Sextuple bond

A sextuple bond is a type of covalent bond involving 12 bonding electrons and in which the bond order is 6. The only known molecules with true sextuple bonds are the diatomic dimolybdenum (Mo₂) and ditungsten (W₂), which exist in the gaseous phase and have boiling points of 4,639 °C (8,382 °F) and 5,930 °C (10,710 °F) respectively.

Theoretical analysis

Roos et al argue that no stable element can form bonds of higher order than a sextuple bond, because the latter corresponds to a hybrid of the s orbital and all five d orbitals, and f orbitals contract too close to the nucleus to bond in the lanthanides.^[1] Indeed, quantum mechanical calculations have revealed that the dimolybdenum bond is formed by a combination of two σ bonds, two π bonds and two δ bonds. (Also, the σ and π bonds contribute much more significantly to the sextuple bond than the δ bonds.)^[2]

Although no φ bonding has been reported for transition metal dimers, it is predicted that if any sextuply-bonded actinides were to exist, at least one of the bonds would likely be a φ bond as in quintuply-bonded diuranium and dineptunium.^[3] No sextuple bond has been observed in lanthanides or actinides.^[1]

For the majority of elements, even the possibility of a sextuple bond is foreclosed, because the d electrons ferromagnetically couple, instead of bonding. The only known exceptions are dimolybdenum and ditungsten.^[1]

Quantum-mechanical treatment

The formal bond order (FBO) of a molecule is half the number of bonding electrons surplus to antibonding electrons; for a typical molecule, it attains exclusively integer values. A full quantum treatment requires a more nuanced picture, in which electrons may exist in a superposition, contributing fractionally to both bonding and antibonding orbitals. In a formal sextuple bond, there would be $P = 6$ different electron pairs; an effective sextuple bond would then have all six contributing almost entirely to bonding orbitals.

Molecule	FBO	EBO^[1]
Cr₂	6	3.5
[PhCrCrPh]	5	3.5
Cr₂(O₂CCH₃)₄	4	2.0
Mo₂	6	5.2
W₂	6	5.2
Ac₂	3	1.7
Th₂	4	3.7
Pa₂	5	4.5
U₂	6	3.8^[4]
[PhUUPh]	5	3.7
[Re₂Cl₈]^2-	4	3.2

In Roos et al's calculations, the effective bond order (EBO) could be determined by the formula $EBO=\left({\frac {1}{2}}\right)\sum _{p=1}^{P}(\eta _{b,p}-\eta _{ab,p})-c$ where $η b$ is the proportion of formal bonding orbital occupation for an electron pair $p$ , $η ab$ is the proportion of the formal antibonding orbital occupation, and $c$ is a correction factor accounting for deviations from equilibrium geometry.^[1] Several metal-metal bonds' EBOs are given in the table at right, compared to their formal bond orders.

Dimolybdenum and ditungsten are the only molecules with effective bond orders above 5, with a quintuple bond and a partially formed sixth covalent bond. Dichromium, while formally described as having a sextuple bond, is best described as a pair of chromium atoms with all electron spins exchange-coupled to each other.^[5]

While diuranium is also formally described as having a sextuple bond, relativistic quantum mechanical calculations have determined it to be a quadruple bond with four electrons ferromagnetically coupled to each other rather than in two formal bonds.^[4] Previous calculations on diuranium did not treat the electronic molecular Hamiltonian relativistically and produced higher bond orders of 4.2 with two ferromagnetically coupled electrons.^[6]

Known instances: dimolybdenum and ditungsten

Laser evaporation of a molybdenum sheet at low temperatures (7 K) produces gaseous dimolybdenum (Mo₂). The resulting molecules can then be imaged with, for instance, near-infrared spectroscopy or UV spectroscopy.^[7]

Both ditungsten and dimolybdenum have very short bond lengths compared to neighboring metal dimers.^[1] For example, sextuply-bonded dimolybdenum has an equilibrium bond length of 1.93 Å. This equilibrium internuclear distance is significantly lower than in the dimer of any neighboring 4d transition metal, and suggestive of higher bond orders.^[8]^[9]^[10] However, the bond dissociation energies of ditungsten and dimolybdenum are rather low, because the short internuclear distance introduces geometric strain.^[1]^[11]

Dimer	Force constant (Å)^[10]	EBO^[10]
Cu₂	1.13	1.00
Ag₂	1.18	1.00
Au₂	2.12	1.00
Zn₂	0.01	0.01
Cd₂	0.02	0.02
Hg₂	0.02	0.02
Mn₂	0.09	0.07
Mo₂	6.33	5.38

One empirical technique to determine bond order is spectroscopic examination of bond force constants. Linus Pauling investigated the relationships between bonding atoms^[12] and developed a formula that predicts that bond order is roughly^[13] proportional to the force constant; that is, $k_{e}=n\cdot k_{e}^{(1)}$ where $n$ is the bond order, $k e$ is the force constant of the interatomic interaction and $k e (1)$ is the force constant of a single bond between the atoms.^[14]

The table at right shows some select force constants for metal-metal dimers compared to their EBOs; consistent with a sextuple bond, molybdenum's summed force constant is substantially more than quintuple the single-bond force constant.

Like dichromium, dimolybdenum and ditungsten are expected to exhibit a ¹Σ_g⁺ singlet ground state.^[15]^[16] However, in tungsten, this ground state arises from a hybrid of either two ⁵D₀ ground states or two ⁷S₃ excited states. Only the latter corresponds to the formation of a stable, sextuply-bonded ditungsten dimer.^[8]

Ligand effects

Although sextuple bonding in homodimers is rare, it remains a possibility in larger molecules.

Aromatics

Theoretical computations suggest that bent dimetallocenes have a higher bond order than their linear counterparts.^[17] For this reason, the Schaefer lab has investigated dimetallocenes for natural sextuple bonds. However, such compounds tend to exhibit Jahn-Teller distortion, rather than a true sextuple bond.

For example, dirhenocene is bent. Calculating its frontier molecular orbitals suggests the existence of relatively stable singlet and triplet states, with a sextuple bond in the singlet state. But that state is the excited one; the triplet ground state should exhibit a formal quintuple bond.^[17] Similarly, for the dibenzene complexes Cr₂(C₆H₆)₂, Mo₂(C₆H₆)₂, and W₂(C₆H₆)₂, molecular bonding orbitals for the triplet states with symmetries D_6h and D_6d indicate the possibility of an intermetallic sextuple bond. Quantum chemistry calculations reveal, however, that the corresponding D_2h singlet geometry is stabler than the D_6h triplet state by 3–39 kcal/mol, depending on the central metal.^[18]

Oxo ligands

Both quantum mechanical calculations and photoelectron spectroscopy of the tungsten oxide clusters W₂O_n (n = 1-6) indicate that increased oxidation state reduces the bond order in ditungsten. At first, the weak δ bonds break to yield a quadruply-bonded W₂O; further oxidation generates the ditungsten complex W₂O₆ with two bridging oxo ligands and no direct W-W bonds.^[19]

References

^ ^a ^b ^c ^d ^e ^f ^g Roos, Björn O.; Borin, Antonio C.; Laura Gagliardi (2007). "Reaching the Maximum Multiplicity of the Covalent Chemical Bond". Angewandte Chemie International Edition. 46 (9): 1469–72. doi:10.1002/anie.200603600. PMID 17225237.
^ Bursten, Bruce E.; Cotton, F. Albert; Hall, Michael B. (September 1980). "Dimolybdenum: nature of the sextuple bond". Journal of the American Chemical Society. 102 (20): 6348–6349. doi:10.1021/ja00540a034. ISSN 0002-7863.
^ Bursten, Bruce E.; Ozin, Geoffrey A. (August 1984). "X.alpha.-SW calculations for naked actinide dimers: existence of .vphi. bonds between metal atoms". Inorganic Chemistry. 23 (18): 2910–2911. doi:10.1021/ic00186a039. ISSN 0020-1669.
^ ^a ^b Knecht, Stefan; Jensen, Hans Jørgen Aa.; Saue, Trond (January 2019). "Relativistic quantum chemical calculations show that the uranium molecule U2 has a quadruple bond" (PDF). Nature Chemistry. 11 (1): 40–44. Bibcode:2019NatCh..11...40K. doi:10.1038/s41557-018-0158-9. ISSN 1755-4330. PMID 30374039. S2CID 53112083.
^ Goodgame, Marvin M.; Goddard, William A. (February 1981). "The "sextuple" bond of chromium dimer". The Journal of Physical Chemistry. 85 (3): 215–217. doi:10.1021/j150603a001. ISSN 0022-3654.
^ Gagliardi, Laura; Roos, Björn O. (2005-05-17). "Quantum Chemical Calculations Show that the Uranium Molecule U2 Has a Quintuple Bond". ChemInform. 36 (20): 848. Bibcode:2005Natur.433..848G. doi:10.1002/chin.200520001. ISSN 0931-7597.
^ Kraus, D.; Lorenz, M.; Bondybey, V. E. (2001). "On the dimers of the VIB group: a new NIR electronic state of Mo₂". PhysChemComm. 4 (10): 44–48. doi:10.1039/b104063b.
^ ^a ^b Borin, Antonio Carlos; Gobbo, João Paulo; Roos, Björn O. (April 2010). "Electronic structure and chemical bonding in W2 molecule". Chemical Physics Letters. 490 (1–3): 24–28. Bibcode:2010CPL...490...24B. doi:10.1016/j.cplett.2010.03.022. ISSN 0009-2614.
^ Efremov, Yu.M; Samoilova, A.N; Kozhukhovsky, V.B; Gurvich, L.V (December 1978). "On the electronic spectrum of the Mo2 molecule observed after flash photolysis of Mo(CO)6". Journal of Molecular Spectroscopy. 73 (3): 430–440. Bibcode:1978JMoSp..73..430E. doi:10.1016/0022-2852(78)90109-1. ISSN 0022-2852.
^ ^a ^b ^c Jules, Joseph L.; Lombardi, John R. (March 2003). "Transition Metal Dimer Internuclear Distances from Measured Force Constants". The Journal of Physical Chemistry A. 107 (9): 1268–1273. Bibcode:2003JPCA..107.1268J. doi:10.1021/jp027493+. ISSN 1089-5639.
^ Joy, Jyothish; Jemmis, Eluvathingal D. (2017). "A halogen bond route to shorten the ultrashort sextuple bonds in Cr2 and Mo2". Chemical Communications. 53 (58): 8168–8171. doi:10.1039/c7cc04653g. ISSN 1359-7345. PMID 28677703. S2CID 206066221.
^ Hardcastle, F. D. (2016-01-01). "A General Valence-Length Correlation for Determining Bond Orders: Application to Carbon-Carbon and Carbon-Hydrogen Chemical Bonds". Journal of the Arkansas Academy of Science. 70. doi:10.54119/jaas.2016.7009. ISSN 2326-0505.
^ Lombardi, John R.; Davis, Benjamin (2002-06-01). "Periodic Properties of Force Constants of Small Transition-Metal and Lanthanide Clusters". Chemical Reviews. 102 (6): 2431–2460. doi:10.1021/cr010425j. ISSN 0009-2665. PMID 12059275. Pauling showed that the force constant is approximately proportional to the bond order...Note that the term 'bond order' as used here is not the same as the usual chemical definition [i.e., 1/2(no. of bonding electrons - no. of antibonding electrons) or better a function of the electron density]. This might more accurately be termed the 'vibrational bond order' since it is experimentally determined.
^ Johnston, Harold S. (1966). Gas Phase Reaction Rate Theory. Ronald Press Company. ISBN 978-0-608-30060-3.
^ Merino, Gabriel; Donald, Kelling J.; D'Acchioli, Jason S.; Hoffmann, Roald (2007). "The Many Ways To Have a Quintuple Bond". J. Am. Chem. Soc. 129 (49): 15295–15302. doi:10.1021/ja075454b. PMID 18004851.
^ Borin, Antonio Carlos; Gobbo, João Paulo; Roos, Björn O. (January 2008). "A theoretical study of the binding and electronic spectrum of the Mo2 molecule". Chemical Physics. 343 (2–3): 210–216. Bibcode:2008CP....343..210B. doi:10.1016/j.chemphys.2007.05.028. ISSN 0301-0104.
^ ^a ^b Xu, Bing; Li, Qian-Shu; Xie, Yaoming; King, R. Bruce; Schaefer, Henry F. (2010-02-17). "Metal−Metal Quintuple and Sextuple Bonding in Bent Dimetallocenes of the Third Row Transition Metals". Journal of Chemical Theory and Computation. 6 (3): 735–746. doi:10.1021/ct900564p. ISSN 1549-9618. PMID 26613304.
^ Sun, Zhi; Schaefer, Henry F.; Xie, Yaoming; Liu, Yongdong; Zhong, Rugang (September 2013). "Does the metal–metal sextuple bond exist in the bimetallic sandwich compounds Cr2(C6H6)2, Mo2(C6H6)2, and W2(C6H6)2?†". Molecular Physics. 111 (16–17): 2523–2535. Bibcode:2013MolPh.111.2523S. doi:10.1080/00268976.2013.798434. ISSN 0026-8976. S2CID 94537427.
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