvent molecules. • The charge distribution of the solute polarizes the solvent producing a reaction potential. • The reaction potential of solvent alters the solute. • This interaction is represented by a solvent reaction potential introduced into the Hamiltonian. • Interactions must be computed self consistently • Also know as self consistent reaction field (SCRF) methods due to Onsager’s seminal paper3 • Significantly cheaper than explicit solvent models, especially if FMM can be utilized • Cannot model specific interactions such as hydrogen bonds

9. Continuum Solvation Categories • Generalized Born Approximation (GBA) • Multipole Expansion (MPE) methods • Apparent Surface Charge (ASC) methods • Image Charge (IMC) methods • Nothing new since 19942 • Finite Element Methods (FEM) • Superceded by Boundary Element Method (BEM) • Finite Difference Methods (FDM) • Superceded by BEM

10. Generalized Born Approximation • Ion of charge q in a spherical cavity of radius a • Widely used in biochemistry community • Allows for partial charges • Equal solvation energy for positive and negative ions • Neglects cavitation and dispersion energy • Born radii, i, are not well defined

11. PCM – Polarizable Continuum Model • Shape of cavity determined by shape of solute • Overlapping van der Waals spheres (PCM and CPCM) (all atom or united atom) • Solvent accessible surface • Isodensity surface (IPCM, SCIPCM) • Electrostatic potential from solute and polarization of solvent must obey Poisson equation • Polarization of solvent calculated numerically • FE or FD solution of the Poisson equation • Apparent surface charge method • Generalized Born / surface area

12. Multipole Expansion Methods • Aka Kirkwood-Onsager Model (SCRF=Dipole) • Solute with dipole, , in a spherical cavity of radius a. • Easily generalized for multipole expansions • Multipole expansions are slow to converge

13. Multipole Expansion Methods • QM requires a new potential term in F • Allows solute to respond to the reaction potential resulting from polarization of the solute • MPE easily rolled into the SCF/CPHF equations • Very sensitive to the cavity radius a • Determine a from the molecular volume [Volume and iop(6/44=4)]

14. Surface Definitions

15. Apparent Surface Charge (ASC) methods • The polarization of the solute’s charge distribution, M, must obey Poisson equation • On the cavity surface, , two jump conditions exist • From the second jump condition, the apparent surface charge, (s), can be defined

16. Boundary Element Method • BEM used to solve ASC equation •  approximated by tesserae small enough to consider (s) almost constant within each tessera • A set of point charges, qk, are defined based on the local value of (s) in a tessera of area Ak • Adaptable for linearized Poisson-Boltzmann applications: nonzero ionic strength solvents • FMM speed up BEM calculations

17. ASC Methods: PCM • The Polarizable Continuum Model (PCM) is the oldest ASC method. • The PCM surface charge is • Three major formulations • DPCM (SCRF=PCM) • IPCM (SCRF=IPCM) • SCIPCM prone to stability issues (SCRF=SCIPCM) • CPCM = COSMO with k=0.5 (SCRF=CPCM) • IEFPCM = IVCPCM = SS(V)PE recommended method (SCRF=IEFPCM)

18. Gaussian Output for PCM SCF Done: E(RHF) = -98.569083211 A.U. after 5 cycles Convg = 0.4249D-05 -V/T = 2.0033 S**2 = 0.0000 -------------------------------------------------------------------- Variational PCM results ======================= <psi(f)| H |psi(f)> (a.u.) = -98.568013 <psi(f)|H+V(f)/2|psi(f)> (a.u.) = -98.573228 Total free energy in solution: with all non electrostatic terms (a.u.) = -98.569083 -------------------------------------------------------------------- (Polarized solute)-Solvent (kcal/mol) = -3.27 -------------------------------------------------------------------- Cavitation energy (kcal/mol) = 5.34 Dispersion energy (kcal/mol) = -3.08 Repulsion energy (kcal/mol) = 0.34 Total non electrostatic (kcal/mol) = 2.60 --------------------------------------------------------------------

19. References • C. J. Cramer, “Essentials of Computational Chemistry,” 2002, John Wiley & Sons Ltd. (ISBN 0-471-48551-9) • Chem. Rev.2005, 105, 2999. • J. Am. Chem. Soc. 1936, 58, 1486.