Handbook of Solvents - George Wypych - ChemTech - Ventech!

488 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
ary conditions, the dielectric constant ε of the liquid within the cutoff radius is calculated by
applying the following expression, where V is the volume of the system, k the Boltzmann
constant and T the temperature:
2
4π
M
⎛ ε−1⎞2εrf + 1
= ⎜ ⎟
9 kVT ⎝ 3 ⎠ 2ε
+ ε
rf
[8.127]
In the application of eq. [8.127] careful attention has to be paid to the choice of ε rf, the
dielectric constant to use for the continuum part of the system. Usually, rapid convergence
in calculations is obtained by choosing ε rf ≈ε.
A different approach to the calculation of the dielectric constant via computer simulations
is given by the polarization response method, i.e., to determine the polarization response
of a liquid to an applied electric field E 0. If is the average system dipole moment
per unit volume along the direction of E 0, ε can be calculated using the following expression:
4π
P ⎛ ε−1⎞2εrf + 1
= ⎜ ⎟
0 9 E ⎝ 3 ⎠ 2ε
+ ε
rf
[8.128]
Even if this second approach is more efficient than the fluctuation method, it requires
the perturbation of the liquid structure following the application of the electric field.
As an example, such a methodology has been applied by Essex and Jorgensen 109 to the
calculation of the dielectric constant of formamide and dimethylformamide using Monte
Carlo statistical mechanics simulations (see Section 8.7.2 for details). The simulation result
for dimethylformamide, 32±2, is reasonably in agreement with respect to the experimental
value, 37. However, in the case of formamide, the obtained value, 56±2, underestimates the
experimental value of 109.3. The poor performance here addresses the fact that force field
models with fixed charges underestimate the dielectric constant for hydrogen-bonded liquids.
Other methodologies exist for the calculation of the static dielectric constant of pure
liquids by means of computer simulations. We would like to recall here the use of ion-ion
potentials of mean force, 110 and an umbrella sampling approach whereby the complete probability
distribution of the net dipole moment is calculated. 111
Computer simulations are not the only methods which can be used to calculate the dielectric
constant of pure liquids. Other approaches are given by the use of integral equations,
in particular, the hypernetted chain (HNC) molecular integral equation and the
molecular Ornstein-Zernike (OZ) theory (see Section 8.7.1 for details on such methodologies).
The molecular HNC theory gives the molecular pair distribution function g(12) from
which all the equilibrium properties of a liquid can be evaluated. It is worth recalling here
that the HNC theory consists for a pure liquid in solving a set of two equations, namely, the
OZ equation [8.76] and the HNC closure relation [8.78] and [8.80].
In order to solve the HNC equations, the correlation functions are written in terms of a
mnl
series of rotational invariants Φμv ( 12 ) , which are defined in terms of cosines and sines of
the angles defining the relative orientation of two interacting particles 1 and 2: