Variations in the Nature of Triple Bonds: The N2, HCN, and HC2H

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Variations in the Nature of Triple Bonds: The N, HCN and HCH Series Lu T Xu, and Thom H. Dunning J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b03631 • Publication Date (Web): 14 Jun 2016 Downloaded from http://pubs.acs.org on June 18, 2016

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The Journal of Physical Chemistry

Variations in the Nature of Triple Bonds: The N2, HCN and HC2H Series Lu T. Xu† and Thom H. Dunning, Jr.*,†,‡ Department of Chemistry, University of Illinois at Urbana-Champaign, 600 S. Mathews Avenue, Urbana, Illinois 61801 USA ABSTRACT: The inertness of molecular nitrogen and the reactivity of acetylene suggest there are significant variations in the nature of triple bonds. To understand these differences, we performed generalized valence bond as well as more accurate electronic structure calculations on three molecules with putative triple bonds: N2, HCN and HC2H. The calculations predict that the triple bond in HC2H is quite different than the triple bond in N2, with HCN being an intermediate case but closer to N2 than HC2H. The triple bond in N2 is a traditional triple bond with the spins of the electrons in the bonding orbital pairs predominantly singlet coupled in the GVB wave function (92%). In HC2H, on the other hand, there is a substantial amount of residual CH(a4Σ–) fragment coupling in the triple bond at its equilibrium geometry with the contribution of the perfect pairing spin function dropping to 82% (77% in a full valence GVB calculation). This difference in the nature of the triple bond in N2 and HC2H may well be responsible for the differences in the reactivities of N2 and HC2H.

1. INTRODUCTION As noted by Bodner et al.,1 there are a variety of theoretical constructs upon which our understanding of chemistry is built and not all are self-consistent. For example, they noted that the reactivity of acetylene is usually attributed to the presence of a triple bond in the molecule. Yet, the presence of the triple bond in N2 is also used to explain the inertness of this molecule. This is the case in spite of the fact that the triple bond in HC2H is stronger than that in N2. This puzzle could be resolved if there was a fundamental difference in the nature of the bonding in N2 and HC2H. Molecular orbital theory, which regards both molecules as having three doubly occupied bond pairs, provides little insight into this difference. In a recent paper we characterized the nature of the bonding in the homonuclear diatomic pnictogen molecules, N2, P2 and As2, using generalized valence bond (GVB) theory.2 We found that, around its equilibrium geometry, N2 was well described by the perfect pairing spin function corresponding to one σ and two π bonds. However, the weight of the perfect pairing spin function decreases from N2 to P2 and As2, with the quasi-atomic and atomic spin couplings becoming relatively more important further down 1

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L. T. Xu and T. H. Dunning, Jr.

the column. In fact, in As2 the weight of the quasi-atomic spin coupling is essentially the same as that of the perfect pairing spin coupling. In the quasi-atomic spin function, the spins of the electrons in the σ bonding orbitals are singlet coupled (paired), but those of the electrons in the π orbitals on each atom are high spin coupled as they are in the atom, with the high spin fragments then being coupled into a singlet. Thus, the electronic structure of P2 and As2 is more complicated than implied by the traditional Lewis pair triple bond model, which may well contribute to the enhanced reactivity of P2 and As2 vis-à-vis N2. Recently, Danovich et al.3 reported GVB calculations on acetylene, HC2H, finding that the perfect pairing spin function accounts for only 77% of the wave function. This is in marked contrast to N2, where the perfect pairing spin function accounts for 92% of the wave function and is even less than found for the heavier homonuclear diatomic pnictogen molecules, namely, 88% (P2) and 87% (As2).2 Subsequently, Dunning et al.4 noted that HC2H was also an outlier in the H3C–CH3, H2C=CH2 and HC≡CH sequence of single, double and triple carbon-carbon bonds. They found that non-dynamical (dynamical) correlation contributed 9.7 (15.9) kcal/mol to the carbon-carbon bond energy in C2H6 and, as expected, a much larger 25.2 (30.2) kcal/mol in C2H4, but only 26.3 (21.0) kcal/mol in C2H2. The change from H3C–CH3 to H2C=CH2 is consistent with the increase in the number of electrons in the carbon-carbon bond as well as the decrease in the bond distance. The more modest increase in the contribution of non-dynamical correlation along with the significant decrease of dynamical correlation from H2C=CH2 to HC≡CH, on the other hand, is surprising. Both of the above findings suggest that the nature of the triple bond in acetylene, the prototype for triply bonded organic molecules, may be more complicated than previously thought. A better understanding of the nature of the triple bond in N2 and HC2H is clearly needed. To this end, we investigated the electronic structure of the triple bond in the series of molecules N2, HCN and HC2H using GVB theory, which accounts for non-dynamical correlation, as well as more accurate electronic structure calculations that included the effects of dynamical correlation. All three of these molecules are considered to have traditional triple bonds, a result that is easily rationalized by considering their formation from the quartet states of the fragments:

2


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Variations in the Nature of Triple Bonds: The N2, HCN and HC2H Series

N(4S) + N(4S) → N2(X1Σg+)

(1a)

CH(a4Σ–) + N(4S) → HCN(X1Σ+)

(1b)

CH(a4Σ–) + CH(a4Σ–) → HC2H(X1Σg+)

(1c)

This series of molecules not only allows us to characterize the nature of the triple bond in each of these species, but it also allows us to understand how the successive replacement of the N(4S) atom in the X2 molecule by the CH(a4Σ–) fragment affects the nature of the triple bond. In Section 2, we briefly describe the GVB wave functions for N2, HCN and HC2H, focusing primarily on the (6-electron, 6-orbital) GVB description of the triple bond. In Section 3, we present the results of GVB calculations and analyze the nature of the triple bond in these three molecules. In Section 3.1, we present and discuss the results of the “triple bond” (6-electron, 6-orbital) GVB calculations on N2, HCN and HC2H, and in Section 3.2 we present and discuss the “full valence” (10-electron, 10-orbital) GVB calculations on HC2H that include the electrons in the CH bonding orbitals. We conclude in Section 4.

2. THEORETICAL AND COMPUTATIONAL CONSIDERATIONS As a reference wave function, the generalized valence bond (GVB) wave function strikes an appealing balance between accuracy and interpretability. The GVB wave function is more accurate than the Hartree-Fock (HF) wave function, including most (if not all) of the non-dynamical atomic and molecular correlation effects. Further, since the GVB wave function is variationally optimized, it includes important mean-field effects, e.g., hybridization, polarization and delocalization of the atomic orbitals as well as changes in the spin coupling of the electrons in the active orbitals. The former means that the GVB wave function captures many multiconfiguration valence bond effects in a single orbital product, while the latter means the GVB wave function does not predetermine a specific type of spin coupling, and hence bonding scheme, as does the HF wave function. As a result, the GVB wave function describes molecules that are poorly described by HF wave functions, e.g., C2, where De(HF) = 18.3 kcal/mol and De(GVB) = 112.6 kcal/mol versus De(expt’l) = 145.2 kcal/mol.5 Finally, the GVB wave function can be 3



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interpreted in terms of concepts familiar to many chemists, e.g., bond pairs, lone pairs, polarization, delocalization, etc., although new concepts have also arisen, e.g., recoupled pair bonding6,7,8 and throughpair interactions.9,10 2.1. GVB Wave Functions for N2, HCN and HC2H. The 6-electron, 6-orbital GVB wave function for

the triple bond in the ground states of the three molecules of interest (N2, HCN, HC2H) is:

Ψ GVB = Aˆφv1φv1φv2φv2ϕ a1ϕ a2ϕ a3ϕ a4ϕ a5ϕ a6 αβαβ Θ60,0

(2)

In Eq. (2), (S = 0, M = 0) for the singlet state of these three species and, for simplicity, the doubly occupied core orbitals on carbon and nitrogen have been omitted. The doubly occupied valence orbitals,

(φv1 , φv2 ) , are: (i) the polarized 2s-like orbitals on the nitrogen atom in N2, (ii) the CH bond orbital and the polarized 2s-like lone pair orbital on the nitrogen atom in HCN, and (iii) the two CH bond orbitals in HC2H. In the nitrogen atoms and/or CH fragments, the six active orbitals, (ϕ a1 − ϕ a6 ) , correspond to the (2pz, 2px, 2py) orbitals on the nitrogen atom and/or the (2s+/2s–, 2px, 2py)-like orbitals on the carbon atom. The (2s–, 2s+) orbitals are the carbon 2s lobe orbitals; see Figure 2 in Ref. 7. There are five linearly independent spin functions for a six-electron singlet state: 5

Θ 60,0 = ∑ c0,0;k Θ 60,0;k

(3)

k=1

We use the Kotani spin functions in Eq. (3) with the five spin functions, k = 1–5, being symbolically represented by Θ0,0;1 = (αααβββ), Θ0,0;2 = (ααβαββ), Θ0,0;3 = C (αβααββ), Θ0,0;4 = (ααββαβ), and

Θ0,0;5 = (αβαβαβ). These spin functions define the paths taken in the branching diagram11 (for the functional forms for these spin functions, see the Supporting Information in Ref. 2). The Kotani spin functions are orthonormal, so the square of a spin coupling coefficient, wk = (c0,0;k )2 , represents the contribution of spin function “k” to the GVB wave function without the ambiguity that arises when nonorthogonal spin bases, such as the Rumer spin basis,11 are used. If the GVB orbitals are ordered as bond pairs, the weight of the perfect pairing (PP) spin function,

ΘPP = Θ0,0;5 , indicates the extent to which the molecule can be considered as having three well defined 4


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singlet-coupled pairs, i.e., a traditional triple bond. Thus, in N2, wPP = w5 = 0.919,2 indicating that this molecule is indeed well described as having a triple bond. In fact, for most molecules we find that wPP is dominate near the equilibrium geometry. However, this is not universally true as we showed in our recent study of C2, where wPP = 0.67 at Re.5 If wPP is not dominant, it may be possible to find a better representation of the GVB wave function by rearranging the order of the orbitals to obtain a spin function with a larger weight. Although the GVB orbitals and energy are invariant to the ordering of the orbitals, the spin coupling coefficients, {c0,0;k } , depend on the orbital order. In Refs. 2 and 5, we showed how this property of the GVB wave function could be used to gain insights into the nature of the bonding in these molecules. We will use this same analysis in the current paper where we will consider three different orbital orderings: •

Paired ordering. If the orbitals are ordered as sequential σ and π bond pairs, the perfect pairing spin function, Θ 60,0;5 = (αβαβαβ), describes a molecule with a traditional triple bond. We referred to this ordering as the “Molecular” ordering in Refs. 2 and 5.

•

Quasi-atomic ordering. If the orbitals are ordered so that the two σ bonding orbitals are first, the two π orbitals on the left atom are next, and the two π orbitals on the right atom are last, the

Θ 60,0;3 = (αβααββ) spin function describes a molecule with a traditional σ bond with the spins of the electrons in the π orbitals on the two atoms high spin coupled as they are in N(4S) and/or CH(a4Σ–). •

Atomic ordering. If the orbitals are ordered so that the orbitals on the left atom are ordered first and those on the right atom are ordered last, the Θ 60,0;1 = (αααβββ) spin function describes a molecule with the spins of all of the electrons on the two atoms coupled as they are in N(4S) and/or CH(a4Σ–).

In each of these cases, the weight of the spin function (wk, k = 5, 3, or 1) in the GVB wave function provides a measure of how well the molecule is described by the given bonding scheme. Further, if the energy of the restricted GVB wave function using only a specific orbital ordering and spin function (e.g., “Paired” and Θ60,0;5 ) is calculated, the increase in the energy relative to the full GVB energy provides an energetic measure of the error associated with the given bonding description. A restricted GVB wave 5



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function that includes only the Θ 60,0;1 spin function is equivalent to a spin-extended Hartree-Fock (SEHF) wave function.12 Although our focus is on the GVB description of the bonding between the two heavy atoms in N2, HCN and HC2H, we also report and analyze GVB calculations on HC2H that include the electrons in the CH bond orbitals in the GVB wave function – the full valence GVB wave function (10 electrons, 10 orbitals). In this case the doubly occupied orbitals in Eq. (2), ( φv1 , φv2 ), are each described by a pair of orbitals and the spin function αβαβ Θ 60,0 is replaced by Θ10 0,0 . There are forty-two (42) linearly independent spin functions for a ten-electron singlet state.11 Again, most of the spin function coefficients (weights) tend to be small but the dominant coefficients (weights) again provide insight into the nature of the bonding in the molecule. Although we use the GVB notation in this paper, the full GVB wave function is identical to the spincoupled valence bond (SCVB) wave function of Gerratt, Cooper, Raimondi, Karadakov, and coworkers. 13 , 14 The calculations reported herein were, in fact, made possible by the methodology developed by Cooper et al. 15-19 and implemented in the CASVB module in Molpro.20 2.2 Computational Details. The potential energy curves, equilibrium geometries, dissociation energies, and other properties of the ground states of N2, HCN and HC2H were computed at various levels of theory: HF, GVB, MRCI (CASSCF+1+2 based on a valence CAS wave function21-23 ) with the quadruples (+Q) correction, 24 and coupled cluster singles and doubles with perturbative triples [CCSD(T)].25 An augmented correlation consistent basis set of quadruple-zeta quality (aug-cc-pVQZ) was used for the hydrogen, carbon and nitrogen atoms.26 All of the calculations in this work were performed with the Molpro suite of quantum chemical programs.20

3. RESULTS AND DISCUSSION Over the past decade we have established a protocol that enables us to discover trends and anomalies in the physical and chemical properties of molecules even in the absence of a complete set of experimental data. This protocol builds on the remarkable advances in electronic structure theory over the past twenty-five years, which enables us to use predictions from high-level electronic structure methods, 6


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e.g., MRCI and coupled cluster methods with high quality basis sets, 27,28 as proxies for the experimental data, all the while comparing the calculated results with the known experimental data to double-check the validity of the protocol. We then use generalized valence bond (GVB) calculations to gain insights into the trends and anomalies in the data from the high level calculations. Use of this protocol has been critical in investigating the physical and chemical properties of molecular species containing second row (Na-Ar) elements where solid experimental data for many of the species was missing. Although it is less important in the current case—N2, HCN and HC2H are, by comparison, well characterized—we will nonetheless continue to follow this protocol to ensure consistency with our previous studies. 3.1. Triple Bond (Six Electrons, Six Orbitals) GVB Calculations. The results of HF, GVB, MRCI+Q, and CCSD(T) calculations on N2, HCN, and HC2H are summarized in Table 1. To place the calculated dissociation energies (De) on these three molecules on the same footing, we calculated De relative to the asymptotes in Eq. (1). As expected, the MRCI+Q and CCSD(T) calculations yield equilibrium geometries and bond energies in good agreement with the experimental data: errors in Re of 0.001–0.003 Å and errors in De of 4–5 kcal/mol. The source of these errors is well known: incompleteness of the basis set, neglect of core-valence correlation effects, etc. However, this level of accuracy is sufficient to establish trends in the triple bonds of these molecules. For the bond lengths between the heavy atoms, CCSD(T) calculations predict an essentially linear increase as the nitrogen atom is replaced by the CH radical: ΔRe = 0.056 Å (N2→HCN) and 0.050 Å (HCN→HC2H). The variation in the bond strengths are more pronounced: ΔDe = 17.9 kcal/mol (N2→HCN) and 25.2 kcal/mol (HCN→HC2H). The GVB wave function recovers almost the same fraction of the bond energy in N2 and HCN, 76% and 73%, respectively, but in HC2H the trend is reversed and the GVB wave function recovers 84% of De. Like the variations found earlier in the C2H6–C2H4–C2H2 series,4 these differences suggest that the triple bond in HC2H differs from that in N2 and HCN, especially the former. The GVB orbital diagrams for N2, HCN and HC2H are given in Figure 1. GVB wave functions provide a much better description of multiply bonded molecules than HF wave functions, a direct result of the smaller overlap of the 2pπ-like orbitals on each of the atoms. As a result, we find that the errors in the 7



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GVB predictions of the equilibrium bond lengths (Re) and strengths (De) for the bond between the heavy atoms are much smaller than for the HF predictions. For example, the errors in the bond lengths are: 0.002 Å versus 0.032 Å (N2), -0.002 Å versus 0.030 Å (HCN), and -0.006 Å versus 0.024 Å (HC2H), while the errors in the bond strengths are: 57.0 kcal/mol versus 106.4 kcal/mol (N2), 64.3 kcal/mol versus 98.1 kcal/mol (HCN), and 44.9 kcal/mol versus 71.5 kcal/mol (HC2H). The errors in the GVB predictions are largely due to the neglect of dynamical correlation. The GVB wave function is characterized by three quantities: the orbitals, the orbital overlaps, and the spin coupling coefficients (or weights). We plot the GVB orbitals and list the orbital overlaps in Figure 2, while the spin function weights are given in Table 2. Consider the orbitals first. As can be seen, the three sets of orbitals look similar, although the orthogonality tails on the nitrogen σ bonding orbitals due to their interaction with the doubly occupied, polarized 2s-like orbital on that atom are much more pronounced than those associated with the interaction of the carbon σ bonding orbital with the doubly occupied CH bonding orbitals in HCN and HC2H. The overlaps between the σ and π bonding orbitals in N2 and HCN are quite similar with the overlaps in HC2H being only modestly larger. Other than these modest differences, there is little in the orbitals to indicate a significant difference in the nature of the triple bond in N2, HCN and HC2H. An examination of spin coupling weights is more revealing. In the “Paired” column in Table 2, we give the weights of the spin functions with the bonding pairs arranged sequentially. The weight of the perfect pairing spin function, w5, decreases by a modest amount from N2 to HCN (0.034), but by nearly twice that amount from HCN to HC2H (0.062). The weight of the PP spin function of HC2H reported here, 0.823, is slightly larger than that from the calculations of Danovich et al.,3 0.773, because the current calculations only include the electrons and orbitals involved in the triple bond (see Section 3.2 for the results of the full valence GVB calculations). Nonetheless, a value of 0.823 indicates that the bonding in HC2H is significantly different than that in the other two species, particularly in N2. The anomalous behavior of the bonding in HC2H is further reinforced by the fact that the weight of the Q60,0;5 = (αααβββ ) spin function with the three orbitals on each atom ordered as they are in the CH(a4Σ–) fragment (the “Atomic” ordering), 0.874, is larger than the weight of the PP spin function with the bond pairs ordered 8


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sequentially, 0.823, a reversal of the trend found in N2 and HCN. Finally, unlike P2 and As2,2 the weight of the dominant spin function, (αβααββ), with the “Quasi-atomic” orbital ordering and the σ bond pair is much less than the other two weights in HC2H. Clearly, HC2H retains more of the coupling associated with the two quartet fragments than does N2 or HCN. The anomalous behavior of the triple bond in HC2H is further reinforced by the errors in the restricted GVB energies; see Table 3. With the “Paired” orbital ordering, the error in the restricted GVB(k=5) energy is 5.81 kcal/mol, while the error in the restricted GVB(k=1) energy for the “Atomic” orbital ordering is slightly less, 5.70 kcal/mol. This is in distinct contrast to N2 and HCN where the error in the restricted GVB(k=5) energy with the “Paired” ordering is much smaller than the error in the restricted GVB(k=1) energy with the “Atomic” ordering, by 6.40 kcal/mol in N2 and 3.27 kcal/mol in HCN. In their paper on C2, Cooper et al.29 advocated the use of spin correlation matrices to analyze the GVB wave function for C2 as these matrices are independent of the spin bases used in the calculation. The spin correlation matrices for an orbital pair, ( ϕ ai , ϕ aj ), can be used to quantify the degree of singlet and triplet coupling associated with the spins of the electrons in that pair. We directly computed the fraction of singlet coupling for an orbital pair by ordering the ( ϕ ai , ϕ aj ) pair first in the GVB wave function and then summing the weights of the spin functions that have αβ in the first two positions (w3 + w5 for Θ60,0 ); the fraction of triplet coupling is then just one minus the fraction of singlet coupling. For the PP spin function, the fraction of singlet coupling is exactly 1.0 if both orbitals are in a singlet-coupled pair and the triplet coupling is exactly 1.0 if they are in different singlet coupled pairs. Thus, these fractions provide an orbital pair by orbital pair accounting of the deviation from the PP spin coupling. These numbers are given in Table 4 for all three molecules as percentages. For N2 the spins of the electrons in the (σA, σB) bond pair are singlet coupled in 94.9% of the wave function. This drops to 91.6% in HCN and to just 85.6% in HC2H. This drop in singlet coupling of the (σA, σB) bonding pair is a direct consequence of the decreasing importance of the PP spin function in the GVB wave function. The trend for the (πxA, πxB) pair is similar with the singlet character declining from N2 to HCN to HC2H. Interestingly enough, the rate of decrease is less for the (πxA, πxB) pair than for the (σA, σB) pair, so that in both HCN and HC2H the amount of singlet character in the (πxA, πxB) pair is 9



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slightly larger than that in the (σA, σB) pair, by 1% in HCN and by 3.9% in HC2H. In contrast to the (σA, σB) and (πxA, πxB) pairs, the (σA, πxA) and (πxA, πyA) pairs are predominately triplet coupled, with the percentage of triplet coupling increasing from N2 to HCN to HC2H, in line with the increasing importance of the “Atomic” spin coupling in this same sequence. What emerges from the GVB calculations on N2, HCN and HC2H is a description of the electronic structure of the HC2H molecule that is at odds with the traditional depiction of the bonding in this molecule, namely, a triple bond just like that in N2. In fact, as noted by Bodner et al.,1 chemists are schizophrenic about triple bonds: “Imagine what might happen if a junior or senior chemistry major tried to explain the reactivity of acetylene to a student taking organic chemistry. The odds are good that the reactivity of acetylene would be attributed to the C≡C triple bond. Suppose that later that day the same undergraduate chemistry major was tutoring a student in general chemistry who was struggling with the descriptive chemistry of nitrogen. When asked to explain why the N2 molecule is virtually inert at room temperature, it is almost a virtual certainty that the N≡N triple bond would be invoked. At the end of the day, our chemistry major would go back to his or her room feeling good about what he or she had done, without realizing that the same phenomenon—the existence of a triple bond—had been used to explain why one of a pair of isoelectronic molecules is fairly reactive and the other is chemically inert.” The above statements are made despite the fact that the triple bond in HC2H is stronger than that in N2: 273.2 versus 228.4 kcal/mol. This value for the HC≡CH bond strength is relative to the CH(a4Σ–) + CH(a4Σ–) limit, but the triple bond in HC2H is also stronger than the N2 bond even when the former is calculated relative to the CH(X2Π) + CH(X2Π) limit, namely, De = 235.4 kcal/mol. The difference in the reactivity of the triple bonds in N2 and HC2H suggests that these two triple bonds are inherently different. The GVB calculations reported here support this proposition. The triple bond in N2 is a traditional triple bond with the spins of the electrons in the bonding orbital pairs predominantly singlet coupled. In HC2H, on the other hand, there is a substantial amount of residual CH(a4Σ–) fragment coupling in the triple bond at its equilibrium geometry. Although characterizing the 10


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impact of this residual fragment coupling on the reactivity of acetylene requires further study, the difference in the coupling of the electron spins involved in the triple bond clearly shows that there is a significant difference in the nature of the triple bonds in N2 and HC2H. In fact, similar differences were observed between N2 and (P2, As2), and (P2, As2) are also far more reactive than N2.2 As has long been discussed in the chemical literature, multiple bonds may be viewed as (σ, π) bonds or as equivalent bent bonds (ω bonds). For example, Slater used bent bonds to explain the bonding in C2H4 and C2H2,30 while Mulliken used (π, π*) orbitals to explain the electronic spectra of C2H4 and C2H2.31-33 Although these two representations are equivalent in Hartree-Fock theory,34,35 this is not the case in GVB theory—the corresponding wave functions will have different energies. In 1986 Palke reported that the GVB(PP) wave function for C2H4 gave a lower energy with bent bonds than with (σ, π) bonds.36 Subsequently, Schultz and Messmer,37 Karadakov et al.,38 and Oligaro et al.39 showed that fully optimized GVB wave functions with bent bonds gave the lower energy for C2H2; Ogliaro et al. also found that the bent bond representation gave a lower energy in N2. Using the cc-pVTZ basis set,26 Oligaro et al. reported an energy difference of 6.2 kcal/mol for C2H2 and 5.9 kcal/mol for N2. Even more interesting, Karadakov et al. found that the weight of the PP spin function increased from 0.83 for (σ, π) bonds to 0.88 for bent bonds. Further work will be needed to determine how the triple bonds in N2, HCN and HC2H differ using the bent bond representation of the triple bonds. 3.2 Full Valence (Ten Electrons, Ten Orbitals) GVB Calculations. Because of the above findings on the nature of the triple bond in acetylene, we performed full valence (10-electron, 10-orbital) GVB calculations on HC2H that included the CH bonds. To facilitate comparison of the triple bond and full valence GVB calculations, the CH bond pair orbitals are ordered first with the triple bond orbitals following. The orbital identifiers for the five bond pairs are: σHA(1sHA), σCA–(2sCA–), σHB(1sHB), σCB+(2sCB+), σCA+(2sCA+), σCB–(2sCB–), πxA(2pxCA), πxB(2pxCB), πyA(2pyCA), and πyB(2pyCB) with the atomic designations in parentheses (see Figure 1). The results of the full valence GVB calculations are summarized in Tables 5–7. These results are, in general, consistent with those from the (six-electron, six-orbital) triple bond GVB calculations. There are, however, quantitative differences. In the full valence GVB calculation wPP = w42 is smaller than in the 11



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triple bond calculation, 0.773 versus 0.823, a result that is consistent with the value of wPP reported by Danovich et al.3 The PP weight in HC2H, although significantly larger than that in C2, 0.669,5 is also much smaller than what would be expected for a molecule that is well described by five sets of Lewis bond pairs. Since wPP is calculated in the Kotani spin basis, which is orthonormal, the PP weight is well defined and does not suffer from the ambiguities associated with the use of Rumer spin functions.3 The same trend is found for the “Atomic” orbital ordering and associated spin function (αβαβαααβββ): wPP = 0.794 (full valence) versus 0.874 (triple bond), with the decrease being slightly larger for this restricted orbital/spin coupling wave function than for the “Paired” orbital/PP spin coupling wave function. The overlaps listed in Table 5 for the full valence GVB calculation show a significant decrease in the overlap between the two CC σ bonding orbitals, Sσ

σ CB–

CA+

, relative to the triple bond calculation: 0.847

versus 0.906. This suggests a measurable change in the carbon σ bonding orbitals associated with the triple bond as a result of including the CH bonds in the GVB wave function. There is a large inter-pair overlap, which is energetically unfavorable because of the Pauli Principle, associated with the two carbon σ bonding orbitals on the same atom: | Sσ

CA–

σCA+

| = 0.277. The resulting interaction is likely one of the

reasons for the change in the CC σ bonding orbitals associated with the triple bond. The change in the π overlaps, S π π , on the other hand, are minor, 0.684 versus 0.688. It is, of course, not surprising that the π A B

bonding orbitals are little perturbed by the GVB description of the CH bonds. In Table 6 we report the energies and errors associated with various restricted GVB calculations. The calculated GVB(k=42) energy is 9.01 kcal/mol above that from the full valence GVB wave function. This is an increase of 3.20 kcal/mol from the triple bond calculation (5.81 kcal/mol). If we add to the PP spin function, the spin function that includes the other singlet coupling of the electrons in the π orbitals, (αβαβαβααββ), the error decreases slightly to 7.78 kcal/mol. If we now include all of the spin functions which have αβαβ in the first four positions, i.e., all of those spin functions that keep the spins of the electrons in the CH bond orbitals paired, the error drops to 3.34 kcal/mol, which is close to the difference in the error between the triple bond and full valence GVB calculations. This is one sign that the full valence and triple bond wave functions for HC2H provide similar descriptions of the triple bond. This finding is further reinforced by analyzing the singlet and triplet coupling of selected orbital pairs 12


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for the triple bond and full valence GVB wave functions, which are listed in Table 7. Note, e.g., the strong singlet coupling of the CH bond pair, 95.9%. Clearly the CH bonds in H2CH are well described as Lewis bond pairs. One other interesting feature is the amount of singlet coupling between the (σCA–, σCA+) lobe orbitals on each carbon atom, 22.0%. This likely contributes to the change in the Sσ

σ B–

A+

overlap

noted above. It is also interesting that the amount of singlet character for the (σCA–, πxA) orbital pair, where the carbon σ orbital is associated with the CH bond, is nearly twice that for the (σCA+, πxA) pair, where the carbon σ orbital is associated with the CC bond. This is consistent with the importance of the “Atomic” description of the triple bond in HC2H. Other than these changes, the differences between the triple bond and full valence GVB wave function are modest. In summary, although there are quantitative differences between the descriptions of the triple bond in HC2H provided by the triple bond and full valence GVB calculations, the overall description of acetylene remains the same—the electronic structure of HC2H is not well described by a set of three Lewis bond pairs between the two carbon atoms. Although the difference is not as dramatic as that found earlier for C2,5 it nonetheless indicates that a Lewis pair description of the triple bond in HC2H is inadequate. In contrast to the CC triple bond, the two CH bonds in HC2H are well described by Lewis bond pairs.

4. CONCLUSIONS GVB calculations on N2, HCN and HC2H indicate that the triple bond in HC2H differs from that in N2 with HCN being an intermediate case, although closer to N2. The difference is manifested in the difference in the coupling of the spins of the electrons in the triple bonds of these molecules. The triple bond in N2 is a traditional set of (σ, π) bonds with the spins of the electrons in the bonding orbital pairs predominantly singlet coupled. In HC2H, on the other hand, there is substantial residual CH(a4Σ–) fragment coupling in the triple bond at its equilibrium geometry with the weight of the perfect pairing structure dropping from 92% in N2 to just 82% in HC2H [triple bond (6-electron, 6-orbital) GVB calculations]. Full valence (10-electrons, 10-orbitals) GVB calculations on HC2H, although yielding somewhat different numerical results, lead to essentially the same conclusion although the weight of the perfect pairing structure drops to 77%. 13



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Chemists have long known that not all triple bonds are alike. The inertness of N2 is usually attributed to its strong triple bond. On the other hand, the reactivity of HC2H is also attributed to its triple bond, despite the fact that the triple bond in HC2H is stronger than that in N2. The finding that there is a significant amount of residual CH(a4Σ–) fragment coupling in the HC2H triple bond that is not present in N2 may well be a significant factor in the enhanced reactivity of HC2H. Similar differences were observed between N2 and (P2, As2), and (P2, As2) are also far more reactive than N2. Further studies will be needed to define the role played by the quartet fragment coupling on the reactivity of the triple bond of HC2H versus that of N2. Further studies will also be needed to characterize the differences in the bent bond representations of the triple bonds for these three molecules.

AUTHOR INFORMATION Corresponding Author *

Email: [email protected]; [email protected]; Telephone: 206-616-1439.

Present Address †

(L.T.X.) Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 104 S.

Wright Street, Urbana, Illinois, 61801, USA; NASA Ames Research Center, Moffett Field, California, 94035, USA. ‡

(T.H.D.Jr.) Northwest Institute for Advanced Computing (NIAC), Pacific Northwest National

Laboratory, c/o University of Washington, Sieg Hall, 3960 Benton Lane NE, Seattle, Washington 98195, USA; Department of Chemistry, University of Washington, Seattle, Washington 98195, USA.

Funding This work was supported by funding from the Distinguished Chair for Research Excellence in Chemistry at the University of Illinois at Urbana−Champaign.

Notes The authors declare no competing financial interests. 14


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ACKNOWLEDGEMENTS We thank the other members of our research group for many valuable discussions. This work was supported by funding from the Distinguished Chair for Research Excellence in Chemistry at the University of Illinois at Urbana-Champaign.

15



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Page 16 of 26


Table 1. Equilibrium bond distances (Re, in Å), dissociation energies of the bond between the heavy atoms (De, in kcal/mol), and total energies (Ee, in hartrees) for the ground states of N2, HCN and HC2H from HF, 6-electron GVB, MRCI+Q, and CCSD(T) calculations. GVB wave function: 6-electrons, 6-orbitals; basis set: aug-cc-pVQZ.

N 2a

HCN

HC2H

Re

D eb

Ee

Re(H–CN)

Re(HC–N)

D eb

Ee

Re(H–C2H)

Re(HC–CH)

D eb

Ee

HF

1.066

122.0

-108.994933

1.057

1.123

142.4

-92.916915

1.054

1.179

201.7

-76.855875

GVB

1.096

171.4

-109.073564

1.056

1.155

176.2

-92.988973

1.054

1.209

228.3

-76.918041

MRCI+Q

1.101

223.8

-109.406619

1.062

1.158

240.6

-92.301027

1.063

1.207

267.2

-77.210714

CCSD(T)

1.101

224.1

-109.407243

1.068

1.157

242.0

-93.303457

1.064

1.207

267.2

-77.210997

1.09768c

228.4d

1.065825e

1.153193e

246.4f

1.0605g

1.2033g

273.2f

Expt’l a

From Ref. 2.

b

De’s are relative to: N(4S) + N(4S), CH(a4Σ–) + N(4S), and CH(a4Σ–) + CH(a4Σ–), respectively.

c

Huber, K.P.; Herzberg, G. Constants of Diatomic Molecules; Van Nostrand Reinhold: New York, 1979.

d

Lofthus, A; Krupenie, P. H.; The Spectrum of Molecular Nitrogen. J. Phys. Chem. Ref. Data 1977, 6, 113-307.

e

Woods, R. C. Microwave Spectroscopy of Molecular Ions in the Laboratory and in Space. Phil. Trans. R. Soc. London A 1988, 324, 141-146.

f

From experimental data for atomization energies in Computational Chemistry Comparison and Benchmark Database, National Institute of Standards and Technology (© 2013 U.S. Department of Commerce).

g

Strey, G.; Mills, I. M. Anharmonic Force Fields of Acetylene. J. Mol. Spectros. 1976, 59, 103-115. 16


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Table 2. Spin function weights from (6-electron, 6-orbital) GVB calculations with three orbital orderings for N2, HCN and HC2H. Equilibrium geometry from MRCI/avqz calculations. Orbital Order:

Paired

Quasi-atomic

Atomic

N2 w1 (αααβββ)

0.0169

0.0169

0.7722

w2 (ααβαββ)

0.0085

0.0339

0.0298

w3 (αβααββ)

0.0300

0.8407

0.0895

w4 (ααββαβ)

0.0254

0.0000

0.0814

w5 (αβαβαβ)

0.9192

0.1085

0.0271

HCN w1 (αααβββ)

0.0279

0.0279

0.8185

w2 (ααβαββ)

0.0139

0.0557

0.0204

w3 (αβααββ)

0.0317

0.8166

0.0612

w4 (ααββαβ)

0.0418

0.0000

0.0749

w5 (αβαβαβ)

0.8847

0.0999

0.0250

HC2H w1 (αααβββ)

0.0479

0.0479

0.8741

w2 (ααβαββ)

0.0239

0.0958

0.0097

w3 (αβααββ)

0.0334

0.7692

0.0291

w4 (ααββαβ)

0.0718

0.0000

0.0654

w5 (αβαβαβ)

0.8229

0.0872

0.0218

17



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Page 18 of 26


Table 3. Energies and energy differences (relative to the full GVB calculations) for restricted (6-electron, 6-orbital) GVB calculations with the three orbital orderings and associated spin functions (k) for N2, HCN and HC2H. Equilibrium geometry from MRCI/avqz calculations. Orbital Order: Spin Function:

Paired

Quasi-atomic

Atomic

k=5

k=3

k=1

-109.066870

-109.058550

-109.056666

4.15

9.37

10.55

-92.980996

-92.973703

-92.975807

4.96

9.55

8.23

-76.908632

-76.902534

-76.908807

5.81

9.64

5.70

N2 EGVB(k) (hartrees) ΔEGVB(k) (kcal/mol)

HCN EGVB(k) (hartrees) ΔEGVB(k) (kcal/mol)

HC2H EGVB(k) (hartrees) ΔEGVB(k) (kcal/mol)

18


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Table 4. Percent of singlet and triplet character for selected orbital pairs in N2, HCN and HC2H. GVB wave function: (6-electrons, 6-orbitals). N2 Orbital Pair

HCN

%Singlet %Triplet

HC2H

%Singlet %Triplet

%Singlet %Triplet

(σA, σB)

94.9

5.1

91.6

8.4

85.6

14.4

(πxA, πxB)

94.4

5.6

92.6

7.4

89.5

10.5

(σA, πxA)

11.7

88.3

8.6

91.4

5.1

94.9

(πxA, πyA)

10.8

89.2

10.0

90.0

8.7

91.3

Table 5. Energies, orbital overlaps, and spin function weights, for full valence (10-electron, 10-orbital) GVB calculations on HC2H. Equilibrium geometry from MRCI/avqz calculations. Orbital Order:

Paired

EGVB (hartrees)

Atomic -76.952854

Intra-pair Overlaps Sσ

σ B–

0.847

Sπ

π

0.684

A+

A B

Sσ

A–

0.818

σ HA

Unique Inter-pair Overlaps Sσ

σ HA

-0.043

Sσ

σ HA

0.146

Sσ

σA+

-0.277

Sσ

σB+

0.036

Sσ

σHB

-0.012

Sσ

σB+

0.026

A+

B–

A–

HA

HA

A–

w1 w2

(αααααβββββ) (ααααβαββββ)

0.0005 0.0003 19


0.0005 0.0000


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


w3 w4 w5 w6 w7 w8 w9 w10 w11 w12 w13 w14 w15 w16 w17 w18 w19 w20 w21 w22 w23 w24 w25 w26 w27 w28 w29 w30 w31 w32 w33 w34 w35 w36 w37 w38 w39

(αααβααββββ) (ααβαααββββ) (αβααααββββ) (ααααββαβββ) (αααβαβαβββ) (ααβααβαβββ) (αβαααβαβββ) (αααββααβββ) (ααβαβααβββ) (αβααβααβββ) (ααββαααβββ) (αβαβαααβββ) (ααααβββαββ) (αααβαββαββ) (ααβααββαββ) (αβαααββαββ) (αααββαβαββ) (ααβαβαβαββ) (αβααβαβαββ) (ααββααβαββ) (αβαβααβαββ) (αααβββααββ) (ααβαββααββ) (αβααββααββ) (ααββαβααββ) (αβαβαβααββ) (ααααββββαβ) (αααβαβββαβ) (ααβααβββαβ) (αβαααβββαβ) (αααββαββαβ) (ααβαβαββαβ) (αβααβαββαβ) (ααββααββαβ) (αβαβααββαβ) (αααβββαβαβ) (ααβαββαβαβ)

0.0018 0.0004 0.0021 0.0000 0.0073 0.0024 0.0065 0.0021 0.0006 0.0057 0.0000 0.0374 0.0000 0.0037 0.0012 0.0032 0.0010 0.0003 0.0028 0.0000 0.0187 0.0001 0.0002 0.0003 0.0002 0.0348 0.0000 0.0110 0.0036 0.0097 0.0031 0.0009 0.0085 0.0000 0.0561 0.0001 0.0001 20


0.0087 0.0027 0.0080 0.0001 0.0145 0.0045 0.0133 0.0017 0.0013 0.0062 0.0001 0.7945 0.0002 0.0005 0.0003 0.0002 0.0011 0.0002 0.0027 0.0000 0.0121 0.0034 0.0005 0.0081 0.0001 0.0364 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000 0.0002 0.0000 0.0580 0.0001 0.0000

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w40 (αβααββαβαβ) w41 (ααββαβαβαβ) w42 (αβαβαβαβαβ)

0.0000 0.0001 0.7733

0.0001 0.0000 0.0193

Table 6. Energies and energy differences for selected restricted (10-electron, 10orbital) GVB(k) calculations on HC2H.

Equilibrium geometry from MRCI/avqz

calculations. Orbital Order: k EGVB(k)(hartrees) ΔEGVB(k)(kcal/mol)

Paired

Atomic

42

28, 42

14, 23, 28, 37, 42

14

-76.938488

-76.780799

-76.947535

-76.938662

9.01

7.78

3.34

8.91

Table 7. Percent of singlet and triplet character for selected orbital pairs in HC2H. GVB wave functions: triple bond (6-electrons, 6orbitals) and full valence (10-electrons, 10-orbitals). (6e, 6o) Orbital Pair

(10e, 10o)

%Singlet %Triplet

%Singlet %Triplet

100.0a

0.0

95.9

4.1

(σCA–, σCA+)

–

–

22.0

78.0

(σCA–, πxA)

–

–

12.9

87.1

(σCA+, σCB–)

85.6

14.4

88.1

11.9

(πxCA, πxCB)

89.5

10.5

86.7

13.3

(σCA+, πxCA)

5.1

94.9

6.9

93.1

(πxCA, πyCA)

8.7

91.3

7.8

92.2

(σHA, σCA–)

a

In the (6e, 6o) GVB calculations the CH bonds are described by doubly occupied molecular orbitals, which correspond to 100% singlet coupling.

21



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Figure 1. GVB orbital diagrams for the ground states of N2, HCN and HC2H. Pictured are the (2px, 2py, 2pz) orbitals of nitrogen, the (2s–, 2s+, 2px, 2py) orbitals of carbon, and the 1s orbital of hydrogen. Dots represent orbital occupancies. Electron/orbital pairs joined by solid lines are predominately singlet coupled; dashed lines in HC2H indicate a decrease in this coupling pattern.

22


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Figure 2. GVB bond orbitals for the triple bond in N2, HCN and HC2H. The overlaps between the (σ, π) bonding orbitals are: N2 (0.891, 0.686), HCN (0.890, 0.684), and HC2H (0.906, 0.688).

23



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Dunning, Jr., T. H., Jr.; Xu, L. T.; Takeshita, T. J. Fundamental Aspects of Recoupled Pair Bonds. I. Recoupled Pair Bonds in Carbon and Sulfur Monofluoride. J. Chem. Phys. 2015, 142, 034113-13.

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Dunning, T. H., Jr.; Takeshita, T. Y.; Xu, L. T. Fundamental Aspects of Recoupled Pair Bonds. II. Recoupled Pair Bond Dyads in Carbon and Sulfur Difluoride. J. Chem. Phys. 2015, 142, 034114-10.

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Lindquist, B. A.; Dunning, Jr., T. H. The Nature of the SO Bond of Chlorinated Sulfur-Oxygen Compounds. Theor. Chem. Acc. 2014, 133, 1443-13.

10. Takeshita, T. Y.; Dunning, T. H., Jr. Generalized Valence Bond Description of Cahlcogen-Nitrogen Compounds. I. NS, F(NS), and H(NS). J. Phys. Chem. A 2014, 119, 1446-1455. 11. Pauncz, R. The Construction of Spin Eigenfunctions. An Exercise Book; Kluwer Academic/Plenum Publishers: New York, U.S.A., 2000. 12. Mayer, I. The Spin-Projected Extended Hartree-Fock Method. Adv. Quantum Chem. 1980, 12, 190-262. 13. Gerratt, J.; Cooper, D. L.; Karadakov, P. B.; Raimondi, M. Modern Valence Bond Theory. Chem. Soc. Rev. 1997, 87-100. 14. Gerratt, J. Scientific Publications of J. Gerratt. Int. J. Quantum Chem. 1999, 74, 77-81. 15. Thorsteinsson, T.; Cooper, D. L.; Gerratt, J.; Karadakov, P. B.; Raimondi, M. Modern Valence Bond Representations of CASSCF Wavefunctions. Theor. Chem. Acta 1996, 93, 343-366. 16. Thorsteinsson, T.; Cooper, D. L. Exact Transformation of CI spaces, VB Representations of CASSCF Wavefunctions and the Optimization of VB Wavefunctions. Theor. Chim. Acta 1996, 94, 233-245. 17. Thorsteinsson, T.; Cooper, D. L.; Gerratt, J.; Raimondi, M. Symmetry Adaptation and the Utilization of Point Group Symmetry in Valence Bond Calculations, Including CASVB. Theor. Chim. Acta 1997, 95, 131-150. 18. Cooper, D. L.; Thorsteinsson, T.; Gerratt, J. Fully Variational Optimization of Modern VB Wave Functions Using the CASVB Strategy. Int. J. Quantum Chem. 1997, 65, 439-451.

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Variations in the Nature of Triple Bonds: The N2, HCN, and HC2H

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