Docstoc

Statistical Models of Solvation

Document Sample
Statistical Models of Solvation Powered By Docstoc
					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)!
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!      (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)
    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(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
          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

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:4
posted:4/22/2011
language:English
pages:16