Introduction

The concept of ‘chemical bonding’ is ubiquitous and central to the understanding of almost all aspects of Chemistry. Nonetheless an unequivocal definition of a chemical bond remains elusive till date. On the contrary, the issue whether or not a given pair of atoms in a molecule is to be considered chemically bonded still gives rise to debatable controversies, and consequently fosters a dynamic field of research [1, 2]. This is argued to emanate from the lack of a well-defined quantum mechanical operator that can yield an accurate expectation value corresponding to a chemical bond [1,2,3,4,5]. To this end, in the Quantum Theory of Atoms in Molecules (QTAIM) developed by Bader, a quantum mechanical description of a chemical bond was attempted on the basis of the topological density operator \(\left( {\hat{\rho }\left( {{\varvec{r}}} \right)} \right)\) formalism [1,2,3,4,5]. In the context of QTAIM, it is imperative to realize that \(\hat{\rho }\left( {{\varvec{r}}} \right)\) is a quantity, a Dirac observable, with experimentally detectable expectation value. In the QTAIM a quantum mechanical description of the phenomenon of chemical bonding between a pair of nuclei is presented in terms of what is known as the ‘bond path’ based on the topological parameters of the electron density between them [1,2,3,4,5,6]. This picture is indeed not only capable of reproducing the Lewis (dot) structure of a molecule but also describing as well as quantifying the entire interaction spectrum between the atoms in a chemical species (molecule, radical or ion), so to say, the weak interactions like hydrogen bonding, van der Waals interactions etc. are also included within the purview of QTAIM [1,2,3,4,5,6,7,8,9,10,11,12,13]. Consequently, the QTAIM description of a chemical species furnishes an authentic platform to analyze its stability or instability, as opposed to the controversial notions surrounding a chemical bond. In the QTAIM approach, the presence of a bond path between a given pair of nuclei in the equilibrium geometry of a molecule is argued to yield a necessary and sufficient condition to validate the presence of a bonding interaction between them because a bond path is the line of maximum electron density connecting the nuclei in comparison to any neighboring line, that is, the line conforming to the maximum negative potential energy density (the virial path) between the nuclei; a bond path thus always reflects stabilization in the atomistic picture [1,2,3,4,5,6]. It is interesting to note that the QTAIM formalism has been successfully applied to account for various intra- and inter-molecular short bond paths in which the inter-nuclear separation is smaller than the sum of the van der Waals radii of the constituent atoms [6, 7, 14,15,16], for example, the H···H interaction between two non-covalently bonded hydrogen atoms in biyphenyl and analogous compounds [6, 7] making a stabilizing contribution up to ~ 41.84 kJ mol−1 instead of being denoted as “non-bonded steric repulsion” [6, 7], closed-shell non-covalent inter-molecular interactions like O⋅⋅⋅O and X⋅⋅⋅X (X = halogen) interactions [7, 14] and so on. However, the interpretation of the stabilizing contributions to the energy of the concerned molecular species within the QTAIM formalism has been intensely criticized in another school of thought according to which the presence of a bond path is not to be inevitably connected with a stabilizing (attractive) interaction between the associated pair of nuclei [17,18,19,20,21]. As a result, a comprehensible and tenable interpretation of such interactions accompanying a short contact between the pair of nuclei remains inadequate till date despite the significant volume of research devoted to the effect [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21].

In the present work a detailed QTAIM analysis has been applied to explore the intramolecular hydrogen bonding (IMHB) interactions present in three analogous molecules, namely, 9-hydroxy-2,4-dihydro-1H-phenalen-1-one (9HP1O), 2-hydroxy-9,9a-dihydro-1H-phenalen-1-one (2HP1O), and 3-hydroxy-1H-phenalen-2(4H)-one (3HP2O) (Scheme 1). The importance of various types of hydrogen bonding interactions in a vast arena of chemistry, biology, and physics in regulating the structure, property, and functionality of various molecular species is well-known [22,23,24,25,26,27].

figure 1

Results and discussion

Geometry optimization

The optimized geometries of 9HP1O, 2HP1O, and 3HP2O (as obtained from calculations at B3LYP/6-311++G(d,p) level) are presented in Scheme 1 and the relevant geometrical parameters surrounding the IMHB framework are summarized in Table 1 (the scheme of numbering of atoms followed in the study is indicated in Scheme 1). Table 1 also summarizes the energies of the Od–H1···Oa IMHB (designated as \(E_H^{\left( {\text{r}} \right)}\) in \({\text{kJ mol}}^{ - 1}\)) in the studied molecules as assessed from computing the difference in energy of the intramolecularly hydrogen bonded closed conformation from the respective open (non-hydrogen bonded) conformations in which the hydroxyl functional moiety (Od–H1 group) is rotated 180° away from the plane of the IMHB. However, it is important to note in this context that the IMHB energy resulting from the aforementioned method of calculation does not return an accurate account of the IMHB energy given that the energies of the non-hydrogen bonded conformations are obtained from partially relaxed calculations with the underlying assumption of disregarding any stabilizing/destabilizing effect on the equilibrium geometries other than the IMHB [10,11,12,13, 28].

Table 1 Geometrical parameters surrounding the IMHB framework in the equilibrium optimized geometries of the studied molecules, and the IMHB energies calculated at different levels of theory
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Topological analysis based on QTAIM

According to the QTAIM [1,2,3,4,5,6,7,8, 13, 29, 30], the distribution of electron density \(\left( {\rho \left( {{\varvec{r}}} \right)} \right)\) between a given pair of nuclei principally governs the nature of interaction between them, and the presence of critical points (that is, points at which the gradient of \(\rho \left( {{\varvec{r}}} \right)\) vanishes) serves as a characteristic signature to this end. So to say, the geometric identification of the critical points (a maximum, a local minimum or a saddle point) is assayed from the second derivatives of the electron density (the Hessian matrix whose trace gives the Laplacian of \(\rho_c\), that is, \(\nabla^2 \rho_c\) (\(\rho_c\) denotes \(\rho \left( {{\varvec{r}}} \right)\) at the critical point) [1,2,3,4,5,6,7,8, 13, 29, 30]. In the QTAIM the nature of interaction between two atoms is thus categorically identified by the presence of a bond path (BP), and the existence of a critical point [1,2,3,4,5,6,7,8, 13, 29, 30].

According to the Quantum Theory of Atoms-In-Molecules [1,2,3,4,5,6,7,8, 29, 30], the gradient vector of electronic charge density \(\rho\) do not cross the interatomic surface bounding a quantum subsystem, a condition of zero flux in gradient of \(\rho\,\left({{\text{that is}},\nabla \rho } \right)\) through the surface [1,2,3,4,5,6,7,8, 29, 30]. This condition of zero flux in gradient of \(\rho\) is satisfied by a surface passing through the critical points (CPs) only (and not by any arbitrary surface), that is, points at which \(\nabla \rho = 0\) [1,2,3,4,5,6,7,8, 29, 30]. The electronic charge density \(\rho\) has the characteristic property of exhibiting local maxima at the nuclear positions only, and \(\rho\) decreases in any direction away from the nuclei [1,2,3,4,5,6,7,8, 29, 30]. The contour maps of electron density in the plane of the molecular plane for the studied molecules are shown in Fig. 1 (the contour maps of gradient norm of electron density for the studied molecules are shown in the Supporting Information: Fig. S1 (for 9HP1O), Fig. S2 (for 3HP2O), and Fig. S3 (for 2HP1O)). This specific property of electron density necessitates that the traces in the gradient vector field of \(\rho\) (in a given plane of the surface) will point in the direction of the nuclei and will be terminated at the nuclei [1,2,3,4,5,6,7,8, 29, 30] (shown in the Supporting Information: Fig. S1 (for 9HP1O), Fig. S2 (for 3HP2O), and Fig. S3 (for 2HP1O)). The nuclei in the gradient vector field of \(\rho\) are thus the only ‘attractors’, and the ‘basin’ of an attractor in three-dimensional space refers to the region of termination of the gradient trace, each basin thus represents a single nucleus [1,2,3,4,5,6,7,8, 29, 30]. The interbasin paths in the contour maps of \(\rho\) are also shown in Fig. 1.

Fig. 1
figure 1

Contour maps of electron density in the molecular plane of a 9HP1O, b 3HP2O, and c 2HP1O (the interbasin paths are denoted by the blue lines)

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A molecular graph corresponding to a molecular geometry is the circuit of ‘atomic interaction lines’ connecting pairs of neighboring attractors (nuclei) along which \(\rho\) attains is maximum value compared to all other neighboring lines [1,2,3,4,5,6,7,8, 29, 30], that is, an ‘atomic interaction line’ represents the line of concentration of electron density pairs of neighboring attractors (nuclei) and are known as bond paths (BPs) in a bound system [1,2,3,4,5,6,7,8, 29, 30]. A BP is thus said to represent the ‘necessary and sufficient condition’ [29] for a chemical bond between the nuclei. The QTAIM-based topological parameters pertinent to the Od–H1···Oa IMHB in the studied molecules (in the respective optimized geometries) are summarized in Table 2 along with the molecular graphs of the ground-state equilibrium geometries of 9HP1O, 2HP1O, and 3HP2O being presented in Fig. 2. The Od–H1···Oa IMHB interactions in the studied molecules are indicated by the presence of a bond critical point (BCP) as well as a ring critical point (RCP) within the quasi-ring comprising the IMHB system.

Table 2 Summary of AIM parameters (in au at the BCP) corresponding to the H-bond in the studied compounds
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Fig. 2
figure 2

Molecular graphs of the equilibrium geometries of 9HP1O, 2HP1O, and 3HP2O showing the atomic interactions in the structures. Color scheme: carbon atoms are in gray spheres, oxygen atoms are in red spheres and hydrogen atoms are in white spheres. The small red dots denote the BCPs and the small yellow dots indicate the RCPs (color figure online)

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The relevant discussions in this context are presented below:

(i) First, it is noted that the electron density \(\left( {\rho_c } \right)\) at the BCP corresponding to the IMHB is considerably greater than the Popelier upper threshold of 0.04 au (range 0.002–0.04 au) for conventional hydrogen bonding interactions [31] in 9HP1O and 3HP2O. In addition, the Laplacian of \(\rho_c\), that is, \(\nabla^2 \rho_c\) at the BCP of the IMHBs in 9HP1O and 3HP2O is also found to exhibit positive deviations from the proposed upper threshold of ~ 0.13 au (range ~ 0.024 to 0.13 au) for conventional hydrogen bonding interactions. Collectively, these observations substantiate the presence of intramolecular hydrogen bonding (Od–H1···Oa) interactions in 9HP1O and 3HP2O with noticeable degree of covalence. However, in 2HP1O, \(\rho_c\) as well as \(\nabla^2 \rho_c\) is found to be within the proposed ranges of the respective parameters with values slightly greater than the lower threshold as stated above. This in turn indicates a reduced degree of covalence in the IMHB of 2HP1O. This argument is found to be corroborated from the markedly shorter IMHB distances (and greater IMHB energies) in 9HP1O and 3HP2O in comparison to that in 2HP1O (Table 1). In this context, it is pertinent to note that the ellipticity values \(\left( \varepsilon \right)\), a measure of preferential accumulation electron density in the plane of the bond path [3,4,5,6], at the BCPs of the H1···Oa IMHBs in 9HP1O and 3HP2O are appreciably small indicating cylindrical symmetry of the accompanying hydrogen bonds whereas a considerably larger value of \(\varepsilon\) at the BCP of the H1···Oa IMHB in 2HP1O shows significant deviation from cylindrical symmetry of the hydrogen bond in 2HP1O (Table 2).

(ii) According to the QTAIM approach the Laplacian of \(\rho_c\), that is, \(\nabla^2 \rho_c\) (\(\rho_c\) being \(\rho \left( {{\varvec{r}}} \right)\) at the critical point) is related to the electronic kinetic energy density \(G_c\) (that is, \(G\left( {{\varvec{r}}} \right)\) at the BCP) and electronic potential energy density \(V_c\) (that is, \(V\left( {{\varvec{r}}} \right)\) at the BCP) via the virial equation as follows [2,3,4,5, 10,11,12,13]:

$$\frac{1}{4}\nabla^2 \rho \left( {{\varvec{r}}} \right) = 2G\left( {{\varvec{r}}} \right) + V\left( {{\varvec{r}}} \right)$$
(1)

Given that \(G\left( {{\varvec{r}}} \right) > 0\) and \(V\left( {{\varvec{r}}} \right) < 0\), the predominant electronic energy density contribution (kinetic or potential) at the locus \({{\varvec{r}}}\) is indicated from the sign of the Laplacian. As a result, \(\nabla^2 \rho \left( {{\varvec{r}}} \right)\) (Laplacian of electron density) is crucial in defining the local regions of concentration \(\left( {\nabla^2 \rho \left( {{\varvec{r}}} \right) < 0} \right)\) or depletion \(\left( {\nabla^2 \rho \left( {{\varvec{r}}} \right) > 0} \right)\) of electron density [2, 31, 32]. The contour maps of \(\nabla^2 \rho \left( {{\varvec{r}}} \right)\) for the studied molecules along with the interbasin paths are displayed in Fig. 3 (Fig. 3a for 9HP1O, Fig. 3b for 3HP2O, and Fig. 3c for 2HP1O).

Fig. 3
figure 3

Contour maps of the Laplacian of electron density in the plane of the molecules: a 9HP1O, b 3HP2O, and c 2HP1O (the interbasin paths are denoted by the blue lines) (color figure online)

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At the BCP the above equation can be represented as

$$\frac{1}{4}\nabla^2 \rho_c = 2G_c + V_c$$
(2)
$$H_c = G_c + V_c$$
(3)

Here, \(H_c\) represents the total electronic energy density at the BCP.

From the data collected in Table 2 a positive sign of the Laplacian (at the BCPs of the H1···Oa IMHBs) appears to reveal a predominance of the kinetic energy density at the BCPs, which is suggestive of accumulation of the electron density \(\rho \left( {{\varvec{r}}} \right)\) toward the nuclei, typical of electrostatic (closed shell) interactions, as opposed to a negative Laplacian which is customarily argued to reflect a covalent (shared) interaction in which the electron density remains concentrated within the internuclear spatial region [2,3,4,5, 10,11,12,13].

However, it is important to note in this context that the total electronic energy density at the BCP \(\left( {H_c } \right)\) is conventionally accredited as a more authentic parameter than the Laplacian \(\left( {\nabla^2 \rho_c } \right)\) for evaluation of the nature of the bond path as well as quantification of its strength [3,4,5, 11, 31, 32]. A dominant electrostatic (closed shell) interaction is indicated by \(H_c > 0\), that is, \(\left| {V_c } \right| < \left| {G_c } \right|\) (because \(G_c\) is positive and \(V_c\) is negative), while a negative \(H_c \left( {H_c < 0} \right)\) conforms to a covalent (shared) interaction [3,4,5, 11, 31, 32]. A rationale for this apparent contradiction between \(H_c\) and the Laplacian is argued as follows. A positive Laplacian \(\left( {\nabla^2 \rho_c > 0} \right)\) corresponds to \(\left| {V_c } \right| < 2\left| {G_c } \right|\) which can give rise to two different cases, namely, \(\left| {V_c } \right| < \left| {G_c } \right|\) and hence \(H_c > 0\) implying electrostatic (closed shell) interaction, and \(\left| {V_c } \right| > \left| {G_c } \right|\) but \(\left| {V_c } \right| < 2\left| {G_c } \right|\) and hence \(\left( {H_c < 0} \right)\) (though \(\nabla^2 \rho_c > 0\)) implying covalent (shared) interaction. The data compiled in Table 2 show that the condition \(\left| {V_c } \right| > \left| {G_c } \right|\) but \(\left| {V_c } \right| < 2\left| {G_c } \right|\) leading to \(H_c < 0\) does indeed correspond to the cases of 9HP1O and 3HP2O. Such conditions are known to reflect contributions from electrostatic as well as covalent interactions [33, 34] because of resonance assistance to the IMHB [10,11,12,13, 33,34,35,36]. On the contrary, in 2HP1O the aforesaid condition (that is, \(\left| {V_c } \right| > \left| {G_c } \right|\)) is not seen to be satisfied leading to a positive \(H_c\) (Table 2) whereby indicating an electrostatic (closed shell) interaction resulting in the formation of the IMHB. The lack of resonance assistance to the IMHB in 2HP1O is also corroborated from a discernibly longer IMHB distance and smaller IMHB energy in 2HP1O (Table 1).

(iii) A semi-quantitative estimation of the IMHB energies in the studied molecules based on the QTAIM topological parameter, namely, the electronic potential energy density \(V_c\) (that is, \(V\left( {{\varvec{r}}} \right)\) at the BCP), has been carried out via the relationship \(E_{HB}^{\left( {{\text{AIM}}} \right)} \approx \frac{1}{2}V_c\) [31, 32, 37, 38], and the results are presented in Table 1. In corroboration to the above discussion it is seen that the IMHB energy of 9HP1O and 3HP2O is substantially greater than that of 2HP1O. However, Emamian et al. [39] recently proposed that for a neutral species the binding energy for hydrogen bonding interaction is reliably predicted by a refined equation \(E_{HB - r} \left( {{\text{kcal mol}}^{ - 1} } \right) = - 223.08 \times \rho_c \left( {{\text{au}}} \right) + 0.7423\) [39] (with a mean absolute percentage error of ~ 14.7%) as against the commonly used relationship \(E_{HB}^{\left( {{\text{AIM}}} \right)} \approx \frac{1}{2}V_c\) [31, 32, 37, 38], which is argued to return results with larger errors [39]. The hydrogen bond energies calculated using this refined equation (as denoted by \(E_{HB - r}^{\left( {{\text{AIM}}} \right)}\)) for the studied molecules are presented in Table 1 along with the values obtained from the relationship \(E_{HB}^{\left( {{\text{AIM}}} \right)} \approx \frac{1}{2}V_c\) for a direct visual comparison between the results.

It is worthwhile to state in this context that the IMHB system results in the formation of a six-membered quasi-ring in 9HP1O as contrary to five-membered quasi-rings in 3HP2O and 2HP1O (as is also evidenced from the presence of a ring critical point (RCP) within the concerned quasi-rings involving the IMHB circuits in the studied molecules, Fig. 2). However, it is important to note that the IMHB energies in 9HP1O and 3HP2O are fairly comparable in magnitude to one another, and are significantly greater than that in 2HP1O. This result is supported from the observation of a significant degree of covalence in the IMHB interactions in 9HP1O and 3HP2O (\(\left| {V_c } \right| > \left| {G_c } \right|\) but \(\left| {V_c } \right| < 2\left| {G_c } \right|\) leading to \(H_c < 0\), topological parameters that reveal the contributions from electrostatic and covalent interactions because of resonance assistance to the IMHB) as against the case of 2HP1O in which the IMHB interaction is principally derived from an electrostatic origin (\(\left| {V_c } \right| > \left| {G_c } \right|\) leading to \(H_c > 0\)). Collectively, these results on the differential degrees of covalence in IMHB in the studied molecules reveal the inequivalence of substitution pair positions in the studied phenalene derivatives [37].

Noncovalent Interaction (NCI)

The scatter plots showing the variation of reduced density gradient \(\left( {{\text{RSG}}, s = \frac{1}{{2\left( {3\pi^2 } \right)^{1/3} }}\frac{{\left| {\nabla \rho } \right|}}{{\rho^{4/3} }}\left( {{\text{au}}} \right)} \right)\) with \({\text{sign}}\left( {\lambda_2 } \right)\rho \left( {{\text{au}}} \right)\) for all the studied molecules are shown in Fig. 4 (\({\text{sign}}\left( {\lambda_2 } \right)\) denotes the sign of the second Hessian eigenvalue (that is, \(\lambda_2\)) when \(\nabla^2 \rho\) is decomposed as \(\nabla^2 \rho = \lambda_1 + \lambda_2 + \lambda_3\), the \(\lambda\)-terms being the eigenvalues of the Hessian (second derivative) matrix of electron density [14, 29, 40, 41]). While a covalent bond is typically characterized by \(\lambda_1 < 0, \lambda_2 < 0 {\text{and }}\lambda_3 > 0\) with an overall negative Laplacian of electron charge density, a noncovalent interaction denotes \(\lambda_2 < 0\) (bonding) or \(\lambda_2 > 0\) (non-bonding) [40, 41]. Thus, the sign of \(\lambda_2\) forms a rational indicator to denote the nature of noncovalent interactions [40, 41]. The low density and low gradient spikes occurring at negative values in Fig. 4 are suggestive of the occurrence of stabilizing noncovalent interactions such as H-bonding (with the spikes occurring at positive values being indicative of destabilizing interactions such as steric crowding) [40, 41]. Covalent bonds denoted by a saddle point in the electronic charge density (the BCP [14, 29, 40, 41]) have the characteristic value of \(s = 0\) (large values of \(\rho\) are obtained in regions in the neighborhood of the nuclei) [14, 29, 40, 41]. A plot of \(s {\text{vs}}. \rho\) shows a noncovalent interaction by a spike in the region of low values of \(s\) as well as \(\rho\) in which the value of \(\rho\) denotes the strength of interaction. However, a plot of \(s {\text{vs}}. \rho\) does not produce an unequivocal indicator of the noncovalent interactions because different interactions like H-bonding and steric repulsion also tend to appear in the same region of \(\rho\) and \(s\) [14, 29, 40, 41]. In order to circumvent this disadvantage, the second derivatives of \(\rho\) is considered because the strength of an interaction is denoted by low or high electron density [14, 29, 40, 41] whereas the sign of the Laplacian \(\left( {\nabla^2 \rho } \right)\) provides a useful indicator to distinguish between different types of interactions [14, 29, 40, 41]. Thus in Fig. 4, that is, plot of \(s {\text{vs}}.{\text{ sign}}\left( {\lambda_2 } \right)\rho\), the H-bonding interactions (stabilizing noncovalent interactions) are denoted by spikes of values of \(s\) as well as \(\rho\) where \(s < 0\), while the destabilizing noncovalent interactions (e.g., steric repulsions) are denoted by spikes of low values of \(\rho\) and low values of \(s\) where \(s > 0\) [14, 29, 40, 41].

Fig. 4
figure 4

Reduced density gradient vs. sign \(\left( {\lambda_2 } \right)\rho\) plots for a 9HP1O, b 3HP2O, and c 2HP1O

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The reduced density gradient isosurfaces (Fig. 5) for the studied molecules reveal stabilizing O–H···O IMHB interactions by the blue elliptical slabs (while the destabilizing steric crowding leads to red ellipses and relatively weaker interactions, such as van der Waals interactions are denoted by light brown/green ellipses) [40]. A corroboration of the strengths of the IMHBs in the studied molecules is directly visualized from the reduced density gradient isosurfaces (Fig. 5) because a relatively stronger IMHB interaction in 9HP1O and 3HP2O than in 2HP1O is indicated by a blue elliptical ring while the same in 2HP1O is indicated by a cyan elliptical slab (the color code is included in Fig. 5 for a visual reference).

Fig. 5
figure 5

NCI isosurface plots (isovalue = 0.5) of a 9HP1O, b 3HP2O, and c 2HP1O. The top panel shows the color code for visualization of various types of interactions (color figure online)

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Conclusion

The O−H⋅⋅⋅O intramolecular hydrogen-bonding (IMHB) interactions in a series of phenalene derivatives, 9HP1O, 2HP1O, and 3HP2O, have been studied based on the QTAIM quantum chemical method and it is found that the differential degrees of covalence play an important role underlying the differential IMHB energies in the studied molecules. The IMHB interactions in 9HP1O and 3HP2O are found to be assisted by electrostatic as well as covalent contributions leading to a greater IMHB energy in comparison to that in 2HP1O in which the IMHB is found to be predominantly electrostatic in nature (closed shell interaction). The results of QTAIM analyses for all the studied molecules are also found to be suitably corroborated by the results of NCI analysis including a visual inspection of the results. Based on these results it may also be implied that the study of the IMHB energy could serve as a parameter to indicate the inequivalence of the substitution pair positions in the phenalene derivatives.

Computational methods

Geometry optimizations

The optimization of the studied molecular geometries, namely, 9HP1O, 2HP1O, and 3HP2O (Scheme 1) has been performed on Gaussian 09 suite of programs [42] using the B3LYP hybrid functional and 6-311++G(d,p) basis set imposing no symmetry constraints (that is, a triple-ζ quality basis set for valence electrons equipped with diffuse functions essential for calculations with molecular geometries containing lone pair(s) of electron [43]). The absence of imaginary frequency in the optimized equilibrium geometries has been employed to assess the nature of the stationary points [13, 44, 45]. Furthermore, other DFT functional, namely, CAM-B3LYP [17], and the Møller-Plesset perturbation theory (MP2) have been employed in the study to emphasize on the non-covalent interactions present in the studied molecules. Furthermore, the non-hybrid GGA-functional LC-ωPBE [46], that is, the long range corrected version of PBE [47], and the Minnesota M06-2X functional [48] which is known to account for non-covalent interactions are also included in the calculations [49].

Quantum theory of atoms-in-molecules (QTAIM) calculations

The QTAIM calculations have been performed on the AIM2000 suite of programs [50] using the wavefunction files obtained for the optimized equilibrium geometries from calculations at B3LYP/6-311++G(d,p), CAM-B3LYP/6-311++G(d,p), MP2/6-311++G(d,p), LC-ωPBE/6-311++G(d,p) and M06-2X/6-311++G(d,p) levels (on the Gaussian 09 suite).

The plots of the electron density \(\left( \rho \right)\), the Laplacian of the electron density \(\left( {\nabla^2 \rho } \right)\), and the reduced density gradient (s) have been created using the Multiwfn software package [51, 52]. The isosurfaces to visualize the noncovalent interactions (NCI) have been generated on the Visual Molecular Dynamics (VMD), v. 1.9.3 package [53].