Not Carbon s–p Hybridization, but Coordination Number Determines C−H and C−C Bond Length

Abstract A fundamental and ubiquitous phenomenon in chemistry is the contraction of both C−H and C−C bonds as the carbon atoms involved vary, in s–p hybridization, along sp3 to sp2 to sp. Our quantum chemical bonding analyses based on Kohn–Sham molecular orbital theory show that the generally accepted rationale behind this trend is incorrect. Inspection of the molecular orbitals and their corresponding orbital overlaps reveals that the above‐mentioned shortening in C−H and C−C bonds is not determined by an increasing amount of s‐character at the carbon atom in these bonds. Instead, we establish that this structural trend is caused by a diminishing steric (Pauli) repulsion between substituents around the pertinent carbon atom, as the coordination number decreases along sp3 to sp2 to sp.

Abstract: Af undamental and ubiquitous phenomenon in chemistry is the contraction of both CÀHa nd CÀCb onds as the carbon atoms involved vary,i ns -p hybridization, along sp 3 to sp 2 to sp. Our quantum chemical bonding analyses basedo nK ohn-Sham molecular orbital theory show that the generally accepted rationaleb ehind this trend is incorrect. Inspection of the molecular orbitals and their corresponding orbitalo verlapsr eveals that the above-mentioned shortening in CÀHa nd CÀCb onds is not determinedb ya ni ncreasinga mount of s-character at the carbon atom in these bonds. Instead, we establish that this structuralt rend is caused by ad iminishing steric (Pauli)r epulsion between substituents aroundt he pertinent carbon atom, as the coordinationn umber decreases along sp 3 to sp 2 to sp.
The geometrical properties of organic (and inorganic) molecules are, in general, explained using hybridization theory, which wasi ntroducedb yL inus Pauling in 1931. [1,2] Ac ase in point is the fundamental and ubiquitous phenomenoni n chemistry that CÀHa nd CÀCb onds contract as the carbon atoms involved vary,i ns -p hybridization, along sp 3 to sp 2 to sp. Archetypal examples are the CÀHa nd CÀCb onds in ethane, ethene, ethyne and propane, propene, propyne, respectively.H ybridization theory ascribes the shortening of CÀH and CÀCb ond lengths along sp 3 to sp 2 to sp to the increasing percentage of s-character of the hybrid orbital at the pertinent carbon, as s-orbitalsa re more contracted to the nucleust han p-orbitals, thus givingr ise to an optimal bond overlap at a shorter interatomic distance. [3,4] This model is generallya ccepted and appearsi nmost (physical) organic chemistryt extbooks. [5][6][7][8] Herein, we show that the above standard model is incorrect. Through detailed quantum chemicalb onding analyses of a series of representative, archetypal model systems ( Figure 1), we have been able to reveal that the above-mentioned shortening in CÀHa nd CÀCb onds is not relatedt oa ni ncreasing amount of s-character at the carbon atom in these bonds. Instead,w ef ind that ad iminishing steric (Pauli)r epulsion between substituents aroundt he carbon atom constitutes the physicalm echanism behind the universal trend in molecular structure, as the number of substituents around the pertinent carbona tom decreases from 4t o3t o2a long sp 3 to sp 2 to sp hybridization, respectively.O ur findings are based on the analysis of the CÀHa nd CÀCb onding mechanismsi nas ystematic series of model systems featuring sp 3 -, sp 2 -, and sp-hybridized CÀHa nd CÀCb onds in saturated and unsaturated hydrocarbons (Figure 1), using the quantitative molecular orbital (MO) modelc ontained in Kohn-Sham density functional theory (DFT) [9][10][11] at BP86/TZ2P [12][13][14] in combination with am atching canonical energy decomposition analysis( EDA) as implemented in the ADF program. [15,16] Our findings are both,n ovel to the extent that they are paradigm-changing anda lso suitably consistentw itht he well-known role of steric repulsioni no ther contextso fm olecular structure, such as,t he stereochemical arrangemento fs ubstituents around ac entrala tom or the dependence of bond distances on the steric bulk around the bond in question. [17][18][19][20][21][22][23][24] Not unexpectedly,o ur DFT computationsr eproduce the aforementioned trend of as horteningo ft he CÀHa nd CÀC bond lengths as we go along sp 3 to sp 2 to sp hybridization of the carbon atom involved in such bonds ( Table 1). The CÀH bond length decreases along ethane (R 3 CÀH, 1.099 ), ethene (R 2 CÀH, 1.091 ), and ethyne( RCÀH, 1.070 )w hile the corre-spondingC ÀCs ingle bond lengths decrease along propane (R 3 CÀCH 3 ,1 .533 ), propene (R 2 CÀCH 3 ,1 .500 ), and propyne (RCÀCH 3 ,1 .456 ). Note that in all cases, bond shortening correlatesw ith bond strengthening as reflected by the increasei n bond dissociation energy (BDE; DE = ÀDE BDE )a long sp 3 to sp 2 to sp hybridization.I no rder to analyze the origin of the trend in bond strengths in more detail, we decompose the bond energy DE according to the activation strainm odel (ASM) of reactivity [Eq. (1)]: [25][26][27][28] Here, the strain energy DE strain is the penalty that needs to be paid for deforming the fragments from their equilibrium structure to the geometry they adopt at the equilibrium CÀX( X= H, CH 3 )b ondl ength. On the other hand, the interaction energy DE int accounts for all mutuali nteractions that occur between the deformed fragments. In all cases, the magnitude and trend in CÀHa nd CÀCb ond dissociation energiesa ppear to be determinedb yt he interaction energies DE int .The strain energy DE strain has only little influence on the calculated bond energy DE and does not affect the overall trend in relative bond strengths. They originate from the fact that, upon the formation of an ew CÀHo rC ÀC bond, the other substituents around ac arbon atom involved in the new bond bend away in order to reduce the otherwise even more destabilizing steric (Pauli)r epulsion. This destabilizing effect is more pronouncedw hen more substituents are aroundt he carbon.T hus, DE strain is most destabilizing for ethane (R 3 C-H) and propane (R 3 C-CH 3 )i nw hich the intrinsically planar R 3 CC radical undergoes pyramidalization. [29] The geometrical deformations of the stericallyl ess crowded R 2 CC and RCC radicalf ragments in, for example, ethene( R 2 C-H) ande thyne (RC-H) are less severe and, therefore, lead to lower strain energies.
In order to pinpoint the differences between the interaction energies, we have analyzed the various CÀHa nd CÀCb onds in great detail by decomposingt he DE int into three physically meaningful terms using the canonical energy decomposition analysis(EDA) scheme [Eq. (2)]: [9] In Eq. (2), DV elstat is the classical electrostatic interaction between the unperturbed charge distributions of the (deformed) reactants. The Pauli repulsion DE Pauli comprises the destabilizing interaction between occupied orbitals due to the Pauli exclusion principle and is an excellent descriptor of steric repulsion. Finally,t he orbital interaction DE oi includes the formation of the electron-pairb ond between two singly occupied molecular orbitals (SOMOs) and orbitalr elaxation (i.e.,c harge transfer and polarization).  Note that the decomposed interaction energy terms depicted in Table 1s trongly depend on the CÀHa nd CÀCb ond distance. Therefore, the differences between these energy terms along the hybridizations eries must be interpretedw ith special precaution because they emerge not only from the original variation in the intrinsic bondingp roperties but also from the concomitant geometricalr elaxation which affects the original trends. [25,26] In order to solely focus on the trend in the intrinsic bondingp roperties of our model systems, we have decomposed the interaction energy:( i) as af unction of the CÀHa nd CÀCb ond distance;w hile (ii)keeping R n CC and HC or CH 3 C fragments fixed in the equilibrium geometry and valence electron configuration of the overall systems, i.e.,R n CÀHa nd R n CÀCH 3 (n = 1,2,3).T he former ensures ac onsistent comparison of the energy terms at any bond distance whereas the latter prevents any other geometricalr elaxation within the fragments to mask primary changes in the energy terms. Note that this measure guarantees that none of the primarye ffects in the interaction energy terms is absorbed into the strain term whichr emains constant.
Prior to discussingt he decomposedi nteraction energy terms as af unctiono ft he bond length, we first examine the orbitalo verlap integralsc orresponding to the CÀHa nd CÀC electron-pair bonds (Figures 2a andb ;s ee Figure 1c for molecular orbital diagram). The larger the overlap between the SOMOs of the two fragments, the more stabilizing the corresponding electron-pair bondingo rbitali nteraction. [30] Thus, the point at which the SOMO-SOMOo verlap reaches am aximum is often considered as an essential factor in determining the equilibrium bond length. [1,2] These maxima follow as imilar trend as the equilibrium bond lengths themselves, that is, as the fragments approacht owards each other,t he SOMO-SOMO overlap achieves its maximum earlier,a talongerb ond distance, in the case of the sp 3 -hybridized R 3 CC than for the sp 2hybridized R 2 CC than for the sp-hybridized RCC.T his observation correspondsw ellw itht he spatial extent of thed ifferent hybridized SOMOs,w hich, in line with thec urrent rationale, steadily decreasesf romt he sp 3 -hybridizedR 3 CC to thes p-hybridized RCC (Figure2c),i no ther words, theS OMOo fR 3 CC is closer to the grey vertical line than theSOMOofRC · .T hiscan also be seen in thezoom-in of Figure 2d:ifone approaches thecarbonnucleus from infinity,t he orbital function of R 3 CC reaches the value of 0.05 au earlier,i .e.,f urther away from the carbon nucleus, compared to the R 2 CC and RCC analogues. In addition, R 3 CC also has as maller orbital amplitude close to the carbon nucleus compared to R 2 CC and RCC (Figure 2d), giving rise to less orbital overlap for the former as seen in Figures 2a and 2b.
Note, however,t wo striking phenomena: (i)all equilibrium CÀH, and CÀC, bond distances differ significantly from the distance at which the bond overlaps achieve their maximum, CÀ Hb onds are in facta ll longer;a nd (ii)the contractiono fC ÀH, and also CÀC, bonds as the carbon atoms involved vary,i ns -p hybridization, along sp 3 to sp 2 to sp, is significantly smaller than the variation in the distance at whicht he corresponding bond overlaps achieve their maximum (see verticall ines and dots in Figures 2a and b). Thus, despite the fact that the positions of the maximum SOMO-SOMO overlap display the ex-pected trends, other physical mechanisms are crucial for achieving the actual equilibriumb ond distances.
Our energy decomposition analysis as af unctiono ft he CÀX (X = H, CH 3 )d istance, shows that, in contrastt op resent-day textbookk nowledge,t he orbital interactions DE oi are not responsible fort he stronger and shorter sp-hybridized CÀHa nd CÀCb ond ( Figures 3a and b). The interaction energy DE int follows the trend discussed earlier,i .e.,b onds involving sp 3 -hybridized carbon atoms are weaker and have al ess stabilizing DE int than their sp-hybridized analogs. Strikingly,t he orbital interactions DE oi ,h owever,s how an opposite behavior:f rom sp 3to sp 2 -t os p-hybridized carbon atom in CÀXb onds, the DE oi curvesb ecome decreasingly stabilizing, although the difference between sp 3 and sp 2 hybridization is only marginal. Thist rend stems from the shrinking contribution of orbitalr elaxation which relieves the Pauli repulsion, especially at shorter CÀX distances at which closed-shell-closed-shell repulsion becomes very large. Thus, if it were for the orbitali nteractions alone, CÀ Xb onds would become longer, not shorter,a long sp 3 ,s p 2 ,a nd sp hybridization of the carbona tom. [31] The electrostatic attraction DV elstat follows as imilar trend alongt he series as DE oi ,i .e., the curvesb ecomed ecreasingly stabilizing and shallow along sp 3 ,s p 2 ,a nd sp and, therefore, also favor elongation of the CÀ Xb ond distance along this series.
We now identify DE Pauli as the decisive factor in determining the equilibrium bond length because the only differenceb etween the sp 3 -a nd sp 2 -hybridized CÀXb onds lies in this repulsive term, which is less destabilizing for the latter.T hisd ifference allows the fragments to approache ach other more closely,l eading to shorter bond distances. Continuing to the bonds involving an sp-hybridized carbon atom,t here is ar emarkably large drop in Pauli repulsion DE Pauli .T his effect partly compensates the weakening of DE oi and, especially for CÀC, also of DV elstat .I ti st herefore the change in DE Pauli that determines the longerb ond lengths in sp 3 -hybridized CÀHa nd CÀCb onds comparedt ot heir sp 2 -a nd sp-hybridized analogs. We recall that this phenomenon is also displayed in the EDA terms corresponding to the equilibrium geometries ( Table 1). As shown, ah ighly destabilizing DE Pauli induces an elongation of the CÀX bond, which, in turn, reduces all EDA terms, including the DE Pauli .N evertheless, the DE Pauli of the longer sp 3 -hybridized CÀ Xb ond is more destabilizing than the less hybridized counterparts, indicating that it is this term that governs the observed lengtheningo ft he CÀXb ond.
The relation between the Pauli repulsion and the number of sterically hindering substituents becomes even more evident when,i nn umerical experiments,w ee xplicitly change the size and number of substituents (see Ta ble S1). Increasings ubstituent size leads to al onger sp 3  Importantly,o ur analyses also shed light on the nature, especially the orbital energy,o ft he s*-orbital of the sp n -hybridized R n CÀXb onds (n = 3, 2, 1; X = H, alkyl, halogen,e tc.), which is of direct relevance for understanding varioust ypes of reactionsa nd supramolecular aggregates featuring these bonds. [32][33][34] We find that the s*-orbital of R n CÀXb onds be- comes increasingly more stabilized, on going from sp 3 to sp 2 to sp carbon centers, again,d ue to ar eduction in the number of substituents aroundt he pertinent carbon atom. The s*-orbital of the sp n -hybridized R n CÀHb ond lowers in energy alongn = 3, 2, 1, from 1.7 eV for R 3 CÀHt o1 .4 eV for R 2 CÀHt o1 .0 eV for RCÀH, respectively,b ecauset he R n CC SOMO becomesg radually more stabilized. This behaviorc an be ascribed to two phenomena: (i)the R n CC SOMO is R n ÀC 2s antibonding, which reduces as R n decreases from n = 3t o2t o1 ,d ue to less orbital overlap;( ii)the R n CC SOMO is also R n ÀC 2p bondinga nd its bondingc apability,i .e.,o rbital overlap, becomes stronger as R n can align betterw ith C 2p along this series (see Supporting Information Discussion1 and Figure S1 for ad etailed molecular orbitalanalysis).
To conclude, we have shown that, in contrastt ot he present-day paradigm, the contraction of CÀHa nd CÀCb ond lengths on going from sp 3 to sp 2 to sp carbon centers, originates from ad iminished Pauli repulsion, the magnitude of which is directly relatedt ot he steric proximity between the substituents aroundt he pertinent carbon atom. The orbital interaction, which was up to this point seen as the driving force, shows behavior that counteracts the observed trend in bond strength and, consequently,i sn ot responsible for the decreasing bond length.