Basis Set Calculations of Heavy Atoms

Round 1

Reviewer 1 Report

This paper considers an important question of using single-electron
basis in the accurate calculations for many-electron atoms.
It suggests a new type of the basis and illustrates its usefulness
by calculating the QED corrections to the energies and E1 transition
amplitudes and the RPA corrections to the hyperfine structure of Cs.
The results are good and should be published.

However, the paper has some shortcomings which should be addressed
before the publication.

1. The orbitals of the new basis are defined in Eq. (5). No further
details are given (is it assumed in (5) that n'=n+1?).
It would be useful to compare new basis with what was used in [13].
As I understand, the main difference is that derivative (dP/dr) is
used in (5) while wave function (P) is used in [13]. The rest might
be the same. Is this correct? The rest includes the procedure of
orthonormalization. Any basis used in the calculations for
many-electron atoms must be orthogonal to core states and states
in the lower continuum (positron states). The orthonormalization
is done for the basis in [13]. But no statement about this is made
about new basis.

2. Next section considers calculations of the quantum electrodynamic
(QED) corrections in Cs using new basis. However, it is not clear
why any basis at all is needed for this. The authors do have a procedure
to generate the Dirac-Fock (DF) orbitals. The same procedure can be
used to include the QED corrections. The inclusion of the QED
corrections can be reduced to simple modification of the DF potential.
This would be the easiest and very accurate way to include the QED
corrections. In contrast, diagonalization of the matrix on a basis,
is completely different procedure which brings new problems, such
as e.g. the completeness of the basis.

3. In contrast to the QED case the use of matrix diagonalization for
the calculation of the RPA corrections is an useful method.
The RPA corrections to the hyperfine structure constants of low
states of Cs were calculated many times by many authors. It would
be more useful to compare the results of present work with
previous calculations than just compare two results obtained
with the use of the same basis but different size.

4. It is not clear the usefulness of the local screening potentials
in the calculations of the QED corrections. Any accurate calculations
for many-electron atoms start from the DF approximation. This is due
to its convenience in the calculation of the correlations. The QED
corrections are small compared to the correlation corrections.
The calculations of QED corrections should not disturb the calculations
of the correlations by replacing the DF potential by anything else.

Author Response

We thank the referee for the comments and suggestions. Below we give a point-by-point reply to all the issues raised in the report. Original comments are given in italic, and changes to the text of the article are highlighted in bold.

1. The orbitals of the new basis are defined in Eq. (5). No further details are given (is it assumed in (5) that n'=n+1?). It would be useful to compare new basis with what was used in [13]. As I understand, the main difference is that derivative (dP/dr) is used in (5) while wave function (P) is used in [13]. The rest might be the same. Is this correct? The rest includes the procedure of orthonormalization. Any basis used in the calculations for many-electron atoms must be orthogonal to core states and states in the lower continuum (positron states). The orthonormalization is done for the basis in [13]. But no statement about this is made about new basis.
   The referee is right: the difference with the basis set from Ref. [13] is that the derivative is used instead of the function. Of course, the orthogonalization is necessary in both cases. In the old variant the function r*P has a wrong asymptotic at the origin, but correct behavior is restored after orthogonalization. Here it is correct from the beginning, which allows to change the orbitals near the origin more efficiently. For the virtual orbitals the index n’ does not have a meaning of the principal quantum number, so it is arbitrarily assigned. To avoid confusion, we changed the notation and used index v instead of n’. We added corresponding explanations below Eq.(5). We also added the comparison with the calculation of the RPA corrections by Duba et al (1989). Most other authors do not single out RPA contribution.
2. Next section considers calculations of the quantum electrodynamic (QED) corrections in Cs using new basis. However, it is not clear why any basis at all is needed for this. The authors do have a procedure to generate the Dirac-Fock (DF) orbitals. The same procedure can be used to include the QED corrections. The inclusion of the QED corrections can be reduced to simple modification of the DF potential. This would be the easiest and very accurate way to include the QED corrections. In contrast, diagonalization of the matrix on a basis, is completely different procedure which brings new problems, such as e.g. the completeness of the basis.
   We agree with the referee that it is more accurate to add model QED potential on the stage when the DF equations are solved. However, sometimes it may be useful to have the same basis set for the calculations with and without QED corrections. In this case we are sure that correlation corrections are treated identically, which allows to single out QED effects in the complex multiconfigurational calculation.

3.  In contrast to the QED case the use of matrix diagonalization for the calculation of the RPA corrections is an useful method. The RPA corrections to the hyperfine structure constants of low states of Cs were calculated many times by many authors. It would be more useful to compare the results of present work with previous calculations than just compare two results obtained with the use of the same basis but different size.

   Here we are interested in the speed of the convergence, so we studied how the results depend on the size of the basis set. When we compare with other calculations the differences may be caused by the different models of the distributions of the charge and magnetization inside the nucleus (the BR and the BW effects respectively). The differences in the results between our two basis sets are smaller than these effects. We added this explanation in the end of second paragraph on page 4.
   

4. It is not clear the usefulness of the local screening potentials in the calculations of the QED corrections. Any accurate calculations for many-electron atoms start from the DF approximation. This is due to its convenience in the calculation of the correlations. The QED corrections are small compared to the correlation corrections. The calculations of QED corrections should not disturb the calculations of the correlations by replacing the DF potential by anything else.

   We absolutely agree on this point with the referee. As we have demonstrated here, all local potentials give incorrect results for the states with l>0. This is why the effective QED operators are so useful in atomic physics. We need local screening potentials to compare our results with ab initio QED calculations, which cannot be done with the DF potential. We emphasized this point at the end of Sec 3.2 of the new version of the paper. It is also discussed in the Conclusions.

Reviewer 2 Report

Please see the attachment

Comments for author File: Comments.pdf

Author Response

We thank the reviewer for comments and suggestions. Below is a point-by-point reply to the issues raised in the report. Initial comments are in italic and changes to the text of the paper are highlighted in bold.

(1) In Eq. (4), ⌊?⌋≪1, how to determine an appropriate value of ? for different s, p, d, f, …, functions? 

One can probably write some analytical estimate for epsilon. However, an accurate estimate should include contribution from the relaxation of the self-consistent field caused by the perturbation. We do it numerically by the iterative diagonalization of the DF Hamiltonian with the perturbation on the basis set, which includes these virtual (orthogonalized) orbitals.

(2) In Line 73, ‘We see that …’, here seems to miss ‘see Figure 2’. 

Yes, we added reference to Figure 2.

(3) In Lines 72-73, the B7 and B30 basis sets are defined. Here, I guess, B7 has 7+6+4+3*7=38 functions, where 1-7s, 2-7p, 3-6d are DF functions, while 8-14s, 8-14p, and 7-13d are virtual orbitals that are generated based on ansatz Eq. (5), and B30 has the same DF functions, but with more virtual orbitals, 1.e., 8-30s, 8-30p, 7-29d. I don’t know if I am right, but this is what I read from Lines 67-72. 

Yes, this is true, only the number of relativistic partial waves is 5 (s1/2, p1/2, p3/2, d3/2, & d5/2), so the total number of orbitals is 62, not 38. We added additional explanation to this paragraph. 

(4) In Table I, ? has already been used to represent another quantity before. 

To avoid confusion, we changed \varepsilon to \epsilon when we speak about binding energies and added its definition to the caption and in the text.

(5) In Table II, we found that the results under This work (DF) and Ref. [20] and Ref. [21] that used different electric correlation implementations and basis sets are very close. Does this imply that the calculation of QED contributions (to valence binding energy, E1 transition amplitudes, and hyperfine structure constants) is not so sensitive to electronic correlation effects and choice of basis set? 

In this work we calculated QED corrections in the DF approximation. Correlation corrections were not included. The authors of Refs. [20,21] calculated both QED corrections and correlation corrections to the binding energies and amplitudes. As far as we understand, their QED corrections are also calculated in the DF approximation and all three groups used the V^{N-1} approximation. Therefore, the small differences in the results can be attributed to the differences in the basis sets.

Reviewer 3 Report

The accurate calculations of the hyperfine structure and PNC effects rely highly on wavefunctions that have convergence issues with B-splines for mixed basis sets. The authors suggested a method to expand the large component of DF orbitals in the basis set by adding a differential term in the virtual orbital. The differential ansatz and results as discussed in the paper are indeed interesting in my opinion and should be published. I wonder if the authors any insights to offer regarding two questions:

1. The differential ansatz the authors proposed considers the first order differential term. A natural question would be the consideration of higher order terms. Is it feasible or any improvement expected to consider higher order terms? For example, any improvement might be possible to consider higher order terms in Figure 1.

2. What is the transferability of the method to other elements? Especially to transition metals. The studied Cs element as the Alkali metals could be arguably less complicated than transition metals.

The following are some comments that need to be addressed.

1. Eqn reference, such as in page 3, line 64 ,“...with the respective scaled orbitals(5)” to  “...with the respective scaled orbitals as Eqn(5)”, and a few other places.

2. Figure 2 is not referenced. It would be nice to see the author provide some explanation to Figure 2(a) the unmatched oscillation at 250 for nodes of the radial grid.

3. I would suggest the author provide a summary of errors for Table 3 comparison of results for different orbitals.

4. Perhaps change Table 2 column header “Ab initio” to “Ab initio (KS)” or “KS” for better readability

5. In page 2, line 31, “It is known,... then calculation of the energies”, “then” to “than”

Author Response

We thank the reviewer for comments and suggestions. Below is a point-by-point reply to the issues raised in the report. Initial comments are in italic and changes to the text of the paper are in highlighted bold.

The accurate calculations of the hyperfine structure and PNC effects rely highly on wavefunctions that have convergence issues with B-splines for mixed basis sets. The authors suggested a method to expand the large component of DF orbitals in the basis set by adding a differential term in the virtual orbital. The differential ansatz and results as discussed in the paper are indeed interesting in my opinion and should be published. I wonder if the authors any insights to offer regarding two questions:

1. The differential ansatz the authors proposed considers the first order differential term. A natural question would be the consideration of higher order terms. Is it feasible or any improvement expected to consider higher order terms? For example, any improvement might be possible to consider higher order terms in Figure 1.
   We agree that it is natural to think about higher derivatives. There are two potential obstacles on this route. First, differentiation of the numerical orbitals reduces their smoothness. Second, such orbitals may be linear dependent with other orbitals in the basis set. Our tests show that both problems are real. At present we think that higher derivatives are not very useful, but we continue different tests.
2. What is the transferability of the method to other elements? Especially to transition metals. The studied Cs element as the Alkali metals could be arguably less complicated than transition metals.
   This is a very interesting question. We think that this approach can be useful for treating correlations in atoms with open d and f shells. We plan to do multiconfigurational calculations with these new basis sets for polyvalent atoms, including transition metals and lanthanides. We added discussion of this issue to the text (see second paragraph of the Conclusions). 

The following are some comments that need to be addressed.

1. Eqn reference, such as in page 3, line 64 ,“...with the respective scaled orbitals(5)” to  “...with the respective scaled orbitals as Eqn(5)”, and a few other places.
   We added ‘Eq.’ in the places where we cite equations by their number.
2. Figure 2 is not referenced. It would be nice to see the author provide some explanation to Figure 2(a) the unmatched oscillation at 250 for nodes of the radial grid.
   We added the reference to Figure 2.
   Oscillations instead of the smooth behavior is a typical effect of a finite basis set. When the length of the basis set increases, the amplitude of the oscillations decreases. We added this comment to the text in the top of page 4.
3. I would suggest the author provide a summary of errors for Table 3 comparison of results for different orbitals.
   The summary is added in the end of the first paragraph in Sec. 3.2
4. Perhaps change Table 2 column header “Ab initio” to “Ab initio (KS)” or “KS” for better readability
   Done.
5. In page 2, line 31, “It is known,... then calculation of the energies”, “then” to “than”
   Done.

Reviewer 4 Report

Although dozens of approaches to the computation of atomic structure have been developed, and computing resources have expanded by orders of magnitude over the first practical approximations, there still are limitations that one would prefer to surpass on the path to higher accuracy, as well as shortcomings that one might avoid by clever computation strategies. The authors have chosen a new way of formulating their ansatz, aiming for specific improvements and
optimisations. This topic clearly falls into the scope of the journal "Atoms" and is most welcome.

The high scientific qualification and expertise of the authors is beyond doubt. There is a slight smudge on the manuscript that is caused by structural differences between Russian and English. In English, "a" and "the" are frequently used articles; while these do not exist in Russian, their absence betrays many foreign language authors. It is recommended to have colleagues with a higher proficiency in English check the text and remove those blemishes. (for example, line 2 should
read "... is a very difficult computational problem, and an optimization ..." Some formulations of text sound a bit un-English, but a certain level of that should be permitted. To cure that would require an expert in the physics field to find better expressions in English - which exceeds the qualifications of this reviewer. Maybe the colleague who tries to straighten
out the problem of  "a" and "the" (mostly the few missing ones) can have an eye and ear also for the text flow.

Line 30, delete the comma

Line 31 and 139, "than" (comparison), not "then" (temporal sequence)

The exposition of the ansatz and the demonstration of sample cases are fine.

Slight Russian flavour, ought to be smoothed out among colleagues.

Author Response

We thank the reviewer for comments and suggestions. Below is a point-by-point reply. Initial remarks are in italic and changes to the thext of the paper are in bold.

Although dozens of approaches to the computation of atomic structure have been developed, and computing resources have expanded by orders of magnitude over the first practical approximations, there still are limitations that one would prefer to surpass on the path to higher accuracy, as well as shortcomings that one might avoid by clever computation strategies. The authors have chosen a new way of formulating their ansatz, aiming for specific improvements and optimisations. This topic clearly falls into the scope of the journal "Atoms" and is most welcome.

We thank the reviewer for this assessment.

The high scientific qualification and expertise of the authors is beyond doubt. There is a slight smudge on the manuscript that is caused by structural differences between Russian and English. In English, "a" and "the" are frequently used articles; while these do not exist in Russian, their absence betrays many foreign language authors. It is recommended to have colleagues with a higher proficiency in English check the text and remove those blemishes. (for example, line 2 should read "... is a very difficult computational problem, and an optimization ..." Some formulations of text sound a bit un-English, but a certain level of that should be permitted. To cure that would require an expert in the physics field to find better expressions in English - which exceeds the qualifications of this reviewer. Maybe the colleague who tries to straighten out the problem of  "a" and "the" (mostly the few missing ones) can have an eye and ear also for the text flow.

We changed the sentence in the abstract as suggested and tried our best to add proper articles where necessary.

Line 30, delete the comma
Done.
Line 31 and 139, "than" (comparison), not "then" (temporal sequence)
Done.
The exposition of the ansatz and the demonstration of sample cases are fine.

Thanks.

Slight Russian flavour, ought to be smoothed out among colleagues.

We tried our best to improve the language throughout the paper. We do not have native speakers among the coauthors, so some Russian flavor may still be present.

Round 2

Reviewer 1 Report

I am totally satisfied with the changes made by the authors. The manuscript can be published now.

Reviewer 2 Report

The authors have addressed my questions nicely. I agree with the publication of this paper. 

Reviewer 3 Report

No additional comments