Chapter 2. Theory
2.2 Treatments on HB Interactions
We briefly review here the treatment of hydrogen bonding interactions in the successive improvements of the COSMO based model, leading to the current MESP approach. In the case where the two segments in contact may form a hydrogen bond, Klamt [29]
suggested that an additional energy gain (favorable) between the two segments should be accounted for. Therefore, the segment interactions (Eq. 2.1-9) are modified to be
Δ𝑊(𝜎𝑚, 𝜎𝑛) = 𝐶𝐸𝑆(𝜎𝑚+ 𝜎𝑛)2+ 𝐶ℎ𝑏max[0, 𝜎acc− 𝜎hb]min[0, 𝜎don+ 𝜎hb] (2.2-1) where σhb (= 0.0084 e/Å2) is the cutoff charge density above which the segments are considered to be capable of forming a hydrogen bond. The σacc and σdon are the larger (positive, associated with hydrogen bonding acceptor atoms) and smaller (negative, associated with hydrogen bonding donor atoms) values of σm and σn (usually within 0.025 e/Å2), respectively. The coefficient 𝐶ℎ𝑏 is a parameter accounting for the strength of hydrogen bonding interactions.
density greater than the cutoff, σhb, and the strength of the interaction is proportional to the product of the charge excesses of the donor and acceptor segments. Hsieh et al. [27]
refined the description of hydrogen bonding interactions by differentiating the contributions from different hydrogen bonding groups: segments on a hydroxyl groupurface (OH) and segments on the other hydrogen bonding surface (OT), such as on ethers, carbonyls, amines, etc. In other words, the σ-profile is composed of non-hydrogen bonding (nhb) part, the OH part, and the OT part:
𝑃𝑖(𝜎) = 𝑃𝑖𝑛ℎ𝑏(𝜎) + 𝑃𝑖𝑂𝐻(𝜎) + 𝑃𝑖𝑂𝑇(𝜎) (2.2-2) with
𝑃𝑖𝑂𝐻(𝜎) =𝐴𝑖𝑂𝐻𝐴(𝜎)
𝑖 × 𝑃𝐻𝐵(𝜎) (2.2-3)
𝑃𝑖𝑂𝑇(𝜎) =𝐴𝑖𝑂𝑇𝐴(𝜎)
𝑖 × 𝑃𝐻𝐵(𝜎) (2.2-4)
where 𝐴𝑖𝑂𝐻(𝜎) and 𝐴𝑖𝑂𝑇(𝜎) are the surface area of a certain charge density of OH and OT of species i respectively. The 𝑃𝐻𝐵(𝜎) is an empirical function to reduce the probability of a segment having low charge density to form a hydrogen bond and can be written as
𝑃𝐻𝐵(𝜎) = 1 − exp (2𝜎𝜎
02) (2.2-5)
with σ0=0.007 e/Å2.
The reduced part from the HB-level σ-profile are re-added to the non-hydrogen bonding part of σ-profile, i.e.,
𝑛ℎ𝑏(𝜎) 𝑂𝐻(𝜎) 𝑂𝑇(𝜎)
where the superscripts s and t stand for the type of segments and the coefficient 𝐶ℎ𝑏 depends on the type of donor-acceptor segments.
𝐶ℎ𝑏(𝜎𝑚𝑡, 𝜎𝑛𝑠) = {
𝐶𝑂𝐻−𝑂𝐻 if 𝑠 = 𝑡 = OH and 𝜎𝑚𝑡 × 𝜎𝑛𝑠 < 0 𝐶𝑂𝑇−𝑂𝑇 if 𝑠 = 𝑡 = OT and 𝜎𝑚𝑡 × 𝜎𝑛𝑠 < 0 𝐶𝑂𝐻−𝑂𝑇 if s = OH, t = OT and 𝜎𝑚𝑡 × 𝜎𝑛𝑠 < 0 0 otherwise
(2.2-8)
Note that Eqs. 2.2-1 and 2.2-7 use different criteria for describing the strength of HB interactions. In Eq. 2.2-1, the strength of HB is determined based on the charge density of the donor and acceptor segments, whereas in Eq. 2.2-7 the strength is determined based on the types of hydrogen bonding surfaces, in addition to charge density. 𝐶ℎ𝑏 in both Eq.
2.2-1 and Eq. 2.2-7 are used to assess the strength of HB interactions. 𝐶ℎ𝑏 can be obtained from the fitting of VLE or LLE, more derails will be discussed in Chapter 3.
Instead of differentiating the hydrogen bonding surfaces, Chen and Lin [12] proposed to restrict the hydrogen bonding segments to those lying in the direction of lone pairs of the donor atoms (e.g., O, N, and F) and the OH, NH, and FH bonds. The lone pair vector, determined based on the VSEPR theory, and OH, NH, and FH bond vectors are used to find a hydrogen bond center (hbc) on the molecular surface, as shown in Figure 2.2-1 (a).
Furthermore, the HB-surface regions are identified as those located within a certain cutoff distance (RcutHB) of the hydrogen bond center, as shown in Figure 2.2-1 (b) and (c).
(a)
(b)
(c)
-0.025 -0.025 e/Å2
With the directional hydrogen bonding approach, the strength of segment interactions is then determined as the following:
ΔW(σm, σn) = (AES + BES
T2 )(σm + σn)2+ Chbmax[0, σacc - σhb]min[0,σdon + σhb] (2.2-9)
Note that the temperature dependence is introduced (as in the 2010 model) and the form of the hydrogen bonding interaction is the same as in the original model. After applying this modification in describing hydrogen bonding interactions, progress has been made.
In addition, it is worth to mention that the number of adjustable parameters is actually decreased, compared with the 2010 model. (see Table 3.1-1)
Rather than using the VSEPR theory, here we propose to use the MESP to determine the lone pair direction [34, 37-40]. The vector pointing from the donor atom to the local minimum in the MESP is considered as the lone pair direction, and its intersection on the molecular surface is the hydrogen bond center. The local potential minima can be found by searching the values in the potential field determined in quantum chemical calculations (evaluated on a 3-dimensional spatial grid). Each of the local minima found is regarded as a lone pair site. All other subsequent calculations are the same as those in the VSEPR approach (illustrated in Figure 2.2-2).
HB surface on O/N/F HB surface on H
(a) (b) (c)
Figure 2.2-2 (a) The hydrogen bond centers found by VSEPR and MESP for water. The green balls stand for the hydrogen bond centers from MESP while the yellow balls stand
for that in VSEPR. (b) The HB surface region of water from COSMO-SAC (DHB)/VSEPR. (c) The HB surface region within RcutHB water from COSMO-SAC (DHB)/MESP. Note that the hydrogen bond centers for proton donors are unchanged
since the only difference between the two methods is in the treatment of proton acceptors.
One advantage of MESP over VSEPR in determining the lone pair vector is illustrated in Figure 2.2-3 for HF, DMSO and methylisocyanate. For these compounds, it is not able to obtain the specific lone pair vectors by VSEPR. Through MESP, one can obtain lone pair directions without any auxiliary bond directions. More details about auxiliary bond directions and the difficulty of arranging lone pair positions are discussed in Appendix A.
The hydrogen bond centers determined from MESP for these compounds are illustrated in Figure 2.2-3.
(a) (b) (c)
Figure 2.2-3 The hydrogen bond centers determined from MESP for (a) HF, (b) DMSO and (c) methylisocyanate. The green balls stand for the hydrogen bond centers and the
black ones stand for electrostatic potential local minima found by MESP. The VSEPR fails to provide the lone pair direction for these compounds.
The difference in the treatment of hydrogen bonding segments is reflected in the σ-profile
“fingerprint” of the molecule. Figure 2.2-4 compares the water σ-profiles resulting from different COSMO-based models. To make it easier to recognize the different results of each COSMO-based models, the σ-profiles are separated into two parts, the contribution of non hydrogen-bonding segments (non hydrogen-bonding level σ -profile) and the contribution of hydrogen-bonding segments (hydrogen-bonding level σ -profile). Note that one can obtain the same σ-profile after sum up the two contributions for the same molecule. It can be seen that different treatments result in the same hydrogen bonding σ-profiles for extreme charge densities (e.g., |σ |00.016 e/Å2), indicating all 4 methods consider the most polarized segments as hydrogen bonding surfaces. At |σ| around 0.015 e/Å2, the 2002, VSEPR and MESP approaches identify a higher portion of surface regions as a hydrogen bonding surface. The most significant differences in the hydrogen bonding σ-profiles are observed for |σ|<0.014 e/Å2. In general, the 2002 treatment identify the
directional hydrogen bonding methods, VSEPR and MESP, identify the smallest regions as being hydrogen bonding surfaces.
(a)
(b)
Figure 2.2-4 σ-profile of water (a) non hydrogen-bonding level (b) hydrogen-bonding 0
-0.025 -0.015 -0.005 0.005 0.015 0.025
PiAi(Å2)
-0.025 -0.015 -0.005 0.005 0.015 0.025
PiAi(Å2)
σ (e/Å2)