Complex Electronic Configurations

In the MCHF approach a number of superposed configurations are chosen and the mixing coefficients (weights of the configurations) and also the radial parts of the wave functions are varied. This method does not depend on choice of the basis set and both analytical and numerical wave functions may be used. However, MCHF calculations for complex electronic configurations would require variation of a large number of parameters, which needs powerful computers. Problems may also occur with the convergence of the procedure [45]. 

Clebsch-Gordan coefficients have already occurred several times in our considerations in the Introduction (formula (2)) while generalizing the quasispin concept for complex electronic configurations, while defining a relativistic wave function (formulas (2.15) and (2.16)), in the addition theorem of spherical functions (5.5) and in the definition of tensorial product of two tensors (5.12). Let us discuss briefly their definition and properties. There are a number of algebraic expressions for the Clebsch-Gordan coefficients [9, 11], but here we shall present only one ... 

We shall not consider here the more complex cases if necessary, they may be found in [9, 11] or deduced utilizing methods described there. The expressions for matrix elements of the majority of energy operators (1.16) in terms of radial integrals and transformation matrices or 3n/-coefficients for complex electronic configurations may be found in [14]. 

As was already mentioned, in theoretical atomic spectroscopy, while considering complex electronic configurations, one has to cope with many sums over quantum numbers of the angular momentum type and their projections (3nj- and ym-coefficients). There are collections of algebraic formulas for particular cases of such sums [9, 11, 88]. However, the most general way to solve problems of this kind is the exploitation of one or another versions of graphical methods [9,11]. They are widely utilized not only in atomic spectroscopy, but also in many other domains of physics (nuclei, elementary particles, etc.) [13],... 

Thus, for complex electronic configurations there is a large variety of possible coupling schemes. Therefore, the combination of theoretical and experimental investigations is of great importance for the interpretation of such spectra (see also Chapter 12). 

These equations can be used to establish additional analytical relationships when dealing with the matrix elements of operators of physical quantities in the case of complex electron configurations. 

Operators of electronic transitions, except the third form of the Ek-radiation operator for k > 1, may be represented as the sums of the appropriate one-electron quantities (see (13.20)). Their matrix elements for complex electronic configurations consist of the sums of products of the CFP, 3n./-coefficients and one-electron submatrix elements. The many-electron part of the matrix element depends only on tensorial properties of the transition operator, whereas all pecularities of the particular operator are contained in its one-electron submatrix element. 

As was shown in Chapter 26, the pecularities of relativistic operators of electronic transitions are confined in their one-electron submatrix elements. Therefore formulas (26.7), (26.10), (26.12), (26.13), (26.15), (26.17)-(26.19), (26.21) and (26.23) are equally applicable for both the relativistic Ek- and Mk-transitions between complex electronic configurations. This is denoted by the subscripts e, m at the operator 0 k The same holds for sum rules (26.8), (26.9), (26.24)-(26.28). Therefore, we have only to present the appropriate expressions for the submatrix elements of non-relativistic operator (4.16) of Mk-transition in general cases of complex electronic configurations. For Mk-transitions between the levels of a shell of equivalent electrons the following formula is valid ... 

Here n ensures the normalization condition of the wave function. The use of the concept of irreducible tensorial operators in this approach [220] opens up the possibility of exploiting such a method for complex electronic configurations however, so far it has been applied only to light atoms and ions. 

MBPT significantly improves the electron transition wavelengths, line and oscillator strengths, transition probabilities as well as the lifetimes of excited levels. Therefore, it seems promising to generalize such an approach to cover the cases of more complex electronic configurations having several open shells, even with n > 2. 

The accuracy of oscillator strengths and transition probabilities in the best cases, depending on the method employed, approaches 10 or even a few per cent. However, in many cases, particularly for neutral or weakly ionized atoms having complex electronic configurations, discrepancies may be much larger (about 50-100% or even more). The largest errors, as a rule, are caused by cancellation effects. 

Let us also briefly discuss some developments in accounting for correlation effects while performing calculations of spectral properties of complex many-electron atoms and ions. The method of superposition of configurations based on the transformed radial orbitals, briefly described in Chapter 29, has demonstrated its effectiveness particularly for complex electronic configurations. Code for transformed radial orbitals is described in [11]. There the following transformed radial orbitals are recommended ... 

Part 2 is devoted to the foundations of the mathematical apparatus of the angular momentum and graphical methods, which, as it has turned out, are very efficient in the theory of complex atoms. Part 3 considers the non-relativistic and relativistic cases of complex electronic configurations (one and several open shells of equivalent electrons, coefficients of fractional parentage and optimization of coupling schemes). Part 4 deals with the second-quantization in a coupled tensorial form, quasispin and isospin techniques in atomic spectroscopy, leading to new very efficient versions of the Racah algebra. 

---