Ab initio charge density analysis of [([B.sub.6]C).sup.2-] and [B.sub.4][C.sub.3] species--how to describe the bonding pattern?

Abstract: In this study, a detailed topological charge density analysis based on the quantum theory of atoms in molecules (QTAIM) developed by Bader and co-workers, has been accomplished (using the B3LYP method) on the C[B.sub.6.sup.2-] anion and three planar isomers of the [C.sub.3][B.sub.4] species, which had been first proposed by Exner and Schleyer as examples of molecules containing hexacoordinate carbon atoms. The analysis uncovers the strong (covalent) interactions of boron atoms as well as the "nondirectional" interaction of central carbon atom with those peripheral atoms. On the other hand, instabilities have been found in the topological networks of [([B.sub.6]C).sup.2-] and [B.sub.4][C.sub.3](para) species. A detailed investigation of these instabilities demonstrates that the topology of charge density has a floppy nature near the equilibrium geometries of the species under study. Thus, these species seems to be best described as complexes of a relatively concrete ring containing boron or carbon atoms and a central carbon atom that is confined in the plane of the molecule, but with nondirectional interactions with the surrounding atoms.

Key words: hypervalency, hexacoordinate carbon, quantum theory of atoms in molecules, charge density analysis, ab initio methods.

Resume : Faisant appel a la mrthode B3LYP et en se basant sur la theorie quantique des atomes dans les molecules developpee par Bader et ses collaborateurs, on a fait une analyse detaillee de la densite de charge topologique de l'anion C[B.sub.6.sup.2-] et de trois isomeres plans de l'espece [C.sub.3][B.sub.4] qui ont ete originalement proposees par Exner et Schleyer comme exemples de molecules contenant des atomes de carbone hexacoordines. L'analyse met en evidence les fortes interactions covalentes des atomes de bore ainsi que l'interaction sans direction de l'atome centre de carbone avec les atomes peripheriques. On a par ailleurs mis en evidence des instabilitrs dans les reseaux topologiques des especes [([B.sub.6]C).sup.2-] et [B.sub.4][C.sub.3](para). Une etude detaillee de ces instabilites demontre que la topologie de la densite de charge est de nature souple pres des geometries d'equilibre des especes etudiees. Il semble donc que la meilleure facon de decrire ces especes soit de les assimiler a des complexes d'un noyau relativement concret contenant du bore ou des atomes de carbone et un atome de carbone central qui est confine dans le plan de la molecule mais qui a des interactions non directionnelles avec les atomes qui l'entourent.

Mots cles : hypervalence, carbone hexacoordine, theorie quantique des atomes dans les molecules, analyse de la densite de charge, methodes ab initio.

[Traduit par la Redaction]

Introduction

During the last two decades of the previous millennium, a real revolution took place in theoretical chemistry. Due to advances in both computational implementation (software) of reliable quantum chemical methods and hardware facilities, the role of theoretical chemistry has been changed from an almost descriptive tool to a powerful predictive mean. Nowadays, it is a routine procedure to investigate the properties of molecules prior to synthesis by employing high level ab initio quantum chemical methods. So, this unprecedented opportunity in combination with a chemist's imagination makes it possible to designate novel molecules before any experiments. On the other hand, by modeling a single molecule in a computer, it is possible to investigate species that in usual laboratory conditions have no thermodynamic stability. Therefore intermediates, transient and unstable species, and even transition states could be studied with much less effort (rather than using experimental procedures) using current theoretical methods.

In this regard, strange carbon compounds (strange in comparison to traditional ideas in classical organic chemistry) have always been the central attention of predators of odd molecules. Historically, the pioneering work done by Schleyer and co-workers (1-3) almost 20 years ago may be viewed as the first systematic research (using ab initio quantum chemical methods) in this field. Since then, a wealth of studies has been conducted to investigate novel, unprecedented hypervalent carbon compounds. In 1997, Olah and Rasul (4) reviewed different previous calculations on penta-to hepta-coordinate carbon-containing species. In the same period, reports on new experimental data confirmed that odd carbon compounds are not hypothetical entities imagined by theoretical chemists, but real molecules, some even with bulk stability (5-7).

In such an atmosphere, Exner and Schleyer (8) proposed a new class of boron--carbon compounds, which they believed to be the first examples of planar hexacoordinate carbon compounds. Since that pioneering work, new theoretical studies have been conducted to dig more into the nature of these and related similar compounds (9-11). Not only in these studies has a special attention been paid on the bonding pattern of cited compounds, but the bonding mode of these species is also the main concern of our study.

Although these and related studies (for an up-to-date review of the relevant literature, see ref. 11) confirm the great power of theoretical approaches that are unprecedented during the whole history of chemistry, but such an approach brings delicate technical problems regarding the connection of the theoretical quantum chemical calculations with the usual chemical concepts. Questions regarding the mode of bonding in these odd molecules can be used as the best examples of these approaches. How can we propose or classify the bonding patterns in these molecules? How is it possible to make a relationship with the experimentally known species? And more importantly: Could traditional approaches in common bonding models (based and developed on well-known species) be successfully extended to embed these odd species?

Traditionally, and indeed in the great part of the history of chemistry, chemists have developed the bonding patterns for experimentally known compounds to construct a model that connects the different aspects of the chemical and physical properties of the molecule under study in a logical manner. So, in most cases (but not all), bonding models have been developed to "describe" experimental facts. Consequently, it seems that the bonding models have been employed more for "classification" of the known facts rather than predictive tools in chemistry. In a more detailed view, it is reasonable to claim that the "molecular structure hypothesis" along with the "bonding scheme" of the constituent atoms or functional groups of the target molecule, "coded" the chemical and even some physical information in a compact form. So, a question naturally arises: What does one really ask of a theory of bonding model using only the facts that are obtained from the computed expectation values of quantum mechanical operators? It seems to us that this and similar related questions are crucial in the extension of chemical bonding models to extreme domains as exemplified above.

Therefore, our concern in this paper is the role of the rigorous quantum theory of atoms in molecules (QTAIM) developed by Bader and co-workers (12) in the description of the bonding pattern of the species under study. Since charge density and related functions are the main inputs of QTAIM, in this study we will concentrate on charge densities and relevant topological features of the four species described in the original study (8), namely C[B.sub.6.sup.2-] and three isomers of [C.sub.3][B.sub.4] (see Fig. 1), and leave other aspects of QTAIM study to another paper. Although these species are intrinsically interesting, our main motivation in this study is not solely confined to these species and they serve as prototypes for applying the general strategy of QTAIM to give answers to some of methodological questions mentioned previously.

[FIGURE 1 OMITTED]

Computational details

Density functional calculations have been performed on [([B.sub.6]C).sup.2-], ortho-[B.sub.4]C3, meta-[B.sub.4[C.sub.3], and para-[B.sub.4][C.sub.3] species (see Fig. 1) at the B3LYP/6-311+G(d) level (the same computational level was used in the original study (8)), without any spatial symmetry constraints. In the first step, the equilibrium geometries were located and then frequency calculations were done to establish the local minima. To compare the C--C, B--B, and B--C bonded interactions in the cited species with those in the known chemical species, the molecules [C.sub.2][H.sub.6], [C.sub.2][H.sub.4], [B.sub.2][H.sub.4] ([D.sub.2d]) [B.sub.2][H.sub.2], (linear), B[H.sub.2]C[H.sub.3] (bisected), and BHC[H.sub.2] were chosen and their local minimum geometries were obtained at the same level of calculations. Then QTAIM's 3D partitioning scheme of charge densities was performed using the wave function of each molecule. All DFT calculations were carried out using PC GAMESS (version 6.4) (13, 14) and QTAIM calculations with MORPHY99 (15-20). The useful utilities implemented in the ChemCraft (version 1.4 beta) program (21) were optimized for handling the outputs of GAMESS.

Results and discussion

Although the optimized geometries of the species under study were reported in the original account (8) using another computational package a reoptimization was carried out to check the consistency. The results of the geometry optimizations are offered in Table 1 (the numbering scheme from Fig. 1 is used). With no exceptions, the internuclear distances of all the species under study coincide with the original study within 0.001 [Angstrom]. The internuclear distances are important since they are usually employed to decide whether certain atoms are "bonded" or not (8). As will be discussed (vide infra), this is not the approach followed in this study.

The corresponding state functions of the optimized geometries have been used to construct electronic charge densities. The results of the topological analysis of charge densities are qualitatively depicted in Fig. 2, with the usual emphasis on general topological characters and the corresponding quantitative results are in Table 2. The standard topological analysis and nomenclature were employed in this study, therefore, the details of the analysis have not been discussed in this paper and only some primary descriptions are made in following paragraphs (for details and a comprehensive discussion of relevant analysis, see ref. 12).

[FIGURE 2 OMITTED]

According to QTAIM, the bonded atoms are determined by duo-gradient paths of charge density function, which originate from a bond critical point (a point where the gradient of charge density vanishes and the Hessian matrix of charge density has two negative and one positive eigenvalues) and terminate on corresponding atomic nuclei. As has been discussed by Bader in detail (12, 22), these gradient paths, which at equilibrium geometry are called "bond paths", must be regarded as the "universal indicators of bonded interactions" even when this contradicts descriptions based on classical physics or chemical intuition. The details of methodological problems and controversies regarding this choice have been discussed elsewhere (23). So, in this paper we follow the standard QTAIM proposal and admit the bond paths as a necessary and sufficient condition for the two atoms so-linked to be bonded to one another (22).

Even a brief look at Figs. 1 and 2 clearly denotes that there are differences in the overall bond path networks in the four species under study. Particularly, the number of bonded atoms to the central carbon atom vary. On the other hand, a detailed inspection of Fig. 2 demonstrates that, in the case of [([B.sub.6]C).sup.-2] and para-[B.sub.4][C.sub.3] species, we are faced with unusual molecular graphs (we will use the phrase "molecular graph" to distinguish the topological networks (graphs) at equilibrium geometry with those corresponding to nonequilibrium geometries). Before discussing the detailed topological analysis of the individual members of the molecular set selected in this paper, this point necessitates special attention and further elaboration.

Stable vs. unstable molecular graphs

As has been discussed fully elsewhere (12), the stable topological networks are invariant under some finite geometrical changes of nuclei (the nuclear excursions). So, for a finite domain of nuclear geometrical changes, the "equivalent" topological network is retained. The word equivalent means that in the two topological networks corresponding to two neighboring nuclear geometries, the number and type of critical points and also the gradient paths that originate or terminate from these critical points remains the same. In the technical literature, this topological equivalence is termed homomorphism. On the other hand, under sufficiently steep deformations of nuclear geometry, new topological networks will appear. According to catastrophe theory, as implemented in the dynamical theory of charge density evolution relative to nuclear geometrical variations, the two networks interconverted abruptly (for details and a comprehensive discussion see ref. 12). Accordingly, there is a certain nuclear geometry for which the topological network does not belong to either of the two networks mentioned previously. This topological network may be imagined as a borderline between the two stable topological domains. By implementation of a mathematical theorem on structural stability as first derived by Palis and Smale (24) (and then implemented on the topological evolution of charge density relative to changes of nuclear geometry (12)), the borderline unstable topological networks may be classified systematically. Without going into detail, this classification leads to two distinct groups of unstable topological networks. In the first class, one encounters topological networks with degenerate critical points (critical points with ranks lower than three) and the other class contains networks with critical points whose connection causes the "nontransverse" intersect of stable and unstable manifolds (for details, see ref. 12). The first type of unstable topological network (with degenerate critical points) has been named bifurcation points, whereas the second type has been named conflict structures. The nuclear geometries corresponding to these two types of topological networks may be viewed as a natural borderline between the two topologically stable domains. So, with a small perturbation of nuclear geometry corresponds to an unstable topological network, this network will disappear and a new topological structure with a finite domain of stability will appear. In this regard, Fig. 2 vividly depicts the conflict topological networks at equilibrium geometries of the [([B.sub.6]C).sup.2-] and para[B.sub.4][C.sub.3] species. In the case of the molecular graph of [([B.sub.6]C).sup.2-], the bond paths that originated from bond critical points (3,4,5,8,9,11), from one end have been terminated (normally) in the nucleus of central carbon atom, but from the other end have been terminated in another group of bond critical points (2,1,6,7,10,12), respectively. Similarly, in the case of the topological network of para-[B.sub.4][C.sub.3] the bond paths that originated from the bond critical points (3,4), from one end have been terminated in the nucleus of the central carbon atom, but from the other end have been terminated to another group of bond critical points (2,10), respectively. Both cases exemplify conflict structures. These topological instabilities have delicate effects on the bonding description of associated species and will be considered in detail in this paper.

Topological analysis

Table 2 offers the quantitative information regarding the amount of charge density, the values of the Laplacian of charge density ([[DELTA].sup.2][rho]), the ellipticities ([epsilon] = ([[lambda].sub.1]/[[lambda].sub.2]) - 1), and the corresponding [[lambda].sub.1] and [[lambda].sub.2] values at bond critical points (BCPs) of all four species ([[lambda].sub.1] and [[lambda].sub.2] are the two negative eigenvalues of the Hessian matrix of charge density. By convention, [[lambda].sub.2] is the smaller curvature. Ellipticity provides a measure of the extent to which charge is preferentially accumulated in a given plane. For details, see ref. 12 and refs. cited therein). Table 3 has also been constructed to compare the results of Table 2 with a standard set. Table 3 contains some geometrical as well as topological characters (like those offered in Table 2) for the set of molecules [C.sub.2][H.sub.6], [C.sub.2][H.sub.4], [B.sub.2][H.sub.4] ([D.sub.2d]), [B.sub.2][H.sub.2] (linear), B[H.sub.2]C[H.sub.3] (bisected), and BHC[H.sub.2]. This set has been selected as a prototype standard reference for C--C, B--C, and B--B single and double bonds (of course, single and double bonds in a classical context). Although such selection and corresponding comparison is inevitably arbitrary and since this is a common strategy in similar considerations, we also adapted this strategy in our study. Accordingly, with this standard set and the information available from Table 2 and Fig. 2, it is now possible to make a detailed analysis of the charge density of the species under study.

The [([B.sub.6]C).sup.2-] anion

A detailed inspection of Tables 2 and 3, as well as Fig. 2, demonstrates some interesting patterns regarding the bonding mode of [([B.sub.6]C).sup.2-]. The amount of charge density in BCPs connecting boron atoms to each other is only 0.150 au, which is a bit lower than those in the [B.sub.2][H.sub.4] and [B.sub.2][H.sub.2 ]species. On the other hand, the ellipticity is 0.583, which is between the ellipticity of the B--B bonded interaction in the [B.sub.2][H.sub.4] and [B.sub.2][H.sub.2] species. Also, the sign of the Laplacian of electronic charge density is negative like those for the [B.sub.2][H.sub.4] and [B.sub.2][H.sub.2] molecules. These negative values are indicator of a "shared" interaction, which is usually interpreted as presence of a "covalent" bond (12). As is evident from Fig. 2, there are no direct bonded interactions between boron atoms and the central carbon atom. The topological characters of other BCPs, as well as ring critical points (RCPs), may be used to investigate the instability of the topological network of [([B.sub.6]C).sup.2-].

In the case of the topological network of [([B.sub.6]C).sup.2-] at equilibrium geometry, we encountered a conflict structure as discussed previously. The data in Table 2, also confirm the unstable nature of this topological network. The comparison of the amounts of charge density of BCPs (3,4,5,8,9,11) and RCPs reveals small differences (less than 0.01 au), so the overall charge density is relatively flat in a hypothetical ring imagined around the central carbon nucleus. This flatness of charge density may be interpreted as an indication of the ease of change in topological structure (12). On the other hand, the large ellipticities that are associated with BCPs (3,4,5,8,9,11) also confirm this conclusion. The large ellipticities are another indicator of the readiness of topological changes (12). Indeed, a more detailed inspection of the corresponding negative eigenvalues of the Hessian matrix of charge density ([[lambda].sub.1] and [[lambda].sub.2]) from Table 2 demonstrates the near vanishing of the negative curvature of the BCPs ([[lambda].sub.2] = -0.001) along an axes in the plane of the ring and perpendicular to the bond path, and the same near vanishing of the correspondingly directed positive curvature at the RCPs ([[lambda].sub.2] = 0.005). To establish these changes unequivocally, small arbitrary perturbations of geometrical parameters have been made to demonstrate the abrupt onset of topological changes in the molecular graph of [([B.sub.6]C).sub.2-].

Figure 3 depicts the topological networks of this molecule after some small, arbitrary shifting in nuclear geometrical parameters. In the first step, with a small 0.02 [Angstrom] shift of the central carbon (C7) nucleus toward a line bisecting a hypothetical line going through B--B (X direction), the molecular graph of [([B.sub.6]C).sup.2-] has been changed dramatically and a new topological network has emerged. Accordingly, from this new graph (shown in Fig. 3a), it is evident that there are two BCPs, which are connected from one side to the nucleus of the central carbon and from the other side to another BCP, so, this molecular graph is also a new conflict structure (must be distinguished from the original conflict structure). In a second step, a 0.2 [Angstrom] shift of the central carbon nucleus in the same direction was performed and as is evident from Fig. 3b, this new topological structure is now a stable topological network. The same procedure (shifting of central carbon nucleus) was accomplished for two other directions, namely one directly toward one of the boron nuclei of the peripheral ring (Y direction) and the other in a perpendicular, out-of-plane shifting (Z direction). Figures 3c-3f depict the resultant topological networks. In all cases, dramatic changes of molecular graphs have taken place following the applied geometrical changes. Except for Fig. 3c, all other topological networks are conflict structures (with the same criterion described previously). What is more interesting in considering these graphs, is that there are two or even three "distinct gradient paths" connecting the central carbon nucleus and one of the boron nuclei of the peripheral ring, as is evident from Fig. 3. This is an unusual and possibly a novel trait regarding the common topological networks considered previously (12). Although these are only a selected set of possible graphs (with a constraint search in selected directions of geometrical parameters space), even with this set and taking into account the nuclear excursion during the vibrations, it is evident that [([B.sub.6]C).sup.2-] does not have a "unique" topological structure around its equilibrium geometry.

[FIGURE 3 OMITTED]

On the other hand, with external energy pumping into the vibrational degrees of freedom (vibrationally "hot" molecules), not only the low-lying vibrational states, but also the higher (excited) vibrational levels, may also gain considerable population. So, the high amplitude vibrations may take place and as previous analysis has demonstrated (with controlled shifting of the central carbon from 0.02 to 0.2 [Angstrom]) new topological regimes will be available to the molecule (since explicit consideration of nuclear motion has not yet been incorporated in QTAIM, this is an ad hoc, but reasonable assumption). The other interesting feature of the topological networks of Fig. 3 is the bonded pattern of corresponding atoms. By changing the mode of geometrical perturbations of the central carbon, bonded interactions have been created and annihilated according to the amount and also the direction of the corresponding geometrical shift. For instance, whereas B5--B6 internuclear distances are the same in Figs. 3a and 3b, in Fig. 3a there is a bonded interaction between them, whereas the same bonded interaction has been ruptured in the topological network that is depicted in Fig. 3b. The same feature may be traced in the other topological networks, which are depicted in Fig. 3. Since the molecular vibrations are usually considered using normal mode analysis (25) (from nuclear dynamics analysis in the context of Born-Oppenheimer approximation) and this implies complex relative motions of nuclei, therefore, with the mixture of different normal modes and possible contributions of excited vibrational states with corresponding normal modes, it seems hard (if not impossible) to attribute a single or even a well-defined set of molecular graphs to the [([B.sub.6]C).sup.2-] anion. So, it seems that [([B.sub.6]C).sup.2-] anion must be viewed topologically as a "floppy" molecule.

The three isomers of [B.sub.4][C.sub.3]

In the case of [B.sub.4][C.sub.3], we are faced with numerous possible isomers, but this study is confined to the three isomers that were considered in the original study (8) and are depicted in Fig. 1. They have been named according to the relative position of the two carbon atoms in the peripheral ring of atoms around the central carbon atom, applying the usual classical nomenclature, namely ortho-[B.sub.4][C.sub.3], meta-[B.sub.4][C.sub.3], and para[B.sub.4][C.sub.3]. As noted previously and as is evident from Fig. 2, the molecular graphs of these three species differ remarkably. These differences, as well as the similarities, will be considered separately.

In the case of the ortho-[B.sub.4][C.sub.3] molecule, we are faced with a stable molecular graph. The central carbon atom is bonded from one side to the two carbon atoms, whereas from the other side, there are two other bonded interactions with boron atoms. So, all three types of bonding couples, namely B--B, C--C, and B--C, coexist in this molecule. The comparison of topological characters from Tables 2 and 3 reveals some interesting patterns. In all cases, except the C4--C5 bonded interaction, the amount of charge density at BCPs are smaller than those in reference double and even single bond BCPs, although, with no exception, they are all in the range of shared interactions. On the other hand, the Laplacians of charge density at BCPs of C--C and B--B bonded interactions are negative, also indicating shared interactions, whereas the same quantity is positive at BCPs of B--C bonded interactions. This positive value of the Laplacian of charge density seems to be in contradiction with the shared interaction proposed according to the charge density of corresponding BCPs. But, it must keep in mind that the internuclear distance of B--C (1.46 and 1.50 [Angstrom]) in this molecule is shorter than the internuclear distance of B--C in B[H.sub.2]C[H.sub.3] (1.55 [Angstrom]), and so it is probable that the shortening of B--C bonds in the ortho-[B.sub.4][C.sub.3] molecule causes a transition to the "internal" regime of the positive Laplacian of charge density (see ref. 26 and refs. cited therein). As considered in detail by Costales et al. (26), the lengthening and shortening of internuclear distances will motivate some universal changes of charge density distributions. One of these characteristics is the appearance of a positive Laplacian of charge density at BCPs (indicator of closed shared interactions) in molecules, where there is no doubt on their shared interaction (like the [N.sub.2] molecule). These positive values appear at large and small internuclear distances (usually out of the range of normal equilibrium internuclear distances), whereas at intermediate distances the usual negative value is retained. So, it is reasonable to assume that if the equilibrium internuclear distance of certain nuclei decreases considerably (like the B--C in ortho-[B.sub.4][C.sub.3] molecule), then a positive value of the Laplacian of charge density at BCP is not unexpected. The values of ellipticities are also worth mentioning in this analysis. In spite of small ellipticities in C4--C5 and B--C bonded interactions, large, unusual ellipticites (in comparison to those in Table 3) were observed at other BCPs. As mentioned in the case of the [([B.sub.6]C).sup.2-] anion, the large unusual value of ellipticity is an indicator of the proximity of a topological instability. Also, in this case one may find the near vanishing of the negative curvature of the BCPs ([[lambda].sub.2] < -0.09) along an axes in the plane of the ring and perpendicular to the bond path and the same near vanishing of the correspondingly directed positive curvature at the RCPs ([[lambda].sub.2] < 0.08). Accordingly, the comparison of the charge density values at BCPs, which possess the large of ellipticity values, with the charge density values at RCPs, confirms the proximity of topological instability. The difference between the amounts of charge density at BCPs (1,4,2,6,3,5) with neighboring RCPs (1,3,2) is between 0.002 and 0.012 au, which reveals the relative flatness of charge density around the central carbon nucleus. This feature is reminiscent of the charge density distribution of the [([B.sub.6]C).sup.2-] anion as discussed previously. So, the presence of a multiple of topological networks regarding the nuclear excursion in a vibrational normal mode analysis is not unexpected.

Some of these topological characteristics of ortho-[B.sub.4][C.sub.3] seem to be shared with the two other isomers. In the case of the meta-[B.sub.4][C.sub.3] isomer, the amounts of charge density at BCPs are comparable with those in the ortho-[B.sub.4][C.sub.3] isomer. The sign of the Laplacian of charge density is also like those in the ortho-[B.sub.4][C.sub.3] isomer, although the sign of this function at the two BCPs of the B--C bonded interactions (B2--C7 and B3--C7) are slightly negative. As is evident from Fig. 2c, this may be due to a large deformation of corresponding bond paths from a straight line. On the other hand, large ellipticities have been observed again, particularly at the BCPs of the B2--C7 and B3--C7 bonded interactions. The proximity of the RCPs (1, 3) to the BCPs (1, 2) reveals the origin of these large ellipticities. One may also observe the near vanishing of the negative curvature of the BCPs ([[lambda].sub.2] = -0.024) along an axes in the plane of the ring and perpendicular to the bond path and the same near vanishing of the correspondingly directed positive curvature at the RCPs ([[lambda.sub.2] = 0.019). Comparison of the amounts of charge density at these RCPs and BCPs confirms that the difference is even smaller than 0.001 au, and so we are near to an unstable topological network with a "degenerate critical point", which may be created from the coalescence of a RCP and a BCP (see ref. 12 and refs. cited therein). So, this topological graph is also on the edge of the stability--instability regime. In the case of para-[B.sub.4][C.sub.3], in contrast to the two other isomers, a conflict structure exists at equilibrium geometry like those in the [([B.sub.6]C).sup.-] anion. However, in the case of bonded interactions, namely B--B, B--C, and C--C, the amounts of charge density at the corresponding critical points is almost similar to those found in ortho-[B.sub.4][C.sub.3], meta-[B.sub.4][C.sub.3], and [([B.sub.6]C)sup.2-] species. In all species considered, with the minor exception of B2--C7 and B3-C7 in meta-[B.sub.4][C.sub.3], the sign of the Laplacian of charge density at BCPs corresponding to all C--C and B--B bonded interactions are negative and those belonging to B--C bonded interactions are positive. The large ellipticity values of the BCPs (3,4) also confirm the topological instability of the molecular graph. Again, the difference between the amounts of the charge density of BCPs (3,4) and neighboring RCPs (1,2,3,4) is only 0.004 au, and combined with the near vanishing of the negative curvature of the BCPs ([[lambda].sub.2] = -0.010) along an axes in the plane of the ring and perpendicular to the bond path and the same near vanishing of the correspondingly directed positive curvature at the RCPs ([[lambda].sub.2] = 0.027), may be interpreted as the possibility of coalescence of BCPs and RCPs and creation of degenerate critical points is probable. Like the [([B.sub.6]C).sup.2-] anion, some selected deformations have been applied to the equilibrium geometry of this molecule and the corresponding topological networks are depicted in Fig. 4. Again, dramatic changes have taken place in the topological networks. Some of these structures, such as Figs. 4a, 4b, 4e, and 4f, are also unstable conflict structures. So, according to the reasoning that has been offered in the case of the [([B.sub.6]C).sup.2-] anion, the vibrational dynamics of nuclei prevent the attribution of a single definite molecular graph to the para-[B.sub.4][C.sub.3] molecule around equilibrium geometry, another example of a topologically floppy molecule.

[FIGURE 4 OMITTED]

Conclusion and remarks

This study reveals some of the interesting features of the topological characters of the charge density in the species under consideration and clearly indicates that a nontrivial, possibly floppy topological pattern exists in this series of molecules because of the extraordinary sensitivity of the topological characters of these species relative to the small changes in their geometries. In this regard, it is worth mentioning that such a topological floppiness is not unique and has been reported previously in other sets of molecules (27-29). So, what can be said about their bonding pattern?

In a recent paper entitled "Where to draw the line in defining a molecular structure", Bader et al. (30) discussed in detail an interesting case, namely bonding between Mn, Si, and H atoms in the adduct resulting from the reaction between HSi[Cl.sub.3] and Cp[(CO).sub.2] Mn molecules. Since there were conflicting opinions regarding the most appropriate bonding pattern of this adduct (31, 32), the QTAIM clear-cut answer regarding the associated bonding pattern based solely on quantum mechanics of proper open systems seems a remarkable achievement. So, at least if one may neglect (with or without reasonable physical justification) the nuclear motion and hence just describe the topology of the charge density of a certain point in geometrical space or corresponding PES, QTAIM could offer unambiguous results. Unfortunately, this study did not encounter such a well-defined case. Left alone, the conflict structures of [([B.sub.6]C).sup.2-] and para-[B.sub.4][C.sub.3] molecules, which are inherently topologically unstable structures, and even in the case of ortho-[B.sub.4][C.sub.3] and meta-[B.sub.4][C.sub.3] molecules with stable topological structure, there are clues that nuclear excursions may cause geometrical deformations that make available new topological domains different from those corresponding to the equilibrium geometry. So, it seems to us that explicit consideration of the nuclear motion in the framework of QTAIM could be helpful for a better understanding of the bonding pattern of these molecules. Although this possibility for a general extension of QTAIM is under study, (2) even the analysis accomplished in this paper reveals certain points that are worth mentioning.

Intuitionally, one expects that in such hexacoordinate species, the nucleus of the central carbon atom should be connected to the nuclei of peripheral atoms with six distinct bond paths, but Figs. 2-4 clearly demonstrate that none of the topological networks corresponding to the species under study have a central carbon atom with six or more bond paths connecting them to all six atoms in the peripheral ring. It is interesting to note that the same topological analysis that has been done on the other celebrated hexacoordinate molecule, namely C[Li.sub.6], came to another conclusion and confirmed that the central carbon atom, which is confined in an octahedral structure, has indeed six bond paths connecting the carbon atom to six lithium atoms (33). (The same molecular graph was retained using more extended basis sets and even taking into account the correlation effects by employing sophisticated DFT methods)) So, it seems to us that we are faced with unusual, "possibly" hexacoordinate species in this study.

On the other hand, the topological characteristics of the BCPs of the [([B.sub.6]C).sup.2-] anion unequivocally established the shared interaction between boron atoms and also the flat charge density around the nucleus of the central carbon. This seems to indicate that no firm "directional interaction" takes place between the central carbon and the boron atoms. Although the free rotation of a single atom is not of physical significance, if the charge density distribution of the other extended planar boron rings encompassing the small molecular clusters resembles the [([B.sub.6]C).sup.2-] anion, then a nearly free rotation of the central cluster must be expected. Although according to the best of our knowledge no study of the topological characters of charge density distributions have been yet performed on such extended boron--carbon clusters, the ab initio calculations (11) have been conducted on such species and indeed have been estimated to have a very low barrier for the rotation of the central carbon clusters encompassing the ring of boron atoms. The future direct examination of the topological features of the charge density of these species will establish the possible common trait among them. This study has been restricted only to the detailed analysis of charge density distributions and corresponding topological networks. The evaluation of atomic expectation values, as well as analysis of localization/delocalization indices, will be done in a subsequent paper.

Acknowledgments

Special thanks to Professor Kirk A. Peterson from Washington State University for his very useful comments and Dr. Paul Lode Albert Popelier from the University of Manchester Institute of Science and Technology, Manchester, UK (UMIST) for helping us to access the MORPHY99 software and sending the related materials. We are also grateful to Professor R.EW Bader and two other reviewers for their useful technical comments and to the respected editor, since his final comments on the linguistic aspects of this paper helped us to improve the manuscript.

Received 9 October 2005. Published on the NRC Research Press Web site at http://canjchem.nrc.ca on 17 May 2006.

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(2) Sh. Shahbazian. Manuscript under preparation.

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C. Foroutan-Nejad. Department of Chemistry, Faculty of Science, University of Tehran, Tehran, Iran.

G.H. Shafiee and A. Sadjadi. (1) Department of Chemistry, Islamic Azad University of Kazeroon, P.O. Box 73135-168, Kazeroon, Fats, Iran.

Sh. Shahbazian. Department of Chemistry, Faculty of Science, Shahid Beheshti University, P.O. Box 19395-4716, Evin, Tehran, Iran.

(1) Corresponding author (e-mail: [e-mail] or [e-mail]).

Table 1. The internuclear distances ([Angstrom]) between C--C, B--C,
and B-B nuclei at the B3LYP/6-311+G * level.

C-C                     Internuclear   B-C      Internuclear
nuclei                  distance       nuclei   distance

[([B.sub.6]C).sub.2-]
--                      --             Bl-C7    1.5928
--                      --             B2-C7    1.5928
--                      --             B3-C7    1.5928
--                      --             B4-C7    1.5928
--                      --             B5-C7    1.5928
--                      --             B6-C7    1.5928

O-[B.sub.4][C.sub.3]
C4-C5                   1.3747         Bl-C7    1.4984
C4-C7                   1.5266         B2-C7    1.4984
C5-C7                   1.5266         B3-C4    1.4591
                                       B3-C7    1.5550
                                       B6-C5    1.4591
                                       B6-C7    1.5550

m-[B.sub.4][C.sub.3]
C4-C7                   1.4968         Bl-C7    1.4940
C6-C7                   1.4968         B2-C7    1.5178
                                       B2-C6    1.4869
                                       B3-C7    1.5178
                                       B3-C4    1.4869
                                       B5-C4    1.4601
                                       B5-C6    1.4601
                                       B5-C7    1.6044

P-[B.sub.4][C.sub.3]
C2-C7                   1.4430         B1-C2    1.4882
C4-C7                   1.4430         B1-C7    1.5548
                                       B3-C4    1.4882
                                       B3-C7    1.5548
                                       B5-C4    1.4882
                                       B5-C7    1.5548
                                       B6-C2    1.4882
                                       B6-C7    1.5548

C-C                     B-B           Internuclear
nuclei                  nuclei        distance

[([B.sub.6]C).sub.2-]
--                      B1-B2         1.5928
--                      B1-B3         1.5928
--                      B2-B6         1.5928
--                      B3-B4         1.5928
--                      B4-B5         1.5928
--                      B5-B6         1.5928

O-[B.sub.4][C.sub.3]
C4-C5                   B1-B2         1.6235
C4-C7                   B1-B3         1.6171
C5-C7                   B2-B6         1.6171

m-[B.sub.4][C.sub.3]
C4-C7                   B1-B2         1.6152
C6-C7                   B1-B3         1.6152

P-[B.sub.4][C.sub.3]
C2-C7                   B1-B3         1.5834
C4-C7                   B5-B6         1.5834

Table 2. The topological characters of the BCPs and RCPs of charge
density distributions at the B3LYP/6-311+G * level.

CP numbering scheme     Type of CP   [[rho].sub.b] (au)

[([B.sub.6]C).sup.2-]
1,2,6,7,10,12           BCP                0.150
3,4,5,8,9,11            BCP                0.143
1,2,3,4,5,6             RCP                0.140
12 BCPs                 6 RCPs

O-[B.sub.4][C.sub.3]
1,4                     BCP                0.157
2,6                     BCP                0.147
3,5                     BCP                0.214
7,9                     BCP                0.181
8                       BCP                0.292
1,3                     RCP                0.145
2                       RCP                0.205
9 BCPs                  3 RCPs

m-[B.sub.4][C.sub.3]
1,2                     BCP                0.150
3,4                     BCP                0.229
5                       BCP                0.162
6,7                     BCP                0.179
8,9                     BCP                0.187
1,3                     RCP                0.150
2                       RCP                0.139
9 BCPs                  3 RCPs

p-[B.sub.4][C.sub.3]
1,6,7,9                 BCP                0.175
2,10                    BCP                0.153
3,4                     BCP                0.150
5,8                     BCP                0.248
1,2,3,4                 RCP                0.146
10 BCPs                 4 RCPs

                         [[nabla].sup.2]
CP numbering scheme     [[rho].sub.b] (au)   [epsilon]

[([B.sub.6]C).sup.2-]
1,2,6,7,10,12                 -0.234            0.583
3,4,5,8,9,11                  -0.048          188.730
1,2,3,4,5,6                    0.356
12 BCPs

O-[B.sub.4][C.sub.3]
1,4                            0.526            3.099
2,6                           -0.154            5.842
3,5                           -0.099            2.486
7,9                            0.248            0.123
8                             -0.626            0.019
1,3                            0.249
2                              0.088
9 BCPs

m-[B.sub.4][C.sub.3]
1,2                           -0.088            5.736
3,4                           -0.202            0.919
5                              0.536            1.723
6,7                            0.130            0.335
8,9                            0.136            0.047
1,3                            0.174
2                              0.415
9 BCPs

p-[B.sub.4][C.sub.3]
1,6,7,9                        0.199            0.193
2,10                          -0.208            0.939
3,4                           -0.079           14.657
5,8                           -0.283            0.808
1,2,3,4                        0.392
10 BCPs

CP numbering scheme     [[lambda].sub.1] (au)   [[lambda].sub.2] (au)

[([B.sub.6]C).sup.2-]
1,2,6,7,10,12                  -0.167                  -0.105
3,4,5,8,9,11                   -0.155                  -0.001
1,2,3,4,5,6                    -0.160                   0.005
12 BCPs

O-[B.sub.4][C.sub.3]
1,4                            -0.222                  -0.054
2,6                            -0.160                  -0.023
3,5                            -0.317                  -0.091
7,9                            -0.269                  -0.240
8                              -0.456                  -0.447
1,3                            -0.146                   0.035
2                              -0.306                   0.082
9 BCPs

m-[B.sub.4][C.sub.3]
1,2                            -0.159                  -0.024
3,4                            -0.325                  -0.169
5                              -0.218                  -0.080
6,7                            -0.288                  -0.215
8,9                            -0.280                  -0.267
1,3                            -0.169                   0.019
2                              -0.138                   0.132
9 BCPs

p-[B.sub.4][C.sub.3]
1,6,7,9                        -0.259                  -0.217
2,10                           -0.155                  -0.080
3,4                            -0.152                  -0.010
5,8                            -0.349                  -0.193
1,2,3,4                        -0.154                   0.027
10 BCPs

Table 3. Internuclear distances ([Angstrom]) and topological characters
of the BCPs between C-C, B-B, and B-C nuclei in a standard set of
molecules at the B3LYP/6-311+G ** level (only singlet ground states
have been considered).

                             Type of      Internuclear
Molecule                   bonded atoms     distance

[H.sub.3]CC[H.sub.3]           C-C           1.531
[H.sub.2]CC[H.sub.2]           C-C           1.329
[H.sub.2]BB[H.sub.Z] (a)       B-B           1.629
HBBH (b)                       B-B           1.523
[H.sub.2]BC[H.sub.3] (c)       B-C           1.554
HBC[H.sub.2]                   B-C           1.376

Molecule                     CP type      [[rho].sub.b] (au)

[H.sub.3]CC[H.sub.3]           BCP             0.238
[H.sub.2]CC[H.sub.2]           BCP             0.344
[H.sub.2]BB[H.sub.Z] (a)       BCP             0.179
HBBH (b)                       BCP             0.195
[H.sub.2]BC[H.sub.3] (c)       BCP             0.189
HBC[H.sub.2]                   BCP             0.248

                            [[nabla].sup.2]
Molecule                   [[rho].sub.b] (au)   [epsilon]

[H.sub.3]CC[H.sub.3]            -0.531            0.000
[H.sub.2]CC[H.sub.2]            -1.028            0.332
[H.sub.2]BB[H.sub.Z] (a)        -0.476            0.000
HBBH (b)                        -0.517            0.835
[H.sub.2]BC[H.sub.3] (c)        -0.285            0.298
HBC[H.sub.2]                    -0.107            0.550

(a) The point group of the optimized geometry is [D.sub.2d].

(b) The optimized geometry is linear.

(c) The optimized geometry is bisected.