TREATMENT OF SOLVATION EFFECTS
For an overview, see: Quantum Chemistry of Organic Compounds, Mechanisms of Reactions, V.I. Minkin, B. Ya. Simkin, and R.M. Minyaev, Springer-Verlag, New York, 1990, pp. 88-105
Most organic reactions take place in solution. Analysis of the mechanism, kinetics, & thermodynamics of the reaction should include the effect of the medium (solvent effect). For reviews of experimental studies of reaction rate dependence on the medium, see:
F.M. Menger, Tetrahedron, 39, 1013 (1983)
S. Thea and A. Williams, Chem. Soc. Rev. , 15, 125 (1985)
Example of solvent effect: alkali hydrolysis of alkylhalogenides and esters in water and in the gas phase. The gas phase reaction is 20 orders of magnitude faster.
Because of the large number of solvent molecules that would have to be included in a calculation, it is impossible to use quantum mechanics to directly calculate the solute-solvent interactions. Approximations must be made.
To calculate solvation effects, a typical procedure is find the wavefunction & the energy of the solute in the field of the solvent. This is the type of approach used by the AM1/SM2 (AM1/Solvent Model 2) method in Spartan.
Macroscopic (Continuum Solvent) Approximation:
The medium is assumed to exert a weak polarizing effect on the electronic structure of the solute. This is a good approximation for molecules without conjugated bonds & for solutions in which solute-solvent hydrogen bonds are not formed, solute-solvent charge transfer complexes are not formed, or cases in which the solvent does not form a stable adduct with the solute
Total energy of system (solute + solvent) is:
E = Eo + Es,
where Eo is the quantum mechanical energy of the isolated solute molecule & Es is the solvation energy of the solute.
Es = Ees + Edisp + Ecav + Erep,
where
Ees is the electrostatic energy of interaction of the constant and induced dipole moment of the solute with those of the solvent
Edisp is the dispersion energy
Ecav is the energy of formation of a cavity in the dielectric medium where the solute is inserted
Erep is the energy of repulsion of the valence non-bound atoms (large for intermolecular distances less than the sum of van der Waals radii; usually ignored)
When ions are solvated by polar solvents, Ees dominates; when polar molecules are solvated by polar solvents, Ees, Edisp, & Ecav are of similar magnitude. Usually Ees & Edisp are negative, & Ecav is positive. Often Ees can be used to give a qualitative estimate of Es.
From Kirkwood Theory, the electrostatic energy of the interaction of a solute molecule with a solvent assumed to be an isotropic dielectric with dielectric permitivity e is:
° N N
E
es = 1/2 · (n+1) (1-e) / ([n+(n+1)e] · · Qj Qk (rjrk)n / a2n+1n=0 j=1 k=1
x P
n (cos Qjk),where a is the radius of a spherical cavity around the solute, Q
j are the atomic point charges of the N atoms of the solute, & the Pn are the Legendre Polynomials that describe the monopole, dipole, quadrupole,... interactions.Since the terms quickly decrease in size, only the first four are important to include in the calculation.
Molecules generally aren't spherical; some approximation is introduced by this assumption, as well as the choice of the radius of the spherical cavity.
Therefore, this formula gives a relative, rather than absolute, estimate of E
es. Should be used to compare relative values for a series of structurally similar molecules with roughtly equal volume (conformers, tautomers)For n=0, get the Born Eqn. for ion solvation. This term is zero for neutral molecules:
E
0 = -1/2 Q2/a (1 - 1/e)For n=1, get the Onsager dipole energy (a model of the solvent reaction field):
E
1 = -1/2 [ 2(e-1) m2 ] / [2 (e+1) a3]Model Hamiltonians in the Macroscopic Approximation:
The above treatment assumes that E
0 is constant, i.e. that the electronic structure of the solute doesn't change as it passes into the solvent. In fact, this change can be significant, especially for a highly polarizable molecule. For a more exact treatment, construct a model Hamiltonian for the solute:H = H
0 + U,where H
0 is the Hamiltonian of the isolated molecule & U is the operator that describes the interaction of the solute with the medium. If all possible interactions were included in U, this would be the supermolecule approach. Instead only electrostatic interactions are included.Self Consistent Reaction Field
Here, the model Hamiltonian is
H = H
0 - m g <y|m|y>where
g depends on the macroscopic parameters of the medium & on the structure of the first solvation shells,
y is the wavefunction of the solute, &
m is the dipole moment operator.
To find the electronic and geometrical structure of the solute in the presence of solvent, must solve:
H
y = E(R) y,where H = H
0 - m g <y|m|y>.MOLECULAR ELECTROSTATIC POTENTIAL (MEP)
MEP is the potential generated by the charge distribution of the molecule.
Sum of multipole moments of molecule
Scalar quantity
Indicative of chemical reactivity - nucleophilic & electrophilic sites indicated by MEP contour maps
CALCULATION OF THE MEP:
At point r, the MEP, V(r), is given by
V(r) = · Z
a / |Ra - r| - º r(r') / |r' - r| dr',a
where
r
(r') is the electron density at r', Za is the charge on nucleus a located at Ra. The first term is due to the nuclear charge; the second, to the electronic distribution.The electrostatic energy of interaction with a point charge Q located at r is:
Q V(r).
Q is taken as a point charge of +1 which allows V(r) to be called an electrostatic energy (i.e. energy of interaction of the charge distribution with a proton)
V(r) given above is a static potential - the charge distribution is that of an isolated state not perturbed by other interactions. If the molecule interacts with another, then charge transfer & polarization effects must be taken into account..
Calculation of the MEP, as well as derivation of "potential-derived" point charges, has been programmed into Gaussian92 & MOPAC 6.0
APPROXIMATIONS:
Semiempirical MEP's do not always give the same relative energy ordering of the minima as ab initio MEPs.
Potential-derived point charges are atom-centered point charges which are fit to reproduce the (ab initio or semiempirical) MEP. Give an accurate representation of whatever MEP is used in the fit.
Mulliken pint charges do NOT give an accurate representation of the MEP.
REPRESENTATIONS:
2-dimensional contour plots
3-dimensional surface plots