Not Guilty on Every Count: The “Non‐Innocent” Nitrosyl Ligand in the Framework of IUPAC′s Oxidation‐State Formalism

Abstract Nitrosyl–metal bonding relies on the two interactions between the pair of N–O‐π* and two of the metal's d orbitals. These (back)bonds are largely covalent, which makes their allocation in the course of an oxidation‐state determination ambiguous. However, apart from M‐N‐O‐angle or net‐charge considerations, IUPAC′s “ionic approximation” is a useful tool to reliably classify nitrosyl metal complexes in an orbital‐centered approach.


Computations
Structure optimisation and numerical frequency analyses on the DFT level were performed by Orca 4.2.1. [2] Computational details for the cobaltate in 1 The recently published Ref. [4] deals with {CoNO} 8 species and their reduced forms. There, the ground state of a diamagnetic SPY-5 {CoNO} 8 complex was described as a singletbiradical with the true singlet 6.02 kcal mol −1 above the biradical (Table 1 of the reference).
For 1, this finding was checked in terms of a broken-symmetry approach. As a result, we found dependence on the method. With the GGA functional BP86 the BS approach fell back to the singlet, whereas the hybrid functional B3LYP ended up with a singlet-biradical ground state, some 6 kJ mol −1 more stable than the true singlet and an overlap of the corresponding MOs of Sαβ = 0.81.
The TBPY-5 isomer 1' is a local minimum on the singlet's potential energy surface, ca. 80 kJ mol −1 above the SPY-5 ground state in terms of a B97-D3-ZORA/def2-TZVP+CPCM(∞) calculation. Figure S1 shows the molecular structure. The wavenumber of the N-O stretch is 1739 cm −1 .  Figure S2 shows the frontier orbitals from CASSCF(8,7)/def2-TZVP+CPCM(∞) calculations on 1 and 1'. Figure S2. The MOs of the active space of CASSCF(8,7) calculations on 1 (right) and 1' (left), isovalue 0.06 a.u.; the population is given in parentheses, the arrows represent the ground state's leading 2222000 configuration (79 % contribution for both species). Note the higher antibond population for the π-bonds. Static correlation is larger for π-bonds due to the generally lower overlap compared to σ-bonds. The value for 4 was verified by its shift to 1467 cm −1 for 15 NO. The wavenumbers of the N-O stretches were taken for a [Fe(CN)5(NO)] 2− from Ref. [5] , for b [Fe(H2O)5(NO)] 2+ from Ref. [6] . The reference line is a fit according to Badger's rule applied on the free NO +/0/− species:  /cm −1 = 856.3 × (d/Å − 0.558) −3/2 . Calculated N-O distances were used for a better inclusion of {FeNO} 7 compounds. As explicated in Ref. [6] , experimental N-O distances often appear too short due to the slightly tilted NO-group's precession about the Fe-N axis (note the large ellopsoid of the O-atom in Figure 6). For the other species, d(exp) and d(calc) coincide within narrow limits. Structure optimisation and numerical frequency analyses were performed on the B97-D3-ZORA/def2-TZVP+CPCM(∞) level of theory for the metal-containing species, and in a CASSCF(all valence electrons, all valence orbitals) approach for the free NO −/0/+ species, i.e. CASSCF (10/11/12,8).

Details for
Right: force constants of the N-O bonds were extracted from the Hessian of the same calculation (using the orca_vib routine of Orca 4.2.1) as above as a function of the QTAIM charges of 1-4, 1', a, b; the line is a fit for the values of the free NO +/0/− species by means of a combined Badger-Gordy approach of the form f = a (q + b) 3/2 with a = 3.292 and b = 2.844. Badger's formula is referenced in the main text, for Gordy's fit see Ref. [7] . The QTAIM charges were calculated by means of Multiwfn, version 3.6. [8] The frontier orbitals of the vanadate 4 Figure S3 shows the frontier orbitals of the [V(NO)(tea)] − species 4. Note the marked extent of static correlation in terms of the antibond population. The frontier orbitals of the chromate 2 Figure S4 shows the frontier orbitals of the [Cr(fpin)2(NO)] 2− species 2. Note the marked extent of static correlation in terms of the antibond population. Figure S4. The frontier MOs of a CASSCF(5,7) calculation on 2, isovalue 0.06 a.u.; the population is given in parentheses, the arrows represent the ground state's leading 2210000 configuration (72 % contribution).

The spin population of the chromate 2
The spin values depend on the method. For the nitrosyl ligand in 2 we found:

Details for Scheme 2
As an approximation to the NO-character of a bond the gross population of the diatomic NO fragment was used.

A. Computational details
The wave function and electron densities of all systems have been computed at the BP86-D3(BJ)/def2-tzvp and CASSCF/def2-tzvp including CPCM(water) solvent effects using Gaussian09. [1] In the case of the perfluoro-ligands, the F atoms have been described with the def2-svp basis in the CASSCF calculation due to computational limitations.

B. Local spin analysis
Local spins can be extracted from singlet correlated wavefunctions, identifying the presence of effective unpaired electrons due to correlation. In this approach, the overall < S 2 > value (zero, in this case) is exactly decomposed into one-and two-center contributions as The < S 2 > AA terms account for the presence of local spin in the atom or fragment A, while the < S 2 > AB terms account for the effective spin-spin couplings between the local spins. They are positive if the local spins on A and B are parallel, and negative otherwise. The local spin formulation ensures zero local spins for restricted single-determinant wavefunction, so that the electron pairing in conventional bonds is clearly distinguished from antiferromagnetic interactions, even for a pure singlet without spin density.
In the case of a perfect diradical system, with perfectly localized radical centers on A and B, the expected local spin values would be < S 2 > AA = < S 2 > BB = ¾. That is, the < S 2 > value one would obtain for the isolated radical center. Since the overall < S 2 > is zero for a singlet, the diatomic spin contributions that would be obtained are < S 2 > AB = -3/4.
In the case of the Co-NO (1) and Co-NO (1´) species, the local spin analysis obtained from the CASSCF calculations show that only the Co and NO moieties exhibit meaningful local spin. The results obtained can be gathered on 2x2 matrices with the local spin values of Co and NO in the diagonal, and the diatomic spin coupling in the off-diagonal: The obtained local spins are significantly below the 0.75 value expected for a diradical. The analysis shows only moderate diradicaloid character of the CASSCF wavefunction, slightly larger for the co-linear 1' species. Nevertheless, the CASSCF description of the system does contain the partial diradicaloid character that KS-DFT methods detected via two (closed-shell and broken-symmetry) states close in energy.

C. Spin-resolved effective orbitals
Let us consider a system with n orthonormalized occupied molecular orbitals ) (r i   , i = 1, 2,.., n of a given spin case (alpha or beta), and a fuzzy division of the 3D-space (atom-in-molecule definition) into N at atomic domains  A defined e.g., by a continuous atomic weight function Let us for each atom A (A = 1, 2,..., N at ) form the n × n Hermitian matrix Q A with the elements The matrix Q A is essentially the net atomic overlap matrix in the basis of the molecular orbitals (MO-s) We diagonalize the Hermitian matrix Q A by the unitary matrix U A : It can be shown that every 0  A i  , as is the case for a proper overlap matrix. For each atom A we obtain where n A is the number of non-zero eigenvalues The occupation number of each effective atomic orbital (eff-AO) is given by the eigenvalues The sum of the occupation number of the n A eff-AOs is the net population of the atom A for the given spin case: Gross atomic populations associated to each eff-AOs can be derived from the atomic overlap of the localized MO-s (3) The thus defined gross populations of the eff-AOs add up to the total atomic population derived from the underlying atom-in-molecule method used. It is worth mentioning that in case a disjoint approach is used, such as QTAIM, the eff-AOs net and gross populations are fully equivalent, since in this particular case The shape and occupation number of the eff-AOs faithfully reproduce the core and valence shells of the atoms; those with occupation numbers close to 1 are associated to core orbitals or lone pairs, whereas those with smaller but significant occupation are identified with the atomic orbitals directly involved in the bonds.
The remaining eff-AO-s are marginally occupied and have no chemical significance. For most atoms the number of hybrids with significant occupation number always coincide with the classical minimal basis set, except for those that exhibit hypervalent character.
Notice that the eff-AOs and their occupation numbers can be obtained in the framework of 3D-space analysis even in the absence of an underlying atom-centered basis set, i.e., for plane wave calculations. [5] Another relevant aspect is that the eff-AOs can be easily obtained for any level of theory, provided a firstorder density matrix is available (in the case of Kohn-Sham DFT the latter is approximated by the usual Hartree-Fock-like expression). As noted by Mayer, [10] the eff-AOs of a given atom A can also be obtained from the diagonalization of the matrix PS A , where P is the LCAO density matrix and S A is the intra-atomic overlap matrix in the actual AO basis. This permits the straightforward generalization to correlated wave functions, from which the P matrix is usually available.

D. Effective oxidation states (EOS) analysis
The information provided by the eff-AOs and their occupation numbers is used to derive the most appropriate electron configuration of the atoms within the molecule. The integer electrons (alpha and beta, separately) are distributed among the atoms by comparing the occupations of the eff-AOs on different atoms, rather than independently rounding them to the nearest integer. Such strategy also underlines the fact that the OS depends on all atoms of the system and of course on the total number of electrons Moreover, when the number of atoms of the system is large, accidental pseudo degeneracies of the occupation numbers of the eff-AOs are likely to occur, which hinders the assignment of oxidation states. Note that one is usually interested in the oxidation state of the transition metal atoms and the formal charge of their ligands. Hence, a slightly more involved but more efficient strategy is a hierarchical approach, by which molecular fragments are defined before the eff-AO analysis in a first iteration. That is, instead of eff-AOs we obtain effective fragment orbitals (EFOs) by using fragment weight functions of the form where the sum runs for all atoms of molecular fragment P. The effective oxidation states (EOS) analysis, [3] after molecular fragments have been defined, goes as follows: (i) the alpha eff-AOs that are significantly populated are collected for all fragments, (ii) the eff-AOs are sorted according to decreasing occupation number, and (iii) integer alpha electrons are assigned to the eff-AOs of the fragments with higher occupation number, until the number of alpha electrons is reached. Then, proceed analogously for the beta electrons.
By this procedure an effective electronic configuration is obtained for each atom/fragment. The EOS of each atom/fragment is simply given by the difference between its atomic number and the number of alpha and beta electrons that have been assigned to it. This scheme can be safely applied to basis sets including effective core potentials, simply by readily assigning the electrons described by the atomic core potential to the given atom.
and then That is, the overall R(%) index is the minimum value obtained for either the alpha or beta electrons. The larger the R(%) value the closer the overall assignment of the EOS is to the actual electronic structure of the system. Note that R(%) can take values formally from 0 to 100, where values below 50% indicate that the assignment of the electrons has not followed an aufbau principle according to the occupation numbers of the eff-AOs. The latter avenue can be used to measure to which extent the molecular system conforms with a given set of oxidation states, rather than which are the most appropriate formal oxidation states.
If the frontier eff-AOs for any spin case are degenerate (same occupation number) and belong to different fragments, a value of R=50% would be obtained. In that case, however, one may choose to assign halfelectron to each of the two atoms/fragments involved (or, in general, a fraction of the last m electrons that must be distributed among n d degenerate eff-AOs), to accommodate e.g. genuine mixed-valence situations.  0.600 0.567 CASSCF(7,12) results, as described in [8] for species b. CASSCF(6,6) results for species a Figure S8: EFO of the NO * hybrid orbitals and d-type hybrids on the central metal atom for species 1.
EFO gross occupations in red are considered unoccupied upon EOS analysis. Last occupied and first unoccupied EFOs marked in bold.