Handbook of Solvents - George Wypych - ChemTech - Ventech!

11.1 Theoretical treatment of solvent effects 649
is the reduced number density of the solvent. In the two last equations, a M and a S denote the
intrinsic radii of the solute and solvent molecules, respectively, and ρ is the number density
of the solvent. In the case of an ellipsoidal solute cavity, the SPT cavity formation energy
has been given by the following equation 10
2
2
⎧⎪
⎛ αy ⎞ a ⎡ βy
⎛ y ⎞ ⎤ ⎛ a ⎞ ⎫⎪
ΔGcav = RT⎨1−ln( 1−
y)
+ ⎜ ⎟ + ⎢ + γ⎜
⎟ ⎥ ⎜
⎝1−y⎠
aS⎣⎢1−y
⎝1−y
⎜
⎟ ⎬
⎩⎪
⎠ ⎦⎥
⎝aS⎠
⎭⎪
[11.1.13]
where αβ , and γ denote the geometrical coefficients and a is the characteristic length of the
ellipsoid (the major semi-axis). The scaled particle theory has been extended to dilute solutions
of arbitrary shaped solutes and has been successfully applied for the calculation of the
solvation free energy of hydrocarbons in aqueous solutions. 11
For most practical applications that involve the lowest excited states of the molecules,
the increase in the cavity size during the excitation of the solute molecule would not be accompanied
with a significant energetic effect. However, it may be important to account for
the so-called Pauli repulsion between the solute electronic system and the surrounding medium.
This interaction will force the solute electrons to stay inside the cavity and not to penetrate
into the dielectric continuum (consisted of electrons, too) that surrounds it. The Pauli
repulsion has been modeled by the respective model potentials, e.g., by expanding the potential
in spherical Gaussian shells as follows: 12
2
[ β ( ) ]
PR ∑
i
i exp i o, i
V = b − r −r
[11.1.14]
where bi are the weight factors,β i the exponents and r0,i the radii of spherical shell functions.
In general, the electrons in the solvent cavity could be treated as confined many-electron
systems. 13
11.1.3 THEORETICAL TREATMENT OF SOLVENT ELECTROSTATIC
POLARIZATION ON ELECTRONIC-VIBRATIONAL SPECTRA OF
MOLECULES
The origin of the solvatochromic shifts in the electronic spectra is related to the change in
the electrostatic and dispersion forces between the solvent and the chromophoric solute
molecule in the ground and in the excited state, respectively. The semiclassical approach to
the treatment of the respective effects is based on the assumption that the solute and the solvent
molecules are sufficiently separated to neglect the overlap between the electron distribution
of these two molecular systems. The wave function for the whole system can then be
approximated as the product of the wavefunctions of each individual system, i.e., the solute
and individual solvent molecules:
Ψ=ψ ψ ψ ψ ν
0 0 0 ( )
� [11.1.15]
s( 1)
s( 2)
s( n) a
0 0
where ψ 1)
ψ ψ , ,� , etc. are the wavefunctions of the respective solvent molecules in
0
s( s(2) s(n)
the ground state and ψ ν ( )
a is the wavefunction of the solute molecule in the ν-th state. The
antisymmetry of the total electronic wavefunction is ignored as the individual molecules are
assumed separated enough not to allow the electron exchange. This approximation may not
be valid in the case of strong semichemical interactions between the solute and the solvent