Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic Cases
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsReport on manuscript atoms-3609663 entitled "Some remarks concerning matrix elements with different effective charges: non-relativistic and relativistic cases", by H. O. Di Rocco and J. C. Aguiar
The article deals with the calculation of hydrogenic matrix elements within the framework of the screened hydrogenic model, in both the non-relativistic and relativistic cases. The article compiles formulas that are not new, but presents them in a general context, with different screened charges for the initial and final configurations, and by exploiting the fact that they can be expressed as Laplace transforms. In the relativistic case, an approximation allowing the use of only a single component is mentioned.
In section 7 (transition from LS coupling to jj coupling), I don't see any mention of the 9j coefficient. I understand that the authors don't really need it for their argument, but I still find it surprising.
It seems to me that the formalism presented here can also be used while taking quantum defect theory into account. I have a question on this topic: what is more important for obtaining the most realistic results with a hydrogenic model, using screened charges or quantum defects (i.e., effective principal quantum numbers)? How are these two concepts related?
The article also contains many useful elements that are not always specified in the literature on the subject, such as the correspondence between the conventions used for the Laguerre polynomials.
Formula (27), involving the hypergeometric function, is new to me and definitely deserves to be highlighted. It would be interesting if the authors could comment of the potential practical interest of such a formula, in terms of numerical evaluation.
Finally, the authors propose an application of their formula to the calculation of electron-impact excitation cross sections within the framework of the Plane Wave Born approximation, which is pretty convincing.
I am sure that this article will be useful to many specialists in atomic physics. It is concise, rigorous, easy to read, and well written. It has the merit of offering useful formulas that synthesize various works on the subject, and it also contains new elements, which is no small feat for such an academic topic!
I recommend it without hesitation for publication in the journal Atoms.
Author Response
Report: atoms-3609663
Title: Hydrogenic matrix elements with different effective charges: Non-Relativistic and Relativistic cases by H. O. Di Rocco and J. C. Aguiar
Responses to Referee #1
The article deals with the calculation of hydrogenic matrix elements within the framework of the screened hydrogenic model, in both the non-relativistic and relativistic cases. The article compiles formulas that are not new, but presents them in a general context, with different screened charges for the initial and final configurations, and by exploiting the fact that they can be expressed as Laplace transforms. In the relativistic case, an approximation allowing the use of only a single component is mentioned.
Q1: In Section 7 (transition from LS coupling to jj coupling), I noticed no mention of the 9j coefficient. While I understand that the authors may not require it for their arguments, its omission is surprising.
A1: We have added a short subsection (7.2) discussing the 9j symbols.
Q2: It seems that the formalism presented here could also be applied to include quantum defect theory. I have a specific question: what is more critical for achieving the most realistic results with a hydrogenic model, using screened charges or quantum defects (i.e., effective principal quantum numbers)? How are these two concepts related?
A2: In section 8 we have now added to explore the relationship between screened charges and quantum defects.
The article also includes many useful details that are often overlooked in the literature, such as the conventions used for Laguerre polynomials.
Q3: Formula (27), involving the hypergeometric function, is new to me and deserves attention. It would be interesting if the authors could discuss its potential practical applications, particularly regarding numerical evaluations.
A3: A brief note has been added, acknowledging the vast scope of hypergeometric functions and our current inability to provide a definitive response. We have included new references pointing to classical texts on these functions.
Finally, the authors demonstrate the application of their formulas to calculate electron-impact excitation cross sections within the Plane Wave Born approximation, which is highly convincing.
This article will undoubtedly benefit many specialists in atomic physics. It is concise, rigorous, accessible, and well-written. It provides useful formulas that synthesize various works on the subject while also offering new insights—no small feat in such an academic field. I wholeheartedly recommend it for publication in Atoms.
Reviewer 2 Report
Comments and Suggestions for AuthorsThe manuscript
“Some Remarks Concerning Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic Cases” by Di Rocco and Aguiar
is devoted to an interesting subject in atomic physics namely to the analytical representation of matrix elements that have important applications in atomic physics of isolated atoms and also in collisional-radiative codes (related to atoms in dense plasmas).
In particular, the paper investigates the H-like radial matrix element for arbitrary initial and final states and arbitrary exponent beta.
The presentation, however, looks rather incomplete and the manuscript requests serious revisions before publication:
1) The authors should write down explicit expressions for the matrix elements: it is not clear form eq. 6 what is the final analytical expression. In fact eq. 7 is finally replaced by eq. 8 but how then eq. 8 is used in eq. 6 is not so obvious.
The authors should also provide an explicit expression for the binominal coefficients in eqs. (10,11) for non-integer arguments (that arrive for real values of beta in eq. (1)) and simplify the final expression. Will that not end up with Gamma functions of non-integer arguments ?
2) The statement 1 in the “Conclusion” seems not to be correct, in fact, an entirely analytical expression based on Gamma-functions for the arbitrary case of initial and final states and arbitrary beta has been published previously: Open Physics, vol. 21, page 100241 (2024). This reference has to be included at appropriate places in the manuscript and the authors need to discus in what respect their findings are different from the already published ones (eq. 9 in Open Physics, vol. 21, page 100241 (2024)).
It is also not clear to what extend the proposed expressions (eqs. 6-11) are more advantageous and simpler than those provided in (Open Physics, vol. 21, page 100241 (2024)) given the fact that highly efficient analytical expressions exist for the Gamma function.
The authors need to add a corresponding discussion in their manuscript and refine their conclusions.
3) For the relativistic case, the authors should likewise provide an explicit analytical expression for the matrix elements and not just say that it is similar to the non-relativistic case (which was already not well presented).
4) Likewise, there is missing explicit expressions for the generalized oscillator strengths. I guess there should be explicit expression at least for the dipole approximation, however, given the context of the manuscript considering general matrix elements, at least some other orders/terms of the Bessel functions in eq. 37 should be considered.
5) The case of the more general expression eq. 34 should also be considered more explicitly.
After the above-mentioned points have been included and upgraded references have been mentioned and discussed the manuscript might be suitable for publication.
I would also suggest then to change the title of the manuscript just to: “Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic Cases”.
Author Response
Report: atoms-3609663
Title: Hydrogenic matrix elements with different effective charges: Non-Relativistic and Relativistic cases by H. O. Di Rocco and J. C. Aguiar
Responses to Referee #2
The manuscript addresses an interesting topic in atomic physics: the analytical representation of matrix elements with significant applications in atomic physics of isolated atoms and collisional-radiative models for dense plasmas. Specifically, it explores H-like radial matrix elements for arbitrary initial and final states and arbitrary exponent β.
However, the presentation feels incomplete, and the manuscript requires significant revisions before publication:
Q1a: The authors should provide explicit expressions for the matrix elements, as the final analytical expression is unclear in Eq. (6). Additionally, Eq. (7) is replaced by Eq. (8), but it is not obvious how Eq. (8) is used in Eq. (6).
A1a: The text has been revised for clarity. Note that Eq. (7) is not replaced by Eq. (8); only the integral in Eq. (7) is substituted by Eq. (8).
Q1b: Explicit expressions for the binomial coefficients in Eqs. (10) and (11) for non-integer arguments (arising from real values of β in Eq. (1)) should be provided. Simplifying the final expression may lead to Gamma functions of non-integer arguments.
A1b: This has been clarified. A new paragraph explains cases involving dense plasmas with exponents like 3/2. Additionally, new references related to the mean value ⟨r³/²⟩ have been included.
Q2a: Statement 1 in the conclusion seems incorrect, as entirely analytical expressions based on Gamma functions for arbitrary initial and final states with arbitrary β have been published previously (Open Physics, vol. 21, page 100241, 2024). This reference should be included, and the authors must discuss how their findings differ.
A2a: The reference has been included, and we thank the reviewer for pointing it out. While the computational equivalence of expressions from various authors is acknowledged in the introduction, we believe our formula offers simplicity in special cases, such as Z₁ = Z₂ and n₁ = n₂.
Q2b: It is unclear how Eqs. (6–11) are more advantageous or simpler than those in Open Physics (2024), considering the availability of efficient analytical expressions for the Gamma function.
A2b: Please refer to the response above.
Q3: For the relativistic case, explicit analytical expressions for the matrix elements should be provided instead of merely stating their similarity to the non-relativistic case (which itself is not well-presented).
A3: We believe Eq. (30), derived from Eqs. (19), (25), and (26), provides a straightforward approach to the matrix element. Making it more explicit would result in an overly complex expression.
Q4: Explicit expressions for generalized oscillator strengths are missing. While the dipole approximation is essential, given the context of general matrix elements, additional terms of the Bessel functions in Eq. (37) should be considered.
A4: A note has been added, referencing J.-C. Pain and D. Benredjem (HEDP, 2021 with corrections in 2024). These corrections, identified by one of the authors, were acknowledged by the mentioned researchers.
Q5: The general expression in Eq. (34) should also be addressed explicitly.
A5: This was already considered in Subsection (7.2), where the most common DLS cases in jj coupling are explicitly detailed.
Once these points are addressed and updated references are included, the manuscript may be suitable for publication.
Additionally, I suggest revising the title to: “Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic Cases”.
Response: The title has been updated accordingly in this revised version.
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsThe revised manuscript
“Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic Cases” by Di Rocco and Aguiar
has taken a few of the suggested corrections into account.
However, the appreciations of formulas from other references lacks clarity.
- a) As the authors state correctly now, their derivation of the matrix element formula was obtained assuming beta to be integer. A discussion is added now after eq. (11), that non-integer values are important in dense plasma physics and a correspondence of factorials to Gamma functions is proposed. This is not a strict derivation and therefore, the work published by Sanchez [ref. 11] and Li [13] are “superior” and should not be disqualified in the conclusion just by “..the approaches presented by the authors mentioned in the introduction are also valid”.
In fact, after reinspection of ref. 11 and 13 I find that these formulas are the only strict derived formulas for non-integer beta values and any combination of upper and lower quantum numbers and that ref. 13 seems to be the only one that is strictly derived for any beta value including any effective nuclear charges of upper and lower states.
Correspondingly, the authors should qualify in the manuscript their eq. (11) as an alternative formula that could have some advantages for simplified analytical expression for special cases:
Add before paragraph 2.1 a phrase like: In ref. 11, 13 a strict derivation of a closed formulas for any real value of beta and any combination of upper and lower state quantum numbers are given in terms of Gamma functions while in ref. 3 this type of representation is generalized for any values of effective nuclear charges of upper and lower states.
2) As expressions (30, 31) are not very explicit (albeit requested), the authors should add the reference JQSRT, vol 117, 123 (2013) where some more detailed relativistic studies have been presented.
3) Before eq. (37) the authors of the present manuscript state that authors of ref. 14 used this equation…. this is not wrong but not really correct either. In fact this expression is used by very many authors, at least refs. 7 and 9 should be added here before 14.
The next point is the usefulness of eq. (37) which has a long history….I suggest to add at least reference PRA, vol. 53, 2425 (1996) to point out some developments in the direction of effective Gaunt factors.
4) Conclusion, last phrase of point 1 should be modified:
“The formula (11), derived for integer beta values, which allows….discrete-discrete case, while in ref. 11, 13 generalized formulas for any real values of beta and different effective nuclear charges have been derived. Unlike other expression…..n1=n2. (skip sentence “if we focus….”).
With these final corrections that paper can be recommended for publication.
Author Response
Report: atoms-3609663
“Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic Cases” by Di Rocco and Aguiar
has taken a few of the suggested corrections into account.
However, the appreciations of formulas from other references lacks clarity.
1- a) As the authors state correctly now, their derivation of the matrix element formula was obtained assuming beta to be integer. A discussion is added now after eq. (11), that non-integer values are important in dense plasma physics and a correspondence of factorials to Gamma functions is proposed. This is not a strict derivation and therefore, the work published by Sanchez [ref. 11] and Li [13] are “superior” and should not be disqualified in the conclusion just by “..the approaches presented by the authors mentioned in the introduction are also valid”.
In fact, after reinspection of ref. 11 and 13 I find that these formulas are the only strict derived formulas for non-integer beta values and any combination of upper and lower quantum numbers and that ref. 13 seems to be the only one that is strictly derived for any beta value including any effective nuclear charges of upper and lower states.
Correspondingly, the authors should qualify in the manuscript their eq. (11) as an alternative formula that could have some advantages for simplified analytical expression for special cases:
Add before paragraph 2.1 a phrase like: In ref. 11, 13 a strict derivation of a closed formulas for any real value of beta and any combination of upper and lower state quantum numbers are given in terms of Gamma functions while in ref. 3 this type of representation is generalized for any values of effective nuclear charges of upper and lower states.
Response: 1-a) Dear Reviewer,
We sincerely appreciate the time and effort you have dedicated to evaluating our manuscript. Addressing your questions, we have incorporated the following clarifications into Section 2: “We have verified that expression (11) is entirely adequate for calculating radial integrals with non-integer β. It serves as an alternative formula that offers potential advantages for simplified analytical expressions in specific cases, as demonstrated in Subsection 2.1. Additionally, references [11] and [13] provide further context on this topic.
A rigorous derivation of closed-form expressions for any real value of β and any combination of upper and lower state quantum numbers is now presented using Gamma functions. Furthermore, in Ref. [13], this representation has been extended to encompass any values of the effective nuclear charges associated with the upper and lower states.”
2) As expressions (30, 31) are not very explicit (albeit requested), the authors should add the reference JQSRT, vol 117, 123 (2013) where some more detailed relativistic studies have been presented.
Response: 2) Following the referee's suggestions, we have incorporated in section 5 the following text: Ruano et al. [34] presented a com prehensive study of relativistic screened hydrogenic radial integrals. Later, J. C. Pain [35] refined this approach by incorporating Clebsch-Gordan coefficients.
3) Before eq. (37) the authors of the present manuscript state that authors of ref. 14 used this equation…. this is not wrong but not really correct either. In fact this expression is used by very many authors, at least refs. 7 and 9 should be added here before 14.
The next point is the usefulness of eq. (37) which has a long history….I suggest to add at least reference PRA, vol. 53, 2425 (1996) to point out some developments in the direction of effective Gaunt factors.
Response: 3) In section 6 we have incorporated the following text:
An alternative method for calculating excitation cross sections involves the well-known Van Regemorter formula, which utilizes effective Gaunt factors [38]. A development along these lines was presented by Fisher et al. [39].
4) Conclusion, last phrase of point 1 should be modified:
“The formula (11), derived for integer beta values, which allows….discrete-discrete case, while in ref. 11, 13 generalized formulas for any real values of beta and different effective nuclear charges have been derived. Unlike other expression…..n1=n2. (skip sentence “if we focus….”).
Response: 4) The changes have been made following the referee's suggestions, for which we are sincerely grateful for their time and effort.
With these final corrections that paper can be recommended for publication.
Round 3
Reviewer 2 Report
Comments and Suggestions for AuthorsThe 2nd revised version has considered essentially all the referee’s comments and recommendations and is therefore recommended for publication.
