Valence orbitals, “core-like

Valence orbital Xij is the lowest energy solution of equation 9.23 only if there are no core orbitals with the same angular momentum quantum number. Equation 9.23 can be solved using standard atomic HF codes. Once these solutions are known, it is possible to construct a valence-only HF-like equation that uses an effective potential to ensure that the valence orbital is the lowest energy solution. The equation is written... 

An important advantage of ECP basis sets is their ability to incorporate approximately the physical effects of relativistic core contraction and associated changes in screening on valence orbitals, by suitable adjustments of the radius of the effective core potential. Thus, the ECP valence atomic orbitals can approximately mimic those of a fully relativistic (spinor) atomic calculation, rather than the non-relativistic all-electron orbitals they are nominally serving to replace. The partial inclusion of relativistic effects is an important physical correction for heavier atoms, particularly of the second transition series and beyond. Thus, an ECP-like treatment of heavy atoms is necessary in the non-relativistic framework of standard electronic-structure packages, even if the reduction in number of... 

As the atom becomes larger, the number of basis functions needed to describe it increases as well. However, since one is most interested in the valence shell where most of the action occurs, the increasingly larger number of inactive or core functions become more and more of a nuisance. One cannot simply omit them as the valence orbitals would then collapse into smaller core orbitals (which are of much lower energy). One solution is development of core pseudopotentials or effective core potentials (ECP) which eliminate the need to include core functions explicitly, yet keep the valence functions from optimizing themselves into core orbitals ° . Such pseudopotentials are commonly used in elements of the lower rows of the periodic table, like Br or I. 

The Fock-like operator F incorporates contributions from the core and valence orbitals,... 

In principle any function can be perfectly expressed by plane waves because the set of plane waves is complete. However, the position-independent property of plane waves makes the expressions for some sharp peak-like functions very difficult. In these cases an enormous number of plane-waves is needed to obtain near-convergent results. For this reason, the rapidly oscillating core orbitals of an atom are difficult to treat using plane wave basis sets. Because valence orbitals are orthogonal to the core orbitals, even the valence orbitals are difficult to be expressed when core orbitals are included in the calculations. In order to avoid the difficulties in expressing the core orbitals with plane wave basis set, it is necessary to freeze the core electrons and to model the integral effects of core electrons and nuclei with pseudopotentials. 

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. 

Hyperfine couplings, in particular the isotropic part which measures the spin density at the nuclei, puts special demands on spin-restricted wave-functions. For example, complete active space (CAS) approaches are designed for a correlated treatment of the valence orbitals, while the core orbitals are doubly occupied. This leaves little flexibility in the wave function for calculating properties of this kind that depend on the spin polarization near the nucleus. This is equally true for self-consistent field methods, like restricted open-shell Hartree-Fock (ROHF) or Kohn-Sham (ROKS) methods. On the other hand, unrestricted methods introduce spin contamination in the reference (ground) state resulting in overestimation of the spin-polarization. 

The dominance of ionic bonding and the -i-3 oxidation state in 4f complexes is attributed to the core-like nature of the valence 4f orbitals [1]. From the parent Ln electronic configuration, [Xe]6s 5d 4F (n= 1-14), the 6s and 5d electrons are easily ionized to give [Xe]4T Ln configurations. For several of the Ln series, notably Yb, Sm, and Eu, the molecular chemistry of the +2 oxidation state is rich and well... 

The original version of the FD HF program employed one grid of points of constant density to represent all orbitals and potentials. In the case of heavy systems where both tightly contracted atomic-like core orbitals and rather diffused, valence, orbitals are present this restriction has to be relaxed. Recently the multiple grids in variable have been introduced which allow to further reduce the number of grid points (19). 

With the Coulomb and exchange parts of the MP discussed so far, the core-like solutions of the valence Fock equation would still fall below the energy of the desired valence-like solutions. In order to prevent the valence-orbitals collapsing into the core during a variational treatment and to retain an Aufbau principle for the valence electron system, the core-orbitals are moved to higher energies by means of a shift operator... 

The permanganate anion MnC)4 has a tetrahedral geometry with a Td point group symmetry, Fig. 1. At each oxygen atom, we have a core like 2s orbital as well as two 2-pn and one 2pffvalance orbitals. The total number of valence electrons is 32 of which 8 are in core type 2s levels and the remaining 24 in molecular orbitals spanned by 2pn, 2pa and 3d. We show in Fig. 2 the levels represented by orbitals made up of 2pn, 2pa and 3d. 

This simple model depends on the properties of a deformable core, and is only likely to be significant for heavier, polarisable metals. In addition, there are the directional influences of the valence orbital overlaps, which have been widely explored, and, in general, favour cis geometry in the d-block. For example, the Extended Huckel calculations of Tatsumi and Hoffmann show the dominant role of 4d-n orbitals in stabilising the cis geometry in MoO + [75]. In this... 

The operator / has an obvious physical interpretation it is a generalized fock-like operator involving the average potential generated by the one-particle density matrix of tpo- Since we have chosen our orbitals to be natural, the density matrix is diagonal. For the core orbitals, the diagonal elements are unity, while for the valence orbitals they are the occupancy of these orbitals ... 

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