On the Nature of the Three-Electron Bond

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On the Nature of the Three-Electron Bond David Danovich, Cina Foroutan-Nejad, Philippe Charles Hiberty, and Sason Shaik J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b11919 • Publication Date (Web): 16 Jan 2018 Downloaded from http://pubs.acs.org on January 19, 2018

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The Journal of Physical Chemistry

On the Nature of the Three-Electron Bond David Danovich*a, Cina Foroutan-Nejad*b, Philippe C. Hiberty*c, Sason Shaik*a

a. Institute of Chemistry, Hebrew University of Jerusalem, 9190401 Jerusalem, Israel b. CEITEC – Central European Institute of Technology, Masaryk University, Kamenice 5/A4, CZ-62500, Brno, Czech Republic c. Laboratoire de Chimie Physique, UMR CNRS 8000, Groupe Théosim, Université de ParisSud, 91405 Orsay Cédex, France

ABSTRACT

We analyze the properties of fifteen 3-electron bonds, which include σ-3-electron-bonds, such as dihalide radical anions and di-Noble gas radical cations, π-3-electron-bonds as in hydrazineradical cation, and doubly-π-(3e)-bonded species such as O2, FeO+, S2, etc. The primary analytical tool is the breathing-orbital valence-bond (BOVB) method, which enables us to quantify the charge-shift resonance energy (RECS) of the 3 electrons, and the bond dissociation energies (De). BOVB is tested reliable against MRCI calculations. Our findings show that in all 3-electron bonds, none of the VB structures has by itself any bonding. In fact, in each VB structure, the 3 electrons maintain Pauli repulsion, while the entire bonding energy arises from

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the resonance due to charge shift between the two or more constituent VB structures. Hence, 3ebonds are charge-shift bonds (CSB). The CSB character is probed by calculating the Laplacian (L) of the 3e-bond. Thus, much like the CSBs in electron-pair bonds, such as F2 or the central bond in [1.1.1]propellane, here too L is positive, thus showing the excess kinetic energy of the shared-density due to the Pauli repulsion in the 3-electron VB structures. The RECS values for 3electron bonds are invariably larger than the corresponding bond energies. For the doubly-π(3e)-bonded species, RECS is very large, exceeding 100 kcal mol-1. As such, it is fitting to conclude that σ- and π-3-electron-bonds find their natural place in the CSB family along with two-electron CSBs, with which they share identical energetic and topological characteristics. Experimental manifestations/tests of 3e-CSBs are proposed.

1. Introduction Two-center three-electron-bonded radicals have attracted considerable attention in view of their importance in free-radical chemistry,1-3 biochemistry,4 intrazeolite photochemistry,5,6 and bioinorganic enzymology.7-10 In the rest of this manuscript, we refer to these species as 3e-bonds. 3e-bonds were initially described by Pauling using valence bond (VB) theory,11,12 in which he showed that the stability of the species arises due to a resonance between two Lewis structures that are mutually related by charge transfer (charge shift). This is shown in equations (1), (2), and (3) for 3e-bonded species involving cations, anions, and radicals, respectively. A•+ :B ↔A: •B+

(1)

A• :B– ↔A:– •B

(2)

A: •B ↔A•+ :B–

(3)

Subsequently, it has been pointed out that a significant resonance energy requires a similar stability of the two resonating Lewis structures.4,5 Clark13 has shown that the 3e-bond energy of


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ions decreases exponentially with the difference between the ionization potentials of A and B in equation (1), or with the difference between the electron affinities of A and B in equation (2). Consequently, many of the known 3e-bonds are homonuclear, or at least involving two atoms of similar IPs or EAs. Moreover, 3e-bonds are seldom observed in neutral species, because the charge transfer in equation (3) is generally strongly endothermic, especially in the gas phase. In the molecular orbital (MO) framework, such molecules are seen to be held together by a three-electron configuration (symbolized as ∴), in which one bonding MO is doubly occupied while the corresponding antibonding MO is singly occupied, leading to a net bond order of ½. Experimentally, 3e-bonds are abundant and well characterized.1-4,14-41 Numerous (R2S∴SR2)+ radical cations,14-20 (RS∴SR)– radical anions,19,21-26 and R2S∴SR neutral radical19,21,24,27,28 have been identified, as well as N∴N,29-33 P∴P,34,35 As∴As,34,36 Se∴Se,37,38 Te∴Te,38 I∴I,39,40 Cl∴NH3,41 and more generally all kinds of X∴Y 3e-bonds (X, Y = N, S, P, halogen, etc.). Experimental and theoretical studies of such 3e-bonds between transition metals have recently been reviewed.42 On the theoretical side, Clark13 and Radom43 have carried out systematic calculations on series of radical cations involving 3e-bonds between atoms of the first and second rows of the periodic table, with hydrogen atoms as substituents. The inclusion of electron correlation is essential for the calculation of 3e-bonding energies. The Hartree-Fock error is nonsystematic and always large, sometimes of the same order of magnitude as the bonding energy itself or even larger. Thus, most of the studies were performed at post-Hartree-Fock levels.26,44-61 The MP2 level is satisfactory and provides geometries and bonding energies in good agreement with higher orders of perturbation theory.43,62] Importantly, the necessary electron correlation is entirely of dynamic nature, not static. Examining the nature of 3e-bond MO configuration and mapping it to the corresponding VB


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wave function easily explain the essence of the dynamic correlation. Indeed, starting from the Hartree-Fock wave function of a homonuclear molecule Aa∴Ab, which displays a 3e-bond between respective atomic orbitals χa and χb, leads to the resonating VB description Aa• :Ab ↔ Aa: •Ab as in equations (1) or (2) above. This description is exemplified in the corresponding wave functions of a σ-type 3e-bond in equations 4a-c below: (4a) σ = χa + χb

(4b)

σ∗ = χa - χb

(4c)

where the normalization constants are dropped and φi stands for the “spectator” MOs not directly involved in the 3e-bond. Thus, both VB and MO theories converge to the very same description of 3e-bonding in which two Lewis structures mutually related by charge-shift represent 100% of the wave function. Of course, at this computational level, both wave functions possess a common set of orbitals (ϕi, χa, χb), which are optimized for an average neutral situation in which the charge (or atomic occupancy) is equally shared by Aa and Ab. On the other hand, the VB description offers a significant refinement that conserves the form of the wave function, but allows at the same time each VB structure to have its own specific set of orbitals, as in equation 5, with ϕi’, χa’, χb’ orbitals being different from ϕi, χa, χb. (5) Here, ΨBOVB is calculated by the “Breathing Orbital Valence Bond” method (BOVB,63-66 see the Methods section). The difference between ΨBOVB and ΨHF corresponds to dynamic correlation,


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which is included in the BOVB procedure, and is retrieved in MO computations through postHartree-Fock treatment (e.g. Möller-Plesset perturbation). Chemical bonds dominated by charge fluctuation constitute a well-established family among two-electron bonds. Such bonds, called charge-shift bonds (CSBs) are characterized by exceptionally large resonance energies arising from the mixing of the covalent and ionic components of the bond wave function.66-71 Thus, in homopolar as well as heteropolar CSBs, the resonance energy is the major component of the bonding energy, while the covalent coupling is of minor importance and in some notable cases, it is altogether repulsive at all distances. Archetypal CSBs are the F2 molecule or the central C-C bond in [1.1.1]propellane. In other words, CSBs owe their stability to the resonance fluctuation of the electron pair between the two centers (as represented in Scheme 1a). This situation is different than in classical covalent or polar covalent bonds, in which the extra stabilization due to charge fluctuation is of minor importance, and bonding is brought about mostly by the covalent structure, A•–•B in Scheme 1a.

Scheme 1. (a) Fluctuations of the electron pair in a two-electron bond A–B; the resonance energy, associated with the fluctuation, is the major cause for bonding in a CSB, whereas the bonding energy of a classical covalent bond is dominated by the stabilization due to spin-pairing in A•–•B. (b) Fluctuation of the electron pair in an A∴B 3e-bond.


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Leaving apart the ionic bonds, two-electron CSBs are readily characterized in the framework of VB theory, by estimating the magnitude of the covalent-ionic resonance energy, RECS in equation (6), RECS = E(A•–•B) – E(A•–•B ↔ A+ B– ↔ A– B+)

(6)

where A•–•B and A+ B– ↔ A– B+ are the respective covalent and ionic components of a twoelectron A–B bond. CSB is not a mere theoretical construct of VB theory; a variety of other experimental and theoretical signatures characterize these bonds. Density difference maps, available experimentally, show a group of “no-density bonds” that coincides with the CSB family as defined by VB.72-74 In the Atoms-in-Molecules (AIM) approach,75-78 homopolar CSBs are most easily recognized by their unique electron-density features. Thus, CSBs possess, on the one hand, significant electron density ρ(rc) at the bond critical point71,78 -- which is a feature of shared density -- and on the other hand a positive Laplacian (L), which is a feature of repulsive closed-shell interactions and ionic bonds.70,75-78 In the Electron-Localization Function (ELF) approach, CSB is characterized by a basin population much, much smaller than 2.0e–, or altogether by a lack of a disynaptic basin (bonding basin), and by a large value of the variance, characterizing an intense electron fluctuation.69,71,79 Back to 3e-bonds, it can be seen that their wave functions, of the type A: •B ↔ A• :B (omitting the charges), is of a pure charge-shift (CS) type, especially in the homonuclear case A = B where both Lewis structures have equal weights. Therefore, unless either form A: •B or A• :B possesses significant bonding by itself -- which qualitative VB considerations shows to be impossible80 -one may safely anticipate that all the bonding energy of such bonds is provided by resonance due to charge shifting between the two structures, which is the very definition of charge-shift bonding (CSB).


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Following the above qualitative analysis, our aim in this paper is to ascertain whether or not 3e-bonds exhibit indeed the characteristic features of CSBs. This will be done by means of accurate ab initio VB computations, of the BOVB type.63-65 The specific questions we wish to answer are the following: (i) Are the limiting Lewis structures that compose the 3e-bond wave functions individually bonded or repulsive? (ii) Are the resonance energies associated with charge shifting between the two Lewis structures significant, and close to the bonding energies or larger? (iii) Do the 3e-bonds display positive Laplacians at their bond critical points like the two-electron CSBs? To answer these questions, we investigated a series of representative radical cations and anions, displaying 3e-bonds of σ as well as π types, using the BOVB method that has amply proven to provide accurate energetics with extremely compact wave functions. The choice of VB methodology is justified by its ability to compute the energy of each single Lewis structure, and the resonance energy arising from their mixing. Finally, the AIM theory will be used to calculate the Laplacians of the density at the critical points of the bonds.

2. Target Systems To examine the 3e-bonds and characterize their features, we selected a set of 15 molecules, which are depicted in Schemes 2a-e. Schemes 2a and 2b involve diatomic σ-type 3e-bonds, A∴A, which in MO theory involve σ2σ*1 electronic configurations. Scheme 2a shows the classical dihalogen 3e-bonds, 1-3,81-83 generated by a combination of a halide anion and a halogen atom. Scheme 2b involves three (Ng∴Ng)•+ cationic 3e-bonds, 4-6, generated by the union of a Noble gas (Ng) atom and its corresponding radical cation.[84] Scheme 2c involves polyatomic 3e-bonds, 7 and 8, one is (HO∴OH)•–, which is a σ-type, while the other is a


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hydrazine radical cation, which possesses a π-type 3e-bond, having a π2π*1 MO configuration. In turn, Scheme 2d collects five diatomic species, 9-13, which are isoelectronic and all possessing double π-type 3e-bonds in two perpendicular planes and an underlying σ-bond.85 These species will be referred to hereafter as doubly-π-(3e)-bonded species. Finally, Scheme 2e involves two experimental systems, 14 and 15, containing (S∴S)•+ and (N∴N)•+ σ-type 3e-bonds.4,30,31

Scheme 2: Target 3e-bonded species: (a) σ-Type (X∴X)•– dihalide species, 1-3. (b) σ-Type (Ng∴Ng)•+ di-noble gas species, 4-6. (c) σ-Type (HO∴OH)•–, 7, and π-type (H2NNH2)•+ species, 8. (d) Doubly-π-(3e)-bonded species, 9-13. As exemplified by the cartoon for O2, all the remaining species possess double-π-(3e)-bonds in two mutually perpendicular planes and an underlying σ-bond. (e) σ-Type (S∴S)•+ and (N∴N)•+ bicyclic and tricyclic species, 14 and 15.


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As such, the chosen target species will provide us with trends in the bonding energy and charge-shift resonance energies (RECS) of the σ- and π-type 3e-bonds, and the dependence of these quantities on the identity of the bond constituents, the number of bonds, and the charges of the species. The doubly-π-(3e)-bonded diatomic species (Scheme 2d) are also triplet diradicals, which may reveal how the charge-shift resonance energies control their respective chemical behaviors. For example, the O2 molecule was computed before, using VB calculations,85-87 which led to the conclusion that the charge-shift resonance energy of the two π-type 3e-bonds, Scheme 3, is a large quantity that exceeds the resonance energy of benzene. The recent thermochemical value, derived by Borden and Hoffmann et al,88 is ~100 kcal mol-1, which accounts for the kinetic inertness of O2. The thermochemical estimate is highly significant since it shows that the chargeshift resonance energy (RECS) is in principle a quasi-observable, and that it affects the chemical behavior of the molecules.

Scheme 3. The four VB structures that are responsible for the kinetic stability of the doubly-π(3e)-bonded O2 Molecule. Shown also is the thermochemically estimated[88] charge-shift resonance energy.


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Furthermore, as we will show the RECS is the root cause of the bonding in 3e-bonds. Thus, even in those cases where the 3-electron bond dissociation energies (De) cannot be determined or are negative (see later in: 11 and 12), we shall calculate the quasi-observable RECS quantity. In addition to the energy quantities, we shall present the values of the corresponding Laplacians in the respective bond critical points.70,75-78 As shown in the past, this quantity typifies charge-shift bonds of the electron-pair type.66-71,89 All in all, the target systems depicted in Scheme 2 will tell a substantial story.

3. Methods We have carried out many test calculations for the post Hartree-Fock and VB calculations, using different basis sets. The various results are summarized in the supporting information (SI) document submitted along with this paper. Here, in the manuscript, we describe only the results labeled as “Method/aug-cc-pVTZ”, where “Method” correspond to multi-reference configuration interaction (MRCI),[90] coupled-cluster method[91] which includes singles, doubles, and perturbational triples, CCSD(T), and the breathing orbital VB (BOVB) method.63-65,92 3.1. Post Hartree-Fock Methods. The geometries and De values were determined for all the species in Scheme 2 initially by post Hartree-Fock calculations of two types, using MOLPRO suit of programs.93 Wherever possible we used the MRCI method, with the Davidson corrections94 (typically 1-3 kcal mol-1) of the De values. The CCSD(T) method was used for species where MRCI encountered convergence difficulties (e.g., species 8 and 13 in Scheme 2). The so obtained De values will serve as benchmark for testing the BOVB method.


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3.2. Basis Sets. We tested basis sets ranging from cc-pVTZ95 all the way to aug-cc-pV6Z.96 Our results showed that for the anionic species, MRCI/aug-cc-pVTZ and CCSD(T)/aug-ccpVTZ gave reliable results (e.g., the respective De results for F2– were 0.5 kcal mol-1 of each other). Therefore, the aug-cc-p-VTZ basis set was used uniformly in the study, including in the VB calculations. 3.3. Valence Bond Methods. The VB calculations were carried out using the XMVB software,97-99 based on the optimized geometries using the post Hartree-Fock methods. The VB calculations were restricted to BOVB, which is the most compact VB method that involved also good accuracy.92 The BOVB calculations were tested using different basis sets, the BOVB/augcc-pVTZ results matched those of CCSD(T) and MRCI at the same basis set. Among the BOVB methods, the most accurate ones are D-BOVB and SD-BOVB.100 In DBOVB the active electrons/orbitals are treated by BOVB with orbitals strictly localized on a single atom or fragment, while the inactive shell orbitals (e.g., lone pairs, etc.) are used as delocalized (D) MOs, which undergo optimization during the BOVB procedure. In the SDBOVB method, the inactive shell orbitals are delocalized, and in addition, the doubly occupied atomic hybrid orbitals (HAOs) in the active shell undergo splitting to reduce the electronelectron repulsion of the bonding electrons. The BOVB requirement that the active orbitals be localized on the atoms with which they are associated is not a constraint, but an orbital definition that is consistent with the explicit inclusion of all relevant Lewis structures in the calculation. Thus, the BOVB has all the degrees of freedom necessary for a proper description of the 3e-bond. It allows the orbitals of each VB structure to take freely shape and size on their respective atoms, while the resonance between the two structures takes care of the “delocalization”. Because of these features the BOVB method


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leads to bond energies in good agreement with experiment and high-level post-Hartree-Fock calculations. In contrast, the use of delocalized orbitals is consistent with wave functions in which some Lewis structures are not immediately apparent but are nevertheless involved in an implicit way. For example, the generalized valence bond (GVB) description of a 2-electron A-B bond is formulated as a formally covalent coupling between two delocalized orbitals. However, this GVB wave function implicitly contains the ionic forms A+ :B– and A:– B+, which appear in the explicit-VB wave function with HAO-centered structures.80 Similarly, a 3e-bond [A∴B]–, described by an MO configuration σ2σ*1, where both σ and σ* orbitals are delocalized, implicitly contains the two Lewis structures A:– •B and A• :B– (as shown at the outset of the paper). Thus, explicitly involving these two Lewis structures and at the same time letting the orbitals be delocalized in a BOVB calculation would be inconsistent. For example, in a 2e-bond, this would lead to double-counting of the VB structures, leading to artifacts, as has been demonstrated for F2.101 The advantage of the BOVB description over VB methods, in which the orbitals are delocalized, such as GVB, is that BOVB takes dynamic electron correlation into account without increasing the number of VB structures, thus leading to higher quantitative accuracy.

Scheme 4. BOVB orbital types for 3e-bonds: (a) Hybrid atomic orbitals (HAOs) and BOVB or D-BOVB structures for σ- and π-types single-3e-bonds, (A∴A). The subscripts l and r signify the location of the electron pair, on the left or right hand sides, respectively. (b) HAOs and SDBOVB structures. Here the doubly occupied HAOs are split into inner and outer lobes. This


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splitting reduces the electron-electron repulsion. The additional coupling modes, III’ and IVr’, complete the resonance energy for the corresponding 3e-bond.

To illustrate the BOVB orbitals we depict some corresponding cartoons in Scheme 4. Scheme 4a shows the BOVB left- and right- HAOs for σ- and π-types 3e-bonds (I, Ir and II, IIr). It is seen that due to the freedom of the orbitals to fit in size and shape to the local charges (or electron occupancy), the orbitals of the doubly occupied HAOs will breathe out and become “larger”. The inactive shell constitutes of delocalized (D) MOs, which account for the remaining electron pairs in the molecules, e.g., the π-lone pairs in F2–, or the σ-N-N and N-H bonds in hydrazine radical cation, 8 in scheme 2. These MOs are also allowed to breathe. Scheme 4b shows the SD-BOVB orbitals. Here, it is seen that the doubly occupied HAO, e.g. I (Scheme 4a), is being split into inner and outer lobes. The resulting VB structure with the split HAO is depicted as III. This structure brings about radial electron correlation and as such it reduces the electron-electron repulsion.[102] The same applies to structure IVr. It is possible however to augment this structure-set by adding III’ and IVr’. These additional structures arise


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from the fact that we now have three singly occupied HAOs, and hence the three-electron system should be described by two coupling modes.103 One of these modes couples the electrons in the inner and outer lobes on the same atom as in III and IVr, and the other mode couples the electron in the outer lobe of one atom to the electron in the inner lobe of the second (during optimization the lobes automatically adapt themselves to get the best energy).

Scheme 5. The VB structure set for the treatment of the doubly-π-(3e)-bonded species, 9-13.

The treatment of doubly-π-(3e)-bonded species requires significantly more VB structures.85 Firstly, the double-π-(3e)-bond has four fundamental structures (shown already in Scheme 3 above). If however, we consider that the σ-bond will be appreciably affected by the two π-(3e)bonds which surround the σ-bond, then this bond should be added to the active sub-shell. All in all, for this system of 8-electrons (6π and 2σ) in 6-HAOs, there are 105 canonical VB structures.


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However, as was shown in the previous treatment of the O2 molecule,85 12 of these structures reproduce the dissociation energy curve and the respective De to within 99.9% of the whole set. These 12 VB structures are depicted in Scheme 5, and in order to emphasize that all these structures possess triplet spin states, we use the labels Ti to enumerate them. The fundamental structures are T1-T4 discussed above in Scheme 3. These structures represent the two π-(3e)-bonds by the two neutral and two ionic structures, which were used before, to discuss the RECS[double-π-(3e)-bonds] in O2.85,88 T5 and T6 pair up the π-triplet electrons (of T1 and T2) to singlet, and instead create triplet in the σ-bond; these structures are essential only for obtaining a correct dissociation limit. Structures T7-T12 describe the six ionic structures of the σ-bond in T1-T4. It is apparent that the double-π-(3e)-bond is different than the single π- and σ-(3e)-bonds, even in terms of the fundamental structures of the respective bonds (T1-T4 vs. I and Ir). 3.4. Laplacian Calculations. The Laplacians of the 3e bonds at their bond critical points (rc)[78] was determined using program AIMAll version 17.01.25104 and the aug-cc-pVTZ basis set.[95] In all cases we took only the electron density due to the 3-electron bond. Thus, for example, for 8 in Scheme 2 we considered only the density of the electrons in the π2π*1 subshell. In addition to the Laplacian of the full 3e-bond state, we calculated the Laplacians of the single VB structures. For example, for 8 we looked at the 3e-density of II (Scheme 4) by inputting the BOVB wave function into the AIM program. For the total BOVB wave function, we have two nonorthogonal structures, and as such, the resonance density (akin to overlap population) contributes also a corresponding Laplacian quantity, denoted henceforth as L(res) 85

4. Results


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4.1. Post Hartree-Fock Results. Figures 1 and 2 show the bond lengths (R) and bond energies (De) of the various species, 1-13, calculated at the MRCI/aug-cc-pVTZ level with Davidson correction (except for 8, which was calculated using UCCSD(T)/aug-cc-pVTZ). For species 14 and 15, we calculated only the Laplacian values, which will be discussed later. The following trends are apparent: (a)

The De values moderately decrease down a column of the periodic table (see 1-6 in Figure 1, and 9 vs. 10 in Figure 2), in line with the bond length variation. The high De value for 8-10 and 13 reflect dissociation of the 3e-bonds as well as of the underlying σbonds. For example, the rotational barrier (Figure 1) for 8, which breaks only the corresponding π-(3e)-bond is 41.4 kcal mol-1 which provides lower limit approximation for the De(π-(3e)) value of the corresponding bond.

(b)

It is seen that the π-(3e)-bond in 8 is stronger than the σ-(3e)-bonds for 1 and 7. Based on the subsequent discussion, this is most likely a combination of weaker 3e-Pauli repulsion and stronger RECS of the π-(3e)-bond.

(c)

The cationic 3e-bonds (4-6) have larger De values than the anionic ones (1-3).

(d)

The total De values for 9 and 10 displaying double-π-(3e)-bonds (Figure 2) exhibit similar trend and decrease down the column in the periodic table. In species 13, which includes the transition metal, the De value is somewhat smaller than in 12. But this does not provide us with a true measure of the corresponding double-π-(3e)-bonds, because 9, 10 and 13 involve also an underlying σ-bond.


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(e)

Species 11 and 12 are outstanding, in the sense that they are local minima, having barriers for dissociation, and their De values are highly negative. The basis of the later feature is clear – the repulsion between the two positive charges is very large and is released during the dissociation. What is then the raison d’être of the corresponding local minima?

Figure 1. MRCI/aug-cc-pVTZ calculated bond lengths (R in Å) and bond energies (De with Davidson correction, in kcal mol-1) of 3-electron bonded species 1-8 (8 was calculated using UCCSD(T)/aug-cc-pVTZ). The state symmetry (term symbol) is shown near each species. For 8, De was calculated relative to the molecule at inter atomic (fragment) distance of 20 Å, corrected by the triplet-singlet energy gap of NH2+ that accounts for the S→T demotion energy of NH2+ at dissociation (29.6 kcal mol-1 at the UCCSD(T) level).


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Figure 2. MRCI/aug-cc-pVTZ calculated bond lengths (R in Å) and bond energies (Davidson’s corrected De in kcal mol-1), and term symbols for species 9-13 which possess double-π-(3e)bonds. The value in the parentheses for O2 corresponds to MRCI/aug-cc-pV6Z and for FeO+ to MRCI/aug-cc-pV5Z basis sets. De values in italic font are experimental values.[105,106] Species 11 and 12 which feature negative De values, are local minima with barriers (∆E‡, in kcal mol-1) for dissociation.

4.2. Valence Bond Results. The trends and puzzles discussed above will be clearer when we discuss the VB results. Let us turn to inspect first the results.


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Figure 3 shows the De values calculated at the various BOVB levels. Comparison of these values to the De data obtained in Figures 1 and 2, at the MRCI level, shows reasonably close values and similar trends and puzzles.

Figure 3. BOVB/aug-cc-pVTZ calculated bond energies (De in kcal mol-1) at the geometry optimized values obtained by MRCI/aug-cc-pVTZ. For species 1-3 and 6 the values correspond respectively to BOVB; SD-BOVB. For 9-12, the De values are obtained at the BOVB level (with 12 VB structures). For 13, the De value is the Davidson corrected MRCI/aug-cc-pV5Z result (see Figure 2). For 8, De was calculated as the stretching energy of the molecule from equilibrium distance to 20 Å, corrected by the triplet-singlet energy gap of NH2+ that accounts for the S→T demotion energy of NH2+ at dissociation (taken as the UCCSD(T) datum, 29.6 kcal mol-1).


19


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It is seen that for the dihalide anions, Cl2– and Br2–, the D-BOVB level underestimates the De values by ~ 4 kcal mol-1, and the 4-structure SD-BOVB model (Scheme 4b) is required to achieve results closer to the MRCI method. For the two smaller Ng2+ species (Ng = He, Ne) DBOVB is sufficient and even BOVB yields very close results to MRCI (see SI). However, for Ar2+ even D-BOVB underestimates De by 3 kcal mol-1 (see Table S13 in SI). Using SD-BOVB again improves the De values and brings them close to MRCI. All the species which possess double-π-(3e)-bonds, 9-13, were computed at the BOVB level with 12 structures. For 13, we do not have a De value, since at long distances the Fe+ species did not converge to its ground state. The corresponding MRCI/aug-cc-pV5Z Davidson corrected value of De is 84.5 kcal mol-1. Nevertheless, we have calculated the FeO+ species at the equilibrium distance with BOVB (12 structures) and we therefore possess all the properties of the molecule.

5. Discussion Valence bond (VB) calculations and theory provide insight into the nature of the 3e-bond in the target species in Scheme 2. As such, the discussion will focus on a VB analysis of the bonding, with support coming from AIM theory, and specifically the Laplacian (∇2ρ(rc)) at the bond critical point (rc).[78] Henceforth we shall refer in the text to the Laplacian as L. 5.1. The Nature of the 3e-Bond by VB Theory. In order to probe the nature of these bonds, we show in Figure 4 the potential energy curves calculated at a common BOVB/aug-cc-pVTZ level, for the six typical cases, comprising the dihalides and di-Noble gases radical ions. For each case we show the full BOVB wave function (two structures, as in Scheme 4a), using a blue energy curve, and the reference single structure, using a red energy curve.


20

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(a)

(b)


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Figure 4. BOVB/aug-cc-pVTZ energy curves for six species. For each species, the 2-structure BOVB state is the blue curve, whereas the single-structure wave function is shown by the red curve: (a) the dihalide species, (X∴X)• –, 1-3; and (b) the di-Noble gas species, (Ng∴Ng)•+, 4-6.

It is seen that in all the cases, the single structure has a tiny long-range minimum due to charge-induced dipole interaction (1.1-1.4 kcal mol-1), but otherwise the curve is repulsive. As shown in Scheme 6a, the nature of the curve for the single-structure is due to the Pauli repulsion (EPauli) of the three electrons which occupy a common space in two overlapping HAOs.80,92 Mixing of the two single structures (Scheme 6b) brings about resonance energy stabilization. As mentioned in the Introduction, since this stabilization energy is due to resonance fluctuation of the three electrons, we refer to it as charge-shift resonance energy, RECS.


22

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Scheme 6. Key energy terms in 3-electron bonds (A∴A). (a) The Pauli repulsion, EPauli, due to the overlap of the two hybrid AOs (HAOs) in a single structure, e.g., A: •A. (b) The mixing of the two single structures, brings about charge-shift resonance energy, RECS, which stabilizes the 3-electron bond.

The bond energy (De) of the 3-electron bond is then given by the balance between RECS and the Pauli repulsion at the equilibrium distance, equation 7: De = |RECS| - ∆EPauli

(7)

It is apparent from Figure 4 that the RECS is significantly larger than the Pauli repulsion, leading therefore to De of the order of 30-60 kcal mol-1 for all the six species. We can further project that the same applies to the species where the 3e-bond(s) is (are) held by an underlying σ-bond, as in species 8-13 in Scheme 2. We may therefore conclude that the 3e-bond originates in the chargeshift resonance energy of the interacting VB structures. Table 1 summarizes the RECS, De and L (∇2ρ) data for the 3-electron bonds in species 18, 14 and 15, using a common level, BOVB-aug-cc-pVTZ.

Table 1: 3e-Bonded Species (along with their numbers in Scheme 2), the corresponding ChargeShift Resonance Energies (RECS in kcal mol-1), the Bond Energies (De in kcal mol-1), the Total


23


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Laplacian Values of the 3e-Bonds (∇2ρ(3e-bond) in ea0-5 units), the Laplacians of a Single VB Structure (∇2ρ(1-st.)), Laplacians of the resonance density (∇2ρ(res))a and the Electron Density at the Bond Critical Point (ρ(rc) in ea0-3 units). Speciesb

RECS

Dec

∇2ρ(rc)(3e-bond)d

∇2ρ(rc) (1-st)

∇2ρ(rc) (res)

ρ(rc)

F2• – (1)

43.8

30.4

+0.1791/(+0.1392)

+0.2660

-0.3529

0.0510

Cl2• – (2)

34.9

24.1

+0.0437/(+0.0282)

+0.0677

-0.0917

0.0340

Br2• – (3)

31.4

23.0

+0.0275/(+0.0172)

+0.0526

-0.0777

0.0287

He2•+ (4)

80.8

53.4

+0.5633/(+0.4994)

+0.8901

-1.2169

0.1753

Ne2•+ (5)

50.7

34.9

+0.2970/(+0.2505)

+0.4614

-0.6258

0.0709

Ar2•+ (6)

40.0

26.8

+0.0638/(+0.0472)

+0.0597

-0.0556

0.0427

(HO)2• – (7)

25.7

20.1

+0.0515/(+0.0498)

+0.0815

-0.1115

0.0232

N2H4•+ (8)

56.1

133.7

+0.0352/(-0.0452)

+0.1886

-0.3420

0.0619

∆Erot =41.4 ‡

(CH2)6S2•+(14)

-

-/(+0.0128)

-

-

0.0204

(CH2)9N2•+(15)

-

-/(+0.0229)

-

-

0.0266

a) Laplacians of the resonance density (∇2ρ(res)) were calculated as in Ref. 70. b) In parentheses are the numbers of the corresponding species in Scheme 2. c) De was calculated at the BOVB level as in Figure 3. d) Values in parenthesis are for the B3LYP density of the 3ebond. Out of parenthesizes is for the 2-structure BOVB density.

It is seen that for all species where both quantities could be determined, the RECS is larger than the corresponding De values, thus underscoring the above conclusion that RECS fully originates the 3e-bonding, while each structure by itself suffers from Pauli repulsion. A similar conclusion was reached by use of energy decomposition analysis (EDA)107 of the 3e-bond in (H2S∴SH2)+.[108] Thus, as was demonstrated by Bickelhaupt et al,108 the 3e-bond energy is a balance between the Pauli repulsion of the two α-spin electrons located on the two sulfur atoms,


24

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while the interaction of the β electron in the H2S: lone pair with the vacant β spin orbital on the H2S•+ radical-cation site takes care of the charge-shift resonance energy. This EDA-based description of 3e-bonding is similar to Linnett’s VB description.[109] Clearly, the origin of 3ebonding according to the EDA-analysis is seen to be equivalent to the VB-based formulation in equation 7, in terms of Pauli repulsion and RECS. The other data in Table 1 correspond to Laplacian (L) values (∇2ρ) at the bond critical points. The Laplacian is defined by equation 8, which shows that L is a sum of two terms, one G(rc) is a kinetic energy density at the critical point, while the other V(rc) is the corresponding potential energy density. (8) In eq. 8, V(rc) is negative, while G(rc) is positive. Whenever 2G(rc) < |V(rc)|, L will be negative, which is a typical situation for covalent bonds where the shared-electron density is stabilized by the excess potential energy. In contrast, L will be positive when 2G(rc) > |V(rc)|, such that the shared density is dominated by Pauli repulsion, which raises the kinetic energy of this electron density beyond the limit allowed by the virial theorem.71 With this background, we can turn to discuss the L values in Table 1. The total L(3e-bond) values determined by BOVB/(B3LYP), respectively, are rather close, and are generally positive, except for 8, which exhibits a small negative B3LYP value (-0.0452) vs. a small positive BOVB value (0.0352). We note that since the B3LYP values of L are based on a particular combination of MOs, these values are less reliable than the BOVB values that are obtained for two unique sets of localized Hybrid AOs, which are allowed to change from one structure to the other. As such, 3e-bonds are typified by positive L(3e-bond) values, which means that the shared density of the bond in the bonding region is dominated by the positive kinetic energy (G(rc) in eq. 8) beyond


25


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the limit permitted by the virial theorem. This positive L originates of course in the large Pauli repulsion, evidenced in Figure 4, of the reference VB structure (e.g., A: •A in Scheme 6), as can be found in the L(1-st.) values, which are much more positive than the full L(3e-bond) values. As we argued before,71,89 when the shared density suffers from Pauli repulsion, the virial ratio of the kinetic (T) to potential (V) energies is tipped off-balance, and a mechanism is required that lowers the kinetic energy at the bonding region and achieves equilibrium where the ratio T/V = ½ is restored in the molecule. The mechanism is the charge-shift resonance energy, which is a negative kinetic energy. Indeed, what keeps down the positive values of the full L(3e-bond) is the Laplacian of the resonance density, L(res),70 which is invariably negative. The charge-shift resonance energies in Table 1 change in an orderly fashion. Firstly, the RECS values correlate with the L(res) values (correlation coefficient, r2 = 0.854). Secondly, as seen from the series 1-3 and 4-6, RECS decreases down a column in the periodic table, while by comparing 1 and 7, RECS appears to increase as we move from left to right in a period. For electron-pair bonds, we showed that RECS correlates with the average electronegativity of the bond constituents.[71,89] The only electronegativity scale that includes Noble gases is L. C. Allen’s.[110] Indeed in Allen’s scale the Noble gases (except for He, which is usually omitted) have the highest electronegativity in each period. The reason is that Allen’s scale is based on the average state ionization energy (IE) of the atom, and Noble gases have the highest IE values in their respective periods. Indeed, a plot of RECS vs. IEA, in Figure 5 shows an approximate linear correlation between the two quantities for species 1-6, which includes both radical cations and radical anions. This correlation further demonstrates the similar origins of the RECS quantities in 2e- and 3e-bonds and the associated two respective families of charge-shift bonds.


26

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Figure 5. A plot of |RECS| of A∴A vs. the ionization energy, IEA, for: (a) A∴A species except for A = He, and (b) all A∴A species including A = He.

5.2 The Nature of the Double-π-(3e)-Bond by VB Theory. Table 2 collects the RECS, De, L(rc) and ρ(rc) data for species 9-13. Each species has three sub-entries; the first is the total RECS[π+σ] due to the π-(3e)-bonds combined with the covalent-ionic mixing in the σ-bond, as defined as in equation 9: RECS(π+σ) = E(T1) – E(T1-T12)

(9)

The second is the RECS(π-(3e)-bonds) given by equation 10, and the third is the RECS(cov-ion, σ), defined by equation 11. RECS(3e-π-bonds) = E(T1,T7,T8) – E(T1-T12)

(10)

RECS(cov-ion, σ) = E(T1-T4) – E(T1-T4, T7-T12)

(11)

Based on the respective data in Table 2, it is apparent that the total RECS, involving the resonance energies of the σ- and the two π-(3e)-bonds, is very large, and so are the RECS(π-(3e)bonds) values. Judging from the comparisons of 9 to 10, and 11 to 12, we can see that the RECS values decrease down a column in the periodic table, in line with the correlation with the


27


Page 28 of 50

ionization energy of the constituent atom, as we present above in Figure 5 for the single 3ebonds.

Table 2: Doubly-π-(3e)-bonded Species (along with their numbers in Scheme 2), the corresponding Charge-Shift Resonance Energies (RECS), the Total Bond Energies (De), the Total Laplacian Values of the 3e-Bonds (∇2ρ(3e-bonds)), the Laplacians of a Single VB Structure (∇2ρ(1-st)) and the Electron Density at the Bond Critical Point (ρ(rc)). Species

O2 (9)

RECS (kcal mol-1)

De (kcal mol-1)

∇2ρ(rc)

∇2ρ(rc)

(3e-bond)d

(1-st) 1.3612

0.3164

0.1396

0.1140

1.2524

0.2247

0.2736

0.0955

–i

0.1276

RECS(π+σ): 150.8a

107.6

1.0584g

RECS(3e-π): 111.2b

120.1e

1.0744h

75.3

0.0856g

ρ(rc)d

RECS(2e-σ): 39.6c S2 (10)

RECS(π+σ): 70.5a

0.0888h

RECS(3e-π): 57.1b RECS(2e-σ): 13.3c F2++ (11)

RECS(π+σ): 134.9a

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On the Nature of the Three-Electron Bond

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