Computational Approaches Towards Modeling Finite Molecular Assemblies: Role of Cation-π, π–π and Hydrogen Bonding Interactions

Abstract

The current review focuses on theoretical approaches for various kinds of noncovalent interactions such as cation-π, π–π stacking, and hydrogen bonding which govern the formation of finite molecular assemblies. Cation-π interactions were shown to be arguably the strongest of noncovalent interactions through a series of systematic computations and their comparison with experiments. The major factors affecting cation-π interaction, including the role of solvation, nature and size of systems and regioselectivity for cation attack have been discussed using theoretical studies. The mutual dependence of cation-π interactions with the neighboring non bonded interactions, such as stacking and hydrogen bonding has been explained. Cooperativity in systems containing cation-π interactions has been quantified. Relevance of cation-π and π–π interactions in function and structure of biological molecules and materials has also been dealt with. The role of quantum chemical calculations and molecular dynamics simulations in understanding the structure and energetics of nonbonded interactions is explored.

18.1 Introduction

Chemists have made outstanding progress in understanding bonding principles which primarily explore how molecules make and break bonds thereby accounting for stability and reactivity of molecules. Covalent bonds result from the interaction of atoms leading to the formation of molecules under certain conditions. The molecules formed have properties completely different from those of the original systems [1]. Noncovalent interactions were first recognized by J. D. van der Waals in the later part of nineteenth century helping him to reformulate the equation of state for real gases [2]. These interactions lead to the formation of molecular clusters while covalent interactions lead to the formation of a classical molecule. Interactions observed between any two entities whose valencies are satisfied form noncovalent interactions. In the last couple of decades the study of material formation from molecules has gained considerable attention. Thus the importance and significance of noncovalent interactions has been widely recognized by material scientists and biologists. Noncovalent interactions are known to act at long distances of several angstroms unlike covalent bonds. Although these interactions are traditionally considered to be weak, their strength covers a substantial range from a few kJ/mol to several hundreds of kJ/mol depending on the type of interaction. Cation-π interactions, hydrogen bonding, π–π interactions, hydrophobic interactions and van der Waals forces are representative of different kinds of noncovalent interactions.

Nature has adopted noncovalent interactions in its fold for several crucial processes specifically in context of biomolecules. The synergistic interplay between different kinds of nonbonded interactions is relevant to maintain the structure of important bio-macromolecules like DNA and protein and in retaining the fidelity of information processing needed for normal life (Fig. 18.1).

Fig. 18.1

Representation of important noncovalent interactions observed in small molecules and larger macromolecules

The gecko lizard is a classic example of the effect of dispersion based noncovalent interactions creating an incredible ability of the geckos to climb rapidly up smooth vertical surfaces. Full et al. [3] show how a large animal such as a gecko can fully support its substantial body weight by the noncovalent interactions between the few hundred thousands of keratinous hairs, or setae, on their feet and the surface using a two-dimensional micro-electro-mechanical systems force sensor and a wire as a force gauge. Adhesive force values support the hypothesis that individual seta operate by van der Waals forces. The role of noncovalent interactions in the process of molecular recognition process has attracted substantial interest in recent times [4]. Several early studies have demonstrated how cation-π and π–π interactions in the active sites of numerous enzymes involved in catalysis, ion channels as well as ligand recognition drive the functional aspects in these molecules [5–12].

Finite functional assemblies evolved over millions of years have been a source of inspiration for scientist’s world over. Pioneering studies by J. M. Lehn and coworkers lead to the development of numerous chemical self-organizing systems, furthering the generation of supramolecular architectures from their components assembled through important noncovalent interactions like hydrogen-bonding and ligand-metal ion recognition processes. These supramolecules include double and triple helicates, circular inorganic helices, multi compartmental nanocylinders, grid-type entities, ordered polymetallic arrays etc. [4, 13, 14]. Important studies revealing the implication of self assembly in large complex biological molecules involved generation of a closed circular helicate assembly analogous to viral DNA [13]. Proteins form remarkably intricate structures by component self-assembly, such as the icosahedral framework combining 180 subunits in the cowpea chloritic mottle virus [15]. These frameworks create highly specific pockets of chemical space that can induce selective reactivity [16, 17]. The role of weak interactions in controlling the three dimensional structure of macromolecules and supramolecular assemblies has thus been well recognized [18–20].

The experimental determination of noncovalent interactions relies heavily on various spectroscopy methods. Theoretical studies have also played a vital part in understanding different kinds of noncovalent interactions. Stacked nucleic acid base pairs which model DNA have been subjected to ab initio calculations to determine the theoretical methods suitable to generate reliable characteristics [21]. Besides stacking interactions numerous theoretical and experimental studies have been performed on water and other molecular clusters to understand hydrogen bonding and other noncovalent interactions [22, 23]. The interactions of a number of π systems with different ligands including rare gas atoms and complex clusters of water and methanol as model systems was reviewed by Kim et al. These studies reveal how interaction energies of these π-complexes are dependent on both the nature of the ligand and the π-system and establish how in all cases, the repulsive interactions have a vital role in governing the observed geometry [24]. Dougherty’s group has established the generality of cation binding to π face of an aromatic structure through gas phase measurements and studies of model receptors in aqueous medium [5–8]. The importance of cation-π interactions in chemistry and biology particularly in proteins have been demonstrated especially with the cationic side chains of basic amino acids and the amino acids phenylalanine, tyrosine and tryptophan [5]. Several prototypical cation-π systems, including structures of relevance to biological receptors and prototypical heterocyclic systems important in medicinal chemistry have also been thoroughly studied and the dominant role played by the electrostatic component in cation-π interaction was clearly delineated with quantum chemical calculations. They demonstrated that the electrostatic potential (ESP) evaluated at a single point above the center of a substituted aryl ring predicts the strength of the cation-π interaction, in particular more negative ESPs indicate stronger interactions [6]. Using the ability to design and evaluate low-molecular weight model systems while surveying several PDB crystal structures alongside Gokel et al. confirm that arenes can serve as π donors for alkali metal cations [25]. Recent studies by Houk et al. reveal π-polarization models of cation-π interactions are flawed and that substituent effects arise primarily from direct through-space interactions with the substituents [26]. Studies modeling not only DNA base pairs but also amino-acid pairs at the CCSD(T)/CBS limit yield accurate interaction energies and demonstrate how wave function theories seem to reliably explain hydrogen bonding and stacking interactions while DFT fails to describe dispersion bound stacking interactions [27]. Sherill et al. performed an assessment of theoretical methods for nonbonded interactions where a comparison of complete basis set limit coupled-cluster potential energy curves for the benzene dimer, methane dimer, benzene-methane, and benzene-H₂S was undertaken with various spin component scaled methods SCS-MP2, SCS-CCSD, DFT methods corrected for dispersion (DFT-D) and meta-generalized-gradient approximation functionals (M05-2X and M06-2X) [28]. They suggest that a combination of general approximations, which significantly reduce computational time and newer approximate electronic structure methods, provide fairly reliable results for the nonbonded interactions making trustworthy computations for much larger chemical systems a possibility. Elaborating on the chemical variety and energy span of numerous kinds of hydrogen bonds the concept of a ‘hydrogen bridge’ has been suggested recognizing a hydrogen bonding interaction without borders considering variation in its relative covalent, electrostatic, and van der Waals content [29]. Special type of cation-π interactions, metal ligand aromatic cation-π interaction (MLACπ) where ligands are coordinated to a metal interacting with aromatic groups been observed in several metalloproteins, DNA and RNA have been reviewed by Zaric from a computational perspective [30]. The field of noncovalent interactions has thus been a subject of extensive and thorough study motivating analysis from varied perspectives.

An atomic level comprehension of condensed phase structure warrants a clear understanding of the role, strength and relevance of noncovalent interactions and how they mutually influence each other. In the following discussions we present a detailed view on three prominent classes of nonbonded interactions namely cation-π interactions, π–π stacking and hydrogen bonding while emphasizing their contemporary relevance and significance (Fig. 18.2). In our efforts to employ rigorous computations on the medium sized molecules which involve noncovalent interactions several interesting features were unraveled. We have focused on metal ion interactions as well as molecular clusters in chemical and biological systems.

Fig. 18.2

Major factors which govern the three noncovalent interactions, cation-π, π–π and hydrogen bonding

The primary emphasis in this review is on cation-π interactions. The relative preferences of different kinds of alkali and alkaline earth metals for binding to different aromatic systems, the dependence on size of different π systems and solvation have been dealt with in detail. Besides cation-π interactions we present studies on hydrogen bonded clusters and π–π stacked systems of different sizes. Two databases, cation aromatic interaction database and the aromatic-aromatic interaction database have been developed based on our studies. An interesting observation is the mutual enhancement in interaction strength when noncovalent interactions work in concert with each other. The extent of cooperativity in these interactions is analyzed with an emphasis on its effect on controlling structure using different model systems. The role of quantum chemical calculations and molecular dynamics simulations in understanding the structure and energetics of these interactions is studied.

18.2 Noncovalent Interactions

18.2.1 Cation-π Interactions

Cation-π interactions are amongst the strongest noncovalent interactions ranging from 5 to 80 kJ/mol. They include interactions between any aromatic group (may be part of amino acid side chain) and any cation (alkali, alkaline earth metals, transition metals and cations such as NH ⁺₄ , NMe ⁺₄ , SH ⁺₃ , OH ⁺₃ etc.). We focus primarily on cation-π interactions of alkali and alkaline earth metals in our studies. The bonding observed in cation-π interactions of transition metals are quite different from those seen with alkali and alkaline earth metals due to the presence of d orbitals in these metals. Although they are important in biology they do not fall under the purview of interactions mentioned here and are out of the scope of the current review. Electrostatic interaction and induction are two important components for the metal ion-aromatic interaction.

Benzene, fluorobenzene, anisole, nucleobases and nitrogen heterocycles have been used as model aromatic groups in numerous studies [31–34]. Dougherty et al. have pursued studies on cation-π interactions involving alkali and alkaline earth metal cations that are significant in biological macromolecular structure and function [5–8]. Dunbar et al. calculated the Na⁺, K⁺ affinities of phenylalanine, tyrosine, and tryptophan using kinetic method [35]. The bond dissociation energies of cation-π complexes of anisole and alkali metal cation has been determined by Rodgers et al. using the collision-induced dissociation technique and also by theoretical studies at MP2(full)/6-311+G** level [36]. They report the energetics of Na⁺, K⁺ complexes with aromatic amino acids such as phenylalanine, tyrosine, and tryptophan using the guided ion beam tandem mass spectrometry method. Zhu et al. have employed B3LYP/6-311++G** calculations to systematically explore the geometrical multiplicity and binding strength for the alkali and alkaline earth metal complexes with nucleobases (namely adenine, cytosine, guanine, thiamine, and uracil) [37]. HF and MP2 computations on the interaction of mono- and divalent metal ions with nucleobases have been reported by Sponer et al. [38]. Garau et al. [39] have studied the nonbonded interactions of different anions with benzene using a topological analysis of the electron density and molecular interaction potential with polarization (MIP) energy partition scheme calculations. Further quantitative estimation of cation-π and anion-π interactions was carried out, emphasizing the changes in the aromaticity of the ring upon complexation and charge-transfer. Studies on the dependence of basis set quality, electron correlation, and structural variations on the interaction of the alkaline-earth metal divalent cations with benzene were undertaken by Tsuzuki et al. [40]. Ikuta has described the interaction between the monovalent cations (Li⁺, Na⁺, and K⁺) with anthracene and phenanthrene molecules at the hybrid DFT [41]. Thus very subtle factors such as the size of the system and change in the nature of bond formation can substantially modulate the strength of the nonbonded interactions.

In the context of evaluating the major factors which play a role in cation-π interactions we have considered the impact of various kinds of cations such as alkali, alkaline earth metal cations and their site of attack, analyze the size effect of π systems on the nature of cation-π interactions, understand the role of substituents in aromatic rings and importantly gauge the impact of solvent on the behavior of the interaction as well as on individual metal (Fig. 18.2). All the aspects mentioned above have been tackled using different model systems which mentioned below.

• 1. Two model studies are taken into account while exploring the role of different kinds of cations. First cation interaction with aromatic side chain motifs of four amino acids (viz., phenylalanine, tyrosine, tryptophan and histidine) [42] were investigated followed by a study where the structural and energetic preferences of π, σ and bidentate cation binding to aromatic amines (Ph–(Ch₂)_n–NH₂, n = 2–5) is theoretically studied [43].

• 2. π and σ complexation of various heteroaromatic systems which include mono-, di-, and tri substituted azoles, phospholes, azines and phosphinines with various metal ions, viz. Li⁺, Na⁺, K⁺, Mg²⁺, and Ca²⁺ [44]

• 3. Exploring the size dependence of cyclic and acyclic π systems on strength of interaction [45, 46] and

• 4. Solvation of metal ions as well as cation-π complexes [47–51].

In the following section we provide an overview of the four factors mentioned above, revealing the diverse behavior of cation-π interactions.

18.2.1.1 Computational Details

Stabilization of all the mentioned types of noncovalent complexes is due to favorable energy. It implies that the energy of a complex is lower than the sum of the energies of its separated subsystems if a complex is formed in vacuum. However the situation in presence of a solvent particularly in the aqueous phase is quite different. For our studies on noncovalent interactions, the strength of interaction is measured in terms of interaction energy. For a binary cation-aromatic complexes, the interaction energy (IE) was calculated as the difference of the total energy of the complex and sum of the energies of the aromatic system and the metal taken

$$ {\text{IE}} = {{\text{E}}_{\text{complex}}} - ({{\text{E}}_{{{\text{aromatic}}\,{\text{system}}}}} + {{\text{E}}_{\text{cation}}}) $$

(18.1)

In case of ternary complexes involving cation-π (IB) and π–π (BB) interactions, the interaction energies in the ternary complex was calculated based on equations mentioned below

$$ \Delta {{\text{E}^{\prime}}_{\text{IB}}} = {{\text{E}}_{\text{IBB}}} - ({{\text{E}}_{\text{BB}}} + {{\text{E}}_{\text{I}}}) + {\text{BSSE}} \vspace*{-3pt}$$

(18.2)

$$ \Delta {{\text{E}^{\prime}}_{\text{BB}}} = {{\text{E}}_{\text{IBB}}} - ({{\text{E}}_{\text{IB}}} + {{\text{E}}_{\text{B}}}) + {\text{BSSE}} $$

(18.3)

Where, E_IBB, E_BB, E_IB, E_B and E_I are the total energies of the ternary, binary and monomeric systems. The interaction energy for molecular clusters was evaluated using the following equation

$$ {\text{IE}} = {{\text{E}}_{{{\text{cluster}} - {\text{energy}}}}} - {\text{n}} * {{\text{E}}_{\text{monomer}}} $$

(18.4)

Where E_{cluster-energy} – Total energy of water cluster and E_monomer – Total energy of a single monomer unit. To evaluate basis set super position error the Boys and Bernadi method has been applied for the purpose of counterpoise correction (CP) in the different systems wherever mentioned [52]. All calculations reported in the model systems have been performed using Gaussian 03 suite of programs [53]. Based on Bader’s AIM (Atoms-in Molecule) analysis we have calculated the electron density values at critical points as mentioned in specific cases [54].

18.2.1.2 Impact of Different Cations and Preferential Site of Binding to Aromatic Group

A systematic analysis of cation (M=H⁺, Li⁺, Na⁺, K⁺, Mg²⁺, Ca²⁺, NH ⁺₄ , NMe ⁺₄ ) binding with different aromatic side chains is undertaken [42]. Scheme 18.1 gives a representation of all the model systems considered for study. The regioselectivity aspect of cation binding to aromatic side chain motifs and protons has been investigated. The regioisomers of protonated complexes assess the relative propensity of various sites for proton attachment. The covalent binding of proton to the aromatic ring carbon atoms is contrastingly different to the other metal cations as well as ammonium ions which are found to form cation-π and cation–heteroatom interactions. The NH ⁺₄ and NMe ⁺₄ ions have shown N–H…π interaction and C–H…π interaction with the aromatic motifs. The interaction energies of N–H…π and C–H… π complexes are higher than hydrogen bonding interactions; thus, the orientation of aromatic side chains in protein is effected in the presence of ammonium ions. However, the regioselectivity of metal ion complexation is controlled by the affinity of the site of attack. In the imidazole unit of histidine the ring nitrogen has much higher metal ion (as well as proton) affinity as compared to the π-face, facilitating the in-plane complexation of the metal ions. The interaction energies increase in the order of benzene-M < toluene-M < para-hydroxy benzene-M < methyl indole-M < methyl imidazole-M for all the metal (M) ions considered. Similarly, the interaction energies with the model systems decrease in the following order: Mg²⁺ > Ca²⁺ > Li⁺ > Na⁺ > K⁺ = NH⁴⁺ > NMe⁴⁺. The bond lengths between ring atoms show a definite increase on formation of cation-π complex while the bonds between ring atom and substituent group are noticeably shortened. An important observation is also that is in presence of an alternative basic group, the covalent interaction appears to overtake the cation-π interaction. Thus, the proton and metal ion complexation in biological systems with aromatic motifs can be substantially different.

Scheme 18.1

The two principle interactions in proteins with metal ions are cation-π interactions and σ interactions of metal with amines [55–57], besides the coordination of the metal ion with the side chain of acidic residues. Cation-π interactions are competitive with cation-σ interactions within the same molecules and this plays an important role in stabilizing chelating conformations [58]. To probe into this aspect of regioselectivity and preferential binding of the cation in the same molecule MP2/aug-cc-pVTZ calculations were carried out [43]. Three distinct binding preferences, namely, monodentate binding in π and σ fashions to aromatic and amine groups, respectively, and the bidentate mode of binding of Li⁺, Na⁺ and Mg²⁺ ions with aromatic amines (Ph–(CH₂)_n–NH₂, n = 2–5) were considered.

The model systems devised to examine the binding strength of the interactions where the aromatic and amine motifs are not interconnected are shown in Schemes 18.2 and 18.3. The main questions addressed here were regarding the relative preference of Li⁺ and Mg²⁺ to bind to an aromatic ring in a π fashion and of amines to bind in a σ fashion, the differences in conformations of complexes involving metal compared with those of protonated ones and the extent of structural reorganization required to achieve bidentate confirmations.

Scheme 18.2

Scheme 18.3

The results obtained in this study reveal how Li⁺ and Na⁺ have displayed a consistently higher propensity to bind with the amine group compared to the aromatic group. In contrast, Mg²⁺ binds more strongly to the π systems compared to the amine group. From the mono- to bidentate, the chelation gain in the binding energy for Mg²⁺ is about three to four times greater than that of Li⁺ and Na⁺. Cation-π interactions seem to show a higher dependence on the charge of the metal ion compared to that of the cation interaction with lone-pair-bearing molecules. The monodentate binding of metal ions with −NH₂ has a small variation in the interaction energies as the spacer chain length increases. While the binding of Mg²⁺ to an aromatic ring is sensitive to its side-chain orientation and its length, Li⁺ and Na⁺ complexation is independent of spacer chain length and orientation. Structural reorganization due to Mg²⁺ complexation is slightly higher than that due to Li⁺ or Na⁺ complexation. Thus, the divalent metal ion complexation leads to a significant variation in the macromolecular structure and the function. The charge on the metal ion depends on the side-chain length and the mode of complexation of metal ions with the aromatic amine (mono or bidentate). Regioselectivity and the nature of cation involved thus play a vital role in determining the effective strength of cation-π interactions.

18.2.1.3 σ vs π Binding of Cations to Heteroaromatic Systems

The preferential binding mode of alkali and alkaline earth metal cations to aromatic rings in presence of different substituted aromatic groups has been investigated [44]. While exploring the metal ion binding to the heteroaromatics the following issues were addressed: (a) the relative strength of σ and π binding modes, (b) the regioselectivity of metal ion binding, and (c) all possible minima of metal ion and ring complexes. The presence of several crystal structures which have Li⁺ and Na⁺ bound to phosphorus systems in CSD (Cambridge Structural Database) and PDB (Protein Data Bank) [59] motivated choosing cation-π complexes of phosphorus containing heteroaromatics, while highlighting the contrasts in the structural and binding affinities between the nitrogen and phosphorus containing heterocyclic systems. The heterocyclic systems studied include mono-, di-, and trisubstituted azoles, phospholes, azines and phosphinines.

Azoles and azines form stronger σ complexes in sharp contrast to the phospholes and phosphinines which form stronger π complexes with the metal ions Li⁺, Mg²⁺, and Ca²⁺. With Na⁺ and K⁺ there is little difference between the σ and π complexation energies for phosphorus heteroaromatics. The nitrogen heterocyclic system 4H- [1, 2, 4] triazole and pyridazine formed the strongest σ complex among the five- and six-membered heteroaromatic systems considered. The σ and π complexation energy of azoles and azines is found to decrease as the heteroatom substitution increases in the ring. In contrast, the complexation energies of both phosphole and phosphinines show little dependence on the number of phosphorus atoms in the ring. The complexation energy of a given heteroaromatic system with various metals followed the order Mg²⁺ > Ca²⁺ > Li⁺ > Na⁺ > K⁺. Among the chosen metals, Mg²⁺ show a higher propensity to bind to the phosphorus systems while forming a π complex. The phosphinine Mg²⁺ complexes were found to have comparable complexation energy to benzene Mg²⁺ complex. The metal preferred to bind in the bidentate fashion to azoles and azines, whereas for the phospholes and phosphinines, no such binding preference was observed. For both azoles and phospholes, the metal binds away from the electron-deficient heteroatom. Thus, a very important contrast between nitrogen- and phosphorus containing heteroaromatics is revealed through this study.

18.2.1.4 Cation-Aromatic Database

Important studies by Gallivan and Dougherty [8] report a quantitative survey of cation-π interactions in high-resolution structures in the Protein Data Bank (PDB). Based on an energy criterion for identifying significant side chain interactions, they conclude how the geometry is biased toward one that would experience a favorable cation-π interaction when a cationic side chain is near an aromatic side chain. Energetically significant cation-π interactions are available from their program CaPTURE [8]. Besides several other studies focus on using different criterion such as angle, distance etc. in order to understand the propensity of different cation-π interactions in the PDB [60]. In most of these studies, the cation is the side chain protonated nitrogen of the basic amino acid residues. An understanding of the metal ion interactions however is rather scarce. Understanding the nature of metal cation-π interactions is key to realize and model various processes in metalloproteins. Towards this end we developed an exhaustive database (Cationic-Aromatic Database) of Metal-aromatic motifs present in the PDB based on relevant geometrical criteria so as to help in screening and developing new methods for identifying and ranking these interactions [61]. Metal-aromatic motifs include both σ as well as π interaction geometries in the database.

A statistical analysis of this database reveals that the aromatic side of the histidine moiety prefers to bind in a σ fashion, while the rest show a propensity to bind in π-fashion. The predominance of σ-type interaction in His moiety may be traced to the presence of electron deficient nitrogen atom. Considering cation–π distance the following trend is observed Trp > Phe > Tyr. Even though most of the cation–aromatic interactions are contributed by basic amino acid residue cations Lys, Arg, and His, metal ions too have significant number of cation–aromatic interactions (cation–σ and cation–π). Coming to cations of basic amino acid residues, Arg cation forms interactions within cation–π interaction distance range. Among all the metal cations studied, Zn followed by Fe show more cation–aromatic interactions. An on-line tool has been incorporated in the database, which furnishes all the cation–π interactions for any new protein. This database is available in the public domain.

18.2.1.5 Size of System

The size of the π-system chosen has important implication on the structural and functional aspects of metal binding. To explore the size effect calculations were performed on the cation–π complexes of Li⁺ and Mg²⁺ with the π-face of linear and cyclic unsaturated hydrocarbons [45]. In the case of the acyclic π-systems, we started with the simplest system, e.g.: ethylene followed by buta-1,3-diene, hexa-1,3,5-triene, and octa-1,3,5, 7-tetraene with 2, 3 and 4, conjugated π units, respectively. These linear systems with two and more number of π units can have various conformations wherein the π units can have cis, trans or a combination of both cis and trans orientations. Similarly for cyclic systems cyclobutadiene, benzene, cyclooctateraene, naphthalene, anthracene, phenanthrene and naphthacene have been included. Thus a wide range of sizes for aromatic systems have been covered.

Similar to the earlier case where the role of differing cations was noted, here the impact of the variation in the interaction energy of linear and cyclic conjugated π-systems as a function of the size of the π-system is probed. The interaction energies depend on the size of the π-system, with larger molecules exhibiting higher complexation energy. The increase in the interaction energy can also be correlated to the strain induced in the system upon metal ion complexation and also upon charge transfer. In acyclic systems, which have higher flexibility to reorient the structure upon metal ion complexation, the stabilization energy is higher than in cases where the π-system is highly distorted from the idealized planar form. The electrostatic interaction seems to be a major factor and there is some correlation between the interaction energy and the charge transfer. However, this study does not show any quantitative correlation between interaction energy and the cation–π distance. Thus, the size of the π-system which can be estimated as the number of double bonds present (n) shows a dramatic increase when n goes from 1 to 9. This increase is uniform both in cyclic and acyclic systems and thus the number of double bonds in conjugation may be taken as a general signature, to estimate the cation–π binding at least in the gas phase.

In the context of assessing the size effect we also reviewed the impact of curvature of polycyclic systems in terms of their role in cation-π interactions. B3LYP/6-311+G** calculations were performed to assess the effect of curvature and remote electronic perturbations on the cation–π interactions of a large series of aromatic hydrocarbons and their hetero analogs shown in Scheme 18.4 [46].

Scheme 18.4

In all cases, except corannulene, the π-system is structurally and electronically modified aromatic six-membered ring. The metal ions (Li⁺ and Na⁺) bind to both the faces of the buckybowls arising to two possibilities for π-complexes; convex face binding is preferred over concave binding in all the cases by about 1–4 kcal/mol. Both the bowl and planar forms yield similar binding energies, indicating that the curvature of the buckybowls has very little effect on the complexation energies. The strength of cation binding to the six-membered ring is mainly controlled by electronic factors, while the curvature plays only a marginal role. Heterosumanene or heterotrindene has a very high complexation energy compared to other compounds when X═NH. The interaction energies observed in this class of compounds exhibit a wide range from 25–59 to 15–43 kcal for Li⁺ and Na⁺ ions, respectively. Importantly, the curvature and flexibility of the curved surfaces are virtually undisturbed upon metal ion complexation.

18.2.1.6 Solvation

Biological systems exist in aqueous phase and are influenced by presence of a solvent not only in terms of retaining structure but also from a functional view point. Modeling studies incorporate a solvent to model its impact on strength of different noncovalent interactions. The solvent effect in quantum chemical calculations on model systems is addressed by including an implicit solvent or by explicit addition of water molecules at various positions to gauge the solvent effect on nature of nonbonded interaction. Dougherty et al. have shown that cation-π interactions are frequently found on the surfaces of proteins and exposed to aqueous solvation in a study which projects the importance of solvent on cation-π interactions [62]. An analysis, of 2,878 energetically significant cation-π interactions present in the data set of 593 non homologous proteins shows that 20% of the aromatic amino acid residues have at least 20% of their surface exposed to water, whereas 70% of all cationic amino acids expose more than 20% of their surface to water. Vaden et al. investigated the hydrated Na⁺, K⁺ ion complexes with benzene and phenol using infrared spectroscopy, wherein hydrated ions interact with the π-system of benzene as well as the phenol oxygen atom [63]. According to their analysis in hydrated environments, cation-π interactions are size-selective toward the hydrated ions. Earlier computational studies on the influence of water molecules on cation-π interactions revealed that the strength of the cation-π interactions gets substantially reduced when solvated with water [64].

An analysis performed on the structures of PDB and CSD clearly demonstrates that a higher number of cation-π interactions exist in the distance range of 3–4 Å from the cation to the centroid of the aromatic system. While structural parameters obtained by X-ray analysis are normally in excellent agreement with computations on single molecules in the gas phase, the role of the environment on the geometric parameters appears to be rather critical for cation-π interactions. A quick look at a cation-aromatic database built reveals that the frequency of cation-π interactions is relatively high in cation-π distances around 3.5–4.5 Å. These bond lengths are well over 1–2 Å longer than the optimized geometries obtained for smaller model systems using reliable quantum chemical methods. To identify possible reasons for this disparity between bond distances from x-ray crystal structures and theoretical studies, the effect of explicit solvation of the cation-π system where the first solvation shell of cations is saturated with water is considered. This provides a realistic description of the first solvation shell, not only in a manner relevant for biomolecules, but also to mimic the saturation of metal ion coordination (Fig. 18.3).

Fig. 18.3

Impact of solvation on metal, strength of a noncovalent interaction and relevance of site solvated

As part of the effort to gauge the relevance of solvation using theoretical methods we initially performed studies on hydrated metal ion (Li⁺, Na⁺, K⁺, Mg²⁺, Ca²⁺) complexes with benzene (cation-π) as model systems (Scheme 18.5) [47]. The geometrical parameters and interaction energies of these complexes were evaluated using B3LYP/6-31G (d,p) level of theory.

Scheme 18.5

This study reveals that the strength of interaction of an arene, modeled by benzene, with a fully solvated metal ion is nearly half of the gas-phase complex for K⁺, while it reduces to almost one-fifth for the divalent Mg²⁺. There is a stepwise decrease in the strength of cation-π interactions as the metal ion is solvated and thus the cation-π strength is actually much smaller in the condensed phase. However, all energies are still higher than the interaction energy of the water-benzene complex (∼3 kcal/mol). Further coordination of metal ions with water molecules results in lengthening the cation-π distance. Interestingly, while the potential in the gas phase for the cation-π may be very tight, it appears to be loose in the condensed phase. This is precisely the reason for the adaptation of a large span of cation-π distances that are observed in protein databases. Thus there exists a tremendous disparity in the behavior of cation-π interactions in gas and condensed phases.

Sequential attachment of water molecules via the explicit solvation mode to Li⁺, K⁺ and Mg²⁺ complexes with benzene was pursued [48]. This study reveals how cation-π interaction energy is sensitive to the site of solvation of cation-π systems, and size and charge of the metal ion. Compared to Mg²⁺ and Li⁺, K-π interaction energies are more competitive with metal-water interaction energies. The approach of the water molecules determines how the cation-π strengths are altered upon solvation, the strength is attenuated when water molecules selectively surround the metal ion while it is enhanced upon selectively solvating the π system. Thus, solvation of metal ions lowers the interaction energy and causes lengthening of the cation-π distance, while when water molecules selectively approach the π system a starkly contrasting effect results, i.e., increase of interaction energy and shortening of the cation-π distance. Although the qualitative observation is virtually similar in the three cations (Li⁺, K⁺, and Mg²⁺) studied, the solvent-assisted augmentation and attenuation of the cation-π strength depending on the face of metal ion attack is more dramatic in the case of K⁺ ion. RVS analysis indicates that the major contributions to the cation-π interaction energy are coming from the POL (polarization) and CT (charge transfer) energy terms of benzene. The topological analysis of electron density distribution within Bader’s atoms in molecules theory (AIM) consolidates how depending on the site of solvation, cation-π interaction energy becomes stronger or weaker. The electron density values at BCPs confirm the C–H· · ·O bridge type hydrogen bonding between the oxygen atom of water and the C–H group of benzene.

A more recent study done towards understanding impact of solvation on cation-aromatic interaction involved quantum chemical calculations done on the binding of hydrated Li⁺, Na⁺, K⁺, Mg²⁺, Cu⁺, and Zn²⁺ metal ions with biologically relevant heteroaromatics such as imidazole and methylimidazole [49]. This study reveals how alkali, alkaline earth metal, and transition metal ions binding to imidazole and methyl imidazole motifs, which model the histidine side chain, have a strong preference to bind to the lone pair of nitrogen. The water molecules virtually fill the first solvation shell to start the explicit solvation process. The metal ions show higher propensity to bind to the imidazole motif compared to water or benzene. Histidine thus is the most ubiquitous amino acid residue in metal binding sites. The study carefully analyzes hydrogen-bond-donating abilities of the N–H group of histidine upon unsolvated and solvated metal ion binding to its other nitrogen [N(3)] center. The study ensures that the first solvation shell is satisfied for all metal ion complexes. The presence of solvent molecules at the N(1) position of imidazole or methylimidazole enhances the metal ion binding at the N(3) position, and the binding of metal ion at the N(3) position strengthens the hydrogen bonding at the N(1) position. The shift in the vibrational frequencies of N–H indicates that the presence of metal ion at the N(3) position strengthens the hydrogen bond at the N(1) position. The study demonstrates a higher cooperative effect in the hydrogen bonding due to transition metal ions compared to alkali and alkali earth metals.

There is also a high interest in the individual hydration characteristics of metal ions. The study of hydrated metal ions in gas phase provides a connection between the essential chemistry of the isolated ion and that in the solvent. Solvated ions also appear in high concentrations in living organisms, where their presence or absence can fundamentally alter the functions of life. In fact, the structure and dynamics of solvation shells have a large impact on any chemical reaction of metal ions in solution. Solvation of alkali and alkaline earth metal ions in particular has stimulated considerable theoretical interest. Different experimental techniques such as high-pressure mass spectrometry (HPMS), collision induced dissociation (CID) using guided ion beam mass spectrometry, blackbody infrared radiative dissociation (BIRD) kinetics, and electrospray ionization (ESI) with Fourier transform mass spectrometry have been adopted to study water solvation of alkali and alkaline earth metal ions [65–67]. A combined experimental and theoretical investigation on the solvation of Ca²⁺ with water molecules by Armentrout et al. reveals how the sequential binding energies are changed by the addition of each water molecule [68]. While reporting the predominance of electrostatic energies on the binding of alkali metal cations with water molecules Kim et al. high how the sum of induction and dispersion energies are almost canceled out by exchange-repulsion energy [69]. Merrill et al. have evaluated the performance of effective fragment potential method (EFP) with 6-31+G(d) basis set to the description of solvation in simple metal cationic systems [70]. The variation in energetic boundary between the first and the second solvation shell based on size of metal ions was explored using dipole moment and polarizabilities of M(H₂O)₁₋₈ (M = Be²⁺, Mg²⁺, Ca²⁺, and Zn²⁺) clusters by Pavlov et al. [71] using density functional theory. Glendening et al. have reported the binding energies of alkali and alkaline earth metal ions with water molecules using the HF and the MP2 methods [72]. According to their analysis, the HF method provides a reasonable description of cation-water interactions for small (n = 1−3) clusters, whereas it is not adequate for large clusters involving water-water hydrogen bonding. M(H₂O)_n [M = Mg²⁺, Ca²⁺, Sr²⁺, Ba²⁺ and Ra²⁺] clusters seem to favor structures in which all water molecules directly coordinate to the di-cation in highly symmetric arrangements.

Thus there is a huge amount of current interest in metal ion solvation. We pursued a systematic study wherein water molecules were added to metal ions [M = Li⁺, Na⁺, K⁺, Be²⁺, Mg²⁺ and Ca²⁺] and the conformational space of these hydrated metal ion complexes [M(H₂O)n; n = 1−6] explored using ab initio and density functional theory methods with a range of basis sets [50]. This benchmark study not only provides insight into the nature of solvation of metal ions but also establishes the method and basis set dependency of this solvation. As hydration of metal ions is a topic of great interest, it is necessary to identify theoretical methods that can satisfactorily reproduce experimental results at the lowest computational cost. Experimental sequential binding energies of hydration of Li⁺, Na⁺, and K⁺ were taken as reference values to evaluate the computational procedures. In those cases where experimental results of sequential binding energies for all of the hydrated complexes involving divalent ions is unavailable high level G3 energies were employed as reference values to assess various levels employed in the study. The following set of results was generated from this benchmarking study.

Triple-ζ basis set with B3LYP or MP2 method seems to serve best for correct identification of the lowest energy conformer of hydrated metal ions involving more than four water molecules. No single level (at B3LYP and MP2) is found to model consistently hydration of all the metal ions chosen for this study. Sequential binding energies at various levels of theory follow a definite trend. For alkali metal cations, MP2 and CCSD(T) perform consistently well with both double- and triple-ζ basis sets (∼6.5 kJ/mol average deviation from experimental results). The performance of MP2 and CCSD(T) seems to be promising with an average deviation of 2–10 kJ/mol for alkaline earth metal ions, and MP2(FULL)/6-31+G(d) is found to be reliable. For both Be²⁺(H₂O)_n and Ca²⁺(H₂O)_n complexes, the MP2 method shows much better agreement with G3 compared to B3LYP functional. Considering the fact that experimental values are unavailable, validation of the performance of these routine levels of theory is not straightforward. The variation in the M–O distance is higher for B3LYP than MP2 method upon addition of each water molecule. As the size of metal-water cluster increases, the charge on metal ion decreases monotonically.

Another study on impact of solvation, considers the dissociation preference of metal-cycopentadienyl (M–Cp) complexes into either radicals or ions in presence and absence of solvent [51]. In this study two plausible dissociation pathways of half sandwich complex of selected main group metallocenes has been considered and nature of these paths as a function of metal ion and solvent has been explored (Scheme 18.6).

Scheme 18.6

The types of systems are considered in this study include a metal (M = Li, Na, K, Mg and Ca) interacts with a single cyclopentadienyl unit to form a half sandwich structure (M–Cp). In order to gauge the impact of presence of solvent on dissociation of M–Cp complex, PCM optimization was performed with water as implicit solvent. Substantially high values of DE (dissociation energy in kcal/mol) in case of gas phase calculation are noted when ions are involved ranging between (171 and 122 kcal/mol for neutral complexes of Li–Cp > Na–Cp > K–Cp; Mg–Cp > Ca–Cp). The corresponding DE in solvent phase calculations for ions shows a total reversal with much lower dissociation energies ranging between (29 and 56 kcal/mol). A drastic decrease in dissociation energy DE_ions of the complexes is observed on comparing aqueous and gas phase, with all neutral complexes showing ∼90–150 kcal/mol lesser dissociation energy and the +1 complexes of Mg and Ca having ∼350–250 kcal/mol lower dissociation energy values in solvent phase. In contrast to this the dissociation energy calculated when radicals are involved DE_rad, represented as DE_rad (gas) and DE_rad (water), do not show such a drastic variation from gas (between ∼81 and 35 kcal/mol) to solvent phase (between ∼73 and 33 kcal/mol). While all metals show a clear preference to dissociate into ions in solvent phase, the Mg and Ca complexes having +1 charge on the complex show surprisingly identical preference to dissociate into M²⁺ and Cp⁻ or M^+. and Cp^. (M = Mg, Ca) Therefore, there appears to be an equal probability in these cases for dissociation by either competing pathways. Thus a marked preference for dissociation of the complex as radicals in gas phase and as ions in solvent phase can be inferred from these PCM studies.

Thus we have explored several aspects of solvation in terms of how they affect cations and cation-π interaction. Although the extent of effect which solvent has on each system is quite case specific, it remains an important factor in reliable quantum chemical calculations of model systems. We demonstrate how even subtle and minor variations in solvation seem to have a significant alteration in the manifestation of the interaction as seen by studies of different model systems.

18.2.2 π–π Interactions

The importance of stacking interactions is made obvious by its ubiquitous presence in DNA, the most important nucleic acid in a majority of life forms. While hydrogen bonding has been considered for a long time as the more predominant interaction while stabilizing nucleic acid structure, the role of π–π stacking is no less important. Jennings et al. reveal how relatively weak intramolecular edge-to-face interactions between aromatic rings can affect or determine the conformation of organic molecules in the solid state and in solution. They show how experimental estimates indicate that these interactions are energetically attractive by ~1.5 kcal/mol but disfavored in solution by entropic factors due to the restricted internal mobility. Hence, these interactions are more manifest at low temperature in solution or in crystal structures where conformational entropy effects are negligible [73]. Several experimental and theoretical modeling studies work on stacking interactions primarily in terms of evaluating their role in supramolecular assembly [74–76]. Though the issue of supramolecular stacking leading to stabilization of macromolecules is well grounded in experimentally determined structures, there is not much information available on the energetics of stabilization [12, 77]. The aromatic side chains anchored on the peptide backbone and several CH–π and π–π stacking interactions involving them in the hydrophobic pocket are implied to be functionally important [78, 79]. The benzene dimer has served as a prototype model system to model both CH–π and π–π interactions [80–82]. Several major bottlenecks exist in treating large molecules which are bound through π–π or dispersive interactions. The conventional DFT functionals [83] fail while MP2 calculations are known to overestimate the attraction by dispersion even at the basis set limit [84]. Calculations with CCSD(T) are shown to be more reliable compared to MP2 method particularly for benzene dimer as evidenced from several key studies [85–87]. Although the accuracy of CCSD(T) method is indisputable the prohibitive expense of these calculations on larger systems makes it difficult to perform them. Basis set superimposition error (BSSE) also becomes a stumbling block for obtaining reliable energetics, especially when one includes polarization and diffuse functions in the basis set and at MP2 level. We have however applied MP2 method for larger systems as it is computationally feasible and less expensive when compared to higher levels of other correlated methods, viz., CC, CI, etc. (Fig. 18.4a).

Fig. 18.4

Molecular clusters used as model systems (a’, b’ and c’) to study π–π, CH–π and hydrogen bonding interactions observed in several natural systems (a, b, and c)

The relative strengths of π–π and CH–π interactions have been modeled using clusters of benzene (Bz)_n, n = 2−8, at MP2/6-31++G** level of theory based on a linear scaling method called MTA (Molecular Tailoring Approach) using a divide and conquer approach [88]. Although several lower scaling MP2 methods [89, 90] (which do not involve fragmentation) have been recently applied to large molecular systems we adopt a linear scaling strategy for MP2 through MTA. This work provides a proof of concept for MTA where reasonably accurate energetics for large benzene clusters is evaluated. The final energies and corresponding complexation energies for these clusters indicate that compact structures are more stable than the linear stacked ones. While the existence of more number of interactions in compact structures than in linear ones may seem a pretty obvious conclusion to make we also notice how compact structures reveal a greater predominance of CH–π interactions in preference to π–π interactions. MTA-based calculations also reveal that long-range interactions in benzene rings do not play a significant role in energetics and contributions from important two- and three-body terms (in which the distance between two benzene rings is less than 4 Å) are adequate to produce faithful energy estimates. Although a substantial contribution of BSSE is present even on employing MTA at the MP2/6-31++G** level of theory the results are virtually identical to the standard counterpart in dealing with impact of basis set or BSSE. Also it is noteworthy that the trends of uncorrected and BSSE-corrected energies are similar to each other.

π–π stacking interactions have also been employed as a model system to gauge the impact of basis set superposition error (BSSE) on geometry of the molecules [91]. The fact that BSSE contributes a significant percentage of the interaction energy of the nonbonded interactions has received broad consensus [92–96]. The impact of BSSE on affect the potential energy surface of complexes bound by hydrogen bonding has been elucidated by Dannenburg et al. [97]. Studies on the effect of BSSE on the geometry of the stacked systems however have been relatively scarce, with a few groups having thrown some light on this aspect [98–100]. Hermida-Ramón et al. [100] have carried out a detailed study on the effect of BSSE on the structure, energy and vibrational frequencies of benzene–benzene and benzene–naphthalene dimers. Berski et al. [101] also have examined the effect of BSSE on the stacking interactions of the water dimer in a very systematic manner. Their study reveals that BSSE effect may be substantial on the geometry as well as energy, which appears to depend critically on the orientation and the distance separating the van der Waals complexes. More recently, Hobza et al. present studies on the effect of BSSE on the structure, energetics, and other properties of a large number of noncovalently bound molecules, including those bound exclusively through dispersive interaction [99, 102]. Saeki et al. [103] have found a critical dependence in the structural and energetic aspects, with emphasis on the shifts in frequencies of the spectrum upon using counter poise correction based on their studies on the structures and energies of naphthalene dimers at MP2 level of theory.

With the objective of evaluating how BSSE manifests in the stacking interaction of larger π-systems, we modeled the stacking interactions of bowl shaped molecules, sumanene and corannulene (Fig. 18.5). The contrasts in the crystal packing of corannulene and sumanene, lead us to examine the most approachable way to tackle the problem. While the crystal structure of sumanene is packed in a stacking fashion [104], corannulene structure shows a CH–π type of interaction [105]. A comparison of impact of BSSE correction in smaller stacked systems as against larger buckybowls dimer systems is made to estimate impact of BSSE as a function of size of system. Coming to the benzene dimer the interaction energies obtained are very similar on the geometries obtained with and without employing CP correction, despite the presence of substantial geometric differences.

Fig. 18.5

Role of basis set super position error on energy and geometry for π-stacked dimers

An analysis of the results in this study crucially reveals how counterpoise optimized geometries at MP2 level with basis sets higher than double ζ quality is in much better agreement with higher level compared with the geometry obtained without employing the counterpoise correction scheme in the optimization. For most of the medium-sized basis set, which require a BSSE correction, need to employ the BSSE correction not only to obtain a reliable energy but also geometry. It is obvious that the best quality basis set needs to be employed for getting reliable structural, energetic, and other properties.

The extent of cation-π interaction seems to dramatically increase as the size of the π-system increases. Subsequently we ventured on exploring BSSE’s effect on larger planar aromatic analog like naphthalene and two of the prototypical buckybowls corannulene and sumanene. Consistent with the results of the benzene dimer, we see that for the larger systems also, the π–π distance computed for the CP geometry is longer than that for the uncorrected geometry. Incorporation of the CP correction during the optimization may be crucial to predict more reliable geometrical parameters for the larger aromatic systems, especially as the most practical basis sets on these systems are not sufficiently large. The interaction energies predicted on the CP corrected PES is slightly larger than that predicted on the uncorrected surface. This difference indicates magnitude of error caused due to basis set superposition. A key observation from this study is essentiality of modeling of the noncovalent interactions in the counterpoise corrected PES and that the counter poise correction has significant impact not only on the energetics of the system, but also to obtain reliable intermolecular geometrical parameters. π–π interactions thus have substantial impact in biological molecules especially proteins. The type of interactions in clusters containing aromatic moieties includes both CH…π and π–π stacking. To capture these important noncovalent interactions in biomolecules we have generated a database of all aromatic-aromatic interactions from PDB “Aromatic–Aromatic Interactions Database” (A2ID). This is available in the public domain and it comprises of all types of π–π networks and their connectivity pattern present in proteins [106].

18.2.3 Hydrogen Bonding

One of the most important noncovalent interactions is the hydrogen bond. It plays an essential role in the properties of various materials such as synthetic polymers, biomolecules, and molecular solids and fluids [107, 108]. Hydrogen bonding is also responsible for the conformational preferences of a large number of molecules and produces significant modifications in the kinetics and mechanism of enzymatic reactions. The formation of matter from molecules is governed by the way in which non–covalent interactions operate inter alia. Hydrogen bonding is arguably the most extensively studied among all the non–covalent interactions. The stability of water clusters, based on the arrangement of individual molecules in different phases has been widely explored [109].

Formamide and water are the simplest molecules usually chosen as models for studying biological systems exhibiting the peptide type of bonding as in proteins. Hydrogen bonding complexes of formamide such as formamide-water, formamide-methanol can thus serve as model systems for protein-water and protein-solvent interactions. On account of the simplicity of this model, the characterization of the hydrogen bond interactions between water and formamide has been of considerable interest to experimentalists and theoreticians.

In the current review we focus on two kinds of model systems, first hydrogen bonded molecular clusters of water and the second is hydrogen bonded 1:1 complexes formed between formamide and water molecules. Coming to the case of water clusters we have undertaken a computational analysis of three linear forms of water cluster arrangements to gauge their relative stability and reactivity [110]. Constrained microenvironment of organic and metal organic host lattices give way to multitudes of hydrogen bonded water cluster arrangements [111]. Thus quite exotic structures which are not minima in the gas phase have a high probability to occur in supramolecular architecture and there is a need to look at such structures Theoretical calculations have been performed from dimer to eicosamer of water clusters (Fig. 18.4b) using HF, DFT and MP2 methods. The caged clusters of (H₂O)_n; n = 2−20 obtained from the Cambridge Structural Database (CSD) [112] have been widely employed in earlier ab initio investigations and we employed them as a standard reference. The impact of various levels of theory and basis sets employed in the calculations along with the effect of basis set superposition error on complexation energy of the water clusters is explored. The cationic (H₂O) ⁺_n and anionic (H₂O)_n− radical counterparts of the four water cluster arrangements were generated. A comparison of calculated ionization potential (IP) and electron affinity (EA) of the corresponding optimized ionic radicals was used to get an indication of the relative propensity of various water cluster arrangements towards ionization.

The primary aim of this study was to evaluate cooperativity of hydrogen bonding in water clusters, an elaborate discussion of which is made in the subsequent section. Other significant results we obtained from this exhaustive study are summarized in this section. The caged water clusters seen in W3D arrangement are more stable than the other three linear arrangements at HF, B3LYP and MP2 levels of theory. The average electron density values (ρ) obtained at the bond critical points using AIM analysis follow the trend W3D > W2DH > W2D > W1D and exhibit a high correlation with complexation energy of all the arrangements. It is observed that the electron density values at the bond critical point of the hydrogen bonds present towards the centre of the linear chain of W1D, W2D and W2DH arrangements are marginally higher in magnitude when compared to terminal H–bonds. PCM based implicit solvation yields lower complexation energy in the solvated clusters than corresponding gas phase geometries. In all the four arrangements W2DH clusters show highest propensity to form cation and anion radicals. Thus, the neutral and ionized water clusters adopt preferentially caged and helical arrangements respectively.

For the study on 1:1 complexes of formamide and water, systematic calculations using HF, B3LYP, MP2 and CCSD(T) methods have been performed [113]. Three stable structures are considered on the potential energy surface of formamide and water system. The optimized geometric parameters and interaction energies for various isomers at different levels are estimated. The IR frequencies, intensities, and frequency shifts have also been reported. The geometrical parameters of formamide at B3LYP/aug-cc-pVDZ method are in good agreement with experimental studies and other reported values from calculations at higher level of theory. Out of the three stable structures which are considered study we find the cyclic double bonded structure to be most stable while hydrogen bonding to the carbonyl group seems to be slightly more energetically favorable compared to the complex which is hydrogen bonded to the amide group. The geometrical parameters (non-counterpoise corrected) of formamide-water complexes obtained from B3LYP method are close to higher level (MP2) method and experimental values. Similar trend was observed with geometries obtained through CP optimization. Scaled and non scaled harmonic frequencies (non-CP) of formamide-water complexes demonstrate that B3LYP method frequencies are closer to experimental values when compared with HF and MP2 results. Thus the two studies mentioned above give a computational exploration of relative preferences of arrangement of hydrogen bonded molecules.

18.3 Cooperativity

Considering the strong interest we have in cation-π interactions one of our primary objectives is to see how much is the impact of cation-π on neighboring nonbonded interactions. The discussions in the foregoing sections clearly demonstrate that the structure, energy and site and facial preference of cation- π interactions could be dramatically altered by a range of factors. Considering such dramatic changes in the structure and strength of cation-π interactions are due to the neighboring nonbonded interactions we felt that it is appropriate to ask how nonbonded interactions mutually influence each other. A cursory look at the current literature reveals that cooperativity is a widely used term, especially in biology. This cooperative behavior has served as the first clue towards discovery of conformational transition and allosteric interactions in proteins. The best known example in biology for cooperativity is offered by allosteric oxygenation of the haemoglobin molecule [114–116].

Cooperativity however is not limited to ligand binding processes and allosteric proteins. In 1957, Wen et al. were the first to discuss the importance of many-body effects in water in their description of the cooperativity of hydrogen bonds [117]. They postulate that the formation of hydrogen bonds in water is predominantly a cooperative phenomenon, so that, in most cases, when one bond forms several will form, and when one bond breaks then, typically a whole cluster will dissolve. Other early studies consider factors governing the influence of a first hydrogen bond on the formation of a second one by the same molecule or ion [118]. The contribution of many body effects in both aqueous and non aqueous systems have been extensively reviewed [119]. Suhai explored cooperative effects in hydrogen bonding based on the structural and electronic properties of ice and hydrogen-bonded periodic infinite chains of water molecules using first principles quantum mechanical modeling [120]. An enhancement of the binding energy per hydrogen bond by 47% was reported for an infinite polymer compared to water dimer. Dannenberg et al. explored the geometric and vibrational properties of numerous hydrogen bonded model systems including chains of formamide, acetic acid, nitroanilines, urea etc. [121]. The studies performed on these model systems indicate that the electrostatic model for pair wise interactions to be inadequate to describe the major component of cooperativity in case of long chains. Thus quite a few studies have focused on cooperativity observed in hydrogen bonded systems. However studies considering rigorous quantitative estimation of cooperativity or anti cooperativity in systems with cation-π interactions are rather limited.

To explore this vital issue of cooperativity which reflects the nonadditive behavior of noncovalent interactions we have considered different model systems. In the first case we seek to check for presence of cooperativity among cation-π and hydrogen bonding [122] and cation-π and π–π interactions [123]. In the second case we quantify cooperativity in systems with noncovalent interactions other than cation-π interaction, for example in hydrogen bonded clusters of water, formamide and acetamide [110], clusters of π–π stacked benzene [124] etc. (Fig. 18.6).

Fig. 18.6

(a) Demonstration of cooperativity between cation-π interaction and hydrogen bonding (b) Cooperativity between cation-π and π–π interactions working in concert with each other (c) Cooperativity in polar clusters of water and formamide (d) Additive nature of cooperativity in benzene clusters

The common coexistence of M–π and π–π interactions in biology and chemistry has been explored in order to garner a better understanding of how one kind of noncovalent interaction affects the strength of another [82]. This influence is typically described in terms of cooperativity and anti cooperativity in bonding. The occurrence of M–π–π (M = Li⁺, K⁺, Na⁺, Mg²⁺, and Ca²⁺) interactions in the Cambridge Structural Database (CSD, CSD V5.26) and Brookhaven Protein Data Bank (PDB) databases was chosen as subject of study to check for veracity of mutual influence of noncovalent interactions over one other [59]. Different forms of benzene dimers (PD-parallel displaced, S-stacked and T shaped) were also subjected to ab initio calculations as part of this study, in order to establish the relative preference of differently oriented aromatic moieties (represented by benzene) to bind to each other. The PDB was searched for metal ion containing structures with less than 3 Å resolution and R-value <0.3. To remove redundancy, the structures with greater than 90% sequence identity were eliminated from the above dataset. This resulted in a total of 1,941 protein structures containing the 5 metal ions, which were then subjected to analysis. We choose to present the results for 7 Å cutoff, based on results of earlier reported studies. The database analysis reveals the following points: (a) There exists a high occurrence of M–π and M–π–π interactions in chemistry and biology. (b) The prevalence of M–π–π is seen to be higher in most cases or comparable to that of exclusively M–π configurations. (c) The number of M–π–π motifs present is only marginally lower than the number of metal ions available. The non availability of enough metal ions appears to be a reason that a small percentage of π–π interactions are metal ion assisted. (d) In proteins, the aromatic amino acid side chains seem to prefer the PD–M and TB–M orientations

A combined approach of database analysis and computational study was then undertaken. All the initial geometry optimizations for benzene dimer were done at the MP2/6-31G* level. When M = Li⁺, there is a substantial increase in the π–π interactions for the S and PD type, while the T-shaped stacking has a higher interaction energy only in the TB–M orientation. Essentially the same trends have been noticed for M = Na⁺ and K⁺, albeit to a minor extent. Surprisingly, the enhancement of π–π interaction in the presence of metal ion is quite substantial, especially when the metal has a higher charge. This study also reveals that Mg²⁺ ion enhances the strength of the π–π interactions by almost more than fivefold in three configurations of benzene, S–M, PD–M, and TB–M. Thus, with the π–π interaction energy value around 17 kcal/mol, their strength of stabilization is quite substantial. Hence, the strengths of subtle π–π interactions transform to substantial, under the influence of a di-cationic metal. Importantly, the interaction between the two benzene rings in most orientations is enhanced to almost a comparable extent. It indicates that the metal ions stabilize the π–π interactions in most orientations. The influence that the M ion has on the first ring could be the major source of the enhanced interaction. Interestingly, interaction of the metal ion with a single benzene molecule is much lower compared to the interaction of the metal.

The study thus reveals that M–π and π–π interactions work in concert, and the subtle π–π interaction become substantial in the presence of a metal ion. We find the metal ion assisted π–π interaction strengths may become comparable in magnitude to that of the hydrogen bonding interaction. This study triggered our interest to explore further the role of cooperative effect of noncovalent interaction in the 3D aggregation of supramolecular entities and biomolecules.

Quantum chemical calculations were performed to gauge the effect of cation-π and hydrogen bonding interactions on each other, with M-phenol-acceptor (M = Li⁺ and Mg²⁺; acceptor (A) = H₂O, HCOOH, HCN, CH₃OH, HCONH₂ and NH₃) taken as model ternary systems. (Fig. 18.6a) Direct assembly of the ternary complex from its constituents phenol (P), metal cation (M), and hydrogen bond acceptor (A) proceeds with reaction energy ΔE_APM, which can be calculated as the energy difference between the ternary complex and the energies of components P, M, and A. The extent to which the hydrogen bonding and cation-π interaction act in concert in these ternary systems can be deduced quantitatively by comparing the overall reaction energy ΔE_APM with the three individual interaction energy terms as in (18.5):

$$ {{\text{E}}_{\text{coop}}} = \Delta {{\text{E}}_{\text{APM}}} - \Delta {{\text{E}}_{\text{AP}}} - \Delta {{\text{E}}_{\text{MP}}} - \Delta {{\text{E}}_{{{\text{M}} - {\text{A}}}}} $$

(18.5)

The “cooperativity energy” E_coop amounts to −2.77 kcal/mol for the example of M = Li⁺ and A = H₂O. The degree of cooperativity may alternatively be quantified by comparing hydrogen bonding energies in the absence of Li⁺ (ΔE_AP = −6.18 kcal/mol) and in the presence of this cation (ΔE_c,_A−PM = −8.95 kcal/mol). The complexation of phenol with Li⁺ or Mg²⁺ in either π- or σ-fashion strengthens the phenol-H₂O hydrogen bonding interaction energy by about 2 and 7 kcal/mol in Li⁺ and Mg²⁺ complexes respectively. Essentially the same trends are found when a range of additional acceptors (HCO₂H, HCN, CH₃OH, HCONH₂ and NH₃) are also considered. In all these cases an increase in the hydrogen bonding interactions with phenol can be observed upon complexation with a metal ion. Although the hydrogen bond interaction energy increase is almost 3 kcal/mol with Li⁺, it is about 4–9 kcal/mol for Mg²⁺.

The cooperativity of cation–π and π–π interactions has also been demonstrated taking a series of mono-valent cationic species such as, Li⁺, Na⁺, K⁺, NH ⁺₄ ,PH ⁺₄ , OH ⁺₃ and SH ⁺₃ where systematic quantum chemical studies were performed to estimate the effects of cation–π and π–π interaction on each other in cation–π–π systems (Fig. 18.6b) [123]. In this study all possible orientations of onium ions to form cation–π complexes have been explored and the most stable conformation has been taken further to evaluate its effect on π–π interaction. The results from this study indicate a notable increase of 2–5 kcal/mol in the π–π interaction energy in the presence of the cations. The cation–π interaction energy is also enhanced in the presence of π–π interaction albeit to a smaller extent. The wide varieties of cations (which include both metal and inorganic ions) employed here underline the generality of the results.

Cooperativity in case of molecular clusters may be defined in terms of energetics as a nonadditive property of interacting monomers, where the interaction energy per monomer depends on the degree of aggregation (Fig. 18.6c). A case in point being the universal example of cooperativity observed in water. Hydrogen bonding here works in a highly cooperative fashion; i.e., the cumulative strength of networks of hydrogen bonds is larger than the sum of the individual bond strengths when they work together. Water has been subject to wide range of studies, where the mode of aggregation of the individual molecules in different cluster arrangements and their relative stabilities have been explored using not only high level ab initio calculations but also molecular dynamics approaches [109]. We mentioned in the prior section on hydrogen bonding, the impact of different arrangements of water molecules in different kinds of clusters as a function both of size of cluster (n = 2−20) and method of computation used. The same model systems have also been used to explain cooperativity in hydrogen bonding. The ratio of complexation energy per hydrogen bond from decamer to dimer [H₂O_10:2] and eicosamer to decamer [H₂O_20:10] is calculated to quantify the strength of hydrogen bond with increase in cluster size and is used as an indicator of cooperativity that is seen in the water clusters.

Out of a wide variety of basis sets which have been employed for single point calculations on the B3LYP/6-311+G* optimized geometries, aug–cc–pVTZ shows maximum ratio of complexation energy as the cluster size increases from dimer to decamer in case of W1D, W2D and W2DH clusters. In case of W3D, this ratio is maximum with 6-31G** followed by 6-31G* basis set. The complexation energy ratios ([H₂O_10:2], [H₂O_20:10]) of BSSE corrected single point calculations with 6-311+G* basis set for W1D (1.33, 1.040), W2D (1.380, 1.050), W2DH (1.352, 1.040) and W3D (1.913, 1.030) shows that the strength of complexation energy increases by 33%, 38%, 35% and 91% respectively as cluster size increases from 2 to 10. However, as the cluster size increases from 10 to 20, the increase in strength of hydrogen bonding is in the range of 4–6% only. This is valid in all the calculations where split valence basis sets have been used. A similar trend is observed when correlation consistent basis sets are used. Thus the non additivity of hydrogen bond strength (cooperativity) is much more evident as cluster size increases from 2 to 10 rather than from 10 to 20, where the augmentation of hydrogen bond strength is only marginal for all the four arrangements. The clear inference from the above mentioned results is the presence of definite amount of cooperativity in hydrogen bonded water clusters where the addition of a subsequent monomer to a hydrogen bonded cluster augments the strength of existing interactions. The mode of arrangement, in particular, how these interactions are arranged also plays a crucial role on the strength of interaction.

Besides water clusters the concept of cooperativity in hydrogen bonding is explored using theoretical calculations in several other molecular clusters including formamide [121, 125], acetamide, N methyl acetamide [126, 127] etc. as test cases mimicking the protein secondary structure. In order to detect its presence and quantify the extent of cooperativity in a set of benzene clusters (Fig. 18.6d), the complexation energy per interaction between a pair of benzene molecules (CE/(n−1), n = number of benzene monomers in the cluster) is compared from dimer to hexamer [111]. The results obtained in this study indicate a near uniform CE/(n−1) value of ∼−3.3 kcal/mol from dimer to hexamer in case of parallel displaced stacked clusters at MP2/6-31G* level of theory. The corresponding value of CE/(n−1) is ∼−6.5 kcal/mol at MP2/6-31++G** level of theory for the same structures. The addition of another benzene molecule does not seem to enhance the strength of each individual interaction. This is quite in contrast to earlier reported studies dealing with cation-π, π–π, and hydrogen bonding interactions. In those clusters where CH–π interactions are more prominent, clusters considered in this instance being tri-05, tet-10, pen-09 and hex-09 a similar absence of cooperativity is observed. The CE per interaction values here are in the range of −2.7 to −4.6 kcal/mol at MP2/6-31G* and consistently about a couple of kcal/mol higher at MP2/6-31++G** level. An inspection of all the clusters and their CEs indicate that the interaction energies are additive irrespective of the number or type (C–H…π or π–π) of interactions. Hence we may infer that stacking interactions in benzene clusters are essentially additive in nature. The cooperativity observed in clusters of polar molecules is substantially due to induction interactions which play an important role in determining their structure. As electrostatic interactions contribute little in case of benzene clusters, the resultant small induction interactions may be the cause for negligible cooperativity that is observed here.

Thus based on the above mentioned studies we infer a wide variety in the manifestation and extent of cooperativity in the systems analyzed. While hydrogen bonded systems demonstrate significant cooperativity, it a does not appear to contribute much to clusters dominated by CH–π and π–π interactions. However there seems to be quite a substantial enhancement in strength of other noncovalent interactions for instance in the mutual presence of cation-π and hydrogen bonding and even π–π and hydrogen bonding interactions together.

18.4 Correlation and Dispersion

For modeling the nonbonded interactions it is important to employ a method that judiciously accounts for the electrostatic and London dispersion forces. Traditionally the HF method is unsuitable to model noncovalent interactions and methods employing dynamic electron correlation become indispensible. Earlier studies reveal that it is important to employ highly electron correlated methods such as couple-cluster to accurately model nonbonded interactions [128, 129]. However considering the prohibitive cost involved in such calculations in most cases MP2 appears to be the only viable alternative. There are even more economical variants of the MP2 method such as RIMP2 which have proved to be fair in modeling nonbonded interactions. However one of the major bottlenecks in modeling nonbonded interactions is the failure of popular Becke’s functional such as B3LYP. Recently the dispersion corrected DFT-functionals have become a good alternative and another possibility is the new Minnesota functional such as M05, M05-X, M05-2X which appears to fair well for modeling nonbonded interactions [130, 131]. The dispersion corrected DFT functionals include those corrected by empirical or exchange-hole derived dispersion coefficients, DFTs parameterized to reproduce dispersion behavior, or DFTs used with dispersion-correcting potentials [83]. The spin-component-scaled MP2 (SCS-MP2) approach developed by Grimme is also a widely used improvement over MP2 for the π-stacked dimers, reducing the binding energy by ∼10 kJ/mol [89, 90, 132].

18.5 Materials

Besides biomolecules, the field of material science is an area where noncovalent interactions play a pivotal role, which is actively pursued currently using theoretical and high level computational approaches. Nanomaterials form a class of materials, involving research at the interfaces of chemistry, biology, materials science, physics, and engineering [133]. Computation is extensively employed in nanostructures with implication in nano biotechnology, nanofabrication and self directed-assembly. The potential value of several materials hitherto unrecognized for example zeolites, clatharates etc. for adsorption and storage of CO₂, H₂, and other gases impacting environment based on their ability to form noncovalent interactions have been explored using computational tools. Materials with possible utility in drug delivery including dendrimers [134], funtionalized single walled nanotubes [135] and cyclodextrins [136] etc. are currently being modeled extensively using quantum chemical approaches. The importance of noncovalent interactions in controlling the catalytic activity of the hydrogen oxidation, oxygen reduction and methanol oxidation reactions on platinum surfaces in alkaline electrolytes serve as an example of the current interest in this field [137]. Studies on interactions of small molecules and metal atoms on surface of branched and linear polycyclic aromatic hydrocarbons mimicking grapheme reveal the growing importance of noncovalent interactions in nanomaterials [138]. The feasibility for application of high level ab initio computational methods on large molecules have long been a major bottleneck owing to their large size, conformational flexibility and limitation to model real time conditions Several low scaling methods as well as combined approaches like QM/MM techniques and MD simulations have now evolved as efficient tools to counter this problem.

18.6 Molecular Dynamics

Several early reviews on MD simulations of biomolecules and in chemistry such as those by Karplus et al. [139] and van Gunsteren et al. [140] give an exhaustive coverage of methodology and salient features involved in dynamics calculations while illustrating their utility in understanding functionality of proteins. MD simulations seek to understand the properties of assemblies of molecules in terms of their structure and the microscopic interactions between them. Molecular dynamics simulation consists of the numerical, step-by-step, solution of the classical equations of motion. For this purpose the forces acting on the atoms are calculated, and these are usually derived from a potential energy based on complete set of 3N atomic coordinates. The size of the configurational space that is accessible to the molecular system, and the accuracy of the molecular model or atomic interaction function or force field that is used to model the molecular system are two basic problems one encounters while performing MD calculations. The broadness and level of sophistication of this technique can hardly be understated (Fig. 18.7).

Fig. 18.7

The need for molecular dynamics (MD) calculations

Ab initio molecular dynamics methods which form part of an extension of traditional MD and electronic structure calculations compute the forces acting on the nuclei from electronic structure calculations that are performed on the fly as the molecular dynamics trajectory is generated. The approximation involved in the MD process is shifted from the level of selecting the model potential to the level of selecting a particular approximation for solving the Schrodinger equation. One such approach which includes Car-Parrinello [141] and Born-Oppenheimer molecular dynamics schemes is the CPMD package [142] where the electronic structure approach is dealt by Kohn Sham density functional theory within a plane wave pseudo potential implementation and the Generalized Gradient Approximation [143].

Besides CPMD several other powerful packages such as CASTEP [144], CP-PAW [145], VASP [146] etc. form part of ab initio molecular dynamics programs now available based on a similar idea. The breadth of this technique is demonstrated by the range of fields where it finds applications such as solids, polymers and materials like silicon, surfaces, interfaces, adsorbates, amorphous substances, clusters, fullerenes, nanotubes, zeolites and several biomolecules etc.

18.7 Biological Relevance

The discussions in the preceding sections emphasize the relevance of noncovalent interactions including hydrogen bonding and stacking in stabilizing and retaining supramolecular structure [110, 124]. The role of cation-π interactions in the function and structure of enzymatic proteins also can hardly be understated. In particular metal ions play a pivotal role in the function of metalloproteins. Protein crystal structures from PDB are a great source of the information to understand the structure-activity relationship. Metal histidine complexes are highly abundant in nature, and protein crystal structure data indicates that such complexes are available with essentially most metals. Metal ions not only determine the catalytic activity of proteins but also determine the stability of DNA duplex, depending on the modes of binding and the nature of the metal ions. An extremely important functional and regulatory role is played by metal ions in biomacromolecules [147]. Many histidine-rich sites in proteins bind to transition metal ions thereby controlling protein folding and are important for the enzymatic capabilities, such as hydrolysis, dioxygen transport, and electron transfer. A few examples of metalloproteins, where cation-aromatic interactions are crucial and the metal ions are partially solvated and bound to histidine are given below. These examples show the myriad ways in which metal ions play an important role in biological systems. Metal ions Cu²⁺, Zn²⁺, and Fe³⁺ are known to be involved in the oxidation process of methionine of amyloid β peptide by complexing to the imidazole of the same peptide [148]. Carbamoyl phosphate synthetase (CPS) from Escherichia coli involved in the catalytic formation of carbamoyl phosphate has two polypeptide chains where the larger chain provides the active site and the smaller chain catalyzes the hydrolysis of glutamine to glutamate and ammonia. In the larger chain, carbamoyl phosphate is synthesized from two molecules of Mg²⁺ATP, one molecule of bicarbonate, and one molecule of glutamine [149]. Carbonic anhydrases catalyze the hydration and dehydration of carbon dioxide and bicarbonate, respectively. The α-class of anhydrase present mostly in vertebrates, has a Zn ion which is tetravalently coordinated to three imidazole rings and one water molecule in the active site Carbonic anhydrase isozyme IX belongs to the α-class and it has been found that in some cancers over expression of carbonic anhydrase isozyme IX promotes growth and metastatis of the tumor, making it a good target for cancer treatment. In this protein, water acts as the fourth ligand and as a critical component. It has been suggested that the mechanistic pathway for activating the water depends on the identity of the other three ligands coordinated to zinc ion and their spacing [150]. Superoxide dismutases (SODs) are a class of enzymes that catalyze the dismutation of superoxide into oxygen and hydrogen peroxide. They are an important antioxidant defense in nearly all cells exposed to oxygen. Among three major families of SODs copper-zinc superoxide dismutases (CuZnSOD) are most common in eukaryotes. In CuZnSOD the active metal is Cu; i.e., Cu plays an essential role in enzymatic behavior, whereas Zn atom acts as a structural stabilization factor [151]. Thus it should be amply clear that they metal ions play an extremely diversified and important role in determining biological structure and function. Clearly, the effect and role of solvated water in metal ion complexes is profound and is a topic of outstanding importance and relevance in understanding biological processes.

Site directed mutagenesis studies on the structure of a chitinase complex (PDB:1H0G) show that mutation of aminoacids Y240 and W252 to alanine which are greater than 12 Å away from the ligand show a great loss in activity. W470 and F190 too which are more than 5 Å away from the active site mutations exhibit activity loss. Mutation of these residues, results in breaking of the network which could be one of the reasons for the decreased activity [152]. Mutation studies on HIV-1 reverse transcriptase (PDB: 1TKT), at position Y181 and Y188 lead to high-level resistance to first generation non-nucleoside inhibitors [153]. This resistance was mainly attributed to the loss of extensive stacking interactions of the side chains of Y181C and Y188C mutants with the aromatic rings of the inhibitors. These observations however warrant more site directed mutagenesis studies in which the residues are mutated to other aromatic and non-aromatic residues. Thus analysis from studies on aromatic-aromatic interactions in crystal structures from PDB lead us to postulate that a mutation resulting in breaking of π–π networks could result either in destabilizing the protein or in activity loss [106, 124]. Studies on kinases, including choline kinase [154] and p38 kinase [155] which are important targets in treatment of diseases such as cancer, have also helped to reinforce the role and relevance of cation-π interactions in the active site of enzymes. The above mentioned discussion forms a strong base to explain the biological relevance of noncovalent interactions. It is but the tip of the iceberg with a vast number of large biological molecules yet to be explored by rigorous application of quantum chemical studies quantifying the contribution of noncovalent interactions in them.

18.8 Outlook

A multitude of nonbonded interactions working in “concordance or discordance” are usually involved in bestowing the structure, function and dynamics of supramolecular assemblies. Quantum mechanical calculations on x-ray structures play a very important role in understanding the factors responsible for stabilization/destabilization of these assemblies (Fig. 18.8).

Fig. 18.8

Interplay between different factors is essential for understanding noncovalent interactions

Although the nonbonded interactions are synonymous with weak interactions some of them are not weak. The total stabilization achieved by several nonbonded interactions is greater than the sum of individual stabilization. Thus nonbonded interactions have a profound influence on other neighboring interactions and on operating cooperatively generate an amplified effect. It is also important to employ reliable molecular dynamics simulations along with quantum chemical approaches to get picture of time dependant behavior of different model systems studied. In this chapter a detailed explanation of several strategies employed effectively on noncovalent interactions is presented. Subtle variations in different factors are shown to play a key role on the strength of weak interactions. This includes solubility, nature and size of systems considered, computational method applied to evaluate interaction and so forth. We demonstrate using numerous examples, how weak attractive intramolecular and intermolecular interactions between aromatic rings, cations and in differently bonded clusters can play a significant role in determining the preferred conformation of molecules. The field of noncovalent interactions presents a lot of scope for inter disciplinary research on account of huge complexity inherently present in it and computations emerge as an extremely vital method to tackle this complexity.