A Versatile Unitary Transformation Framework for an Optimal Bath Construction in Density-Matrix Based Quantum Embedding Approaches
Mauricio Rodríguez Mayorga
Round 1
Reviewer 1 Report
In this work, the authors present an extension of the Block-Householder method (used to produce bath orbitals within a density matrix embedding scheme) and use different types of cost functions to impose constraints on the sets of parameters that need to be optimized. In general, the paper is clear and presents many interesting results that will contribute to the construction of (better) embedding schemes in the future. Thus, I am convinced that is worth recommending it for publication after minor revision. There are some points that need to be clarified.
1) Is the prime in Eq. 1 correct? Do the authors annihilate an electron with sigma' spin and create it with sigma spin?
2) In line 81, Ne is not defined, which I guess is the number of orbitals of the environment.
3) In Eq. 4, Where is the dependence w.r.t. the l index?
4) In line 91, N in R_{N Ni} is not defined.
5) In Eq. 5, the Id. notation for the identity matrix is not a conventional notation and would be better to use boldface I.
6) In line 96, there is a typo, and an 'e' is missing in 'the number'
7) In Eq. 10, is the minus sign absorbed by the \tilde{t} quantity defined in Eq. 11?
8) In line 126, is there a typo and it should be through instead of though?
9) In Fig. 4 it might be easier for the reader to use Block-Householder in the caption.
10) In lines, 303-303, I do not understand if the authors want to provide the asymptotic values or to mention that the performance
of the various cost functions is good. If they mean that the performance is good, I disagree with them seeing the deviations w.r.t. the BA results.
The authors should clarify this phrase.
11) The information contained in Figs. 5 and 7. and the discussions presented between lines 326-340 and 359-368 are not conclusive. I would recommend either expanding these sections to understand the dependence w.r.t. the parameters or removing them completely from the work.
12) In Fig. 6, I can not recognize the black solid line. Is it just the 0 baseline?
13) In line 357, a citation is missing for DaC approaches.
14) In Fig. 8, the ranks are presented using lowercase 'r' in all the rest of the work and the caption should also be presented with lowercase 'r'.
15) Concerning the cost functions, I have two questions: a) regarding Figs. 4 and 6, would it be worth using as a cost function a linear combination of Delta (Eq. 23) and Hartree (Eq. 25)? and b) the authors use the Hartree term to build a cost function, what happens with the exchange? is there any reason to neglect it? the exchange can also be accounted for at the mean-field level. So, why is it not considered?
16) In lines 399 to 402, the authors mention that the oscillations are due to the iterative procedure used to optimize a large number of parameters. What is the number of parameters? I would think that it is due to the shape of the hypersurface rather than the number of parameters that causes these difficulties.
There are a few typos that require a spelling check, but those are minor changes.
Author Response
Response to Reviewer 1 Comments
Open Review ( ) I would not like to sign my review report (x) I would like to sign my review report Quality of English Language ( ) I am not qualified to assess the quality of English in this paper ( ) English very difficult to understand/incomprehensible ( ) Extensive editing of English language required ( ) Moderate editing of English language required (x) Minor editing of English language required ( ) English language fine. No issues detected Comments and Suggestions for Authors In this work, the authors present an extension of the Block-Householder method (used to produce bath orbitals within a density matrix embedding scheme) and use different types of cost functions to impose constraints on the sets of parameters that need to be optimized. In general, the paper is clear and presents many interesting results that will contribute to the construction of (better) embedding schemes in the future. Thus, I am convinced that is worth recommending it for publication after minor revision. There are some points that need to be clarified. We thank the referee for his/her positive feedback. We reply his/her points in the following. Point 1: Is the prime in Eq. 1 correct? Do the authors annihilate an electron with sigma' spin and create it with sigma spin? Response 1: We thank the referee for his/her remark. Indeed, the prime was not correct and has been corrected in the revised manuscript . Point 2: In line 81, Ne is not defined, which I guess is the number of orbitals of the environment. Response 2: Ne is defined in the introduction of the revised manuscript. Point 3: In Eq. 4, Where is the dependence w.r.t. the l index? Response 3: We thank the referee for his/her remark. Indeed, equation 4 was not correct, the subscript l replaces the subscript i for the element of the R matrix. The subscript l is now replaced by an e to refer to orbitals belonging to the environment in the revised manuscript. Point 4: In line 91, N in R_{N Ni} is not defined. Response 4: N is defined in the introduction of the revised manuscript. Point 5: In Eq. 5, the Id. notation for the identity matrix is not a conventional notation and would be better to use boldface I. Response 5: The notation Id. is replaced by I in the revised manuscript. Point 6: In line 96, there is a typo, and an 'e' is missing in 'the number' Response 6: Corrected. Point 7: In Eq. 10, is the minus sign absorbed by the \tilde{t} quantity defined in Eq. 11? Response 7: The minus sign absorbed is now clarified in equation 11. Point 8: In line 126, is there a typo and it should be through instead of though? Response 8: Corrected. Point 9: In Fig. 4 it might be easier for the reader to use Block-Householder in the caption. Response 9: The Block-Householder transformation method is now simplified to the acronym BHt throughout the manuscript and in the legends. Point 10: In lines, 303-303, I do not understand if the authors want to provide the asymptotic values or to mention that the performance of the various cost functions is good. If they mean that the performance is good, I disagree with them seeing the deviations w.r.t. the BA results. The authors should clarify this phrase. Response 10: We thank the referee for his/her remark. There is indeed an error, the asymptotic behavior of the kinetic energy and the double occupation is not correctly reproduced for highly correlated regimes. A comment has been added in lines 314-316 to explicitly clarify this result. Point 11: The information contained in Figs. 5 and 7. and the discussions presented between lines 326-340 and 359-368 are not conclusive. I would recommend either expanding these sections to understand the dependence w.r.t. the parameters or removing them completely from the work. Response 11: Figure 5 highlights an important point: the fact that some angles are independent of the relative repulsive force U/t and also of the cost functions used. It is thought that this behavior is due to the symmetries (topological or electron-hole) of the system, although proving these assumptions would require further investigation beyond the scope of this work. Next, we suggest improving the numerical optimization by varying only those angles that change significantly. Lines 333-342 have been rearranged in the revised manuscript to make the message clearer.Concerning Figure 7, we show that the angles changing significantly with U/t differ between the IB and NIB cases for moderately and strongly correlated regimes. This information is not very relevant, and the section concerning the commentary on Figure 7 has now been removed.
Point 12: In Fig. 6, I can not recognize the black solid line. Is it just the 0 baseline? Response 12: To avoid any ambiguity, baselines have been removed for all figures Point 13: In line 357, a citation is missing for DaC approaches. Response 13: The citation corresponding to the DaC approaches is already specified in the introduction (line 73 of the revised manuscript). We believe it might be redundant to include it again. Point 14: In Fig. 8, the ranks are presented using lowercase 'r' in all the rest of the work and the caption should also be presented with lowercase 'r'. Response 14: Corrected. Point 15: Concerning the cost functions, I have two questions: a) regarding Figs. 4 and 6, would it be worth using as a cost function a linear combination of Delta (Eq. 23) and Hartree (Eq. 25)? and b) the authors use the Hartree term to build a cost function, what happens with the exchange? is there any reason to neglect it? the exchange can also be accounted for at the mean-field level. So, why is it not considered? Response 15: We thank the referee for their pertinent comment. Regarding question a), the referee is correct in stating that a linear combination of cost functions can be envisioned. However, it adds additional parameters (the weights) that need to be fixed arbitrarily or based on physical conditions. We leave this question open for future work. Concerning question b), the Fock contribution at half-filling is nil in the Hubbard model because the on-site bi-electronic operator only acts on pairs of electrons with opposite spins. Point 16: In lines 399 to 402, the authors mention that the oscillations are due to the iterative procedure used to optimize a large number of parameters. What is the number of parameters? I would think that it is due to the shape of the hypersurface rather than the number of parameters that causes these difficulties. Response 16: We thank the referee for their pertinent comment. In the case of 5 baths, there are 12 variational parameters that lead to numerical difficulties, probably due to the complex shape of the hyper-surface, as mentioned by the referee. Indeed, as exemplified in Figure 3 with only N_b = 3, the hyper-surface to be explored is already non-trivial, showing many local minima as well as flat zone. These difficulties are unfortunately expected to increase with the number of parameters. To be clear, we modified lines 411-412 in the revised manuscript. Comments on the Quality of English Language There are a few typos that require a spelling check, but those are minor changes. Submission Date 26 July 2023 Date of this review 07 Aug 2023 15:49:29Reviewer 2 Report
This paper presents extensions of established quantum embedding schemes and tests them for the one-dimensional Hubbard model. The idea is to find a suitable unitary transformation that effectively decouples an impurity from the environment; the impurity then only interacts with a small bath, whereas the environment is treated in a simplified manner. Here, the transformation uses the Block-Householder algorithm, which was proposed in earlier work (of which one of the present authors was a coauthor, see Ref 30). What is new here is that the bath is allowed to be more complex than just a single site or orbital, and the resulting freedom is used for variational optimization. Indeed, improvements are found compared to prior implementations of the Block-Householder algorithms, but the method is also more costly.
All in all, this is a nice theoretical and computational study, which covers new ground in an area of topical interest (quantum embedding schemes). The work is competently executed and well presented. I recommend publication.
Here are a few comments.
1. In equations (1), (2) and (3) there are spurious commas which make it look as if there is a prime (') after a spin index. These commas should be removed.
2. Something is wrong in equation (4). There is no label "l" in the formula which would refer to the cluster.
3. In lines 108/109: the construction only holds if a square matrix is invertible. Is that always guaranteed? What if it isn't?
4. It is unclear how equation (22) follows from equation (20). It seems to me as if there is a term missing. And how did
gamma^{jT}_{Ne} gamma^1_{Ne} turn into ||gamma^1_{Ne}||?
5. Equation (25) is the "square of the Hartree (i.e. mean field) contribution energy between the cluster and the environment". In this expression, the interaction term involves a product of the density matrices. Doesn't this mean that this is actually the Hartree-Fock energy, and not just the Hartree energy?
6. Figure 6 presents the relative error of the ground state energy per site, calculated with various embedding schemes. I wonder how this compares to a direct mean-field (Hartree-Fock) solution of the 1D Hubbard chain? This data should be straightforward to include.
7. Generally speaking, the embedding schemes perform well for small to moderate U, but the errors become quite large for stronger correlations. This is, of course, not unexpected, but I wonder whether this is the typical accuracy achievable with quantum embedding schemes. In other words, is this the best one can hope to do? How about the SVD approach?
The paper needs some proofreading. Here are a few things I noticed:
line 81: as little as possible
line 94: as follows:
line 98: considering
line 135: non-local
line 167: leaves (instead of leads)
line 277: minima
line 313, 315, 316: constraint
Author Response
Response to Reviewer 2 Comments
Open Review (x) I would not like to sign my review report ( ) I would like to sign my review report Quality of English Language ( ) I am not qualified to assess the quality of English in this paper ( ) English very difficult to understand/incomprehensible ( ) Extensive editing of English language required ( ) Moderate editing of English language required (x) Minor editing of English language required ( ) English language fine. No issues detected Comments and Suggestions for Authors This paper presents extensions of established quantum embedding schemes and tests them for the one-dimensional Hubbard model. The idea is to find a suitable unitary transformation that effectively decouples an impurity from the environment; the impurity then only interacts with a small bath, whereas the environment is treated in a simplified manner. Here, the transformation uses the Block-Householder algorithm, which was proposed in earlier work (of which one of the present authors was a coauthor, see Ref 30). What is new here is that the bath is allowed to be more complex than just a single site or orbital, and the resulting freedom is used for variational optimization. Indeed, improvements are found compared to prior implementations of the Block-Householder algorithms, but the method is also more costly. All in all, this is a nice theoretical and computational study, which covers new ground in an area of topical interest (quantum embedding schemes). The work is competently executed and well presented. I recommend publication. Here are a few comments. We thank the referee for his/her positive feedback. We reply his/her remarks and comments in the following. Point 1: In equations (1), (2) and (3) there are spurious commas which make it look as if there is a prime (') after a spin index. These commas should be removed. Response 1: We agree with the referee that the commas might be confusing. We move them apart to avoid confusion in the revised manuscript. Point 2: Something is wrong in equation (4). There is no label "l" in the formula which would refer to the cluster. Response 2: We thank the referee for his/her remark. Indeed, equation 4 was not correct, the subscript l replaces the subscript i for the element of the R matrix. The subscript l is now replaced by an e to refer to orbitals belonging to the environment. Point 3: In lines 108/109: the construction only holds if a square matrix is invertible. Is that always guaranteed? What if it isn't? Response 3: The referee is wright. Indeed, for one and two impurities, we can show that the roots of the characteristic polynomial are different from 0, and tend towards 0 when the fragment tends to be disconnected. In the other cases,it cannot be said the matrix is invertible. In practice, the choice of the impurity fragment is done such that it is always invertible. Point 4: It is unclear how equation (22) follows from equation (20). It seems to me as if there is a term missing. And how did gamma^{jT}_{Ne} gamma^1_{Ne} turn into ||gamma^1_{Ne}||? Response 4: The referee is wright. Indeed, equation (22) was wrong. It has now been corrected in the revised manuscript. Point 5: Equation (25) is the "square of the Hartree (i.e. mean field) contribution energy between the cluster and the environment". In this expression, the interaction term involves a product of the density matrices. Doesn't this mean that this is actually the Hartree-Fock energy, and not just the Hartree energy? Response 5: The Fock contribution at half-filling is nil in the Hubbard model because the on-site bi-electronic operator only acts on pairs of electrons with opposite spins. Point 6: Figure 6 presents the relative error of the ground state energy per site, calculated with various embedding schemes. I wonder how this compares to a direct mean-field (Hartree-Fock) solution of the 1D Hubbard chain? This data should be straightforward to include. Response 6: In Fig. 1 of the reply letters, we compare results from Fig. 6 with the restricted mean-field approach. Because the Fock-exchange contribution is nil in the particular case of the half-filled Hubbard model, mean-field results are very poor at describing the energy of the fundamental state, especially for moderately to strongly correlated regimes. Point 7: Generally speaking, the embedding schemes perform well for small to moderate U, but the errors become quite large for stronger correlations. This is, of course, not unexpected, but I wonder whether this is the typical accuracy achievable with quantum embedding schemes. In other words, is this the best one can hope to do? How about the SVD approach? Response 7: Within the Block-Householder transformation, the asymptotic behavior close to the atomic limit is shown to be recovered up to three impurities (and three bath sites, i.e., within a 6-site cluster) in Ref [32] of the revised manuscript. In this contribution, we show that optimization of the unitary transformation can already improve results at this limit without expanding the size of the cluster. The expected improvement, however, strongly depends on the cost function. We leave for future work the design of cost functions specifically aimed at improving the U/t >> 1 regime without deteriorating results at other regimes. Finally, an SVD-type approach in our case shows similarities to the Block-Householder method, as demonstrated in Ref. [24], and therefore does not provide satisfying results at the atomic limit. A comment has been added to explicitly clarify this result in lines 314-317.Reviewer 3 Report
In the present manuscript entitled "A versatile unitary transformation framework for an optimal bath construction in density-matrix based quantum embedding approaches", by Quentin Marécat and Matthieu Saubanère additional variational parameters are introduced in the Block-Householder transformation for improving the bath orbitals' construction. Although I like the main idea of this method, I think that the description of this method is not clear enough and that the quality of the manuscript is poor at this point. There are several unclear points across the manuscript that I divided into major and minor issues:
Major issues:
* On line 17: "... exponentially scaling cost ..." what is the magnitude that is supposed to scale with system size?
* The authors used abstract terms such as fragment, bath, cluster, etc. I strongly suggest drawing a schematic representation of these terms in the same spirit as in Figure 1 of the Arxiv manuscript 2306.07641 by some of the present authors.
* On line 30: " ... to treat localized electron correlation ... " Is this a dynamic or static correlation?
* The notation used by the authors is misleading in several parts of this work, for instance, the c_{i \sigma} operators have some expression in Eq. 2 but this expression is changed in Eq. 3. This needs to be clarified.
* Notice that variables are used prior to their definition, as an example, N_i, N_b, and N_e are used on line 80 but they are first defined until line 178. Please, define all variables, operators, matrices, ... as they appear in the text. As another example "N" on line 178 does not seem to be defined. Please, check the definition of all variables.
* We encourage the authors to explicitly write what Hamiltonian operator they will be solving in the present work, the one on Eq. 1, 10, 13, or 14?
* On lines 163-164: "... we would like to control the number of bath orbitals, independently of the number of impurity orbitals ..." I wonder if would it also make sense to just increase the number of impurity orbitals in the original Block-Housholder transformation with the consequent increase in the number of bath orbitals? Would the results of the original BH transformation improve?
* We strongly recommend improving the notation for the matrices X across the entire manuscript, just as a simple example, what is the difference between X^{jT} and X^{1} in Eq. 22.
* What is the analytical expression of the "set of angles" mentioned on line 198?
* I guess the purpose of Figure 2 is to clarify the relationship between the original and the modified HS vectors (in this work). However, the meaning of this figure is still obscure to me. A more concise explanation is needed for this figure. What is the message the authors are trying to convey with this figure?
* What is the physical meaning of the \Delta buffer zone?
* The authors are introducing a set of angles that are variationally optimized which makes this method more expensive than the original HS transformation. As I mentioned previously, would it make sense to just increase the number of impurity-bath orbitals in the original HS transformation without the additional optimization scheme proposed in this work? How would the results be compared?
* A more detailed and concise explanation of Figure 3 is needed where are the axes? What is shown on the bar scale?
* On line 295: "... a damping of 60% ..." where is the damping used and how?
* Lines 296-349 encompass long paragraphs. I strongly recommend splitting them into shorter paragraphs with the specific cases for U/t mentioned across the text.
* Is there any reason to use the two ratios U/t and U/(U+4t)? Could authors stick with only one of them? At some point, line 311 for instance, authors used both ratios to define the same regime.
* I found cumbersome the explanation of the results in Figures 4-9. My main concern was in the naming of the methods. For example, in Figure 4, Block refers to the original BS method or the modified BS version proposed in this work by the authors? If this is the modified version, could you include the results of the original BS method as well? In the same figure, what are the \Delta, DM matching and Hartree results? We strongly suggest creating delimited subsections for each of the methods in the Theory section where the names of these methods are explicitly defined. Right now the methods look entangled to me, for instance, as I mentioned before I don't clearly see what are the new results with the method of the authors and the results with existing methods. Authors must strive to describe in detail existing methods and the proposed method in this paper. In particular, the names of the methods must be explicitly written as they are described in the text. These comments hold for all mentioned figures.
* In Figure 4, the relative error for all methods is around 20% for highly correlated systems, does it mean that all methods used here are not suitable for describing these systems?
* How expensive is the proposed BS transformation in this paper compared to existing methods including the original BS transformation? Could authors provide any scaling behaviour?
Minor issues:
* English grammar
* Define abbreviations as they appear in the text, for instance, 1RDM on line 5
English grammar must be thoroughly checked. For instance, British English is mixed with American English.
Phrases like: "... of the optimal construction of the orbitals of the bath" on lines 2-3 don't sound well. There are other sentences across the manuscript that need to be checked.
Author Response
Response to Reviewer 3 Comments
Open Review (x) I would not like to sign my review report ( ) I would like to sign my review report Quality of English Language ( ) I am not qualified to assess the quality of English in this paper ( ) English very difficult to understand/incomprehensible (x) Extensive editing of English language required ( ) Moderate editing of English language required ( ) Minor editing of English language required ( ) English language fine. No issues detected In the present manuscript entitled "A versatile unitary transformation framework for an optimal bath construction in density-matrix based quantum embedding approaches", by Quentin Marécat and Matthieu Saubanère additional variational parameters are introduced in the Block-Householder transformation for improving the bath orbitals' construction. Although I like the main idea of this method, I think that the description of this method is not clear enough and that the quality of the manuscript is poor at this point. There are several unclear points across the manuscript that I divided into major and minor issues: We thank the referee for his/her positive feedback. We trust that we have addressed any remaining ambiguities in the revised manuscript. Major issues: Point 1: On line 17: "... exponentially scaling cost ..." what is the magnitude that is supposed to scale with system size?Response 1: We thank the referee for their comment, and indeed, the beginning of the referee's comment regarding the exponential scaling cost was unclear. Here, we are referring to the computational cost associated with a straightforward resolution of the Schrodinger equation, which scales exponentially with the size of the system (Wave function approach). This exponential scaling computational cost makes numerical solutions impractical for systems containing around twenty electrons. In contrast, reduced-quantity-based theories such as DFT or rDMFT scale with a polynomial cost, typically of order 3-4, albeit at the expense of drastic approximations. Being at the edge of Wave function and reduced quantities formalisms makes Dac approaches interesting. The introduction has been adequately modified.
Point 2: The authors used abstract terms such as fragment, bath, cluster, etc. I strongly suggest drawing a schematic representation of these terms in the same spirit as in Figure 1 of the Arxiv manuscript 2306.07641 by some of the present authors. Response 2: We add a schematic picture to pedagogically introduce the meaning of the different terms used. Point 3: On line 30: " ... to treat localized electron correlation ... " Is this a dynamic or static correlation? Response 3: In the context of the Hubbard model, electronic correlation is typically discussed without distinguishing between static and dynamic correlation. There doesn't appear to be a compelling reason why the embedding Hamiltonian should better capture the effects of dynamic correlation rather than static correlation between the fragment and the environment in the global system. Point 4: The notation used by the authors is misleading in several parts of this work, for instance, the c_{i \sigma} operators have some expression in Eq. 2 but this expression is changed in Eq. 3. This needs to be clarified. Response 4: We thanks the referee for his comment. In the revised manuscript, we have changed the ordering of equations 2 and 3 for a sake of clarity. Point 5: Notice that variables are used prior to their definition, as an example, N_i, N_b, and N_e are used on line 80 but they are first defined until line 178. Please, define all variables, operators, matrices, ... as they appear in the text. As another example "N" on line 178 does not seem to be defined. Please, check the definition of all variables. Response 5: We apologize for such mistake that have been corrected in the revised version of the manuscript. All the acronyms, variables, operators and matrices have been carefully checked and are now properly defined, in particular N, N_i, N_b and N_e are defined in the introduction. Point 6: We encourage the authors to explicitly write what Hamiltonian operator they will be solving in the present work, the one on Eq. 1, 10, 13, or 14? Response 6: In the revised manuscript, we have specified that it is the effective and truncated Hamiltonian in Eq. 14 that is numerically diagonalized. Point 7: On lines 163-164: "... we would like to control the number of bath orbitals, independently of the number of impurity orbitals ..." I wonder if would it also make sense to just increase the number of impurity orbitals in the original Block-Housholder transformation with the consequent increase in the number of bath orbitals? Would the results of the original BH transformation improve? Response 7: It does make sense. The increase in the number of impurities for the BH transformation is studied in Ref. [32], demonstrating that the results systematically improve as the fragment size increases. In this work, we present a more general transformation where the generalized single impurity equations are compared with the BH method for systems of identical size, and therefore with the same numerical complexity. Thus, the case with 1 impurity and 3 bath sites is compared with the BH transformation with 2 impurities (and consequently 2 bath orbitals), which is simply a special case of the generalized transformation, as specified at the beginning of the Results and Discussion section. However, the aim of this work is to go beyond the current embedding methods, which are based on a singular value decomposition of the cluster-environment elements of the 1RDM.A comment has been added to explicitly clarify this result in lines 314-317.
Point 8: We strongly recommend improving the notation for the matrices X across the entire manuscript, just as a simple example, what is the difference between X^{jT} and X^{1} in Eq. 22. Response 8: In order to provide more rigorous notation, it was decided to introduce so-called "pythonic" notation, which makes a clear distinction between matrices and vectors, and simplifies the reading of the equations. Although these equations are more rigorous, they do make the notations more cumbersome. Point 9: What is the analytical expression of the "set of angles" mentioned on line 198? Response 9: There is no analytical expression. The angles can take on any values, and will satisfy equations (21) and (22). Point 10: I guess the purpose of Figure 2 is to clarify the relationship between the original and the modified HS vectors (in this work). However, the meaning of this figure is still obscure to me. A more concise explanation is needed for this figure. What is the message the authors are trying to convey with this figure? Response 10: We have now simplified the paragraph explaining the figure to get straight to the point, and added a paragraph summarizing the advantages of spherical representation. Point 11: What is the physical meaning of the \Delta buffer zone? Response 11: We believe that the delta buffer-zone to give a measure of the one-body disentanglement between the fragùent and the environment by measuring the froebenius norm of the fragment-environment 1-RDM. Point 12: The authors are introducing a set of angles that are variationally optimized which makes this method more expensive than the original HS transformation. As I mentioned previously, would it make sense to just increase the number of impurity-bath orbitals in the original HS transformation without the additional optimization scheme proposed in this work? How would the results be compared? Response 12: The answer to this question is given in the response 7. Point 13: A more detailed and concise explanation of Figure 3 is needed where are the axes? What is shown on the bar scale? Response 13: We provide complementary details of the meaning of figure 3 in lines 262-263 and in the caption. Point 14: On line 295: "... a damping of 60% ..." where is the damping used and how? Response 14: Damping is an arbitrary numerical parameter introduced to prevent drastic changes in the 1-RDM during the DaC process. The choice of this parameter is discussed in Ref. [32]. Point 15: Lines 296-349 encompass long paragraphs. I strongly recommend splitting them into shorter paragraphs with the specific cases for U/t mentioned across the text. Response 15: As the paragraph is rather dense, we have decided to split it into 2, with the first section describing the various common trends in the methods proposed for the different schemes, and the second section describing the specific features of each cost function. Point 16: Is there any reason to use the two ratios U/t and U/(U+4t)? Could authors stick with only one of them? At some point, line 311 for instance, authors used both ratios to define the same regime. Response 16: The convention U/(U+4t) allows us to better account for the competition between electron delocalization within the non-interacting bandwidth of 4t and the local repulsion force U, enabling us to treat the weakly and strongly interacting regimes equally. However, the U/t convention is simpler to interpret. To avoid the current confusion, we have added double axes to each figure, ensuring that the figures and results are accessible and interpretable regardless of the convention. Point 17: I found cumbersome the explanation of the results in Figures 4-9. My main concern was in the naming of the methods. For example, in Figure 4, Block refers to the original BS method or the modified BS version proposed in this work by the authors? If this is the modified version, could you include the results of the original BS method as well? In the same figure, what are the \Delta, DM matching and Hartree results? We strongly suggest creating delimited subsections for each of the methods in the Theory section where the names of these methods are explicitly defined. Right now the methods look entangled to me, for instance, as I mentioned before I don't clearly see what are the new results with the method of the authors and the results with existing methods. Authors must strive to describe in detail existing methods and the proposed method in this paper. In particular, the names of the methods must be explicitly written as they are described in the text. These comments hold for all mentioned figures. Response 17: To enhance the readability of the figures, we have provided the names of the different cost functions at the beginning of the Results and Discussion section. Additionally, we have clarified that the Block-Householder transformation method is used as a comparison to the various proposed methods. In the figures, the designed cost functions are represented by colored lines, while the Block-Householder transformation method is indicated by gray lines. Point 18: In Figure 4, the relative error for all methods is around 20% for highly correlated systems, does it mean that all methods used here are not suitable for describing these systems? Response 18: The asymptotic behavior of the kinetic energy and double occupancy is not correctly reproduced for highly correlated regimes. However, within the Block-Householder transformation, it has been demonstrated that the asymptotic behavior close to the atomic limit is recovered with up to three impurities (and three bath sites, i.e., within a 6-site cluster) as shown in Ref [32] of the revised manuscript. In this contribution, we show that optimization of the unitary transformation can already improve results at this limit without expanding the size of the cluster. The expected improvement, however, strongly depends on the cost function. We leave for future work the design of cost functions specifically aimed at improving the U/t >> 1 regime without deteriorating results in other regimes. A comment has been added to explicitly clarify this result in lines 314-317. Point 19: How expensive is the proposed BS transformation in this paper compared to existing methods including the original BS transformation? Could authors provide any scaling behaviour? Response 19: There are two contributions to the numerical complexity. The first corresponds to obtaining the ground state of the embedded cluster, which scales exponentially as O(4^(Nc)) with the number of orbitals in the cluster. The second corresponds to the unitary transformation of the 1RDM, which scales as O(N^3) with N being the number of orbitals in the system and must be repeated during the minimization process. Consequently, for large clusters where 4^Nc exceeds N^3, the numerical complexity is the same as the BH transformation. Minor issues: * English grammar * Define abbreviations as they appear in the text, for instance, 1RDM on line 5 All comments have now been taken into account. Comments on the Quality of English Language English grammar must be thoroughly checked. For instance, British English is mixed with American English. Phrases like: "... of the optimal construction of the orbitals of the bath" on lines 2-3 don't sound well. There are other sentences across the manuscript that need to be checked. Submission Date 26 July 2023 Date of this review 04 Aug 2023 14:59:52Round 2
Reviewer 3 Report
The present version of the manuscript "A versatile unitary transformation framework for an optimal bath construction in density-matrix based quantum embedding approaches", by Quentin Marécat and Matthieu Saubanère shows considerable improvement w.r.t. the initial version. The Theory part, for instance, displays better consistency in the notation for the equations. The captions for figures, Figure 3 for example, are now more instructive. Results and Discussions section also improved its readability. The authors addressed the comments I made in a concise manner. For these reasons, I accept this manuscript in its present form.
