REVIEW 4 major objections 3 minor 1 references
Semi-analytic Evaluation of 1, 2 and 3-Electron Coulomb Integrals with Gaussian expansion of Distance Operators W= R$_{C1}^{-n}$R$_{D1}^{-m}$, R$_{C1}^{-n}$r$_{12}^{-m}$, r$_{12}^{-n}$r$_{13}^{-m}$
T0 review · 4 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read A Gaussian expansion of $|r|^{-u}$ turns all the title's Coulomb integrals into sums of elementary Gaussian integrals, for any real powers $n$ and $m$.
desk verdict Plausible semi-analytic scheme for generalized Coulomb integrals, but the promised validation against known cases is absent and the 'any real n,m' claim needs a divergence caveat. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the Gaussian expansion of the Coulomb operator, $|r|^{-u} \approx \sum_{k=0}^{L}\sum_{i=1}^{M} C_{ik} r^{2k} \exp(-A_{ik} r^2)$, with $L=1$ recommended. Its role is to eliminate the square root $|r|=\sqrt{x^2+y^2+z^2}$ by replacing it with even powers $r^{2k}$, so that products of Gaussians remain Gaussians. The paper constructs the coefficients in two ways: a least-squares fit of $1/r$ on the interval $(0.04,50)$ with $M=50$ terms, the only concrete fit tabulated, and modified Taylor expansions in $s=r^2$ around $r_0=1$ that match derivatives while allowing boundary pivoting. Once the expansion is inserted, the Boys-function and erf arithmetic of standard formulas is replaced by sums over Gaussian terms; the remaining steps are routine Gaussian product formulas, binomial re-centering of polynomial factors, and rotation of the quadratic form to remove cross terms.
What would settle it
Compute the two-electron integral $\int\int \exp(-p|r_1-R_P|^2-q|r_2-R_Q|^2)|r_1-r_2|^{-1/2}\,dr_1dr_2$ numerically and by the Gaussian-expansion formula with a fitted $u=1/2$ expansion; if the summed expansion deviates by more than the claimed fit accuracy while the $u=1$ case passes, the claim that arbitrary real $n,m$ are covered is not established.
Extended reading notes
Core claim
The central claim is that if one fixes a Gaussian expansion of $|r|^{-u}$ as a sum of terms $c_i r^{2k_i} e^{-a_i r^2}$, then every integral in the title reduces to sums of products of one-dimensional Gaussian integrals. Because $r^{2k}$ is an even power under the substitution $r = \sqrt{x^2+y^2+z^2}$, the square root inside $|r|$ disappears, and no Laplace transform restricted to integer $n$ is needed. The paper shows this for one-electron density integrals (Eq. 11), two-electron integrals (Eq. 12), and three-electron integrals with $r_{12}^{-n}r_{13}^{-m}$, where mixed terms such as $x_1x_2$ are removed by rotating the quadratic form in the exponent. The same expansion serves both nucleus-electron and electron-electron distances, so the coefficient set can be reused. The known $n,m=0,1,2$ results are recovered as special cases, which the paper treats as validation.
Load-bearing premise
Everything rests on the premise that a finite fit of $|r|^{-u}$ over a limited range of distances stays accurate wherever the Gaussian electron clouds give the integral significant weight; the paper shows this only for $u=1$ between 0.04 and 50 bohr, and gives no error bound for other powers or other distance ranges.
Editorial extensions
If this is right
- For any real $n$ and $m$, the same Gaussian-product and diagonalization steps evaluate the one-, two-, and three-electron integrals named in the title; changing $u$ only changes the precomputed expansion coefficients.
- The known cases $(n,m)=(1,0)$, $(0,1)$, $(1,2)$ and the three-electron $r_{12}^{-1}r_{13}^{-1}$ integral are recovered as special cases, giving an internal consistency check.
- Non-integer and negative powers become usable, opening correlation corrections and operators such as $r_{12}/r_{13}$ to the same treatment.
- The Boys-function/erf arithmetic in standard formulas is replaced by sums over Gaussian terms, trading special functions for a larger number of elementary integrals.
- Because the expansion coefficients depend only on $u$ and the fitting interval, one fitted set can serve both nucleus-electron and electron-electron distance operators across many geometries and basis sets.
Reading between the lines
- The method's practical reach will be set by how well $u\neq 1$ fits perform; the paper tabulates only $u=1$, so the fractional-power claim is a testable extension rather than a demonstrated fact.
- For very diffuse Gaussians or very large inter-particle separations, the integration weight can extend beyond the fitted interval $(b_1,b_2)$, so accuracy may degrade even where the $u=1$ area check looks good.
- The same Gaussian expansion plus orthogonal diagonalization should carry over to four-electron operators such as $r_{12}^{-n}r_{34}^{-m}$, with larger symmetric matrices replacing the $2\times2$ rotation.
- Precomputing coefficient sets once per $u$ would let existing Gaussian-basis codes adopt the scheme without changing their integral drivers.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a semi-analytic scheme for evaluating one-, two-, and three-electron Coulomb-type integrals in which the distance operator is one of W = R_{C1}^{-n} R_{D1}^{-m}, R_{C1}^{-n} r_{12}^{-m}, or r_{12}^{-n} r_{13}^{-m}. The central idea is to approximate the radial factor r^{-u} on a finite interval by a linear combination of Gaussians and r^2-weighted Gaussians (Eqs. 8-10), then substitute this expansion into the integral. Because Gaussian products and exponent shifts are elementary (Appendices 3-5), the resulting integrals reduce to sums of elementary one-dimensional Gaussian integrals. The paper provides one explicit least-squares fit for u=1 on (0.04,50) with coefficients listed in Appendix 2, and it argues that this extends the known analytic cases n,m=0,1,2 to arbitrary real powers. No actual Coulomb integral is computed numerically with the expansion, despite the introduction announcing that the n,m=0,1,2 cases "serve as test." The algebraic steps are standard and mostly correct, but the central accuracy claim is not validated.
Significance. If the method were validated, it would provide a unified semi-analytic route to Coulomb-like integrals with non-integer and higher powers, which could be useful for correlation corrections, R12-type theories, and approximate Hamiltonians. The paper also collects useful Gaussian product and rotation formulas in the appendices, and it provides explicit fit coefficients for the u=1 case, which is a reproducible concrete artifact. The main mathematical construction is transparent: the approximation is openly fitted, so there is no hidden circularity in the derivation itself; the fit, however, is the load-bearing approximation. The significance is currently limited by the absence of numerical validation against exact integrals and by an unaddressed divergence for powers n>=3 with Gaussian factors centered on the nucleus. These issues are fixable, but they are essential to the paper's central claim.
major comments (4)
- [Section 'A good fit for 1/r^u' and Appendix 2] The introduction states that the known n,m=0,1,2 cases "serve as test," but the manuscript never performs this test. Equations (3)-(7) list the exact results for those cases, yet no numerical comparison is made between those formulas and the Gaussian-expansion results. This is load-bearing because the paper's claim of an "alternative solution" for the known cases and of controlled accuracy for general (n,m) depends on such a comparison. I request a table comparing, for representative geometries, the Gaussian-expansion evaluations of Eqs. (11) and (12) against the exact values from Eqs. (3)-(6), including error columns.
- [Eq. (8) and Section 'Evaluation of generalized Coulomb integrals with Gaussian expansion'] The fit r^{-u} ≈ Σ c_i r^{2k} exp(-a_i r^2) is determined on the finite interval (b1,b2) = (0.04,50), but the Coulomb integrals are over all space. The paper reports only the area deviation and pointwise deviation inside that interval. For r < 0.04, where r^{-u} is largest for u>0, no error bound is given; the added correction C1 exp(-A1 r^2) + C2 exp(-A2 r^2) improves the fit but does not quantify the error, and cancellation among the large alternating coefficients in Appendix 2 (some ~10^7) can amplify small local errors. The paper needs an error estimate for the all-space integrals, or at least a convergence test as the fit interval and term count are varied.
- [Eqs. (11)-(12) and abstract] The claim that the method works for "any real (n,m)" is too broad. For u >= 3, the integral ∫ exp(-p|r1-RP|^2) |r1-RC|^{-u} d^3r1 diverges when RC = RP, because the integrand behaves as r1^{-u} near the nucleus. The Gaussian expansion, however, returns a finite value since each term r^{2k} exp(-a r^2) is integrable. The paper mentions singularities of the incomplete gamma function only in a passing parenthesis, but the abstract and the main claims do not state the restriction u < 3. The scope must be restricted accordingly (or a regularization must be introduced), and the abstract amended.
- [Section 'A good fit for 1/r^u' and Section 'Evaluation of generalized Coulomb integrals'] Although the method is presented for arbitrary real u, the only fitted coefficients provided are for u=1. No numerical demonstration is given for non-integer u or for u = 2, despite those cases being central to the claim of a general procedure. The derivation shows how the integrals would be evaluated once a fit is available, but it does not show that the least-squares fit performs adequately for other u values. At minimum, the manuscript should provide fits or a fitting procedure with error measures for the u values actually used in the applications it cites.
minor comments (3)
- [Throughout] The text contains numerous typographical errors and formatting artifacts (e.g., garbled subscripts in displayed formulas, missing equation numbers, and repeated symbols) that make the derivation harder to follow than necessary. A careful copyedit is recommended.
- [References [9] and [10]] References [9] and [10] are generic web links to Wolfram Alpha and Wikipedia. The integrals and identities used should be cited to standard textbooks or primary sources with page numbers, so that the reader can verify the statements independently.
- [Appendix 2 and Section 'A good fit for 1/r^u'] The text says the effective M0 is 35 because two correction Gaussians are added, but the listed coefficients in Appendix 2 still show M0=33 and M2=17. The relationship between the tabulated coefficients and the claimed effective count should be clarified.
Circularity Check
No circularity: the Gaussian expansion coefficients are explicitly fitted to the operator being replaced, and the subsequent integral evaluation is an exact Gaussian-product consequence of that fit, not a disguised prediction.
full rationale
The derivation chain is transparent: Eq. (8) defines |r|^-u ≈ Σ c_i r^{2k} exp(-a_i r^2); the coefficients are obtained by least-squares fit on (b1,b2) (or by Taylor-type point matching) specifically to that target function, and the paper is explicit that 'a good fit is fundamental.' The Coulomb integrals are then evaluated exactly as linear combinations of standard Gaussian integrals via the product rule, completing the square, and elementary integrals. No fitted parameter is renamed as a prediction: the final integrals inherit the quality of the openly fitted expansion, which is an approximation rather than a circular identity. The announced check against known n,m ∈ {0,1,2} integrals is not carried out, and no error bound is given beyond the single u=1 fit; these are validation/accuracy gaps, not circular steps. The self-citations ([2], [5]) are contextual (known analytic cases, a two-electron solution) and are not load-bearing for the new derivation. Nothing in the argument reduces to its own inputs by construction.
Assumptions & free parameters
free parameters (4)
- Gaussian expansion coefficients {c_i} for u=1 =
Listed in the text: 33 values for i=1..M0 and 17 values for i=M0+1..M, e.g., c_1=26.7712
- Gaussian exponents {a_i} for u=1 =
a_i ranges from 312.5 down to 0.0004, chosen from inflection and maximum points
- Small-r correction coefficients (C1,A1,C2,A2) =
(1340.0, 500000.0, 155.0, 5000.0) or (96.0, 5000.0, 10.5, 1250.0)
- Fit interval and term counts (b1,b2,M,M0,M2) =
(0.04, 50.0), M=50, M0=33, M2=17
assumptions (4)
- domain assumption The Gaussian expansion (Eqs. 8-10) accurately represents |r|^-u on the integration range for arbitrary real u
- domain assumption Electron densities are linear combinations of Gaussian-type orbitals (Eq. 2)
- standard math Gaussian product theorem and multivariate Gaussian integral (Eq. 17) are correct
- standard math Incomplete gamma functions and error functions can be evaluated numerically
Cite this review
Pith. "Pith review of Semi-analytic Evaluation of 1, 2 and 3-Electron Coulomb Integrals with Gaussian expansion of Distance Operators W= R$_{C1}^{-n}$R$_{D1}^{-m}$, R$_{C1}^{-n}$r$_{12}^{-m}$, r$_{12}^{-n}$r$_{13}^{-m}$." pith.science (2026). https://pith.science/paper/O2IDLDKK
@misc{pith2026191002987,
author = {Pith},
title = {Pith review of: Semi-analytic Evaluation of 1, 2 and 3-Electron Coulomb Integrals with Gaussian expansion of Distance Operators W= R$_C1^-n$R$_D1^-m$, R$_C1^-n$r$_12^-m$, r$_12^-n$r$_13^-m$},
year = {2026},
howpublished = {\url{https://pith.science/paper/O2IDLDKK}},
note = {Machine review of arXiv:1910.02987}
}
read the original abstract
The equations derived help to evaluate semi-analytically (mostly for k=1,2 or 3) the important Coulomb integrals Int rho(r1)...rho(rk) W(r1,...,rk) dr1...drk, where the one-electron density, rho(r1), is a linear combination (LC) of Gaussian functions of position vector variable r1. It is capable to describe the electron clouds in molecules, solids or any media/ensemble of materials, weight W is the distance operator indicated in the title. R stands for nucleus-electron and r for electron-electron distances. The n=m=0 case is trivial, the (n,m)=(1,0) and (0,1) cases, for which analytical expressions are well known, are widely used in the practice of computation chemistry (CC) or physics, and analytical expressions are also known for the cases n,m=0,1,2. The rest of the cases - mainly with any real (integer, non-integer, positive or negative) n and m - needs evaluation. We base this on the Gaussian expansion of |r|^-u, of which only the u=1 is the physical Coulomb potential, but the u.ne.1 cases are useful for (certain series based) correction for (the different) approximate solutions of Schrodinger equation, for example, in its wave-function corrections or correlation calculations. Solving the related linear equation system (LES), the expansion |r|^-u about equal to SUM(k=0toL)SUM(i=1toM) Cik r^2k exp(-Aik r^2) is analyzed for |r| = r12 or RC1 with least square fit (LSF) and modified Taylor expansion. These evaluated analytic expressions for Coulomb integrals (up to Gaussian function integrand and the Gaussian expansion of |r|^-u) are useful for the manipulation with higher moments of inter-electronic distances via W, even for approximating Hamiltonian.
Figures
Reference graph
Works this paper leans on
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[1]
1.: A.Szabo, N.S.Ostlund: Modern Quant. Chem.: Intro. Adv. Electr. Struct. Theory, 1982, McMillan, NY. 2.: S.Kristyan, AIP Conference Proceedings 1978, 470030-1 to 6 (2018), see also https://arxiv.org/ and https://chemrxiv.org for kristyan. 3.: W.Klopper, F.R.Manby, S.Ten-No, E.F.Valeev, Int. Rev. in Physical Chemistry, 25, 427–468 (2006). 4.: S.Reine, T....
work page 2018
Reviewed August 14, 2026 · model on record in the stance chip above.
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