Comment on “A quantitative definition of hypervalency” by M. C. Durrant, Chem. Sci., 2015, 6, 6614

Consideration is given to (electronically) hypervalent increased-valence structures, which possess 2c–1e bonds, fractional 2c–2e bonds, and usually normal 2c–2e bonds.


Introduction
Durrant 1 has provided a quantitative denition of hypervalency via reference to atomic charges, which were obtained either from experimental or from theoretical electron densities using Bader's quantum theory of atoms in molecules (QTAIM) methodology. 3 Without calculation it was not usually possible to conclude whether a given molecule was hypervalent. For example, the hypercoordinate molecules CLi 6 and SiH 6 2À were calculated 1 to be hypervalent and non-hypervalent, respectively. 4 For this comment on hypervalence (primarily for electronrich molecules), we use and discuss types of hypervalent VB structures that were not considered in ref. 1, and which, since 1968, 8,9 have been designated as "increased-valence" structures without expansion of valence shells. They involve 2-centre, 1electron (2c-1e) bonds and fractional 2c-2e bonds (with bondnumbers less than unity, and represented by thin bond lines), 6, 8,9 and usually non-fractional 2c-2e bonds.
In Schemes 1, 2 and 4-6, increased-valence structures for PCl 5  When relevant atomic orbitals (AOs) overlap, Lewis-type VB structures for electron-rich species can be stabilized via 1-electron delocalizations from separate lone-pair AOs into 2-centre bonding MOs or bond orbitals (BOs), as is shown, for example, in structures 1c for PCl 5 in Scheme 1, and 2a, 4a, 5a and 6a for O 3 , SO 4 2À , NO 3 À and N 2 O 4 in Schemes 2 and 4-6. The resulting VB structures, 1d, 2b, 4b, 5b and 6b possess (thin-bond line) fractional 2c-2e bonds and 2c-1e bonds, as well as normal 2c-2e bonds. By inspection, one can see that more electrons participate in bonding in VB structures 1d, 2b, 4b, 5b and 6b than does occur in the Lewis Kekulé structures 1c, 2a, 4a, 5a and 6a. Therefore 1d, 2b, 4b, 5b and 6b are examples of "increased-valence " structures 6,8,9 without expansion of the valence shell. Because relative to the octet Lewis structures, increased-valence structures involve additional electrons in both nearest-neighbour and non-neighbour bonding, increased-valence structures are hypervalent relative to the Lewis structures. 6, 8,9 Some properties of increased-valence structures With Heitler-London AO type wavefunctions for 2c-2e bondsfor example a(1)b(2) + b(1)a(2) for the 2c-2e A-B bondincreased-valence structures for electron-rich molecules summarize resonance between two types of Lewis structures, namely the familiar, standard/Kekulé type Lewis octet structures, and "long-bond"/formal bond/singlet diradical/Dewar type Lewis (octet) structures. As indicated already, the latter type of Lewis VB structure is not considered in ref. 1. For the O 3 increased-valence structure 2b, these two types of Lewis octet structures (namely structure 3a and structures 3b-3d) are displayed in Scheme 3. None of them is hypervalent, but resonance between them generates hypervalence for the resulting increased-valence structure.
The PCl 5 and O 3 Dewar structures 1b, 3b and 3d do not carry atomic formal charges, and 1b and 3d in particular were not considered by Durrant.
For an increased-valence structure that does not involve a valence shell expansion to provide an additional AO for bonding, none of the component Lewis structures violates the octet rule, but resonance between them to generate the increased-valence structure leads to (an apparent) violation of the octet rule. 6 . As well as structure 4b, another type of increasedvalence structure can be constructed with four fractional 2c-2e bonds + four 2c-1e bonds and oxygen atom formal charges of À1/2. increased-valence structures with 1-electron bonds and fractional 2c-2e bonds provide better insight into the possible origin of some molecular properties than do the hypervalent VB structures of ref. 1. For example: (a) The fractional N-N 2c-2e bond in the increased-valence structure 6b for N 2 O 4 is in accord with the presence of a long, weak N-N single bond. 9 (b) The results of VB calculations for O 3 and related 1,3-dipolar systems from numerous laboratoriessee for example ref. 7d and e and 10 and references thereinindicate that their ground-states possess substantial singlet-diradical character. It arises primarily from the contribution of the Lewis structure 3d of Scheme 3 to the ground-state Lewis structure resonance scheme. In contrast to structure 2c (i.e. structure 5b of ref. 1), increased-valence structure 2b of Scheme 2 reects the diradical character.

S N 2 reactions
With Coulson-Fischer 11 type BOs (a + k 1 b) and (b + k 2 a) replacing the a and b AOs of the Heitler-London wavefunction for a 2c-2e bond, the course of an S N 2 reaction, X À + AY / XA + Y À has been formulated 12 as For it, the increased-valence structures for the (XAY) À reactant-like and product-like complexes, each with an additional bonding electron relative to the AY reactant and XA product, are both hypercoordinate and hypervalent relative to the VB structures for the latter species.

Further comments and conclusion
When 2-centre Coulson-Fisher type BOs, such as the a + k 1 b and b + k 2 a above, are used to accommodate the electrons that form the fractional 2c, 2e bonds in the increased-valence structures, allowance can be made for polarization of these bonds.
In ref. 13, the wavefunctions for 3c-4e VB structures of the types X-A-Y (as would occur in the Durrant structures 2c, 5c and 6c for O 3 , NO 3 À and N 2 O 4 if expansion of the valence shell does not occur) and X A Y, and the Rundle-Hach 14 -Pimentel 15 3c-4e MO conguration have been shown to be special cases of wavefunctions for resonance between the increased-valence structures X A-Y and X-A Y, with 2-centre Coulson-Fischer orbitals, and one variational parameter. 16 Increased-valence structures can also be constructed for: (a) systems that involve 3c-3e bonding units, 17 with non-reactant and non-product ionic structures replacing the singlet-diradical structures of electron-rich systems, and (b) diatomic molecules. 18 Regardless of the method used to construct the wave-functions for increased-valence structures, because they involve the participation of more electrons in bonding than do the Kekulé-type Lewis structures from which they are derived, increased-valence structures are hypervalent relative to these Lewis structures. As indicated above, for electron-rich systems, this is due to the inclusion of singlet diradical structures in the Lewis structure resonance scheme. Also, increased-valence structures involve at least one Pauling three-electron bond as a diatomic component. 9

Appendix: valence electron equivalent parameter g(A) and atomic valencies
In ref. 1, the valence electron equivalent parameter g(A) for atom A is used to determine whether a molecule exhibits hypervalence. 19 Here for the linear, symmetrical, triatomic systems of Table 1, each with one 3c-4e bonding unit, we shall show that the Durrant method 1 to construct g(A) is equivalent to a method that uses a 3c-4e MO conguration with AO overlap integrals omitted.
As well as 5b, another type of increased-valence structure can be constructed, with two fractional 2c-2e bonds and two oxygen atoms with formal charges of À1/2.
The x, a and y are the overlapping AOs on the three atomic centres, and j 1 ¼ x + ka + y and j 2 ¼ x À y are the bonding and non-bonding MOs that can be constructed from them. 14,15 The right-hand side of eqn (1) gives the valence-bond structure 21 (X-A-Y) q , with fractional 2c-2e bonds that arise from double-occupation of two non-orthogonal BOs, and q ¼ À1 or 0.
To determine the value for k, we equate the charge Q A of ref. 1 for atom A to X A À 2k 2 /(k 2 + 2). The X A is the core charge of atom A when the 3c-4e electrons are removed, and 2k 2 /(k 2 + 2) ¼ P aa is the A-atom charge density that arises from the 3c-4e bonding. For the neutral species and anions of Table 1, X A ¼ 2 and 1, respectively.

Durrant's method 1 to construct g(A)
To construct the g(A) parameter for the systems considered in Table 1, initially we follow Durrant's methodology, 1 as described in ref. 1 for CO, and for SCl 4 in the ESI for ref. 1. We use the expanded valence-shell (hypervalent) covalent structure X-A (q) -Y (with two non-fractional 2c-2e bonds) and the ionic structure X (À) A (q+2) Y (À) (with q ¼ X A À 2).
These structures are weighted according to the value of q ¼ X A À 2 so that the QTAIM charge Q A of ref. 1 is reproduced via eqn (2) is then calculated from eqn (4) via eqn (3), in which 10 and 6 are the number of A-atom electrons associated with the covalent and ionic structures.
Three of the calculated values for g(A) reported in Table 1 are greater than 8, which indicates hypervalency 1 for the associated species. However of course different numerical values and conclusions would be obtained with different types of 3c-4e wavefunctions and P(k) functions.

A-Atom valence
In ref. 22, with AO overlap integrals omitted, it is deduced that the A-atom valence (V A ) for the MO conguration of eqn (1) is given by eqn (7), with a maximum value of 1.3726 when k 2 ¼ 2O2.
In Table 1, the MO estimates of the A atom valence for each species exceeds unity, and therefore its A atom is hypervalent. However, regardless of the values of the A-atom valence and the g(A), because of the presence of 2c-1e bonds in addition to the 2c-2e bonds, more electrons participate in nearest-neighbour and non-neighbour bonding 6,8,9 for each of the increasedvalence structures than does occur in any of their component Lewis structures. Therefore all increased-valence structures are electronically hypervalent.

Note added in proof
For PCl 5 , an increased-valence analogue of structure 1a is obtained via the delocalization of two electrons from the Cl À of the Kekulé structure 1c rather than one (as in structure 1c), to give two 1-electron P-Cl bonds, a fractional equatorial 2c-2e P-Cl bond as well as the fractional axial 2c-2e P-Cl bond of structure 1d.