Fully Conjugated Heteroatomic Non- and Quasi-Alternant Polyradicals

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The design and synthesis of polyradicals(oids) are currently attracting much attention due to their potential use in the creation of novel spintronic molecular devices. In this paper, the authors have proposed a design strategy for restricting the lower-bound number of unpaired electrons in the fully π-conjugated diradical(oid)s and tetraradical(oid)s with five-membered non-alternant cyclopentadienyl and quasi-alternant thiophene rings. The authors have predicted and analyzed relative stabilities of (closed-shell or open-shell types of) valence bond forms, considering the balance between the aromatic stabilization energy and the energy of breaking π-bond. The authors also attempted to design tetraradical systems with no apparent terminal radical sources, based on the rules and strategies they established. 
The proposed molecular framework for multiradical(oid)s and the design rules presented are very interesting. Although the referee has several questions/comments on the present version of the manuscript, the overall quality of this paper is considered to be up to the standard for publication in Chemistry.

Questions/Comments

- In lines 169-171 of p.7, the authors explained that, 
"As a consequence, the diradical character of this compound is y0 = 0.34−0.49 and tetraradical character is y1 = 0.013−0.058 according to CASSCF NO occupation numbers of singlet states ..."
The referee thinks that these values were evaluated by substituting the occupation numbers of NOs at the CASSCF level into Yamaguchi's equation. But this is incorrect use of Yamaguchi's equation.
y_i value based on Yamaguchi's equation corresponds to "the occupation number of LUNO+i considering the (perfect-paring type) spin-projection scheme for the triplet spin-contaminant in the broken-symmetry UHF solution". Therefore, if the theory for evaluating the density and NOs is spin-contamination-free, like CASSCF, one should not apply Yamaguchi's equation but employ the occupation number of LUNO+i itself as the value of y_i (i.e., n_LUNO+i = y_i) at the level.
For details, please see the derivation of the original paper by Yamaguchi (ref. 27), or other papers describing the derivation (e.g., Nakano, The Chemical Record, 2017, 17, 27-62). Then, please correct the numbers of y_i at the CASSCF level (just replace them with their original occupation numbers).

- The authors performed geometry optimization at the BLYP level, a pure DFT-GGA. Usually, pure DFT-GGA, like BLYP, tends to predict too small energy gap between occupied and unoccupied levels and too delolicalized geometries for the singlet state case and may be rarely used for geometry optimization of π-conjugated systems. So the referee thinks that there are some special reasons for this choice. Why do the authors employ this level of approximation for the geometries of triplet/quintet states? (Perhaps, is there severe spin-contamination error in the high-spin state solution with the hybrid-DFT that causes problem?)

- Related to the above question, the authors employed geometries of high-spin states whereas the CASSCF results predicted the low-spin ground state for CTs and PTs. Many readers may wonder why the singlet state optimized geometries were not used. 

- The orbital symbols, like C0, C1', D0, D1', D1p R+, R-, L+*, L-*, N, etc. given in Tables of the main text were not well explained, except for L (long). The explanations should be given.

- For all the present systms, CASSCF results indicate the ground state singlet. Some readers (or other reviewers) may be interested in how dynamical correlation treatment will affect the ordering of spin states. The referee understands that such a computation demands high computational resources, and the discussion on the dynanical correlation effect is beyond the scope of this paper, so this is an optional comment, but it may be nice if the authors can comment something on it.

- In p.5, "Table 3. CASSCF(12,12/" should be "Table 3. CASSCF(12,12)/"

Author Response

The design and synthesis of polyradicals(oids) are currently attracting much attention due to their potential use in the creation of novel spintronic molecular devices. In this paper, the authors have proposed a design strategy for restricting the lower-bound number of unpaired electrons in the fully π-conjugated diradical(oid)s and tetraradical(oid)s with five-membered non-alternant cyclopentadienyl and quasi-alternant thiophene rings. The authors have predicted and analyzed relative stabilities of (closed-shell or open-shell types of) valence bond forms, considering the balance between the aromatic stabilization energy and the energy of breaking π-bond. The authors also attempted to design tetraradical systems with no apparent terminal radical sources, based on the rules and strategies they established. 
The proposed molecular framework for multiradical(oid)s and the design rules presented are very interesting. Although the referee has several questions/comments on the present version of the manuscript, the overall quality of this paper is considered to be up to the standard for publication in Chemistry.

Questions/Comments

• In lines 169-171 of p.7, the authors explained that, 
  "As a consequence, the diradical character of this compound is y0 = 0.34−0.49 and tetraradical character is y1 = 0.013−0.058 according to CASSCF NO occupation numbers of singlet states ..."
  The referee thinks that these values were evaluated by substituting the occupation numbers of NOs at the CASSCF level into Yamaguchi's equation. But this is incorrect use of Yamaguchi's equation.
  y_i value based on Yamaguchi's equation corresponds to "the occupation number of LUNO+i considering the (perfect-paring type) spin-projection scheme for the triplet spin-contaminant in the broken-symmetry UHF solution". Therefore, if the theory for evaluating the density and NOs is spin-contamination-free, like CASSCF, one should not apply Yamaguchi's equation but employ the occupation number of LUNO+i itself as the value of y_i (i.e., n_LUNO+i = y_i) at the level.
  For details, please see the derivation of the original paper by Yamaguchi (ref. 27), or other papers describing the derivation (e.g., Nakano, The Chemical Record, 2017, 17, 27-62). Then, please correct the numbers of y_i at the CASSCF level (just replace them with their original occupation numbers).

Answer. We acknowledge the fact that Yamaguchi’s approach does not capture the full picture of the open-shell character in multiconfigurational formalism. To calculate the open-shell character for n number of electrons, one must calculate the collective weight of the configurations with at least n number of unpaired electrons.

However, Yamaguchi’s method can never overestimate the open-shell character with occupation numbers of CASSCF calculations with large active spaces. Hence, this approach, although far from perfect, gives us way not to overestimate the open-shell character and still have some estimate of the degree of open-shell behavior.

 

• The authors performed geometry optimization at the BLYP level, a pure DFT-GGA. Usually, pure DFT-GGA, like BLYP, tends to predict too small energy gap between occupied and unoccupied levels and too delolicalized geometries for the singlet state case and may be rarely used for geometry optimization of π-conjugated systems. So the referee thinks that there are some special reasons for this choice. Why do the authors employ this level of approximation for the geometries of triplet/quintet states? (Perhaps, is there severe spin-contamination error in the high-spin state solution with the hybrid-DFT that causes problem?)

• Related to the above question, the authors employed geometries of high-spin states whereas the CASSCF results predicted the low-spin ground state for CTs and PTs. Many readers may wonder why the singlet state optimized geometries were not used.

Answer. Usually, the density functional theory does not do well for the open-shell low-spin states, hence, to optimize structures as open-shell singlet may fail to reproduce the proper unpaired electron density distribution. This has been shown in our previous works (check the Supporting Information of Phys. Chem. Lett. 2024, 15, 19, 5243–5249 and J. Org. Chem. 2024, 89, 19, 14006–14020).

However, for a given polyradical of order K, the highest spin-state (K + 1 multiplicity) unrestricted Kohn-Sham DFT, gives us good estimate of the electron density distribution for the close-by spin states in the low-energy spectrum of a polyradical. Note that in polyradicals, the electron-density distribution is very similar in different spin states within a low-energy spectrum. Furthermore, when necessary, we are using two different extremes for DFT geometries to make sure that the solutions of electronic structures fall within these extremes. Note that we do not rely on DFT for the energies of the states, but only for the geometry.

As the reviewer 1 states, if BLYP geometries for the singlet ground state is too delocalized, this implies that the open-shell character is (at least) not exaggerated by using such geometry.

 

•  The orbital symbols, like C0, C1', D0, D1', D1p R+, R-, L+*, L-*, N, etc. given in Tables of the main text were not well explained, except for L (long). The explanations should be given.

Answer.  The orbital naming explanations have been added to the manuscript.

 

• For all the present systms, CASSCF results indicate the ground state singlet. Some readers (or other reviewers) may be interested in how dynamical correlation treatment will affect the ordering of spin states. The referee understands that such a computation demands high computational resources, and the discussion on the dynanical correlation effect is beyond the scope of this paper, so this is an optional comment, but it may be nice if the authors can comment something on it.

Answer. In the past works, we have made such tests and “dynamical correlation” in these systems rarely affects the qualitative aspect of the results.  

- In p.5, "Table 3. CASSCF(12,12/" should be "Table 3. CASSCF(12,12)/"

Answer.  Thank you for the correction, this has been implemented in the manuscript.

 

Reviewer 2 Report

Comments and Suggestions for Authors

This paper describes computation of polyradicals containing five-membered rings such as non-alternant cyclopentadineyl and quasi-alternant thiophene rings and stability of the open-shell structure. It seems that this finding is important for designing polyradicals with any number of unpaired electrons and any ground state multiplicity from different classes of organic compounds, which can lead to multifunctional organic materials. Therefore, I feel that it is adequate for publication in Chemistry.

Author Response

This paper describes computation of polyradicals containing five-membered rings such as non-alternant cyclopentadineyl and quasi-alternant thiophene rings and stability of the open-shell structure. It seems that this finding is important for designing polyradicals with any number of unpaired electrons and any ground state multiplicity from different classes of organic compounds, which can lead to multifunctional organic materials. Therefore, I feel that it is adequate for publication in Chemistry.

Answer: We thank the reviewer 2 for the positive evaluation of the manuscript.

Reviewer 3 Report

Comments and Suggestions for Authors

It was a pleasure to read this paper.  I commend the authors for the presentation of their work. I have only minor comments for the authors to consider:

1) The second word of the Introduction should begin with a lowercase p.

2) I would aid the reader to insert from ChemDraw or similar structures of some polyradicals into the Introduction.

3) A suggestion for the authors is to break up long paragraphs into smaller ones at good breaking points.

Author Response

It was a pleasure to read this paper.  I commend the authors for the presentation of their work. I have only minor comments for the authors to consider:

1) The second word of the Introduction should begin with a lowercase p.

2) I would aid the reader to insert from ChemDraw or similar structures of some polyradicals into the Introduction.

3) A suggestion for the authors is to break up long paragraphs into smaller ones at good breaking points.

Answers

• Implemented in the paper
• This suggestion has been considered but to keep the article simple, we did not include more drawings.
• This has been considered and applied to the text where necessary.

We thank the reviewer 3 for the positive evaluation of the manuscript.

Reviewer 4 Report

Comments and Suggestions for Authors

This article presents a theoretical framework for designing fully π-conjugated heteroatomic polyradicals, particularly focusing on diradical(oid)s and tetraradical(oid)s with non-alternant and quasi-alternant five-membered rings, such as cyclopentadienyl and thiophene. By controlling the topology of π-conjugation and leveraging aromatic stabilization, the authors demonstrate how to restrict the number of unpaired electrons and design compounds with high open-shell character. Using theoretical calculation methods like CASSCF, they analyze the electronic structures and low-energy spectra of the designed compounds, revealing significant diradical and tetraradical characters. The study extends the authors' previous theory of rational polyradical design, showing its applicability to non- and quasi-alternant systems. The work highlights the potential for creating multifunctional organic materials with tunable magnetic properties, though experimental validation remains for future research. Overall, this work represented a systematical study and contained sufficient novelty elements, which could be considered for acceptance after solving the following problems:

1. The authors assumed that aromatic resonance energy alone can sufficiently stabilize open-shell structures might not account for other factors like steric effects or other environmental influences, which could alter the compounds' properties. These limitations suggest that the proposed design rules, while insightful, may require further refinement and experimental validation to ensure their broader applicability.
2. The authors assume that the aromatic resonance energy of thiophene rings can be directly compared to benzene rings (e.g., 2.5–75 thiophene rings ≈ 2–3 benzene rings). This equivalence might be controversial, as the aromaticity of thiophene is influenced by the heteroatom (sulfur), which could introduce additional electronic effects not fully accounted for in the model.
3. While the authors briefly mention the use of anti-aromaticity to destabilize closed-shell configurations, it does not thoroughly explore how anti-aromaticity might destabilize open-shell structures as well. This could lead to an incomplete understanding of the energetic trade-offs in polyradical design.
4. The paper focuses on static electronic structures and does not address dynamic effects, such as vibronic coupling or thermal fluctuations, which could significantly impact the stability and reactivity of polyradicals in real-world conditions.
5. To strengthen the validity of the results, the authors could compare CASSCF outcomes with those from other computational methods, such as density functional theory (DFT) with different functionals or multireference perturbation theory (e.g., CASPT2 or NEVPT2). This would help identify potential biases or limitations of the CASSCF approach.

Author Response

This article presents a theoretical framework for designing fully π-conjugated heteroatomic polyradicals, particularly focusing on diradical(oid)s and tetraradical(oid)s with non-alternant and quasi-alternant five-membered rings, such as cyclopentadienyl and thiophene. By controlling the topology of π-conjugation and leveraging aromatic stabilization, the authors demonstrate how to restrict the number of unpaired electrons and design compounds with high open-shell character. Using theoretical calculation methods like CASSCF, they analyze the electronic structures and low-energy spectra of the designed compounds, revealing significant diradical and tetraradical characters. The study extends the authors' previous theory of rational polyradical design, showing its applicability to non- and quasi-alternant systems. The work highlights the potential for creating multifunctional organic materials with tunable magnetic properties, though experimental validation remains for future research. Overall, this work represented a systematical study and contained sufficient novelty elements, which could be considered for acceptance after solving the following problems:

1. The authors assumed that aromatic resonance energy alone can sufficiently stabilize open-shell structures might not account for other factors like steric effects or other environmental influences, which could alter the compounds' properties. These limitations suggest that the proposed design rules, while insightful, may require further refinement and experimental validation to ensure their broader applicability.

Answer: We explore not only aromaticity but also the topological and steric factors to design the compounds with the described open-shell behavior. The environmental infuence is currently out of the scope of the studies, but is indeed important to research in the upcoming studies.

 

2. The authors assume that the aromatic resonance energy of thiophene rings can be directly compared to benzene rings (e.g., 2.5–75 thiophene rings ≈ 2–3 benzene rings). This equivalence might be controversial, as the aromaticity of thiophene is influenced by the heteroatom (sulfur), which could introduce additional electronic effects not fully accounted for in the model.

Answer: We acknowledge that there could be additional electronic effects by the sulfur into the thiophene ring. However, our model coalesces this effect into the value of aromatic resonance energy. This energy is basically a difference between quinoidal (or otherwise non-aromatic) and aromatic configurations of the ring. Hence, we can reason that aromatic resonance energy is still higher in benzene as this is also verified by our calculations based on comparisons of the open-shell characters of studieds compounds with same number of benzene and thiophene rings.

 

3. While the authors briefly mention the use of anti-aromaticity to destabilize closed-shell configurations, it does not thoroughly explore how anti-aromaticity might destabilize open-shell structures as well. This could lead to an incomplete understanding of the energetic trade-offs in polyradical design.

Answer: Note that the principle of the design of open-shell systems is that only more stable π-configuration of the bridging subsystem can be compatible to create the open-shell. Because we need to offset the energy needed to break the π bond that led to the open-shell structure. If the open-shell structure is only compatible with antiaromatic bridging π-subsystem and closed-shell structure allows for non-aromatic bridging π-subsystem, of course, the open-shell character will be diminished to avoid antiaromaticity and also maximize the number of π-bonds. The reason such systems are not shown in the article is because they do not represent any successful way of polyradical design, but only serve as the limit of preferential closed-shell behavior when there is destibilizing resonance energy in the bridging subsystem if the open-shell configuration is assumed which breaks one π-bond per subshell in any case. 

 

4. The paper focuses on static electronic structures and does not address dynamic effects, such as vibronic coupling or thermal fluctuations, which could significantly impact the stability and reactivity of polyradicals in real-world conditions.

Answer: We agree with the reviewer that dynamic effects are not addressed. This is out of scope of this work and can be explored for some systems of practical interest.

Nevertheless, the ground-state open-shell character of the presented compounds is still expected, as the used geometries are local minima and an observed variance in open-shell character between different geometries (RKS or UKS-triplet, for example) is not too much to expect dramatic changes in open-shell behavior upon minor perturbations.  

 

5. To strengthen the validity of the results, the authors could compare CASSCF outcomes with those from other computational methods, such as density functional theory (DFT) with different functionals or multireference perturbation theory (e.g., CASPT2 or NEVPT2). This would help identify potential biases or limitations of the CASSCF approach.

Answer: For the electronic structure calculations of multiconfigurational systems, DFT as a single-configurational method cannot be expected to perform better than CASSCF, especially when the used basis set is sufficiently large and the active space is also large and appropriately chosen. Moreover, DFT is never guaranteed to distinguish between different spin states of same muliplicity. Also, we have shown in our previous publications (Supporting Information of J. Phys. Chem. Lett. 2024, 15, 19, 5243–5249 and J. Org. Chem. 2024, 89, 19, 14006–14020) that energy gaps and qualitatively correct behavior may change significantly upon changing the used exchange-correlation functional.

As for multireference perturbation theory, we have performed the calculations for similar systems in our previous studies and rest of the “dynamic correlation” does not affect the qualitative open-shell behavior dramatically.