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Quantitative Characteristics of Qualitative Localized Bonding Patterns† Dmitry Yu. Zubarev,‡ Dominik Domin,‡,§ and William A. Lester, Jr.*,‡,§ Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, UniVersity of California at Berkeley, Berkeley, California 94720-1460, and Chemical Sciences DiVision, Lawrence Berkeley National Laboratory, Berkeley, California 94720 ReceiVed: July 21, 2009; ReVised Manuscript ReceiVed: September 2, 2009
Patterns of localized and delocalized chemical bonding obtained using the recently proposed adaptive natural density partitioning (AdNDP) provide a qualitative description of electronic structure but miss any quantitative information. Descriptors, such as the electron localization function (ELF), provide quantitative characteristics of bonding and can enhance the usefulness of qualitative patterns. In the present study, we used ELF and a related construct, charge-density-weighted ELF (ELFF), to characterize localized and delocalized bonding in a variety of systems. It is demonstrated that ELFF yields a more detailed description than ELF when used to analyze bonding in aromatic, conflicting aromatic, and antiaromatic systems. Both canonical molecular orbitals (CMOs) and localized multicenter two-electron (nc-2e) bonds obtained in the latter case by AdNDP localization are used to calculate ELFF. Introduction Electronic structure theory relies on various definitions of chemical bonds and their characteristics. The essentially intuitive Lewis definition1 of the chemical bond as a pair of electrons is a cornerstone of the general model of chemical bonding. There are also approaches that use mathematically and physically rigorous procedures to extract chemically relevant representations of electronic structure from wave functions and electron densities.2-13 An important group of methods are localization procedures that rely on unitary transformations of canonical molecular orbitals (MOs).14-16 Methods based on the concept of natural orbitals, such as natural bond orbital (NBO) analysis17,18 and the related adaptive natural density partitioning (AdNDP) analysis,19 have proved to be useful for extracting Lewis-like representations of electronic structure. Various local quantummechanical functions, related to the Pauli exclusion principle, such as the Fermi Hole (FH) mobility function,20-22 electron localization function (ELF),23-25 electron localizability indicator (ELI),26 localized orbital locator,27 and related descriptors,28-31 are becoming popular since they combine conceptual rigor with an informative visual representation. The AdNDP method is a localization technique that represents chemical bonding in terms of n-center two-electron (nc-2e) objects, where n spans the range from one (core electrons and lone-pairs) to the total number of atoms in molecules (completely delocalized bonding). In this way, AdNDP can be used for a seamless description of chemical bonding in aromatic, antiaromatic, and conflicting aromatic systems.19 Bonding patterns obtained in the course of the analysis are easy to read and intuitively understandable, though they do not provide any measure of localization and delocalization. Only the overall pattern of chemical bonding can be assessed and recognized as aromatic/antiaromatic. It seems reasonable to augment the qualitative AdNDP description with a simple quantitative characteristic to increase its functionality. †
Part of the “Benoît Soep Festschrift”. * Corresponding author. E-mail:
[email protected]. ‡ University of California at Berkeley. § Lawrence Berkeley National Laboratory.
The ELF function is a convenient tool for the analysis of chemical bonding as it reveals regions in molecular space where the probability of finding an electron pair is high. From this point of view, ELF preserves the notion of an electron pair as the central element of chemical bonding theory. Topological analysis of ELF is becoming increasingly popular in the characterization of chemical bonding in systems ranging from clusters in the gas phase to solids. ELF can also be used to calculate topological bond order.32 ELF, however, is intrinsically incapable of distinguishing between contributions from σ- and π-components of the wave function. Also, in the analysis of systems with delocalized bonding (aromatic/antiaromatic molecules), ELF results do not immediately suggest the presence of a delocalized electronic subsystem; e.g., the ELF picture of benzene reveals only pairwise C-C and C-H interactions.33,34 Santos et al.34,35 previously proposed calculating ELF for individual MOs to separate σ- and π-bond contributions to ELF and thereby quantitatively characterizing σ- and π-delocalized bonding and the related property of aromaticity. The total ELF basins are not the sum of partial ELF basins in this case because ELF is not decomposable into partial orbital contributions. This approach has been used to evaluate the electronic structure and properties of a silabenzene series and a new aromatic species Si6Li6.36 It is convenient for derivation of an aromaticity scale that can be used to compare aromatic character of systems of very different chemical nature, e.g., organic molecules and main group element clusters. Performance of ELF and other quantitative approaches for the evaluation of delocalized bonding in aromatic molecules has been recently reviewed.37 In the present study, we weighed ELF by charge densities of the respective components of the electronic structure, i.e., MOs and localized nc-2e bonds, to calculate an alternative quantitative descriptor of chemical bonding that we designate ELFF. We show that ELF and ELFF have somewhat different topology for π-bonding components of aromatic and antiaromatic molecules. Also, ELFF turns out to be a more precise quantitative descriptor of delocalized bonding, as will be shown below. The systems studied include a doubly aromatic Al42- cluster with lone pairs and delocalized 4c-2e σ- and π-bonds among its
10.1021/jp906914y 2010 American Chemical Society Published on Web 10/09/2009
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bonding elements, a conflicting aromatic B5- cluster with 2c2e σ-bonds, 3c-2e σ-bonds, and a delocalized 5c-2e π-bond, a doubly antiaromatic B62- cluster with 2c-2e σ-bonds and 3c-2e σ- and π-bonds, cyclobutadiene with 2c-2e σ- and π-bonds, and benzene with 2c-2e σ-bonds and delocalized 6c-2e π-bonds.
ELF F(r)component ) ELF ×
F(r)component F(r)total
(5)
all
∑
ELF F(r)component ) ELF
(6)
component
Theory The ELF concept was introduced by Becke and Edgecombe.23 For single determinant wave functions, ELF is defined in terms of the excess local kinetic energy due to the Pauli exclusion principle, T(F(r)), and the Thomas-Fermi kinetic energy density, Th(F(r)). As usual, the density is defined by the sum of squares of orbitals (Hartree-Fock or Kohn-Sham)
F(r) )
∑ |φi(r)|2
(1)
i
The Thomas-Fermi kinetic energy density of the homogeneous electron gas of density F(r) is defined by
Th(F(r)) ) 2.871F(r)5/3
(2)
The excess local kinetic energy due to the Pauli exclusion principle is given by
T(F(r)) )
1 2
∑ |∇φi(r)|2 - |∇F(r)| 8F(r)
2
(3)
i
and the ELF for single determinant wave functions is defined by
[
ELF ) 1 +
T(F(r))2 Th(F(r))2
]
-1
(4)
The value of ELF is bound between 0 and 1, and regions of space where electrons are highly localized will have large ELF values. The homogeneous electron gas itself has the characteristic ELF value of 0.5. Local maxima of ELF are referred to as attractors. The parts of molecular space associated with individual attractors are called basins and represent core shell electrons, nonbonding valence electrons, and bonding electrons.24,25 Valence basins are characterized by synaptic order, which is the number of core basins (or, essentially, the number of nuclei) with which they share a common boundary. Monosynaptic basins correspond to lone pairs; disynaptic basins correspond to two-center bonds; and polysynaptic basins indicate multicenter bonds. A single ELF isosurface with a small value may contain more than one basin. Increasing the value of the isosurface will cause basins to separate, and the process may be visualized through a bifurcation diagram. The ELF value at which a bifurcation occurs can be interpreted as a measure of the interaction between the different basins that are connected through the bifurcation point. We define the charge-density-weighted ELF (ELFF) as the product of the ELF value at a certain point and the ratio of charge density due to a particular component of the wave function, i.e., a MO or a nc-2e bond, to the total wave function at the same point
This procedure is essentially partitioning by weighting. The weights here have a simple interpretation. Charge density characterizes the probability of finding an electron at a particular position. Therefore, each position can be assigned a probability of being associated with a given component of the wave function. This probability is the ratio of the charge density due to the particular component to that due to the total wave function at the given position. For a property of interest that depends on position and intrinsically does not allow for decomposition, we assume that its probability of being associated with the given component of electronic structure is also the ratio mentioned above. So, this ratio is used as a weighting factor in partitioning the property of interest. Obviously, the sum over all the partial ELFF components will yield the ELF value because the charge density F(r) is additive with respect to the contributions of different components of the wave function. Just like ELF, the ELFF values are restricted to the (0, 1) interval. Partitioning can be performed using canonical MOs, Kohn-Sham orbitals, or localized orbitals, obtained by any localization procedures. The bifurcation points can be characterized by the percentage of the bifurcation ELFF value to the maximum ELFF value. A large percentage value for the bifurcation point indicates a high degree of delocalization of electrons among the different basins connected at the bifurcation point. The AdNDP localization19 is based on the identification of the eigenvectors of n-atomic blocks of the first-order reduced density matrix in the basis of natural atomic orbitals. The eigenvectors with occupation numbers (ON) close to the ideal value 2.00 |e| are regarded as nc-2e bonds. As the central object of bonding is still a pair of electrons, AdNDP essentially provides a generalization of the Lewis concept of chemical bonding. The search for nc-2e bonds is performed in such a manner that the recovered bonding pattern always corresponds to the point group of the molecule after nc-2e bonds are superimposed onto the molecular frame. If some part of the density cannot be associated with a fragment of the system, it is described as a bond involving all the atoms. As a result, AdNDP incorporates both “localized” and “delocalized” elements into the chemical bonding patterns. Computational Details. The geometries of the systems studied and the molecular orbitals were calculated at the B3LYP/ 6-31G* level of theory38-41 using the Gaussian 03 software package.42 The description of chemical bonding in terms of nc2e bonds was obtained from the AdNDP localization procedure.19 The NBO 3.143 software of the Gaussian 03 package was used to generate the first-order reduced density matrix in the basis of natural atomic orbitals (NAO). Charge densities and ELFs were calculated using the TOPMOD software package.44,45 Visualization of the results was performed using the MOLEKEL 5.3 program.46 The calculations were performed in the following manner. AdNDP localization was performed to produce nc-2e bonds. The ELFs and charge densities were obtained from the total wave function. ELF(MO) designates ELF calculated for the groups of MOs contributing to nc-2e bonds with a given n and type of the overlap, i.e., lone pairs, 2c-2e σ-bonds, 2c-2e π-bonds, 3c-
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Figure 1. Geometry and AdNDP nc-2e bonds of (a) Al42- (D4h, 1A1g); (b) B5- (C2V, 1A1); (c) B62- (D2h, 1Ag); (d) C4H4 (D2h, 1Ag); (e) C6H6 (D6h, 1 A1 g).
2e σ-bonds, 3c-2e π-bonds, and so on. This procedure follows the work of Santos et al.34 For the same groups of MOs, the partial charge densities were obtained, and ELFF(MO) was calculated according to eq 5. Furthermore, we used localized AdNDP bonds to produce partial charge densities and to obtain ELFF(AdNDP). Analysis of bifurcations of the partial ELF(MO), ELFF(MO), and ELFF(AdNDP) is presented below, with bifurcation values given as the percentage of the highest ELF value for the particular component. Results and Discussion Geometries of the systems studied and the representations of their electronic structure in terms of AdNDP nc-2e bonds are shown in Figure 1. Al42- (D4h, 1A1g). The Al42- (D4h, 1A1 g) cluster47,48 is the first recognized all-metal doubly aromatic species. The σ-aromaticity of the system is associated with completely delocalized (4c2e) σ-bonds due to valence 2a1g σradial-MO and 1b2g σtangentialMO and π-aromaticity with 4c-2e π-bond due to valence 1a2u π-MO (Figure 1a). Four MOs, 1a1g, 1eu, and 1b1g, contribute primarily to four s-type lone pairs (LP) (Figure 1a). The bifurcations of the partial ELF, ELFF(MO), and ELFF(AdNDP) are shown in Figure 2. The ELF(MO) and ELFF(MO) were calculated for the sets of four MOs 1a1g, 1eu, and 1b1g (Figure 2, section LP), two MOs 2a1g and 1b2g (Figure 2, section 4c-2e σ), and one MO 1a2u (Figure 2, section 4c-2e π). ELFF(AdNDP) was calculated for the groups of nc-2e bonds presented in Figure 1a. For the LP contributions, ELF(MO)LP gives the highest bifurcation value for the separation of the basins of the lone pairs (92%), while the ELFF(MO)LP value is lower (70%) and the ELFF(AdNDP)LP value is the lowest (20%). The ELFF(AdNDP)LP value is most consistent with the nonbonding nature of lone pairs since lone pairs are not expected to delocalize with other lone pairs, chemical bonds, or core electrons. Moreover, while both ELF(MO)LP and ELFF(MO)LP demonstrate noticeable topological features associated with pairwise interactions in addition to clearly distinguishable LP basins, ELFF(AdNDP)LP has only basins of four s-type lone pairs.
Figure 2. Bifurcation analysis of (I) ELF(MO); (II) ELFF(MO); and (III) ELFF(AdNDP) for Al42- (D4h, 1A1g). The bifurcation values are given as percentages of the highest ELF value for the particular component.
The presence of two completely bonding orbitals in the σ-bonding framework of Al42- gives rise to two bifurcations of both ELF(MO)4c-2eσ and ELFF(MO)4c-2eσ. The first bifurcation separates the central 4-synaptic basin (39% for ELF(MO)4c-2eσ and 61% for ELFF(MO)4c-2eσ) from the peripheral basin, while the second bifurcation leads to the reduction of the peripheral basin to four 2-synaptic basins oriented along the sides of the unit (62% and 88%, respectively). The topology of ELFF(AdNDP)4c-2eσ is somewhat different from the former as there is just one bifurcation (88%) leading to the reduction of the basin, including the central and peripheral regions, to four 2-synaptic basins. These basins are oriented in a radial manner unlike the corresponding
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Figure 3. Bifurcation analysis of (I) ELF(MO); (II) ELFF(MO); and (III) ELFF(AdNDP) for B5- (C2V, 1A1). The bifurcation values are given as a percentage of the highest ELF value for the particular component.
Figure 4. Bifurcation analysis of (I) ELF(MO); (II) ELFF(MO); and (III) ELFF(AdNDP) for B62- (D2h, 1Ag). The bifurcation values are given as a percentage of the highest ELF value for the particular component.
ELF(MO)4c-2eσ and ELFF(MO)4c-2eσ basins. This single bifurcation at a high ELFF(AdNDP)4c-2eσ value is consistent with the notion that two completely delocalized σ-MOs associated with σ-aromaticity of Al42- in fact compose a single delocalized (aromatic) system. The most drastic difference in topologies of ELF(MO)4c-2eπ and ELFF(MO/AdNDP)4c-2eπ is observed in the case of the π-bonding framework of Al42-. ELF(MO)4c-2eπ yields a high value (99%) bifurcation leading to four 1-synaptic basins resembling p-type lone pairs on four Al atoms. In contrast, both ELFF(MO)4c-2eπ and ELFF(AdNDP)4c-2eπ reveal a 4-synaptic basin in the π-system of Al42-. This result is consistent with a single multicenter bond. Clearly, as there is just one 4c-2e π-bond in Al42-, a 4-synaptic valence basin is found (see Figure 2, section 4c-2e π). For comparison in the σ-framework of Al42-, there are two delocalized 4c-2e σ-bonds, so instead of two 4-synaptic attractors only one with high bifurcation value (88%) is observed. Similar behavior is encountered in the case of π-bonding in benzene (see below). B5- (C2W, 1A1). The B5- (C2V, 1A1) cluster49,50 has been previously characterized as a conflicting aromatic system with a framework of peripheral 2c-2e σ-bonds.19,51 These five 2c-2e σ-bonds (Figure 1b) are due to 1a1, 1b2, 2a1, 3a1, and 2b2 MOs. The σ-antiaromaticity originates from two MOs: completely bonding 4a1 and partially bonding 3b2 and leads to the formation of two 3c-2e σ-bonds or islands of σ-aromaticity51 (Figure 1b). The π-aromaticity originates from one completely bonding 1b1 MO and leads to the formation of one 5c-2e π-bond (Figure 1b).51 The bifurcations of the partial ELF(MO), ELFF(MO), and ELFF(AdNDP) are shown in Figure 3. The ELF(MO) and ELFF(MO) were calculated for the sets of five σ-MOs 1a1, 1b2, 2a1, 3a1, and 2b2 (Figure 3, section 2c-2e σ), two more σ-MOs 4a1 and 3b2 (Figure 3, section 3c-2e σ), and one π-MO 1b1 (Figure 3, section 5c-2e π). ELFF(AdNDP) was calculated for the groups of nc-2e bonds presented in Figure 1b. The bifurcation values of ELF(MO)2c-2eσ, ELFF(MO)2c-2eσ, and ELFF(AdNDP)2c-2eσ for the wave function components associated with 2c-2e σ-bonds are close to each other (80%, 66%, and 78%, respectively). These values are relatively high and hint toward a certain degree of interaction
between the 2-synaptic basins. The deviation of the occupation numbers of 2c-2e σ-bonds (ranging from 1.87 |e| to 1.98 |e|, Figure 1b) from the exact value of 2.00 |e| characterizes the degree of imperfection of the localized description and can be due to the interaction of the localized bonding elements. The topology of ELF(MO)3c-2eσ, ELFF(MO)3c-2eσ, and ELFF(AdNDP)3c-2eσ calculated for the electronic subsystem, associated with 3c-2e σ-bonds, is qualitatively identical and includes two 3-synaptic basins and five 1-synaptic basins. In the case of ELF(MO)3c-2eσ and ELFF(MO)3c-2eσ, the final separation of 3-synaptic basins (33% and 15%, respectively) occurs before the final separation of 1-synaptic basins (68% and 43%, respectively). In the case of ELFF(AdNDP)3c-2eσ, the 1-synaptic basins separate at very low ELFF(AdNDP)3c-2eσ values (4%), thus clearly distinguishing the major role of 3-synaptic basins as bonding elements. The ELFF(AdNDP)3c-2eσ bifurcation value for the separation of the 3synaptic basins is 74%. The large bifurcation values point out the relatively high degree of interaction of fragments of the σ-antiaromatic system. This interaction can be seen as a consequence of the constraints imposed by the presence of the π-aromatic system and, thus, by the overall conflicting aromatic nature of the species. As in the case of the previously considered Al42-, there is a high-value (100%) bifurcation of ELF(MO)5c-2eπ obtained for the π-bonding framework of the B5- cluster leading to the formation of five irreducible 1-synaptic basins. Again, both ELFF(MO)5c-2eπ and ELFF(AdNDP)5c-2eπ reveal a single 5-synaptic irreducible basin in agreement with the idea that π-bonding in B5- (C2V, 1A1) is due to a single 5c-2e π-bond (Figure 1b). B62- (D2h, 1Ag). The double antiaromaticity of the D2h (1Ag) structure of a B62- cluster has been established in a series of experimental and theoretical studies.52-55 The chemical bonding in this system (Figure 1c) can be described as a combination of six 2c-2e peripheral σ-bonds originating from six MOs (1ag, 1b1u, 2ag, 1b2u, 1b3g, and 2b2u), two 3c-2e σ-bonds originating from 3ag and 2b1u MOs associated with σ-antiaromaticity, and two 3c-2e π-bonds originating from 1b3u and 1b2g MOs associated with π-antiaromaticity.22 The bifurcations of the partial ELF(MO), ELFF(MO), and ELFF(AdNDP) are shown in Figure 4. The ELF(MO) and ELFF(MO) were calculated for the sets of
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six σ-MOs, 1ag, 1b1u, 2ag, 1b2u, 1b3g, and 2b2u (Figure 4, section 2c-2e σ), two more σ-MOs 3ag and 2b1u (Figure 4, section 3c2e σ), and two π-MOs (1b3u and 1b2g, section 3c-2e π). ELFF(AdNDP) was calculated for the groups of nc-2e bonds presented in Figure 1c. The bifurcation values of ELF(MO)2c-2eσ, ELFF(MO)2c-2eσ, and ELFF(AdNDP)2c-2eσ for the wave function components associated with 2c-2e σ-bonds are close to each other (86%, 66%, and 67%, respectively). The decrease of the bifurcation value of ELFF(AdNDP)2c-2eσ from 78% for B5- (C2V, 1 A1) to 67% for B62- (D2h, 1Ag) is accompanied by an increase of the ON of the 2c-2e peripheral bonds in these species (ranging from 1.87 to 1.98 |e|, Figure 1b, and 1.95 to 1.99 |e|, Figure 1c, respectively). Overall, the 2c-2e σ-bond bifurcation values for the same type of the ELF descriptor are transferable between these clusters. The topology of ELF(MO)3c-2eσ, ELFF(MO)3c-2eσ, and ELFF(AdNDP)3c-2eσ calculated for the electronic subsystem, associated with 3c-2e σ-bonds, is qualitatively similar and consistent with the idea of formation of islands of aromaticity in a σ-antiaromatic system. The separation of two 3-synaptic basins in the case of ELF(MO)3c-2eσ and ELFF(MO)3c-2eσ occurs at low bifurcation values (7% and 9%, respectively) before the final separation of six peripheral 1-synaptic basins (75% and 30% bifurcations are shown, respectively). In the case of ELFF(AdNDP)3c-2eσ, 1-synaptic basins separate at very low values (not shown), clearly distinguishing the major role of 3-synaptic basins as bonding elements. The bifurcation value for the separation of 3-synaptic basins is only 24%. The low bifurcation value is consistent with the antiaromatic nature of σ-bonding in a doubly antiaromatic species. The π-antiaromaticity of B62- (D2h, 1Ag) also leads to the formation of islands of aromaticity revealing themselves as two 3c-2e π-bonds (Figure 1c). The first bifurcation of ELF(MO)3c-2eπ, ELFF(MO)3c-2eπ, and ELFF(AdNDP)3c-2eπ (23%, 38%, and 29%, respectively) leads to the separation of two 3-synaptic basins. There is another bifurcation of ELF(MO)3c-2eπ (100%) that further reduces each 3-synaptic basin to a set of three irreducible 1-synaptic basins. Again, 3-synaptic basins of both ELFF(MO)3c-2eπ and ELFF(AdNDP)3c-2eπ are irreducible, in agreement with the presence of two 3c-2e π-bonds in this system (Figure 1c). The low bifurcation value is consistent with the antiaromatic nature of π-bonding in the doubly antiaromatic species. C4H4 (D2h, 1Ag). Bonding in cyclobutadiene C4H4 involves only 2c-2e bonds of σ- and π-type. The σ-bonding framework consists of four C-C and four C-H bonds, (Figure 1d) that are originating from 1ag, 1b1u, 1b2u, and 3ag MOs and 2ag, 1b3g, 2b1u, and 2b2u MOs, respectively. Two 2c-2e π-bonds (Figure 1d) can be seen as a consequence of antiaromaticity of the system associated with 1b3u and 1b2g MOs. The bifurcations of the partial ELF, ELFF(MO), and ELFF(AdNDP) are shown in Figure 5. The ELF(MO) and ELFF(MO) were calculated for the sets of eight σ-MOs 1ag, 1b1u, 1b2u, 2ag, 3ag, 1b3g, 2b1u, and 2b2u (Figure 5, section 2c-2e σ) and two π-MOs 1b3u and 1b2g (Figure 5, section 2c-2e π). ELFF(AdNDP) was calculated for the groups of nc-2e bonds presented in Figure 1d. Topological features of ELF(MO)2c-2eσ, ELFF(MO)2c-2eσ, and ELFF(AdNDP)2c-2eσ for the components contributing to 2c-2e σ-bonds are identical and include four 2-synaptic basins corresponding to C-C bonds and four 1-synaptic basins corresponding to C-H bonds. ELFF(MO)2c-2eσ and ELFF(AdNDP)2c-2eσ demonstrate bifurcations at lower values (first at 51% and 51%, respectively, second at 65% and 66%, respectively) than ELF(MO)2c-2eσ (first at 72% and second at 79%). In the π-bonding framework, ELF(MO)2c-2eπ, ELFF(MO)2c-2eπ, and ELFF(AdNDP)2c-2eπ undergo the first bifurcation at 8%, 24%, and 22%, respectively, leading to two 2-synaptic basins.
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Figure 5. Bifurcation analysis of (I) ELF(MO); (II) ELFF(MO); and (III) ELFF(AdNDP) for C4H4 (D2h, 1Ag). The bifurcation values are given as a percentage of the highest ELF value for the particular component.
Figure 6. Bifurcation analysis of (I) ELF(MO); (II) ELFF(MO); and (III) ELFF(AdNDP) for C6H6 (D6h, 1A1g). The bifurcation values are given as a percentage of the highest ELF value for the particular component.
ELF(MO)2c-2eπ also undergoes the second bifurcation at 100% value that reduces these 2-synaptic basins to two 1-synaptic basins each. In contrast, 2-synaptic basins of ELFF(MO)2c-2eπ and ELFF(AdNDP)2c-2eπ are irreducible. C6H6 (D6h, 1A1g). Benzene C6H6 is a prototypical aromatic molecule, combining six 2c-2e C-C σ-bonds, originating from 1a1g, 1e1u, 1e2g, and 1b2u MOs, six 2c-2e C-H σ-bonds, originating from 2a1g, 1b1u, 2e1u, and 2e2g MOs, and three 6c2e C-C π-bonds, originating from 1a2u and 1e1g MOs. The pattern of the localized bonds is shown in Figure 1e. The bifurcations of the partial ELF(MO), ELFF(MO), and ELFF(AdNDP) are shown in Figure 6. The ELF(MO) and ELFF(MO) were calculated for the sets of 12 σ-MOs 1a1g, 1e1u, 1e2g, 1b2u, 2a1g, 1b1u, 2e1u, and 2e2g (Figure 6, section 2c-2e σ), and three π-MOs 1a2u and 1e1g (Figure 6, section 6c-2e π). ELFF(AdNDP) was calculated for the groups of nc-2e bonds presented in Figure 1e. Topological features of ELF(MO)2c-2eσ, ELFF(MO)2c-2eσ, and ELFF(AdNDP)2c-2eσ for the components contributing to 2c-2e σ-bonds are identical and include six 2-synaptic basins corresponding to C-C bonds and six 1-synaptic basins corresponding to C-H bonds. ELFF(MO)2c-2eσ and ELFF(AdNDP)2c-2eσ bifurcations occur at lower values (57% and 56%, respectively) than ELF(MO)2c-2eσ (74%). Clearly, for each ELF descriptor there is a good transferability of bifurcation values in the σ-framework between C4H4 and C6H6. Also, these values are lower than the bifurcation values for the 2c-2e σ-framework of B5- and B62clusters, suggesting better localization properties of C-C bonds
Qualitative Localized Bonding Patterns in comparison with B-B bonds. As the π-bonding framework of benzene consists of three MOs, one of which is completely bonding and two partially bonding/partially antibonding, it is closer to the formation of p-type lone pairs than π-systems of Al42- and B5-. So, it is reasonable that ELF(MO)6c-2eπ, ELFF(MO)6c-2eπ, and ELFF(AdNDP)6c-2eπ calculated for the π-system of benzene undergo a bifurcation at 90%, 88%, and 88%, respectively, leading to the separation of six 1-synaptic basins, resembling lone pairs. The high value of the bifurcations suggests strong interaction between 1-synaptic basins and is characteristic of delocalized bonding according to Santos et al.34 Concluding Remarks In the present study, we have used a combination of two localization descriptors, AdNDP and ELF, to augment qualitative results of the former with quantitative findings of the latter. This helps to refine understanding the details of chemical bonding by having a value-based assessment of the degree of localization/ delocalization and to preserve the clarity of the interpretation by working with a bonding representation that appeals to chemical intuition. It also enables the comparison of localization patterns for related species. We formulated a simple approach to the characterization of individual components of the chemical bond by weighting ELF by the charge density of a particular bonding component leading to ELFF. Both MOs and AdNDP nc-2e bonds were used to calculate ELFF. The value based on localized orbitals, ELFF(AdNDP), has the advantage of being applicable to systems with both localized and delocalized bonds as it essentially characterizes the quality of the localized description. Following Santos et. al,34,35 it can be used to derive a universal aromaticity scale for comparison of aromaticity in systems of different chemical nature unlike magnetic or geometry-based measures, which involve comparison with a related prototypical system. It is not obvious, however, whether ELFF(AdNDP) can unambiguously identify global aromaticity/ antiaromaticity of conflicting aromatic systems. Also, the new descriptor is not connected to observable properties and remains purely computational. We compared ELFF with previous calculations of ELF for individual MOs. On the basis of the results obtained, ELF and ELFF have in general similar topologies, although in the case of π-bonds ELFF yields objects, such as multisynaptic basins, that agree better with chemical intuition. Also, bifurcation values of ELFF for nonbonding elements, such as lone pairs, are appropriately low. Bifurcation values of ELFF for the same types of bonds demonstrate good transferability between related systems. The sum of ELFF over all bonding components results in an ELF value, unlike the calculations of ELF for individual MOs. Acknowledgment. Dmitry Y. Zubarev was supported by the National Science Foundation under grant NSF CHE-0809969. Dominik Domin and William A. Lester, Jr. were supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences, Geosciences and Biosciences Division of the US Department of Energy, under Contract No. DE-AC03-76F00098. References and Notes (1) Lewis, G. N. J. Am. Chem. Soc. 1916, 38, 762. (2) Coulson, C. A. Proc. R. Soc., London Ser. A-Mathematical Phys. Sci. 1939, 169, 413. (3) Wiberg, K. B. Tetrahedron 1968, 24, 1083.
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