Orthogonal operators: extension to hyperfine structure and equivalent p- and f-electrons

Abstract

Orthogonal operators are a next step in the semi-empirical description of complex spectra. Orthogonality yields optimal independence and thus least correlation between the operators. The increased stability of the fitting process is used to include higher-order many-body as well as fully relativistic effects. The calculated eigenvalues are frequently an order of magnitude more accurate with respect to a conventional semi-empirical approach. The resulting eigenvectors may not only be put to use to calculate transition probabilities and g-factors, but also to calculate hyperfine structure A and B constants. We illustrate our first steps in this field with two examples of first spectra of the iron group elements. The results are compared to current experimental hyperfine structure A-values while strong and weak points of the method are discussed. In particular to be able to deal with lanthanides and actinides, the orthogonal operator method is completed with the definition of operators for equivalent p- and f-electrons.

Graphical abstract

Appendix A: p-shell orthogonal operators

The electrostatic structure of $p^N$ configurations is described by the first-order parameter $O_2$ and the effective higher-order parameter $E_{\alpha }$. The contribution of $F^2(pp)$ is given by:

$$\begin{aligned} O_2=\frac{3}{25} \sqrt{2} \cdot F^2(pp) \end{aligned}$$

(A1)

The magnetic structure is described by the operators $z_p$, $z_{\textrm{c}}$ and $z_3$ for the spin–orbit effects and $z_1$ for the spin–spin interaction; the associated parameters are $\zeta _p$, $A_{\textrm{c}}$, $A_3$ and $A_1$, respectively.

Table 11 Matrix elements of the electrostatic operators in $p^2$

Table 12 Reduced matrix elements of the double-vector operators in $p^2$

Table 13 Multiplication factors w.r.t. [8] for the entries of the three-electron operators $t'_i$ and the four-electron operators $f'_i$

The spin–spin operator $z_1$ has only one nonzero reduced matrix element ${\langle }^{3\!}P \parallel z_1 \parallel ^{3\!\!}P{\rangle }=10\sqrt{3}$. The magnitude of all magnetic operators in $p^2$ equals 12. The physical content of all the $p^N$ operators is given in Eqs. (A1), (A2) and (A3). Unlike the case of the d- and the f-shell, no additional three-particle electrostatic operator appears in $p^3$. For the two-particle magnetic operators $A_{\textrm{c}}$ and $A_3$, one obtains the below first- and second-order contributions:

$$\begin{aligned} A_{\textrm{c}}= & {} \frac{1}{4} \sqrt{2}\left( -5M^0(pp)+\frac{3}{10}P^2(p\rightarrow p')\right) \nonumber \\ A_3= & {} \frac{1}{4} \sqrt{\frac{2}{5}}\left( -37M^0(pp)+\frac{3}{10}P^2(p\rightarrow p')\right) \end{aligned}$$

(A2)

$$\begin{aligned} A_1= & {} -\frac{2}{5} \sqrt{15} \cdot M^0(pp) \end{aligned}$$

(A3)

where the electrostatic spin–orbit (EL-SO) contribution $P^2(p \rightarrow p')$ is given by:

$$\begin{aligned} P^2(p \rightarrow p')=\sum \!\!\!\!\!\!\!\!\!\int _{p \rightarrow p'} \frac{2 \zeta (p p') R^2(pp;pp')}{E_{pp'}} \end{aligned}$$

The occurring $M^0$ integrals are defined in accordance with equation (3) of [12]:

$$\begin{aligned} M^k(ab)=\frac{1}{4}\alpha ^2\! \!\! \int _0^{\infty } \hspace{-10.0pt}\int _0^{\infty } \!\!\! \textrm{d}r_1 \textrm{d}r_2 \,a^2(1)b^2(2) \;\frac{r_2^k}{r_1^{k+3}} \; \varepsilon (r_1-r_2) \end{aligned}$$