Are One-Electron Bonds Any Different from Standard Two-Electron

Aug 8, 2017 - This interference energy analysis has been applied to a large variety of molecules, including diatomics and polyatomics, molecules with ...
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Are One-Electron Bonds Any Different from Standard Two-Electron Covalent Bonds? David Wilian Oliveira de Sousa and Marco Antonio Chaer Nascimento* Instituto de Química, Universidade Federal do Rio de Janeiro Cidade Universitária, CT Bloco A Sala 412, 21941-909 Rio de Janeiro-RJ, Brazil CONSPECTUS: The nature of the chemical bond is perhaps the central subject in theoretical chemistry. Our understanding of the behavior of molecules developed amazingly in the last century, mostly with the rise of quantum mechanics (QM) and QM-based theories such as valence bond theory and molecular orbital theory. Such theories are very successful in describing molecular properties, but they are not able to explain the origin of the chemical bond. This problem was first analyzed in the 1960s by Ruedenberg, who showed that covalent bonds are the direct result of quantum interference between one-electron states. The generality of this result and its quantification were made possible through the recent development of the generalized product function energy partitioning (GPF-EP) method by our group, which allows the partitioning of the electronic density and energy into their interference and quasi-classical (noninterference) contributions. Furthermore, with GPF wave functions these effects can be analyzed separately for each bond of a molecule. This interference energy analysis has been applied to a large variety of molecules, including diatomics and polyatomics, molecules with single, double, and triple bonds, molecules with different degrees of polarity, linear or branched molecules, cyclic or acyclic molecules, conjugated molecules, and aromatics, in order to verify the role played by quantum interference. In all cases the conclusion is exactly the same: for each bond in each of the molecules considered, the main contribution to its stability comes from the interference term. Two-center one-electron (2c1e) bonds are the simplest kind of chemical bonds. Yet they are often viewed as odd or unconventional cases of bonding. Are they any different from conventional (2c2e) bonds? If so, what differences can we expect in the nature of (2c1e) bonds relative to electron-pair bonds? In this Account, we extend the GPF-EP method to describe bonds involving N electrons in M orbitals (N < M) and show its application to (2c1e) bonds. As examples we chose the molecules H2+, H3C·CH3+, B2H4−, [Cu·BH3(PH3)3], and an alkali-metal cation dimer, and we evaluated the components of the electronic energy and density, which account for the formation of the bond, and compared the results with those for the respective analogous molecules exhibiting the “conventional” two-electron bond. In all cases, it was verified that interference is the dominant effect for the one-electron bonds. The GPF-EP results clearly indicate that molecules exhibiting (2c1e) bonds should not be considered as special systems, since one- and two-electron bonds result from quantum interference and therefore there is no conceptual difference between them. Moreover, these results show that quantum interference provides a way to unify the chemical bond concept. bonds) as special cases of this model.3 From the MO side, Mulliken4 put the focus on the molecular structure, suggesting that each molecule should be regarded as a distinct entity and dropping entirely the idea that they should be viewed as atoms held together by valence forces. The modern versions of these models are known by the acronyms GVB (generalized VB)5 or SCVB (spin-coupled VB)6 and HF-MO (Hartree−Fock MO). Both models are nowadays routinely used to study molecular and electronic structure as well as molecular properties. Nevertheless, none of these methods provides an explanation for the nature of the chemical bond. Since atoms and molecules are quantum entities (i.e., their existence cannot be classically predicted), there must a quantum effect responsible for the

1. INTRODUCTION The concept of the chemical bond is central to chemistry and several other related areas of science. In the last years we have experienced a revival of interest in this subject, culminating with the centennial celebration of Lewis’ famous paper in JACS in 19161 in which he proposed that chemical bonds would result from atoms sharing pairs of “valence” electrons. However, he admitted the existence of certain compounds that do not fit into his model, as they have an odd number of electrons. With the rise of quantum mechanics, the structure of molecules could be more profoundly investigated through two distinct independent particle models (IPMs), namely, the Heitler− London valence bond (VB) model2 and the Hund−Mulliken molecular orbital (MO) model. Pauling, who first translated Lewis’ ideas to the quantum-mechanical framework through the VB method, treated odd-electron bonds (one- and three-electron © 2017 American Chemical Society

Received: May 24, 2017 Published: August 8, 2017 2264

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Accounts of Chemical Research formation of a chemical bond, but what is that effect? The first to address this problem was Ruedenberg, in his seminal paper where he proposed that quantum interference between oneelectron states is the effect responsible for the formation of a chemical bond.7 However, this long paper is not easy to grasp even for specialists in the theme. Moreover, in Ruedenberg’s original method for the analysis of a chemical bond, the choice of the atomic orbital basis for calculating the interference was somewhat arbitrary. Consequently, this beautiful paper and the ideas there exposed have never reached the general audience, although some other authors have independently8,9 reached the same conclusion. In 2009, our research group developed the generalized product f unction energy partitioning (GPF-EP) method,10 which provides a way to calculate interference contributions of individual chemical bonds, or groups of bonds, to the total energy of a molecule using atomic orbitals that are uniquely defined within a given basis set, thus avoiding the arbitrariness11 involved in the choice of atomic orbitals in Ruedenberg’s original method.7 This kind of calculation is known currently as interference energy analysis (IEA). IEA was then applied to a variety of molecules,12−22 including diatomics and polyatomics, molecules with single, double, and triple bonds, molecules with different degrees of polarity, linear or branched molecules, cyclic or acyclic molecules, conjugated molecules, and aromatics, in order to verify the role played by quantum interference and the consistency of our analysis. Chemical bonds in homologous series were also investigated to show that bond interference contributions are transferable among equivalent bonds of different molecules. Nonetheless, there are important examples of stable molecules presenting one-electron bonds, and there remains the question of whether quantum interference will be the dominant effect for these systems as well.

(ϕ1 + ϕ2), and the probability of finding the electron at a certain position (r) will be [ϕ1(r) + ϕ2(r)]2. The total electronic density, ρTOTAL, which is given by eq 1 except for a normalization factor, is different from that expected classically (the quasi-classical density, ρQC), which is formed by simply adding the two amplitudes (orbitals) squared separately. There is an additional term, ρINT, which is associated with a pure quantum effect, namely, interference: ρ TOTAL = (ϕ1 + ϕ2)2 = (ϕ12 + ϕ22) + 2ϕ1ϕ2 = ρQC + ρ INT

(1)

The electronic energy can be obtained directly from the density, so it can also be partitioned into quasi-classical and interference components (where by “quasi-classical” we mean the terms of the energy expression that can be interpreted classically, i.e., all terms other than interference): E[total] = E[QC] + E[INT]

(2)

H2+,

In the quasi-classical energy E[QC] is repulsive, and therefore, the chemical bond is due to interference alone. In Figure 1, we plot the electronic density of H2+ and its

2. BACKGROUND: INTERFERENCE AND THE NATURE OF THE CHEMICAL BOND Interference is a characteristic feature of waves, and the most common manifestation of this phenomenon occurs in the wellknown double slit experiment, where the amplitudes of the waves passing through the slits interfere constructively or destructively, giving rise to an interference pattern that is totally different from of the sum of amplitudes of the waves passing through each slit separately. Electrons, because of their dual character, also exhibit the typical interference pattern of waves when submitted to similar experiments. This experiment reveals a fundamental law of the quantum world: whenever a given event (electrons or photons passing through slits, for example) can occur in several alternative and indistinguishable ways, the total amplitude for observing this event is the sum of the amplitudes for each one of them separately, and the probability of observing the event is proportional to the square of the total amplitude. An alternative view of the double slit experiment to describe the role played by interference in chemical bonding, based on the donor−acceptor concept, has been proposed by Weinhold.23 Something analogous occurs in the formation of a molecule.24 Let us take H2+ as an example. In both cases there are two indistinguishable events taking place: in the double slit experiment, a photon can pass through the first slit or the second slit; in H2+, the electron can be attracted by the left proton or the right proton, and to each of these events one can associate an orbital (amplitude) centered on the left (ϕ1) or right (ϕ2) proton, respectively. The total amplitude will be

Figure 1. Electronic density of H2+ along the bond axis and its quasiclassical and interference components.

components along the bond axis. The interference density, ρINT, takes density out of the region near the nuclei and concentrates it in the bond region, making the total density smoother in the bond region. These changes in the electronic density brought about by interference affect the energy E[INT] in the following way: the concentration of density in the region between the nuclei increases the potential energy (V[INT]), and the softening of the density decreases the kinetic energy (T[INT]). Therefore, the main reason for the formation of the chemical bond is a drop in the kinetic energy associated with interference. It should be noted that this does not contradict the virial theorem, which refers to the total potential and kinetic energies. One could imagine the formation of the molecule as a simple twostep process (Figure 2). Consider one hydrogen atom approaching a proton from an infinite distance and the changes in the electronic density promoted by just classical (electrostatic) effects. A contraction of the electronic density occurs,25 causing a large decrease of the potential energy (VQC) and consequently a rise in the kinetic energy (TQC). The total 2265

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antisymmetric product of smaller functions, strongly orthogonal to each other, containing groups of electrons. They are a generalization of the IPM wave functions, including the HFMO and GVB/SCVB ones. The splitting of the wave function into groups is convenient to treat specific subspaces of the molecule separately, for example, core and valence electrons or individual chemical bonds. A more detailed description of the GPF-EP method is available elsewhere.10,28 The GPF-EP program was recently merged with the last version (2.8) of the VB200029 software. Most molecules containing one-electron bonds are formed from atoms of groups 1, 13, and 14 of the periodic table. In group 1, besides the well-known H2+ molecule, there are homoand heteronuclear alkali-metal cation dimers. Concerning elements of groups 13 and 14, mostly boron and carbon, there are many evidences of (generally) charged species containing one-electron σ and π bonds.30−35 Besides those groups, there are few other examples,36 one of them recently reported, a organometallic complex containing a one-electron σ bond between copper and boron resulting in an electrically neutral compound.37 Several molecules were examined, and to illustrate the results, we have selected H2+; the alkali-metal dimers A2+ and their hetero-diatomics AB+; H3C·CH3+ and B2H4− as examples of one-electron σ and π bonds, respectively, between atoms of groups 13 and 14; and [Cu·BH3(PH3)3] as a model of the recently reported copper−boron one-electron bond.37 For these systems, the GPF functions comprise three groups as follows: (a) all of the core electrons in a single group, treated at the HF level; (b) the valence electrons not involved in the one-electron bond, treated at the GVB/PP (perfect-pairing) level; and (c) the one-electron bonds treated at the spincoupled (1,2) level, i.e., one electron and two orbitals. The results were obtained with the cc-pVTZ basis, unless otherwise specified. Potential energy curves (PECs) were constructed along the coordinate of the one-electron bond of each molecule, with intervals of 0.05 Å. The energy partitioning was performed along the curves, evaluating the components of the total energy (E[QC] and E[INT]) and of the total interference energy (T[I] and V[I] = Ven[I] + Vee[I]). In each case, the results are compared to the corresponding ones for the equivalent molecules exhibiting two-electron bonds. Any additional calculations were performed with the software Jaguar

Figure 2. Formation of the chemical bond in the H2+ molecule, showing the energy changes on each step.

balance, however, is just slightly positive (about 4 kcal/mol), resulting in a “quasi-classical molecule” that is not stable. Next, if one allows the rules of the quantum world to operate, interference will displace electronic density to the region between the nuclei, causing, as already seen, a large decrease in the kinetic energy (TINT). The potential energy also increases because of interference (VINT), but not enough to cancel the decrease in kinetic energy. The final total energy of the molecule (De(H2+) = 64 kcal/mol) can only be explained by quantum interference, and the main reason for its stability is the lowering of the interference kinetic energy. At this point one may ask whether this is a general result for systems containing one-electron bonds. In order to answer this question, we have recently extended the GPF-EP method to describe bonds involving N electrons in M orbitals (N < M), based on the developed SCVB(N, M) wave function,26 which is a generalization to the GVB/SCVB picture to describe active spaces with different numbers of electrons and orbitals. The GPF-EP method works with GPF wave functions, proposed by McWeeny,27 each of which consists of the

Figure 3. Electronic density contour maps and the quasi-classical and interference components (GVB/cc-pVTZ level): regions of density accumulation are shown in blue and regions of density depletion in red. 2266

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Figure 4. Energy partitioning plots for the H2 and H2+ molecules.

Table 1. Energy Partitioning Values (in kcal·mol−1) for H2 and H2+ at the GVB/cc-pVTZ Level

7.938 or GAMESS 2014.39 For polyatomic molecules, the potential energy curves were constructed as previously reported:15 at each point of the curve a geometry optimization was carried out at the GVB-PP/cc-pVTZ(-f) level, keeping the analyzed bond distance fixed. Next, a GPF-EP calculation was performed for each optimized geometry at the GVB-PP/ccpVTZ level. Electronic density contour plots were also constructed to verify the effect of interference on the bond formation.

E[total] E[QC] E[INT] T[I] Ven[I] Vee[I] V[I] T[QC] V[QC]

3. RESULTS We start the discussion with the results for H2+ (and H2) because they illustrate well the general behavior of the other systems. These molecules have recently been re-examined by Ruedenberg and co-workers.40 Figure 3 shows the density partitionings for both molecules. The density contours clearly show that the mechanism of formation of the chemical bonds in H2 and H2+ consists of concentration of electronic density in the internuclear region promoted by interference. It is the very same effect in both molecules, independent of the number of electrons involved in bonding. Some differences can be noticed in the plots of Figure 4: (a) for the H2 molecule there is a small but non-negligible quasi-classical contribution (about 16%) to the depth of the potential well. For the H2+ molecule, however, the electrostatic part is repulsive. This can be easily understood by comparing the components of E[QC] in the two molecules, as shown in Table 1. For H2, the quasi-classical potential energy is V[QC] = Vnn + Ven[QC] + Vee[QC], and the Ven[QC] contribution has four terms (because there are two nuclei and two

H2+

H2

−64.17 (−64.42a) 6.25 −70.42 −100.06 29.64 − 29.64 164.10 −157.85

−95.11 (−103.27b) −15.34 −79.77 −97.72 16.75 0.56 17.30 189.94 −205.27

a

Calculated with the analytic wave function (ref 41). bExperimental D0 (ref 42).

electrons). For H2+, V[QC] = Vnn + Ven[QC], but in this case Ven[QC] has only two terms. Hence, there are not enough attraction terms to overcome the nuclear repulsion in H2+, while in H2 there is still a small contribution of electrostatic attraction for the binding in spite of the Vee[QC] contribution. (b) For H2 the interference energy increases monotonically (in absolute value) as the nuclei approach each other, but in H2+ it reaches a maximum and then decreases. In order to understand this difference in behavior, one should not forget that in both cases (one-electron or two-electron bond) interference takes place between two one-electron states. However, in the limit of the united atoms there would be just one possible state for H2+, and the interference would approach zero, while for H2 there 2267

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Buckingham and Rowlands for the difference in stability of Li2 and Li2+. It is also in agreement with the explanation provided by Weinhold and Landis,44 who attributed the difference in the BDEs to the drastic change in the bonding orbital of Li2+ relative to that of Li2, the former exhibiting a much larger sp character (17.6% p) than the latter (2.7% p). Concerning the role of interference and the quasi-classical effects, the observed behavior is opposite to that in H2/H2+: for the neutral molecules E[QC] is repulsive, while for the cations it is a stabilizing factor, as shown in Table 3. Moreover, the similarity between the interference energies in these molecules reveals that, as in H2 and H2+, the difference in the BDEs in both cases is due to electrostatic factors (polarizability and core−valence electron repulsion), implying once more that the phenomenon responsible for the formation of the chemical bonds in the neutral and charged molecules is qualitatively and quantitatively the same. It is worth mentioning that for Li2+ the quasi-classical and interference contributions to the energy are practically the same. In his original work, Ruedenberg pointed out that E[QC] should normally be a small fraction of E[INT], although odd cases where this rule is broken are possible.7 Classical VB calculations suggest that it can happen in lithium-containing molecules.46 However, it is very simple to understand this result from the IEA analysis (Table 2). The absence of electron repulsion in the valence shell of Li2+ and the much lower core− valence repulsion help to increase the E[QC] contribution to the stability of the molecule. The H3C·CH3+ species exhibits a one-electron C−C σ bond analogous to the one in ethane, while H2B−̇ BH2− exhibits a one-electron π bond not present in the neutral molecule. Figure 5 shows the electronic density plots, which confirm the accumulation of charge in the bond region due to interference. Table 4 shows the results of the IEA analysis. Once again, the one-electron bonds in these molecules do not differ from the “conventional” electron-pair bond in their general features: interference is the dominant effect, driven by a decrease in the interference kinetic energy. From the results in Table 4 the most noteworthy difference between the one-electron (C−C)σ bonds in C2H6 and C2H6+ is the value of the interference energy. Both V[I] and T[I] in ethane are much higher than in the cation. This is probably due to the much longer C−C bond (1.973 Å) in C2H6+, which causes less effective orbital overlap. The quasi-classical energy does not contribute energetically to the formation of the bond in those molecules. The interference energy for the (B−B)π bond in B2H4− is larger than that for the (C−C)σ in C2H6+. This is certainly related to the fact that the (C−C) bond length in C2H4+ (1.973 Å) is considerably larger than the (B−B) bond length in B2H4− (1.683 Å). As a last example, we examine the case of a one-electron polar bond. Moret and co-workers recently reported the synthesis and characterization of tris[2(diisopropylphosphino)phenyl]boranecopper(0) (CuTPB), a compound containing a one-electron polar bond between the copper and boron atoms.37 This is an exceptional fact because of the difficulty in isolating compounds exhibiting one-electron polar bonds. This molecule has a total of 95 atoms and 349 electrons, which makes GVB calculations cumbersome. We thus used Cu(PH3)3BH3 as a model to study this bond. The geometry of the complex was kept eclipsed, as in CuTPB. In order to calculate the dissociation curve, a geometry scan was

will be always two one-electron states and the interference will reach its maximum value. The results in Table 1 show that at the respective equilibrium distances (1.06 Å for H2+ and 0.74 Å for H2), the interference energies of the two molecules are similar. Indeed, at the same internuclear distance, the values of E[INT] are practically the same for H2 and H2+, implying that the phenomenon responsible for the formation of the two chemical bonds is qualitatively and quantitatively the same. The fact the bond energy in H2+ is about half (62%, more precisely) that of H2 is just a coincidence. This difference comes from the contribution of the quasi-classical electrostatic factors to the total electronic energy of H2 and has nothing to do with any conceptual difference in the nature of the two chemical bonds. The bonding in the alkali-metal dimers and their heterodiatomics is particularly interesting because, unexpectedly, their cations have higher bond dissociation energies (BDEs) than the respective neutral molecules. This is in contradiction to the MO prediction, as the cations have one less bonding electron than the respective neutral molecules. Buckingham and Rowlands43 have proposed two possible reasons for the higher BDEs of the cations: first, because of the high polarizability of the alkali-metal atom, its valence orbital would be more efficiently distorted in the molecular ion, and second, the core− valence electron repulsion is higher in the neutral molecule. These hypotheses can be directly verified through the GPF-EP method. In order to check the effect of polarization, PECs were calculated for Li2 and Li2+ using a sequence of basis sets: 9s, 9s4p, and 9s4p1d. The 9s4p basis is equivalent to the uncontracted Dunning−Hay basis, and the result with the 9s4p1d basis reproduces almost exactly the one with cc-pVTZ. The core−valence interaction, Vee[QC](c-v), can be directly obtained from the GPF-EP analysis, as it represents the intergroup quasi-classical electron repulsion term. Table 2 shows the effect of the basis set on the energy components for Li2 and Li2+. The addition of polarization Table 2. Effect of the Basis Set on the Energy Components of Li2 and Li2+ at the Respective Equilibrium Geometries (Energies in kcal·mol−1) Li2+

Li2 E[QC] E[INT] T[I] Ven[I] Vee[I] V[I] Vee[QC](c-v)

9s

9s4p

9s4p1d

9s

9s4p

9s4p1d

5.5 −13.5 −19.5 2.1 2.8 4.9 340.8

4.4 −14.6 −22.2 1.2 5.4 6.6 389.7

4.2 −14.5 −22.1 1.0 5.6 6.6 390.3

1.9 −14.4 −17.8 −2.2 2.8 0.6 157.2

−13.8 −14.3 −10.6 −23.7 18.6 −5.1 233.3

−14.6 −14.4 −9.4 −26.9 20.6 −6.3 240.7

functions to the basis set does not produce any significant effect on the description of Li2. For Li2+, however, the effect is quite significant, indicating that the polarization promoted by the Li+ ion represents an important component of the bond energy. Nevertheless, E[I] for Li2+ is larger than that for Li2 even with the 9s basis, and therefore, polarization cannot be the only important factor in the energy balance. In fact, the core− valence repulsion is much higher in Li 2 than in Li 2 + independent of the basis set and should be the compensatory factor for the relative stability of the molecules. Thus, the GPFEP analysis confirms the simple explanation advanced by 2268

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Table 3. Comparison between the Quasi-Classical and Interference Energies (in kcal·mol−1) of Some Alkali Metal Dimers and Cation Dimers, Calculated at the GVB/cc-pVTZ Level; The Last Row Shows the Static Dipole Polarizabilities of the Respective Neutral Atoms M = Li

a

E[QC] E[INT]

4.2 −15.5

E[QC] E[INT] α(M)/a.u.a

−15.0 (51%) −14.3 (49%) 164

M = Na M2 Molecules 5.5 −13.4 M2+ Molecules −9.7 (43%) −12.7 (57%) 162.7

M=K

LiNa

4.6 −9.4

5.3 −14.7

−7.9 (43%) −10.5 (57%) 290.6

−7.3 (34%) −14.0 (66%)

From ref 45.

Figure 5. Contour maps of the electronic density and its quasi-classical and interference components for the C−C σ bonds in C2H6 and C2H6+ (GVB/cc-pVTZ level) and the π bond in B2H4− (GVB/aug-cc-pVTZ level).

Table 4. Energy Partitioning Values (in kcal·mol−1) for the One-Electron Bonds in C2H6, C2H6+ (GVB/cc-pVTZ Level), and B2H4− (GVB/aug-cc-pVTZ Level) E[QC] E[I]σ T[I]σ Ven[I]σ Vee[I]σ V[I]σ

H3C−CH3

H3C·CH3+

2.5 −92.1 −195.3 178.3 −75.1 103.2

0.3 −38.8 −99.8 81.6 −20.6 61.0

performed at the B3LYP/6-31G(2d,p) level followed by singlepoint calculations at the GVB-PP/6-31G(2d,p) level. Because of the relatively small size of the basis set, it is possible that the energy values are not as accurate as for the other molecules. However, for the N2 molecule a systematic study of the influence of the basis size (up to quintuple basis) on the IEA results12 showed that a double-ζ with polarization basis set is good enough to obtain accurate partitioning results. For this more complex molecule, even if the quantitative results may not be so accurate, the qualitative behavior observed for the components of the energy exhibits the same features as for the other one-electron bonds so far analyzed (Figure 6). Again, the only factor responsible for the formation of the bond is the

H2B−̇ BH2− E[QC] E[I]π T[I]π Ven[I]π Vee[I]π V[I]π

16.4 −66.5 −183.2 380.8 −264.1 116.7

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Figure 6. Energy partitioning plot for the Cu−B bond in Cu(BH3)(PH3)3.

Figure 7. Energy partitioning plots for N2, CO (μ = 0.120 D), and LiH (μ = 5.882 D), which exhibit distinct degrees of polarity.

by quasi-classical factors, namely, the difference in the polarizabilities of M and M+ and the difference in the core− valence repulsion. In summary, the formation of one-electron bonds follows the same mechanism as that of (2c2e) bonds since the same phenomenon, i.e., quantum interference, rules the stabilization of systems containing such bonds. Therefore, there is no distinction between one- and two-electron bonds from the conceptual point of view. As a final remark, we point out that quantum interference provides a way to unify the chemical bond concept. Chemical bonds are traditionally classified as pure covalent, polar, or ionic, depending on the difference in the electronegativities of the atoms involved. This criterion is very deceiving as it induces chemists (and scientists of related areas) to believe that there is something fundamentally different between these “types” of bonds. Besides, it does not even resist a simple analysis using diatomic molecules formed by first-row atoms.17 Regardless of the polarity of the bond, the dominant effect is always quantum interference,10,17 as exemplified in Figure 7 for N2, CO, and LiH, which are pure covalent, polar, and ionic bonds, respectively, according to the electronegativity difference criterion. From the perspective of quantum interference, this traditional classification loses its meaning because all of the bonds result from the same effect, and as it takes one orbital from each atom to build up interference, all bonds are covalent.

interference energy, particularly its kinetic component, as the quasi-classical energy is very repulsive.

4. CONCLUDING REMARKS The results of the IEA analysis for the (2c1e) bonds support the assertion that chemical bonds result from quantum interference. In all cases, including those not presented in this Account, we observe the same general trend: chemical bonding occurs because of the concentration of density in the internuclear region driven by quantum interference and the corresponding decrease in the kinetic energy of interference. In comparison with two-electron bonds formed from the same atoms, the IEA analysis shows that the main differences come from the quasi-classical components of the total electronic energy. Normally, one-electron bonds have lower BDEs than the corresponding two-electron bonds, with the exception of the alkali-metal dimers. The case of the alkalimetal dimers is different because of the high polarizability of the atoms involved in bonding. On this account, the form of the valence orbitals in M2+ molecules is slightly different from that in M2, which reflects on changes in the kinetic energy. Nevertheless, the difference in the BDEs of one- and twoelectron bonds has no relation to bond ordersone-electron bonds are not “half-bonds”. Interference is the dominant factor for the formation of the bond and depends basically on the overlapping orbitals. Thus, if the orbitals involved in bonding are similar in the two cases, the difference in the BDEs has to come from the quasi-classical part. Even in the case of the alkali-metal dimers, that difference in energy can be explained 2270

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(8) Woodrow Wilson, C.; Goddard, W. A. Exchange Kinetic Energy, Contragradience, and Chemical Binding. Chem. Phys. Lett. 1970, 5, 45−49. (9) Kutzelnigg, W. The Physical Mechanism of the Chemical Bond. Angew. Chem., Int. Ed. Engl. 1973, 12, 546−562. (10) Cardozo, T. M.; Nascimento, M. A. C. Energy Partitioning for Generalized Product Functions: The Interference Contribution to the Energy of Generalized Valence Bond and Spin Coupled Wave Functions. J. Chem. Phys. 2009, 130, 104102. (11) More recently, Ruedenberg and co-workers tried to circumvent the problem of the choice of the orbitals. See: West, A. C.; Schmidt, M. W.; Gordon, M. S.; Ruedenberg, K. A Comprehensive Analysis of Molecule-Intrinsic Quasi-Atomic, Bonding, and Correlating Orbitals. I. Hartree-Fock Wave Functions. J. Chem. Phys. 2013, 139, 234107. (12) Cardozo, T. M.; Nascimento, M. A. C. Chemical Bonding in the N2 Molecule and the Role of the Quantum Mechanical Interference Effect. J. Phys. Chem. A 2009, 113, 12541−12548. (13) Cardozo, T. M.; Nascimento Freitas, G.; Nascimento, M. A. C. Interference Effect and the Nature of the π-Bonding in 1,3-Butadiene. J. Phys. Chem. A 2010, 114, 8798−8805. (14) Fantuzzi, F.; Cardozo, T. M.; Nascimento, M. A. C. The Role of Quantum-Mechanical Interference and Quasi-Classical Effects in Conjugated Hydrocarbons. Phys. Chem. Chem. Phys. 2012, 14, 5479−5488. (15) Vieira, F. S.; Fantuzzi, F.; Cardozo, T. M.; Nascimento, M. A. C. Interference Energy in C-H and C-C Bonds of Saturated Hydrocarbons: Dependence on the Type of Chain and Relationship to Bond Dissociation Energy. J. Phys. Chem. A 2013, 117, 4025−4034. (16) Cardozo, T. M.; Fantuzzi, F.; Nascimento, M. A. C. The NonCovalent Nature of the Molecular Structure of the Benzene Molecule. Phys. Chem. Chem. Phys. 2014, 16, 11024. (17) Fantuzzi, F.; Nascimento, M. A. C. Description of Polar Chemical Bonds from the Quantum Mechanical Interference Perspective. J. Chem. Theory Comput. 2014, 10, 2322−2332. (18) Fantuzzi, F.; Cardozo, T. M.; Nascimento, M. A. C. Nature of the Chemical Bond and Origin of the Inverted Dipole Moment in Boron Fluoride: A Generalized Valence Bond Approach. J. Phys. Chem. A 2015, 119, 5335−5343. (19) de Sousa, D. W. O.; Nascimento, M. A. C. Is There a Quadruple Bond in C2? J. Chem. Theory Comput. 2016, 12, 2234−2241. (20) Fantuzzi, F.; Cardozo, T. M.; Nascimento, M. A. C. The Nature of the Singlet and Triplet States of Cyclobutadiene as Revealed by Quantum Interference. ChemPhysChem 2016, 17, 288−295. (21) Fantuzzi, F.; de Sousa, D. W. O.; Chaer Nascimento, M. A. Chemical Bonding in the Pentagonal-Pyramidal Benzene Dication and Analogous Isoelectronic Hexa-Coordinate Species. Comput. Theor. Chem. 2017, DOI: 10.1016/j.comptc.2017.03.020. (22) Fantuzzi, F.; Cardozo, T. M.; Nascimento, M. A. C. On the Metastability of Doubly Charged Homonuclear Diatomics. Phys. Chem. Chem. Phys. 2017, 19, 19352−19359. (23) Weinhold, F. Chemical Bonding as a Superposition Phenomenon. J. Chem. Educ. 1999, 76, 1141−1146. (24) Nascimento, M. A. C. The Nature of the Chemical Bond. J. Braz. Chem. Soc. 2008, 19, 245−256. (25) Kutzelnigg, W.; Schwarz, W. H. E. Formation of the chemical bond and orbital contraction. Phys. Rev. A: At., Mol., Opt. Phys. 1982, 26, 2361−2367. (26) Karadakov, P. B.; Cooper, D. L.; Duke, B. J.; Li, J. Spin-Coupled Theory for “N Electrons in M Orbitals” Active Spaces. J. Phys. Chem. A 2012, 116, 7238−7244. (27) McWeeny, R. The Density Matrix in Many-Electron Quantum Mechanics. I. Generalized Product Functions. Factorization and Physical Interpretation of the Density Matrices. Proc. R. Soc. London, Ser. A 1959, 253, 242−259. (28) Fantuzzi, F.; de Sousa, D. W. O.; Nascimento, M. A. C. The Nature of the Chemical Bond from a Quantum Mechanical Interference Perspective. Chemistry Select 2017, 2, 604−619. (29) Li, J.; McWeeny, R. VB2000: Pushing Valence Bond Theory to New Limits. Int. J. Quantum Chem. 2002, 89, 208−216.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +55-21-3938-7563. Fax: +55-21-3938-7265. ORCID

Marco Antonio Chaer Nascimento: 0000-0002-5655-0576 Author Contributions

D.W.O.S. designed the project under the supervision of M.A.C.N.; D.W.O.S. wrote the paper, and M.A.C.N. revised it; D.W.O.S. performed the calculations; M.A.C.N. and D.W.O.S. analyzed the results of all the calculations and established the basis for writing the paper. Notes

The authors declare no competing financial interest. Biographies David Wilian Oliveira de Sousa studied chemistry at the Federal University of Rio de Janeiro, where he received his B.Sc. in 2014. He obtained his M.Sc. in 2016 at the same university, working on the quantum interference perspective of the chemical bond in one-electron bonds and in the C2 molecule. In 2016, he started his Ph.D. studies, extending his previous work to other classes of chemical bonds. Marco Antonio Chaer Nascimento is a Professor of Physical Chemistry at the Federal University of Rio de Janeiro. He studied chemistry at the School of Chemistry of the University of Brazil and received his Ph.D. from California Institute of Technology in 1977. He is a Member of the Brazilian National Academy of Sciences and a recipient of the Golden Medal of the Brazilian Chemical Society and of the Order of Scientific Merit, granted by the President of Brazil. He was a senior visitor at the University of Cambridge, U.K., in 1984 and a visiting associate at the California Institute of Technology in 1996 and Université Henri Poincaré, Nancy I, France, in 2000. He is a member of the editorial board of Progress in Theoretical Chemistry and Physics and the scientific board of the World Association of Theoretical and Computational Chemists (WATOC).

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ACKNOWLEDGMENTS The authors acknowledge FAPERJ, CNPq, and CAPES for financial support.

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REFERENCES

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DOI: 10.1021/acs.accounts.7b00260 Acc. Chem. Res. 2017, 50, 2264−2272

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