PhD Work

Thesis Title: Quantum Chemical Studies on the Mixed Molecular Clusters: Quantitative Estimates of Hydrogen Bond Strength and Cooperativity. 

Thesis Supervisor: Dr. Milind Madhusudan Deshmukh

Molecular Tailoring Approach

Inclusion-Exclusion or Cardinality Principle: Consider two overlapping sets A and B. The total number of elements in set A ∪ B can be calculated as the number of elements in A plus those in B minus the number of elements common to both A and B (A∩B).

(A∪B)  =  (A)  + (B) – (A∩B)

Utilizing this idea, one can estimate the energies of hydrogen bonds (HB) in molecular clusters. For example, consider a water–HF tetramer (cf. Scheme ). In the schematic fragmentation procedure of parent HF-water cyclic tetramer (HF)2(H₂O)2, denoted as M (with HF1 and HF2 as hydrogen fluoride molecules, and W1 and W2 as water molecules), is depicted in the scheme. Suppose we want to estimate the HB energy of HB1. The hydrogen bond HB1 consists of an interaction between the F–H bond of HF1 and the O atom of W1.

By removing W1 and HF1, respectively, two primary fragments F1 and F2 are generated from the parent cluster M. Fragment (F1​∩F2​) that is the HF2···W2 common to both the fragment F1 and F2. This is analogous to the intersection term in the inclusion–exclusion formula. Following this logic, the energy of the parent cluster without the HB1 interaction can be estimated using the following expression:

         E(F1​∪F2​) = E(F1​) + E(F2​) − E(F1​∩F2​)                             …(1)

However, this combination lacks the HB1 interaction energy, as HB1 is not present in any of the three fragments but present in parent cluster M. Thus, by subtracting this estimated energy from the actual energy of the full parent molecular cluster M ​, one can obtain the energy of the hydrogen bond HB1:

  EHB (MTA)  = [E(F1​) + E(F2​) − E(F1​∩F2​)] – EM                          …(2)

Note 1: This EHB (MTA) is the energy of HB1 in the cluster which incorporates the cooperativity effects of other HBs in the cluster. Here in this case the calculated energy of HB1 = 14.5 kcal mol-1  at MP2/aug-cc-pVTZ level of theory.

If HB1 is isolated (chopped out) from the network as a dimer HF1···W1, it lacks the cooperativity contributions. The energy of the isolated dimer can be calculated as sum of monomer energies, minus the dimer energy, so called supramolecular approach:

                                                                                                                                        EHB (Dimer) = [EHF1​ + EW1​] – EHF1···W1                                        …(3)

Note 2: Here, the geometry of the dimer is taken from the cluster to preserve consistency. In this case the calculated dimer energy of HB1 = 9.6 kcal mol-1 at MP2/aug-cc-pVTZ level of theory.

In an interconnected network of hydrogen bonds, the strength of an individual bond may be influenced by the network, either enhancing or diminishing its strength. This phenomenon is called cooperativity. The cooperativity contribution to the HB1 can be calculated as the difference between  and the dimer energy .                     

  EHB (Cooperativity)   =  EHB (MTA)  -  EHB (Dimer)             …(4)

The observed cooperativity contribution towards HB1 is 4.9 9.6 kcal mol-1 at MP2/aug-cc-pVTZ level of theory. This is about 34.4% of the HB energy, indicating that significance of cooperativity contributes to hydrogen bond strength in the cluster.

Note 3: Reliability Check: A reliable computational method should not only provide numerical values but also offer a way to verify their correctness. To assess the reliability of HB energies obtained from the MTA-based procedure, one can use the bond additivity principle. According to this, the parent cluster (M) energy should approximately equal the sum of monomer energies and HB energies:

                            EM ≈ ∑ Emonomers + ∑ EHBs                                                                                 …(5)

The typical error in MTA-based estimates is around 7 milli Hartree. Notably, the average error per hydrogen bond, calculated by dividing the total error by the number of HBs, is generally less than 1 kcal mol-1 for most HF-water clusters. This indicates that the MTA-estimated H-bond energies are highly reliable.

Advantages:

1. Accuracy and Reliability Check: Provides robust estimates of individual hydrogen bond energies with minimal errors.

2. Applicability to Weakly Bound Systems: Effective for systems with weak interactions, including hydrogen bonds (HB), pnictogen bonds (PB), chalcogen bonds (CB), halogen bonds (XB), and more.

3. Basis Set Superposition Consideration: It does not significantly affect the results, with BSSE corrections typically less than 1 kcal/mol.

4. Nuclear Quantum Effects: Negligible impact.

5. Scalability: Well-suited for large molecular systems.

Limitations:

1. Time-Consuming: While computationally efficient, the method can still become time-consuming for large systems or when numerous fragments are involved.

2.  Charge Systems: The method may have reduced accuracy when applied to charged systems or systems with significant charge transfer effects, as these may not be well captured by the approach.