nature ARTICLES chemistry PUBLISHED ONLINE:29 JANUARY 2012|DOI:10.1038/NCHEM.1263 Quadruple bonding in C>and analogous eight-valence electron species Sason Shaik*,David Danovich',Wei Wu2,Peifeng Su2,Henry S.Rzepa3 and Philippe C.Hiberty4 Triple bonding is conventionally considered to be the limit for multiply bonded main group elements,despite higher metal-metal bond orders being frequently observed for transition metals and lanthanides/actinides.Here,using high-level theoretical methods,we show that C2 and its isoelectronic molecules CN,BN and CB (each having eight valence electrons)are bound by a quadruple bond.The bonding comprises not only one o-and two mbonds,but also one weak 'inverted'bond,which can be characterized by the interaction of electrons in two outwardly pointing sp hybrid orbitals.A simple way of assessing the energy of the fourth bond is proposed and is found to be ~12-17 kcal mol for the isoelectronic species studied,and thus stronger than a hydrogen bond.In contrast,the analogues of C2 that contain higher-row elements,such as Siz and Gez,exhibit only double bonding. "he interest in multiple bonding has been on the rise ever since directionality of these hybrids is the main factor that dictates why it was demonstrated that transition metals and lanthanides/ this 'inverted'fourth bond is commonly ruled out by chemists. actinides can form metal-metal bonding in which the However,a recent estimate of the bonding in [1.1.1]propellane24 maximum practical bond order reaches four to six bonds!-8.In shows that such outwardly pointing hybrids may nevertheless main elements,however,the maximum number of bonds between maintain a highly significant bonding interaction.Indeed,if the two atoms has remained three-13,this being composed of one two odd electrons in the outwardly pointing hybrids were very o-and two r-bonds.Nevertheless,there are diatomic molecules weakly coupled,the molecule would have exhibited a diradicaloid such as C2,Siz,CN+and BN,which,by having eight valence elec- character with a closely lying triplet state.However,the diradicaloid trons,could at least formally express quadruple bonding between character is absent25.More compelling is the fact that the corre- the two atoms (H.S.Rzepa,www.ch.imperial.ac.uk/rzepa/blog/ sponding triplet state in which these electrons are unpaired, ?p=3065).One might then ask,no matter how naively,can the lies 26.4 kcal mol above the ground state 5.16,indicating that eight valence electrons (for example,between the two carbon these electrons maintain a significant bonding interaction in the atoms in C2)couple to create four bonds and,if so,what is the ground state.Therefore,one cannot rule out the inference that C2 bonding energy of the putative fourth bond?This is the focus of has a quadruple bond,as depicted by structure 3 in Fig.1b.Here, the present article,which uses valence bond(VB)theory1214 and we test this hypothesis.We present an assessment of the energy of full configuration interaction(FCI)calculations to determine the the fourth bond by means of VB and FCI calculations,and we bonding energy of the fourth bond in C2 and its absence or presence demonstrate the quadruple bonding from the FCI wavefunction. in some of its isoelectronic species. As we shall show,although C2,CN+,BN and CB definitely have C2 has been extensively investigated using a variety of methods, a fourth bond,higher-row analogues such as Si or Ge2 have only which have provided valuable information on its ground state a double bond. (X)and 17 of its excited states15-22.Nevertheless,C,continues to challenge our understanding of bonding!4.A nominal consideration of the bond order in the molecular orbital diagram b:C三C: in Fig.la would suggest a bond order of two23,as in structure 1 in Fig.1b,and,because the 20 and 2o orbitals are both filled, OCCO ↑ the molecule would then have two z-bonds unsupported by an underlying o-bond (or a weak one assuming that 20 is rather 0 c=cd weakly antibonding),and two o lone pairs.In contrast,using sp-hybridized carbons would suggest that it is possible to form a strong triple bond composed of one o-and two z-bonds,with 20g two electrons remaining in the outwardly pointing hybrids,as in Xixg structure 2 (Fig.1b). A recent VB theory study!4 has shown that,by using structure 2, Figure 1|Representations of bonding in C2.a,Molecular orbital diagram. the properties of C can be predicted quite well,and that its two The shapes of the 20u and 3o molecular orbitals,as determined from FCI electrons in the outwardly pointing hybrids are singlet-paired, calculations,are also represented together with their respective energy thus yielding the known singlet ground state The levels.b,Three simplified bonding cartoons. 'Institute of Chemistry and The Lise Meitner-Minerva Center for Computational Quantum Chemistry,Hebrew University of Jerusalem,91904,Jerusalem, Israel,'The State Key Laboratory of Physical Chemistry of Solid Surfaces,Fujian Provincial Key Laboratory of Theoretical and Computational Chemistry,and College of Chemistry and Chemical Engineering Xiamen University,Xiamen,Fujian 361005,China,Department of Chemistry,Imperial College London, South Kensington Campus,London SW7 2AZ,UK,'Laboratoire de Chimie Physique,UMR CNRS 8000,Universite de Paris Sud,91405 Orsay Cedex, France."e-mail:sason@yfaat.ch.huji.ac.il NATURE CHEMISTRY|ADVANCE ONLINE PUBLICATION www.nature.com/naturechemistry 2012 Macmillan Publishers Limited All rights reserved
Quadruple bonding in C2 and analogous eight-valence electron species Sason Shaik1 *, David Danovich1 , Wei Wu2, Peifeng Su2, Henry S. Rzepa3 and Philippe C. Hiberty4 Triple bonding is conventionally considered to be the limit for multiply bonded main group elements, despite higher metal–metal bond orders being frequently observed for transition metals and lanthanides/actinides. Here, using high-level theoretical methods, we show that C2 and its isoelectronic molecules CN1, BN and CB2 (each having eight valence electrons) are bound by a quadruple bond. The bonding comprises not only one s- and two p-bonds, but also one weak ‘inverted’ bond, which can be characterized by the interaction of electrons in two outwardly pointing sp hybrid orbitals. A simple way of assessing the energy of the fourth bond is proposed and is found to be ∼12–17 kcal mol21 for the isoelectronic species studied, and thus stronger than a hydrogen bond. In contrast, the analogues of C2 that contain higher-row elements, such as Si2 and Ge2, exhibit only double bonding. The interest in multiple bonding has been on the rise ever since it was demonstrated that transition metals and lanthanides/ actinides can form metal–metal bonding in which the maximum practical bond order reaches four to six bonds1–8. In main elements, however, the maximum number of bonds between two atoms has remained three9–13, this being composed of one s- and two p-bonds. Nevertheless, there are diatomic molecules such as C2, Si2, CNþ and BN, which, by having eight valence electrons, could at least formally express quadruple bonding between the two atoms (H. S. Rzepa, www.ch.imperial.ac.uk/rzepa/blog/ ?p=3065). One might then ask, no matter how naively, can the eight valence electrons (for example, between the two carbon atoms in C2) couple to create four bonds and, if so, what is the bonding energy of the putative fourth bond? This is the focus of the present article, which uses valence bond (VB) theory12,14 and full configuration interaction (FCI) calculations to determine the bonding energy of the fourth bond in C2 and its absence or presence in some of its isoelectronic species. C2 has been extensively investigated using a variety of methods, which have provided valuable information on its ground state (X1 Sg þ) and 17 of its excited states15–22. Nevertheless, C2 continues to challenge our understanding of bonding14. A nominal consideration of the bond order in the molecular orbital diagram in Fig. 1a would suggest a bond order of two23, as in structure 1 in Fig. 1b, and, because the 2sg and 2su orbitals are both filled, the molecule would then have two p-bonds unsupported by an underlying s-bond (or a weak one assuming that 2su is rather weakly antibonding), and two s lone pairs. In contrast, using sp-hybridized carbons would suggest that it is possible to form a strong triple bond composed of one s- and two p-bonds, with two electrons remaining in the outwardly pointing hybrids, as in structure 2 (Fig. 1b). A recent VB theory study14 has shown that, by using structure 2, the properties of C2 can be predicted quite well, and that its two electrons in the outwardly pointing hybrids are singlet-paired, thus yielding the known singlet ground state X1 Sg þ. The directionality of these hybrids is the main factor that dictates why this ‘inverted’ fourth bond is commonly ruled out by chemists. However, a recent estimate of the bonding in [1.1.1]propellane24 shows that such outwardly pointing hybrids may nevertheless maintain a highly significant bonding interaction. Indeed, if the two odd electrons in the outwardly pointing hybrids were very weakly coupled, the molecule would have exhibited a diradicaloid character with a closely lying triplet state. However, the diradicaloid character is absent25. More compelling is the fact that the corresponding triplet state c 3 Su þ, in which these electrons are unpaired, lies 26.4 kcal mol21 above the ground state15,16, indicating that these electrons maintain a significant bonding interaction in the ground state. Therefore, one cannot rule out the inference that C2 has a quadruple bond, as depicted by structure 3 in Fig. 1b. Here, we test this hypothesis. We present an assessment of the energy of the fourth bond by means of VB and FCI calculations, and we demonstrate the quadruple bonding from the FCI wavefunction. As we shall show, although C2, CNþ, BN and CB2 definitely have a fourth bond, higher-row analogues such as Si2 or Ge2 have only a double bond. 2σg 2σu 3σg 1πu 1πg X1Σg + a b C C C C C C 1 2 3 Figure 1 | Representations of bonding in C2. a, Molecular orbital diagram. The shapes of the 2su and 3sg molecular orbitals, as determined from FCI calculations, are also represented together with their respective energy levels. b, Three simplified bonding cartoons. 1 Institute of Chemistry and The Lise Meitner-Minerva Center for Computational Quantum Chemistry, Hebrew University of Jerusalem, 91904, Jerusalem, Israel, 2 The State Key Laboratory of Physical Chemistry of Solid Surfaces, Fujian Provincial Key Laboratory of Theoretical and Computational Chemistry, and College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, Fujian 361005, China, 3 Department of Chemistry, Imperial College London, South Kensington Campus, London SW7 2AZ, UK, 4 Laboratoire de Chimie Physique, UMR CNRS 8000, Universite´ de Paris Sud, 91405 Orsay Ce´dex, France. *e-mail: sason@yfaat.ch.huji.ac.il ARTICLES PUBLISHED ONLINE: 29 JANUARY 2012 | DOI: 10.1038/NCHEM.1263 NATURE CHEMISTRY | ADVANCE ONLINE PUBLICATION | www.nature.com/naturechemistry 1 © 2012 Macmillan Publishers Limited. 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ARTICLES NATURE CHEMISTRY DOI:10.1038/NCHEM.1263 only classical interactions,which sum to zero4426.In turn,D bond =a ob(ion+ion. is the energy difference(Fig.2b)of the Voc and of bond states. Figure 2c shows another way of estimating D by calculating the Din1/2AEST corresponding triplet state,which uncouples the electrons of the Ψoc fourth bond to a triplet state with two identical spins.As has been bondA-B discussed previously28(see ref.26,p.131),the singlet-to-triplet Yion=(A:B)ion=(A B: excitation of a bond is approximately twice the desired D Figure 2d shows the symbols for the C,states that are involved in this particular singlet-to-triplet excitation. Figure 2e shows the relationship between the total D.and BDE for all the electron pairs in C2.The BDE involves the relaxation of the fragments and their electronic demotion to the corresponding electronic ground states,which in the case of C atoms are the sp Din1/2AEST states.However,D(total)measures the stabilization energy due to bond-pairing of the prepared's states of C,without any effects associated with the relaxation of the fragments electronically 2略d2 (or geometrically if applicable).We shall refer to Dn(total)as the 'intrinsic bonding energy'. The first two entries in Table 1 list the D values for C2 from VB theory.The two methods give values for D of the fourth C-C bond of 14.30 and 11.64 kcal mol-,respectively.We note that the VB-calculated AEsr value (23.28 kcal mol-)is quite 2C(5S) close to the experimental value of the vertical excitation from Xto the c+state (26.4 kcal mol-)15.16.This is expected, D (total) because the VB calculation of AEsr is entirely equivalent to the XlΣt→c∑excitation that uncouples the singlet pair of the fourth-bond electrons to a triplet spin,as shown in Fig.2c,d2 (see also ref..26,pp.57,79,88,188). 2C(3P BDE Using molecular orbital-based FCI computations of AEr is a convenient alternative way to obtain D data.Indeed,our FCI calculations for these states in C show that the ground-state and the triplet-state c are dominated by the 202 202 and Cz (Xig) 23configurations(Supplementary Section II.2).The cal- culated FCI value of AEsr is 29.6 kcal mol,slightly higher than the Figure 2|VB wavefunctions and energy terms.a,The full-bond state experimental value.Using the relation of D to AEsr in Fig.2c,we list (n for bond A-B,with covalent and two ionic( the corresponding Dn values in Table 1 (last two entries),estimated contributions,and the spin arrangement patterns that make up the covalent from the experimental and FCI AEsr quantities.These values are structure.b,Definition of the in situ bond energy (Di)as the energy gap 13.2 and 14.8 kcal mol.Thus,four different methods bracket between and the C stateRE is the covalent-ionic the intrinsic bonding energy of the fourth bond in the range resonance energy.c,Schematic energy diagram,showing D as half the 11.6-14.8 kcal mol energy gap (EsT)between the full bond state(and the triplet state To gauge the relative intrinsic bonding energy of this fourth ()d,Corresponding singlet X and triplet cC,states,needed for C-C bond in relation to the others within the molecule,we used the same VB method and obtained 100.4 kcal mol for the internal calculating D,and their dominant electronic configurations.e,Schematic representation of the dissociation of C>to two C atoms.Di(total)is the (C-C)bond and 94.2 kcal mol for each of the two m-bonds, intrinsic bonding energy due to bond-pairing of the prepared C(S)states, close to previously obtained values'4.Thus,the fourth bond of while the BDE measures the dissociation energy to the ground C(P)states. C2 has an intrinsic bonding energy value that is ~15%of the The promotion energy(AFm is the35s difference (S and Pare internal bonds in the molecule.Although this bond is not of great indicators of angular momentum). strength,it is nevertheless significant and cannot be ignored or dismissed. Results and discussion The fourth bond of C2.The bond dissociation energy(BDE)of one Comparing the quadruple bond in C,to the triple bond in bond in a molecule like C,is meaningless.Hence,we calculate the HCCH.Let us compare the intrinsic bonding energy of C2 to that in situ bond energy,Dn(refs 12,14,24,26),which measures the bonding interaction in a given bond.VB theory enables us to determine Dn for any bond using a reference non-bonding state, Table 1 Values of in situ bond energies (Din,kcal mol) ca so-called quasi-classical (C)state as illuistrated for the fourth bond of diatomic molecules calculated in in Fig.2a,b.Thus,the wavefunction of the bond,bond,is given different ways. in Fig.2a as a combination of a covalent structure,o,and Method Source of D:C2 BN CN CB secondary ionic structures,Vion(refs 12,24,27).The covalent VBSCF/6-31G QC state 143016.97 17.3814.16 structure is stabilized by the resonance energy of its constituent spin-arrangement patterns,one with spin-up/spin-down and the VBSCF/6-31G AFST 11.64 11.46 12.74 11.55 other with spin-down/spin-up.Mixing of the ionic structures into FC/6-31G AEST 14.80 16.64 16.8913.37 the covalent structure further augments the bonding interaction Experimental datum AEST 13.19t 13.72 with covalent-ionic resonance energy. The QC state is one of the spin-arrangement patterns of o, Ref 15. and it is non-bonding as the two odd electrons in it maintain 机et46 NATURE CHEMISTRY|ADVANCE ONLINE PUBLICATION |www.nature.com/naturechemistry 2012 Macmillan Publishers Limited All rights reserved
Results and discussion The fourth bond of C2. The bond dissociation energy (BDE) of one bond in a molecule like C2 is meaningless. Hence, we calculate the in situ bond energy, Din (refs 12,14,24,26), which measures the bonding interaction in a given bond. VB theory enables us to determine Din for any bond using a reference non-bonding state, CQC, a so-called quasi-classical (QC) state12,14,24,26, as illustrated in Fig. 2a,b. Thus, the wavefunction of the bond, Cbond, is given in Fig. 2a as a combination of a covalent structure, Ccov, and secondary ionic structures, Cion (refs 12,24,27). The covalent structure is stabilized by the resonance energy of its constituent spin-arrangement patterns, one with spin-up/spin-down and the other with spin-down/spin-up. Mixing of the ionic structures into the covalent structure further augments the bonding interaction with covalent–ionic resonance energy. The QC state is one of the spin-arrangement patterns of Ccov, and it is non-bonding as the two odd electrons in it maintain only classical interactions, which sum to zero12,14,24,26. In turn, Din is the energy difference (Fig. 2b) of the CQC and of Cbond states. Figure 2c shows another way of estimating Din by calculating the corresponding triplet state, CT, which uncouples the electrons of the fourth bond to a triplet state with two identical spins. As has been discussed previously28 (see ref. 26, p. 131), the singlet-to-triplet excitation of a bond is approximately twice the desired Din. Figure 2d shows the symbols for the C2 states that are involved in this particular singlet-to-triplet excitation. Figure 2e shows the relationship between the total Din and BDE for all the electron pairs in C2. The BDE involves the relaxation of the fragments and their electronic demotion to the corresponding electronic ground states, which in the case of C atoms are the 3 P states. However, Din (total) measures the stabilization energy due to bond-pairing of the ‘prepared’ 5 S states of C, without any effects associated with the relaxation of the fragments electronically (or geometrically if applicable). We shall refer to Din (total) as the ‘intrinsic bonding energy’. The first two entries in Table 1 list the Din values for C2 from VB theory. The two methods give values for Din of the fourth C–C bond of 14.30 and 11.64 kcal mol21 , respectively. We note that the VB-calculated DEST value (23.28 kcal mol21 ) is quite close to the experimental value of the vertical excitation from X1 Sg þ to the c 3 Su þ state (26.4 kcal mol21 )15,16. This is expected, because the VB calculation of DEST is entirely equivalent to the X1 Sg þ c 3 Su þ excitation that uncouples the singlet pair of the fourth-bond electrons to a triplet spin, as shown in Fig. 2c,d29 (see also ref. 26, pp. 57,79,88,188). Using molecular orbital-based FCI computations of DEST is a convenient alternative way to obtain Din data. Indeed, our FCI calculations for these states in C2 show that the ground-state X1 Sg þ and the triplet-state c 3 Su þ are dominated by the 2s2 gp4 u 2s2 u and 2s2 gp4 u2s1 u3s1 g configurations (Supplementary Section II.2). The calculated FCI value of DEST is 29.6 kcal mol21 , slightly higher than the experimental value. Using the relation of Din to DEST in Fig. 2c, we list the corresponding Din values in Table 1 (last two entries), estimated from the experimental and FCI DEST quantities. These values are 13.2 and 14.8 kcal mol21 . Thus, four different methods bracket the intrinsic bonding energy of the fourth bond in the range 11.6–14.8 kcal mol21 . To gauge the relative intrinsic bonding energy of this fourth C–C bond in relation to the others within the molecule, we used the same VB method and obtained 100.4 kcal mol21 for the internal s(C–C) bond and 94.2 kcal mol21 for each of the two p-bonds, close to previously obtained values14. Thus, the fourth bond of C2 has an intrinsic bonding energy value that is 15% of the internal bonds in the molecule. Although this bond is not of great strength, it is nevertheless significant and cannot be ignored or dismissed. Comparing the quadruple bond in C2 to the triple bond in HCCH. Let us compare the intrinsic bonding energy of C2 to that e 2 C (5S) 2 C (3P) C2 (X1Σg +) BDE Din (total) ΔEprom C C C Ψcov = (A B) = (A B) (A B) Ψion,l= (A: B ) Ψbond = a Ψcov + b (Ψion,l + Ψ' ion,r) ΨQC Ψ' QC = (A B: ) a c ΨT Ψbond Dinª1/2ΔEST Dinª1/2ΔEST A B 2σg 2πu 42σu 13σg 1 2σg 2πu 42σu 2 A B b + ΨQC Ψcov Ψbond REcov-ion Dcov A B A B Din d Ψbond C C C C c3Σu A B + Ψ' ion,r Figure 2 | VB wavefunctions and energy terms. a, The full-bond state (Cbond) for bond A–B, with covalent (Ccov) and two ionic (Cion) contributions, and the spin arrangement patterns that make up the covalent structure. b, Definition of the in situ bond energy (Din) as the energy gap between Cbond and the QC state CQC. REcov-ion is the covalent–ionic resonance energy. c, Schematic energy diagram, showing Din as half the energy gap (DEST) between the full bond state (Cbond) and the triplet state (CT). d, Corresponding singlet X1 Sg þ and triplet c3 Su þ C2 states, needed for calculating Din, and their dominant electronic configurations. e, Schematic representation of the dissociation of C2 to two C atoms. Din (total) is the intrinsic bonding energy due to bond-pairing of the prepared C(5 S) states, while the BDE measures the dissociation energy to the ground C(3 P) states. The promotion energy (DEprom) is the 3 P 5 S difference (S and P are indicators of angular momentum). Table 1 | Values of in situ bond energies (Din, kcal mol21 ) for the fourth bond of diatomic molecules calculated in different ways. Method Source of Din C2 BN CN1 CB2 VBSCF/6-31G* QC state 14.30 16.97 17.38 14.16 VBSCF/6-31G* 1 2DEST 11.64 11.46 12.74 11.55 FCI/6-31G* 1 2DEST 14.80 16.64 16.89 13.37 Experimental datum† 1 2DEST 13.19† 13.72‡ – – † Ref. 15. ‡ Ref. 46. ARTICLES NATURE CHEMISTRY DOI: 10.1038/NCHEM.1263 2 NATURE CHEMISTRY | ADVANCE ONLINE PUBLICATION | www.nature.com/naturechemistry © 2012 Macmillan Publishers Limited. All rights reserved.
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 components of the internal triple bond, is significant and cannot be ignored. C2 bond orders and force constants. Interestingly, the doublehybrid 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.
ARTICLES NATURE CHEMISTRY DOI:10.1038/NCHEM.1263 left-hand carbon atom,with a smaller tail on the right-hand carbon,whereas the other,oR,is localized on the right-hand carbon and has a tail on the left-hand atom30.51.The electrons in the and o orbitals are singlet-paired,which in Fig.3 is symbolized by the dotted line connecting the two singly S=(4L{)=0.7749 S=A|)=0.7749 occupied orbitals. The two transformed orbitals for all the molecules are shown in Figure 5I GVB orbital pairs and their overlap S values for the Fig.4 together with S,a measure of their overlap.The larger the internal bonds in C2.a-c,out-of-plane (a),in-plane (b)and internal o overlap in the GVB pairss1,the stronger the respective bond. bond (c)represented by the 2 molecular orbital. The significant overlaps in Fig.4 underscore the conclusion that the bond energies of the fourth bond are significant in all configurations that displayoneless bond that theX ground state. these cases. These factors can flatten the ground state near the equilibrium In fact,it is easily seen that the Vrc wavefunction describes a geometry but will not affect the well depth,and as such will lower quadruple bond.Merely reading equation(2)reveals that the C2 the RFC of C,relative to acetylene,despite the quadruple bonding molecule has an internal triple bond,composed of three strongly in the former. bonding orbitals populated by six electrons,201 and a fourth bond made from the transformed GVB pair.The same Nature of the fourth bond in C,,CN",BN and CB revealed by picture is true for the other molecules;they all have an internal full CI.The fourth bond can be easily understood using VB triple bond made from the 2021 sub-shell,which is augmented language as a hybrid of covalent and ionic structures (Fig.2a), by a GVB pair -g By common knowledge (ref.26,pp.42 similarly to any other bond1427.Although the VB mechanism is and 241,refs 50,51)the GVB wavefunction of a pair like is straightforward,one may wonder what FCI tells us about the equivalent to the localized VB picture in Fig.2a. nature of the fourth bond? In fact,the FCI wavefunction describes all the other bonds in Inspection of the FCI results for the four molecules that exhibit the same manner (Supplementary Sections II.3.2 and II.3.3). quadruple bonding reveals that,in all of them,the FCI wavefunction Thus,in addition to d,which correlates the electrons of the is dominated by a mixture of the fundamental configuration, fourth bond,there are also two negatively signed configurations, and a smaller and negatively signed contribution from the doubly which correlate the electrons in the two internal bonds,for excited one,p (in which two electrons that populate the weakly example 120221m)and 1202201m1),which can be antibonding o orbital in o now populate the previously vacant combined with the fundamental configuration,as in equation(3), bonding or orbital in )For C,,these are the 20,and 30 orbitals to generate a TC wavefunction with two -GVB pairs.The depicted in Fig.la,whereas for heteronuclear diatomics such as only bond that is not correlated in this manner by the FCI CN+,these are 4o and 5o,which are the analogous weakly wavefunction is the internal o(C-C)bond,which is rather well antibonding and weakly bonding orbitals,respectively.These two described by the doubly occupied 2o orbital,and its corresponding configurations constitute ~80%of the total weight of the FCI di-excited configuration is too high to mix appreciably into the wavefunction.The rest of the configurations have much smaller CI wavefunction.Figure 5a,b depicts the GVB pairs for weights and we shall deal with their significance later.Therefore, the in-plane and out-of-plane bonds,and Fig.5c shows the to a first approximation,the FCI wavefunction can be written in internal occ bond represented by the 20 orbital.As such,together terms of the two leading configurations with corresponding with the pair of the fourth bond (Fig.4a),we have a quad- coefficients Co and Cp.For example,for C2 we have the following ruple bond in C.. wavefunction whereΦ。andΦare expressed in their Slater- The same applies to all four first-row molecules studied here determinant representations: (Supplementary Sections II.3.4-II.3.6).Thus,because GVB bond pairs are by definition mixtures of covalent and ionic structures Ψa=C2o112a2元.-Cn2c11)3cg3og+ (Fig.2a)5051,the FCI and VB descriptions of C2,CN+,BN and CB are in fact completely equivalent;both pictures view these (2) molecules as quadruply bonded species.The fourth bond is thus established herein by two independent and high-level compu- where Co=0.828,Cp=0.324.Here,the orbital terms in parenth- tational procedures.Quadruple bonding is indeed possible in first- eses correspond to the closed-shell part of the two configurations, row main-group elements (H.S.Rzepa,www.ch.imperial.ac.uk/ consisting of the filled 20.and doubly degenerate 17 orbitals rzepa/blog/?p=3065). (Fig.1),written schematically.On the other hand,the part that undergoes a change fromΦ。toΦis written explicitly,with the Conclusions bar over the orbital indicating spin B,and the lack of bar indicating We have shown herein,by a combination of FCI and VB calcu- spin a. lations,that the ground or low-lying+singlet states of the mol- Taking now the leading two configurations(TC)and dropping ecules C2,CN+,BN and CB are all quadruply bonded,having the normalization constant,we obtain the following wavefunction: three internal bonds (one o and two and one weak 'inverted' C-C bond.The intrinsic bonding energy of the fourth bond is Ψc=2c11m)202元.-22121)3og3元g (3)bracketed in the range 12-17 kcal mol,which is much stronger than a hydrogen bond,and is certainly stronger than the 6 and where A2=Cp/Co=0.6255.As in the textbook example of the TC bonds in dimers of transition metals and lanthanides/actinides. wavefunction for H2 (ref.26,pp.42 and 241,and refs 50,51)the rc As such,it is a bond,as depicted in 3 in Fig.1b.Thus,our study in equation (3)can also be transformed to a generalized valence shows that quadruple bonding also exists in main group element bond (GVB)wavefunction (Supplementary Sections II.3),with chemistry.Other species that are likely to exhibit quadruple two singly occupied orbitals that are spin-paired to a bond,as bonding include,for example,N2+,NO3+and BO+ illustrated in Fig.3.Thus,combining and subtracting the 2o and One may wonder what might be the experimental manifestations 30,orbitals,the TC wavefunction remains invariant,and we have of the fourth bond?A lack of radical reactivity relative to genuine two semi-localized orbitals.The so-called is localized at the radical or diradical species is perhaps one feature,but there may NATURE CHEMISTRY ADVANCE ONLINE PUBLICATION www.nature.com/naturechemistry 2012 Macmillan Publishers Lmited All rights reserved
configurations that display one lesspbond that theX1 Sg þ ground state. These factors can flatten the ground state near the equilibrium geometry but will not affect the well depth, and as such will lower the RFC of C2 relative to acetylene, despite the quadruple bonding in the former. Nature of the fourth bond in C2, CN1, BN and CB– revealed by full CI. The fourth bond can be easily understood using VB language as a hybrid of covalent and ionic structures (Fig. 2a), similarly to any other bond14,27. Although the VB mechanism is straightforward, one may wonder what FCI tells us about the nature of the fourth bond? Inspection of the FCI results for the four molecules that exhibit quadruple bonding reveals that, in all of them, the FCI wavefunction is dominated by a mixture of the fundamental configuration, F0, and a smaller and negatively signed contribution from the doubly excited one, FD (in which two electrons that populate the weakly antibonding s orbital in F0 now populate the previously vacant bonding s orbital in FD). For C2, these are the 2su and 3sg orbitals depicted in Fig. 1a, whereas for heteronuclear diatomics such as CNþ, these are 4s and 5s, which are the analogous weakly antibonding and weakly bonding orbitals, respectively. These two configurations constitute 80% of the total weight of the FCI wavefunction. The rest of the configurations have much smaller weights and we shall deal with their significance later. Therefore, to a first approximation, the FCI wavefunction can be written in terms of the two leading configurations with corresponding coefficients C0 and CD. For example, for C2 we have the following wavefunction where F0 and FD are expressed in their Slaterdeterminant representations: CFCI = C0 (2s2 g1p2 u1p2 u)2su2su − CD (2s2 g1p2 u1p2 u)3sg3sg + ... (2) where C0 ¼ 0.828, CD ¼ 0.324. Here, the orbital terms in parentheses correspond to the closed-shell part of the two configurations, consisting of the filled 2sg and doubly degenerate 1pu orbitals (Fig. 1), written schematically. On the other hand, the part that undergoes a change from F0 to FD is written explicitly, with the bar over the orbital indicating spin b, and the lack of bar indicating spin a. Taking now the leading two configurations (TC) and dropping the normalization constant, we obtain the following wavefunction: CTC = (2s2 g1p2 u1p2 u)2su2su − l2 (2s2 g1p2 u1p2 u)3sg3sg (3) where l2 ¼ CD/C0 ¼ 0.6255. As in the textbook example of the TC wavefunction for H2 (ref. 26, pp. 42 and 241, and refs 50,51) the CTC in equation (3) can also be transformed to a generalized valence bond (GVB) wavefunction (Supplementary Sections II.3), with two singly occupied orbitals that are spin-paired to a bond, as illustrated in Fig. 3. Thus, combining and subtracting the 2su and 3sg orbitals, the TC wavefunction remains invariant, and we have two semi-localized orbitals. The so-called fL is localized at the left-hand carbon atom, with a smaller tail on the right-hand carbon, whereas the other, fR, is localized on the right-hand carbon and has a tail on the left-hand atom50,51. The electrons in the fL and fR orbitals are singlet-paired, which in Fig. 3 is symbolized by the dotted line connecting the two singly occupied orbitals. The two transformed orbitals for all the molecules are shown in Fig. 4 together with S, a measure of their overlap. The larger the overlap in the GVB pairs51, the stronger the respective bond. The significant overlaps in Fig. 4 underscore the conclusion that the bond energies of the fourth bond are significant in all these cases. In fact, it is easily seen that the CTC wavefunction describes a quadruple bond. Merely reading equation (2) reveals that the C2 molecule has an internal triple bond, composed of three strongly bonding orbitals populated by six electrons, 2sg 2 1pu 4 and a fourth bond made from the transformed fL–fR GVB pair. The same picture is true for the other molecules; they all have an internal triple bond made from the 2s2 1p4 sub-shell, which is augmented by a GVB pair fL–fR. By common knowledge (ref. 26, pp. 42 and 241, refs 50,51) the GVB wavefunction of a pair like fL–fR is equivalent to the localized VB picture in Fig. 2a. In fact, the FCI wavefunction describes all the other bonds in the same manner (Supplementary Sections II.3.2 and II.3.3). Thus, in addition to FD, which correlates the electrons of the fourth bond, there are also two negatively signed configurations, which correlate the electrons in the two internal p bonds, for example |2sg 2 2su 2 1pux 2 1pgy 2 l and |2sg 2 2su 2 1puy 2 1pgx 2 l, which can be combined with the fundamental configuration, as in equation (3), to generate a TC wavefunction with two p–GVB pairs. The only bond that is not correlated in this manner by the FCI wavefunction is the internal s(C–C) bond, which is rather well described by the doubly occupied 2sg orbital, and its corresponding di-excited configuration is too high to mix appreciably into the CI wavefunction. Figure 5a,b depicts the fL–fR GVB pairs for the in-plane and out-of-plane p bonds, and Fig. 5c shows the internal sCC bond represented by the 2sg orbital. As such, together with the fL–fR pair of the fourth bond (Fig. 4a), we have a quadruple bond in C2. The same applies to all four first-row molecules studied here (Supplementary Sections II.3.4–II.3.6). Thus, because GVB bond pairs are by definition mixtures of covalent and ionic structures (Fig. 2a)50,51, the FCI and VB descriptions of C2, CNþ, BN and CB2 are in fact completely equivalent; both pictures view these molecules as quadruply bonded species. The fourth bond is thus established herein by two independent and high-level computational procedures. Quadruple bonding is indeed possible in firstrow main-group elements (H. S. Rzepa, www.ch.imperial.ac.uk/ rzepa/blog/?p=3065). Conclusions We have shown herein, by a combination of FCI and VB calculations, that the ground or low-lying 1 Sþ singlet states of the molecules C2, CNþ, BN and CB2 are all quadruply bonded, having three internal bonds (one s and two p) and one weak ‘inverted’ C–C bond. The intrinsic bonding energy of the fourth bond is bracketed in the range 12–17 kcal mol21 , which is much stronger than a hydrogen bond, and is certainly stronger than the d and f bonds in dimers of transition metals and lanthanides/actinides. As such, it is a bond, as depicted in 3 in Fig. 1b. Thus, our study shows that quadruple bonding also exists in main group element chemistry. Other species that are likely to exhibit quadruple bonding include, for example, N2 2þ, NO3þ and BOþ. One may wonder what might be the experimental manifestations of the fourth bond? A lack of radical reactivity relative to genuine radical or diradical species is perhaps one feature, but there may a b c S= ϕL | ϕR = 0.7749 S= ϕL | ϕR = 0.7749 Figure 5 | fL–fR GVB orbital pairs and their overlap S values for the internal bonds in C2. a–c, p out-of-plane (a), p in-plane (b) and internal s bond (c) represented by the 2sg molecular orbital. ARTICLES NATURE CHEMISTRY DOI: 10.1038/NCHEM.1263 4 NATURE CHEMISTRY | ADVANCE ONLINE PUBLICATION | www.nature.com/naturechemistry © 2012 Macmillan Publishers Limited. All rights reserved.
NATURE CHEMISTRY DOI:10.1038/NCHEM.1263 ARTICLES be others.Here,in articulating such a fundamental feature of chemi- 15.Weltner,W.Ir van Zee,R.I.Carbon molecules,ions,and clusters.Chem.Rev. cal bonding,we hope to promote the search for other 89,1713-1747(1989). experimental manifestations. 16.Boggio-Pasqua,M.,Voronin,A.I.,Halvick,P.Rayez,J-C.Analytical representations of high level ab initio potential energy curves of the C,molecule. Note added in proof:The authors became aware of a further relevant I.Mol.Struct..531,159-167(2000). 17.Abrams,M.L.Sherrill,C.D.Full configuration interaction potential energy paper that they would like to cite:Schleyer,P.v.R.,Maslak,P., curves for theX'∑t,B'△，andB'Σstates of C:a challenge for approximate Chandrasekhar,J.Grev,R.Is a CC quadruple bond possible? methods.J.Chem.Phs.121,9211-9219(2004). 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The VB calculations were carried out for C2.CN,BN and CB at the 26.Shaik,S.S.Hiberty,P.C.A Chemist's Guide to Valence Bond Theory,49-51 VBSCF/6-31G levell4 using the XMVB packages4.The VB structure set includes,as (John-Wiley Sons,2008). before,92 structures,of which 21 involve four electron pairs that all make bonds 27.Shaik,S Danovich,D.Wu,W.&Hiberty,P.C.Charge-shift bonding and its between the atoms(Supplementary Sections I and Scheme S.1).The VBSCF/6-31G manifestations in chemistry.Nafure Chem.1,443-449 (2009). calculations for each species involved the QC reference state as well as the 28.Shaik,S.S.A qualitative valence bond approach to organic reactions,in New Σt(XΣt)andΣt(c3∑)states,all carried out at the FCI/6-31G Theoretical Concepts for Understanding Organic Reactions (eds Bertran,J. optimized bond lengths for the corresponding singlet states(see Csizmadia,G.I.)NATO ASI Series C267(Kluwer Publishers,1989). 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be others. Here, in articulating such a fundamental feature of chemical bonding, we hope to promote the search for other experimental manifestations. Note added in proof: The authors became aware of a further relevant paper that they would like to cite: Schleyer, P. v. R., Maslak, P., Chandrasekhar, J. & Grev, R. Is a CC quadruple bond possible? Tetrahedron Lett. 34, 6387–6390 (1993). Methods The 1 Sþ and 3 Sþ states (X1 Sg þ and c 3 Su þ for homonuclear diatomics) of all the molecules (C2, Si2, Ge2, CNþ, BN and CB2) were calculated at the FCI/6-31G* level using the package MOLPRO-2010.1 (ref. 52). The FCI procedure excluded the core electrons and included 2 × 108 determinants. Force constants (f, in N cm21 ) for both singlet C2 and HCCH were calculated using the equation f = 4p2 c 2 y2 m m = m1 · m2 m1 + m2 where the frequency values y (in cm21 ) were obtained with MOLPRO using MRCI/6-31G*. RFC values were calculated using the Compliance program53 using as an input the results from the Gaussian-03 CCSD(T)/6-31G* calculations (where CCSD(T) is ‘coupled cluster including singles and double with perturbative triples’). MRCI/6-31G* calculations of the intrinsic bonding energy were carried out using CASSCF reference configurations (CASSCF is the ‘complete active space self-consistent field’). For C2 it was found that using two leading configurations in the FCI wavefunction as a basis for MRCI leads to results on a par with FCI. The VB calculations were carried out for C2, CNþ, BN and CB2 at the VBSCF/6-31G* level14 using the XMVB package54. The VB structure set includes, as before, 92 structures, of which 21 involve four electron pairs that all make bonds between the atoms (Supplementary Sections I and Scheme S.1). 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ARTICLES NATURE CHEMISTRY DOI:10.1038/NCHEM.1263 47.Tzeli,D.Mavridis,A.First-principles investigation of the boron and 54.Song.L,Wu,W.,Mo,Y.Zhang.Q.XMVB:an ab initio non-orthogonal aluminum carbides BC and AlC and their anions BC-and AlC-.1.J.Phys. valence bond program (Xiamen University,China,2003). Chem.A105,1175-1184(2001). 48.Grimme,S.Semiempirical hybrid density functional with perturbative second- Author contributions order correlation.J.Chem.Phys.124,034108 (2006). S.S.designed the project,analysed the FCI wavefunctions and wrote the paper. 49.Brandhorst,K.Grunenberg.I.How strong is it?The interpretation of force D.D.performed the VB,MRCI,FCI and bond order calculations.W.W.designed the and compliance constants as bond strength descriptors.Chem.Soc.Rev.37, initial VB calculations of C2.P.S.performed the initial set of VB calculations for C2. 1558-1567(2008). P.C.H.participated in the design of the VB determination of D,in the analysis 50.Coulson,C.A.Fischer,I.Notes on the molecular orbital treatment of the of the FCI wavefunctions,and contributed to writing the manuscript.H.R. hydrogen molecule.Phil Mag.Series 7 40,386-393 (1949). initiated interest in the problem,and explored probes for characterizing the 51.Goddard III,W.A.Harding.L.B.The description of chemical bonding from bonding properties. ab initio calculations.Annu.Rev.Phys.Chem.29,363-396(1978). 52.Werner,H-J.et al MOLPRO,version 2010.1 (University College Cardiff Additional information Consultants Limited,UK). The authors declare no competing financial interests.Supplementary information 53.Brandhorst,K.Grunenberg.J.Efficient computation of compliance matrices accompanies this paper at www.nature.com/naturechemistry.Reprints and permission in redundant internal coordinates from cartesian hessians for nonstationary information is available online at http://www.nature.com/reprints.Correspondence and points.1Chem.Phs.132,184101(2010). requests for materials should be addressed to S.S. 6 NATURE CHEMISTRY ADVANCE ONLINE PUBLICATION www.nature.com/naturechemistry 2012 Macmillan Publishers Limited All rights reserved
47. Tzeli, D. & Mavridis, A. First-principles investigation of the boron and aluminum carbides BC and AlC and their anions BC2 and AlC2. 1. J. Phys. Chem. A 105, 1175–1184 (2001). 48. Grimme, S. Semiempirical hybrid density functional with perturbative secondorder correlation. J. Chem. Phys. 124, 034108 (2006). 49. Brandhorst, K. & Grunenberg, J. How strong is it? The interpretation of force and compliance constants as bond strength descriptors. Chem. Soc. Rev. 37, 1558–1567 (2008). 50. Coulson, C. A. & Fischer, I. Notes on the molecular orbital treatment of the hydrogen molecule. Phil. Mag. Series 7 40, 386–393 (1949). 51. Goddard III, W. A. & Harding, L. B. The description of chemical bonding from ab initio calculations. Annu. Rev. Phys. Chem. 29, 363–396 (1978). 52. Werner, H-J. et al. MOLPRO, version 2010.1 (University College Cardiff Consultants Limited, UK). 53. Brandhorst, K. & Grunenberg, J. Efficient computation of compliance matrices in redundant internal coordinates from cartesian hessians for nonstationary points. J. Chem. Phys. 132, 184101 (2010). 54. Song, L., Wu, W., Mo, Y. & Zhang, Q. XMVB: an ab initio non-orthogonal valence bond program (Xiamen University, China, 2003). Author contributions S.S. designed the project, analysed the FCI wavefunctions and wrote the paper. D.D. performed the VB, MRCI, FCI and bond order calculations. W.W. designed the initial VB calculations of C2. P.S. performed the initial set of VB calculations for C2. P.C.H. participated in the design of the VB determination of Din, in the analysis of the FCI wavefunctions, and contributed to writing the manuscript. H.R. initiated interest in the problem14, and explored probes for characterizing the bonding properties. Additional information The authors declare no competing financial interests. Supplementary information accompanies this paper at www.nature.com/naturechemistry. Reprints and permission information is available online at http://www.nature.com/reprints. Correspondence and requests for materials should be addressed to S.S. ARTICLES NATURE CHEMISTRY DOI: 10.1038/NCHEM.1263 6 NATURE CHEMISTRY | ADVANCE ONLINE PUBLICATION | www.nature.com/naturechemistry © 2012 Macmillan Publishers Limited. All rights reserved