Quantification of thermal ring flexibilities of aromatic and heteroaromatic compounds
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The consequences of thermal fluctuations occurring at room temperatures on the aromatic character of a broad group of compounds were analyzed in three distinct ways. First of all, the ring deformations were modeled along normal coordinates coming from quantum thermo-chemistry computations. The amplitudes of vibrations were estimated according to absorbed energies at room temperature. Alternatively, in-plane and out-of-plane ring deformations were modeled via scanning procedure with partial relaxation of the molecular geometry. The influence of ring deformations on π–electron delocalization was expressed in terms of HOMA values. Besides, the ring deformability was defined as the averaged change of bond angles or dihedral angles constituting the ring that was associated with 1.5 kcal mol-1 increase of the system energy. The molecules structures adopted during vibrations at room temperature can lead to significant heterogeneity of structural index of aromaticity. The broad span of HOMA values was obtained for analyzed five- or six-membered aromatic and heteroaromatic rings. However, the averaged values obtained for such fluctuations almost perfectly match HOMA values of molecule in the ground state. It has been demonstrated that the ring deformability imposed by bond angle changes is much smaller than for dihedral angles with the same rise of system energy. Interestingly in the case of out-of-plane vibrations modeled by scanning procedure there is observed linear correlation between ring deformability and HOMA values. Proposed method for inclusion of thermal vibrations in the framework of π–electron delocalization provides natural shift of the way of thinking about aromaticity from a static quantity to a dynamic and heterogeneous one due to inclusion of a more realistic object of analysis – thermally deformed structures. From this perspective the thermal fluctuations are supposed to be non-negligible contributions to aromaticity phenomenon.
KeywordsAromaticity Benzene Heteroaromaticity HOMA Normal coordinate analysis Nucleic acid bases Thermal vibrations
The π–electron delocalization is a commonly occurring phenomenon [1, 2, 3] leading to unique physico-chemical properties of molecular systems due to characteristic structural, energetic, magnetic and reactivity consequences. There is common consensus [4, 5, 6, 7, 8, 9, 10, 11, 12, 13] that aromaticity is characterized by a collection of properties of a cyclic π-electron system independent of whether the cause of aromatic stability is considered due to the π-electron or σ-electron structure. The multi-dimensionality of this quantity  makes the problem of quantification a non-trivial fact and despite numerous aromaticity measures that were introduced by different authors [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] there is no one unambiguous criterion that can be used for all molecular systems. For example one of the most widely used structural index of aromaticity, harmonic oscillator model of aromaticity (HOMA) [15, 16], assumes the idea of non-alternation of chemical bonds in aromatic systems. However, bond-length alternation (BLA) is not always a univocal indicator of aromaticity as it was demonstrated by Feixas et al. [17, 18]. It has been demonstrated that significant BLA of distorted benzene structures do not affect its aromaticity measured by large negative NICS values. Aihara et al.  confirmed that benzene tends to stay highly aromatic and highly diatropic even if strong bond-length alternation is introduced artificially into the π-electron system and within annulene family benzene π-electron delocalization is the least sensitive to bond-length alternation. Interestingly, two decades ago Aihara  suggested that also aromatic stabilization energy (ASE) is quite insensitive to bond length alternation. On the other hand aromaticity is strongly associated with the rings planarity but even this feature is not a sine qua non condition which was documented by numerous experimental [21, 22, 23, 24, 25, 26] and theoretical investigations [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 27, 28]. Benzene ring can undergo large out-of-plane distortions [21, 22, 23, 24, 25, 26, 27, 28] but its aromaticity and ring-current diamagnetism seem to be quite insensitive to such deformations of the molecule [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Even benzene itself in crystalline state at 20 K is non-planar  which has been attributed to intermolecular interactions in the crystal lattice. Van Zijl et al.  noted that meta-cyclophane is highly diatropic despite its strongly bent benzene ring and observed CC bond lengths of the ring are uniform within an experimental error. Furthermore, on the basis of the magnetic criterion of aromaticity, Rice et al.  reported that paracyclophanes should be classified as aromatic notwithstanding the considerable non-planar distortions of the benzene ring . Polycyclic benzenoid hydrocarbons also appear to retain aromaticity even if they are heavily distorted [33, 34]. Another interesting aspect is related to reaction of aromatic compounds. Recently, Rozgonyi et al.  have used an unbiased ab initio molecular dynamics for demonstration that aromaticity remains the organizing force even at high temperature and the ground-state reaction paths continue to proceed through aromatic configuration. Furthermore, the transition state of a Diels Alder reaction between ethylene and butadiene is formally iso-π-electronic with bond-alternate benzene [36, 37]. This pericyclic transition state also exhibits a large negative NICS value at the central region and should reasonably be regarded as an aromatic specie.
Interestingly aromatic ring elasticity was documented also as the common phenomenon not only to benzene analogues but also characteristic for heterocyclic compounds [38, 39, 40, 41, 42, 43]. This is of particular importance since it provides additional degrees of freedom to biomolecules interacting in native environments. The structural non-rigidity of selected numbers of aromatic and heteroaromatic molecules has been investigated by quantum chemical optimization and quantum chemistry molecular dynamics. Shishkin et al. demonstrated in a series of publications [38, 39, 40, 41, 42, 43] that lowest-lying molecular vibrations stand for high flexibility of the aromatic rings. An energy increase between a planar equilibrium conformation and a non-planar deformed structures can be characterized by rising of the endocyclic torsion angles up to ±20 degrees with an increase of only 1.2-1.8 kcal mol-1 depending on the system . This indicates that such molecular motions, even for highly aromatic compounds such as benzene or adenine, can be the source of non-trivial deformations that can potentially lead to alterations of π–electron delocalization. Despite great interest in the flexibility of aromatic and hetero-aromatic rings no systematic study was performed till now for methodical inclusion thermally available aromatic ring deformations. This paper intends to fill this gap and provides detailed analysis of in-plane and out-of-plane flexibilities of six- and five-membered rings in terms of HOMA index of aromaticity.
This way of modeling of molecular motions overestimates the molecular deformations due to the very nature of quantum oscillator. In the case of classical vibrations the most probable states correspond to oscillations equal to amplitudes. This is not the case for quantum vibrations for which the most probable is the ground state. Thus, alternative ways of structure deformations are also considered by adopting Shishkin approach [23, 39]. It is obvious that vibrations along normal modes can be decomposed into different structural deformations as bond or torsion angles stretching, bending, out-of-plane breathing, etc. The distribution of potential energy (PED) [45, 46, 47] in each internal coordinate is commonly used for determination of such contributions. Among them there are two of the most important deformations that seem to be crucial from the perspective of aromaticity changes. The torsion angles constituting the ring skeleton are responsible for out-of-plane distortions and similarly the in-plane deformations are dependent on the bond angle changes. At room temperature significant amount of kinetic energy is available to molecular motions and this is modeled by allowing for partial relaxation of molecule coordinates. Thus, a series of geometry optimizations were performed with fixing of only one coordinate and allowing for relaxation of the remaining parts. In this procedure each of six (or five) bond or torsion angles constituting the ring were kept frozen and the rest of molecule was allowed to relax. The extension of scanning along such coordinate was controlled by energetic criterion and expansion of in-plane or out-of-plane deformations was continued until 1.5 kcal mol-1 increase of energy was achieved. The averaged values of bond angles or dihedrals corresponding to such energy increase were used as the measure of the ring deformability (RD). It is worth mentioning that the proposed scanning procedure does not include all possible in-plane deformations since there are some modes that do not involve bond angle changes of the ring. For example in the case of benzene molecule for b2u symmetry or the breathing mode of a1g symmetry there are active clamping deformations of hydrogen atoms (or in general side groups). These modes were not studied as separate items, since they were included in the first approach.
Results and discussion
The influence of molecular vibrations on structural index of aromaticity was studied for compounds presented in Fig. 1. There were chosen four distinct groups of structures for preserving high diversity of aromaticity. Due to different substituents connected to the rings the considered compounds can significantly differ in aromatic character. This also allows for analysis of the influence of side groups on the ring flexibility. Indeed the span of HOMA values corresponding to optimized geometries ranges from 0.465 (thymine) up to 0.989 (phenol) for six-membered rings and from -0.403 (NO-fulvene) up to 0.871 (imidazole ring of guanine), for five-membered rings, respectively. Benzene analogues (group I) and fulvene derivatives (group II) represent typical carbon-based rings while heteroaromatic compounds were included in group III (purines) and group IV (pyrimidines).
In the first part the influence of thermal vibrations on aromaticity was estimated according to three procedures described in the methodology section. Then the relationships between HOMA values and ring deformability are discussed. For illustrational purposes more examples of structural deformations of aromatic ring are provided for cyclophanes compounds family. Finally, the correlation between obtained here heterogeneities of rings aromaticities imposed by thermal fluctuations with standard way of aromaticity assessments is presented and discussed.
Rings structure affected by thermal vibrations
In Fig. 2 there are also presented in-plane and out-of-plane rings deformations with inclusion of relaxation energy. The first and direct conclusion is that obtained distributions of HOMA values are much more contracted than ones obtained from normal coordinate analysis. Especially relaxation of molecular geometry accomplished with out-of-plane ring distortions can be easily diminished by mutual changes of other geometric molecular coordinates. The modification of one of dihedral angles is accomplished with alterations in opposite direction of two neighboring torsion angles of the ring if geometry relaxation is allowed. Thus, ring flexibility need not directly lead to strong variation of HOMA values if partial optimization is allowed. However, even in this model majority of aromatic compounds are characterized by relatively broad sets of HOMA values. Especially heteroaromatic compounds are sensitive to changes of bond and torsion angles. For example in case of guanine the reduction of C2-N3-C4 or N3-C4-C5 bond angle by about 5° from the canonical form leads to an increase of aromatic character up to 0.710 with only 1.5 kcal mol-1 raise of the total energy.
Deformability of distorted benzene rings
Quantifications of aromaticity fluctuations
Although aromaticity is a commonly used term its multidimensional character and enumerative nature make its application and interpretation a non-trivial task. This is proved by the enormous number of papers published every year that address this old phenomenon. Usually conjunction of several prepositions are used that must be inclusively fulfilled. Among them the most commonly accepted assumes that aromaticity is correlated with the higher energetic stability (in comparison to non-aromatic analogues), the π-electron delocalization expressed in terms of non-alternation of bond lengths (or related properties as bond orders, electron densities at bond critical points, etc.), ring critical points characteristics or magnetic susceptibilities and also as higher reactivity toward substation rather than addition reactions. However, all these aspects are mainly related to the ground state that corresponds to the global minimum on the configuration hyperspace. It is obvious that real molecules possess many more degrees of freedom that might potentially affect its aromaticity. In this paper one of such aspects is discussed and importance of series of structures corresponding to thermal fluctuations is suggested. This can be an important factor affecting aromaticities. The first observation is that the molecular structures adopted during vibrations at room temperature can lead to significant heterogeneity of structural index of aromaticity. The averaged values obtained for such fluctuations almost perfectly match HOMA values of molecule in the ground state. Although ignoring of the thermal fluctuations in the description of aromaticity does not change the mean values of structural index of aromaticity and leads to essentially the same quantity, the distribution of aromaticity indices can be available. This extends the traditional understanding of aromaticity. The standard deviation expressed as the square root of population variance (SDV) which includes Boltzmann probabilities is small for highly aromatic compounds and increases with reduction of aromaticity. The deformability of the ring imposed by thermal fluctuations can also be expressed as the mean change of bond angles or torsion angles constituting the ring that are correlated with a given increase of the system energy. Here the arbitral value of 1.5 kcal mol-1 level was used since such energy increase can be observed for at least 2% of vibration structures at room temperature. The proposed measure of ring resistance to structure deformation is expressed in degrees. It has been demonstrated that the ring deformability imposed by bond angle changes is much smaller than for dihedral angles with the same rise of system energy. Interestingly only in the case of out-of-plane vibrations modeled by scanning procedure lead to linear correlation with HOMA index. Proposed in this paper method for inclusion of thermal vibrations in the framework of π–electron delocalization provides natural extension of the way of thinking about aromaticity from static quantity to dynamic and heterogeneous one for inclusion of a more realistic object of analysis. From this perspective the thermal fluctuations are supposed to be non-negligible contributions to aromaticity phenomenon.
Results were obtained from part of computational grant no 39 of PCSS (Poznań, Poland). The allocation of computational facilities are greatly appreciated. The valuable discussion with Prof. Oleg V. Shishkin from Institute for Single Crystals, National Academy of Science of Ukraine, 60 Lenina Avenue, Kharkiv 61001, Ukraine is warmly acknowledged.
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