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arxiv: 2605.12781 · v1 · submitted 2026-05-12 · 🪐 quant-ph · physics.chem-ph

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· Lean Theorem

Explicitly Correlated Gaussian Basis Approach to Periodic Systems

Kalman Varga

Authors on Pith no claims yet

Pith reviewed 2026-05-14 20:43 UTC · model grok-4.3

classification 🪐 quant-ph physics.chem-ph
keywords explicitly correlated Gaussiansperiodic solidsvariational electronic structurematrix elementsunfolding theoremhydrogen chainlattice sums
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The pith

Closed-form expressions for matrix elements are derived for variational electronic structure calculations of periodic solids using explicitly correlated Gaussian bases.

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

The paper derives closed-form expressions for all matrix elements needed to variationally calculate the electronic structure of periodic solids with explicitly correlated Gaussian basis functions. These periodic basis functions are formed by summing the Gaussians over all lattice translations, and a generalized unfolding theorem simplifies the double sums into single sums for the overlap, kinetic energy, and Coulomb interactions. When applied to an infinite one-dimensional hydrogen chain, the computed ground-state energy per atom in the thermodynamic limit matches results obtained by extrapolating finite-chain calculations from other many-body methods. This provides a direct approach to periodic systems without relying on finite-size extrapolations.

Core claim

Closed-form expressions for all matrix elements required for variational calculation of the electronic structure of periodic solids have been derived using a basis of explicitly correlated Gaussians. Periodic basis functions are constructed by summing shifted correlated Gaussians over all composite lattice translations, where a generalized unfolding theorem reduces the resulting double lattice sum to a single sum through a unified computational framework for overlap, kinetic energy, and Coulomb potential operators. The formalism is validated on an infinite one-dimensional hydrogen chain, where the ground-state energy per atom in the thermodynamic limit agrees with finite-chain results from其他

What carries the argument

The generalized unfolding theorem, which reduces double lattice sums over translations to single sums for overlap, kinetic energy, and Coulomb operators in the periodic explicitly correlated Gaussian basis.

If this is right

  • Variational calculations of electronic structure can be performed directly for infinite periodic systems without finite-size approximations.
  • The ground-state energy per atom for the one-dimensional hydrogen chain in the thermodynamic limit agrees with extrapolated results from other many-body methods.
  • All required matrix elements for overlap, kinetic energy, and Coulomb potential have closed-form expressions in the periodic basis.
  • The same unified framework applies to the three main operators needed for the variational energy evaluation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The lattice-sum reduction could be generalized to two- and three-dimensional periodic lattices for calculations on realistic crystalline solids.
  • The closed-form matrix elements might support extensions to compute response properties or excited states within the same basis.
  • The approach could be combined with other variational techniques to treat systems with stronger electron correlation.

Load-bearing premise

The generalized unfolding theorem correctly reduces the double lattice sum to a single sum for overlap, kinetic energy, and Coulomb operators in the periodic ECG basis.

What would settle it

A direct computation of the ground-state energy per atom for the infinite one-dimensional hydrogen chain that shows a significant deviation from independent extrapolated values obtained by other many-body methods would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.12781 by Kalman Varga.

Figure 1
Figure 1. Figure 1: Dispersion relation E(k) of the 1D hydrogen chain for three lattice constants L = 2.4 Bohr (black), L = 3.6 Bohr (red), L = 6.4 (green), and L = 7.2 Bohr (blue). The horizontal axis is k · L; the vertical axis is the band energy in Hartree [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
read the original abstract

Closed-form expressions for all matrix elements required for variational calculation of the electronic structure of periodic solids have been derived using a basis of explicitly correlated Gaussians (ECGs). Periodic basis functions are constructed by summing shifted correlated Gaussians over all composite lattice translations, where a generalized unfolding theorem reduces the resulting double lattice sum to a single sum through a unified computational framework for overlap, kinetic energy, and Coulomb potential operators. The formalism has been validated through application to an infinite one-dimensional hydrogen chain, where the ground-state energy per atom computed in the thermodynamic limit is shown to agree with finite-chain results extrapolated by other many-body methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper derives closed-form expressions for all matrix elements (overlap, kinetic energy, and Coulomb) needed for variational electronic-structure calculations of periodic solids using a basis of explicitly correlated Gaussians (ECGs). Periodic basis functions are constructed by summing shifted ECGs over all composite lattice translations; a generalized unfolding theorem is invoked to reduce the resulting double lattice sums to single sums for each operator. The formalism is validated on the infinite one-dimensional hydrogen chain, where the ground-state energy per atom in the thermodynamic limit is reported to agree with extrapolated results from other many-body methods.

Significance. If the derivations hold, the work supplies a parameter-free variational route to periodic systems that incorporates explicit electron correlation via ECGs, a basis class known for rapid convergence in molecular calculations. The reduction of double lattice sums to single sums, if rigorously justified for the long-range Coulomb operator, would enable efficient evaluation without auxiliary fitting or numerical quadrature. The reported agreement on the 1D hydrogen chain provides initial support, though extension to three-dimensional solids and quantitative error analysis would be needed to establish broader utility as a benchmark or production method.

major comments (1)
  1. [Derivation of Coulomb matrix elements (generalized unfolding theorem)] The generalized unfolding theorem (invoked to convert the double lattice sum into a single sum for the Coulomb operator) must be shown to survive the conditional convergence of the 1/r potential under periodic boundary conditions. The manuscript should supply the explicit algebraic steps demonstrating cancellation of surface terms or the incorporation of Ewald regularization, because the same identity that works for short-range overlap and kinetic operators does not automatically extend to the Coulomb case without additional proof.
minor comments (2)
  1. [Abstract] The abstract states agreement with extrapolated many-body results on the 1D hydrogen chain but supplies neither error bars, basis-size convergence data, nor the precise extrapolation procedure, which weakens the reader's ability to judge the numerical support for the central claim.
  2. [Formalism section] Notation for the composite lattice translations and the precise definition of the periodic ECG basis functions should be introduced with an explicit equation early in the formalism section to avoid ambiguity when the unfolding theorem is applied.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment on the generalized unfolding theorem. We address the concern point by point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Derivation of Coulomb matrix elements (generalized unfolding theorem)] The generalized unfolding theorem (invoked to convert the double lattice sum into a single sum for the Coulomb operator) must be shown to survive the conditional convergence of the 1/r potential under periodic boundary conditions. The manuscript should supply the explicit algebraic steps demonstrating cancellation of surface terms or the incorporation of Ewald regularization, because the same identity that works for short-range overlap and kinetic operators does not automatically extend to the Coulomb case without additional proof.

    Authors: We agree that an explicit demonstration for the Coulomb operator is necessary given the conditional convergence of the 1/r interaction. In the revised manuscript we will add a dedicated appendix containing the full algebraic steps. These steps proceed by first inserting the Gaussian factors from the ECGs into the double lattice sum, which renders the sum absolutely convergent; the surface terms then cancel identically by symmetry under the periodic translations before the unfolding identity is applied. The resulting single-sum expression remains closed-form and does not require auxiliary Ewald regularization, consistent with the short-range operators. We will also include a short paragraph clarifying this distinction from bare Coulomb lattice sums. revision: yes

Circularity Check

0 steps flagged

No significant circularity; unfolding theorem is independent mathematical reduction

full rationale

The derivation chain begins with the definition of periodic ECG basis functions as lattice sums of shifted Gaussians, then applies a generalized unfolding theorem to convert the double sum over composite translations into a single sum for the overlap, kinetic, and Coulomb operators. This reduction is presented as an algebraic identity that holds for the specific functional forms of those operators; the resulting closed-form expressions are direct consequences of the theorem plus standard Gaussian integral evaluations, not redefinitions or fits. No parameter is fitted to a data subset and then relabeled as a prediction. The theorem itself is not justified solely by self-citation; it is derived within the paper for the periodic case. Validation against extrapolated finite-chain results supplies an external benchmark rather than a tautology. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the generalized unfolding theorem for lattice sums and the existence of closed-form integrals for ECG operators under periodic boundary conditions; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Generalized unfolding theorem reduces double lattice sum to single sum
    Invoked to simplify overlap, kinetic, and Coulomb matrix elements for periodic basis functions.

pith-pipeline@v0.9.0 · 5386 in / 1133 out tokens · 51062 ms · 2026-05-14T20:43:33.111622+00:00 · methodology

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Reference graph

Works this paper leans on

161 extracted references · 161 canonical work pages

  1. [1]

    II H–III)

    Ewald decomposition (Secs. II H–III). The pe- riodic 1 /r potential is split into a short-range complementary-error-function part evaluated in real space and a long-range smooth part evaluated in reciprocal space, following Eq. (40). This ap- proach is valid for any charge configuration (neu- tral or charged cell) and is the standard method for periodic e...

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  2. [2]

    Direct neutral-cell sum (Sec. III E). When the sim- ulation cell is charge-neutral the lattice sum over 1/r converges absolutely when shells are grouped by charge neutrality. Each matrix element re- duces analytically to a screened Coulomb potential erf(R/σ)/R, with no Ewald splitting parameter κ required. This formulation is algebraically simpler and avo...

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  3. [3]

    Dirac delta convolution (Sec. III F). The 1 /r ker- nel is the convolution of a Dirac delta density with the Green’s function of the Laplacian. The matrix element of δ(3)(ri − rj) gives the pair-contact den- sity between electronsi and j, and the 1/r Coulomb matrix element is recovered by integrating this den- sity against 1/|u|: V (ee) kl = X i<j Z R3 ⟨Φ...

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  4. [4]

    Electron–electron reciprocal-space term The electron–electron reciprocal-space matrix element is (see Appendix D V (ee,G) kl = 4π Ω Skl n−1X i=1 nX j=i+1 X G̸=0 e−G2/4κ2 G2 × X M ωM e iG·PT ij¯rM−σ2 ij,sG2/4. (45) The sum over reciprocal lattice vectors G converges rapidly: the factor e−G2/4κ2 from the Ewald decomposi- tion damps contributions from large-...

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  5. [5]

    Electron–electron real-space term For the real-space sum we define the t-augmented non- linear parameter matrix and its determinant ratio, A(t,ij) kl = Akl + t2(ei − ej)(ei − ej)T , (46) det A(t,ij) kl = det Akl · (1 + t2σ2 ij,s), (47) where the second identity follows from the matrix deter- minant lemma. The reduced quadratic form for image M, real-space...

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  6. [6]

    Reciprocal space: V (eN,G) kl = −4π Ω Skl nX i=1 NnucX I=1 ZI X G̸=0 e−G2/4κ2 G2 × X M ωM e iG·(¯rM,i−RI)−σ2 i G2/4

    Electron–nuclear terms The electron–nuclear interaction is obtained from the electron–electron expressions by the replacements ei − ej → ei, σ 2 ij,s → σ2 i , ¯uij,M → ¯rM,i − RI , (50) together with the charge factor −ZI and a sum over nu- clei. Reciprocal space: V (eN,G) kl = −4π Ω Skl nX i=1 NnucX I=1 ZI X G̸=0 e−G2/4κ2 G2 × X M ωM e iG·(¯rM,i−RI)−σ2 i...

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  7. [7]

    Nuclear–nuclear interaction Because VN N is independent of electronic coordinates it is proportional to the overlap: V (N N) kl = EMadelung · Skl, (53) where EMadelung = 1 2 PNnuc I=1 PNnuc J=1 ′ ZI ZJ /|RI −RJ |Ewald is the standard Ewald nuclear-repulsion energy (prime excludes I = J in the same cell): EMadelung = 1 2 ′X I,J ZI ZJ "X n∈Z3 erfc(κ|RI − RJ...

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  8. [8]

    Ewald self-energy correction The Ewald decomposition introduces an unphysical electronic self-interaction, V (self) kl = − κ√π n − πn(n − 1) 2κ2Ω Skl. (55)

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  9. [9]

    (40) each diverge for a charged or uniform-background system taken in iso- lation

    Cancellation of divergences and the neutral-cell alternative The individual terms V (ee,G) kl , V (eN,G) kl , and the G = 0 background correction in Eq. (40) each diverge for a charged or uniform-background system taken in iso- lation. These divergences cancel exactly in the total Coulomb matrix element: the repulsive electron–electron and attractive elec...

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  10. [10]

    Electron–electron contact: δ(3)(ri − rj) After unfolding and combining the Gaussians, the Fourier representation of the δ-function converts the 3n- dimensional integral into an inverse Fourier transform of a 3D Gaussian in momentum q, with variance σ2 ij,s/4 set by the effective pair width Eq. (30). The result is: ⟨Φk|δ(3)(ri − rj)|Φl⟩ = Skl (πσ2 ij,s)3/2...

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  11. [11]

    (31)) and ¯rM,i is the i-th 3D block of the combined center ¯rM (Eq

    Electron at a point: δ(3)(ri − S) For a fixed observation point S ∈ R3 the same Fourier argument with projector Pi yields: ⟨Φk|δ(3)(ri − S)|Φl⟩ = Skl (πσ2 i )3/2 X M ωM e−|¯rM,i−S|2/σ2 i , (62) where σ2 i = ( A−1 kl )ii (Eq. (31)) and ¯rM,i is the i-th 3D block of the combined center ¯rM (Eq. (26)). As a func- tion of S this is a sum of Gaussians centered...

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  12. [12]

    The lattice of composite trans- lations Ln = {TM : M ∈ Z3n} tiles R3n exactly, i.e

    Setup and notation Let Ωn ⊂ R3n be the fundamental domain (simulation cell for all n electrons). The lattice of composite trans- lations Ln = {TM : M ∈ Z3n} tiles R3n exactly, i.e. R3n = F M∈Z3n(Ωn + TM). The periodized basis func- tions are Φ k(r) =P M ϕk(r − TM)

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  13. [13]

    (A1) Expanding both sums: Okl = X Mk,Ml∈Z3n Z Ωn ϕ∗ k(r − TMk) ˆO ϕl(r − TMl) dr

    Proof of Theorem 1 The matrix element of ˆO in the periodized basis is, by definition, Okl = Z Ωn Φ∗ k(r) ˆO Φl(r) dr. (A1) Expanding both sums: Okl = X Mk,Ml∈Z3n Z Ωn ϕ∗ k(r − TMk) ˆO ϕl(r − TMl) dr. (A2) a. Step 1: Shift the integration variable. For each fixed Mk, apply the change of variables r 7→ r + TMk: Okl = X Mk,Ml Z Ωn+TMk ϕ∗ k(r) ˆO ϕl (r − TMl...

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  14. [14]

    Proof of Theorem 2 (Bloch generalization) The Bloch matrix element is O(kB) kl = X Mk,Ml e−ikB·TMk e+ikB·TMl Z Ωn ϕ∗ k(r − TMk) ˆO ϕl(r − TMl) dr. (A6) Applying Steps 1–2 of the Theorem 1 proof, the Mk-sum again promotes the integral to R3n, leaving the phase eikB·(TMl −TMk) = eikB·TM (with M = Ml − Mk) at- tached to each term, giving Eq. (39). □

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  15. [15]

    In every case Okl =P M IM, and the Gaussian decay of ωM (or its analogue after augmentation) ensures conver- gence

    Classification of operator types and resulting image sums Table IV summarizes how the unfolded integrand IM ≡ R R3n ϕk ˆO ϕl(· − TM) dr depends on the operator class. In every case Okl =P M IM, and the Gaussian decay of ωM (or its analogue after augmentation) ensures conver- gence. Appendix B: Derivation of the Overlap Matrix Element

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  16. [16]

    The product ϕk(r)ϕl(r − 16 Table IV: Unfolded integrand structure by operator class

    Combining the two Gaussians Theorem 1 with ˆO = 1 reduces the overlap to Skl =P M R R3n ϕk(r)ϕl(r − TM) dr. The product ϕk(r)ϕl(r − 16 Table IV: Unfolded integrand structure by operator class. The image weight is ωM = e−dT M eCkldM and Skl is defined in Eq. (27). “Poly M” denotes a polynomial in dM arising from Gaussian moments. Operator class IM structur...

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  17. [17]

    Gaussian integral Integrating over R3n with N = 3n: Z R3n e−(r−¯r)T eAkl(r−¯r)dr = π3n/2 (det eAkl)1/2 = π3n/2 (det Akl)3/2 , (B5) giving Eq. (43). Appendix C: Derivation of the Kinetic Energy Matrix Element

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  18. [18]

    Laplacian of a shifted Gaussian Let r′ = r − sl − TM. The mass-weighted Laplacian acting on ϕl(r − TM) = e−r′T eAlr′ gives X i 1 mi ∇2 i ϕl = h 4 r′TeAleΛeAlr′ − 2 Tr(eΛeAl) i ϕl, (C1) where eΛ = Λ ⊗ I3 and we used Tr( eΛeAl) = Tr(Λ Al) · Tr(I3) = 3Tr(ΛAl)

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  19. [19]

    Using the identity Al = Akl − Ak and cyclic trace properties: 6Tr(ΛAl) − 6Tr(AlΛAlA−1 kl ) = 6Tr(ΛCkl)

    Constant (trace) contribution After applying Theorem 1 (Appendix A, Corollary 1b) to the differential operator ˆO = − 1 2 P i m−1 i ∇2 i , the trace term gives a multiple of the overlap integral. Using the identity Al = Akl − Ak and cyclic trace properties: 6Tr(ΛAl) − 6Tr(AlΛAlA−1 kl ) = 6Tr(ΛCkl). (C2) This simplification is established by writing AlA−1 ...

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  20. [20]

    (C1) requires the Gaussian second moment

    Quadratic contribution The quadratic term in Eq. (C1) requires the Gaussian second moment. With u = r − ¯rM and δM = ¯rM − sl − TM = (A−1 kl Ak ⊗ I3)dM, Z r′TeAleΛeAlr′ e−uT eAkludu = π3n/2 (det Akl)3/2 h 3 2Tr(AlΛAlA−1 kl ) + δT MeAleΛeAlδM i . (C3) Substituting δM = (A−1 kl Ak ⊗ I3)dM: δT MeAleΛeAlδM = dT M(AkA−1 kl AlΛAlA−1 kl Ak ⊗ I3)dM. (C4) Expandin...

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  21. [21]

    Combining and final simplification Assembling the constant and quadratic pieces and ap- plying the trace simplification established above gives Eq. (44). 17 Appendix D: Derivation of the Reciprocal-Space Coulomb Matrix Elements

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  22. [22]

    Fourier-modulated Gaussian overlap We need the integral IM(k) = Z R3n ϕk(r) eikT r ϕl(r − TM) dr. (D1) Using the product formula of Appendix B and completing the square with the linear phase, IM(k) = ωM eikT ¯rM Skl e−kT eA−1 kl k/4, (D2) which follows from the standard identityR RN e−xT M x+ikT xdx = πN/2(det M)−1/2e−kT M −1k/4 (M ≻ 0)

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  23. [23]

    (D3) Summing the reciprocal-space Ewald term from Eq

    Electron–electron reciprocal space Setting k = PijG = (ei − ej) ⊗ I3 · G: kTeA−1 kl k = σ2 ij,s G2, kT ¯rM = G·(¯rM,i−¯rM,j). (D3) Summing the reciprocal-space Ewald term from Eq. (40) over all pairs and images yields Eq. (45)

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  24. [24]

    Electron–nuclear reciprocal space Setting k = PiG and including the nuclear phase e−iG·RI gives kTeA−1 kl k = σ2 i G2 and leads to Eq. (51). Appendix E: Derivation of the Real-Space Coulomb Matrix Elements

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  25. [25]

    Integral representation of erfc /r The identity erfc(κr) r = 2√π Z ∞ κ e−t2r2 dt (E1) converts each real-space Ewald term into an auxiliary Gaussian in r, at the cost of a one-dimensional t-integral

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  26. [26]

    The combined matrix is eA(t,ij) kl = [A(t,ij) kl ] ⊗ I3 with A(t,ij) kl given by Eq

    Augmented nonlinear parameter matrix The factor e−t2|PT ij r+n·L|2 is a Gaussian in r with rank- 1 increment t2PijPT ij. The combined matrix is eA(t,ij) kl = [A(t,ij) kl ] ⊗ I3 with A(t,ij) kl given by Eq. (46). a. Matrix determinant lemma. With w = ei − ej, the rank-1 determinant formula gives Eq. (47). b. Sherman–Morrison formula. (A(t,ij) kl )−1 = A−1 ...

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  27. [27]

    (48), and the 3n-dimensional Gaussian integral yields the prefactor Skl/(1 + t2σ2 ij,s)3/2, leading to Eq

    Completing the square and quadratic form After completing the square the combined exponent evaluates to −Q(t,ij,n) M as given in Eq. (48), and the 3n-dimensional Gaussian integral yields the prefactor Skl/(1 + t2σ2 ij,s)3/2, leading to Eq. (49). The elec- tron–nuclear result Eq. (52) follows from the same deriva- tion with the substitutions (50). Appendix...

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  28. [28]

    By Corollary 1a of The- orem 1, the matrix element of the operator ˆV (ee) =Pn−1 i=1 Pn j=i+1 |ri − rj|−1 (lattice-periodic by overall charge neutrality) satisfies Eq

    Setup and applicability For a charge-neutral simulation cell the bare peri- odic Coulomb sum converges absolutely and no Ewald decomposition is required. By Corollary 1a of The- orem 1, the matrix element of the operator ˆV (ee) =Pn−1 i=1 Pn j=i+1 |ri − rj|−1 (lattice-periodic by overall charge neutrality) satisfies Eq. (34), giving, for each pair (i, j),...

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  29. [29]

    Integral representation of 1/r and evaluation Using 1/r = (2/√π) R ∞ 0 e−t2r2 dt and the n = 0 term of the Ewald appendix (Appendix E): J (ij) M = 2√π ωM Skl Z ∞ 0 exp −t2|¯uij,M|2/(1 + t2σ2 ij,s) (1 + t2σ2 ij,s)3/2 dt. (F2) The substitution s = tσij,s/ q 1 + t2σ2 ij,s maps t ∈ [0, ∞) to s ∈ [0, 1) and gives (1 + t2σ2)−3/2dt = (1/σ)ds, so 2√π Z ∞ 0 e−t2R2...

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  30. [30]

    (49) gives erfc(0) /R = 1/R, recovering the full [0, ∞) integral above

    Connection to the Ewald real-space formula Setting κ = 0 and n = 0 in Eq. (49) gives erfc(0) /R = 1/R, recovering the full [0, ∞) integral above. For a neu- tral cell all n ̸= 0 shells and the entire reciprocal-space sum vanish as κ → 0, so the Ewald formula reduces ex- actly to Eqs. (59)–(60), confirming consistency. Appendix G: Delta-Function Matrix Ele...

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  31. [31]

    Overview The contact operators δ(3)(ri − rj) and δ(3)(ri − S) have closed-form matrix elements in the SCG basis. More importantly, the Coulomb operator 1 /|ri − rj| can be expressed as a weighted integral of δ(3)(ri − rj − u) over u, providing a third independent derivation of the neutral Coulomb result and establishing exact equivalence with Appendix F

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  32. [32]

    Matrix element of δ(3)(ri − rj) After unfolding (Theorem 1, Corollary 1a), the per- image integral is D(ij) M = Z R3n ϕk(r) δ(3)(ri − rj) ϕl(r − TM) dr. (G1) Using the combined Gaussian product and the Fourier representation δ(3)(u) = (2 π)−3R eiq·ud3q with u = PT ijr: D(ij) M = ωM Z d3q (2π)3 eiq·¯uij,M Z R3n e−(r−¯rM)T eAkl(r−¯rM)+iq·PT ij(r−¯rM)dr. (G2...

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  33. [33]

    Matrix element of δ(3)(ri − S) For the operator δ(3)(ri−S) with a fixed vectorS ∈ R3, the projector is Pi = ei ⊗ I3, PT i eA−1 kl Pi = σ2 i I3, and the mean is ¯rM,i. The identical Fourier argument gives D(i,S) M ≡ Z R3n ϕk(r) δ(3)(ri − S) ϕl(r − TM) dr = ωM Skl e−|¯rM,i−S|2/σ2 i (πσ2 i )3/2 , (G5) and ⟨Φk|δ(3)(ri − S)|Φl⟩ = Skl (πσ2 i )3/2 X M ωM e−|¯rM,...

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  34. [34]

    Coulomb potential from the delta-function matrix element The 1/r operator admits the resolution 1 |ri − rj| = Z R3 δ(3)(ri − rj − u) |u| d3u. (G7) 19 Inserting this into the matrix element and exchanging the order of integration: J (ij) M = Z R3 D(ij,u) M |u| d3u, (G8) where D(ij,u) M denotes the matrix element of δ(3)(ri −rj − u). By Eq. (G3) with ¯uij,M...

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  35. [35]

    Evaluation and equivalence The remaining integral is of the standard formR R3 e−|c−u|2/a2 /|u| d3u with c = ¯uij,M and a = σij,s. Substituting u = at and using the standard result Z R3 e−|t−R|2 |t| d3t = π3/2 erf(|R|) |R| , (G11) with R = c/a = ¯uij,M/σij,s: Z R3 e−|c−u|2/a2 |u| d3u (G12) = a3 · 1 a Z e−|t−R|2 |t| d3t = a2π3/2 erf(|c|/a) |c|/a = π3/2σ3 ij...

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  36. [36]

    (G15) Using Eq

    Electron–nuclear Coulomb via δ(ri − S) The electron–nuclear potential 1/|ri−RI | can similarly be written as 1 |ri − RI | = Z R3 δ(3)(ri − S) |S − RI | d3S. (G15) Using Eq. (G5): J (i,I) M = ωM Skl (πσ2 i )3/2 Z R3 e−|¯rM,i−S|2/σ2 i |S − RI | d3S. (G16) Applying identity (G11) with R = (¯rM,i − RI)/σi and S − RI → σit, ¯rM,i − S = −σi(t − R): J (i,I) M = ...

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  37. [37]

    The SCG basis smears each electron-pair contact over a 3D Gaussian of width σij,s/ √

    Structural interpretation Equations (G3) and (G5) reveal the following struc- ture. The SCG basis smears each electron-pair contact over a 3D Gaussian of width σij,s/ √

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  38. [38]

    The Coulomb matrix element is the convolution of this smeared con- tact density with the bare 1/r kernel, giving the screened potential erf( R/σ)/R. The effective range σij,s =q (A−1 kl )ii + (A−1 kl )jj − 2(A−1 kl )ij is the standard devia- tion of the pair coordinate ri −rj in the combined Gaus- sian; a more correlated basis (larger off-diagonal Ak) pro...

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  39. [39]

    The problem: slow convergence for diffuse basis functions Every matrix element derived in the paper reduces, after unfolding (Theorem 1), to a lattice sum of the form I = X M ∈Z3n ωM f(dM), ω M = e−d⊤ M ˜Ckl dM , d M = sk−sl−TM , (J1) where ˜Ckl = Ckl ⊗ I3 with Ckl = AkA−1 kl Al positive defi- nite, and f(dM) is an operator-specific function (a poly- nomi...

    work page

  40. [40]

    Factorization of the sum When Ckl is diagonal, Ckl = diag(c1,

    The diagonal case: Jacobi theta functions a. Factorization of the sum When Ckl is diagonal, Ckl = diag(c1, . . . , cn), and the shift difference is dM = (sk,1 − sl,1 − m1L, . . .), the image weight factorizes over the 3 n Cartesian directions α = 1, . . . ,3n: X M ∈Z3n ωM = 3nY α=1 X mα∈Z e−cα(sk,α−sl,α−mαLα)2 | {z } =: ϑα . (J3) Each factor ϑα is a Jacob...

    work page

  41. [41]

    The multidimensional generaliza- tion of the Jacobi transformation is the Poisson sum- mation formula applied to the Gaussian function g(x) = e−x⊤ ˜Ckl x

    The general case: Poisson summation formula When Ckl is not diagonal (the generic situation for correlated Gaussians with off-diagonal Ak), the sum (J1) does not factorize. The multidimensional generaliza- tion of the Jacobi transformation is the Poisson sum- mation formula applied to the Gaussian function g(x) = e−x⊤ ˜Ckl x. a. Result X M ∈Z3n e−d⊤ M ˜Ck...

    work page

  42. [42]

    Convergence rates Let cmin and c−1 max denote the smallest eigenvalue of Ckl and the largest eigenvalue of C −1 kl respectively

    Convergence of the dual sum and optimal switching a. Convergence rates Let cmin and c−1 max denote the smallest eigenvalue of Ckl and the largest eigenvalue of C −1 kl respectively. The number of significant terms in each representation is: Ndirect ∼ χcut √cmin 3n , N dual ∼ 2χcut √cmax 2π/L 3n , (J11) where L is a representative cell dimension andχ2 cut ≈ 20–

    work page

  43. [43]

    The product Ndirect · Ndual is independent of cmin, confirming the reciprocal nature of the two representa- tions. b. Optimal switching criterion Equating Ndirect = Ndual gives the crossover condition: cmin cmax ≈ π2 L2 . (J12) In practice one evaluates Use direct sum if cmin ≳ π L2 , use dual sum otherwise. (J13)

    work page

  44. [44]

    Table V summarizes the parallel

    Connection to the Ewald method The Poisson summation approach for the image sum is the exact analogue of Ewald summation for the Coulomb lattice sum. Table V summarizes the parallel. The key advantage of the Poisson route for Gaussian ba- sis sets is that the Fourier transform of a Gaussian is again a Gaussian — no auxiliary splitting parameter κ or t-int...

    work page

  45. [45]

    The first sum encodes inter-electronic correlations through the unshifted quadratic ri · rj; the second sum is a product of Gaussians each centered at si

    The alternative basis form A second class of many-electron Gaussian basis func- tions appearing in the literature takes the form ψ(r) = exp  − 1 2 nX i,j=1 Aij ri · rj − nX i=1 βi |ri − si|2   , (K1) where A = ( Aij) ∈ Rn×n is a symmetric matrix of pair-coupling constants (not necessarily positive defi- nite), each βi > 0 is a single-particle Gaussian...

    work page

  46. [46]

    Absorbing the constantP i βi|si|2 into an overall normalization, the exponent of ψ becomes − 1 2 X i,j Aij ri · rj − X i βi|ri|2 + 2 X i βi ri · si

    Reduction to standard quadratic form Expand the single-particle terms: βi|ri − si|2 = βi|ri|2 − 2βiri · si + βi|si|2. Absorbing the constantP i βi|si|2 into an overall normalization, the exponent of ψ becomes − 1 2 X i,j Aij ri · rj − X i βi|ri|2 + 2 X i βi ri · si. (K2) Combining the quadratic parts and introducing the di- agonal matrix Dβ = diag(β1, . ....

    work page

  47. [47]

    (K9) are not unique: Dβ is a free positive-definite diagonal matrix subject only to A = 2 Ak − 2Dβ being the desired pair-coupling matrix (which may have any sign on its diagonal)

    Inverse map: SCG parameters to the alternative form Given an SCG with parameters ( Ak, sk), the alternative-form parameters ( A, Dβ, s) satisfying Eq. (K9) are not unique: Dβ is a free positive-definite diagonal matrix subject only to A = 2 Ak − 2Dβ being the desired pair-coupling matrix (which may have any sign on its diagonal). Choosing Dβ fixes everyth...

    work page

  48. [48]

    S. F. Boys, The Integral Formulae for the Variational Solution of the Molecular Many-Electron Wave Equa- tions in Terms of Gaussian Functions with Direct Elec- tronic Correlation, Proc. R. Soc. London, Ser. A 258, 402 (1960)

    work page 1960

  49. [49]

    Singer, The Use of Gaussian (Exponential Quadratic) Wave Functions in Molecular Problems

    K. Singer, The Use of Gaussian (Exponential Quadratic) Wave Functions in Molecular Problems. I. General Formulae for the Evaluation of Integrals, Proc. R. Soc. London, Ser. A 258, 412 (1960)

    work page 1960

  50. [50]

    Ko los and L

    W. Ko los and L. Wolniewicz, Nonadiabatic Theory for Diatomic Molecules and Its Application to the Hydro- gen Molecule, Rev. Mod. Phys. 35, 473 (1963)

    work page 1963

  51. [51]

    G. W. F. Drake, Second bound state for the hydrogen negative ion, Phys. Rev. Lett. 24, 126 (1970)

    work page 1970

  52. [52]

    G. W. F. Drake and R. A. Swainson, Quantum defects and the 1/n dependence of Rydberg energies: Second- order polarization effects, Phys. Rev. A44, 5448 (1991)

    work page 1991

  53. [53]

    Yan and G

    Z. Yan and G. W. F. Drake, Computational methods for three-electron atomic systems in Hylleraas coordinates, J. Phys. B 30, 4723 (1997)

    work page 1997

  54. [54]

    V. I. Korobov, Coulomb three-body bound-state prob- lem: Variational calculations of nonrelativistic energies, Phys. Rev. A 61, 064503 (2000)

    work page 2000

  55. [55]

    Nakatsuji, H

    H. Nakatsuji, H. Nakashima, Y. Kurokawa, and A. Ishikawa, Solving the Schr¨ odinger Equation of Atoms and Molecules without Analytical Integration Based on the Free Iterative-Complement-Interaction Wave Func- tion, Phys. Rev. Lett. 99, 240402 (2007)

    work page 2007

  56. [56]

    G. G. Ryzhikh and J. Mitroy, Positronic lithium, an electronically stable Li- e+ ground state, Phys. Rev. Lett. 79, 4124 (1997)

    work page 1997

  57. [57]

    Bubin and K

    S. Bubin and K. Varga, Ground-state energy and rela- tivistic corrections for positronium hydride, Phys. Rev. A 84, 012509 (2011)

    work page 2011

  58. [58]

    Bubin and L

    S. Bubin and L. Adamowicz, Non-Born-Oppenheimer study of positronic molecular systems: e+LiH, J. Chem. Phys. 120, 6051 (2004)

    work page 2004

  59. [59]

    Stanke, D

    M. Stanke, D. K¸ edziera, M. Molski, S. Bubin, M. Barysz, and L. Adamowicz, Convergence of exper- iment and theory on the pure vibrational spectrum of heh+, Phys. Rev. Lett. 96, 233002 (2006)

    work page 2006

  60. [60]

    Cencek, J

    W. Cencek, J. Komasa, and J. Rychlewski, Benchmark calculations for two-electron systems using explicitly correlated Gaussian functions, Chem. Phys. Lett. 246, 417 (1995)

    work page 1995

  61. [61]

    K. L. Sharkey, N. Kirnosov, and L. Adamowicz, An algorithm for quantum mechanical finite-nuclear-mass variational calculations of atoms with L = 3 using all- electron explicitly correlated Gaussian basis functions, The Journal of Chemical Physics 138, 104107 (2013)

    work page 2013

  62. [62]

    K. L. Sharkey, N. Kirnosov, and L. Adamowicz, An al- gorithm for non-Born-Oppenheimer quantum mechani- cal variational calculations of N = 1 rotationally excited states of diatomic molecules using all-particle explicitly correlated Gaussian functions, The Journal of Chemical Physics 139, 164119 (2013)

    work page 2013

  63. [63]

    Kirnosov, K

    N. Kirnosov, K. L. Sharkey, and L. Adamowicz, Charge asymmetry in rovibrationally excited HD+ determined using explicitly correlated all-particle Gaussian func- tions, The Journal of Chemical Physics 139, 204105 (2013)

    work page 2013

  64. [64]

    Bubin, M

    S. Bubin, M. Pavanello, W.-C. Tung, K. L. Sharkey, and L. Adamowicz, Born–Oppenheimer and Non- Born–Oppenheimer, Atomic and Molecular Calcula- tions with Explicitly Correlated Gaussians, Chem- ical Reviews 113, 36–79 (2013), pMID: 23020161, http://dx.doi.org/10.1021/cr200419d

    work page doi:10.1021/cr200419d 2013

  65. [65]

    Formanek, K

    M. Formanek, K. L. Sharkey, N. Kirnosov, and L. Adamowicz, A comparison of two types of ex- plicitly correlated Gaussian functions for non-Born- Oppenheimer molecular calculations using a model po- tential, The Journal of Chemical Physics 141, 154103 (2014)

    work page 2014

  66. [66]

    K. L. Sharkey and L. Adamowicz, An algorithm for nonrelativistic quantum-mechanical finite-nuclear-mass variational calculations of nitrogen atom in L = 0, M = 0 states using all-electrons explicitly correlated Gaus- sian basis functions, The Journal of Chemical Physics 140, 174112 (2014)

    work page 2014

  67. [67]

    C. D. Lin, Properties of doubly-excited states of Li- and Be: the study of electron correlations in hyperspherical coordinates, J. Phys. B 16, 723 (1983)

    work page 1983

  68. [68]

    Cencek and W

    W. Cencek and W. Kutzelnigg, Accurate relativistic en- ergies of one- and two-electron systems using Gaussian wave functions, J. Chem. Phys 105, 5878 (1996)

    work page 1996

  69. [69]

    J. M. Richard, Stability of the hydrogen and hydrogen- like molecules, Phys. Rev. A 49, 3573 (1994)

    work page 1994

  70. [70]

    Strasburger, Binding energy, structure and annihi- lation proerties of the positron-LiH molecule complex, studied with explicitly correlated Gaussian functions, J

    K. Strasburger, Binding energy, structure and annihi- lation proerties of the positron-LiH molecule complex, studied with explicitly correlated Gaussian functions, J. Chem. Phys. 111, 10555 (1999)

    work page 1999

  71. [71]

    Cencek, Benchmark calculations for He2+ and LiH molecules using explicitly correlated Gaussian func- tions, Chem

    W. Cencek, Benchmark calculations for He2+ and LiH molecules using explicitly correlated Gaussian func- tions, Chem. Phys. Lett. 320, 549 (2000)

    work page 2000

  72. [72]

    Stanke, D

    M. Stanke, D. K¸ edziera, S. Bubin, M. Molski, and L. Adamowicz, Lowest vibrational states of 4he3he+: Non-born-oppenheimer calculations, Phys. Rev. A 76, 052506 (2007)

    work page 2007

  73. [73]

    Bubin, M

    S. Bubin, M. Stanke, D. K¸ edziera, and L. Adamowicz, Improved calculations of the lowest vibrational transi- tions in heh +, Phys. Rev. A 76, 022512 (2007)

    work page 2007

  74. [74]

    Pachucki and J

    K. Pachucki and J. Komasa, Nonadiabatic corrections to rovibrational levels of h2, J. Chem. Phys.130, 164113 (2009)

    work page 2009

  75. [75]

    Holka, P

    F. Holka, P. G. Szalay, J. Fremont, M. Rey, K. A. Pe- terson, and V. G. Tyuterev, Accurate ab initio deter- mination of the adiabatic potential energy function and the Born-Oppenheimer breakdown corrections for the electronic ground state of LiH isotopologues, J. Chem. Phys. 134, 094306 (2011). 24

    work page 2011

  76. [76]

    Puchalski, J

    M. Puchalski, J. Komasa, P. Czachorowski, and K. Pachucki, Nonadiabatic qed correction to the dis- sociation energy of the hydrogen molecule, Phys. Rev. Lett. 122, 103003 (2019)

    work page 2019

  77. [77]

    H¨ olsch, M

    N. H¨ olsch, M. Beyer, E. J. Salumbides, K. S. E. Eikema, W. Ubachs, C. Jungen, and F. Merkt, Benchmarking theory with an improved measurement of the ionization and dissociation energies of h 2, Phys. Rev. Lett. 122, 103002 (2019)

    work page 2019

  78. [78]

    Horny´ ak, L

    I. Horny´ ak, L. Adamowicz, and S. Bubin, Low-lying2s states of the singly charged carbon ion, Phys. Rev. A 102, 062825 (2020)

    work page 2020

  79. [79]

    Bubin and L

    S. Bubin and L. Adamowicz, Lowest 2s electronic exci- tations of the boron atom, Phys. Rev. Lett. 118, 043001 (2017)

    work page 2017

  80. [80]

    Horny´ ak, L

    I. Horny´ ak, L. Adamowicz, and S. Bubin, Ground and excited 1s states of the beryllium atom, Phys. Rev. A 100, 032504 (2019)

    work page 2019

Showing first 80 references.