Corrections relativistic transitions

Relativistic corrections of order v2/c2 to the non-relativistic transition operators may be found either by expanding the relativistic expression of the electron multipole radiation probability in powers of v/c, or semiclas-sically, by replacing p in the Dirac-Breit Hamiltonian by p — (l/c)A (here A is the vector-potential of the radiation field) and retaining the terms linear in A. Calculations show that in the general case the corresponding corrections have very complicated expressions, therefore we shall restrict ourselves to the particular case of electric dipole radiation and to the main corrections to the length and velocity forms of this operator. 

Nearly every technical difficulty known is routinely encountered in transition metal calculations. Calculations on open-shell compounds encounter problems due to spin contamination and experience more problems with SCF convergence. For the heavier transition metals, relativistic effects are significant. Many transition metals compounds require correlation even to obtain results that are qualitatively correct. Compounds with low-lying excited states are difficult to converge and require additional work to ensure that the desired states are being computed. Metals also present additional problems in parameterizing semi-empirical and molecular mechanics methods. 

This program is excellent for high-accuracy and sophisticated ah initio calculations. It is ideal for technically difficult problems, such as electronic excited states, open-shell systems, transition metals, and relativistic corrections. It is a good program if the user is willing to learn to use the more sophisticated ah initio techniques. 

Indelicato, P. and Lindroth, E. (1992) Relativistic effects, correlation, and QED corrections on Ka transitions in medium to very heavy atoms. Physical Review A, 46, 2426-2436. 

For the lighter elements rei is much less than unity (e.g., rei = 0.15 for Sc) and the relativistic corrections are small. However, these corrections become much larger for heavier elements (e.g., ,ei = 0.58 for Hg, about four times larger than for Sc) and the perturbative correction procedure breaks down completely as rei approaches unity. Thus, while relativistic corrections are largely ignorable for the first transition series, these corrections become of dominant chemical importance in later series, particularly after filling of the lanthanide f shell. 

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... 

There exists another more consistent way of obtaining the electron transition operators. We can start with the quantum-electrodynamical description of the interaction of the electromagnetic field with an atom. In this case we find the relativistic operators of electronic transitions with respect to the relativistic wave functions. After that they may be transformed to the well-known non-relativistic ones, accounting for the relativistic effects, if necessary, as corrections to the usual non-relativistic operators. Here we shall consider the latter in more detail. It gives us a closed system of universal expressions for the operators of electronic transitions, suitable to describe practically the radiation in any atom or ion, including very highly ionized atoms as well as the transitions of any multipolarity and any type of radiation (electric or magnetic). 

However, one may expect that there are some relativistic corrections to the transition operators themselves. 

Let us consider the intercombination transitions. Then, we shall retain only the corrections containing the spin operator in the expansion. To find the form of the operator describing the electric multipole intercombination transitions and absorbing the main relativistic corrections, one has to retain in the corresponding expansion the terms containing spin operator S = a and to take into account, for the quantities of order v/c, the first retardation corrections, whereas, for the quantities of order v2/c2 one must neglect the retardation effects. Then the velocity form of the electric dipole transition probability may be written as follows ... 

Eq. (4.3) with K = 0 may serve as starting point to find the relativistic corrections to the length form of the transition operator. The expansion of the matrix element (4.3) up to terms of order v2/c2 for the case of the... 

While using (4.14) and exact wave functions. This supports the conclusion of Drake [83] that for electric dipole transitions, by considering the commutator of with the atomic Hamiltonian in the Pauli approximation, we obtain Qwith relativistic corrections of order v2/c2 (see (4.18)-(4.20)). However, for many-electron atoms and ions, one has to use approximate (e.g., Hartree-Fock) wave functions, and then this term gives non-zero contribution, conditioned by the inaccuracy of the model adopted. 

Thus, the kind and quantity of relativistic corrections to the length and velocity forms of 1-radiation are different. From this point of view the concept of the equivalency of these forms must be improved both forms will lead to coinciding transition values for the accurate (exact) wave functions only if we account for the relativistic corrections of order v2/c2 to the transition operators (in practice, only for the velocity form). The other conclusion accounting for the relativistic effects leads to qualitatively new results, namely, to new operators, which allow not only improved values of permitted transitions, but also describe a number of lines, which earlier were forbidden. These relativistic corrections usually are very small, but they are very important for weak intercombination lines of light neutral atoms (see Chapter 30). 

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