Atomic orbitals open-shell configuration

He, Be and Ne are examples of atoms with complete orbitals closed-shell configurations). For atoms with incomplete orbitals open-shell configurations) more than one Slater determinant is necessary. 

The UHF formalism becomes inconvenient for open-shell configurations of atoms or molecules with point-group symmetry. Unless specific restrictions are imposed, the self-consistent occupied orbitals fall into sets that are nearly but not quite transformable into each other by operations of the symmetry group. By imposing equivalence and symmetry restrictions, these sets become symmetry-adapted basis states for irreducible representations of the symmetry group. This makes it possible to construct symmetry-adapted /V-clcctron functions, as described in Section 4.4. The constraints in general invalidate the theorems of Brillouin and Koopmans. This restricted theory (RHF) is described in detail for atoms by Hartree [163] and by Froese Fischer [130],... 

Free radicals, or radicals, are defined as atomic or molecular species with unpaired electrons on an otherwise open shell configuration. Because a radical has a half-filled orbital, it easily sucks up an electron from another bond, hence exhibiting highly reactive properties and therefore likely to take part in chemical reactions. The first organic free radical is the triphenylmethyl radical, which was identified by Moses Gomberg in 1900 [34], Radicals play an important role in chemical processes such as combustion, atmospheric chemistry, polymerisation, plasma chemistry, biochemistry, etc. [35],... 

As emphasized in Fig. 2.3, the final natural orbitals of an atomic wavefunction will reflect subtle asymmetries of an open-shell configuration, such as slight differences between p, and p spatial orbitals or between pj, and pj spin-orbitals. However, for many purposes, it is preferable to consider slightly-modified forms of these orbitals that exhibit the expected ffee-atom rotational symmetries of both position and spin space. Such natural atomic orbitals have the advantage of complete rotational invariance with respect to arbitrary choices of coordinate axes in either position or spin space, a highly desirable property for analysis purposes. 

Thus, the main relativistic effects are (1) the radical contraction and energetic stabilization of the s and p orbitals which in turn induce the radial expansion and energetic destabilization of the outer d and f orbitals, and (2) the well-known spin-orbit splitting. These effects will be pronounced upon going from As to Sb to Bi. Associated with effect (1), it is interesting to note that the Bi atom has a tendency to form compounds in which Bi is trivalent with the 6s 6p valence configuration. For this tendency of the 6s electron pair to remain formally unoxidized in bismuth compounds (i.e. core-like nature of the 6s electrons), the term inert pair effect or nonhybridization effect has been often used for a reasonable explanation. In this context, the relatively inert 4s pair of the As atom (compared with the 5s pair of Sb) may be ascribed to the stabilization due to the d-block contraction , rather than effect (1) . On the other hand, effect (2) plays an important role in the electronic and spectroscopic properties of atoms and molecules especially in the open-shell states. It not only splits the electronic states but also mixes the states which would not mix in the absence of spin-orbit interaction. As an example, it was calculated that even the ground state ( 2 " ) of Bij is 25% contaminated by Hg. In the Pauli Hamiltonian approximation there is one more relativistic effect called the Dawin term. This will tend to counteract partially the mass-velocity effect. 

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