Quantum Chemical Study of X@BikPbm, BikPbm∙X, X@SbkSnm, and SbkSnm∙X Clusters

Quantum chemical modeling of electron‐deficient hollow TlkPb12–k and TlkBi20–k shells and related endohedral complexes (k = 1; 2)

“…If n i = 1, then the spinor describes occupied natural localized one‐electron state; if n i ≈ 1 or n i ≈ 0, then the deviation from naturalness or from localization is measured by a dispersion (4) [32, 33]. Two localized spinors │ i › = │ i › and │ i ′ › = │ i ′ › with maximal populations n i ≈ n i ′ ≈ 1 (and with minimal dispersions υ i ≈ υ i ′ ≈ 0) describes the lone or binding electron pair, if the time reversal operator [34] T = ‐ i ′σ y C transforms one spinor into the other: The oxidation number Ξ X of an atom or atomic group X in the [X@Ω] ξ cluster [16] we associate with the cluster charge ξ, the populations of one‐electron states localized in the shell Ω, and the atomic core charges Z A by new formula (): For the parameter ε , we take th...…”

“… Two localized spinors │ i › = │ i › and │ i ′ › = │ i ′ › with maximal populations n i ≈ n i ′ ≈ 1 (and with minimal dispersions υ i ≈ υ i ′ ≈ 0) describes the lone or binding electron pair, if the time reversal operator [34] T = ‐ i ′σ y C transforms one spinor into the other: The oxidation number Ξ X of an atom or atomic group X in the [X@Ω] ξ cluster [16] we associate with the cluster charge ξ, the populations of one‐electron states localized in the shell Ω, and the atomic core charges Z A by new formula (): For the parameter ε , we take the value 0.15, that is, we refer all the states with populations 0.85 ≤ n ω ≤ 1.00 to the localized ones. We borrowed the letter designation Ξ X for the oxidation number [35], but abandoned the formula proposed in it, which for X = Hg, Cd, Yb, Sr, and Ba in X@Pb 12 clusters gave negative oxidation numbers instead of the positive integer values Ξ X = +2 [16]. The oxidation number of an atom is usually understood as ...…”

“…The oxidation number Ξ X of an atom or atomic group X in the [X@Ω] ξ cluster [16] we associate with the cluster charge ξ, the populations of oneelectron states localized in the shell Ω, and the atomic core charges Z A by new formula (6):…”

“…In particular, the symmetrical structures of the free neutral and charged Pb n [1–4] and Bi n [5–10] clusters, free hollow [Pb 12 ] − and [Pb 12 •K] − anions [2, 3], free [(La, Ac, Lu, Lr)@Pb 12 ] + cations [11], [(Ni, Pd, Pt)@Pb 12 ] 2− and [Al@Pb 12 ] + ions in crystals [12–14] were established by experimental and quantum chemical methods. Stable highly symmetric structures have been predicted for free uncharged endohedral complexes in which the Mg, Cd, Hg, or Yb atom [15, 16] are surrounded by a shell of 12 lead atoms. Two pair enantiomeric equilibrium ( C 1 )‐structures and symmetrical equilibrium ( D 3d )‐structure with an energy difference of 0.01 eV were predicted using quantum chemical DFT PBE method for the free uncharged Lu@Pb 12 and Lr@Pb 12 complexes containing unpaired valence electrons [11].…”

“…The substitution of two lead atoms in dianions for two bismuth atoms, according to the results of calculations, gives highly symmetric electrically neutral clusters, for example, ( D 5d )‐Bi 2 Pb 10 cluster [16]—an analog of ( D 5d )‐Bi 2 Sn 10 cluster [18, 19] and ( D 5d )‐(Ni, Pd, Pt)@Bi 2 Pb 10 complexes [16]—analogs of ( D 5d )‐(Pd, Pt)@Bi 2 Sn 10 complexes [17] with zero oxidation numbers of Ni, Pd, and Pt endo ‐atoms. The pair of Bi atoms in these clusters saturated an electronic deficiency of the homoleptic lead shell.…”