NATURE CHEMISTRY DOI:10.1038/NCHEM.1263 ARTICLES CXOX○ weakly antibonding and two are bonding.As such,in their states,Siz and Ge2 have double bonds composed of and mbonds, 30g in line with the reluctance of higher-row molecules to form multiple bonds38-41. 2oT Quadruple bonding in CN,BN and CB.We next turned to CX○CXO the isoelectronic first-row analogues of C2 with eight valence electrons:CN,BN and CB (refs 42-47).For all cases we carried ΨTc=④0-2④知 GVB out FCI calculations to ascertain the nature of the ground or low-lying singlet states,and subsequently also calculated CN, Figure 3 Schematic representation of the transformation of the TC BN and CB by VB theory,using VBSCF/6-31G*(VBSCF wavefunction (equation (3))into a GVB wavefunction.The GVB orbitals are refers to valence bond self-consistent field calculations:see a A-weighted sum and difference of the 2o and 3o molecular orbitals.The Supplementary Sections II.2.2,II.2.3,II.2.4 for FCI and Tables singlet coupling of the corresponding electrons is signified by the dotted line S1-S5 for VBSCF). connecting the orbitals. It is well known that for C,and its isoelectronic first-row analogues the two lowest electronic states,I and types,are of HCCH (in which a triple bond binds the HC fragments)by close in energy2022253442-47.This is what we indeed find,but the reference to Fig.2e.Summing up the calculated D values for the focus of our FCI and VB calculations is on the states,which two bonds,the o bond and the fourth bond of C.,we obtain a are the only possible candidates to have quadruple bonding in the total intrinsic bonding energy of Dn(total)=303 kcal mol, molecules at hand. which is the bonding interaction between the C atoms in their 5s To ascertain the quadruple bonding in these three molecules we states (see similar past analyses in refs 30-33).Thus,according to started with VB theory and followed with FCI.The VB results of the Fig.2e,the Dn (total)value for C2 is given by the sum of BDE ground states for these molecules were analogous to those for and the corresponding promotion energies of the C fragments C..Table 1 shows that the D values for the fourth bond of these from the ground states (P)to the high spin(S)states,which are molecules can be bracketed in the range 11.6-17.4 kcal mol-.As prepared'for bonding.Equation(1)expresses this relationship for already noted,this fourth bond,although weaker than the com- any given molecule: ponents of the internal triple bond,is significant and cannot be ignored. Dn(total)≈BDE+△Eprom (1) C,bond orders and force constants.Interestingly,the double- As FCI is too costly for HCCH,we used multi-reference hybrid density functional theoryis calculated Wiberg bond order of configuration interaction (MRCI)calculations,which gave D C2 is larger than 3(it is 3.714 using the Kohn-Sham density).At the (total)=313.7 kcal mol for C2,but only 252.7 kcal mol for same level,the bond order of the C-C bond in HCCH is 2.998,and HCCH.As such,computationally,the intrinsic bonding energy in N2 it is 3.032(Supplementary Section III).These bond orders for C is larger than for HCCH,in agreement with the relative correlate with the above D(total)values we estimated for C,versus bond multiplicities of 4 versus 3.Using experimental BDEs" HCCH.In contrast,our relaxed force constant (RFC)calculations (146.05 and 236.7 kcal mol for C and HCCH,respectively)and show that HCCH has a larger RFC than C2(Supplementary Section promotion energies (96.4 kcal mol- per C atom35 and 16.7 kcal IV).As one generally expects an increase in RFC with increasing mol-2promotion energy per HC)gives Dn (total)= bond multiplicity,this finding constitutes a puzzle;if indeed 338.9 kcal mol-for C and D(total)=270.1 kcal mol-for there is a fourth bond in singlet C,,then why does the triple bond HCCH.These D(total)values lead again to the conclusion that in acetylene have a larger RFC than the quadruple bond in C,?This the intrinsic bonding interaction in the quadruply bonded C2 is is especially surprising,because the estimated total bonding energy larger than that in the triply bonded HCCH.Furthermore, relative to the 'prepared'fragments (vide supra)for C2 is larger than because the Din(2m)values for C and HCCH12 are virtually the analogous quantity for acetylene.The fact that the relative identical (186-188 kcal mol-),this means that the intrinsic bonding energies are not reflected in the RFCs indicates the bonding energy of the internal occ and inverted fourth bond of existence of factors that soften the potential energy of C2 near C2 combined is significantly larger than the occ bond of HCCH. the minimum.A plausible explanation for such a curve-flattening This value can be further corrected by taking into account the factor is the avoided crossing that occurs between the B state promotion'term due to the different orbitals (rehybridization, and theX ground state21 at a distance (1.6 A)quite close to the size)of the high-spin fragments from their situation in the equilibrium distance.Furthermore,the B state is dominated by molecule relative to the free fragments.With this term,which is 11 kcal mol larger for the two HC fragments than for the a b CN" two C fragments,the resulting o-bonding interaction in C2 is 50-57 kcal mol-higher than for HCCH.This is a strong argument in support of the quadruple bond character of C and its augmented bonding interaction compared with HCCH. S=(4|p》=0.4375 S=(4|R)=0.4585 The states of Siz and Gez.Next we turned to the higher-row analogues of C,,Si,and Ge,.Using FCI,both were found to have two BN CB low-lying triplet ground states and/or),in agreement with experiment for Si,and previous CI results3637.The singlet Xstates of Si and Ge lie significantly higher and are different to the corresponding state for C2(Supplementary Sections II.2.5 and II.2.6).Thus,in their singlet states,Si,and Ge2 give up one of S=(-|R》=0.5283 S=(4|)=0.4731 their bonds,and instead populate the (n+1)o orbital Figure 4 I Semi-localized orbitals,which form the fourth bond and (analogous to 30 in Fig.la).Of the three filled o orbitals,one is their overlap S values.a,C2.b,CN+.c BN.d,CB- NATURE CHEMISTRY ADVANCE ONLINE PUBLICATION www.nature.com/naturechemistry 2012 Macmillan Publishers Limited All rights reserved.of HCCH (in which a triple bond binds the HC fragments) by reference to Fig. 2e. Summing up the calculated Din values for the two p bonds, the s bond and the fourth bond of C2, we obtain a total intrinsic bonding energy of Din (total) ¼ 303 kcal mol21 , which is the bonding interaction between the C atoms in their 5 S states (see similar past analyses in refs 30–33). Thus, according to Fig. 2e, the Din (total) value for C2 is given by the sum of BDE and the corresponding promotion energies of the C fragments from the ground states (3 P) to the high spin (5 S) states, which are ‘prepared’ for bonding. Equation (1) expresses this relationship for any given molecule: Din (total) ≈ BDE + DEprom (1) As FCI is too costly for HCCH, we used multi-reference configuration interaction (MRCI) calculations, which gave Din (total) ¼ 313.7 kcal mol21 for C2, but only 252.7 kcal mol21 for HCCH. As such, computationally, the intrinsic bonding energy for C2 is larger than for HCCH, in agreement with the relative bond multiplicities of 4 versus 3. Using experimental BDEs34 (146.05 and 236.7 kcal mol21 for C2 and HCCH, respectively) and promotion energies (96.4 kcal mol21 per C atom35 and 16.7 kcal mol21 2 P 4 S2 promotion energy per HC34) gives Din (total) ¼ 338.9 kcal mol21 for C2 and Din (total) ¼ 270.1 kcal mol21 for HCCH. These Din (total) values lead again to the conclusion that the intrinsic bonding interaction in the quadruply bonded C2 is larger than that in the triply bonded HCCH. Furthermore, because the Din (2p) values for C2 and HCCH12 are virtually identical (186–188 kcal mol21 ), this means that the intrinsic bonding energy of the internal sCC and inverted fourth bond of C2 combined is significantly larger than the sCC bond of HCCH. This value can be further corrected by taking into account the ‘promotion’ term due to the different orbitals (rehybridization, size) of the high-spin fragments from their situation in the molecule relative to the free fragments. With this term, which is 11 kcal mol21 larger for the two HC fragments than for the two C fragments, the resulting s-bonding interaction in C2 is 50–57 kcal mol21 higher than for HCCH. This is a strong argument in support of the quadruple bond character of C2 and its augmented bonding interaction compared with HCCH. The X1 Sg 1 states of Si2 and Ge2. Next we turned to the higher-row analogues of C2, Si2 and Ge2. Using FCI, both were found to have two low-lying triplet ground states (3 Sg 2 and/or 3 Pu), in agreement with experiment for Si2 and previous CI results36,37. The singlet X1 Sg þ states of Si2 and Ge2 lie significantly higher and are different to the corresponding state for C2 (Supplementary Sections II.2.5 and II.2.6). Thus, in their singlet states, Si2 and Ge2 give up one of their p bonds, and instead populate the (n þ 1)sg orbital (analogous to 3sg in Fig. 1a). Of the three filled s orbitals, one is weakly antibonding and two are bonding. As such, in their 1 Sg þ states, Si2 and Ge2 have double bonds composed of s and p bonds, in line with the reluctance of higher-row molecules to form multiple p bonds38–41. Quadruple bonding in CN1, BN and CB2. We next turned to the isoelectronic first-row analogues of C2 with eight valence electrons: CNþ, BN and CB2 (refs 42–47). For all cases we carried out FCI calculations to ascertain the nature of the ground or low-lying singlet states, and subsequently also calculated CNþ, BN and CB2 by VB theory, using VBSCF/6-31G* (VBSCF refers to valence bond self-consistent field calculations; see Supplementary Sections II.2.2, II.2.3, II.2.4 for FCI and Tables S1–S5 for VBSCF). It is well known that for C2 and its isoelectronic first-row analogues the two lowest electronic states, 3 P and 1 Sþ types, are close in energy20,22,25,34,42–47. This is what we indeed find, but the focus of our FCI and VB calculations is on the 1 Sþ states, which are the only possible candidates to have quadruple bonding in the molecules at hand. To ascertain the quadruple bonding in these three molecules we started with VB theory and followed with FCI. The VB results of the 1 Sþ ground states for these molecules were analogous to those for C2. Table 1 shows that the Din values for the fourth bond of these molecules can be bracketed in the range 11.6–17.4 kcal mol21 . As already noted, this fourth bond, although weaker than the com￾ponents of the internal triple bond, is significant and cannot be ignored. C2 bond orders and force constants. Interestingly, the double￾hybrid density functional theory48 calculated Wiberg bond order of C2 is larger than 3 (it is 3.714 using the Kohn–Sham density). At the same level, the bond order of the C–C bond in HCCH is 2.998, and in N2 it is 3.032 (Supplementary Section III). These bond orders correlate with the above Din (total) values we estimated for C2 versus HCCH. In contrast, our relaxed force constant (RFC)49 calculations show that HCCH has a larger RFC than C2 (Supplementary Section IV). As one generally expects an increase in RFC with increasing bond multiplicity11,49, this finding constitutes a puzzle; if indeed there is a fourth bond in singlet C2, then why does the triple bond in acetylene have a larger RFC than the quadruple bond in C2? This is especially surprising, because the estimated total bonding energy relative to the ‘prepared’ fragments (vide supra) for C2 is larger than the analogous quantity for acetylene. The fact that the relative bonding energies are not reflected in the RFCs indicates the existence of factors that soften the potential energy of C2 near the minimum. A plausible explanation for such a curve-flattening factor is the avoided crossing that occurs between the B′1 Sg þ state and the X1 Sg þ ground state21 at a distance (1.6 Å) quite close to the equilibrium distance. Furthermore, the B′1 Sg þ state is dominated by 2σu 3σg ϕL ϕR Ψ ΦGVB TC = Φ0 – λ2ΦD Figure 3 | Schematic representation of the transformation of the TC wavefunction (equation (3)) into a GVB wavefunction. The GVB orbitals are a l-weighted sum and difference of the 2su and 3sg molecular orbitals. The singlet coupling of the corresponding electrons is signified by the dotted line connecting the orbitals. a C2 CN+ BN CB– S= ϕL | ϕR = 0.4375 S= ϕL | ϕR = 0.4585 S= ϕL | ϕR = 0.5283 S= ϕL | ϕR = 0.4731 b c d Figure 4 | Semi-localized fL–fR orbitals, which form the fourth bond and their overlap S values. a, C2. b, CNþ. c, BN. d, CB2. NATURE CHEMISTRY DOI: 10.1038/NCHEM.1263 ARTICLES NATURE CHEMISTRY | ADVANCE ONLINE PUBLICATION | www.nature.com/naturechemistry 3 © 2012 Macmillan Publishers Limited. All rights reserved.