Statistical Models of SolvationStatistical Models of Solvation

Eva Zurek

Chemistry 699.08

Final Presentation

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

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)

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)

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)[ ]

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

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

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

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).

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∫

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

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)

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

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-.

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

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' )