statistical models of solvation eva zurek chemistry 699.08 final presentation
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![Page 1: Statistical Models of Solvation Eva Zurek Chemistry 699.08 Final Presentation](https://reader031.vdocuments.mx/reader031/viewer/2022033106/56649d255503460f949fbe85/html5/thumbnails/1.jpg)
Statistical Models of SolvationStatistical Models of Solvation
Eva Zurek
Chemistry 699.08
Final Presentation
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MethodsMethods 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
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Statistical Mechanics of FluidsStatistical 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
The probability that 1 is at dr1 … and n is at drn irrespective of the configuration of the other particles is
The probability that any particle is at dr1 … and n is at drn irrespective of the configuration of the other particles is
P(N)(r1,...,rN )=e−βUNdr1...drN
ZN
P(n)(r1,...,rn)=e−βUNdrn+1...drN∫∫
ZN
ρ(n)(r1,...,rn) =N!
(N−n)!P(n)(r1,...,rn)
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Radial Distribution FunctionRadial 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
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)
N!(N−n)!
P(n)(r1,...,rn) =Pn(r1)g(n)(r1,...,rn)
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Radial Distribution FunctionRadial 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 4r2g(r)dr is the number of
particles between r and r+dr around the central particle
g(r) =e−βw(r)[ ]
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Correlation FunctionsCorrelation 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
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Ornstein-Zernike (OZ) Ornstein-Zernike (OZ) EquationEquation 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)+ρ c(r13)∫ h(r23)dr3
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Closure Equations Closure Equations c(r) =htotal(r)−hindirect(r)
=gtotal(r)−1−gindirect(r)+1
=g(r) −gindirect(r)
=e−βw(r )[ ] −e−β w(r )−u(r)[ ][ ]
=e−βw(r )[ ] 1−eβu(r)[ ]( )
=g(r) 1−eβu(r)[ ]( )
g(r12)eβu(r12)[ ] =1+ρ g(r13)[1−e
βu(r13)[ ]][g(r23) −1]dr3∫Percus−Yevick (PY) Equation
g(r12)eβu(r12)[ ] =ρ [g(r13)−1−lng(r13)−βu(r13)][g(r23)−1]dr3∫
Hypernetted−Chain (HNC) Equation
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Thermodynamic Functions from Thermodynamic Functions from g(r)g(r) If you assume that the particles are acting through central
pair forces (the total potential energy of the system is pairwise additive), , then you can calculate pressure, chemical potential, energy, etc. of the system.
For an isotropic fluid
UN(r1,...,rN ) = u(rij)i<j∑
E =32NkT+2πρ g(r)u(r)r2
0
∞
∫ dr
P =ρkT−2πρ2
3Vr3 du(r)
drg(r)dr
0
∞
∫
μ=kTlnρΛ3 +4πρ r2u(r)g(r;ξ)drdξ0
∞
∫0
1
∫
where, Λ =h2
2πmkT
⎛ ⎝ ⎜ ⎞
⎠
12 ;ξ is a coupling parameter which varies between 0 and 1.
(Taking a particle in, ξ=1, and out, ξ=0, of the system).
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Molecular LiquidsMolecular 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
For a linear and non-linear molecule the OZ equation becomes the following, respectively
Ω i ≡(φi,θi,ψ i) .
h(r12) =c(r12)+ρ4π
c(r13)h(r23)dr3∫
h(r12) =c(r12)+ρ
8π2 c(r13)h(r23)dr3∫
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Integral Equation Theory for Integral Equation Theory for MacromoleculesMacromolecules If s denotes solute and w denotes water than the OZ
equation can be combined with a closure to give
This is divided into a dependent and independent part
g(rswΩsw)=exp−βu(rswΩsw)+b(rswΩsw) +ρ
8π2 c(rww' Ωww' )h(rsw'Ωsw' )drw'dΩw'∫⎡ ⎣
⎤ ⎦
g(rswΩsw)=8π2P(Ωsw;rsw)g0 rsw( )
g0(rsw) =k(rsw)exp−βu0(rsw)+b0 rsw( )+ρ c0(rww' )h
0(rsw' )drw'∫[ ]
P(Ωsw;rsw) =e−βw(rswΩsw)[ ]
8π2k(rsw)
k(rsw)=1
8π2 e−βw(rswΩsw)[ ]∫ dΩ
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More ApproximationsMore Approximations
is obtained via using a radial distribution function obtained from MC simulation which uses a spherically-averaged potential.
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,
c0(rww' )
c0(rww' )
w(rswΩsw)=μE(rswΩsw)ε'(rsw)
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The Water ModelsThe Water Models BBL Water:
– Water is a hard sphere, with a point dipole = 1.85 D.
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.
uij =uijhs+uij
SP +uijμ
hard-sphere potentialpotential energy of two dipoles for a given orientation
sticky potential used to mimichydrogen-bond interactions.Attractive square-well potential,dependant upon orientation
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Results for SSD WaterResults 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-.
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Results for BBL WaterResults for BBL Water
Radial distribution function aroundfive molecule cluster of water fromtheory (line) and MC simulation(circles)
Twenty-five molecule cluster of water
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ConclusionsConclusions
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?
– how to calculate and ?
Thanks:– Dr. Paul
c0(rww' Ωww' ) h(rsw' Ωsw' )