Transcript Statistical Models of Solvation

Statistical Models of Solvation
Eva Zurek
Chemistry 699.08
Final Presentation
Methods

Continuum models: macroscopic treatment of the solvent;
inability to describe local solute-solvent interaction;
ambiguity in definition of the cavity

Monte Carlo (MC) or Molecular Dynamics (MD)
Methods: computationally expensive

Statistical Mechanical Integral Equation Theories: give
results comparable to MD or MC simulations;
computational speedup on the order of 102
Statistical Mechanics of Fluids



A classical, isotropic, one-component, monoatomic fluid.
A closed system, for which N, V and T are constant (the
Canonical Ensemble). Each particle i has a potential
energy Ui.
The probability of locating particle 1 at dr1, etc. is
( N)
P

(r1 ,...,rN ) 
U N
dr1 ...drN
ZN
The probability that 1 is at dr1 … and n is at drn
irrespective of the configuration of the other particles is
P (r1 ,...,rn ) 
(n)

e
e
 U N
drn1 ...drN
ZN
The probability that any particle is at dr1 … and n is at drn
irrespective of the configuration of the other particles is
 (n) (r1 ,...,rn ) 
N!
P(n) (r1 ,...,rn )
(N  n)!
Radial Distribution Function



If the distances between n particles increase the correlation
between the particles decreases.
In the limit of |ri-rj| the n-particle probability density
can be factorized into the product of single-particle
probability densities.
If this is not the case then
N!
P (n) (r1 ,...,rn )  P n (r1 )g (n) (r1 ,...,rn )
(N  n)!


In particular g(2)(r1,r2) is important since it can be
measured via neutron or X-ray diffraction
g(2)(r1,r2) = g(r12) = g(r)
Radial Distribution Function



g(r12) = g(r) is known as the radial distribution function
it is the factor which multiplies the bulk density to give the
local density around a particle
If the medium is isotropic then 4pr2g(r)dr is the number of
particles between r and r+dr around the central particle
g(r)  e w(r) 
Correlation Functions


Pair Correlation Function, h(r12), is a measure of the total
influence particle 1 has on particle 2
h(r12) = g(r12) - 1
Direct Correlation Function, c(r12), arises from the direct
interactions between particle 1 and particle 2
Ornstein-Zernike (OZ)
Equation



In 1914 Ornstein and Zernike proposed a division of h(r12)
into a direct and indirect part.
The former is c(r12), direct two-body interactions.
The latter arises from interactions between particle 1 and a
third particle which then interacts with particle 2 directly
or indirectly via collisions with other particles. Averaged
over all the positions of particle 3 and weighted by the
density.
h(r12 )  c(r12 )    c(r13 )h(r23 )dr3
Closure Equations
c(r)  htotal (r )  hindirect (r)
 gtotal (r)  1 gindirect (r)  1
 g(r)  gindirect (r)
e
w(r )
  w(r )u(r) 
e

 e w(r ) 1 eu(r) 

 g(r) 1  eu(r) 
g(r12 )e
u(r12 ) 


 1  g(r13 )[1 e 
u(r13 ) 
][g(r23 )  1]dr3
Percus Yevick (PY) Equation
g(r12 )e
u(r12 ) 
  [g(r13 )  1 ln g(r13 )   u(r13 )][g(r23 )  1]dr3
Hypernetted Chain (HNC) Equation
Thermodynamic Functions from g(r)


If you assume that the particles are acting through central
pair forces (the total potential energy of the system is
u(rij ) , then you can calculate
pairwise additive), UN (r1,...,rN )  
i j
pressure, chemical potential, energy, etc. of the system.
For an isotropic fluid

3
E  NkT  2p  g(r )u(r)r 2 dr
2
0

2p 2 3 du(r )
P  kT 
r
g(r )dr
3V 0
dr
1
  kT ln   4p   r2 u(r)g(r;  )drd
3
0 0
1
2
 h 
where,   
;  is a coupling paramet er which varies between 0 and 1.
2pm kT
2
(T aking a particle in, =1, and out ,  = 0, of t he system).
Molecular Liquids



Complications due to molecular vibrations ignored.
The position and orientation of a rigid molecule i are
defined by six coordinates, the center of mass coordinate ri
and the Euler angles  i  (i , i , i ) .
For a linear and non-linear molecule the OZ equation
becomes the following, respectively

h(r12 )  c(r12 ) 
c(r13 )h(r23 )dr3

4p

h(r12 )  c(r12 )  2  c(r13 )h(r23 )dr3
8p
Integral Equation Theory for
Macromolecules

If s denotes solute and w denotes water than the OZ
equation can be combined with a closure to give



g(rsw  sw )  exp  u(rsw  sw )  b(rsw sw )  2  c(rww'  ww' )h(rsw '  sw ' )drw' d w'


8p

This is divided into a  dependent and independent part
g(rsw sw )  8p 2 P(sw ;rsw )g 0rsw 

g 0 (rsw )  k(rsw )exp  u 0 (rsw )  b 0 rsw    c 0 (rww' )h0 (rsw' )drw'
sw sw 
e
P(sw ;rsw ) 
2
8p k(rsw )
1
w(rsw  sw )
k(rsw )  2  e
d
8p
w(r 
)

More Approximations




c 0 (rww ' ) is obtained via using a radial distribution function
obtained from MC simulation which uses a sphericallyaveraged potential.
c 0 (rww ' ) is used to calculate b0(rsw) for SSD water.
For BBL water b0(rsw) = 0, giving the HNC-OZ.
The orientation of water around a cation or anion can be
described as a dipole in a dielectric continuum with a
dielectric constant close to the bulk value. Thus,
w(rsw  sw ) 
E(rsw sw )
 ' (rsw )
The Water Models

BBL Water:
– Water is a hard sphere, with a point dipole  = 1.85 D.

uij  uijhs  uSP

u
ij
ij
hard-sphere potential sticky potential used to mimic
hydrogen-bond interactions.
Attractive square-well potential,
dependant upon orientation

potential energy of
two dipoles for a
given orientation
SSD Water:
– Water is a Lennard-Jones soft-sphere, with a point
dipole  = 2.35 D. Sticky potential is modified to be
compatible with soft-sphere.
Results for SSD Water



Position of the first peak, excellent agreement.
Coordination number, excellent agreement except for
anions which differ ~13-16% from MC simulation.
Solute-water interaction energy for water differs between
~9-14% and for ions/ion-pairs ~1-24%. Greatest for Cl-.
Results for BBL Water
Radial distribution function around
five molecule cluster of water from
theory (line) and MC simulation
(circles)
Twenty-five molecule cluster of water
Conclusions


Solvation models based upon the Ornstein-Zernike equation
could be used to give results comparable to MC or MD
calculations with significant computational speed-up.
Problems:
– which solvent model?
– which closure?
0
– how to calculate c (rww ww ) and h(rsw sw ) ?
'

Thanks:
– Dr. Paul
'
'
'