Chemical bonding in representative astrophysically relevant neutral,cation,and anion HCnH chains

Ioan B?aldea

Theoretical Chemistry,Heidelberg University,Im Neuenheimer Feld 229,D-69120 Heidelberg,Germany

Keywords: astrophysics,interstellar medium(ISM),carbon chains,Wiberg and Mayer bond order indices

1. Introduction

With 46 members astronomically observed, linear carbon-based chains represent the most numerous class among the 204 molecular species reported in space.[1]They made the object of numerous experimental and theoretical investigations in the past.[2–35]For obvious topological reasons, the chains XCnY wherein the terminal atoms X and Y are monovalent and/or trivalent (e.g., HCnH, HCnN, and NCnN) possess the following property: if even parity members (n=2k)are“normal”closed shell molecules,then odd parity members(n=2k+1) are open shell diradical species andvice versa.Closed shell species are spin singlets, and single and triple bonds alternate in their polyynic-type carbon backbone. By contrast, open shell diradcals are spin triplets exhibiting an intermediate structure switching from polyacetylenic bonding between outermost carbon atoms to cumulenic-like bonding between midmost carbon atoms.

In closed shell chains bond lengths between neighboring carbon-carbon pairs substantially vary. Lengths’ difference amount to?0.15 ?A (cf. Table 4). The alternation of single and triple bonds is fully consistent with chemical intuition. It is the direct consequence of the tetravalent carbon atom in the ideal Lewis picture.Still,assigning bonds’multiplicity merely based on bond length values is problematic. The longest single C–C bond ever reported(1.806 ?A[36])is much longer than“typical” single C–C bonds (～1.43–1.54 ?A[37–39]). These are, in turn, substantially longer than the experimental valued(C2?C3)=1.3633 ?A in triacetylene(cf.Table 4). The latter is in fact closer to the double bond length in ethene(1.3305 ?A).Unless further microscopic details are known, reliable information on bond multiplicity cannot be derived merely from bond lengths. It is especially the nontrivial non-intuitive character of the structure of the open shell diradicals that makes the analysis of chemical bonding by merely inspecting the values of the bond lengths highly questionable.

Building on our recent work wherein bond order indices were introduced in studies on carbon chains of astrophysical interest,[34,35,40–44]we will present below a very detailed natural atomic orbital (NAO) and natural bond order (NBO)analysis,[45]with emphasis on HC6H and HC5H as representatives of the even-numbered and odd-numbered members of the HCnH family. The results for the natural atomic charges are particularly interesting.They provide valuable information on the charge redistribution upon electron removal(ionization)and electron attachment.

Importantly, our results clearly demonstrate that Wiberg valence and bond order indices[46]represent an adequate basis for the quantitative understanding of chemical bonding in carbon chains. By contrast,Mayer index values[47]are completely at odds with chemical intuition.

2. Theoretical methods

The results reported below were obtained from quantum chemical calculations using the GAUSSIAN 16 (Ref. [48])suite of programs. To ensure compatibility with our previous studies[35,41,42,49–53]single-point calculations for chemical bond and electronic properties were done at the CCSD(T)level of theory, wherein coupled-cluster expansions include single and double excitations along with perturbations due to triple excitations.[54]For these calculations, we used basis sets of triple-zeta quality augmented with diffuse functions (Dunning aug-cc-pVTZ[55–57]). Unless otherwise specified (see Tables S1, S2, and S13) the molecular geometries used for single point calculations were relaxed via the B3LYP three-parameter hybrid DFT/HF exchange correlation functional[58–61]and 6-311++G(3df,3pd)Pople’s largest basis sets.[62,63]

For reasons explained elsewhere,[34]we employed unrestricted DFT (UB3LYP) methods and restricted open shell coupled-cluster (ROCCSD(T)) methods to handle open shell species. The cis-trans anion splitting and the electron attachment energies(Table S21)were estimated as zero-temperature limit of differences of the pertaining enthalpies of formation computed by compound model chemistries — G4,[64,65]W1BD,[66]and using complete basis set methods (CBS-QB3 and CBS-APNO)[63,67–69]— because they are more reliable than the computationally inexpensive ?-DFT[49,70]values. For natural atomic orbital(NAO)and natural bond analysis (NBA),[45]we used the package NBO 6.0.[71]on top of GAUSSIAN 16 runs.

Figures 1 and 5 were generated with XCRSYDEN,[72]and figures 2 and 6 with GABEDIT.[73]

3. Results and discussion

3.1. Preliminary remarks

The numerous tables and figures presented below aim at providing the interested reader with a very detailed characterization of the electronic structure and chemical bonding of the specific molecular species considered. Comprehensively analyzing every data reported would make the paper disproportionately long. For this reason, in the discussion that follows we confine ourselves to emphasize the most relevant aspects which are “normally” not documented in existing literature studies.

Although not essential from the present perspective of gaining insight into the chemical bonding in carbonbased chains of astrophysical interest, to avoid misunderstandings, let us start with a technical remark. As previously demonstrated,[25,33,74–76]and also illustrated by our results presented in Table S13 geometry optimization for molecular sizes like those presently considered can be performed at the computationally demanding CCSD(T) level of theory with good basis sets. Nevertheless, most of the electronic and chemical bonding properties reported below were obtained at the CCSD(T)/aug-cc-pVTZ//B3LYP/6-311++G(3df,3pd) level of theory,i.e., CCSD(T)/aug-ccpVTZ single point calculations at geometries optimized via B3LYP/6-311++G(3df,3pd). This rationale is backed by calculations done by us[40–42]and others;[77]they revealed that bond metric and related rotational constants’ data obtained via computationally inexpensive DFT-based optimization better agree with experiment than more sophisticatedab initioapproaches.

In our study,special attention was paid to whether molecular vibrations(read Renner–Teller instability)lower the symmetric equilibrium geometry intuitively expected for HCnH chains. Because too loose geometry relaxation may mask this possibility,we carried out calculations imposing very tight optimization conditions and various exchange-correlation functionals. Results like those presented in Tables S1 and S2 rule out this possibility for the HC6H and HC5H neutral chains and their cations. Giving in the various tables all Wiberg bond indices and natural charges obtained from quantum chemical computations better emphasizes whether the molecular species in question possess symmetric equilibrium structures or not.Molecules exist whose equilibrium geometry is asymmetric notwithstanding their symmetric stoichiometric formula;4,4′-bipyridine is an example thereof(cf.Table 4 in Ref.[78]).

3.2. Wiberg indices versus Mayer indices

Except for the ideal cases wherein the electron charge transfer between atoms is complete (ideal ionic bond) or the neighboring atoms equally share an electron pair (ideal covalent bond), assigning numerical values to the bond multiplicity(=bond order), valence or charge of atoms forming a molecule from the wave function/density matrix obtained by quantum chemical calculations is a highly nontrivial task;the computed electron density is extended over the entire molecule rather than belonging to individual atoms.[79]

In our recent studies,[34,35,40–44]we demonstrated the utility of Wiberg’s bond order indices[46]in quantitatively analyzing the chemical bonding in carbon-based chains of astrophysical interest. They are preferable to the more rudimentary Coulson bond order indices[80]introduced in conjunction with the H¨uckel theory or Mulliken’s,[81]which do not properly describe the bond strength and formal bond multiplicity(“chemist’s bond order”,i.e., half of the difference between the number of electrons occupying bonding and antibonding orbitals).

To avoid confusion, a comment on the Wiberg indices used here and in our previous studies is in order. Historically,they were introduced within the semi-empirical framework of complete neglect of differential overlap (CNDO).[46]However, the values reported by us via GAUSSIAN+NBO combination are not obtained from the CNDO-based one-particle reduced density matrix(as initially done by Wiberg.[46])They are“Wiberg”indices only in the sense that they are computed using Wiberg’s expressions of these indices in terms of the one-particle reduced density matrix. The latter is computed from the ab initio CCSD-based wave function,it is not based on CNDO.

Wiberg indices are not the only valence and bond order indices employed in the literature to quantify chemical bonding in molecules. In our earlier studies[34,35,40–44]we did not motivate our preference for Wiberg indices. To justify this preference, we also show below values of the heavily advertisedab initioMayer bond order indices.[47]In Table 1 we compare Mayer and Wiberg bond order indicesNcomputed for acetylene H–C≡C–H. Atomic valenciesV(obtained by summing elements of the bond index matrix in the NAO basis)are also presented there. As visible in Table 1,the Wiberg values are completely satisfactory. The estimatedN-andVvalues(extremely closed to three and four,respectively)are in excellent agreement with the Lewis representation. The very small deviation(<0.06)from the ideal Lewis value(VC=4)is due to the weak polar character of the C–H bond tracing back to the different electronegativity of the H and C atoms (see numerical values below) also reflected in the natural atomic charges(qH?+0.22,qC=?qH??0.22).

It is especially the independence of the basis sets of the Wiberg values emerging Table 1 that makes the strongest contrast with the Mayer values. As seen there, the Mayer values computed with aug-cc-pVTZ basis sets are completely at odds with elementary chemistry. We chose aug-cc-pVTZ to illustrate the disastrous impact of employing basis sets augmented with diffuse functions on the Mayer values. Still, we showed[40]that employing augmented basis sets in studies on carbon chain anions of astrophysical interest is mandatory,e.g.,calculations without properly including diffuse functions fail to correctly predict both the structure and spin multiplicity of the C4N?anion.[82]Table 1 is just one example that Mayer valence and bond order indices are completely unacceptable for carbon chains. The Mayer bond order indices for the non-problematic triacetylene HC6H molecule (Table 2),for the pentadiynylidene HC5H diradical (Table 3) as well as the Mayer valencies included in other tables presented below convey the same message.

Table 1. The Wiberg and Mayer bond order N and valence V indices for the HC2H0 neutral singlet chain computed at the RCCSD(T)/BS//RB3LYP/6-311++G(3df,3pd) level of theory for the HC2H0 neutral singlet chain (H1–C1 ≡C2–H2). The basis sets (BS)employed are indicated below.

Parenthetically, even if they are not so disastrous, the MayerN- andV-values (2.79 and 3.74, respectively) computed with cc-pVTZ basis sets without diffuse functions inadequately describe the triple C≡C bond and the tetravalent carbon in the elementary textbook HCCH molecule.

Table 2. Wiberg and Mayer bond order indices N computed at the RCCSD(T)/BS//RB3LYP/6-311++G(3df,3pd) level of theory for the HC6H0 neutral chain (H1–C1–C3 ≡C4–C5 ≡C6–H2). The basis sets(BS)are indicated below.

Table 3. Wiberg and Mayer bond order indices N computed at the ROCCSD(T)/BS//UB3LYP/6-311++G(3df,3pd)level of theory for the HC5H0 neutral triplet chain(H1C1C2C3C4C5H2). The basis sets(BS)are indicated below.

3.3. Chemical bonding in HC6H chains

More to the main point,let us first consider the HC6H chains. The Cartesian coordinates for equilibrium geometries of the neutral and charged species are presented in Tables S4,S5,S6,S7. Important insight into their ground state electronic structure can be gained at the MO picture level. The pertaining electronic configurations read as follows:

The neutral HC6H0molecule is a typical closed-shell linear polyyne(cf.Fig.1)whose paired valence electrons in the completely filled highest occupied molecular orbital(HOMO)2π4u(cf. Eq. (1a)) determines a singlet ground state. As depicted in Fig. 2, the calculated HOMO spatial density of the neutral linear HC6H0chain is concentrated between atoms,or more precisely,on every second carbon–carbon bond starting from the molecular ends. This makes the HC6H0a quantum chemistry textbook example wherein carbon–carbon bonds alternate between almost perfectly tetravalent carbon atoms.

Fig.1. Geometries of HC6H chains investigated in the present paper. Like the HC6H0 neutral parent, the HC6H+ cation is linear and therefore not shown here.

Mathematically, this is expressed by the numerical values of the atomic valencies collected in Table 5 and Table S3 and the bond order indices included in Table 6. For all carbon atoms, the computed values for the Wiberg valence depicted in Table S3 and Table 5 are only very slightly different from the value of four in the idealized Lewis representation H–C≡C–C≡C–C≡C–H. Similar to the aforementioned HC2H,the small differences from the Lewis value (<0.07), which are comparable with that for the hydrogen atoms,reveal a very weak polar character of the bonds. These slight departures from the localized Lewis picture arise from the small values of the Rydberg natural bond orbitals and the small differences from the core electrons of the isolated atoms presented in Table 5 and Table S3.

Table 4. Results of B3LYP/6-311++G(3df,3pd)very tight geometry optimization for HC6H chains without imposing symmetry constraints. Bond lengths l between atoms XY(in unit ?A),angles α between atoms ∠XYZ(in unit degrees)and Wiberg bond order indices N .

Albeit HC6H+preserves both the linearD∞hconformation of the neutral parent(Fig.1)and the 2πucharacter of its HOMO (cf. Eq. (1b)), the bond lengths are not similarly affected by electron removal. The single bonds of the cation become shorter while the triple bonds become longer(Table 4 and Fig.4(a). Most affected is the central C3≡C4bond whose Wiberg bond order index decreases by almost 0.5(Fig.3(c));this is more than two times larger than in the case of C6H6+(Table S19 and Fig.S2(e).In accord with intuitive expectation regarding the Coulomb repulsion minimization, our calculations found that the C1and C6atoms, which are most distant of each other, acquire the largest positive charge (Fig. 3(e)).In the same vein, the Coulomb repulsion due to the additional positive charge on the C3and C4atoms correlates with the increase in the C3≡C4bond length. Likewise, the shortening of the C2–C3(or C4–C5) bond is compatible with the Coulomb attraction due to the extra charges of opposite sign on the C3and C4atoms(or C3and C4atoms). Nevertheless,our calculations reveal that variation of the bond lengths is not merely an electrostatic effect. The triple bonds C1≡C2and C5≡C6become longer although the atoms involved acquire extra charges of opposite sign which would imply an additional bond squeezing. Calculations also show that chemical intuition may be problematic even in a closed shell molecule like HC6H;inspection of Figs.3(e)and 3(g)reveals a decreasing in the valence state of all carbon atoms although the extra negative charge of C2and C5has opposite sign to the extra(positive)charge of the other C atoms.

Table 5. Natural atomic charges, numbers of core and Rydberg electrons,and Wiberg and Mayer valencies computed via RCCSD(T)/augcc-pvtz//RB3LYP/6-311++G(3df,3pd) for the HC6H singlet neutral chain.

Table 6. Natural atomic charges, numbers of core and Rydberg electrons,and Wiberg and Mayer valencies computed via ROCCSD(T)/aug-ccpvtz//UB3LYP/6-311++G(3df,3pd)for the HC6H+ cation.

It might be tempting to relate the opposite change of the Wiberg indices of the adjacent carbon-carbon bonds driven by ionization to the alternation of the single and triple bonds in HC6H. If this held true, one could expect a more democratic impact of electron removal in molecules with similar carboncarbon bonds. To demonstrate that this is not the case, let us refer again to C6H6. Notwithstanding the equivalent carbon–carbon bonds of the neutral molecule, ionization only shortens two opposite carbon–carbon bonds (C2C3and C5C6in Fig.S2(b)). Their bond order indices in C6H6+are larger than in C6H60(Fig. S2(d)). The other four carbon–carbon bonds are stretched and the corresponding bond order indices are reduced. That is, the process starting with equivalent (aromatic) carbon–carbon bonds in C6H60ends with nonequivalent carbon–carbon bonds in C6H6+. Two carbon–carbon bonds acquire partial double bond character and four carbon–carbon bonds acquire partial single bond character.

Table 7. Natural atomic charges, numbers of core and Rydberg electrons,and Wiberg and Mayer valencies computed via ROCCSD(T)/aug-ccpvtz//UB3LYP/6-311++G(3df,3pd)for the HC6H?cis anion.

Table 8. Natural atomic charges, numbers of core and Rydberg electrons,and Wiberg and Mayer valencies computed via ROCCSD(T)/aug-ccpvtz//UB3LYP/6-311++G(3df,3pd)for the HC6H?trans anion.

Switching to the HC6H?chain, we should first reiterate[43,44]that, contrary to what previously claimed,[83]the anion is not linear. Calculations[43,44]yielded two nonlinear conformers—more precisely, a cis and a trans isomer(cf.Eq.(1c)and Fig.5)—stable both against molecular vibrations (i.e., all computed vibrational frequencies are real) and against electron detachment(i.e.,positive electron attachment energy EA> 0). The cis-trans energy separation is smaller than the“chemical”accuracy(～1 kcal/mol)expected for the various compound model chemistries used in our calculations(cf. Table S21). Therefore, it is reasonable to assume that in fact they are quasi-isoenergetic and coexist.This should be the more so especially in extraterrestrial environments where dedicated paths of synthesis to generate a given (preferably, cis)conformer are unlikely. We said “preferably” because only the HC6H?cis isomer possesses a permanent dipole moment(cf. Table S21). This makes it a potential candidate for astronomical observation via rovibrational spectroscopy.[43]The HC6H?trans isomer does not have a permanent dipole(μ=0)and cannot be detected by radio astronomy. Inspection of Tables 4, 7, and 8 reveals that, apart from the different atom location with respect to the molecular axis, the cis and trans HC6H?isomers possess properties that do not notably differ from each other.They could be hardly distinguished from each other within the drawing accuracy in Figs. 3 and 4. For this reason, only results for the cis anions are depicted in those figures.

While agreeing with the intuitive expectation that electron addition makes the anion longer than the neutral parent,inspection of the bond metric data(Table 4,and Figs.3(a)and 4(a) reveals that electron addition does not stretch all chemical bonds. Interestingly and unexpectedly at the same, electron addition and electron removal have similar bond squeezing and bond stretching effects. That is, the same bonds that are,e.g., elongated upon electron removal are also elongated upon electron attachment. As depicted in Fig.4(a),the single C2–C3and C4–C5bonds are squeezed by virtually the same amount. Albeit more pronounced than for cation, the C–H and triple C1≡C2,C3≡C4,and C5≡C6bonds of the anion are longer than in the neutral. Counterintuitively,the quantitative changes in the Wiberg bond order indices do not follow the changes in the bond lengths. Notwithstanding the virtually identical squeezing of the single C2–C3and C4–C5bonds,the increase in anion’s bond order indices only amounts one third from that in cation. Moreover, although the stretching of the C–H and triple C1≡C2, C3≡C4, and C5≡C6bonds is more pronounced in anion than in cation,the reduction in the corresponding bond order indices in anions is substantially smaller than in cation(cf.Fig.4(c)).

As intuitively expected, the extra electron migrate towards the HC6H?chain ends (cf. Fig. 4(e)). This increases the fractional valence of the H atoms in the anion while leaving the valence of the C atoms unchanged from the ideal Lewis value of four(cf.Fig.4(g)).

The comparison between the HC6H?chain the C6H6?ring is also interesting. In the latter, the excess electron also migrates towards the outermost H atoms(cf.Fig.S2(g))reducing thereby the Coulomb repulsion. Still, while being stable against molecular vibrations(i.e.,all computed vibrational frequencies are real),C6H6?is not stable against electron detachment;i.e.,its electron attachment energy is negative(EA<0).This behavior can be rationalized in terms of electrostatic repulsion. In the longer HC6H?the Coulomb repulsion is overcompensated by stabilization due toπ-electron delocalization,a fact impossible in the C6H6?anion whose shorter diameter makes repulsion too strong.

3.4. Chemical bonding in HC5H chains

Let us now examine the HC5H chains, whose Cartesian coordinates at energy minimum are presented in Tables S8–S12. The relevant ground state electronic configurations read as follows:

In accord with earlier reports,[3,35]the present quantum chemical study confirmed the D∞hsymmetry of the HC5H0.Our calculations comprise very tight geometry optimization with the widely employed B3LYP,[59–61]PBE0,[84]and M06-2X[85]functionals(cf.Tables S1 and S2). The triplet character of the ground state ?X3Σ?gobtained from calculations confirms the physical intuition. According to Hund’s rule,the two electrons in the half-filled HOMO(2π2u,cf.Eq.(2a))should have parallel spin. Inspection of the HOMO depicted in Fig. 2 reveals that its highest density is concentrated on every second carbon atom starting from the chain ends and not between the carbon atoms, as the case of the HC6H0even member chain.“On atoms”and not“on bonds”;this is the reason why,in general,odd members HC2k+1H are less stable than even members HC2kH.[44]

The comparison between the various panels of Fig.3 reveals that the differences between the properties of the diradical open shell HC5H0triplet and those of the non-radical closed shell HC6H0singlet are substantial. The most salient qualitative difference is,of course,the absence of bond alternation in HC5H, but other differences are also notable. For instance, the fact that, unlike other C atoms, the central C3atom in the HC5H0neutral is positively charged(Fig.3(f).

Fig. 2. MO spatial distributions of the HC6H chains investigated in the present paper: neutral singlet,cation,cis anion,trans anion.

Fig.3. (a)–(b)Bond lengths,(c)–(d)Wiberg bond order indices,(e)–(f)natural atomic charges,and(g)–(h)Wiberg valencies of neutral and charged HC6H and HC5H chains investigated in the present paper. Because differences between cis and trans anion isomers would be indistinguishable within the drawing accuracy,only properties of the cis isomers are depicted here.

Fig. 4. Changes relative to the most stable neutral isomer (singlet HC6H0 and triplet HC5H0) of the properties depicted in Fig. 3: (a)–(b) bond lengths,(c)–(d)Wiberg bond order indices,(e)–(f)natural atomic charges,and(g)–(h)Wiberg valencies of neutral and charged HC6H and HC5H chains investigated in the present paper. Because differences between cis and trans anion isomers would be indistinguishable within the drawing accuracy,only properties of the cis isomers are depicted here.

Table 9. Natural atomic charges, numbers of core and Rydberg electrons,and Wiberg and Mayer valencies computed via ROCCSD(T)/aug-ccpvtz//UB3LYP/6-311++G(3df,3pd)for the HC5H triplet.

Table 10. Natural atomic charges, numbers of core and Rydberg electrons, and Wiberg and Mayer valencies computed via RCCSD(T)/aug-ccpvtz//RB3LYP/6-311++G(3df,3pd)for the HC5H singlet.

Table 11. Natural atomic charges, numbers of core and Rydberg electrons, and Wiberg and Mayer valencies computed via ROCCSD(T)/augccpvtz//UB3LYP/6-311++G(3df,3pd)for the cis HC5H?anion.

Table 12. Natural atomic charges, numbers of core and Rydberg electrons,and Wiberg and Mayer valencies computed via ROCCSD(T)/aug-ccpvtz//UB3LYP/6-311++G(3df,3pd)for the trans HC5H?anion.

Table 13. Natural atomic charges, numbers of core and Rydberg electrons,and Wiberg and Mayer valencies computed via ROCCSD(T)/aug-ccpvtz//UB3LYP/6-311++G(3df,3pd)for the HC5H+cation.

Fig.5. Geometries of HC5H chains investigated in the present paper. Like the HC5H0 neutral triplet parent, the HC5H+ cation is linear and therefore not shown here.

Fig. 6. MO spatial distributions of the HC5H chains investigated in the present paper: neutral triplet,neutral singlet,cation,cis anion,trans anion.

As visible in Fig. 3(h), the most substantial deviation from the Lewis valence value is exhibited by the C3atom,which is nominally almost trivalent in the HC5H0triplet. In fact,our NBO calculations for the triplet state found the lone pair residing on the central C3atom according to the idealized Lewis structure

was previously claimed.[16]Our NBO analysis does not substantiate this claim. In the same vein,we also examined a potential contribution to the HC5H triplet from asymmetric Lewis structures that can a priori come into question

This possibility was also ruled out by our NBO analysis.

By contrast, equations (5a) and (5b) appeared to contribute to the electronic configuration of the singlet HC5H chain computed at the triplet optimum geometry. However,that configuration, which is an admixture of Eqs. (3) and(5), renders the linear singlet chain unstable. It eventually evolves into the nonlinear conformer(?a1A′,Cssymmetry)depicted in Fig.5(b),which is stable against molecular vibrations(i.e., vibrational frequencies are all real). The bent chain end(∠H1C1C2?125?,cf. Table 14) appears to stabilize the antiparticle spins in the lone pair residing on the terminal C1atom. It is then understandable that this asymmetric lone pair significantly weakens the H1C1and the C1C3bonds. In the bent HC5H singlet, the corresponding bond lengths become significantly longer than in the linear HC5H triplet (cf.Fig. 4(b). The reduction with respect to the triplet of the C1C2bond order is considerable; it amounts to about 0.7(cf.Fig.4(d).

As the case of longer carbon-based chains,[34,35,44,76]a terminal C–H function confers the adjacent carbon–carbon bond(C1≡C2and C4≡C5in the HC5H0triplet)a triple bond character.In their turn,triple C≡C bonds enforce single bonds in their vicinity. This is visible in Fig. 3(c). C2–C3and C3–C4are basically single bonds. The values of the bond order indices of these C2–C3and C3–C4bonds are very similar to those of the C2–C3and C4–C5single bonds of the HC6H0polyynic chain (Fig. 3(c)). In this way, all carbon–carbon bonds are exhausted, and there is no room for double carbon–carbon bonds in HC5H0. A cumulenic character can only set in sufficiently deep inside sufficiently long HC2k+1H0triplet chains. The shortest HC2k+1H0triplet chain exhibiting some cumulenic character onset is therefore HC7H0(cf.Fig. 7(b)). We said “some cumulenic” because not even the longer HC9H0triplet chain exhibits a true cumulenic bonding(cf.Fig.7(c)).

Fig.7. Wiberg bond order indices of(a)HC5H0,(b)HC7H0,and(c)HC9H triplet chains illustrating that the carbon backbone can acquire a cumulenic character only in sufficiently long odd-numbered members,too long to be among the candidates to be searched for is space in the next future.

As expected on the basis of Eq. (2(a)), calculations confirmed that the cation HC5H+possesses a2Πuground state whose electronic configuration expressed by Eq.(2(b)). Electron removal does not have much impact on the geometry.Unlike the terminal C–H bonds, which become slightly longer,the carbon–carbon bonds are altogether slightly shorter in the HC5H+cation, which preserves the linear geometry of the neutral parent (Fig. 5(c)). Still, counterintuitively, in spite of the bond length changes with respect to the neutral smaller than those of HC6H(cf.Figs.4(a)and 4(b),the changes in the bond index orders are larger than for HC6H(cf.Figs.4(c)and 4(d)). Figure 4(f)depicts that,similar to HC6H+,the hole created by ionization is also delocalized over the HC5H+chain.Overall,changes in the atomic charges upon electron removal are larger in HC5H than in HC6H(cf.Figs.4(e)and 4(f)). Regarding the valence of the carbon atoms, nontrivially, ionization merely impact on the valence of the central C3atom which effectively behaves as trivalent in the neutral HC5H0triplet chain(Fig.3(f)).

The anions of the HC5H chain are interesting for several reasons. Prior to our recent work,[43,44]the existence of a cis HC5H?anion chain was also claimed.[16]In addition, we reported that a trans HC5H?anion chain also exists (cf. Fig. 5).[43,44]Like the cis isomer, the trans HC5H?chain is also stable both against molecular vibrations(all calculated vibrational frequencies are real) and against electron detachment.[43,44]Similar to the case of HC6H?, apart from the different position of the H atoms relative to the carbon backbone, the structural and bond metric data of the HC5H?cis and trans isomers are very close to each other(cf.Table 14).

As evident from the data for the cis-trans isomerization obtained by several composite models(Table S21),the cis and trans HC5H?chains are,like the cis and trans HC6H?chains discussed above, also almost isoenergetic. So, one can also expect that they coexist.

Table 14 and figure 3(b) reveal that the differences between the lengths of adjacent bonds in the HC5H?chain are significantly smaller than in the HC5H0triplet chain:d(C2C3)?d(C1C2)?0.03 ?Aversus ?0.07 ?A.Based on this similarity between adjacent anion’s bond lengths markedly contrasting with the neutral triplet,reference[16]claimed that HC5H?exhibits cumulenic character. Nevertheless, the inspection of Fig. 3(d) along with the underlying values from Table 14 conveys a different message. The differences in the Wiberg bond order indices of the anion’s adjacent carboncarbon bonds are substantial(N(C1C2)?2.25,N(C2C3)?1.57,cf. Table 14) and do not substantiate a homogeneous cumulenic picture, contrary to what the small differences between adjacent bond lengths may suggest.

The comparison between Figs. 4(a) and 4(b) unravels an interesting difference between the HC5H and HC6H chains. As already noted, the carbon–carbon bonds of HC6H elongated/compressed upon electron removal are also elongated/compressed upon electron attachment (Fig. 4(a)). This is no longer the case in HC5H. Removing an electron from HC5H0squeezes all carbon–carbon bonds. Adding an electron merely squeezes the midmost C2C3and C3C4bonds;the farthest C1C2and C4C5bonds get longer (Fig. 4(b)). And still: amazingly, electron removal and electron addition have a virtually perfect (anti)symmetric impact on the individual charges of the HC5H chain (Fig. 4(f)). That is, if ionization yields a variationδqlof the charge of atomXl(X=C,H),electron attachment gives to a variation?δqlof the same atom.

The inspection of Fig. 3(f) reveals what is perhaps the most striking difference between the HC5H?and HC6H?anion chains. Confirming straightforward intuition,we found in Subsection 3.3 that the spatial distribution of the excess electron in HC6H?is concentrated on the two terminal H atoms(Fig.3(e)). By contrast, figure 3(f)shows that the extra electron preferentially goes to the C1,C3,and C5atoms,a process that is furthermore accompanied by electron depletion on the C2and C4atoms.

Table 14. Results of B3LYP/6-311++G(3df,3pd)very tight geometry optimization for HC5H chains without imposing symmetry constraints. Bond lengths l between atoms XY (in unit ?A),angles α between atoms ∠XYZ(in unit degrees)and Wiberg bond order indices N .

4. Conclusion

We believe that this investigation on the chemical bonding in HCnH chains was rewarding for several reasons.

The present results reiterated and added further support to the fact that monitoring bond lengths alone does not suffice to adequately characterize chemical bonding in carbon chains.Changes in bond order indices upon electron removal or electron addition do not simply (not even monotonically) follow changes in bond lengths. This is an aspect was also emphasized recently in a a different context.[86]

Our NBO analysis does not substantiate general and undifferentiated claims often made previously in the literature that odd-numbered chains HC2k+1H are cumulenes. Figure 7(c)depicts that not even the HC9H chain(that is,a chain whose length is comparable with the longest chain HC9H ever observed astronomically[87]) possesses a genuine cumulenic character.

Overall,the present results for charge redistribution upon ionization and electron attachment clearly discredit simplistic views of ionization as electron removal from one atom (let it be an H atom or a C atom)or electron attachment as electron addition to one atom; the electron is removed from the neutral’s HOMO,which is delocalized,and the electron is added to the neutral’s LUMO,which is also delocalized(Figs.2 and 6). Our results unraveled a subtle interplay between electrostatic interaction andπ-delocalization in HCnH chains that definitely deserves further consideration. As of now, monitoring the natural atomic charges in anion chains turned out to be particularly useful:

(i) Inspection of the natural atomic charges unraveled that electron attachment to the HC6H0chain has an impact on charge redistribution that qualitatively differ from that on the HC5H0chain.

(ii) Based on naive intuition, one may expect that the excess electron attached to a neutral chain migrates towards the chain ends. Sometimes NAO calculations do not confirm this expectation;this happens in HC5H(Fig.4(f)). Sometimes NAO calculations support the intuitive expectation. C6H6belong to this category. This behavior is depicted by the changes in natural atomic charges (Fig. S2(g)); it is also understandable by inspecting the benzene’s LUMO shape (Fig. S1(c)).The changes in natural atomic charges calculated for HC6H also substantiate the aforementioned intuitive expectation;see Fig. 4(c). However, the LUMO shape of HC6H (Fig. 2) can hardly be taken as confirmation of the intuitive expectation in spite of the fact that,after all,HC6H is a“normal”(i.e.,nonradical)closed shell molecule.

(iii) Noteworthily,electron removal and electron addition have a virtually perfectly symmetric impact on the individual atomic charges of the HC5H chain: (Fig. 3(f)). This points towards an unexpected charge conjugation invariance. Invariance properties under particle-hole transformation were previously reported in other one-dimensional systems with strong electron correlations(e.g.,Refs.[88,89]and citations therein)but not in carbon-based chains. This is an important point to be addressed in detail in a separate publication.

With regards to anions,we still want to make the following remark. Basically, a HC6H?chain is a valence anion[90]created by putting an extra electron into a higher unoccupied valence(2πg(shù),cf.Eq.(1c))orbital of a molecule whose highest shell(2π4u,cf.Eq.(1a))is fully occupied. Such an orbital possesses an anti-bonding character, and in most cases the equilibrium geometry of the valence anions strongly departs from that of the neutral parents.[90]Therefore,although contradicting previous work[83]claiming that HC6H?chains preserve the linear shape of the neutral parent, our finding that stable HC6H?chains are nonlinear while linear HC6H?chains are unstable should not be too surprising. On the contrary, a HC5H?chain amounts to put an extra electron into a partially occupied valence orbital (2π2u,cf. Eq. (2a)). It would not be too surprising if this anion inherited the(linear)conformation of the neutral molecule. However, calculations showed that the contrary is true.

Finally, by and large the results presented in this paper unambiguously demonstrated that the appropriate framework to deal with chemical bonding in carbon chains is Wiberg’s;Mayer’s valence and bond order indices turned out to be totally inappropriate.

Acknowledgment

The author thanks Jochen Schirmer for valuable discussions. The author gratefully acknowledges financial support from the German Research Foundation (DFG Grant No. BA 1799/3-2) in the initial stage of this work and computational support by the state of Baden–W¨urttemberg through bwHPC and the German Research Foundation through Grant No. INST 40/575-1 FUGG (bwUniCluster 2.0, bwForCluster/MLS&WISO 2.0/HELIX,and JUSTUS 2.0 cluster).