The Hartree–Fock instability of the SCF wave functions obtained for the quadruply bonded complexes Cr2(O2CH)4 and Mo2(O2CH)4 is studied at the abinitio level. Several cases of singlet, nonsinglet, and nonreal instabilities are found for each of these complexes at the experimental value of the metal–metal bond length. For Cr2(O2CH)4, these instabilities are still found at much shorter Cr–Cr distances. The ’’broken symmetry’’ wave functions corresponding to some of the singlet and nonsinglet instability roots of Cr2(O2CH)4 are computed and analyzed. Each of these broken‐symmetry wave functions is characterized by an important decrease of the bonding character of the metal–metal interaction with respect to the symmetry‐adapted wave function. Singlet instability can thus be considered as an attempt to reduce the strong bonding character imposed by the symmetry constraints to the whole class of D4h quadruply bonded complexes. The case of MoCr(O2CH)4 for which the symmetry of the system is broken from D4h to C4v, is analyzed. An examination of the singlet instability matrices As+Bs obtained for Cr2(O2CH)4 and Mo2(O2CH)4 suggests that the presence of a multiple metal–metal bond strongly favors instability. A single determinant Hartree–Fock wave function might lead in such a case to an erroneous description of metal–metal multiply bonded systems. For this class of molecules, singlet stability equations could provide a simple and accurate procedure to check the reliability of Hartree–Fock wave functions.

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34.
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From the sextuple bond postulated for the naked binuclear cluster from the results of extended Hückel36 and SCF‐‐SW37 calculations performed upon Mo2 to the nonbonding interaction proposed for some binuclear complexes of chromium.38
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In the case of a single σ bond, the characteristic matrix is reduced to the diagonal term related to the σ→σ* replacement [submatrix (a2ub2u)]. This term is positive due to the high value of the Kσσ* integral33 and cannot thus yield instability (Tables IV and V).
40.
The trend towards singlet instability also depends, of course, on the symmetry group of the system. However, the present analysis is valid for every symmetry group involving at least a symmetry plane perpendicular to the metal‐metal axis. The presence of such a plane implies the existence of a singlet instability submatrix analagous to the (a2u,b2u) submatrix obtained for D4h systems (Tables IV and V).
41.
A C4v wave function is obtained from a D4h one by elimination of the constraints associated with symmetry relative to the plane perpendicular to the metal‐metal axis. The D2d wave function associated with the (b2u) subproblem is obtained from further elimination of the constraints generated by the symmetry planes σv bisecting the dihedral angles of the ligand planes and by the rotation axes C2 coplanar to the ligand planes. If [i], i = 1,2 designates one metal atom, the dz2[1] and dxy[1] orbitals can still be transformed into dz2[2] and dxy[2], respectively, by the C2 rotations. However, dxz[1] is no longer connected neither to dyz[1] nor to dxz[2], but only to dyz[2] by the S4 operations. Similarly, dyz[1] is only connected to dxz[2]. The nature of the distortions in the b.s. wave functions of other symmetry is more difficult to visualize since it cannot be expressed easily in terms of localized orbitals. Some insight can however be obtained by mixing the two D4h orbitals involved in some orbital replacements of the considered subproblem for instance, σ and π* or δ and π̄* in the eu subproblem; σ and δ* in the b1u subproblem; π and π̄* in the a1u subproblem.
42.
A different localization scheme for the spin‐polarized d electrons is likely to be found for every UHF b.s. wave function generated by the (a2u,b2u) subproblem, and can be deduced in first approximation from the appropriate eigenvectors of the stability matrix. For instance, the wave function associated with the instability root −0.4035(a2u) leads to the following distribution om Cr1:dxy1(α),dz12(α),dxz1(β),dyz1(β). For the solution associated with the root −0.4030(b2u), the localization only holds on the dxz and dyz electrons, thus yielding the following distributions on Cr1:dxy0.5(α),dxy0.5(β),dz0.52(α),dz0.52(β),dxz1(α),dyz1(β). Distributions one Cr2 are obtained by permuting α and β. In the case of nonsinglet instabilities arising from other submatrices, orbital replacements do not involve bonding and antibonding orbitals localized in the same region of space, so that the spin‐polarized d electrons are equally distributed between Cr1 and Cr2.
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This result was quite unexpected when obtained10 and had not been related at that time to the failure of the Hartree‐Fock approximation in the case of Cr2(O2CH)4.
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For Cr2(O2CH)4, the second b.s. C4v solution exists for distances lower than 1.60 Å (Table I). For MO2(O2CH)4, however, it has already vanished at 2.09 Å, so that it is not possible to predict with certainty the existence of such a solution for the experimental conformation of MoCr (O2CH)4.
48.
This important effect of electron repulsion is itself connected to the “electron‐crowded” character of multiply bonded systems. In order to check wh́ether electron crowded organic systems could similarly display singlet instability in the region of their equilibrium geometry, the stability equations have been applied to acetylene and HCN.52 These systems appeared to be singlet stable for C‐C and C‐N distances lower than ∼1.60 Å, which is rather far from the equilibrium positions (dCC = 1.204 Å for C2H2 and dCN = 1.153 Å for HCN). Thus, the HF instability obtained for multiply bonded M‐M complexes at the equilibrium position seems to be specific of d‐type orbitals. However, a similarity between “electron‐crowded” organic and bimetallic systems might exist to some extent, since Bagus et al. noticed that, in the correlation energy of C2H2 and HCN, the weight of configurations higher than singly and doubly excited is probably unusually high.53 It has been shown already that some of these configurations are of crucial importance for a correct description of the quadruply bonded binuclear complexes.10,24
49.
R.
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37
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A similar result is obtained from SCF‐‐SW calculations51 though this method takes account of electron repulsion.
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52.
M. Bénard (unpublished results).
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P. S.
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63
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54.
The idea and part of the matter for this discussion were suggested by a referee.
55.
E(UHF) = ‐2834.652 a.u.,E(CI) = ‐2834.401 a.u.
56.
L. Noodleman and J. G. Norman, Jr., J. Chem. Phys. (in press).
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