Valence electrons in heavy atoms

The QED theory of atoms is reviewed. The principles of QED, the QED theory of the interelectron interaction and the radiative corrections to the energy levels (the Lamb Shift) are considered. The applications of QED to the light atoms and to valence electrons in heavy atoms are discussed. 

The QED theory of the light atoms, apart from the a-expansion exploits also the expansion in parameters aZ where Z is the charge of the nucleus. Thus it is valid only for aZ < 1. This condition does not hold for the inner electron shells in heavy atoms. Moreover it does not hold even for the valence electrons in heavy atoms due to the singularity of QED operators. For the evaluation of the matrix elements of such operators the small distances of the electron from the nucleus become important. At such distances the effective electron chOTge Zejf for the valence electrons may not be small. Therefore the QED theory without aZ expansion appears to be necessary. Such theory was first... 

For electrons in the multicharged ions or even for the valence electrons in heavy atoms the parameter aZ cannot be considered as small. In case of heavy atoms the reason is that the effective QED potentials of the electron interaction with the nucleus are rather short-range and the interaction occurs when the outer electrons penetrate deeply into the core. Therefore the methods described in Section III are not valid anymore and all-orders in aZ methods axe necessary. 

Radiative corrections for the ns valence electrons in heavy atoms. 

The accuracy of the energy level calculations for ns valence electrons in heavy atoms has drastically improved during the last decade. 

Finally we present results of SE and VP calculations for ns valence electrons in heavy and superheavy atoms with n up to 8 and Z up to 119 [13], [14]. For the calculation of the SE contribution the PWR approach described in Sec.4.2 based on the multiple commutator expansion [71] was used. The corrections are given in Table 1. Since the B-spline approach requires the employment of the local potential, the local approximation to the DHF potential obtained by the direct parametrization [77] was used. The VP contribution was treated in the Uehling approximation. One can expect that the Uehling term will suffice not only for highly charged ions but in screened systems as well. The Uehling potential was corrected for the extended nucleus [78] - [80]. The Uehling potential for the point-like nucleus (233) was replaced by the expression ... 

In contrast, the valence d and f orbitals in heavy atoms are expanded and destabilized by the relativistic effects. This is because the contraction of the s orbitals increases the shielding effect, which gives rise to a smaller effective nuclear charge for the d and f electrons. This is known as the indirect relativistic orbital expansion and destabilization. In addition, if a filled d or f subshell lies just inside a valence orbital, that orbital will experience a larger effective nuclear charge which will lead to orbital contraction and stabilization. This is because the d and f orbitals have been expanded and their shielding effect accordingly lowered. 

However, relativistic quantum mechanics was ignored by chemists for decades because of the erroneous belief that in all atoms, the valence electrons (in the outermost shells) which are primarily responsible for the chemistry moved so slowly that their dynamics was not significantly modified by relativity, although there was no evidence to support this premise, especially in the case of valence electrons of atoms of heavy elements (Z>75). 

Effective core potentials address the aforementioned problems that arise when using theoretical methods to study heavy-element systems. First, ECPs decrease the number of electrons involved in the calculation, reducing the computational effort, while also facilitating the use of larger basis sets for an improved description of the valence electrons. In addition, ECPs indirectly address electron correlation because ECPs may be used within DFT, or because fewer valence electrons may allow implementation of post-HF, electron correlation methods. Finally, ECPs account for relativistic effects by first replacing the electrons that are most affected by relativity, with ECPs derived from atomic calculations that explicitly include relativistic effects via Dirac-Fock calculations. Because ECPs incorporate relativistic effects, they may also be termed relativistic effective core potentials (RECPs). 

The limits of collective quantization in atoms are as yet quite unknown. In light atoms such as He or Be, the shell structure corresponding to principal quantum numbers is clearly marked even though the quantization corresponds to collective rather than independent-particle behavior. In heavy atoms such as Sr and Ba, the shell structure of the valence electrons seems to be blurred because excited states associated with one set of principal quantum numbers... 

In an elastic collision of electrons with heavy neutrals or ions, m < M and, hence, Y = 2m /M, which means that the fraction of transferred energy is very small (y 10 " ). It explains, in particular, why the direct electron impact ionization due to a colhsion of an incident electron with a valence electron of an atom predominates. Simply stated, only... 

A further chemically interesting process involving molecular skeletons is isovalent (isovalence electronic) substitution which does not affect the number of heavy atoms and the number of valence electrons. In connection with the cumulenes the most important isovalent substitution proems is the transition from ketenes to thioketenes which retains the overall geometry of the corre-... 

Spin-dependent operators are required when we wish to account for relativistic effects in atoms and molecules [118, 119]. These effects can roughly be classified as strong and weak ones. The relativistic corrections are especially important in heavy atoms where they play a particularly significant role when describing the inner shells. In those cases, they have to be accounted for from the start, usually relying on Dirac-Hartree-Fock method. Fortunately, in most chemical phenomena, only valence electrons play a decisive role and are satisfactorily... 

C. Thierfelder, P. Schwerdtfeger. Quantum electrodynamic corrections for the valence shell in heavy many-electron atoms. Phys. Rev. A, 82 (2010) 062503. 

The mercury spectrum is even less regular. The electron configuration in Hg consists of two valence electrons outside of a closed-shell core. .. (5s) (5p) (4/) (5d) . The Hg spectrum features that are not anticipated in H or K arise from electron spin multiplicity (i.e., the formation of triplet as well as singlet excited states in atoms with even numbers of valence electrons) and from spin-orbit coupling, which assumes importance in heavy atoms like Hg (Z = 80). The mercury spectrum in Fig. 2.2 has been widely used as a spectral calibration standard. 

Unlike semiempirical methods that are formulated to completely neglect the core electrons, ah initio methods must represent all the electrons in some manner. However, for heavy atoms it is desirable to reduce the amount of computation necessary. This is done by replacing the core electrons and their basis functions in the wave function by a potential term in the Hamiltonian. These are called core potentials, elfective core potentials (ECP), or relativistic effective core potentials (RECP). Core potentials must be used along with a valence basis set that was created to accompany them. As well as reducing the computation time, core potentials can include the effects of the relativistic mass defect and spin coupling terms that are significant near the nuclei of heavy atoms. This is often the method of choice for heavy atoms, Rb and up. 

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