Reparametrization of the Colle–Salvetti formula

We investigate the Colle–Salvetti (CS) formula, the basis of the Lee, Yang and Parr (LYP) correlation functional used in approximate density functional theory. The CS formula is reparametrized using high-accuracy Hartree–Fock (HF) wavefunctions to determine the accuracy of the formula to calculate anions. Fitting to the hydride ion or the two-electron system just prior to electron detachment at the HF level of theory does not, in general, improve the calculated correlation energies using the parameters derived from the CS/LYP method. An analysis of the CS parameters used in the popular LYP functional demonstrates the ingenuity and perhaps fortuitousness of the original formulation by CS.


Comments to the Author(s)
In this article, the authors revisit the fitting of parameters in the Colle-Salvetti (CS) formula. In particular, they replace the original Hartree-Fock (HF) wave functions used by Colle and Salvetti with high-accuracy HF wave functions computed using an approach developed in their group. These high-accuracy wave functions correctly capture the electron-nuclear cusp while remaining the electron-electron correlation energy is described using the CS formula. They then investigate various refitting of the CS equations for neutral, cationic, and anionic nuclei. Their primary results suggest: (i) Refitting the CS formula with high-accuracy HF wave functions can improve the description of the atom to which it was fitted, but is not generally transferable to other atoms; (ii) The original CS formula performs badly for anionic systems, particularly the hydride anion; (iii) Refitting the CS formula to a high-accuracy HF wave function for hydride dramatically improves the correlation energy for H^{-}, but is not transferable to other anions.
Given the significance of the CS formula to density functional approximations such as the LYP functional, revisiting the original CS parameterisation using high-accuracy wave functions for these more exotic systems is timely and novel. However, there are areas of the manuscript that I believe require some further clarification. I would therefore support publication once the reviewers have addressed the comments below: (1) The abstract does not read clearly. In particular, the third sentence starting "It is shown that…" is very unclear and I would suggest the authors consider rewriting it.
(2) Equation 2.15 seems to be missing a "dr" for the first integral (0 to inf) (3) In Section 3b(ii), it concerns me that the authors appear to have found different sets of CS parameters for the same Ri and Rf values using their optimisation method or by starting from the original CS parameters. (e.g. in Table 2 they get a=0.011485, b=0.024418, c~0, d=0.664135, but in the paragraphs below they describe 'Fit 2' with a=0. 01628, b=0.18438, c=0.57594, d=0.80562.) This suggests that the parameter optimisation is starting-point dependent. Can the authors clarify what starting point they use, and can they confirm whether the global best fit has been identified? Understanding the choice of starting point is essential for reproducibility. How would "Fit 1" and "Fit 2" compare to the graphical representation in Figure 3?
(4) The failure of CS for the hydride anion is spectacular, and the variable performance across other anions is rightly highlighted as a point of concern. What is even more surprising is that fitting the CS parameters to the hydride anion does not lead to a transferable approach. The authors describe this as an "over-fitting" issue, but that seems unlikely to account for such a large error. I wonder if the authors can provide more physical insight into this failure? To me, it looks like the failure for anionic systems is a static correlation effect, which can be inferred from the presence of UHF symmetry breaking. In particular, previous work from the authors [doi:10.1098/rsos.181357] has shown that the HF approximation predicts a large error in the oneparticle radial distance <r_1> for hydride (<r_1> = 2.71 in fully-correlated vs <r_1> = 2.5 in HF). If the RHF density is a poor initial approximation, then the CS formula must perform the dual task of relaxing the one-particle energy (by relaxing the orbitals) and capturing the two-electron correlation. This may explain why the hydride-fitted parameters overestimate the correlation energy in the heavier cations, where perhaps the one-particle relaxation is less important. A simple test would be generate the 2nd-order HF density for hydride using the exact one-particle density obtained from the fully-correlated wave function in [doi:10.1098/rsos.181357] and use this to fit the CS parameters. This should leave only electron-electron correlation errors. .0043105] define the critical nuclear charge as the point where the occupied orbital energy becomes zero, corresponding to Zc=0.828161008 . This alternative definition might be considered more physical as it is the point where the electrons suddenly start to detach from the nucleus. Furthermore, the asymptotic behaviour of the radial density becomes exp(-a sqrt(r)), in contrast to the standard exp(-a r). Given these alternative definitions for the nuclear charge, and the potential importance of the long-range density behaviour in fitting the CS formula, the authors should provide a better justification for their choice of Zc. In their current results, it is not surprising that the Zc and H-fitted parameters give such similar results as the RHF wave functions are qualitatively very similar at these two charges, i.e. the electrons are still bound at Zc=1.031177528. It would therefore be interesting to compare the optimised CS parameters at both definitions Zc=1.031177528 and Zc=0.828161008 to understand the significance of the critical binding threshold.
(6) Having identified the challenges of fitting the CS parameters, can the authors comment on potential ways to overcome these challenges?
(7) In Supporting Information Table IV, the caption refers to "The percentage error between exact and calculated values is provided in brackets", but I cannot find any brackets in the table.

Decision letter (RSOS-211333.R0)
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Dear Dr Cox:
Title: Reparametrization of the Colle-Salvetti Formula Manuscript ID: RSOS-211333 Thank you for your submission to Royal Society Open Science. The chemistry content of Royal Society Open Science is published in collaboration with the Royal Society of Chemistry.
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Comments to the Author: (There are no comments.) ********************************************** Reviewers' Comments to Author: Reviewer: 1 Comments to the Author(s) The manuscript by Cox and coworkers revisits the Colle-Salvetti (CS) formula, which underlies the LYP correlation functional and is thus a cornerstone of density-functional theory. In particular, it investigates whether a re-parametrization of the CS formula (which dates back to 1975 and is based on a Hartree-Fock wavefunction obtained by Clementi in 1965) using highly accurate data available today could be beneficial. It then extents this by performing a reparamtrization using anions, and assesses whether this improves the description of electron correlation in anions.
Even though the results are mainly negative (i.e., a re-parametrization does hardly improve compared to the original CS formula), they are of great importance for understanding the success of the LYP functional and to provide guidance towards improving correlation functionals. The authors' analysis is instructive, and discusses these insights very clearly. I particularly like the sensitivity analysis of different parameters in the CS formula / LYP functional.
I only have two minor comments that the authors might address in a revised version of their manuscript: 1) The authors refer to "overfitting" several times when discussing their observation that the accuracy of the CS formula is worse if a fit that is more accurate at larger distances is used. I was a bit confused by this, as I think that "overfitting" usually refers to a case where a very flexible function is fitted to a small data set, resulting in spurious oscillations of the fit. This is not the case here, and the problem is not the fitting procedure, but the underlying data, i.e., the fact that the helium atom might be a poor model. Maybe the authors can find a better term here, or at least add a short explanation of their use of the term "overfitting".
2) I think it might be useful to also include "Fit 2" in Figure 3.
Reviewer: 2 Comments to the Author(s) In this article, the authors revisit the fitting of parameters in the Colle-Salvetti (CS) formula. In particular, they replace the original Hartree-Fock (HF) wave functions used by Colle and Salvetti with high-accuracy HF wave functions computed using an approach developed in their group. These high-accuracy wave functions correctly capture the electron-nuclear cusp while remaining the electron-electron correlation energy is described using the CS formula. They then investigate various refitting of the CS equations for neutral, cationic, and anionic nuclei. Their primary results suggest: (i) Refitting the CS formula with high-accuracy HF wave functions can improve the description of the atom to which it was fitted, but is not generally transferable to other atoms; (ii) The original CS formula performs badly for anionic systems, particularly the hydride anion; (iii) Refitting the CS formula to a high-accuracy HF wave function for hydride dramatically improves the correlation energy for H^{-}, but is not transferable to other anions.
Given the significance of the CS formula to density functional approximations such as the LYP functional, revisiting the original CS parameterisation using high-accuracy wave functions for these more exotic systems is timely and novel. However, there are areas of the manuscript that I believe require some further clarification. I would therefore support publication once the reviewers have addressed the comments below: (1) The abstract does not read clearly. In particular, the third sentence starting "It is shown that…" is very unclear and I would suggest the authors consider rewriting it.
(2) Equation 2.15 seems to be missing a "dr" for the first integral (0 to inf) (3) In Section 3b(ii), it concerns me that the authors appear to have found different sets of CS parameters for the same Ri and Rf values using their optimisation method or by starting from the original CS parameters. (e.g. in Table 2 they get a=0.011485, b=0.024418, c~0, d=0.664135, but in the paragraphs below they describe 'Fit 2' with a=0.01628, b=0.18438, c=0.57594, d=0.80562.) This suggests that the parameter optimisation is starting-point dependent. Can the authors clarify what starting point they use, and can they confirm whether the global best fit has been identified? Understanding the choice of starting point is essential for reproducibility. How would "Fit 1" and "Fit 2" compare to the graphical representation in Figure 3? (4) The failure of CS for the hydride anion is spectacular, and the variable performance across other anions is rightly highlighted as a point of concern. What is even more surprising is that fitting the CS parameters to the hydride anion does not lead to a transferable approach. The authors describe this as an "over-fitting" issue, but that seems unlikely to account for such a large error. I wonder if the authors can provide more physical insight into this failure? To me, it looks like the failure for anionic systems is a static correlation effect, which can be inferred from the presence of UHF symmetry breaking. In particular, previous work from the authors [doi:10.1098/rsos.181357] has shown that the HF approximation predicts a large error in the oneparticle radial distance for hydride ( = 2.71 in fully-correlated vs = 2.5 in HF). If the RHF density is a poor initial approximation, then the CS formula must perform the dual task of relaxing the one-particle energy (by relaxing the orbitals) and capturing the two-electron correlation. This may explain why the hydride-fitted parameters overestimate the correlation energy in the heavier cations, where perhaps the one-particle relaxation is less important. A simple test would be generate the 2nd-order HF density for hydride using the exact one-particle density obtained from the fully-correlated wave function in [doi:10.1098/rsos.181357] and use this to fit the CS parameters. This should leave only electron-electron correlation errors.
(5) The authors should be careful about their choice of the HF critical nuclear charge in Section 3c(ii). They choose the charge Zc = 1.031177528 which they have previously identified as the point where the RHF energy is degenerate with the ionised system. However, other work including [doi:10.1063/1.4871018], [doi:10.1103/PhysRevA.101.062504], and [doi:10.1063/5.0043105] define the critical nuclear charge as the point where the occupied orbital energy becomes zero, corresponding to Zc=0.828161008 . This alternative definition might be considered more physical as it is the point where the electrons suddenly start to detach from the nucleus. Furthermore, the asymptotic behaviour of the radial density becomes exp(-a sqrt(r)), in contrast to the standard exp(-a r). Given these alternative definitions for the nuclear charge, and the potential importance of the long-range density behaviour in fitting the CS formula, the authors should provide a better justification for their choice of Zc. In their current results, it is not surprising that the Zc and H-fitted parameters give such similar results as the RHF wave functions are qualitatively very similar at these two charges, i.e. the electrons are still bound at Zc=1.031177528. It would therefore be interesting to compare the optimised CS parameters at both definitions Zc=1.031177528 and Zc=0.828161008 to understand the significance of the critical binding threshold.
(6) Having identified the challenges of fitting the CS parameters, can the authors comment on potential ways to overcome these challenges?
(7) In Supporting Information Table IV, the caption refers to "The percentage error between exact and calculated values is provided in brackets", but I cannot find any brackets in the table.
Author's Response to Decision Letter for (RSOS-211333.R0) See Appendix A.

RSOS-211333.R0 (Original submission)
Review form: Reviewer 1 Is the manuscript scientifically sound in its present form? Yes

Are the interpretations and conclusions justified by the results? Yes
Is the language acceptable? Yes

Recommendation?
Accept as is

Comments to the Author(s)
The authors have addressed all my previous comments.

Review form: Reviewer 2
Is the manuscript scientifically sound in its present form? Yes

Recommendation? Accept as is
Comments to the Author(s) I thank the authors for their excellent responses and I congratulate them on a very interesting study.

Decision letter (RSOS-211333.R1)
We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.
Dear Dr Cox: Title: Reparametrization of the Colle-Salvetti Formula Manuscript ID: RSOS-211333.R1 It is a pleasure to accept your manuscript in its current form for publication in Royal Society Open Science. The chemistry content of Royal Society Open Science is published in collaboration with the Royal Society of Chemistry.
The comments of the reviewer(s) who reviewed your manuscript are included at the end of this email.
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Reviewers' Comments to Author: Reviewer: 1 Comments to the Author(s) The manuscript by Cox and coworkers revisits the Colle-Salvetti (CS) formula, which underlies the LYP correlation functional and is thus a cornerstone of density-functional theory. In particular, it investigates whether a re-parametrization of the CS formula (which dates back to 1975 and is based on a Hartree-Fock wavefunction obtained by Clementi in 1965) using highly accurate data available today could be beneficial. It then extents this by performing a re-paramtrization using anions, and assesses whether this improves the description of electron correlation in anions.
Even though the results are mainly negative (i.e., a re-parametrization does hardly improve compared to the original CS formula), they are of great importance for understanding the success of the LYP functional and to provide guidance towards improving correlation functionals. The authors' analysis is instructive, and discusses these insights very clearly. I particularly like the sensitivity analysis of different parameters in the CS formula / LYP functional. I only have two minor comments that the authors might address in a revised version of their manuscript: 1) The authors refer to "overfitting" several times when discussing their observation that the accuracy of the CS formula is worse if a fit that is more accurate at larger distances is used. I was a bit confused by this, as I think that "overfitting" usually refers to a case where a very flexible function is fitted to a small data set, resulting in spurious oscillations of the fit. This is not the case here, and the problem is not the fitting procedure, but the underlying data, i.e., the fact that the helium atom might be a poor model. Maybe the authors can find a better term here, or at least add a short explanation of their use of the term "overfitting".
We have made the following changes in the text to address this: Page 6 second para "… found that over-fitting could be problematic" replaced by "… found that a tighter fit could be problematic" Page 11 first para comment deleted Page 11 second para "by not overfitting" replaced with "by relaxing the fit" Page 16 mention of overfitting deleted Page 18 conclusions "The loss of accuracy is attributed to over-fitting;" replaced by "The loss of accuracy is attributed to the tightness of the fit;" 2) I think it might be useful to also include "Fit 2" in Figure 3.
Thank you for this suggestion -Fit 2 has now been included in Figure 3.