# Statistical Models of Solvation by gjjur4356

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```									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
U N
( N)                      e           dr1 ...drN
P        (r1 ,..., rN ) 
ZN

   The probability that 1 is at dr1 … and n is at drn
irrespective of the configuration of the other particles is

P(n) (r1 ,...,rn ) 
 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!
(n) (r1,...,rn )                (n)
P (r1 ,...,rn )
(N  n)!
   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!      (n)              n      (n)
P (r ,...,rn )  P (r1 )g (r1 ,...,rn )
1
(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)
   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(r )  c(r12)   c(r )h(r23)dr3
12                  13
Closure Equations
c(r)  htotal (r )  hindire ct (r)
 gtotal (r)  1 gindire ct (r)  1
 g(r)  gindire ct (r)
w(r )           w(r )u(r) 
e                e
 e
w(r )

1 e
u(r) 


 g(r) 1  e
u(r) 

u(r12 )                                        u(r13 ) 
g(r )e
12                    1  g(r )[1 e
13                                  ][g(r23 )  1]dr3
Percus  Yev ick (PY) Equation
u(r12 ) 
g(r )e
12                 [g(r13 )  1 ln g(r )  u(r13 )][g(r23 )  1]dr3
13

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
pairwise additive), UN (r1 ,..., rN )  j u(rij ) , then you can calculate
i
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
2 
2p             du(r )
P  kT 
3V     r3
0
dr
g(r )dr

1

  kT ln   4p   r2 u(r)g(r;  )drd
3

0 0
1
 h 
2    2
w here,            ;  is a coupling parameter w hich varies betw een 0 and 1.
2pmkT 
(Taking a partic le in,  = 1, and out,  = 0, of the s ystem).
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(r )  c(r12 ) 
12
4p  c(r13)h(r23)dr3

h(r )  c(r12 )  2  c(r )h(r23 )dr3
12
8p       13
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'   
e   sw sw 
w(r     )

P(sw ;rsw ) 
8p k(rsw )
2

1
e
w(r sw  sw )

8p 2 
k(rsw )                             d
More Approximations
    c 0 (rww ' ) is obtained via using a radial distribution function
obtained from MC simulation which uses a spherically-
averaged 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,
E(rsw sw )
w(rsw sw ) 
 ' (rsw )
The Water Models
   BBL Water:
– Water is a hard sphere, with a point dipole  = 1.85 D.

hs
uij  uij  uSP  uij
ij

potential energy of
hard-sphere potential sticky potential used to mimic      two dipoles for a
hydrogen-bond interactions.         given orientation
Attractive square-well potential,
dependant upon 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
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

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