1. Chemistry 6440 / 7440 Models for Solvation
Undergrads should be registered for CHM6440, grads for CHM7440
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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. Explicit vs. Implicit Solvation
 = 78.39

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
FMM scales as N(logN) just like a Fast Fourier Transformation

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
Error in Cramer?

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)]
Volume by default uses a Monte Carlo algorithm. The iop command forces true numeric integration of the density

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
VM = electrostatic potential generated by M
VR = reaction potential generated by the polarization of the dielectric medium
The first condition expresses the continuity of V across  while the second condition involves the discontinuity of the component of the field perpendicular to .
s is a vector normal to the surface with a magnitude equal to the area of the surface. The surface often gets chopped up into chunks that s represents.

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
Self-interaction correction (SIC) allows larger tesserae

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)
n is a unit vector perpendicular to the cavity surface and pointing outward.
SS(V)PE is Chipman’s work and it addresses issues of interpenetration between solute and solvent charge density.

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

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