Core orbital radius

For dipole forbidden transitions, e.g., from a p core state to a 4f final state, higher-order terms in the exponential expansion lead to a finite contribution to the cross section. This is in general small unless q 1/r, where is the core orbital radius, but such conditions may be satisfied both in transmission and reflection geometry (Schnatterly 1979, Grunes and Leapman 1980, Ludeke and Koma 1975), and will be of importance in rare earth electron loss spectroscopy. Thus even within the Born-Bethe regime monopole, quadrupole and octopole transitions may be prominent. 

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... 

Variational one-center restoration. In the variational technique of one-center restoration (VOCR) [79, 80], the proper behavior of the four-component molecular spinors in the core regions of heavy atoms can be restored as an expansion in spherical harmonics inside the sphere with a restoration radius, Rvoa, that should not be smaller than the matching radius, Rc, used at the RECP generation. The outer parts of spinors are treated as frozen after the RECP calculation of a considered molecule. This method enables one to combine the advantages of two well-developed approaches, molecular RECP calculation in a gaussian basis set and atomic-type one-center calculation in numerical basis functions, in the most optimal way. This technique is considered theoretically in [80] and some results concerning the efficiency of the one-center reexpansion of orbitals on another atom can be found in [75]. 

For small nonpolar species such as H2 and N2 the dominant interaction between the Rydberg electron and the nuclear vibrational and rotational motion occurs within a small radius around the ionic core, which is traversed in a fraction of a femtosecond. This short encounter justifies the sudden treatment of vibration and rotation in MQDT theory, while also permitting Bom-Oppenheimer estimates of the necessary quantum defect functions. It is also central to the n-3 scaling law because the core transit time is almost energy independent, while the Rydberg orbit time increases as n3. 

The Octet Rule. To the isomorphism exhibited in Table 1 may be added the radius-ratio rules. The statement in crystal chemistry that rattling is bad , that the number of anions about a cation should not be so great as to create a cavity larger than the cation 72>, corresponds to the statement in covalent chemistry that the number of electron-pairs about an atomic core should not be so great as to exceed the number of low-lying, available, valence-shell orbitals. 

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