GaussianType Probability for Hydrogen Bonding
The screening charge distribution of each segment is represented by the аprofile which is mere the threedimensional charge density distribution projected into a twodimensional histogram. In original COSMOSAC, the аprofile is given by p(a) = A_{t}(a) A_{t} and for mixture,
The screening charge density is divided into 61 equidistant bins (0.003 e/A2 to +0.003 e/A^{2}). In modified COSMOSAC, the bins are divided into 71 (0.0035 e/A^{2} to +0.0035 e/A^{2}). The major modification was done to incorporate hydrogen bonding in a more realistic way. There was a strict cutoff value of charge density (10.00841 e/A^{2}) to define threshold for hydrogen bonding. Thus the Hb contribution is nonzero only if one segment has a negative charge density less than zero and the other has a positive charge density greater than zero. In this way, Hb is limited to segment pairs of opposite charge and larger magnitudes. This contribution also increases directly with the charge densities of the two segments. However, the assumption that charge density alone determines Hb behavior is not fully consistent with experimental observations, which suggest that, for the most part, Hb acceptors are limited to small electronegative atoms (e.g. nitrogen, oxygen and fluorine) and that Hb donors are limited to hydrogen atoms bound to such acceptors. The use of a strict cutoff for Hb behavior is also not physically meaningful. These weaknesses leave it prone to occasional error.
Lin, Chang, Wang, Goddard and Sandler (2004) first proposed the modifications in COSMOSAC and predicted vapor pressure and enthalpy of vaporization. It includes the van der Waals interactions in solvation free energy calculations and also the ideal solvation term and charge averaging correction term in electrostatic contribution. The residual term originally includes misfit and hydrogen bonding energies. They also proposed to use two different аprofiles: one for nonhydrogen bonding part (nonHb part) and another for hydrogen bonding part (Hb part). The concept of separate аprofiles for hydrogen bonding and nonhydrogen bonding segments is also used in mixtures when Wang, Sandler and Chen (2007) did modifications in original COSMOSAC. On the basis of the definitions of hydrogen bonding, they classified molecules into three categories described as follows:
1. Compounds that have approximately neutral segments; nbutane is an example of this type of molecule.
 2. Compounds that have neutral segments for the аprofile of the nonhydrogen bonding part and hydrogen bonding acceptor segments from among the O, N, F atoms that form the Hb аprofile. Nitromethane is an example of this type of molecule.
 3. Compounds that not only have neutral segments for the nonhydrogen bonding аprofile but also have both hydrogen bonding acceptor segments (from the O, N, F atoms) and hydrogen bonding donor segments (from H atoms connected to the O, N, F atoms); both contribute to the hydrogen bonding аprofile. Water is an example of this third type of molecule.
There are three ways to account for hydrogen bonding in the Hb part of the аprofile (Wang et al., 2007).
 1. The simplest case is that acceptor and donor segments in the Hb аprofile are considered to be sufficiently polar to result in hydrogen bonding, regardless of the magnitude of charge density.
 2. The second case is that a separate аprofile is considered for hydrogen bonding only when their charge density exceeds a certain threshold value (e.g. 10.00841 e/A^{2}), which leads to the step function change in probability, which was used in the original COSMOSAC model.
 3. The third case is that the probability of forming a hydrogen bond is based on a continuous probability distribution function of charge density so that for the acceptor and donor segments, higher the charge density, the greater the possibility of forming a hydrogen bond. In this procedure, the Hb аprofile is weighted with the probability density function and the difference between the Hb and weighted Hb profile is added back into the nonHb аprofile.
The expression for the Gaussiantype probability (P^{Hb} (ct) ) is given by Equation 5.11.
where а_{0} = 0.007 e/A^{2} and а values are calculated from Equation 5.8. Figure 5.1 represents how P^{Hb} (ct) changes with а.
Each bin in the Hb аprofile is multiplied (reduced) by its P^{Hb} (ct) value and the portion of the Hb аprofile thus removed is added back into the corresponding bin of the nonHb аprofile. In this way, the overall аprofile is conserved. From a physical standpoint, this approach allows the model to limit Hb interactions to a certain fraction of the available Hb donor and
FIGURE 5.1
P^{H}b (ct) changes with о (e/A^{2}).
acceptor segments. The greater the magnitude of о for a given segment, the more likely that segment is to exhibit Hb. This provides a physically reasonable alternative to merely limiting Hb to segments whose о value exceeds an arbitrary threshold. With this explanation the expressions for the Hb оprofile and nonHb оprofile will be (Equations 5.14 through 5.16, respectively)
where n = A_{t}/a_{eff},
where i is the compound considered. А^{н} (ст) is the summation of hydrogen bonding segment areas, Af^{on}~^{Hb} (ст) is the summation of nonhydrogen bonding areas and A_{t} is the total segment areas. The interaction energy is the summation of misfit energy and hydrogen bonding energy. The concept of a separate oprofile was later rigorously investigated for the atomspecific (O, N, F) hydrogen bonding oprofile. We will cover most of it in the subsequent sections. For now, with the hydrogen bonding and nonhydrogen bonding oprofile, the expression for segment activity coefficients is also modified. The interaction energy is the summation of misfit energy and hydrogen bonding energy. Recalling the original expression for interaction energy,
This equation is rewritten as
where s and t can be Hb (hydrogen bonding) or nonHb (nonhydrogen bonding). o_{m} and o_{n} are the average charge densities. c_{es} is given by
where /_{pol} = 0.6916 is the polarization factor and e_{0} is the permittivity of vacuum. c_{Hb} (_{т}, a^{s}n) is given by
The segment activity coefficient (Г) of segment m with the charge density of o_{m} is determined from the oprofile and is given by Equation 5.21, which is solved selfconsistently.
Finally, the restoring free energy is obtained by Equation 5.22.
For solvent j will be replaced by sol and for pure compound it will be i. So the activity coefficient of a compound will be given by
As already discussed, cavity formation free energy is modelled by the StavermanGuggenheim combinatorial term. So the above equation reduces to
The restoring free energy makes dominant contribution in calculating the activity coefficient model especially for polar systems. The dispersion term, which is important for pure property calculations, has a very small contribution in the activity coefficient calculation, as a result of its cancellation between the solvent and pure solute phases. So, Equation 5.24 reduces to
The temperature dependence of the coefficient of electrostatic interactions is incorporated by Hsieh, Sandler and Lin (2010). This is particularly interesting for the nonhydrogen bonding oprofile and thus electrostatic interactions vary with temperature.
where c_{es} is the temperaturedependent coefficient of misfit energy and A_{es} and B_{es} are two constants whose values are obtained regressing experimental vapor liquid equilibria (VLE) data. In Equation 5.27, T is temperature in K (Table 5.1).
TABLE 5.1
Values of A_{es} 
and B_{es} 

Parameter 
Value 
Unit 
As 
6525.69 
(kcal/mol)(A^{4}/e^{2}) 
^{B}es 
1.4859 x 10^{8} 
(kcal/mol)(A^{4}/e^{2}) 
Source: Hsieh, C.M. et al., Fluid Phase Equilib., 297, 9097, 2010.
The concept of a separate oprofile later subdivided into functional group specific Hb oprofiles. The idea became popular because the strengths of hydrogen bonds differ according to the environments of atoms involved. Hsieh et al. (2010) first differentiated surfaces of hydroxyl groups (A?^{H} (a)) from other hydrogen bonding surfaces (А°^{т} (a)).
where T represents N or F atoms.
The values of c_{Hb} are in (kcal/mol)(A^{4}/e^{2}) and are taken from the literature (Hsieh et al., 2010).
Like oxygen, a separate Hb oprofile for nitrogen containing compounds was later proposed. The primary concern is amino groups where primary, secondary and tertiary amines are possible. All the Hb acceptors have lone pairs (nitrogen has one, oxygen has two and fluorine has three). These lone pairs show high directional specificity of hydrogen bonds and alignment with donor determines the strength for interaction. Thus the oprofile becomes
The values of c_{Hb} are in (kcal/mol)(A^{4}/e^{2}) and are taken from the literature (Hsieh & Lin, 2012).