Multipole Electric, Relativistic Form

Now, by use of formulas (2.23)-(2.26) we are in a position to present in. /-representation all the operators needed. For example, the non-relativistic operator of electric multipole radiation will have the form... 

Starting with Eq. (4.1) after tedious mathematical transformations [18] we finally find the following two general forms of the relativistic expressions for the electric multipole (Ek) transition probability (in a.u.) ... 

Relativistic corrections to various forms of the electric multipole... 

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. 

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 ... 

As was pointed out in Chapter 4, division of the radiation into electric and magnetic is connected with the existence of two types of multipoles, characterized by the parities (—l)fc and (—l)fc+1, respectively. The first ones we have studied quite thoroughly in Chapters 24-26. Here let us consider in a similar way the M/c-transitions. Again, as we have seen in Chapter 4, the potential of the electromagnetic field in this case does not depend on gauge. Therefore only one relativistic expression (4.8) was established for the probability of M/c-radiation, described by the appropriate operator (4.9). The probability of non-relativistic M/c-transitions (in atomic units) is given by formula (4.15), whereas the corresponding non-relativistic operator has the form (4.16). 

EFGs and other electric-field-related properties are dealt with in a somewhat different manner. The EFG and multipole moments are calculated as expectation values with the relevant operators and the electron charge density. To avoid PC errors, if the operators are the four-component versions this charge density has to be the four-component (Dirac) density. The latter differs from the two-component density [24,25] already in order which is the same leading order as the relativistic effects on the properties. In the so-called ZORA-4 (Z4) framework, the relevant operators are kept in their four-component form, and an approximate four-component electron charge density is reconstructed from the two-component ZORA density. As was shown by van Lenthe and Baerends [26], the Z4 method eliminates most of the PC errors in order c , with relatively small residual errors. In a Kohn-Sham (KS) DFT framework with two-component molecular orbitals y>, with occupations the ZORA two-component density is... 

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