Localized-orbital locator (LOL) profiles of chemical

Download PDF
24 downloads 0 Views 357KB Size Report
Comment
trons for chemical bonding is investigated, and a LOL-VSEPR (valence shell ... bonding, kinetic-energy density, localized-electron locator, VSEPR theory.
695

Localized-orbital locator (LOL) profiles of chemical bonding Heiko Jacobsen

Abstract: We examine a recently introduced descriptor of chemical bonding, the localized-orbital locator (LOL), which is based on the kinetic-energy density (τ). Examples are presented for prototypical chemical bonds, such as single, double, and triple bonds, for bonding in transition metal complexes, for three-center two-electron bonds, as well as for hypervalent molecules. The topology of LOL is analyzed in terms of (3,–3) attractors (Γ). The influence of core electrons for chemical bonding is investigated, and a LOL-VSEPR (valence shell electron pair repulstion) relationship is established. Further, we compare LOL to the related electron localization function (ELF). Key words: chemical bonding, kinetic-energy density, localized-electron locator, VSEPR theory. Résumé : On a examiné un descripteur proposé récemment pour la liaison chimique, le positionnement d’une orbitale localizée (POL), qui est basée sur la densité d’énergie cinétique, τ. On présente des exemples de prototypes de liaisons chimiques, telles des liaisons simple, double et triples qui donnent lieu à des liaisons dans des complexes de métaux de transition, pour les liaisons à deux électrons et à trois centres ainsi que pour des molécules hypervalentes. La topologie du positionnement d’une orbitale localizée est analysée en fonction d’attracteurs (3,–3) (Γ). On a examiné l’influence des électrons à coeur sur la liaison chimique et on a établi une relation POL-VSEPR. On a de plus comparé le positionnement d’une orbitale localizée à la fonction apparentée de localization, ELF. Mots-clés : liaison chimique, densité de l’énergie cinétique, positionnement d’un électron localizé, théorie VSEPR. [Traduit par la Rédaction]

Jacobsen

702

Introduction The field of chemical bonding is in the midst of a renaissance, and the nature of the chemical bond remains an active field of ongoing chemical research. Although the “electronic structure revolution” initiated by the work of Lewis resulted in a picture of the chemical bond that by now has become a fundamental part of the freshmen chemistry curriculum, it is evident that many open questions remain in the quest for a unified and complete understanding of the true nature of the chemical bond (1). The complexity of this subject manifests itself in the ongoing discussion whether a description of chemical bonding is to be based on orbital or electron density concepts, molecular orbital theory not to be mistaken by the orbital approach and providing a common link between these two fundamentally different descriptions (2). The success of density functional theory (DFT) of electronic structure (3) has provided new impetus for a densitybased description of chemical bonding, and led to the development of new descriptors for old problems of chemical bonding, such as the electron localization function (ELF) introduced by Becke and Edgecombe (4). The problem of Received 17 January2008. Accepted 7 March 7 2008. Published on the NRC Research Press Web site at canjchem.nrc.ca on 23 May 2008. H. Jacobsen.1 KemKom, Libellenweg 2, 25917 Leck, Nordfriesland, Germany (e-mail: [email protected]). 1

Present address: Tulane University, Department of Chemistry, New Orleans, LA 70118, USA.

Can. J. Chem. 86: 695–702 (2008)

electron localization and delocalization constitutes one of the main aspects of the nature of the chemical bond. For density-based descriptions of chemical bonding, various partitioning schemes, mainly based on topological methods, have been proposed that address this very problem. Bader’s quantum theory of atoms in molecules (QTAIM) analyzes the gradient field of the one-electron density and characterizes the nature of a chemical bond through a topology analysis of the Laplacian distribution ∇2(ρ) of the underlying density, ρ (5). In particular, the topology of the function, L, where L = –∇2(ρ), provides valuable information about the nature of the chemical bond (6). An alternative approach has been proposed by Silvi and Savin, who work with ELF as potential function and partition the molecular space into basins of attractors that have a clear chemical meaning (7). Other recently developed descriptors for electron localization are the electron localizability indicator (ELI) (8, 9) and the localized-orbital locator (LOL) (10, 11), which is the main subject of the present work. Another descriptor that both in spirit and in form is related to LOL, is the local temperature, T (12). Local temperature, T, results from an analysis of thermodynamic elements of DFT, and it has been explored for whether local temperature can be used as a more fundamental descriptor of molecular electronic structure (13). One aspect that makes density approaches towards chemical bonding particularly attractive is the direct relationship with experiment; X-ray charge densities not only reveal molecular structure but also contain information about chemical bonding through Laplacian analysis (14). Other bonding descriptors, such as LOL, are also accessible from experi-

doi:10.1139/V08-052

© 2008 NRC Canada

696

Can. J. Chem. Vol. 86, 2008

mental electron densities. This requires constructing the kinetic-energy density directly from the experimentally determined electron density, which is achieved by using the second-order gradient expansion (15). Despite the fact that the gradient expansion diverges near the nuclei, an experimentally constructed LOL is accessible, since the regions around the nuclei may be omitted without significant loss of information (16). Although much progress has been made in recent years in bond description based on experimental electron densities, the main conclusion that is drawn from experimental work is the same as obtained from theoretical studies: the characterization of the chemical bond is not a closed subject (17). Schmider and Becke (11) have pointed out that LOL gives simple, recognizable patterns in classic chemical bonds, and they expressed their hope that “LOL will prove useful in interpreting the structures of exotic materials, and the classic examples of freshman chemistry texts as well” (10). Becke and Schmider have chosen graphical representations as the most evident way to analyze LOL. However, one has to take into consideration that even a three-dimensional function is not easy to visualize. In practice, one considers 2D-cuts, or plots of iso-surfaces, and often, several plots have to be shown to provide all the significant elements. Here, we use a topology analysis of LOL and characterize this function by determining the positions in space where LOL is maximal. We will briefly elucidate the theoretical background of our analysis before we present our results in more detail.

Theoretical background The central property on which the localization descriptors ELF and LOL are built is the kinetic-energy density, τ: [1]

τ=

1 ∑ (∆ψ i ⋅ ∆ψ *i ) 2 i

We note that like other energy densities, the kineticenergy density is not unique. Two important forms of the kinetic-energy density either depend on the Laplacian of the Kohn-Sham orbitals –τ L–, or are defined as being positive definite, –τ (18). The kinetic energy plays a major role in the description of a chemical bond, since the driving force of covalent bonding is a lowering of the quantum kinetic-energy density by orbital sharing (19, 20). What might look like a violation of the virial theorem can be understood when the kinetic energy of a molecule is partitioned into intra-atomic and interatomic contributions. Upon formation of a covalent bond, the kinetic energy of motion normal to the bond direction increases accordingly; along the bond direction, however, the decrease in kinetic energy predominates. The increase in the kinetic energy is purely intra-atomic and has nothing to do with what happens in the bonding region where the decrease in the inter-atomic kinetic energy is the decisive effect (21). Since the contributions to the kinetic energy from the bond regions usually decrease, it is desirable to use a measure of the local kinetic energy as an indicator of bond effects. The kinetic-energy density, τ, is quite suitable, although it has a few drawbacks. Kinetic-energy densities, τ, are usually monotonic for atoms, and the structural features

of τ are less pronounced than one would wish. Regions with high charge density tend to dominate so strongly that salient features of bonding become invisible. To circumvent this problem, the kinetic-energy density, τ, is compared with some suitable reference value, τ0. Both ELF and LOL take as a reference the homogeneous spin-neutral electron gas with electron density, ρ, and τ0 = (3/10)q(3π2)2/3ρ 5/3. LOL analysis then defines the dimensionless variable, t: [2]

t = τ0/τ

The ratio t is bounded by zero from below, but it is not bounded from above. To lower the semi-infinite range of the variable t, the localized-orbital locator, LOL, which is referred to as ν, is a measure of t in a mapping onto the finite range [0, 1]: [3]

ν=

τ0 / τ t 1 = = 1 + t 1 + τ0 / τ 1 + τ0 / τ

Schmider and Becke (10, 11) have analyzed the localizedelectron locator, ν, in much detail. We briefly summarize five important features of this bonding descriptor: (i) LOL reveals electronic shell structure in a simple and clear manner. (ii) The mapping t → ν preserves topological features of t, since it is monotonically increasing. The value t = 1 corresponds to regions where the local kinetic energy of electrons is the same as for a uniform electron gas with the local density, ρ, and is mapped onto ν = 1/2. Values of ν < 1/2 indicate “fast” electrons that have a higher kinetic energy than expected from their local density. Regions with a larger value of ν are characterized by relatively “slow” electrons. (iii) A covalent bond is expressed in ν as a local maximum between the bound centers. The ν = 1/2 surface enclosing the maximum has an increasingly concave shape in multiple bonds. (iv) The localized-orbital locator, ν, attains large values (above 1/2) in regions where the electron density is dominated by a single localized orbital. (v) Lone electron pairs are resolved as local maxima in the center of a characteristically shaped ν = 1/2 surface. The chemical content of the localized-orbital locator, LOL, is similar to that of the electron localization function, ELF, which is also dependent on the kinetic-energy density, τ. However, whereas ELF is founded on consideration of the electron pair density, LOL simply recognizes that gradients of localized orbitals are maximized when localized orbitals overlap. Moreover, LOL has a simpler interpretation in terms of fast and slow electron regions, and always falls to zero asymptotically, whereas ELF sometimes does not. Borrowing ideas from the topological analysis of electron charge density (5), we locate critical points and in particular (3,–3) attractors, Γ, in the gradient vector field of LOL. Each nucleus-position is associated with one attractor, Γ, but as we shall see, LOL also features attractors, Γ, between connected atoms and in molecular regions associated with lone pairs. We refer to attractors between connected atoms and in lone-pair regions as ΓCA and ΓLP, respectively. It is the Γ topology of LOL and its associated νΓ-values that serve as a © 2008 NRC Canada

Jacobsen

697

Fig. 1. ΓCA and ΓLP attractors and associated νΓ -values of (a) ethane (C2H6), (b) ethylene (C2H4), (c) acetylene (C2H2), (d) methanol (CH3OH), (e) carbon dioxide (CO2), and (f ) carbon monoxide (CO).

criterion to describe the nature of chemical bonding in molecular systems. The topologies of L and ELF have been studied in much detail (6), and it was found that localized pairs of electrons, bonding or non-bonding, are not only evident in ELF but also in L, which is homeomorphic to ELF with few exceptions. It was further noted that critical points other than the (3,–3) attractors also carry valuable information about the nature of a chemical bond. However, a complete topology analysis in terms of L even for a small molecule such as water results in an extended picture of bewildering complexity (22). In the spirit of LOL, which “is a simpler function than ELF” (10), we restrict our analysis to the (3,–3) bonding attractors, Γ.

LOL profiles of chemical bonds Schmider and Becke (10, 11) have analyzed the localizedelectron locator for a variety of systems displaying prototypical single and multiple bonds, such as C2H4, N2, and F2, as well as for cases where the bonding situation is not prototypical and sometimes not straightforward, such as ClF3, Cr(CO)6, and B4H4. They base their analysis on an evaluation of contour line diagrams of LOL in representative molecular planes that contain a particular bond under investigation. Here, we shall construct LOL profiles based on ΓCA and ΓLP attractors. We note that in the figures to follow, bond lines between atoms are derived from standard geomet-

ric connectivity arguments and are not to be mistaken for bond paths as obtained from a topological analysis of the electron charge density. Exemplary single, double, and triple bonds We begin with examples of what are commonly considered prototypical single and multiple bonds and investigate the LOL profiles of ethane (C2H6), ethylene (C2H4), acetylene (C2H2), methanol (CH3OH), carbon dioxide (CO2), and carbon monoxide (CO). Locations and LOL values of ΓCA and ΓLP attractors are depicted in Fig. 1. Ethane displays seven ΓCA attractors, one for each C–H and C–C bond. The LOL values are above 0.8 and indicate that the bonds are dominated by a single localized orbital. The C=C double bond in ethylene is also characterized by the presence of one, and not two ΓCA attractors. The LOL picture still shows one attractor for each bond, independent of the formal bond multiplicity. We note that the νΓ-value at ΓCA(CC) is smaller for the double bond in C2H4 than it is for the single bond in C2H6. The picture obtained from our LOL analysis is in agreement with the common consensus that double bonds are more delocalized than single bonds. This trend continues when we turn to acetylene (C2H2) that formally has a C⬅C triple bond. The νΓ-value at ΓCA(CC) further decreases, and an increase in bond multiplicity leads to an increase in delocalized bonding character and thus to a decrease in νΓ. A similar trend has been found in an ELF analysis, and Bader et al. (6) report ELF values at (3,–3) © 2008 NRC Canada

698

Can. J. Chem. Vol. 86, 2008

Fig. 2. ΓCA and ΓLP attractors and associated νΓ -values of the diatomic molecules (a) F2, (b) Cl2, (c) N2, and (d) P2.

critical points for C–C single, double, and triple bonds as 0.9685 (C2H6), 0.9412 (C2H4), and 0.8906 (C2H2). ELF also contains a mapping of values onto the [0, 1] interval, and it seems that the LOL mapping results in clearer separation of νΓ for bonds of different multiplicities. We further note that for all systems ΓCA(CH) is large with values above 0.9. For the ELF function, it has been recognized that gradient field attractors can be circles and spheres if the system belongs to a continuous symmetry group (23). LOL bears similarities to ELF, and for acetylene, belonging to the continuous symmetry group D∞h, we see the appearance of a ring-shaped ΓCA(CC) attractor, a connected set of (2,–2) critical points. We will further comment on ring-shape attractors at a later point in our discussion. When we proceed to systems having a carbon-oxygen linkage, we now encounter atomic centers that have additional lone-pairs. The LOL profile of methanol not only displays, as expected, three ΓCA(CH), one ΓCA(CO), and one ΓCA(OH) attractor, but it also shows two ΓLP attractors around oxygen. We note for the bonding geometry around oxygen that the νΓ values for O-C and O-H connectors are larger that the νΓ values of the oxygen lone pairs. The bonding structure around oxygen is dominated by bonding domains rather than by lone pair domains. For carbon dioxide, we find that the formal C=O double bond again displays a smaller νΓ-value than the C–O single bond in CH3OH. The oxygen lone pairs appear as ring-shaped ΓLP attractors with νΓ-value smaller than the ΓCA(CO) attractor. Although carbon monoxide belongs to the continuous symmetry group, C∞v, no ring shape attractors are found in the LOL profile of CO. A further requirement for the appearance of a ring shape attractor is the presence of an inversion center that renders the two different atomic centers equivalent. The lone pair attractor at carbon displays a larger νΓ-value (0.676) than the lone pair attractor at oxygen (0.573) and is further separated from the C-nucleus (d = 100 pm) than its counterpart is separated from the O-nucleus (d = 76 pm). This observation is consistent with the dipole moment of CO with its negative end on the carbon atom. We note that describing the electronic structure and the nature of the chemical bond of carbon monoxide is not a trivial task because of its unusual chemical and physical properties. Frenking et al. (24) have presented a detailed bonding analysis of CO as “an exercise in modern chemical bonding theory”. The authors use an evaluation of the graphical representation of the Laplacian distribution, –L, to illus-

trate some physical properties of CO. The LOL-profile of CO faithfully recovers the analysis of Frenking et al. (24), and puts it on a more quantitative basis. Atomic shells and chemical bonds The localized-orbital locator does reflect atomic shell structure, similar to the electron localization function. This feature is partially recovered in the LOL profile of molecules, and we discuss the diatomic molecules F2, Cl2, N2, and P2 as illustrative examples. ΓCA and ΓLP attractors are depicted in Fig. 2. For F2, besides attractors found at the position of the nuclei, we have one valence shell bonding attractor and two ring shaped lone pair attractors. For Cl2, which contains atoms with electron occupancy in the K, L, and M shells, we see one additional ring shape core attractor around each chlorine center. Similar observations are made when comparing the LOL profiles of N2 and P2. For dinitrogen, we find one valence shell ΓCA attractor and two valence shell ΓLP attractors. For diphosphorus, the extended electronic core structure manifests itself in two additional attractors located in close vicinity to each phosphorus atom. Although the dinitrogen molecule is of D∞h symmetry, only one ΓCA(NN) single point attractor is found, and not a ring-shaped ΓCA(NN) attractor as one might have expected in comparison to the isoelectronic and isosymmetric HCCH molecule. Similar observations have been made in an analysis of ELF, and with reference to N2 it was noted that a “disturbing feature is that sometimes, when one would expect separated maxima, just a single maximum appears” (25). However, this somewhat unexpected result can be understood if one considers the fact that the LOL profile of a chemical bond reflects to a major part the character of the highest energy orbital that contributes to that particular bond. The ground state valence configuration of N2 is (2σ +g )2(2σ +u )2(1πg)4(3σ +u )2, and the structure of LOL is dominated by the HOMO, a single σ +g orbital that extends along the atom-connecting line, and that gives rise to the presence of one single ΓCA(NN) attractor. On the other hand, for acetylene, the valence shell configuration is (2σ +g )2(2σ +u )2(3σ +g )2 (1πu)4. The HOMO consists of a set of two degenerated πu orbitals that extend perpendicular to the C-C connecting line and merge together due to the presence of the C∞ -symmetry axis, therefore giving rise to a ringshaped ΓCA(CC) attractor. © 2008 NRC Canada

Jacobsen

699

Fig. 3. ΓCA and ΓLP attractors and associated νΓ -values of (a) chlorine trifluoride, (b) tetraborane(4), and (c) chromium hexacarbonyl.

Non-prototypical bonds Schmider and Becke (11) have presented a LOL analysis for a set of molecules where the bonding situation is not prototypical and sometimes not straightforward. They mainly analyzed LOL via its graphical representation. Here, we select three representative examples out of their set of test molecules, namely ClF3, B4H4, and Cr(CO)6, and examine LOL in terms of Γ attractors. LOL profiles for these molecules are shown in Fig. 3. Chlorine trifluoride (ClF3) is often chosen as representative example for hypervalency. We point out that the concept of hypervalency in connection with the octet rule is often misunderstood and misinterpreted (26). All that can safely be attributed to a hypervalent molecule is the fact that it contains an atom forming more than four electron pair bonds, and that the bonds in hypervalent molecules are very similar to those in corresponding non-hypervalent molecules (27). To remove any connection to the nature of the chemical bond, it had been suggested to use the term hypercoordinate, rather than hypervalent, since this provided an empirical characterization of their experimentally observed molecular structures without the necessity of having to endorse a particular view concerning the theoretical description of the chemical nature of a hypervalent bond (28). The LOL profile of hypercoordinated ClF3 displays three ΓCA(ClF) attractors. In comparison to regular single and multiple bonds as discussed before, these attractors show fairly small νΓ-values. Around each atom, two additional ΓLP attractors are found. The νΓ-values of the lone pair attractors, and in particular those of ΓLP around chlorine are larger than those of the ΓCA attractors. Thus, the bonding structure in ClF3 is dominated by lone pair domains rather than by bonding domains. Distinctive is the fact that ΓLP attractors have

larger νΓ-values than ΓCA attractors. This, is not a property of hypervalent molecules only, but is related to the nature of the particular bonds present in a hypercoordinated molecule. It does, however, indicate the potential for a coordination sphere expansion beyond four electron groups. Schmider and Becke (10) have noted before that the localized-electron locator leads to a bonding picture that reflects the basic rules embodied in the valence shell electron pair repulsion (VSEPR) model of molecular geometry. The present study corroborates this result, and for ClF3 we find a faithful mapping of the ΓCA and ΓLP attractors with the number, relative size, and angular orientation of the bonded and non-bonded electron pair domains assumed in VSEPR (29). Furthermore, the dominance of lone pairs is reflected by the fact that ΓLP attractors have larger νΓ-values than ΓCA attractors. We will further explore the LOL-VSEPR relationship in the next section. Boranes comprise a large group of compounds with the generic formulae of BxHy and contain bonds that are characterized by a deficiency of electrons. Thus, the bonding in boranes has stipulated a wide variety of theoretical studies, including an ELF analysis of closo-boron clusters of the type B4X4 and B6X62– (30). Schmider and Becke (11) have selected B4H4 as the simplest representative of an E-E-E three-center two-electron bond, which is a structural element required to explain the existence of most of the closo clusters. The LOL-profile faithfully represents the presence of B-B-B three-center two-electron bonds. Four attractors Γ are located substantially outside the central boron tetrahedron on top of the center of the triangular faces. For chromium hexacarbonyl as representative example of molecules with metal-ligand bonds, Becke and Schmider (11) note that the LOL picture of Cr(CO)6 is relatively clear © 2008 NRC Canada

700

Can. J. Chem. Vol. 86, 2008

Fig. 4. ΓCA and ΓLP attractors and associated νΓ -values of (a) SH2, (b) SF2, (c) SF4, and (d) SH4.

and suggests a ‘’closed-shell” picture of metal–ligand bonding. The carbonyl groups point their carbon lone pair towards the metal atom but a significant localizationdeformation in that direction does not occur. They point out that “the lone-pair region is rather somewhat thinner compared to the free ligand” and that “in the vicinity of the chromium atom there are local maxima on the C3-symmetry axes of the molecule that point towards the triangular faces of the molecular octahedron”, which “should be considered as belonging to the core region”. The Γ-attractor profile for Cr(CO)6 is consistent with this analysis. The ΓLP(C) attractor is 56 pm away from the carbon atom, compared to 53 pm for the free ligand. The νΓ-value decreases from 0.676 for free CO to 0.664 in the bonded ligand. We further see an octet of ΓLP attractors around chromium located on the C3-symmetry axes. These attractors are 40 pm away from the metal center, which is the same separation as reported by Schmider and Becke (11) for local maxima in vicinity of the transition metal center. LOL, core electrons, and VSEPR A comparison of sulfur fluorides with the corresponding hydrogen sulfides demonstrates the importance of core electrons for chemical bonding and elucidates the LOL-VSEPR relationship. LOL profiles of SH2, SF2, SF4, and SH4 are depicted in Fig. 4. For SH2, we find two ΓCA(SH) and two ΓLP(S) attractors. We note that the νΓ-value for the bond attractors is larger than that for the lone pair attractors; the electronic structure of SH2 is dominated by bond pairs, rather than lone pairs. In addition, we find two core attractors Γcore around sulfur due to its extended electronic shell-structure. We note that the

electronic core is accordingly profiled upon formation of a chemical bond. If we proceed to SF2, we again find two ΓCA(SF) attractors, νΓ = 0.569, and two ΓLP(S) attractors, νΓ = 0.716. In addition, we see lone pair attractors around F, as well as core attractors. Also, for sulfur, Γcore(SH2) and Γcore(SF2) are of similar size. What is important is the fact that for SF2 the νΓ-value for the lone pair attractors is larger than that for the bond attractors, and the electronic structure of SF2 is dominated by lone pair domains rather than bond pair domains. A similar picture is found for SF4; the dominant attractor around sulfur again is the lone pair attractor ΓLP(S) with νΓ = 0.716. Hydrogen does not possess an electronic core, and as a consequence its nucleus is located inside the bond region of any X–H bond. The ΓCA(XH) bond maximum is close to the H atom and might even merge with the nucleus position. The absence of an electronic core at hydrogen renders X–H bonds highly localized, and in SH2 for example, the degree of localization measured by LOL is larger than any lone pair localization around sulfur. The importance of core orbitals for a bond description based on descriptors of electron localization has been recognized before. For example, the electron localization function, ELF, shows too-high values when computed from valence densities only instead of using the total density (31), and for transition metal systems the ELF analysis needs to involve the atomic core regions as well to reveal a more detailed insight into the bonding situation (32). The recognition that bonds towards the coreless H are especially localized also has structural consequences. It has been known for a long time that SH4, a hypothetical molecule, the existence of which remains subject of theoretical © 2008 NRC Canada

Jacobsen

investigations (33), does not adapt a C2v see-saw structure as predicted by VSEPR theory, but rather possesses a C 4vgeometry (34), Marsden and Smart (35) have offered an explanation for the origin of the unusual C4v-structure predicted for SH4 based on atomic charge arguments. Here, we suggest an explanation based on LOL values and VESPR theory. The valence shell electron pair repulsion (VSEPR) model has for many years provided a useful basis for understanding and rationalizing molecular geometry and has gained widespread acceptance as a pedagogical tool. In its original formulation, the fundamental rule states that pairs of electrons in a valence shell adopt an arrangement that maximizes their distance apart (36). A recent extended version uses the concept of electron domains, the space occupied by valence shell electrons or electron pairs, and places emphasis on the relative sizes and shapes of these domains (27). It has been noted that the topologies of ELF and L exhibit a remarkable correspondence with the number and arrangement of the localized electron domains assumed in the VSEPR model of molecular geometry (6), and Schmider and Becke (10, 11) have shown that the same holds true for LOL as well. VESPR geometries are derived from the realization that pairs of electrons tend to occupy positions in space that minimize repulsions and maximize the distance of separation between them. Electron groups are classified as lone pairs (lp) and bond pairs (bp), and it is generally assumed that lone pairs occupy more space than bonding electron pairs, and that the electron pair repulsion decreases as lp←lp > lp←bp > bp←bp. We propose that the νΓ-value of (3,–3) Γ-attractors can be taken as a measure for the spatial requirement and repulsion potential of an electron group. An intial geometry is constructed according to the points-on-a-sphere model of VSEPR, in which all electron groups are treated equivalent. Then, geometries are determined based on νΓ-value of (3,–3) Γ-attractors. Both SF4 and SH4 have five electron domains and therefore take on a geometry derived from a trigonal bipyramid. As we noted before, the ΓLP-attractor of SF4 has the largest νΓ-value, and the lone pair is therefore located in an equatorial position. As a result, SF4 adopts a C2v disphenoidal geometry. For SH4, the ΓLP-attractor has the smallest νΓ-value, and the lone pair therefore initially adopts an axial position. The remaining four ΓCA-attractors rearrange into a geometry in which the distance of separation between them is maximized. As a consequence, SH4 takes on a C4v-structure, derived from a square pyramidal arrangement of electron groups around sulfur. This structural rearrangement further creates an additional vacant coordination space for additional electron localization. The positions of the Γ attractors in SF4 and SH4 provide support for our arguments (compare Figs. 4c and 4d). We also note that SH4 displays a second ΓLP-attractor in opposite position to the dominant lone-pair domain, thus utilizing the additional coordination space for electron localization. We conclude this section with a brief comparison of ELF and LOL-values. Bader et al. (6) have reported ELF values for SH2, SF2, and SF4, and their results together with present data are compiled in Table 1.

701 Table 1. Values of (3,–3) attractors in ELF and LOL for SH2, SF2, and SF4, and radial distances, rELF and rLOL, to the S atom (in pm). Molecule

ELF

SH2

0.9694 0.9998 —b 0.8178 0.9961 0.8348 0.8178

SF2 SF4

a b

rELF 99 134 —b 99 95 87 111

LOL

rLOL 77

0.657 0.879 0.716 0.569 0.684 0.589 0.560

117 75 83 74 80 82

Typea n b n b n b to Fe b to Fa

n denotes a non-bonded maximum, b a bonded one. No values reported in ref. 6.

The ELF and LOL analyses give qualitatively comparable results. SH2 shows larger attractor values for bonding domains as opposed to sulfur fluorides. Lone pairs are in closer vicinity to the sulfur center than bond pairs. However, the LOL values are spread out more in the [0, 1] array of values, and νΓ-values for different type of electron groups are better separated in LOL than they are in ELF. The localized-orbital locator, LOL, seems to provide a clearer and more decisive picture than the electron localization function, ELF.

Computational details Density functional calculations have been carried out with the Gaussian03 system of programs (41). The molecular density used in topology analyses was obtained from pure density functional calculations within the generalized gradient approximation as proposed by Perdew, Burke, and Ernzerhof, PBE (42). A valence triple-ζ basis augmented by one polarization function (TZVP) was used for all atoms (43). Densities used in topology analyses were taken from optimized gas-phase structures. Topological analyses of molecular electron densities employed a modified version of the program MORPHY (44). 3-D pictures were generated using the graphical package Jmol (45).

Conclusion Our analysis of molecules with prototypical as well as unusual chemical bonds has shown that the localized-orbital locator, LOL, represents a conceptually simple, but efficient tool in characterizing a chemical bond. In particular, the analysis of LOL in terms of (3,–3) Γ-attractors provides an easy access to meaningful bond profiles. The chemical content of LOL is similar to that of the electron localization function, ELF, but a comparison of both methods suggests that the picture of a chemical bond generated by LOL is more decisive and easier to interpret. The topology of LOL clearly displays the locations of the classic VSEPR electronic groups. LOL nicely illustrates the role of core orbitals, which are an essential element of chemical bonding. Not only does LOL faithfully represent atomic shell structure, it further emphasizes the influence of core electrons on valence localization. Extracting chemical information from molecular electron density contributions continues to be a main topic in bond© 2008 NRC Canada

702

ing analysis, and more rigorous but also more computational demanding model based on maximum probability domains are currently being developed (37, 38). Compared to more sophisticates approaches, LOL has the advantage of being more accessible not only with respect to the computational approach but also with respect to an intuitive interpretation of the results. Besides the exemplary studies of Schmider and Becke (10, 11), only a few applications of LOL to molecular problems of chemical bonding and chemical reactivity have appeared in the literature (39, 40), which all utilize graphical representations of LOL. The location of Γ attractors and the evaluation of νΓ-values puts the LOL analysis on quantitative basis and facilitates the clear formulation of the nature of a chemical bond. It is hoped that the present work will promote the recognition of the localized-electron locator as a valuable tool in the analysis of chemical bonding.

Acknowledgements KemKom acknowledges Professor L. Cavallo for granting access to the MoLNaC Computing Facilities at Dipartimento di Chimica, Università di Salerno.

References 1. G. Frenking and S. Shaik. J. Comput. Chem. 28, 1 (2007). 2. F. Cortés-Guzmán and R.F.W. Bader. Coord. Chem. Rev. 249, 633 (2005). 3. W. Kohn, A.D. Becke, and R.G. Parr. J. Phys. Chem. 100, 12974 (1996). 4. A.D. Becke and K.E. Edgecombe. J. Chem. Phys. 92, 5397 (1990). 5. R.F.W. Bader. Atoms in molecules: A quantum theory. Oxford University Press, Oxford. 1990. 6. R.F.W. Bader, S. Johnson, T.-H. Tang, and P.L.A. Popelier. J. Phys. Chem. 100, 15398 (1996). 7. B. Silvi and A. Savin. Nature (London), 371, 683 (1994). 8. M. Kohout. Int. J. Quantum Chem. 97, 651 (2004). 9. M. Kohout. Faraday Discuss. 135, 43 (2007). 10. H.L. Schmider and A.D. Becke. J. Mol. Struct: THEOCHEM, 527, 51 (2000). 11. H.L. Schmider and A.D. Becke. J. Chem. Phys. 116, 3184 (2002). 12. P.W. Ayers, R.G. Parr, and A. Nagy. Int. J. Quantum Chem. 90, 309 (2002). 13. P.W. Ayers and A. Nagy. J. Chem. Phys. 126, 144108 (2007). 14. T.S. Koritsanszky and P. Coppens. Chem. Rev. 101, 1583 (2001). 15. V. Tsirelson. Acta Crystallogr. B58, 632 (2002). 16. V. Tsirelson and A. Stash. Acta Crystallogr. B58, 780 (2002). 17. P. Coppens. Angew. Chem. Int. Ed. 44, 6810 (2005). 18. J.P. Perdew and L.A. Constantin. Phys. Rev. B, 75, 155109 (2007). 19. T. Bitter, K. Ruedenberg, and W.H.E. Schwarz. J. Comput. Chem. 28, 411 (2007). 20. K. Ruedenberg and M.W. Schmidt. J. Comput. Chem. 28, 391 (2007).

Can. J. Chem. Vol. 86, 2008 21. W. Kutzelnigg. Angew. Chem. Int. Ed. Engl. 12, 546 (1973). 22. P.L.A. Popelier. Coord. Chem. Rev. 197, 169 (2000). 23. S. Noury, F. Colonna, A. Savin, and B. Silvi. J. Mol. Struct. 450, 59 (1998). 24 G. Frenking, D. Loschen, A. Krapp, S. Fau, and S.H. Strauss. Comput. Chem. 28, 117 (2007). 25. A. Savin. J. Mol. Struct: THEOCHEM, 727, 127 (2005). 26. R.J. Gillespie and B. Silvi. Coord. Chem. Rev. 233, 53 (2002). 27. S. Noury, B. Silvi, and R.J. Gillespie. Inorg. Chem. 41, 2164 (2002). 28. W.B. Jensen. J. Chem. Educ. 83, 1751 (2006). 29. R.J. Gillespie and E.A. Robinson. Angew. Chem. Int. Ed. Engl. 35, 495 (1996). 30. A. Burkhardt, U. Wedig, H.G. von Schnering, and A. Savin. Z. Anorg. Allg. Chem. 619, 437 (1993). 31. M. Kohout and A. Savin. J. Comput. Chem. 18, 1431 (1997). 32. M. Kohout, F.R. Wagner, and Y. Grin. Theor. Chem. Acc. 108, 150 (2002). 33. A. Wittkopp, M. Prall, P.R. Schreiner, and H.F. Schaefer. Phys. Chem. Chem. Phys. 2, 2239 (2000). 34. Y. Yoshioka, J.D. Goddard, and H.F. Schaefer. J. Chem. Phys. 74, 1855 (1981). 35. C.J. Marsden and B.A. Smart. Aust. J. Chem. 47, 1431 (1994). 36. R.J. Gillespie. Molecular geometry. Van Nostrand Reinhold, London. 1972. 37. E. Cances, R. Keriven, F. Lodier, and A. Savin. Theor. Chem. Acc. 111, 373 (2004). 38. A. Scemama, M. Caffarel, and A. Savin. J. Comput. Chem. 28, 442 (2007). 39. H. Jacobsen and J.P. Donahue. Can. J. Chem. 84, 1567 (2006). 40. H. Jacobsen and M.J. Fink. Inorg. Chim. Acta, 360, 3511 (2007). 41. M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, J.A. Montgomery, Jr., T. Vreven, K.N. Kudin, J.C. Burant, J.M. Millam, S.S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G.A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J.E. Knox, H.P. Hratchian, J.B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R.E. Stratmann, O. Yazyev, A.J. Austin, R. Cammi, C. Pomelli, J.W. Ochterski, P.Y. Ayala, K. Morokuma, G.A. Voth, P. Salvador, J.J. Dannenberg, V.G. Zakrzewski, S. Dapprich, A.D. Daniels, M.C. Strain, O. Farkas, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J.V. Ortiz, Q. Cui, A.G. Baboul, S. Clifford, J. Cioslowski, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R.L. Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A. Nanayakkara, M. Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, C. Gonzalez, and J.A. Pople. Gaussian 03 [computer program]. Revision C.02. Gaussian, Inc., Wallingford, Conn. 2004. 42. J.P. Perdew, K. Burke, and M. Ernzerhof. Phys. Rev. Lett. 77, 3865 (1996). 43. A. Schäfer, C. Huber, and R. Ahlrichs. J. Chem. Phys. 100, 5829 (1994). 44. P.L.A. Popelier. Comput. Phys. Commun. 93, 212 (1996). 45. Jmol: an open-source Java viewer for chemical structures in 3D. Available from http://www.jmol.org.

© 2008 NRC Canada