Subshell occupancy

Extending (8) to transition elements poses an obvious problem, because it is often not clear how many s and d electrons should be included. Allen et al. addressed this by means of a computational technique that yields fractional subshell occupancies [66], which can then be combined with spectroscopic data. However, this ignores the role of interpenetration involving lower-lying subshells, which can be significant [40,67] for representative elements as well. 

The effective subshell occupation in this model is therefore... 

Consider now a perturbation between a cr7r47r ni state and a method presented in Section 3.2.3 that the 7t37t configuration gives rise to two E states from 7r+(7r-)27r,+ and (7r+)27r-7r subshell occupancies. The 1E- and 1E+ wavefunctions, properly symmetrized with respect to the crv operator acting only on the spatial part of the wavefunction, must be constructed as linear combinations of the four possible A = 0, E = 0 Slater determinants ... 

Assuming that this equation has been solved and that these optimal orbitals are used in calculating H), then the partial derivatives with respect to the subshell occupations q nl) can be studied. One obtains... 

It is noteworthy that Rydberg orbital occupancies on the central atom (rY, final column of Table 3.29) are relatively negligible (0.01-0.03e), showing that d-orbital participation or other expansion of the valence shell is a relatively insignificant feature of hyperbonded species. However, the case of HLiH- is somewhat paradoxical in this respect. The cationic central Li is found to use conventional sp linear hybrids to form the hydride bonds, and thus seems to represent a genuine case of expansion of the valence shell (i.e., to the 2p subshell) to form two bonding hybrids. However, the two hydride bonds are both so strongly polarized toward H (93%) as to have practically no contribution from Li orbitals, so the actual occupancy of extra-valent 2pu orbitals ( 0.03< ) remains quite small in this case. 

N2 = 2. For carbon, the subshells are expansions of 6, respectively, 4, Slater-type functions, that is, V = 6, v2 = 4. Because of the spherical averaging of pc and pv, the occupancies of orbitals with the same n and l values are the same, regardless of their m values. In other words, the electrons in a subshell are evenly distributed among the orbitals with different values of the magnetic quantum number m. 

If the orbitals belonging to different subshells are assumed to have the same one-electron cross sections, the integrated ionization cross section of a particular subshell is simply proportional to the occupancy of that orbital in the subshell in the molecule. 

Relationships between coefficients gk for different degrees of occupation of the subshells (the cases of almost and completely filled subshells) are described by equalities (20.35), (20.36) and (20.38). Therefore, for magnetic interactions one has additionally to consider such conditions only in the case of coefficient dk. Bearing in mind that k acquires only odd values and that the submatrix elements of operator Tk are diagonal with respect to seniority quantum number v, we find... 

Ceulemans considers a dn electron state, split by an octahedral field into the e and t2 levels, so that all the n electrons are in the t2 subshell. In the notation of Sugano et al. [27], rjj(t2SrMsMr) is the multi-electronic wavefunction, with SMs irrep labels for the total spin and rMr irrep labels in the octahedral group for the orbital state. We use a real orbital basis in which all njm factors take their simplest possible forms and suppress S, r and Mr below. It takes six electrons (three pairs each of opposed spin) to fill this t2 subshell. Ceulemans [7] particle-hole conjugation operator 0() has the effect of conjugating the occupancies within this subshell, and of... 

Electrons within a subshell for which l > 0 tend to avoid pairing within the same orbital. This rule is Hund s rule and reflects the relatively greater electrostatic repulsion between two electrons in the same orbital as compared with occupancy of two orbitals having differing values for mi. 

Fe has six 3d electrons in the only unfilled subshell. The maximum unpairing occurs with double occupancy of one of the available d subshells and single occupancy of the remaining four. There will be 4 unpaired electrons. 

In Cu compounds of high covalence no precise decision can be reached regarding the oxidation state of the copper (53), while in ionic systems reference to the occupancy of the 3d subshell is sufficient. Thus, the oxidation states given in I—IY are of merely formal nature. The interesting phenomenon in this scheme is the redox mesomerism of Cu which implies that the chelated metal ion could have different biochemical actions. Type I would represent the reversible oxygenation as found in haemo-cyanin, type II would be the superoxide dismutation, provided OI-really is the substrate, and types III and IV are represented by the catalatic and oxidative action displayed by a considerable number of copper proteins (polyphenol oxidases, amine oxidases etc.). The biochemical specificity of each chelated copper is more of less given by the macromolecular ligands. 

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