Valence‐Shell Electron‐Pair Repulsion Theory Revisited: An Explanation for Core Polarization

Valence‐shell electron‐pair repulsion (VSEPR) theory constitutes one of the pillars of theoretical predictive chemistry. It was proposed even before the advent of the concept of "spin", and it is still a very useful tool in chemistry. In this article we propose an extension of VSEPR theory to understand the core structure and predict core polarization in the main‐group elements. We show from first principles (Electron Localization Function analysis) how the inner‐ and outer‐core shells are organized. In particular, electrons in these regions are structured following the shape of the dual polyhedron of the valence shell (3rd period) or the equivalent polyhedron (4th and 5th periods). We interpret these results in terms of "hard" and "soft" core character. All the studied systems follow this trend, providing a framework for predicting electron distribution in the core. We also show that lone pairs behave as "standard ligands" in terms of core polarization. The predictive character of the model was tested by proposing the core polarization in different systems not included in the original set (such as XeF4 and [Fe(CN)6]3−) and checking the hypothesis by means of a posteriori calculations. From the experimental point of view, the extension of VSEPR to the core region has consequences for current crystallography research. In particular, it explains the core polarization revealed by high resolution X‐ray experiments.

Keywords: chemical bonding; electron localization function; electronic structure; quantum crystallography; VSEPR theory

Core values: Here, the extension of valence‐shell electron‐pair repulsion (VSEPR) theory to predict the electron distribution of the core region is explored. The results show that there is an intimate and predictable relationship between the valence and core shells, broadening the scope of VSEPR to the core regions (see figure). In this way, light is shed on the factors governing core polarization, which are important in current crystallography research.

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The development of simple models that facilitate a reliable prediction and rationalization of molecular structures and properties is of great importance in current chemical research.[[1]] In this regard, a good example of the predictive character of a simple energy model is the well‐known valence‐shell electron‐pair repulsion (VSEPR) theory. Specifically, it allows molecular geometries to be predicted in terms of the repulsion between electron pairs.[3] This theory has proven to work well for main‐group‐based systems,[4] with only some exceptions, especially for complexes bearing a d0 central atom.[[5]] Hence, it constitutes, in conjunction with other models, such as Lewis structures and electronegativity, a basic pillar of "chemical intuition" and structural analysis in chemistry.[[7]]

When developing the concepts behind VSEPR, the repulsion between electron pairs was used to explain the organization of the valence ligands and lone pairs. However, as its name states, this theory is generally applied to the valence shell, and less attention has been paid to the core electrons.[[9]] Herein we show that the principles that govern VSEPR theory can be applied to explain the spatial distribution of all the electrons, that is, both valence and core electrons. This generalization has obvious epistemological consequences, allowing further understanding of the electronic structure of the core regions to be obtained. Moreover, we also highlight its utility in the interpretation of new accurate crystallographic data.[11]

Whereas VSEPR theory was postulated relying mainly on chemical intuition and the empirical observation of a great number of compositions and stoichiometries, nowadays we count on tools that are able to reveal electron localization from first principles. One such tool is the Electron Localization Function (ELF).[12] The ELF is a measure of the probability of finding electron pairs with opposite spins, which makes it an ideal tool for the identification of Lewis entities (the ELF maxima typically appear at bonds and lone pairs). It has proven to be very useful in the understanding and prediction of chemical properties,[15] as well as in the study of a wide variety of reactions, ranging from organic to biochemical and organometallic, among many others.[, [18]] Within the framework of this contribution, we should highlight the work performed by Gillespie and co‐workers, who were able to explain the distortions of different metallic compounds from the VSEPR geometry by applying the ELF to study the interactions between the valence shell ligands and the outer‐core regions.[[9], [19]]

This type of analysis holds an inherent interest for the interpretation of new and more accurate spectroscopies. High‐energy[[20]] and high‐pressure[22] processes have led to interesting novel chemical phenomena in which electrons vacate the core or even leave the atom. This leads to a need to understand the subtle electron structure of the core. As an example, the inclusion of core deformations in the resolution of X‐ray data is a current field of development in crystallography.[23] Crystallographic structures are usually resolved by resorting to a multipolar approximation.[24] In this approach, the cores are kept frozen, whereas the valence deforms to fit the experimental structure factors by using multipoles.[25] However, it was recently proposed that for the appropriate resolution of accurate data, the core regions should also be allowed to deform.[11] Herein, we further confirm the existence of such deformations from a theoretical point of view, by reproducing and explaining the core deformations reported for α‐silicon.[26]

This paper is organized as follows. First, we introduce the concept of the dual and equivalent polyhedra, which will allow us to set the geometrical basis of the model. Then, we analyze the maxima of the ELF in the core region of a set of VSEPR‐geometry molecules with the central elements belonging to the representative groups of the periodic table, so as to unravel the connection between the core organization and the valence shell distribution. In particular, we interpret the localization patterns that arise in a number of VSPER molecular geometries, with the central atoms belonging to the 3rd, 4th, and 5th periods. Within these, we first focus on the core structure of molecular systems with stoichiometries AX6, AX4, AX5, and AX3 (in decreasing order of symmetry). Then, we evaluate the effect that valence‐shell distortions have on the core structure. The consistency of the results has allowed us to propose some inductive general rules that enable the organization of the core to be predicted, and these are summarized at the end of the paper. Finally, the predictive power of the model is verified by 1) explaining the experimentally determined core polarization of α‐silicon[26] and 2) predicting the core polarization of some test systems that are outside the group we used for elaborating the model.

Historically, the organization of the outer core in metals has been studied by means of an analysis of the electron density maxima within the quantum theory of atoms‐in‐molecules (QTAIM) approach.[27] A principle known as ligand‐opposed charge concentration (LOCC) was put forward. It states that the core localization maxima are located in the opposite direction to the ligands so as to minimize the repulsion between electron pairs, as exemplified in Figure  a for tetrahedral and octahedral geometries.[28] Nonetheless, a mere opposition of electron pairs does not always explain the organization of the electron localization maxima predicted by the ELF. For example, for a tetrahedral system such as SiF4, the core ELF maxima have the shape of an inverted tetrahedron (in agreement with the LOCC principle). However, for an octahedral system such as SF6, the ELF localization maxima do not lead to LOCCs (which would result in an octahedron with the same orientation as the ligands, as depicted in Figure  a). On the contrary, a cubic organization is observed. This leads to a reformulation of the principle guiding the outer‐core organization. In this contribution, we show that in general, core organizations do not respond to an electron localization opposed to the ligands, but to a general organization in terms of the overall geometrical disposition of the ligands and the period to which the central atom belongs. This leads to the appearance of the dual and the equivalent polyhedra.

The dual polyhedron (DP) is the one that results from placing a vertex in the middle of the face of the original polyhedra (see Figure  b). This principle allows the shape of the localization maxima observed in the two previous examples to be explained, because the dual polyhedron of an octahedron is a cube (and vice versa), and that of a tetrahedron is another tetrahedron with an inverted orientation (see Figure  b). These results anticipate the core structure to be intimately related to the valence shell, and thus, the pertinent extension of VSEPR theory to the core regions. If this is the case, what happens as we go to the inner core? Whereas the previous few studies focused on the outer core,[9] the concept of the dual polyhedron has encouraged us to analyze the whole core structure of central atoms belonging to the 3rd, 4th, and 5th periods. As previously stated, we have paid special attention to the relationship between the ligand distribution in the valence shell and the core structure, and how distortions in the valence shell affect the core organization.

First, we considered simple octahedral molecules (Oh symmetry). The details of the calculations are presented in the Experimental Section. In particular, we began by investigating the AX6(Z) geometries (A=S, P, and Cl, X=H, F, and Cl, and Z represents the pertinent molecular charge, see Table S5 in the Supporting Information). Note that because the ELF respects the symmetry of the system, it only shows one maximum situated at the core position for the first core shell (2nd period elements). Hence, the ELF can only be used to analyze the outer‐core polarization of central atoms having at least two core shells, that is, those that belong to the 3rd period onwards. In all the studied cases, the core region reflected the repulsion with respect to the valence shell, leading to the corresponding dual polyhedron: The localization maxima in the core for this set of molecules have the shape of a perfect cube (see Figure  a for the case of PF6−). To unravel whether this is a common trend for the outer core, the central atoms were substituted by those belonging to the 4th and 5th periods. Specifically, we considered selenium, arsenic, bromine (4th row), tellurium, antimony, and iodine (5th row). To our surprise, for these systems, a different kind of core polarization was observed. Namely, the outer core directly polarizes following the valence shell, that is, the ELF maxima in the core are ligand‐oriented, having the shape of the equivalent polyhedron (EP). As a consequence, octahedra are observed for the first core shell in the 4th and 5th row elements, as exemplified in Figure  b,c for AsF6− and SbF6−, respectively (see Table S5 in the Supporting Information for more examples). Because relativistic effects may be important in these systems,[29] we applied the relativistic zeroth‐order regular approximation (ZORA) in 5th row calculations, as explained in the Experimental Section.

We believe this finding paves the way to the extension of the chemical hard and soft character to the core regions. Atoms belonging to the 3rd period would exhibit a hard core character, and thus counter‐polarize with respect to the electron pairs in the valence shell (DP). In contrast, the outer core of atoms of the 4th row on would exhibit a soft core character. Hence, they undergo direct polarization (EP) with respect to the valence ligands. Note that the inner‐core shells are also organized in terms of the dual and equivalent polyhedra (see the whole core structures depicted in Figure ).

The previous results prompted us to extend the analysis to tetrahedral molecules (Td symmetry). In particular, the general stoichiometries AX4(Z) with central atoms belonging to the 3rd (A=Si, Al, and P), 4th (A=Ge, Ga, and As), and 5th (A=Sn, In, Sb) periods and with X=H, F, and Cl, and the pertinent molecular charges were considered (see Table S6 in the Supporting Information). Interestingly, the same general behavior as for the AX6 systems was observed. Namely, direct counter‐polarization happened for the 3rd row atoms. The dual polyhedron of the ligand shell was reproduced in the core, giving an inverted tetrahedron, as depicted in Figure  a for SiCl4. In contrast, the 4th and 5th row atoms undergo direct polarization. As a consequence, the equivalent polyhedron was obtained in the core region, that is, a tetrahedron oriented in the same direction as the ligand shell (see Figure  b for GeCl4). Again, the same kind of behavior, that is, organization in terms of the EP and the DP, was found in the inner core. As presented in Figure  c for germanium (4th row), the outer core has the shape of the EP of the valence shell, whereas the inner‐core localization maxima form the DP of the outer core (see Figure S1 for additional details).

Less symmetrical valence‐shell geometries were also evaluated. In particular, we considered trigonal‐bipyramidal and triangular molecules, both with D3h symmetry. For the trigonal bipyramids, we analyzed AX5(Z) stoichiometries (with A=Si, P, and S for the 3rd row, A=Ge, As, and Se for the 4th row, and A=Sn, Sb, and Te for the 5th row, X=H, F, and Cl, and the pertinent charges, as shown in Table S7 in the Supporting Information). Noteworthy, the ELF maxima in the outer core of the 3rd row atoms have the shape of an inverted triangle (see Figure  a for PH5). That is, the core localization maxima adopt the shape of the DP of an equatorial triangle. As for the previous systems, this situation inverts when moving to the 4th row atoms, which directly polarize with respect to the valence shell (see Figure  b). Thus, the electron pairs in the outer core adopt the shape of a trigonal bipyramid (EP). Also in agreement with the previous results, the inner cores of the 4th and 5th row atoms are structured in terms of the EP and DP, as shown in Figure S2.

The same principles were shown to hold for the triangular molecules AX3(Z) (A=Al and Si for the 3rd row, A=Ga and Ge for the 4th row, and A=In and Sn for the 5th row, with X=H, F, and Cl, and the appropriate charges; see Table S8 in the Supporting Information). In the molecular geometries with a central 3rd row atom, the outer core counter‐polarizes, adopting the shape of a trigonal bipyramid with the central vertex opposed to the valence shell ligands (DP) and the two apical vertices perpendicular to the central molecular plane (see Figure  c). As depicted in Figure  c, the expected DP polyhedron in AlH3 (and in general AX3 systems) is formed. However, it is accompanied by another two axial localization regions. It should be noted that within the D3h group, the axial and equatorial maxima form two different reducible representations. The appearance of these "extra" maxima in addition to the predicted ones requires further analysis. As expected, from the 4th row on, the outer core undergoes direct polarization (see Figure  d). Thus, the electron localization maxima have the shape of a triangle oriented towards the ligands (EP). The inner‐core shells are also structured in terms of the DP and EP, as shown in Figure S3.

The possibility of inducing core deformations was also analyzed. First, we investigated the effect of varying the central‐atom–ligand distances. Nonetheless, attempts to invert the core polarization by expanding and compressing the ligand shell did not lead to core structural changes. This result prompted us to analyze the effect of introducing symmetry variations. For this, we substituted one of the ligands in the valence shell by a different one. This procedure is represented in Figure  a for the transformation of SiF4 into HSiF3. As a consequence, the molecular geometry evolves from tetrahedral (Td symmetry) to trigonal pyramid (C3v symmetry). The geometry of the localization maxima in the core was modified according to the variations in the valence shell. In particular, we obtained the DP of a trigonal pyramid (a distorted tetrahedron), which is another trigonal pyramid pointing in the opposite direction. Moreover, we evaluated this effect for 4th row atoms by substituting silicon by germanium. As expected, the outer core underwent direct polarization, and the electron localization maxima had the same geometry as the valence shell, that is, a trigonal pyramid (see Figure  a). It is worth noting that this effect is transferred to the inner core, the localization maxima of which have the geometry of the DP of the outer core, that is, another trigonal pyramid. In this way, we show that modifications and distortions in the valence shell significantly affect all the core regions.

Another way to induce core modifications by means of geometry distortions without changing the VSEPR group is by substituting a ligand by a lone pair (and introducing two more electrons into the central atom). Figure  b shows the evolution of SiF4 to PF3. Note that the lone pair has the same effect as the ligand variation. Namely, the original tetrahedron in the core of SiF4 transforms into a trigonal pyramid as a consequence of the formal substitution of a fluoride ligand by a lone pair. The same effect is observed when taking into account 4th and 5th row atoms. For the particular case of the substitution of phosphorus by arsenic, the outer core directly polarizes towards the valence ligands and the lone pair, forming the EP, that is, an elongated trigonal pyramid pointing in the same direction as the original one. Then, as for the previous approach (fluorine substitution by hydrogen), the lone‐pair effect in the outer core also influences the inner core, with the appearance of a complementary elongated trigonal pyramid. Because the lone pair lacks a core, this shows that the effects hereby observed are directly related to electron–electron interactions, and not to a crystal field effect arising from the external potential (i.e. by the cores of the ligands).

Motivated by the previous results, we tested the ability of the model proposed herein to explain the core polarization experimentally determined in α‐silicon.[26] The extension of the multipolar approximation to the core has shown that the silicon outer core is polarized in the shape of a tetrahedron, which is in an inverted position with respect to the tetrahedron formed by the first coordination sphere. This result is in agreement with our model, according to which the silicon atom undergoes counter‐polarization with respect to the bonded ligands (as previously shown for SiX4 geometries, see Figure  a). Thus, the silicon core polarization has the shape of the DP of the valence ligands, that is, an inverted tetrahedron. To further confirm this result, we considered the model system reported in the original paper, Si(SiH3)4, and analyzed the ELF maxima in the silicon core region.[26] Note that the expected behavior (an inverted tetrahedron in the silicon core) was observed, as depicted in Figure . The agreement between the experimental results and the theoretical calculations further supports the validity of the theory.

At this point, it may be convenient to summarize the general trends that can be derived from the previous results. We have found that, for the studied systems, all the central elements (which belong to the same periodic table row) behave equally in terms of core polarization, independently of the ligands that constitute the valence shell. More specifically, in all cases, the outer core of atoms belonging to the 3rd period undergoes counter‐polarization, and the core localization maxima exhibit the structure of the dual polyhedron of the ligand shell. We interpret this behavior to be a consequence of the tendency of the system to minimize repulsive interactions between electrons in the valence and the outer‐core shell. Note that according to our results, the core electrons do not necessarily form electron pairs. As such, we can observe, for example, a cube as a localization pattern for the outer core of the elements of the 3rd period, which does not lead to two electrons per vertex.

For the 4th and 5th periods, the outer‐core structure directly polarizes in the direction of the valence shell polyhedron. Thus, the localization maxima in the core have the shape of the direct polyhedron of the ligand shell. This highlights the softer nature of the core in 4th and 5th row atoms. This concept of "hard" and "soft" cores is in agreement with the relationship expected between volume, compressibility, and conceptual DFT hardness.[32] Bigger atoms are expected to be easier to deform and polarize,[33] also leading to less compressible materials.[[34]]

The coherence of the results support the predictive character of the polarization model we propose. Hence, it should be possible to predict the core structure of representative elements just by taking into account the position in the periodic table and the coordination number.

In this regard, we tested the predictive ability of the model by applying it to the structures of three different systems that are not related to those considered for developing the previous discussion. In particular, we selected XeF4 and [Fe(CN)6]3−, the core structures of which are shown in Figure  a,b (see Table S11 in the Supporting Information for the Cartesian coordinates). The choice of XeF4 was based on the fact that it has two lone pairs. Thus, this allows us to test whether we can extend the results obtained for one‐lone‐pair systems to those with more lone pairs. Because xenon belongs to the 5th period, we would expect direct polarization of the outer core. The molecular geometry has the shape of a perfect square plus two lone pairs, which together form a compressed octahedron, as depicted in Figure  a. Then, according to the previous results, the expected geometry for the outer core is a slightly compressed octahedron, which is the one that was found when performing the calculation (see Figure  a).

After that, we attempted to predict the core structure of a system bearing a transition metal. For that, we selected [Fe(CN)6]3−, as a well‐known example of a metallic complex for which VSEPR theory holds.[36] Because iron belongs to the first‐transition‐metal row, we expect counter‐polarization in the outer core. Thus, taking into account that the molecular geometry is octahedral, we expect that the localization maxima in the outer core have the shape of the DP of that octahedron, that is, a cube. This hypothesis was checked by performing the calculation, and, effectively, a cube shape was revealed for the iron outer core, as shown in Figure  b. It is important to note that the main analysis performed in this work was carried out with main‐group elements. Hence, we are wary of extending these trends to systems bearing transition metals, and more systems should be studied to establish general and consistent trends.

Finally, we checked our ability to predict the core polarization in a crystal structure, as a test for crystallographic core polarization studies. We selected stishovite, a high‐pressure polymorph of silica.[37] This system has a tetragonal P42/mnm symmetry, in which the silicon is in an elongated octahedral coordination environment. Then, because silicon belongs to the 3rd period, it should counter‐polarize with respect to the valence shell and the core localization maxima should have the shape of an elongated cube (the DP of an elongated octahedron). To test this prediction, we constructed a cluster model, similar to the one considered for the analysis of α‐silicon (see Figure S7 and Table S11 in the Supporting Information). Our prediction is in agreement with the calculation (see Figure  c).

We have shown that the same physical principles that led to the development of VSEPR theory, that is, the molecular geometries adapting to minimize repulsion between electron pairs, can be used for predicting the core structure. Moreover, we have applied this model to understand in more detail the organization of electrons in atoms with several core shells. Two different behavior patterns have been identified. On the one hand, for nuclei belonging to the 3rd period, the electron localization maxima in the core orient toward the center of the polyhedral faces defined by the valence ligands, so as to minimize electron repulsions. In contrast, for the 4th and 5th periods, several core shells appear, and the outer one orients toward the ligands, undergoing direct polarization. This observation suggests a pertinent extension of the chemical concepts of hardness and softness to the core region: 3rd row atoms would be harder than those belonging to the 4th and 5th periods. As a consequence, the spatial disposition of the outer‐core electrons of elements belonging to the 3rd period is that that minimizes electron repulsions with the valence electrons. In contrast, the outer core of atoms in the 4th and 5th rows undergoes direct polarization toward the valence shell. In addition, we have demonstrated that there is an intimate relationship between the ligand shell and the core, as valence distortions are transferred to all the inner‐core shells. Finally, we have shown that lone pairs behave as "standard" ligands in terms of core organization. The principle described herein not only confirms the existence of core polarization, but also explains the experimentally found core spatial distributions in α‐silicon. Hence, these basic principles can be used to help understand and predict core polarization in accurate X‐ray data.

Geometry optimizations and wave function calculations for ELF analyses were obtained through DFT calculations by using the Gaussian 09 program package.[38] The CAM‐B3LYP exchange‐correlation functional was applied[39] in conjunction with the DGDZVP basis set, so as to be able to explicitly reproduce the electronic structure of the inner core of all the considered elements (up to the 5th period). The validity of the previous methodology was checked by using other functionals (B3LYP, PBE, and PBE0[46]) in combination with bigger basis sets (6‐311G(d,p), cc‐pVTZ,[49] and def2‐TZVP[50]). We also applied Hartree–Fock (HF) to analyze whether the absence of electron correlation has a significant effect on the results. As explained in the Supporting Information (Tables S1–S4), the results obtained by varying the computational conditions led to consistent results. The calculation of the ELFs and the analyses of their critical points were performed by using the PROMOLDEN code. Images were produced with VESTA.[53]

As relativistic effects may have an influence on the core structures of atoms belonging to the 4th and 5th periods, relativistic calculations were performed on a set of molecules so as to confirm the validity of nonrelativistic calculations. We considered the zeroth‐order regular approximation (ZORA) as implemented in the Orca 4.0 suite.[54] An all‐electron ZORA‐def2‐TZVP basis set was used in all the calculations[50] in conjunction with the auxiliary SARC/J basis. We considered representative molecules of all the geometries included in the main discussion, with central atoms belonging to the 3rd, 4th, and 5th rows. In particular, the following sets of molecules were chosen: AX3: AlF3, GaF3, and InF3; AX4: SiCl4, SiF4, GeF4, and SnF4; AX5: PF5, AsF5, and SbF5; AX6: SH6, SeH6, and TeH6. In all cases, the same results as for non‐relativistic calculations were obtained for the whole core structure of the 3rd and 4th periods and the outer core of the 5th period. Specifically, the inner core of InF3, SnF4, and SbF5 molecules presented inverted polyhedra as for nonrelativistic calculations. As a result, to provide accurate and reliable results, molecules involving 5th row central atoms were calculated by using relativistic calculations.

J.M. gratefully acknowledges the financial support from the Spanish Ministerio de Ciencia, Innovación y Universidades (FPU14/06003 and EST17/00161) and from the Universidad de Zaragoza, the Fundación Bancaria Ibercaja, and Fundación CAI (CB 6/17). J.C.‐G. expresses her gratitude to Sorbonne Université association CALSIMLAB and to the French Agence Nationale de la Recherche (ANR) within the Investissements d'Avenir program under reference ANR‐11‐IDEX‐0004‐02. All the authors thank the computational resources and technical expertise provided by the Laboratoire de Chimie Théorique (Sorbonne Université). We also thank Emilie Huynh, Hoai Dang, and Daniel l′Anson who participated in the early stages of the work. J.M. is especially grateful to Alberto Pérez‐Bitrián for helpful scientific discussions.

The authors declare no conflict of interest.

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