{"id":"e87b5431-57f4-4280-9c21-8988d85f6d3f","arxiv_id":"2608.07205","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A radial-distribution-function framework computes collapse-induced spontaneous emission rates for arbitrary noise and gives material-specific predictions for germanium and xenon.","lead":"This preprint derives a general formula for spontaneous radiation from atoms driven by collapse-model noise, and evaluates it for germanium and xenon using first-principles electron distributions. It finds that realistic atomic structure suppresses the predicted emission at low energies, which could sharpen experimental searches for collapse physics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (39) double-counts electron pairs: with normalized P_αβ, ∫P from Eq. (39) equals 2N_e(N_e−1), not N_e(N_e−1), so the E→0 CSL cancellation and the finite-energy spectra in Eq. (35) are inconsistent unless the code silently renormalizes.","rationale":"I read the manuscript in good faith. The derivation from the stochastic Schrödinger equation to Eq. (35) is coherent and recovers previous limits; the product approximation for the electron-electron pair density is a real approximation, but it is supported by the Si/Ar benchmark and does not by itself undermine the qualitative low-energy cancellation. The sharpest load-bearing defect is the normalization of the electron-electron RDF: as written, Eq. (39) and Eq. (34) are mutually inconsistent, and this inconsistency directly feeds the E→0 cancellation claim and the finite-energy spectra. The reader's weakest_assumption identified the same factor-2 issue; I therefore agree. The proposed check—printing the integral of P(r) from the public code for Ar—settles whether this is merely a typo in the text or a computational error. If the code renormalizes P(r), the conditional acceptance remains appropriate; if not, the central claim is wrong. I recommend keeping the current CONDITIONAL verdict rather than accepting the paper as final, because this normalization must be stated and verified.","tokens_in":12781,"tokens_out":11585,"duration_ms":111138,"concrete_test":"Run the published code for argon (the Fig. 2 benchmark): print the raw P(r) computed from Eqs. (39)–(43), its integral, and P(k=0). For the ordered-pair normalization of Eq. (34), the integral must be N_e(N_e−1)=306; if it is 612, Eq. (47) is not implemented. Also independently evaluate Eq. (35) at E→0 for argon; the bracket must vanish. If the code renormalizes to 306, state that explicitly and correct Eq. (39) by removing the factor 2 or restricting the sum, then re-check whether the Ge/Xe figures change.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (35) places the whole low-energy phenomenology in the sinc-transformed electron-electron RDF P(k). Below Eq. (40), Pαβ(r) is the intracule density from product densities; with normalized shell densities, its integral over r is 1. For ordered pairs the paper requires the total P(r) to integrate to N_e(N_e−1) in Eq. (34). Equation (39), P(r)=2 Σ_{αβ} n_α(n_β−δ_{αβ})P_{αβ}(r), then integrates to 2[N_e^2−N_e]=2N_e(N_e−1). With N_P=N_e=Z, the E→0 limit of Eq. (35) becomes Z^2+Z−2Z^2+2Z(Z−1)=Z^2−Z, not zero as claimed in Eq. (47). The finite-energy spectra and the comparison to Ref. [27] are shifted by the same ambiguity. If the public code renormalizes P(r) to Eq. (34), the qualitative cancellation survives but Eq. (39) is not the implemented formula; if it does not, the central CSL cancellation is an artifact of double counting. The manuscript must reconcile this and print the actual integral of P(r) for the benchmark atoms.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript derives a general semiclassical expression for the spontaneous photon emission rate induced by collapse noise, valid for arbitrary (factorizable) noise correlations, and reformulates it in terms of radial distribution functions (RDFs). The central formula, Eq. (35), gives dΓ/dE = (1/E)[N_P² + N_e − 2N_P R(k) + P(k)], where R(k) and P(k) are collapse-kernel-modulated sinc transforms of the nuclear–electron and electron–electron RDFs. Electron RDFs are obtained from DFT (GPAW) orbitals within a product approximation for the two-particle density, decomposed over shells and benchmarked against hydrogenic analytical results and literature intracule data for Si and Ar. Rates are evaluated for Ge and Xe for the CSL and DP models, for white and colored noise. The main physical results are that the CSL rate for neutral atoms vanishes as E→0 due to charge-correlation cancellation, that the RDF-based rates stay below the 1/E asymptote, and that unphysical oscillations of the earlier clamped-electron approximation (Ref. [27]) are washed out by realistic radial densities. I confirm that the derivation from Eq. (14) to Eq. (24) is internally consistent and that the stress-test concern about the normalization of P(r) lands: the factor-of-two mismatch between Eqs. (39) and (34) is real and is load-bearing for the paper's central claims.","tokens_in":13066,"tokens_out":33391,"duration_ms":277011,"significance":"If the normalization issues are resolved, the paper offers a clean separation between model-dependent content (the kernel f(r)) and atomic-structure content (the RDFs), with concrete Ge and Xe predictions that are directly relevant to the VIP, MAJORANA, and XENONnT analyses. The derivation is internally consistent; the recovery of the 1/E asymptote at high energies and the E→0 cancellation for neutral atoms provide two physically meaningful checks; and the manuscript ships public code plus benchmarks (hydrogenic analytic case, Si/Ar Hartree–Fock intracule data, and a relativistic-sensitivity test). The falsifiable qualitative predictions — the CSL E→0 cancellation and the suppression of clamped-electron oscillations — are genuinely useful for model discrimination. The significance is currently capped by two issues: as printed, the electron–electron RDF normalization is inconsistent by a factor of two, which directly affects Eq. (47) and Fig. 5, and the quantitative claims of the Abstract ('more robust, material-dependent experimental constraints') rest on two approximations whose rate-level impact is asserted but not quantified.","major_comments":[{"comment":"The normalization of the electron–electron RDF is inconsistent by a factor of two as printed. Eq. (39) defines P(r) = 2 Σ_αβ n_α(n_β − δ_αβ) P_αβ(r), with P_αβ(r) normalized to unity (the benchmark in Fig. 1 and Eq. (45) has ∫P_1s,1s dr = 1). The double sum Σ_αβ n_α(n_β − δ_αβ) equals N_e(N_e − 1), so Eq. (39) integrates to 2N_e(N_e − 1), contradicting the ordered-pair normalization ∫P dr = N_e(N_e − 1) stated in Eq. (34). Inserting the printed normalization into Eq. (35) removes the E→0 cancellation claimed in Eq. (47): for a neutral atom the limit becomes (1/E)[Z² + Z − 2Z² + 2Z(Z − 1)] = Z(Z − 1)/E ≠ 0, and the finite-energy spectra of Fig. 5 and the claim that the RDF rate 'remains consistently below the 1/E limit' would be shifted by the same ambiguity. The manuscript must reconcile Eqs. (39) and (34), report the numerically computed ∫P(r)dr for Ge, Xe, and the Si/Ar benchmarks of Fig. 2, and re-verify Eq. (47) and Fig. 5 with the corrected factor or with an explicitly justified convention. If the public code internally renormalizes P(r) to satisfy Eq. (34), this must be stated, because Eq. (39) would then not be the implemented formula.","section":"Sec. III, Eqs. (34), (39), and Sec. IV, Eqs. (46)–(47)"},{"comment":"The E→0 argument also requires an explicit normalization convention for the kernel f(r), independently of the factor-of-two issue. In Eq. (35) the contact terms N_P² and N_e originate from δ(r)-peaked pair distributions and therefore carry a factor f(0) (see Eqs. (30) and (32)), whereas the transforms R(k) and P(k) in Eqs. (36)–(37) carry the full f(r). The sentence near Eq. (46) stating that the cancellation 'corresponds to the absence of spatial modulation between emitter pairs, namely f(r) = 1' conflates the E→0 limit of the sinc factor, sin(kr)/(kr) → 1, with a normalization of f. As printed, the limit in Eq. (46) is (1/E)[N_P² + N_e − 2N_P f(0)N_e + f(0)∫P dr], which vanishes for a neutral atom only if ∫P dr = N_e(N_e − 1) and f(0) = 1. Please state explicitly that all rates, and the 1/E comparison curves in Fig. 5, are normalized by f(0), and re-derive Eqs. (46)–(47) under that convention.","section":"Sec. IV, Eqs. (35)–(47)"},{"comment":"The two approximations entering the quantitative Ge/Xe predictions are asserted to be subleading but are not quantified at the level of the rate. (i) The product approximation in Eq. (40) neglects exchange and Coulomb correlations; the Si/Ar comparison in Fig. 2 validates the first moment of P(r) at the 1% level, but the quantity entering the rate is the modulated transform P(k) of Eq. (37), which at intermediate energies is sensitive to the short-range part of P(r) where exchange corrections concentrate. I request a rate-level comparison: compute dΓ/dE for Si and Ar using the DFT-product P(r) and, alternatively, the Hartree–Fock intracule data of Ref. [39], and report the resulting difference in the rate curves. (ii) The isolated-atom approximation neglects interatomic correlations, which are non-negligible at the low-energy end of the claimed 1–100 keV range: at E = 1 keV, k ≈ 0.51 Å⁻¹, so nearest-neighbor separations in crystalline Ge (2.45 Å) and liquid Xe (about 4.4 Å) contribute sinc factors of order 0.3–0.8. A quantitative estimate, for instance using static structure factors for Ge and liquid Xe, is needed to support the claim that these effects are subleading and to justify the Abstract's statement of 'more robust, material-dependent experimental constraints.'","section":"Sec. III–IV (product approximation, Eq. (40); isolated-atom approximation)"}],"minor_comments":[{"comment":"Evaluating −∇²D(r) from Eq. (3a) gives a prefactor proportional to ℏ²λ/(m₀²σ²)(1 − r²/6σ²)e^{−r²/4σ²}, whereas Eq. (26) prints σ³ in the denominator; please verify whether this is a typo or a different normalization of D_CSL than the one in Eq. (3a).","section":"Eq. (26), CSL branch"},{"comment":"As printed, L_α = √(l_α/4π) vanishes for s-shells (l_α = 0), which would make G^(ℓ)_αβ = 0 for the 1s–1s contribution and contradict the successful benchmark in Fig. 1; the definition is presumably √((2l_α+1)/4π) and should be corrected.","section":"Eq. (42)"},{"comment":"The comparison of the RDF, 1/E, and Ref. [27] curves is made in arbitrary units; please state the normalization convention (for example, a common value at a reference energy, or normalization by f(0)) so that the claims 'consistently below the 1/E limit' and the cRDF curves are well defined.","section":"Fig. 5"},{"comment":"Please state explicitly the normalization of the shell pair density (∫P_αβ(r)dr = 1 in the present implementation) so that the counting in Eq. (39) can be checked directly and unambiguously.","section":"Near Eq. (40)"},{"comment":"Please clarify whether the Kohn–Sham densities ρ_α(r) entering Eqs. (38)–(40) are spherically averaged; this matters for the interpretation of R(r) and P(r) as radial distributions.","section":"After Eq. (38)"},{"comment":"The sentence ending '...or an effective description [3] Spontaneous collapse models...' is missing a period after the citation [3].","section":"Introduction, first paragraph"},{"comment":"Equation (28) uses a Lorentzian spectral factor for exponentially decaying time correlations; the Abstract's 'arbitrary noise' claim is therefore restricted to noises with factorized spatial/temporal structure and exponentially correlated temporal dependence, and this limitation should be stated where the generality claim is made.","section":"Sec. II, Eq. (28)"},{"comment":"The Data Availability statement notes that 'embargo periods may apply'; please indicate the expected release date for the data underlying Fig. 5, since the public code and data together are central to the reproducibility claims.","section":"Data Availability Statement"}],"recommendation":"major_revision","confidential_remarks":"Dear Editor, the decisive issue I could not resolve from the text is whether the code implementing Eq. (39) renormalizes P(r) to the ordered-pair normalization of Eq. (34) or uses the formula literally. If it uses the formula literally, the central CSL cancellation (Eq. (47)) and the Fig. 5 spectra are computed with the wrong normalization and the paper's main claim fails; if it renormalizes, the printed central equation is wrong and must be corrected. Either way this is a revision-blocking point, and the authors should be asked to print the actual integral of P(r) as computed for Ge, Xe, Si, and Ar. I would also draw your attention to the increment of novelty: the E→0 cancellation is explicitly attributed by the authors to their earlier Ref. [27], and the present work's contribution is the RDF-based evaluation and the smoothing of the clamped-electron oscillations; this is a legitimate but narrower increment than the Abstract's wording suggests. Finally, the Data Availability statement mentions possible embargoes; for a reproducibility-oriented paper with public code, an explicit release date would be preferable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper before you invest time: the general arbitrary-noise rate formula and the RDF reformulation are genuinely useful, but there is a normalization inconsistency in Eq. (39) that undermines the central low-energy cancellation claim. I checked the stress-test note and it holds up. With P_αβ normalized as a probability density, Eq. (39) integrates to 2Ne(Ne−1), not the Ne(Ne−1) required by Eq. (34). Plugging that into Eq. (46) gives Z^2−Z instead of zero, so Eq. (47) does not follow from the written equations. This is load-bearing because the paper sells the vanishing CSL rate at E→0 as a key result and builds the CSL/DP discrimination on it.\n\nThat said, the core of the paper is solid. The derivation from the noise correlation to Eq. (35) is clean and general, and the recovery of the 1/E behavior and previous clamped-electron results is a good consistency check. Replacing clamped emitters with radial distribution functions from DFT is a genuine step forward, and the benchmarks against Si and Ar (1% agreement in first moments) inspire confidence. The code and data availability statements are a plus. The paper is honest about what came from Ref. [27] and does not fit any collapse-rate data, so there is no circularity.\n\nThe other soft spots are secondary. The product approximation for the two-particle density neglects exchange and Coulomb correlation; the authors say this is minor but do not quantify it. The isolated-atom approximation is asserted, not demonstrated, to be subleading in the 1–100 keV range. Both deserve an error estimate, but neither is fatal if the normalization is fixed.\n\nIf the public code renormalizes P(r) to the Eq. (34) convention, the qualitative results likely survive and this is a typo-level fix. But the manuscript must say so explicitly and show the actual integral of P(r) for Ge and Xe. As written, the paper is not internally consistent. I would still send it to peer review — the framework deserves referee time — with a request that the authors reconcile Eq. (39) with Eq. (34) and add a quantitative statement about the pair-density approximation.","headline":"Useful new RDF-based framework for collapse-induced radiation, but Eq. (39) double-counts electron pairs and the central E→0 cancellation is not justified as written.","tokens_in":13630,"tokens_out":5154,"would_cite":true,"duration_ms":43941,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Atomic distances now set collapse-model x-ray spectra.","keywords":["collapse models","spontaneous radiation","radial distribution functions","CSL model","Diósi–Penrose model","atomic correlations","intracule integrals","x-ray emission constraints"],"falsifier":"Compute the electron–electron RDF $P(r)$ for xenon with a correlated many-body method (e.g., configuration interaction) and re-evaluate Eq. (35) at photon energies 1–100 keV; if the resulting rate deviates from the product-approximation prediction by more than the percent-level relativistic correction quoted in the paper, the central cancellation mechanism is not robust. A direct experiment would measure the x-ray spectrum of a high-purity Ge or Xe detector in an ultra-low-background setup: the RDF model predicts a smooth rate always below $(N_P^2+N_e)/E$, while the clamped-electron model of Ref. [27] predicts oscillations above it.","tokens_in":12551,"feed_emoji":"⚛️","tokens_out":6958,"duration_ms":62309,"temperature":0.7,"pith_summary":"Collapse models predict that the stochastic noise responsible for wavefunction localization also shakes charged particles, producing spontaneous radiation that experiments search for in ultra-low-background setups. This paper derives a general formula for that emission rate for an arbitrary noise field, in which the atomic contribution is encoded entirely in radial distribution functions of the emitters—nucleus–electron and electron–electron distance densities—while the collapse model enters only through a spatial modulation kernel. Applied to germanium and xenon, the formula reproduces the familiar 1/E scaling at high energies, stays below 1/E in the 1–100 keV region, and removes the artificial oscillations of earlier clamped-electron treatments. For neutral atoms the rate automatically vanishes at zero energy, a cancellation that follows from charge neutrality and the normalization of the RDFs.","feed_headline":"Atomic distances set collapse-model x-ray spectra","feed_subtitle":"A new formula ties spontaneous x-ray emission to atomic distance distributions, with rates staying below the 1/E limit.","key_machinery":"The machinery has three parts. First, a semiclassical derivation from the stochastic potential correlations $C(r,\\omega)$ leads to the pair-sum rate (24), proportional to $(\\sin kr/kr)(-\\nabla^2 C)$. Factoring $C$ into spatial and temporal parts defines the model-dependent kernel $f(r) = (1/6\\pi^2\\epsilon_0 c^3)(-\\nabla^2 D(r))$—Gaussian for CSL, Gaussian-like with different prefactor for Diósi–Penrose—and the white-noise rate (27). Second, the atomic structure is lifted into two total radial distribution functions: $R(r)$ for nucleus–electron distances and $P(r)$ for electron–electron distances, with normalizations $\\int R = N_e$ and $\\int P = N_e(N_e-1)$. Third, the electron–electron RDF is built from orbital wavefunctions via the product approximation $\\rho_{\\alpha\\beta}(r_1,r_2) \\approx \\rho_\\alpha(r_1)\\rho_\\beta(r_2)$, expanded into angular and radial intracule integrals with 3j-symbol angular factors and Legendre-polynomial radial integrals. The paper benchmarks this implementation against analytic hydrogenic results and published Si/Ar intracule data.","core_discovery":"The paper's central result is Eq. (35): $d\\Gamma/dE = (1/E)[N_P^2 + N_e - 2N_P R(k) + P(k)]$, where $N_P$ and $N_e$ are the numbers of protons and electrons, and $R(k)$ and $P(k)$ are the collapse-kernel-modulated sine transforms of the nucleus–electron and electron–electron radial distribution functions, Eqs. (36)–(37). The first bracket reproduces the known incoherent $(N_P^2+N_e)/E$ behaviour; the two additional terms are the finite-size correlation corrections. When the collapse kernel $f(r)$ is constant on atomic scales, as in CSL with its $10^3$ Å correlation length, the normalizations (34) force the rate to vanish as $E\\to0$ for any neutral atom, giving a parameter-free cancellation. For Ge and Xe, the RDF-based rate remains consistently below the $1/E$ asymptote, whereas the clamped-electron approximation of Ref. [27] develops oscillations that exceed it. The authors take this as evidence that realistic radial densities distribute the spectral weight more smoothly and yield a more physical low-energy spectrum.","pith_inferences":["A natural extension is to molecules and condensed phases: the same pair-sum formula would incorporate intermolecular RDFs at lower energies, where the isolated-atom premise may fail—this is testable by including a liquid-xenon structure factor.","The product approximation's short-range error could be probed by comparing with explicitly correlated wavefunctions for small atoms first; if exchange corrections alter $P(k)$ at $k$ corresponding to 1–10 keV, the low-energy downturn will shift.","The vanishing at $E=0$ has the flavour of a sum rule; if it holds beyond the semiclassical derivation, the fully quantum treatment should also exhibit it, which would make the low-energy rate shape a robust test of mass-proportional collapse.","Since the rate is proportional to $Z$ and $Z^2$ terms with cancellations, high-$Z$ targets like Xe emphasize the correlation terms; scanning targets across a range of $Z$ would isolate the $R(k)$ and $P(k)$ contributions experimentally."],"forward_implications":["At high energies the sinc factors oscillate rapidly and average out, so the general formula automatically returns the established $(N_P^2+N_e)/E$ scaling used in past bounds.","The $E\\to0$ cancellation, a corollary of charge neutrality and the RDF normalizations, turns the low-energy CSL rate off completely for neutral atoms, sharpening the energy window where constraints apply.","Because the DP kernel $f(r)$ varies over atomic distances while the CSL kernel is essentially flat, the spectral shapes differ measurably, so the 1–100 keV window can discriminate between the two models.","The smoother RDF-based spectrum removes the unphysical oscillations of clamped-electron models, giving experiment a monotone, material-dependent prediction to test against."],"supporting_citations":[{"why":"Provides the clamped-electron approximation and low-energy cancellation that this work supersedes.","marker":"[27]"},{"why":"Supplies the master equation and spatial correlation functions D(r) for CSL and DP from which the kernel f(r) is derived.","marker":"[31]"},{"why":"Establishes the semiclassical derivation and the high-energy (N_P^2+N_e)/E rate that Eq. (35) must reproduce.","marker":"[23]"},{"why":"Defines the intracule integral and gives the analytical 1s-1s RDF used to benchmark the numerical implementation.","marker":"[36]"},{"why":"Provides the two-particle density representation used in Eq. (40) for the electron-electron RDF.","marker":"[37]"},{"why":"Gives the angular-radial decomposition of the intracule integral used to compute P_{alpha beta}(r).","marker":"[38]"},{"why":"Supplies reference atomic intracule data against which the silicon and argon RDFs are benchmarked.","marker":"[39]"},{"why":"Provides the density-functional electronic structure code used to generate the orbital densities entering R(r) and P(r).","marker":"[34]"}],"fun_headline_variants":["Atomic correlations quench collapse radiation at low energy","Charge correlations suppress collapse-model x-ray emission","Atomic structure reshapes collapse radiation spectrum","New formula ties atomic distances to collapse x-ray rates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The quantitative predictions stand on the product approximation for the two-electron density, which ignores exchange and Coulomb correlation in the short-range electron pair distribution; if that approximation is inaccurate, or if the normalization factor 2 in Eq. (39) is inconsistent with $\\int P = N_e(N_e-1)$, the correlation terms $P(k)$ and hence the predicted rates change. A second load-bearing premise is the isolated-atom treatment, which neglects interatomic correlations and is asserted, not demonstrated, to be subleading in the 1–100 keV window.","fun_headline_variants_meta":{"raw":{"variants":["Atomic correlations quench collapse radiation at low energy","Charge correlations suppress collapse-model x-ray emission","Atomic structure reshapes collapse radiation spectrum","New formula ties atomic distances to collapse x-ray rates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000225,"raw_usage":{"total_tokens":1475,"prompt_tokens":965,"completion_tokens":510,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":581,"completion_tokens_details":{"reasoning_tokens":453}},"tokens_in":581,"tokens_out":510,"duration_ms":5674,"temperature":1.0,"reasoning_tokens":453,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T12:53:52.889884+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the electron–electron RDF $P(r)$ for xenon with a correlated many-body method (e.g., configuration interaction) and re-evaluate Eq. (35) at photon energies 1–100 keV; if the resulting rate deviates from the product-approximation prediction by more than the percent-level relativistic correction quoted in the paper, the central cancellation mechanism is not robust. A direct experiment would measure the x-ray spectrum of a high-purity Ge or Xe detector in an ultra-low-background setup: the RDF model predicts a smooth rate always below $(N_P^2+N_e)/E$, while the clamped-electron model of Ref. [27] predicts oscillations above it.","supporting_citations":[{"cited_title":"Aprile and XENON Collaboration, The European Physical Journal C84, 784 (2024)","cited_arxiv_id":null,"evidence_quote":"Provides the clamped-electron approximation and low-energy cancellation that this work supersedes."},{"cited_title":"Donadi, K","cited_arxiv_id":null,"evidence_quote":"Establishes the semiclassical derivation and the high-energy (N_P^2+N_e)/E rate that Eq. (35) must reproduce."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the intracule integral and gives the analytical 1s-1s RDF used to benchmark the numerical implementation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the two-particle density representation used in Eq. (40) for the electron-electron RDF."},{"cited_title":"Benesch, International Journal of Quantum Chemistry 6, 181 (1972), tex.copyright: Copyright©1972 John Wiley & Sons, Inc","cited_arxiv_id":null,"evidence_quote":"Gives the angular-radial decomposition of the intracule integral used to compute P_{alpha beta}(r)."},{"cited_title":"Calais and P.-O","cited_arxiv_id":null,"evidence_quote":"Supplies reference atomic intracule data against which the silicon and argon RDFs are benchmarked."},{"cited_title":"Hansen and I","cited_arxiv_id":null,"evidence_quote":"Provides the density-functional electronic structure code used to generate the orbital densities entering R(r) and P(r)."}],"review_version":1}