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REVIEW 2 major objections 6 minor 43 references

Transcorrelated Theory with Pseudopotentials

T0 review · 2 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read First two pseudopotential commutators give transcorrelated theory chemical accuracy.

desk verdict A genuine, honest methods advance — the first nonlocal pseudopotential treatment in transcorrelated theory — whose chemical-accuracy claims rest on an unquantified three-body commutator truncation that the authors flag but do not bound. read the letter →

arxiv 2412.05885 v2 pith:SF7ZW33N submitted 2024-12-08 physics.chem-ph

Kristoffer Simula , Evelin Martine Corvid Christlmaier , Maria-Andreea Filip , J. Philip Haupt , Daniel Kats , Pablo Lopez-Rios , Ali Alavi This is my paper · ORCID
classification physics.chem-ph
keywords transcorrelatedmethodsJastrowfactorpseudopotentialseffectivecorepotentialsexplicitlycorrelatedcoupledclusterFCIQMCchemicalaccuracy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Transcorrelated methods improve basis-set convergence by applying a similarity transformation to the Hamiltonian with a Jastrow factor, a function of interelectronic distances, and their practical reach has been limited by the cost of optimizing that factor. This paper shows that replacing core electrons with pseudopotentials cuts the variational Monte Carlo variance to roughly 8–30% of the all-electron value, making the optimization much cheaper. The catch is that nonlocal pseudopotentials do not commute with the Jastrow factor, so the similarity transformation no longer stops after the second commutator. The paper derives the required commutator terms, truncates at the first two non-zero ones, and shows that xTC-CCSD(T)(PP-2) reaches chemical accuracy for first-row ionization energies, atomization energies, and N2 and F2 dissociation curves. This opens a route toward transcorrelated calculations on larger molecules, transition metals, and solids.

What carries the argument

The machinery is the nested commutator expansion $\hat H_{\mathrm{TC}} = e^{-J}\hat H e^{J} = \hat H + [\hat H,J] + \frac{1}{2!}[[\hat H,J],J] + \cdots$, applied to the nonlocal pseudopotential $\hat V_{\mathrm{eff}}(r) = V_{l_{\max}}(r) + \sum_{l=0}^{l_{\max}-1} V_l(r)\sum_{m=-l}^{l} |Y_{lm}\rangle\langle Y_{lm}|$. The pseudopotential's angular projection operator means it does not commute with the Drummond–Towler–Needs Jastrow factor (a sum of one-, two-, and three-body correlation functions), so the paper must compute the one- and two-body commutator terms $\Pi$ and $\Gamma$ of Eqs. (11)–(12), evaluating the spherical projection numerically on an icosahedral grid with $N_s = 12$ points. Terms involving three distinct electrons ($i > j > m$) from the pseudopotential commutators are dropped, while the three-body kinetic-energy commutator terms are handled by the xTC approximation. The result is a second-quantized transcorrelated Hamiltonian with one-, two-, and three-body pieces whose pseudopotential correction is the tensor $P^{pq}_{rs}$ entering alongside the kinetic-energy corrections $K$ and $L$.

What would settle it

Compute the dropped three-body pseudopotential commutator terms (the $\sum_{i>j>m}$ part of Eq. (9)) for a first-row atom or for N$_2$ and show that they change the energy by more than the claimed sub-milliHartree level; alternatively, run the full commutator series with the three-body terms retained and compare against PP-2.

Watch

Extended reading notes

Core claim

The central result is that a pseudopotential-based transcorrelated Hamiltonian is reliable only if the nonlocal pseudopotential's non-commutation with the Jastrow factor is corrected through at least the first two non-zero commutators, the PP-2 level. In the all-electron case the similarity transformation terminates exactly at the second commutator; with a nonlocal effective potential it does not, and the authors truncate the series by dropping the three-body $i > j > m$ terms under the assumption that valence electrons near the core make them negligible. With that approximation, the corrected transcorrelated Hamiltonian makes xTC-CCSD(T)(PP-2) chemically accurate for the first-row ionization energies Be–F with both eCEPP and ccECP pseudopotentials, for atomization energies of CN, CO, CF, N2, O2, F2, H2O, and CO2 (eCEPP at AVQZ and ccECP at AVTZ), and for xTC-FCIQMC dissociation curves of N2 and F2 at AVQZ. Higher commutator orders change energies by about 1 mHa or less, which the authors take as evidence that PP-2 is sufficient.

Load-bearing premise

The method assumes that the three-body pseudopotential commutator terms dropped from the series are negligible for valence electrons near the core; the paper provides no numerical estimate of their size.

Editorial extensions

If this is right

  • With PP-2, xTC-CCSD(T) reaches chemical accuracy for first-row ionization energies (Be–F) in triple- and quadruple-zeta bases with both eCEPP and ccECP pseudopotentials.
  • Atomization energies of the eight-molecule test set reach chemical accuracy with xTC-CCSDT(PP-2) using eCEPPs in AVQZ and with ccECPs in AVTZ.
  • xTC-FCIQMC(PP-2) reproduces the N2 and F2 dissociation curves to chemical accuracy in AVQZ, apart from the most compressed N2 bond lengths.
  • Because pseudopotentials cut the VMC variance to roughly 8–30% of the all-electron value, Jastrow optimization becomes cheaper, making larger molecules, transition metals, and solid-state systems more accessible.
  • The small size-inconsistency of the combined Jastrow treatment (~1–2 mHa) is removed by the separated Jastrow treatment, which the paper verifies gives identical equilibrium results.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The PP-2 truncation should be re-tested for transition metals and heavier elements, where pseudopotentials have more angular-momentum channels and the nonlocal commutators are likely larger.
  • The non-monotonic basis-set behavior of ccECPs (better at AVTZ than AVQZ for both xTC and F12) suggests the pseudopotential and basis set are not fully converged together; joint reoptimization could remove the artifact.
  • One could estimate the dropped three-body pseudopotential commutator terms by comparing PP-2 against a version that keeps them within the xTC treatment, giving a direct numerical check of the core assumption without all-electron calculations.
  • For strongly correlated dissociation like N2, the need for a multideterminant trial wave function in Jastrow optimization hints that single-reference VMC may be the next bottleneck when applying this method to bond breaking.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper presents an extension of the transcorrelated (TC) method to calculations with nonlocal pseudopotentials (PPs). The similarity transformation of the Hamiltonian with a Jastrow factor is derived for the PP operator, and the resulting commutator terms are evaluated numerically in a modified TCHINT code. The method is tested on first-row atoms and molecules, with ionization potentials, atomization energies, and N2/F2 dissociation curves computed using xTC-CCSD(T), xTC-CCSDT, and xTC-FCIQMC. The central claims are that the PP-2 approximation (including the first two commutators of the PP with the Jastrow factor) is necessary and sufficient to reach chemical accuracy for the studied properties, and that the use of PPs substantially reduces the variance of the Jastrow-optimization VMC calculation.

Significance. If the central claims hold, this work is an important step toward applying transcorrelated methods to heavier elements and periodic systems, where the all-electron approach is costly. The paper provides a clear account of the commutator integrals, a practical numerical integration scheme, and benchmark data across several properties. The demonstrated variance reduction and the availability of optimized Jastrow parameters in the Supplementary Material are valuable. However, the chemical-accuracy claims rest on an unquantified truncation of the PP commutator series, and the combined Jastrow treatment introduces a size-consistency error that is on the order of the accuracy target. These issues make the presented results conditional rather than definitive.

major comments (2)
  1. [Pseudopotentials in the Transcorrelated Hamiltonian, Eqs. (9)-(10)] The truncation of the PP commutator series by dropping all three-body terms (the sums over i>j>m in Eq. 9) is not numerically justified. The paper correctly notes that for nonlocal PPs the BCH series does not terminate at the second commutator, but then it removes the three-body contributions under an assumption of negligible valence-electron density near the core. No estimate of these terms is provided. The convergence test in Fig. 3 and Table 4 (varying the number of retained commutators, n=0-4) does not probe this truncation because all n drop the same three-body terms; it only tests convergence of the two-body part. For first-row atoms, valence electrons have significant probability inside the nonlocal core region, so the omitted terms may be comparable to the 1-2 mHa chemical-accuracy target. The authors should provide a numerical estimate of the dropped three-body PP commutator terms for a representative system (e.g., Be or B), ideally by computing them explicitly or by comparing PP-2 results with an implementation that includes them, and show that they are below the accuracy target.
  2. [Dissociation energies] The combined Jastrow treatment used for the dissociation curves introduces a size-consistency error of -1.5(2) mHa for N2 at the largest bond length, as reported in the text. This error is at the same scale as the chemical-accuracy threshold (1.6 mHa), and the paper does not quantify how it affects the error curves in Figs. 9 and 10. Since dissociation curves are one of the three principal benchmark classes, the authors should either recompute key points with the separate Jastrow treatment (which they state eliminates the error) or provide a quantitative estimate of the size-consistency error along the computed curves to confirm that the chemical-accuracy claims are unaffected.
minor comments (6)
  1. [Introduction] The phrase 'cusps, 1 in the wave function' should have a space before the citation, and the reference formatting should be consistent throughout.
  2. [Pseudopotential approximation] In the sentence 'for to simplify presentation', the word 'the' is missing; it should read 'to simplify the presentation'.
  3. [Evaluation of transcorrelated Hamiltonian with pseudopotentials] In Eq. (13), the summation index 'i' is reused for both the electron index and the grid-point index, which is confusing; a different index (e.g., 'k') should be used for the spherical grid points.
  4. [Evaluation of transcorrelated Hamiltonian with pseudopotentials] The acronym 'NECP' is used in the text but never defined; presumably it denotes the number of grid points under pseudopotential influence, but it should be spelled out at first use.
  5. [Notation] The notation 'xTC-CCSD(T)(PP-0)' is ambiguous because the parentheses could be misread as subtraction; using a clearer notation such as 'xTC-CCSD(T)-PP0' would improve readability.
  6. [Conclusions] The phrase 'a feature than can provide useful' should be 'a feature that can be useful'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: PP commutator theory is derived from the nonlocal PP operator and benchmarked against external experimental/HEAT data; the unquantified 3-body truncation is a reliability risk, not a circular reduction.

full rationale

The paper's central derivation is self-contained: the transcorrelated pseudopotential Hamiltonian in Eqs. (8)-(12) is obtained by explicit commutators of the nonlocal PP operator with the Jastrow factor, and no target observable enters as a fitted input. The pseudopotentials are fixed literature objects (Trail-Needs eCEPPs and Bennett-Mitas ccECPs), and the Jastrow parameters are variance-optimized on Hartree-Fock wave functions rather than fitted to ionization potentials, atomization energies, or dissociation curves. Accuracy is judged against experimental ionization energies, the HEAT database, and experimental dissociation curves, all external benchmarks. The paper does rely on the authors' prior xTC approximation and TC formalism, but those are independently published methods; the present numerical claims concern the additional PP commutator contributions and are not forced by those citations. The dropped three-body PP commutator terms (the i > j > m sums in Eq. 9) are an unquantified truncation and a legitimate correctness concern, and the PP-n convergence scan does not test them because all n drop the same three-body terms. However, that is an unverified approximation about the completeness of the Hamiltonian, not a circular step: no defined quantity reduces by construction to the claimed chemical-accuracy benchmarks.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on the truncation of PP commutators to two-body terms, the xTC approximation from prior work, and the accuracy of literature pseudopotentials. No new physical entities are introduced; the only new objects are operator commutator terms in the effective Hamiltonian.

free parameters (3)
  • Jastrow cutoff radii = u: 4.5 Bohr, chi: 4 Bohr, f: 4 Bohr
    Chosen globally for all systems studied; they set the range of correlation terms and affect accuracy, though they are not fitted to the target energies.
  • FCIQMC initiator threshold = 3
    Chosen for the dissociation curve calculations; it affects stochastic convergence and the reported energies.
  • Spherical grid size for PP projections = 12 icosahedron vertices
    Used for numerical evaluation of nonlocal PP action; tests on denser grids showed no significant change, but it remains a numerical approximation parameter.
assumptions (6)
  • standard math Similarity transformation by exp(-J) preserves the eigenvalues of the Hamiltonian.
    Invoked in Theory, Eq. (7), as the foundation of the transcorrelated method.
  • ad hoc to paper Three-body terms of the pseudopotential commutators are negligible, so the commutator series can be truncated to two-body terms.
    Stated after Eq. (9): the sums over i > j > m are ignored under the assumption that three-body interactions of valence electrons close to the core are negligible. No numerical estimate of the dropped terms is given.
  • domain assumption The xTC generalized normal ordering treatment of the three-body kinetic-energy commutator is accurate.
    Adopted from Ref. 15 and used in the Calculations section to reorganize the three-body terms of Eq. (10).
  • domain assumption eCEPP and ccECP pseudopotentials accurately represent the valence Hamiltonian outside the core region.
    Taken from Refs. 22 and 23; the paper does not revalidate the pseudopotentials themselves.
  • domain assumption The Hartree-Fock determinant is an adequate reference for Jastrow variance optimization for these systems.
    Used throughout except for N2 dissociation, where a multideterminant reference is used; the adequacy is inferred from the resulting accuracy.
  • domain assumption The 12-point icosahedral spherical grid with random rotations gives converged pseudopotential projections.
    Adopted in the Calculations section; tests on denser grids are reported as not changing results significantly.

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Cite this review

Pith. "Pith review of Transcorrelated Theory with Pseudopotentials." pith.science (2026). https://pith.science/paper/SF7ZW33N

@misc{pith2026241205885,
  author       = {Pith},
  title        = {Pith review of: Transcorrelated Theory with Pseudopotentials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SF7ZW33N}},
  note         = {Machine review of arXiv:2412.05885}
}
read the original abstract

The transcorrelated (TC) method performs a similarity transformation on the electronic Schr\"odinger equation via Jastrow factorization of the wave function. This has demonstrated significant advancements in computational electronic structure theory by improving basis set convergence and compactifying the description of the wave function. In this work, we introduce a new approach that incorporates pseudopotentials (PPs) into the TC framework, significantly accelerating Jastrow factor optimization and reducing computational costs. Our results for ionization potentials, atomization energies, and dissociation curves of first-row atoms and molecules show that PPs provide chemically accurate descriptions across a range of systems and give guidelines for future theory and applications. The new pseudopotential-based TC method opens possibilities for applying TC to more complex and larger systems, such as transition metals and solid-state systems.

Figures

Figures reproduced from arXiv: 2412.05885 by the authors.

Figure 1
Figure 1. Vl(r) of eCEPPs (solid lines) and ccECPs (dashed lines) of C, N, O, and F. Black dashed lines show the Coulombic potential of the nucleus. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Parameters of the off-diagonal values of [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. xTC-CCSD(T)(PP-n) energies with eCEPPs (figure (a), AVQZ basis) and ccECPs (figure (b), PVQZ basis) for the neutral and ionised states of the first row el￾ements considered, shown as a function of the degree of PP commutators n. Results with n = 0, 1, 2, 3, 4 are shown. Results are presented relative to CBS estimate as ∆E = E(xTC-CCSD(T)(PP-n)) − E TZ-QZ CBS (CCSD(T)), see Eq. (14) 17 [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Ionisation energies (IPs) Ei = Eatom − Eion for the first row elements, using (a) eCEPPs and (b) ccECPs. The energies are presented as the discrepancy with the experi￾mental ionisation energies, 39 so that the presented energies are ∆E = E basis method − Eexp. The X-ax…
Figure 5
Figure 5. Figure 5: shows the energies of the molecules CN, CO, CF, N2, O2, F2, H2O, and CO2, ob￾tained with eCEPPs (a) and ccECPs (b) with CCSD(T) and xTC-CCSD(T)(PP-n) methods with n = 0 and n = 2. The energies are displayed relative to E TZ-QZ CBS (CCSD(T)). The results are shown as a …
Figure 6
Figure 6. Figure 6: Atomization energies of the molecules CN, CO, CF, N [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]
Figure 7
Figure 7. Figure 7: Atomization energies of a test set of molecules, evaluated with eCEPPs as [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]
Figure 8
Figure 8. Figure 8: Results evaluated as in Fig. 7, but with ccECPs and AVXZ (X=D,T,Q) basis [PITH_FULL_IMAGE:figures/full_fig_p028_8.png]
Figure 9
Figure 9. Figure 9: N2 FCIQMC (black crosses) and xTC-FCIQMC(PP-2) (red circles) dissociation curves as a function of bond length with eCEPPS, computed in AVDZ (top row), AVTZ (middle row) and AVQZ (bottom row) basis sets. MRCI-F12 energies are evaluated in AVQZ basis set in all of the im…
Figure 10
Figure 10. Figure 10: F2 xTC-FCIQMC(PP-2) and MRCI-F12 energies as a function of bond length with eCEPPS, computed in AVDZ (top row), AVTZ (middle row) and AVQZ (bottom row) basis sets. MRCI-F12 energies are the same in all plots, and are evaluated in AVQZ basis set. Results are presented …

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.