Analysis of electron delocalization in aromatic systems: individual molecular orbital contributions to para-delocalization indexes (PDI)

Our research group has recently defined two new aromaticity indexes based on the analysis of electron delocalization in aromatic species using the quantum theory of atoms-in-molecules. One of these indexes is the para-delocalization index (PDI) that measures the electronic delocalization between para-related carbon atoms in six-membered rings. In this paper, we show that this index can be partitioned into individual molecular orbital contributions. We have applied this PDI decomposition to several polycyclic aromatic hydrocarbons showing that this partitioning provides new insight into the origin of aromaticity.

Electron localization function at the correlated level

The electron localization function (ELF) has been proven so far a valuable tool to determine the location of electron pairs. Because of that, the ELF has been widely used to understand the nature of the chemical bonding and to discuss the mechanism of chemical reactions. Up to now, most applications of the ELF have been performed with monodeterminantal methods and only few attempts to calculate this function for correlated wave functions have been carried out. Here, a formulation of ELF valid for mono- and multiconfigurational wave functions is given and compared with previous recently reported approaches. The method described does not require the use of the homogeneous electron gas to define the ELF, at variance with the ELF definition given by Becke. The effect of the electron correlation in the ELF, introduced by means of configuration interaction with singles and doubles calculations, is discussed in the light of the results derived from a set of atomic and molecular systems.

Comparison of the AIM delocalization index and the Mayer and fuzzy atom bond orders

In this paper the behavior of three well-known electron-sharing indexes, namely, the AIM delocalization index and the Mayer and fuzzy atom bond orders are studied at the Hartree-Fock level. A large number of five-membered ring molecules, containing several types of bonding, constitute the training set chosen for such purpose. A detailed analysis of the results obtained shows that the three indexes studied exhibit strong correlations, especially for homonuclear bonds. The correlation is somewhat poorer but still significant for polar bonds. In this case, the bond orders obtained with the Mayer and fuzzy atom approaches are normally closer to the formally predicted bond orders than those given by the AIM delocalization indexes, which are usually smaller than those expected from chemical intuition. In some particular cases, the use of diffuse functions in the calculation of Mayer bond orders leads to unrealistic results. In particular, noticeable trends are found for C-C bonds, encouraging the substitution of the delocalization index by the cheaper fuzzy atom or even the Mayer bond orders in the calculation of aromaticity indexes based on the delocalization index such as the para-delocalization index and the aromatic fluctuation index.

Properties of Aromaticity Indices Based on the One-Electron Density Matrix

Proper normalization of two previously published indices yields aromaticity measures that, when computed within the Hückel molecular orbital (HMO) approximation, closely match the topological resonance energies per pi electron of aromatic annulenes and their ions. The normalized indices, which quantify aromaticity of individual rings in polycyclic systems, are equally applicable to homocyclic and heterocyclic compounds and can be readily computed from 1-matrices calculated at any level of electronic structure theory. However, only the index ING, derived from the Giambiagi formula, produces proper ordering of aromaticities of heterocyclic compounds, provided it is calculated from all-electron wavefunctions in conjunction with the atoms in molecule (AIM) partitioning. Its values are shown to be strongly affected by electron correlation effects. Because of its apparent inability to distinguish between anti- and nonaromatic systems, ING should only be employed for aromatic species.

Electron sharing indexes at the correlated level. Application to aromaticity calculations

Electron sharing indexes (ESI) have been applied to numerous bonding situations to provide an insight into the nature of the molecular electronic structures. Some of the most popular ESI given in the literature, namely, the delocalization index (DI), defined in the context of the quantum theory of atoms in molecules (QTAIM), and the Fuzzy-Atom bond order (FBO), are here calculated at a correlated level for a wide set of molecules. Both approaches are based on the same quantity, the exchange-correlation density, to recover the electron sharing extent, and their differences lie in the definition of an atom in a molecule. In addition, while FBO atomic regions enable accurate and fast integrations, QTAIM definition of an atom leads to atomic domains that occasionally make the integration over these ones rather cumbersome. Besides, when working with a many-body wavefunction one can decide whether to calculate the ESI from first-order density matrices, or from second-order ones. The former way is usually preferred, since it avoids the calculation of the second-order density matrix, which is difficult to handle. Results from both definitions are discussed. Although these indexes are quite similar in their definition and give similar descriptions, when analyzed in greater detail, they reproduce different features of the bonding. In this manuscript DI is shown to explain certain bonding situations that FBO fails to cope with. Finally, these indexes are applied to the description of the aromaticity, through the aromatic fluctuation (FLU) and the para-DI (PDI) indexes. FLU and PDI indexes have been successfully applied using the DI measures, but other ESI based on other partitions such as Fuzzy-Atom can be used. The results provided in this manuscript for carbon skeleton molecules encourage the use of FBO for FLU and PDI indexes even at the correlated level.

Aromaticity of Distorted Benzene Rings: Exploring the Validity of Different Indicators of Aromaticity

The effect of three in-plane (bond length alternation, bond length elongation, and clamping) and three out-of-plane deformations (boatlike, chairlike, and pyramidalization) on the aromaticity of the benzene molecule has been analyzed employing seven widely used indicators of aromaticity. It is shown that only the aromatic fluctuation index (FLU) is able to indicate the expected loss of aromaticity because of distortion from the equilibrium geometry in all deformations analyzed. As FLU has been shown previously to fail in other particular situations, we conclude that there is not yet a single indicator of aromaticity that works properly for all cases. Therefore, to reach safer conclusions, aromaticity analyses should be carried out employing a set of aromaticity descriptors on the basis of different physical manifestations of aromaticity.

The aromatic fluctuation index (FLU): a new aromaticity index based on electron delocalization

In this work, the aromatic fluctuation index (FLU) that describes the fluctuation of electronic charge between adjacent atoms in a given ring is introduced as a new aromaticity measure. This new electronic criterion of aromaticity is based on the fact that aromaticity is related to the cyclic delocalized circulation of pi electrons. It is defined not only considering the amount of electron sharing between contiguous atoms, which should be substantial in aromatic molecules, but also taking into account the similarity of electron sharing between adjacent atoms. For a series of rings in 15 planar polycyclic aromatic hydrocarbons, we have found that, in general, FLU is strongly correlated with other widely used indicators of local aromaticity, such as the harmonic-oscillator model of aromaticity, the nucleus independent chemical shift, and the para-delocalization index (PDI). In contrast to PDI, the FLU index can be applied to study the aromaticity of rings with any number of members and it can be used to analyze both the local and global aromatic character of rings and molecules.

Aromaticity Measures from Fuzzy-Atom Bond Orders (FBO). The Aromatic Fluctuation (FLU) and the para-Delocalization (PDI) Indexes

In the past few years, there has been a growing interest for aromaticity measures based on electron density descriptors, the para-delocalization (PDI) and the aromatic fluctuation (FLU) indexes being two recent examples. These aromaticity indexes have been applied successfully to describe the aromaticity of carbon skeleton molecules. Although the results obtained are encouraging, because they follow the trends of other existing aromaticity measures, their calculation is rather expensive because they are based on electron delocalization indexes (DI) that involve cumbersome atomic integrations. However, cheaper electron-sharing indexes (ESIs), which in principle could play the same role as the DI in such aromaticity calculations, can be found in the literature. In this letter we show that PDI and FLU can be calculated using fuzzy-atom bond order (FBO) measures instead of DIs with an important saving of computing time. In addition, a basis-set-dependence study is performed to assess the reliability of these measures. FLU and PDI based on FBO are shown to be both good aromaticity indexes and almost basis-set-independent measures. This result opens up a wide range of possibilities for PDI and FLU to also be calculated on large organic systems. As an example, the DI and FBO-based FLU and PDI indexes have also been calculated and compared for the C60 molecule.

Bonding in Methylalkalimetals (CH3M)n (M = Li, Na, K; n = 1, 4). Agreement and Divergences between AIM and ELF Analyses

The chemical bonding in methylalkalimetals (CH(3)M)(n)() (M = Li-K; n = 1, 4) has been investigated by making use of topological analyses grounded in the theory of atoms in molecules (AIM) and in the electron localization function (ELF). Both analyses describe the C-M bond as an ionic interaction. However, while AIM diagnoses a decrease of ionicity with tetramerization, ELF considers tetramers more ionic. Divergences emerge also when dealing with the bonding topology given by each technique. For the methylalkalimetal tetramers, the ELF analysis shows that each methyl carbon atom interacts through a bond pair with each of the three hydrogen atoms belonging to the same methyl group and through an ionic bond with the triangular face of the tetrahedral metal cluster in front of which the methyl group is located. On the other hand, the AIM topological description escapes from the traditional bonding schemes, presenting hypervalent carbon and alkalimetal atoms. Our results illustrate that fundamental concepts, such as that of the chemical bond, have a different, even colliding meaning in AIM and ELF theories.