The Hartree–Fock instability of the SCF wave functions obtained for the quadruply bonded complexes Cr_{2}(O_{2}CH)_{4} and Mo_{2}(O_{2}CH)_{4} is studied at the *a**b* *i**n**i**t**i**o* 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 Cr_{2}(O_{2}CH)_{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 Cr_{2}(O_{2}CH)_{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 *D*_{4h} quadruply bonded complexes. The case of MoCr(O_{2}CH)_{4} for which the symmetry of the system is broken from *D*_{4h} to *C*_{4v}, is analyzed. An examination of the singlet instability matrices *A*^{s}+*B*^{s} obtained for Cr_{2}(O_{2}CH)_{4} and Mo_{2}(O_{2}CH)_{4} suggests that the presence of a *m**u**l**t**i**p**l**e* 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.

## REFERENCES

*Applications of Electronic Structure Theory*, edited by H. F. Sehaefer III (Plenum, New York, 1977).

*The Quantum Mechanics of Many‐Body Systems*(Academic, New York, 1961).

*a*denotes one of the σ, π, and δ orbitals and $a*$ the corresponding antibonding orbital.

*27*this result justifies the assumption made in a previous work

^{10}to perform CI over the bonding configuration, but

*not*over the nonbonding configuration, thus allowing the relative order of these two configurations to be reversed in terms of energy (Fig. 1).

^{3(a)}

^{10}These orders of magnitude were found to be similar for the more extended set of integrals of the type $(ii*|jj*)i\u2260j,$ where

*i*and

*j*can be δ, π, π̄, or σ. Since one of these integrals is involved four times in each off‐diagonal term of the considered $As+BS$ submatrix, their contribution is sufficient to explain its uncommon structure.

^{36}and SCF‐$X\alpha $‐SW

^{37}calculations performed upon $Mo2$ to the nonbonding interaction proposed for some binuclear complexes of chromium.

^{38}

^{33}and cannot thus yield instability (Tables IV and V).

*i*], $i\u2009=\u20091,2$ designates one metal atom, the $dz2[1]$ and $dxy[1]$ orbitals can still be transformed into $dz2[2]$ and $dxy[2],$ respectively, by the $C2\u2033$ 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 $\pi *$ or δ and $\pi \u0304*$ in the $eu$ subproblem; σ and $\delta *$ in the $b1u$ subproblem; π and $\pi \u0304*$ in the $a1u$ subproblem.

*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 $\u22120.4035(a2u)$ leads to the following distribution om $Cr1:$ $dxy1(\alpha ),$ $dz12(\alpha ),$ $dxz1(\beta ),$ $dyz1(\beta ).$ For the solution associated with the root $\u22120.4030(b2u),$ the localization only holds on the $dxz$ and $dyz$ electrons, thus yielding the following distributions on $Cr1:$ $dxy0.5(\alpha ),$ $dxy0.5(\beta ),$ $dz0.52(\alpha ),$ $dz0.52(\beta ),$ $dxz1(\alpha ),$ $dyz1(\beta ).$ 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 $Cr2.$

^{10}and had not been related at that time to the failure of the Hartree‐Fock approximation in the case of $Cr2$ $(O2CH)4.$

*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 $\u223c1.60\u2009\xc5,$ which is rather far from the equilibrium positions ($dCC\u2009=\u20091.204\u2009\xc5$ for $C2H2$ and $dCN\u2009=\u20091.153\u2009\xc5$ 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}

^{51}though this method takes account of electron repulsion.