Chemistry 6440 / 7440
Models for Solvation
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
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
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.
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.
“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
Explicit vs. Implicit Solvation
 = 78.39
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
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
Generalized Born Approximation
• Ion of charge q in a spherical cavity of radius a
qiq j
  1
GP  
  i,
j 1 2 f GB
f GB 
rij2   ij2 e
Dij
 ij   i j 
0.5
Dij 
rij2
2 
2
ij
• 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
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



 [ (r ) (r )]  4 M (r )
• Polarization of solvent calculated numerically
– FE or FD solution of the Poisson equation
– Apparent surface charge method
– Generalized Born / surface area
Multipole Expansion Methods
• Aka Kirkwood-Onsager Model (SCRF=Dipole)
• Solute with dipole, , in a spherical cavity of
radius a.
2
1  2( -1)  
GP   

3
2  (2 +1)  a
• Easily generalized for multipole expansions
1 L l L l
GP       M lm fllm m  M lm 
2 l  0 ml l   0 m    l 
• Multipole expansions are slow to converge
Multipole Expansion Methods
• QM requires a new potential term in F
V  r  R
2(  1)
R

3
(2  1)a
• 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)]
Surface Definitions
Apparent Surface Charge (ASC) methods
• The polarization of the solute’s charge distribution,
M, must obey Poisson equation
[ (r)V (r)]  4M (r) V (r)  VM (r)  VR (r)
• On the cavity surface, , two jump conditions exist
[V ]  Vin  Vout  0 on 
 V 
 V 
[V ]  

 0 on 


 n  in
 n  out
• From the second jump condition, the apparent
surface charge, (s), can be defined
 s) 2
V (r)  
d s  VR (r)
rs

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
 s k )Ak
qk
V (r) ; 

r  sk
k
k r  sk
• Adaptable for linearized Poisson-Boltzmann
applications: nonzero ionic strength solvents
• FMM speed up BEM calculations
ASC Methods: PCM
• The Polarizable Continuum Model (PCM) is the
oldest ASC method.
• The PCM surface charge is
  
 s) 
VM  V in

 n
• 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)
Gaussian Output for PCM
SCF Done:
E(RHF) =
Convg
=
S**2
=
-98.569083211
0.4249D-05
A.U. after
-V/T =
5 cycles
2.0033
0.0000
-------------------------------------------------------------------Variational PCM results
=======================
<psi(f)|
H
|psi(f)>
<psi(f)|H+V(f)/2|psi(f)>
(a.u.) =
-98.568013
(a.u.) =
-98.573228
(a.u.) =
-98.569083
Total free energy in solution:
with all non electrostatic terms
-------------------------------------------------------------------(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
--------------------------------------------------------------------
References
1.
2.
3.
C. J. Cramer, “Essentials of Computational
Chemistry,” 2002, John Wiley & Sons Ltd. (ISBN 0471-48551-9)
Chem. Rev. 2005, 105, 2999.
J. Am. Chem. Soc. 1936, 58, 1486.