Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic Cases

Abstract

This work explores the evaluation of hydrogenic matrix elements for non-relativistic and relativistic cases under the Screened Hydrogenic Model (SHM). It focuses on scenarios where the initial and final states have different effective charges $Z_1\ne Z_2$, deriving closed-form solutions for particular cases $n_1=n_2$ and $Z_1=Z_2$. In addition, analytical expressions for radial matrix elements $⟨n^{\prime }{l}^{\prime }\left|r^{\beta }\right|nl⟩$ and their relativistic counterparts are presented. These are applicable for discrete–discrete transitions and allow simplifications for specific configurations using Laplace transforms. The study discusses generalizations of SHM for calculating cross-sections in hot and dense plasmas, employing the Plane Wave Born Approximation (PWBA). It also addresses the transition from LS to jj coupling for matrix elements, providing rules for such transformations.

1. Introduction

In various applications of hot and dense matter, one of the most common approaches to calculate the kinetics of the plasma population is the collisional radiative (CR) model. In this approach, the Screened Hydrogenic Model (SHM) can be implemented using different effective charges for the initial, $Z_i$, and final, $Z_f$, states. This strategy allows the establishment of closed formulas that facilitate the calculation of the different elements of the matrix.

$$P_{\alpha ,{\alpha }^{\prime }}^{\left(\beta \right)}=〈{\alpha }^{\prime }\left|r^{\beta }\right|\alpha 〉,$$

(1)

for both the discrete–discrete and discrete-continuous cases. The physical argument for this approach is that if the degree of ionisation is sufficiently high, the interaction between the nucleus and the electrons is much larger than the interelectron interaction; in practice, this is true for $Z_c\gtrsim 7-8$.

In Equation (1), we denote by $\alpha$ the set of parameters $\left(n,l,j,Z\right)$ corresponding to the wavefunction $\left|nl\right.〉$ (non-relativistic) or $\left|nlj\right.〉$ (relativistic), similarly for ${\alpha }^{\prime }$.

The various contexts in which the SHM can be used are evident in references [1,2,3], as well as in articles published in various journals. For example, Benita Cerdan’s thesis utilizes the Collisional Radiative Model of the Average Atom, based on a relativistic screened atomic model [4]. A set of rules for obtaining the screening constants $s_{nlj}$, such that $Z_{nlj}=Z-s_{nlj}$, was provided by one of us in analytical form [5]; the results are almost indistinguishable from another set, provided by Mendoza et al., based on calculations using relativistic codes [6]. Despite the approximation involved, this methodology is much more efficient than using Hartree–Fock methods [7] or Dirac–Fock methods [8], and sufficiently adequate given the complexity of the models to which it is applied, not only the CR model but also radiation hydrodynamics codes. In general, the vast majority of works related to SHM focus on the case where $Z_i=Z_f$, except for a few, as noted by Upcraft [9]. In the work of Khandelwal [10], a hypergeometric function is used, which does not provide much assistance in the more general case. For example, it is not possible to derive, from the final expression, the case where $Z_i=Z_f$ and $n_i=n_f$. The same happens in the work of Sánchez et al. [11] and in one of our works [12]. The Sánchez formula was modified for the non-hydrogen ions by Li et al. using several definitions of the effective nuclear charge [13]. It is worth noting that a work in the same spirit, for the case of electron-impact excitation cross-sections, was published by Pain and Benredjem [14], involving the spherical Bessel function, $j_t\left(Kr\right)$, where $\mathbf{K}={\mathbf{k}}^{\prime }-\mathbf{k}$ is the negative of the momentum transferred from the free electron to the atom. When $K\to 0$, $j_t\left(Kr\right)\sim Kr/3$, thus providing, in principle, what is necessary for the calculation of $P_{\alpha ,{\alpha }^{\prime }}^{\left(1\right)}\equiv 〈n^{\prime }{l}^{\prime }\left|r^1\right|nl〉$, directly related to the calculation of the non-relativistic oscillator strength of the type $f_{n{n}^{\prime }}$, not the more general case $f_{\gamma J,{\gamma }^{\prime }{J}^{\prime }}$ [7]. In Section 5, we will briefly address this point.

In this work, we will present the following results:

(1)

We will calculate the matrix elements $〈n^{\prime }{l}^{\prime }\left|r^{\beta }\right|nl〉$ (non-relativistic case) and $〈n^{\prime }{l}^{\prime }{j}^{\prime }\left|r^{\beta }\right|nlj〉$ (relativistic case) for the general case where $Z_1\ne Z_f$;

(2)

As a corollary, we will see that the chosen form allows for the straightforward derivation of particular cases using the fact that a certain Laplace transform is known, as we will see later in Equation (10);

(3)

We will make use of a correction in the relativistic case that allows us to work only with the large components;

(4)

We will mention some generalizations regarding the use of the Plane Wave Born approximation (sometimes used in the simplest way possible) for the more general cases of the line strength;

(5)

We will present a straightforward way to transition from expressions for the matrix elements $LS$ to $jj$ using a few simple rules.

2. Non-Relativistic Matrix Elements Using SHM

In the central field model, the angular part of the matrix elements is solved exactly using Angular Momentum Theory. On the other hand, the radial wave functions can be expressed interchangeably in terms of confluent hypergeometric functions or Kummer functions, $M\left(\alpha ,\beta ,x\right)$, associated Laguerre functions, $L_s^t\left(x\right)$, or in the explicit form of polynomials multiplied by an exponential [7]. From the perspective of the final evaluation, all forms are equivalent. However, depending on the chosen representation, it can be easier or harder to deduce specific cases. For instance, the matrix elements (at least for the discrete case) could be calculated using an expression that does not allow analytical derivation. In this work, at least for the non-relativistic case, we will use the generalized Laguerre polynomials. It should be noted that there is no universal way to define these polynomials, as their definitions can vary between authors, as shown in [15].

We will use the form:

$$P_{nl}\left(r\right)=N_{\alpha }{r}^{l+1}exp\left(-Zr/n\right)L_{n-l-1}^{2l+1}\left(\rho \right),$$

(2)

with

$$N_{\alpha }\equiv N\left(n,l,Z\right)={\left({\displaystyle \frac{2Z}{n}}\right)}^{l+1}{\left[{\displaystyle \frac{Z(n-l-1)!}{n^2\left[\right(n+l)!]}}\right]}^{1/2},$$

(3)

and $\rho =2Zr/n$. For $L_{n-l-1}^{2l+1}\left(\rho \right)$, we will explicitly use:

$$L_{n-l-1}^{2l+1}\left({\displaystyle \frac{2Zr}{n}}\right)=\sum _{q=0}^{n-l-1}{\displaystyle \frac{{\left(-1\right)}^q}{q!}}\left({\displaystyle \genfrac{}{}{0pt}{}{n+l}{n-l-1-q}}\right){\left({\displaystyle \frac{2Zr}{n}}\right)}^q,$$

(4)

instead of the more commonly used $L_{n+l}^{2l+1}\left(\rho \right)$, which is frequently found in textbooks and articles. The relationship between the two forms can also be found in ref. [15]. Using the expression in (4), the Laplace transform can be written as:

$$\begin{array}{cc}\mathcal{L}\left[x^{p+\alpha }{L}_n^{\alpha }\left(x\right)\right]\equiv \int e^{-sx}{x}^{p+\alpha }{L}_n^{\alpha }\left(x\right)dx\hfill & \\ & \hfill ={\displaystyle \frac{\left(p+\alpha \right)!}{s^{p+\alpha +n+1}}}\sum _{k=0}^n{\left(-1\right)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n+\alpha }{k+\alpha }}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{p}{k}}\right){\left(s-1\right)}^{n-k},\end{array}$$

(5)

with $p\ge 0$, $\alpha >-1$, $s>0$.

It is worth noting that the term ${\left(s-1\right)}^{n-k}$ in Equation (5) provides the opportunity to simplify particular cases, which are not possible using other forms of $P_{nl}\left(r\right)$, to the best of our knowledge.

To compute (1), we define $s=\left(\left(Z_1{n}_2+Z_2{n}_1\right)/\left(n_1{n}_2\right)\right)$ and, therefore:

$$P_{\alpha ,{\alpha }^{\prime }}^{\left(\beta \right)}\equiv 〈n^{\prime }{l}^{\prime }\left|r^{\beta }\right|nl〉=N_{{\alpha }_1}{N}_{{\alpha }_2}{\int }_0^{\infty }{e}^{-sr}{r}^{l_1+l_2+2+\beta }{L}_{n_1-l_1-1}^{2l_1+1}\left({\displaystyle \frac{2Z_{1}r}{n_1}}\right)L_{n_2-l_2-1}^{2l_2+1}\left({\displaystyle \frac{2Z_{2}r}{n_2}}\right)dr.$$

(6)

Since in the previous expression the Laguerre polynomials are not expressed in terms of the integration variable r, but rather in terms of the variables $\left(2Z_{1}r/n_1\right)$ and $\left(2Z_{1}r/n_1\right)$, the most straightforward approach is to separate $\left(2Z_1/n_1\right)$ on one side and r on the other. Using the expansion for $L_{n_1-l_1-1}^{2l_1+1}\left({\rho }_1\right)$ provided in Equation (4), we obtain the following expression, equivalent to (6)

$$\begin{array}{c}{P}_{\alpha ,{\alpha }^{\prime }}^{\left(\beta \right)}=N_{{\alpha }_1}{N}_{{\alpha }_2}\sum _{q=0}^{n_1-l_1-1}{\displaystyle \frac{{\left(-1\right)}^q}{q!}}\left({\displaystyle \genfrac{}{}{0pt}{}{n_1+l_1}{n_1-l_1-1-q}}\right){\left({\displaystyle \frac{2Z_1}{n_1}}\right)}^q\\ \times {\int }_0^{\infty }{e}^{-sr}{r}^{l_1+l_2+2+\beta +q}{L}_{n_2-l_2-1}^{2l_2+1}\left({\displaystyle \frac{2Z_{2}r}{n_2}}\right)dr.\end{array}$$

(7)

By substituting $t=2Z_{2}r/n_2$, where $r=\left(n_2/\left(2Z_2\right)\right)t$ and $dr=\left(n_2/\left(2Z_2\right)\right)dt$, and defining:

$$\sigma ={\displaystyle \frac{Z_1{n}_2+Z_2{n}_1}{2Z_2{n}_1}},\sigma -1={\displaystyle \frac{Z_1{n}_2-Z_2{n}_1}{2Z_2{n}_1}},$$

we find that the integral (not all expression) in Equation (7) becomes:

$${\left({\displaystyle \frac{n_2}{2Z_2}}\right)}^{l_1+l_2+3+\beta +q}{\int }_0^{\infty }{e}^{-\sigma t}{t}^{l_1+l_2+2+\beta +q}{L}_{n_2-l_2-1}^{2l_2+1}\left(t\right)dt,$$

(8)

or, essentially, the Laplace transform:

$${\left({\displaystyle \frac{n_2}{2Z_2}}\right)}^{l_1+l_2+3+\beta +q}\mathcal{L}\left[t^{l_1+l_2+2+\beta +q}{L}_{n_2-l_2-1}^{2l_2+1}\left(t\right)\right].$$

(9)

Using (5) with $x=l_1-l_2+\beta +q+1$, $p=2l_2+1$, $m=n_2-l_2-1$, we have:

$$\mathcal{L}\left[t^{l_1+l_2+2+\beta +q}{L}_{n_2-l_2-1}^{2l_2+1}\left(t\right)\right]={\displaystyle \frac{\left(l_1+l_2+2+\beta +q\right)!}{{\left[\left(Z_1{n}_2+Z_2{n}_1\right)/\left(2Z_2{n}_1\right)\right]}^{l_1+n_2+2+\beta +q}}}$$

$$\times \sum _{k=0}^{n_2-l_2-1}{\left(-1\right)}^k{\left({\displaystyle \frac{Z_1{n}_2-Z_2{n}_1}{2Z_2{n}_1}}\right)}^{n_2-l_2-1-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n_2+l_2}{2l_2+1+k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{l_1-l_2+\beta +q+1}{k}}\right),$$

(10)

and then we can finally express the value of $P_{\alpha ,{\alpha }^{\prime }}^{\left(\beta \right)}$ as follows. From Equations (8)–(10), it follows that:

$$\begin{array}{c}{P}_{\alpha ,{\alpha }^{\prime }}^{\left(\beta \right)}={\displaystyle \frac{2^{l_1+l_2+2}{Z}_1^{l_1+3/2}{Z}_2^{l_2+3/2}{\left[n_2/\left(2Z_2\right)\right]}^{l_1+l_2+3+\beta }}{n_1^{l_1+2}{n}_2^{l_2+2}{\left[\left(Z_1{n}_2+Z_2{n}_1\right)/\left(2Z_2{n}_1\right)\right]}^{l_1+n_2+2+\beta }}}\\ \times \sqrt{{\displaystyle \frac{\left(n_1-l_1-1\right)!\left(n_2-l_2-1\right)!}{\left(n_1+l_1\right)!\left(n_2+l_2\right)!}}}\\ \times \sum _{q=0}^{n_1-l_1-1}{\displaystyle \frac{{\left(-1\right)}^q}{q!}}\left({\displaystyle \genfrac{}{}{0pt}{}{n_1+l_1}{n_1-l_1-1-q}}\right){\left({\displaystyle \frac{2Z_1{n}_2}{Z_1{n}_2+Z_2{n}_1}}\right)}^q\left(l_1+l_2+2+\beta +q\right)!\\ \times \sum _{k=0}^{n_2-l_2-1}{\left(-1\right)}^k{\left({\displaystyle \frac{Z_1{n}_2-Z_2{n}_1}{2Z_2{n}_1}}\right)}^{n_2-l_2-1-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n_2+l_2}{k+2l_2+1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{l_1-l_2+\beta +q+1}{k}}\right).\end{array}$$

(11)

The different particular cases from the general result (11) can be derived in the next subsection, especially the cases where both $Z_1=Z_2$ and $n_2=n_1$.

Remark: In Equations (10) and (11) we have assumed that $\beta$ is an integer. However, more general cases have been considered by other authors. For example, Blanchard addresses the radial matrix element of Equation (1) and considers (in his words): “…and $\beta$ are any complex number for which the integral is defined” [16]. Similarly, Shertzer states that $\beta \ge -\left(l+l^{\prime }+2\right)$, but $\beta$ is not restricted to integer values [17].

An important case in plasma physics arises when using the ion-sphere approach, as done by Rosmej and collaborators [18]. They generated a fit to the potential due to free electrons in self-consistent ion-sphere calculations, with one of the terms proportional to $r^{3/2}$. The same issue was later discussed by Iglesias [19] and by Pain [20]. The latter author relied on the general expressions of Shertzer, recently discussed. Consequently, non-integer powers of $\beta$ are of particular interest in the context of atomic physics for dense plasmas. Therefore, it is necessary to generalize the aforementioned equations by making the following modifications: replace the factorials with their corresponding Gamma functions $\mathrm{\Gamma }$.

$$x!\to \mathrm{\Gamma }\left(x+1\right)$$

(12)

and therefore the binomial coefficients:

$$\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)={\displaystyle \frac{n!}{k!\left(n-k\right)!}}\to {\displaystyle \frac{\mathrm{\Gamma }\left(n+1\right)}{\mathrm{\Gamma }\left(k+1\right)\mathrm{\Gamma }\left(n-k+1\right)}}.$$

(13)

We have verified that expression (11) is entirely adequate for calculating radial integrals with non-integer $\beta$. It is an alternative formula that offers potential advantages for simplified analytical expressions in specific cases, as will be shown in the following Section “Non-Relativistic Particular Cases”. It should be noted that references [11,13] provide further context on this topic. A rigorous derivation of closed-form expressions for any real value of $\beta$ and any combination of upper and lower state quantum numbers is presented in terms of Gamma functions. Furthermore, in Ref. [13], this representation is extended to accommodate any values of the effective nuclear charges of the upper and lower states.

Non-Relativistic Particular Cases

It is well known that there are no general explicit expressions for obtaining the radial matrix elements $〈n,l\left|r^{\beta }\right|n^{\prime },l^{\prime }〉$ in general, not even in the case where $Z_i=Z_f$. By explicit, we mean a clear dependence of the matrix elements on the quantum numbers n, n’, l, l’,$\beta$.

The first researcher to approach the calculation of $P_{\alpha ,{\alpha }^{\prime }}^{\left(\beta \right)}$, though without fully succeeding, was Gordon, as mentioned in the classical work of Bethe-Salpeter [21]. However, Gordon was only able to obtain values for $n\ne n^{\prime },$ which sufficed for hydrogenic elements. However, when employing the Screened Hydrogenic Model (SHM), where, for instance, $E\left(np\right)\ne E\left(ns\right),$ etc., a more general formulation is required to compute, for example, the so-called intra-shell transitions. Beyond the numerous studies conducted over the years, we will cite some of them in order of publication: Badawi et al. [22], Blanchard [16], Matsumoto [23], Shertzer [17], Morales et al. [24], Hey [25], Blaive et al. [26], and Bautista-Moedano et al. [27]. The work of Shertzer [17] presents particular cases where $n_1=n_2$ (which Gordon did not address), but Shertzer could not resolve the cases where $n_1\ne n_2,$ which Gordon did solve.

We will not specify the different particular cases but will instead focus solely on the case where both $Z_1=Z_2$ and $n_1=n_2.$ Referring to the general solution given by Equation (11), we see that the factor

$${\left({\displaystyle \frac{Z_1{n}_2-Z_2{n}_1}{2Z_2{n}_1}}\right)}^{n_2-l_2-1-k},$$

(14)

appears, with $k=\left[0,n_2-l_2-1\right]$. The only nonzero term occurs when $k=n_2-l_2-1$, reducing the entire sum to the term

$${\left(-1\right)}^{n_2-l_2-1}\left({\displaystyle \genfrac{}{}{0pt}{}{l_1-l_2+\beta +q+1}{n_2-l_2-1}}\right),$$

(15)

so that the expectation values are given by

$$\begin{array}{cc}〈nl\left|r^{\beta }\right|nl〉={\displaystyle \frac{{\left(-1\right)}^{n-l-1}}{2n}}\hfill & {\left({\displaystyle \frac{n}{2Z}}\right)}^{\beta }\hfill \\ & \hfill \times \sum _{q=0}^{n-l-1}{\displaystyle \frac{{\left(-1\right)}^q}{q!}}\left({\displaystyle \genfrac{}{}{0pt}{}{n+l}{n-l-1-q}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\beta +q+1}{n-l-1}}\right)\left(2l+2+\beta +q\right)!,\end{array}$$

(16)

Using this expression, it is straightforward to obtain, for example, the results given by other authors, particularly those of Shertzer [17], Blaive et al. [26] and others.

3. Relativistic Expressions

For cases where the ions present are of high Z, it is necessary to use the relativistic point of view [8,28], both for the wave functions and for the operators. The same is true in the case of warm and dense matter [1,2]. In this section, for the sake of simplicity, we will not use the operators corresponding to a relativistic treatment, which can be found in the references indicated at the beginning of this section. The radial functions can be found, in full detail, in various references. In this work, we take those of Mizushima [29], with the normalization

$${\int }_0^{\infty }\left({\left|F_{nj}\right|}^2+{\left|G_{nj}\right|}^2\right)r^{2}dr=1,$$

(17)

in this notation, $F_{nj}$ is the long component and $G_{nj}$ is the small component.

Defining this way

$$j_{+}=l+1/2,j_{-}=\left|l-1/2\right|,$$

$${\lambda }_{\pm }={\left[{\left(j_{\pm }+1/2\right)}^2-{\alpha }^2{Z}^2\right]}^{1/2},$$

and

$$\kappa =\left(l-j\right)\left(2j+1\right)\equiv {\left(-1\right)}^{l+s+j}\left(j+1/2\right),$$

therefore

$${\kappa }_{\pm }=\mp \left(j_{\pm }+1/2\right);$$

to which are added

$$n_{+}^{\prime }=n-j_{+}-1/2=n-l-1,$$

$$n_{-}^{\prime }=n-j_{-}-1/2=n-l,$$

$$N_{\pm }={\left(\left[n^2-2n_{\pm }^{\prime }\left(j_{\pm }+1/2-{\lambda }_{\pm }\right)\right]\right)}^{1/2};$$

and the expression of self-energies

$${\mathcal{E}}_{nj}=\mu c^2{\left[1-{\left({\displaystyle \frac{\alpha Z}{N}}\right)}^2\right]}^{1/2},$$

which we will use explicitly later, in Equation (26).

Radial Part of Eigenfunctions

These are normalized according to Equation (17),

$$F_1\left(r\right)=M(-n^{\prime }+1,2\lambda +1;2Zr/N)\mathrm{y}{F}_2\left(r\right)=M\left(-n^{\prime },2\lambda +1;2Zr/N\right),$$

(18)

begin $M\left(a,b,x\right)$ Kummer functions. We will have, then

$$F_{nj}\left(r\right)=C_{nj}^F\left\{-n^{\prime }{F}_1\left(r\right)+\left(N-\kappa \right)F_2\left(r\right)\right\}{\left({\displaystyle \frac{2Zr}{N}}\right)}^{\lambda -1}exp\left(-{\displaystyle \frac{Zr}{N}}\right),$$

(19)

(see (29)) and

$$G_{nj}\left(r\right)=C_{nj}^G\left\{n^{\prime }{F}_1\left(r\right)+\left(N-\kappa \right)F_2\left(r\right)\right\}{\left({\displaystyle \frac{2Zr}{N}}\right)}^{\lambda -1}exp\left(-{\displaystyle \frac{Zr}{N}}\right),$$

(20)

where

$$C_{nj}^F={\left[C\left(1+{\displaystyle \frac{{\epsilon }_{nj}}{\mu c^2}}\right)\right]}^{1/2},$$

(21)

and

$$C_{nj}^G={\left[C\left(1-{\displaystyle \frac{{\epsilon }_{nj}}{\mu c^2}}\right)\right]}^{1/2},$$

(22)

with

$$C={\left[{\displaystyle \frac{\mathrm{\Gamma }\left(2\lambda +n^{\prime }+1\right)}{{\left\{\mathrm{\Gamma }\left(2\lambda +1\right)\right\}}^2\mathrm{\Gamma }\left(n^{\prime }+1\right)4N\left(N-\kappa \right)}}{\left({\displaystyle \frac{2Z}{N}}\right)}^3\right]}^{1/2}.$$

(23)

Then

$$\begin{array}{c}{F}_{nj}^2\left(r\right)+G_{nj}^2\left(r\right)=\\ 2C\left\{n^{\prime 2}{F}_1^2\left(r\right)+{\left(N-\kappa \right)}^2{F}_2^2\left(r\right)-2\left({\epsilon }_{nj}/\mu c^2\right)n^{\prime }\left(N-\kappa \right)F_1\left(r\right)F_2\left(r\right)\right\}\\ \times {\left({\displaystyle \frac{2Zr}{N}}\right)}^{2\lambda -2}exp\left(-{\displaystyle \frac{2Zr}{N}}\right),\end{array}$$

(24)

such that Equation (17) is fulfilled.

4. A Useful Approximation

In calculating the effects due to the Breit interaction and quantum electrodynamics, it is essential to use both $F_{nj}\left(r\right)$ and $G_{nj}\left(r\right)$. This can be useful when studying K-shell spectroscopy in hot plasmas: Stark effect, Breit interaction and QED corrections [30], but we will not address this in this article. In our case, since it is always true that $F_{nj}^2\left(r\right)\gg G_{nj}^2\left(r\right)$, we can replace the integrand $F_{nj}^2\left(r\right)+G_{nj}^2\left(r\right)$ with a function of the same form as the large component, which we will denote as $P_{nlj}\left(r\right)$, such that

$$P_{nlj}^2\left(r\right)\approxeq F_{nj}^2\left(r\right)+G_{nj}^2\left(r\right).$$

(25)

Undoubtedly, $P_{nlj}\left(r\right)$ (or, more specifically, $C_{nj}^F$ or C) must be multiplied by a (small) factor. Developing as in (24), we obtain a rather cumbersome expression dependent on r, which simplifies significantly when $j=l+1/2$. In this case, we must replace C with

$$C^{new}\to {\displaystyle \frac{2C^{old}}{(1+\epsilon )}},\mathrm{with}\epsilon ={\left(1-{\left({\displaystyle \frac{\alpha Z}{N}}\right)}^2\right)}^{1/2}\approx 1,$$

(26)

with $\epsilon <1$, this implies that $C^{new}>C^{old}$, thus compensating for the $G_{nj}^2\left(r\right)$ term, which was neglected when setting $P_{nlj}\left(r\right)$ to have the same form as the large component.

5. Multipolar Radial Integrals

In previous sections, we have seen that the hydrogenic radial functions $P_{nl}^{hid}\left(r\right)$ can be written using associated Laguerre polynomials, Kummer functions or directly in the developed forms, as a product of a polynomial and an exponential. The relativistic case is completely analogous.

We now write the radial functions introduced in the previous paragraph. They can be expressed in the developed form1

$$P_{nlj}\left(r\right)=C_{nlj}\sum _{v=0}^{n^{\prime }}{f}_v{r}^{v+\lambda }exp(-Zr/N),$$

(27)

where

$$f_v={(-1)}^v{\displaystyle \frac{\left(2\lambda \right)!{(2Z/N)}^{v+\lambda -1}{n}^{\prime }!}{v!(2\lambda +v)!(n^{\prime }-v)!}}(v-n^{\prime }+N-\kappa ),$$

(28)

and $C_{nlj}$ are given by (21), eventually transformed using (26).

It is remarkable that Equation (27), with its coefficients given by Equation (28), which is merely an alternative way of summing the two Kummer functions in Equation (19), can be expressed using the following generalized hypergeometric function:

$$\begin{array}{c}{P}_{nlj}\left(r\right)={\displaystyle \frac{N{C}_{nlj}}{2Z}}{\left({\displaystyle \frac{2Zr}{N}}\right)}^{\lambda }{e}^{-Zr/N}(-n^{\prime }+N-\kappa )\\ {\times }_2{F}_2([-n^{\prime },1-n^{\prime }+N-\kappa ],[2\lambda +1,-n^{\prime }+N-\kappa ],2Zr/N),\end{array}$$

(29)

but now normalized such that2

$${\int }_0^{\infty }{P}_{nlj}^2\left(r\right)dr=1.$$

(30)

We have verified, for high values of Z, by appropriately correcting with Equation (26), that the normalization condition is perfectly satisfied. To the best of our knowledge, an expression like Equation (29) has not been previously published.

At present, we are not in a position to address the practical potential of Formula (29). Indeed, the treasure trove of hypergeometric functions is overwhelming, as can be verified by consulting the manuals [31,32] or [33]. It is difficult to trace whether, for instance, integrals involving functions of the type $2F2$ multiplied by $r^{\beta }$ have been previously published.

A general property of these $P_{nlj}\left(r\right)$ functions is that, denoting $P_{nl}$ as the non-relativistic functions, $P_{nlj}^{+}$ as the relativistic function for $j=j_{+}$ and $P_{nlj}^{-}$ for $j=j_{-}$, we find

$$P_{nlj}^{+}\approx P_{nl},\mathrm{and}{P}_{nlj}^{-}\approx c{r}^l{e}^{-Zr/N}+P_{nl},$$

since $n_{+}^{\prime }=n-l-1$ and $n_{-}^{\prime }=n-l$. With reference to integrals

$${\int }_0^{\infty }{P}_{n^{\prime }{l}^{\prime }{j}^{\prime }}{r}^{\gamma }{P}_{nlj}\left(r\right)dr,$$

(31)

the result we will derive below will be valid regardless of the effective charges of the initial and final states. Writing the radial functions as in (27) and (28), we will have, with a slight change of notation,

$${\int }_0^{\infty }{r}^{\gamma }{F}_i\left(r\right)F_t\left(r\right)dr=C_i{C}_t\sum _{i=0}^{n_i^{\prime }}\sum _{t=0}^{n_t^{\prime }}{a}_i{b}_t{\int }_0^{\infty }{r}^{i+t+{\lambda }_i+\lambda t+\gamma }exp\left(-X_{it}r\right)dr,$$

with $X_{it}=Z_i/N_i+Z_t/N_t$. Taking into account the value of the integral

$${\int }_0^{\infty }{e}^{-\alpha x}{x}^{n}dx={\displaystyle \frac{\mathrm{\Gamma }\left(n+1\right)}{{\alpha }^{n+1}}}(={\displaystyle \frac{n!}{{\alpha }^{n+1}}}\mathrm{for}\mathrm{integer}n),$$

the final result is:

$${\int }_0^{\infty }{r}^{\gamma }{F}_i\left(r\right)F_t\left(r\right)dr=C_i{C}_t\sum _{i=0}^{n_i^{\prime }}\sum _{t=0}^{n_t^{\prime }}{a}_i{b}_t{\displaystyle \frac{\mathrm{\Gamma }\left(i+t+{\lambda }_i+\lambda t+\gamma +1\right)}{X_{it}^{i+t+{\lambda }_i+{\lambda }_t+\gamma +1}}}.$$

(32)

The expectation values

$$〈r^{\gamma }〉={\int }_0^{\infty }{r}^{\gamma }{P}_{nlj}^2\left(r\right)dr$$

(33)

will be found as a particular case of expression (32). It is worth noting that the theoretical calculation of $〈r^{\gamma }〉$ using large and small components has a large tradition (see, for example, Ref. [8]). A comprehensive study of relativistic screened hydrogenic radial integrals was presented by Ruano et al. [34]. Later, J. C. Pain [35] refined this approach by incorporating Clebsch–Gordan coefficients.

The value of $\lambda$ that appears in the above equations is, as we have seen, non-integer (in general, differing little from an integer, except for the 1s orbital and for very large net charges). This means that, unlike the non-relativistic case, the definite integrals cannot be calculated in a closed analytic form. What we will do here is assume that the $\lambda$ values are integers and then, in the final result, substitute their relativistic values.

The factorials that will appear in the final results (with non-integer $\lambda$) are calculated using Stirling’s formula [36]:

$$n!\sim \sqrt{2\pi n}{\left({\displaystyle \frac{n}{e}}\right)}^n\left(1+{\displaystyle \frac{1}{12n}}+{\displaystyle \frac{1}{288n^2}}-{\displaystyle \frac{139}{51840n^3}}\right)$$

(34)

with $e\approx 2.7183$.

For the case of Li-like ions, the matrix elements $P_{\alpha ,{\alpha }^{\prime }}^{\left(\beta \right)}$ for the $2s-2p$ transitions can differ, between the non-relativistic and relativistic cases, by up to 3% for Z = 36 (Kr), 6% for Z = 54 (Xe) and 16% for Z = 79 (Au). Since the oscillator strengths depend on ${\left(P_{\alpha ,{\alpha }^{\prime }}^{\left(\beta \right)}\right)}^2$, these numbers become 6%, 12% and 32%, respectively. For Na-like ions, on the other hand, the discrepancies are smaller, around 8% for ${\left(P_{\alpha ,{\alpha }^{\prime }}^{\left(\beta \right)}\right)}^2$ in the case of Z = 54 and 20% for Z = 79.

As mentioned above, the use of relativistic operators is not taken into account in this work.

6. Use of the SHM for Dense and Hot Plasmas

As can be seen in Refs. [1,4], collisional-radiative (CR) models require a vast number of collisional and radiative rates. These can be computed using either the LS or the jj coupling schemes. The same applies to statistical treatments of levels, lines and intensities when the number of transitions reaches millions or even billions [2], as we will discuss later on. In this section, we address a few specific topics.

6.1. Electron-Impact Excitation Cross Sections

The SHM, with different screening parameters for initial and final states, can be used to calculate excitation cross sections within the Plane-Wave Born Approximation (PWBA). This approach requires the generalized oscillator strength [7]:

$$g{f}_{J{J}^{\prime }}\left(K\right)={\displaystyle \frac{\mathrm{\Delta }E}{K^2}}\sum _t(2t+1){〈\gamma J∥\sum _m{j}_t\left(K{r}_m\right)∥{\gamma }^{\prime }{J}^{\prime }〉}^2,$$

(35)

where $\mathbf{K}={\mathbf{k}}^{\prime }-\mathbf{k}$ is the momentum transferred from the free electron to the atom. Using Rydbergs ($Ry$) as energy units, we set $k=\sqrt{\epsilon }$, with $\epsilon$ being the kinetic energy of the free electron.

The evaluation of the matrix element in LS coupling reads:

$$S_{LS}^{1/2}\equiv D_{LS}=D_1{D}_2\dots D_7〈l_i∥j_t\left(Kr\right){\mathbf{C}}^{\left(t\right)}∥l_k〉\equiv D_{LS}^{\mathrm{ang}}〈l_i∥j_t\left(Kr\right){\mathbf{C}}^{\left(t\right)}∥l_k〉,$$

(36)

involving 6-j symbols and fractional parentage coefficients (cfp). The angular factor $D_{LS}^{\mathrm{ang}}$ is the same used in the calculation of oscillator strengths (here denoted simply as $gf$). The only difference is in the radial integrals: for oscillator strengths, we use the one-electron radial matrix element

$$〈l_i∥r^t{\mathbf{C}}^{\left(t\right)}∥l_k〉,$$

(37)

while in the present case we use

$$〈l_i∥j_t\left(Kr\right){\mathbf{C}}^{\left(t\right)}∥l_k〉={(-1)}^t\sqrt{[l_i,l_k]}\left(\begin{array}{ccc}{l}_i& t& l_k\\ 0& 0& 0\end{array}\right){\int }_0^{\infty }{P}_i\left(r\right)j_t\left(Kr\right)P_k\left(r\right)dr.$$

(38)

The calculation of $D_{LS}^{\mathrm{ang}}$ is generally complex, but the expressions simplify significantly for configurations such as $l_1^n{l}_2^{k-1}\to l_1^{n-1}{l}_2^k$, $l_1^n\to l_1^{n-1}{l}_2$ or $l_1{l}_2\to l_1{l}_2^{\prime }$, provided closed shells are excluded [7], as we will see next.

The calculation of the hydrogenic case of excitation $n_a{l}_a\to n_b{l}_b$ (independent of J) was considered, among others, by Cowan [7], Upcraft [9] and Pain and Benredjem [14], using:

$$gf\left(K\right)={\displaystyle \frac{\mathrm{\Delta }E}{K^2}}(2l_a+1)(2l_b+1)\sum _t(2t+1){\left[\left(\begin{array}{ccc}{l}_a& t& l_b\\ 0& 0& 0\end{array}\right){\int }_0^{\infty }{P}_a\left(r\right)j_t\left(Kr\right)P_b\left(r\right)dr\right]}^2,$$

(39)

with $|l_a-l_b|\le t\le l_a+l_b$, and $\mathrm{mod}(l_a+l_b+t+2)=0$. Here, $j_t\left(Kr\right)$ denotes the spherical Bessel function of order t. In this approximation, $gf\left(K\right)$ can be expressed analytically, which is useful for computing excitation rates:

$$N_e\langle v\sigma \rangle =N_e{\int }_{\mathrm{\Delta }E}^{\infty }v\sigma \left(\epsilon \right)F\left(\epsilon \right)d\epsilon ,$$

(40)

where $F\left(\epsilon \right)$ is the energy distribution function of the plasma electrons.

An explicit expression for $gf\left(K\right)$ in terms of the quantum numbers $\left(n,l,Z_{eff}\right)$ can be found in Ref. [14] (published in 2021), taking special care to account for the correction that appeared in 2024. An alternative method for calculating excitation cross sections involves the well-known Van Regemorter formula, which utilizes effective Gaunt factors [37]. A development along these lines was presented by Fisher et al. [38].

As anticipated in the introduction, Equation (39) becomes equivalent–when $Kr\to 0$–to the calculation of weighted oscillator strengths $g{f}_{n,n^{\prime }}$, though not for transitions of the form $\gamma J\to {\gamma }^{\prime }{J}^{\prime }$. For such detailed cases, one must use the more general expressions in Equation (36), although the radial integrals can still be computed using screened hydrogenic wavefunctions.

If jj coupling is needed, then the corresponding angular factor $S_{jj}^{1/2}\equiv D_{jj}$ should be computed using the methods presented in the following sections.

6.2. Relative Advantages of the PWBA

Nowadays, various computational codes are available for computing different kinds of cross sections, particularly electron-impact excitation cross sections. Some of these are listed in [14]. However, when integrating atomic processes under non-local thermodynamic equilibrium (NLTE) conditions into fully integrated radiation-hydrodynamics simulations, fast and reasonably accurate computations are needed. This is where PWBA-type methods and other semi-empirical expressions, such as the well-known Van Regemorter formula (with a suitably chosen Gaunt factor), become particularly useful.

By reasonably accurate, we mean that such formulas—being easy to implement—allow for realistic estimates of coarse plasma quantities such as the average ionization or ionic populations [39].

Moreover, in the case of PWBA-type calculations, it is possible to explore the influence of various plasma density effects—especially those due to electrons—on the cross sections [14].

7. From LS to jj Matrix Elements

Since many works related to WDM use non-relativistic expressions [1,2,3] (e.g., simplified expressions for excitation and ionization by electron impact). Here, we present a few rules for converting LS matrix elements to jj matrix elements. As a summary of what was published in [40], we will focus on introducing expressions related to radiative transitions, which were not published there and are presented here for the first time3.

7.1. Necessary Modifications

In the aforementioned reference, the expressions of atomic theory in the LS coupling, extensively and clearly developed in Cowan’s book [7]4, were analyzed. It was shown that if the expressions are in LS, going to jj is straightforward. The reciprocal case is not as simple, as the angular spin moments do not appear in the jj formulation, making their subsequent inclusion more complicated. Recall that all coefficients in the theory, corresponding to both energies and transition probabilities, are expressed through quantities such as $3nj$ symbols, reduced matrix elements of various tensor operators (e.g., $〈l_1∥C^{\left(k\right)}∥l_2〉$, etc.) and fractional parentage coefficients.

Let us begin by recalling some relationships for various matrix elements of the normalized spherical harmonics $C_q^{\left(k\right)}={\left(4\pi /\left(2k+1\right)\right)}^{1/2}{Y}_{kq}$. For non-relativistic functions, the relationship is [7]:

$$〈l_1∥C^{\left(k\right)}∥l_2〉={\left(-1\right)}^{l_1}{[l_1,l_2]}^{1/2}\left(\begin{array}{ccc}{l}_1& k& l_2\\ 0& 0& 0\end{array}\right),$$

(41)

whereas for the relativistic case, it is [8,41]:

$$〈j_1∥C^{\left(k\right)}∥j_2〉={\left(-1\right)}^{j_1+1/2}{[j_1,j_2]}^{1/2}\left(\begin{array}{ccc}{j}_1& j_2& k\\ -1/2& 1/2& 0\end{array}\right)\mathrm{\Pi }\left(l_1+l_2+k\right),$$

(42)

where $\mathrm{\Pi }\left(x\right)$ is the parity: $\mathrm{\Pi }\left(x\right)=1$ if x is even and $\mathrm{\Pi }\left(x\right)=0$ if x is odd. In some texts, the matrix element on the left-hand side is denoted $〈{\kappa }_1∥C^{\left(k\right)}∥{\kappa }_2〉$, where

$$\kappa ={\left(-1\right)}^{l+s+j}\left(j+1/2\right)=\left(l-j\right)\left(2j-1\right).$$

(43)

In Appendix A, it is shown how, starting from Equations (41), (A1) and (A2), we can arrive at Equation (42).

The above results apply to the calculation of various matrix elements, which are computed using Racah Algebra. To transition from $LS$ formulas (from Ref. [7]) to $jj$ formulas ([41] and/or [28]), the following changes must be made:

1.

Ignore all spin angular momentum symbols $(s,S,\mathrm{etc}.)$ in factors like ${\delta }_{S{S}^{\prime }}$, and nullify the values of $(s,S,\dots )$ in exponents like ${\left(-1\right)}^{\mathrm{something}}$, as well as in $S_{6j}$ symbols that contain any spin angular momentum $(s,S,\mathrm{etc}.)$. If a $S_{6j}$ involving spins has all its elements zero, then $S_{6j}=1$ (this is a general property of $S_{6j}$ symbols).

2.

Replace all symbols $l,L,\overline{L},\dots$ with their respective $j,J,\overline{J},\dots$, preserving their meaning (e.g., if $\overline{L}$ indicates the orbital angular momentum of the parent, $\overline{J}$ will indicate the total angular momentum of the parent, etc.).

3.

If the expression in LS coupling contains matrix elements of the form $〈l_1∥C^{\left(k\right)}∥l_2〉$, replace them directly with $〈j_1∥C^{\left(k\right)}∥j_2〉$ as in Equation (42).

4.

Alternatively, based on Equations (41) and (42), if the combination

$${\left(-1\right)}^{l_1}\left(\begin{array}{ccc}{l}_1& k& l_2\\ 0& 0& 0\end{array}\right)$$

(44)

appears, replace it with

$${\left(-1\right)}^{j_1+1/2}\left(\begin{array}{ccc}{j}_1& j_2& k\\ -1/2& 1/2& 0\end{array}\right),$$

(45)

along with the aforementioned changes.

5.

In matrix elements of the unit tensor operator $U^{\left(k\right)}$, $〈l^N∥U^{\left(k\right)}∥l^N〉$, expressed in terms of $6j$ symbols $\left(S_{6j}\right)$ and fractional parentage coefficients (cfp), make the changes described in Steps 1–3 and replace LS-cfp $\left(l_i^{n-1}\overline{{\alpha }_i}\overline{L_i}\overline{S_i}\parallel l_i^n{\alpha }_i{L}_i{S}_i\right)$ por los jj-cfp $\left(j^{n-1}\overline{\alpha }\overline{J}\parallel j^n\alpha J\right)$,

6.

In the case of double tensor operators $〈l^N∥V^{\left(k1\right)}∥l^N〉$ (essential for evaluating spin-orbit interaction in LS coupling), note that if “S” terms are removed, then $〈l^N∥V^{\left(k1\right)}∥l^N〉$ transforms into $〈l^N∥U^{\left(k\right)}∥l^N〉$, which is replaced by elements of the form $〈j^N∥U^{\left(k\right)}∥j^N〉$ or equivalent expressions, as presented in Ref. [28].

7.

For radiative transitions (transition probabilities $A_{ij}$, or oscillator strengths $f_{ij}$), we encounter a fundamental expression given by the reduced matrix element ${\mathbf{P}}_{\alpha {\alpha }^{\prime }}^{\left(1\right)}$, where $\alpha ,{\alpha }^{\prime }$ denotes, for brevity, the values $n,l,j$, etc. Thus, we write ${\mathbf{P}}_{\alpha {\alpha }^{\prime }}^{\left(1\right)\mathrm{no}-\mathrm{rel}}$ or ${\mathbf{P}}_{\alpha {\alpha }^{\prime }}^{\left(1\right)\mathrm{rel}}$, as appropriate:

$${\mathbf{P}}_{\alpha {\alpha }^{\prime }}^{\left(1\right)\mathrm{no}-\mathrm{rel}}\equiv 〈nl\left|\left|\mathbf{r}\right|\right|n^{\prime }{l}^{\prime }〉\equiv 〈l∥C^{\left(k\right)}∥l^{\prime }〉I_{\alpha {\alpha }^{\prime }}^{\left(1\right)\mathrm{no}-\mathrm{rel}},$$

(46)

where

$$I_{\alpha {\alpha }^{\prime }}^{\left(1\right)\mathrm{no}-\mathrm{rel}}\equiv {\int }_0^{\infty }{P}_{nl}r{P}_{n^{\prime }{l}^{\prime }}dr.$$

(47)

Using the above rules, we can directly write:

$${\mathbf{P}}_{\alpha {\alpha }^{\prime }}^{\left(1\right)\mathrm{rel}}\equiv 〈nlj\left|\left|\mathbf{r}\right|\right|n^{\prime }{l}^{\prime }{j}^{\prime }〉\equiv 〈j∥C^{\left(k\right)}∥j^{\prime }〉I_{\alpha {\alpha }^{\prime }}^{\left(1\right)\mathrm{rel}},$$

(48)

where $I_{\alpha {\alpha }^{\prime }}^{\left(1\right)\mathrm{rel}}$ is the relativistic generalization of $I_{\alpha {\alpha }^{\prime }}^{\left(1\right)\mathrm{no}-\mathrm{rel}}$ (see below: Section 7.3).

Since all the final expressions in Ref. [7] are expressed in terms of $6j$ symbols and the reduced matrix elements ${\mathbf{P}}_{\alpha {\alpha }^{\prime }}^{\left(1\right)}$, we will present our results in the jj cases under consideration similarly.

7.2. More Complex Matrix Elements: The 9jCoefficients

As mentioned in the last sentence of the previous paragraph, we do not require $3nj$ symbols beyond the $6j$ symbols. However, from both a notational and even conceptual standpoint, $9j$ symbols often appear in various contexts. Notable examples include the transformation between $LS$ and $jj$ couplings, many cases of configuration interaction and matrix elements of the type

$$〈\left(l_1^{w_1}{L}_1{S}_1\right)\left(l_2^{w_2}{L}_2{S}_2\right)LS∥\widehat{H}∥\left(l_1^{w_1}{L}_1^{\prime }{S}_1^{\prime }\right)\left(l_2^{w_2}{L}_2^{\prime }{S}_2^{\prime }\right)L^{\prime }{S}^{\prime }〉$$

or its relativistic counterpart. However, in this article, we will not delve into the $9j$ symbols. As explained in Reference [7], the $9j$ symbol is primarily of notational utility; for computational purposes, it can be fully expressed in terms of $6j$ symbols (see Equation (5.37) in that book).

7.3. Plane Wave Collision Strengths and Radiative Transitions

In this section, we transform the results derived in Ref. [7] to the $jj$ case. The fundamental quantity in the theory is the reduced dipole matrix element $D_{\beta {\beta }^{\prime }}=〈\beta J∥P^{\left(1\right)}∥{\beta }^{\prime }{J}^{\prime }〉$ (in the representation $\beta {\beta }^{\prime }$). The most general case can be represented as general transition arrays [7].

$$l_1^{w_1}\dots l_i^n\dots l_j^{k-1}\dots l_q^{w_q}-l_1^{w_1}\dots l_i^{n-1}\dots l_j^k\dots l_q^{w_q},$$

(49)

and is presented in the form

$$D_{\mathcal{LS}}=D_1{D}_2\dots D_7{P}_{l_i{l}_j}^{\left(t\right)};$$

(50)

Next, we will refer exclusively (in general terms) to the angular part $D_{\mathcal{LS}}^{\mathrm{ang}}=D_1{D}_2\dots D_7$, as the angular component remains the same for the calculation of both PWB collision strengths and oscillator strengths. These components must be multiplied by either ${\mathbf{P}}_{\alpha {\alpha }^{\prime }}^{\left(1\right)\mathrm{no}-\mathrm{rel}}$ (or ${\mathbf{P}}_{\alpha {\alpha }^{\prime }}^{\left(1\right)\mathrm{rel}}$) for oscillator strength calculations, or by an integral of the form:

$$\mathcal{I}(t,\beta ,c,K)={\int }_0^{\infty }{e}^{-cr}{j}_t\left(Kr\right)r^{\beta }dr,$$

as shown in [14].

7.4. The Simplest Case

As an example of the changes to be made, let us consider the simplest case: the $l-l^{\prime }$ transition in hydrogen-like atoms. Based on Equation (14.46) in Ref. [7], for a single-electron system $\left(L\to l,S\to s,\mathrm{etc}.\right)$, we obtain:

$$D_{LS}={\left(-1\right)}^{l+s+j^{\prime }+1}{\left[j,j^{\prime }\right]}^{1/2}\left\{\begin{array}{ccc}l& j& s\\ j^{\prime }& l^{\prime }& 1\end{array}\right\}{\mathbf{P}}_{\alpha {\alpha }^{\prime }}^{\left(1\right)\mathrm{no}-\mathrm{rel}},$$

(51)

In the relativistic case, the situation becomes more complex, as shown in Refs. [8,28]. There, several radial integrals appear, which will not be detailed here. Grant expresses:

$$M_{\alpha {\alpha }^{\prime }}^e\left(\omega ;G_k\right)=M_{\alpha {\alpha }^{\prime }}^e\left(\omega ;0\right)+G_k{M}_{\alpha {\alpha }^{\prime }}^l\left(\omega \right),$$

(52)

where $M_{\alpha {\alpha }^{\prime }}^e\left(\omega ;0\right)$ (the integral in the Coulomb gauge) corresponds, in the Pauli approximation, to the velocity electric dipole matrix element, while the longitudinal component $M_{\alpha {\alpha }^{\prime }}^l\left(\omega \right)$ leads, in the same approximation, to the length electric dipole matrix element (Babushkin gauge: $G_k=\sqrt{\left(k+1\right)/k}$). Thus, if we choose to work in this latter case, we write:

$$D_{jj}={\left(-1\right)}^{j+j^{\prime }+1}{\left[j,j^{\prime }\right]}^{1/2}\left\{\begin{array}{ccc}j& j& 0\\ j^{\prime }& j^{\prime }& 1\end{array}\right\}{\mathbf{P}}_{\alpha {\alpha }^{\prime }}^{\left(1\right)\mathrm{rel}},$$

(53)

and we evaluate ${\mathbf{P}}_{\alpha {\alpha }^{\prime }}^{\left(1\right)\mathrm{rel}}$ using the expressions above (48). If detailed evaluation, including both large and small components, is required, we indicate it using $M_{\alpha {\alpha }^{\prime }}^e$. On the other hand, using Equation (26.7) from Ref. [28] (already in $jj$-coupling) with $N_1=0$ and $J_1=0$, we recover the previous result apart from notational differences.

It may be useful to explicitly consider the special case:

$$\left\{\begin{array}{ccc}j& j& 0\\ j^{\prime }& j^{\prime }& 1\end{array}\right\}=\left\{\begin{array}{ccc}j& j^{\prime }& 1\\ j^{\prime }& j& 0\end{array}\right\}={\displaystyle \frac{{\left(-1\right)}^{j+j^{\prime }+1}}{{\left[j,j^{\prime }\right]}^{1/2}}},$$

(54)

which simplifies, considering (48), to:

$$D_{jj}={\left(-1\right)}^{j+1/2}{[j,j^{\prime }]}^{1/2}\left(\begin{array}{ccc}j& j^{\prime }& k\\ -1/2& 1/2& 0\end{array}\right)I_{nlj,n^{\prime }{l}^{\prime }{j}^{\prime }}^{\left(1\right)\mathrm{rel}}.$$

(55)

7.5. Other Relatively Simple Cases

These cases are crucial for both the study of spectra and the computation of various statistical moments in the analysis of superconfigurations [2]. A frequently encountered case, denoting closed shells as $\left(cs\right)$, is:

$$\left(cs\right)l_1^{w_1}{l}_2-\left(cs\right)l_1^{w_1}{l}_2^{\prime },$$

(56)

where the result for LS coupling is given by Equation (14.62) of Ref. [7]:

$$D_{LS}^{ang}={\delta }_{\alpha ,{\alpha }^{\prime }}{\delta }_{S{S}^{\prime }}{\left(-1\right)}^{S+J^{\prime }+L_1+l_2^{\prime }}{[J,J^{\prime },L,L^{\prime }]}^{1/2}\left\{\begin{array}{ccc}L& S& J\\ J^{\prime }& 1& L^{\prime }\end{array}\right\}\left\{\begin{array}{ccc}{L}_1& l_2& L\\ 1& L^{\prime }& l_2^{\prime }\end{array}\right\}.$$

(57)

Applying these rules to the case:

$$\left(cs\right)j_1^{w_1}{j}_2-\left(cs\right)j_1^{w_1}{j}_2^{\prime },$$

we have:

$$D_{jj}^{ang}={\delta }_{{\alpha }_1{J}_1,{\alpha }_1^{\prime }{J}_1^{\prime }}{\left(-1\right)}^{J^{\prime }+J_1+j_2^{\prime }}[J,J^{\prime }]\left\{\begin{array}{ccc}J& 0& J\\ J^{\prime }& 1& J^{\prime }\end{array}\right\}\left\{\begin{array}{ccc}{J}_1& j_2& J\\ 1& J^{\prime }& j_2^{\prime }\end{array}\right\}.$$

(58)

Further simplifications and detailed considerations follow similar patterns, as described in Ref. [28].

7.6. Applications of LS and $jj$ Expressions in Hot Plasma

In the study of atomic properties in hot plasmas, where a multitude of ions must be treated using the concept of superconfigurations, statistical techniques are employed to avoid detailed line-by-line analysis, providing moments of distributions (e.g., energies, transitions). This topic is extensively covered in [2] and references therein. Specifically, in the detailed study of statistical properties of transition arrays, these arrays are analyzed in both LS and $jj$ coupling schemes. Various cases have been tabulated, and their statistical parameters, such as variance, are explicitly detailed (see Chapter 4 of [2], particularly Tables 4.4, 4.5, 4.8, 4.9 and 4.10).

Expressions reveal that LS coupling involves numerous $3j$ symbols of the form:

$$\left(\begin{array}{ccc}l& k& l^{\prime }\\ 0& 0& 0\end{array}\right),$$

while $jj$ coupling employs symbols of the form:

$$\left(\begin{array}{ccc}j& k& j\\ 1/2& 0& -1/2\end{array}\right).$$

Similarly, all $6j$ symbols such as:

$$\left\{\begin{array}{ccc}l& l& k^{\prime }\\ l& l& k\end{array}\right\}$$

transform into:

$$\left\{\begin{array}{ccc}j& j& k^{\prime }\\ j& j& k\end{array}\right\}.$$

This confirms that the rules introduced in earlier sections have also been derived by other authors in the context of dense and hot plasmas (WDM: warm and dense matter).

8. Screened Charges and Quantum Defects

As we know, any departure from the purely Coulombian potential can be described by suitably modifying the Bohr formula, either in the denominator $\left(n\to n^{*}=n-\delta \right)$:

$$T_{nl}=I-E_{nl}={\displaystyle \frac{Z_c^2}{{\left(n^{*}\right)}^2}}Ry={\displaystyle \frac{Z_c^2}{{\left(n-\delta \right)}^2}}Ry,$$

(59)

or in the numerator $\left(Z\to Z_{eff}=Z-\sigma =Z_c+p\right)$:

$$T_{nl}=I-E_{nl}={\displaystyle \frac{{\left(Z-\sigma \right)}^2}{n^2}}Ry\equiv {\displaystyle \frac{{\left(Z_c+p\right)}^2}{n^2}}Ry.$$

(60)

In Equations (59) and (60), I represents the ionization energy, and $E_{nl}$ denotes the energy of each level $\left|nl\right.〉$, measured relative to the ground state of each ion.

In the first case, useful when dealing with the alkaline elements and the excited electron states, $n^{*}$ is called the effective quantum number while $\delta$ is the quantum deffect. $Z_c=Z-N+1$ is called the net core charge (is the charge felt by a highly excited electron).

The second case will be used in the systematization of isoelectronic sequences. This term refers to a sequence of ions having the same number of electrons N. Spectra of ions of different elements having fixed N tend to be very similar in general structure, especially for highly ionized atoms. $Z_{eff}$ is called the effective charge, $\sigma$ is denominated screening parameter and p is the penetration parameter.

In our opinion, when employing a hydrogenic model from one perspective or another, we believe that for the treatment of highly ionized atoms ($Z_c\gtrsim 6-7$), it is more fruitful to use effective charges for the following reason. Over time, various procedures have been proposed to estimate effective charges [5,6], as mentioned earlier. Regarding the use of quantum defects, it is worth highlighting the model by Kostelecky and Nieto [42], who extended the Bates–Damgaard method. This method is widely used to estimate transition probabilities based on experimental energy level values. The combined use of both concepts was addressed by the present authors in Reference [43].

9. Conclusions

In this work, we have presented several expressions that, in some cases, generalize and, in others, simplify the calculation of hydrogenic matrix elements when the initial and final states have different effective charges. These results are summarized as follows:

(1)

Equation (11) was originally derived for integer values of $\beta$, enabling the computation of $〈n^{\prime }{l}^{\prime }\left|r^{\beta }\right|nl〉$ in the discrete–discrete case. In contrast, Refs. [11,13] extend this approach by deriving generalized formulas for arbitrary real values of $\beta$ and varying effective nuclear charges. Unlike other expressions, this formula possesses the following property: it enables a straightforward deduction of particular cases, especially when $Z_1=Z_2$ and $n_1=n_2$.

(2)

We have introduced a simplified relativistic expression that focuses exclusively on the large component $\left(F_{nlj}\left(r\right)\right)$ and includes an easy-to-use correction for its normalization. We present its explicit form and the calculation of relativistic matrix elements in terms of $\mathrm{\Gamma }$ functions $\mathrm{\Gamma }\left(i+t+{\lambda }_i+{\lambda }_t+\gamma +1\right)$.

(3)

A noteworthy contribution is related to the calculation of the generalized oscillator strength $g{f}_{J{J}^{\prime }}\left(K\right)$ for excitations from a state $\left|LSJ\right.〉$ to another state $\left|L^{\prime }{S}^{\prime }{J}^{\prime }\right.〉$, including its expression in the jj coupling scheme.

(4)

Finally, as an application of certain rules presented in Ref. [40], we explicitly derive several expressions for calculating the line strength $S^{1/2}\equiv D$ in the jj coupling scheme.