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Toward PK-Free Molecular Hamiltonians: Two-Center Sturmian Integrals and Coupled Composition

Josh Loutey

Published Jul 9, 2026• 0 citations
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Abstract

The composed natural geometry framework (Paper 17) constructs molecular Hamiltonians by partitioning electrons into single-center blocks and coupling them via Phillips–Kleinman (PK) pseudopotentials. This produces structurally sparse qubit Hamiltonians—$O(Q^{2.5})$ Pauli scaling, 51–1{,}712$×$ fewer terms than Gaussian baselines (Paper 14)—but PK limits accuracy to 5.3–19.4% equilibrium bond length error. PK cannot be improved: six modification attempts have failed (v2.0.6–23), and the overcounting diverges at higher angular momentum. We report a negative result (Track CB, v2.0.37): replacing PK with explicit cross-block electron-electron integrals without corresponding cross-center nuclear attraction integrals produces an unbalanced Hamiltonian (29% error, worse than PK's 15%). The root cause is identified: two-center one-body integrals $\langle \psi_{nlm}^A | V_B | \psi_{n'l'm'}^A \rangle$, where an orbital on center $A$ feels the nuclear potential of center $B$, are required for energy balance but absent from the single-center framework. We observe that these missing integrals are precisely the Shibuya–Wulfman integrals of the Coulomb Sturmian literature [ref], which are computed via Fock's momentum-space projection onto $S^3$—the same conformal map that underlies the GeoVac framework (Paper 7). The mathematical foundation is therefore already in place. We review the convergence properties of the Shibuya–Wulfman expansion, identify the Coulomb resolution technique [ref] as a potentially faster alternative, and outline the ``coupled composition'' architecture that would result from incorporating two-center integrals. If the angular expansion converges in few terms at molecular bond lengths, the coupled Hamiltonian would be PK-free, structurally sparse, and significantly more accurate than the current composed framework. This paper presents the balanced coupled architecture, documents its mathematical foundations, and reports computational benchmarks across three molecules (LiH, BeH$_2$, H$_2$O). We report computational results from the first three phases. The convergence concern is resolved: the multipole expansion of the cross-center potential terminates exactly at $L_{\max} = 2l_{\max}$ by Gaunt selection rules (33 one-body terms at $n_{\max}=2$). The balanced Hamiltonian has 878 Pauli terms (Conjecture 1 confirmed). Four-electron FCI gives $R_{\rm eq}$ error of 7.0% and $D_e = 0.037$ Ha—the only bound configuration among all four coupling schemes tested (Conjecture 2 not confirmed in strict sense, but the comparison crosses FCI regimes). At $n_{\max} = 3$ ($Q = 84$), single-point FCI gives 0.20% energy error (down from 1.8%) with analytical integrals (incomplete gamma functions, machine precision), confirming convergence. The $R_{\rm eq}$ drift is structural to the block decomposition ($+0.053$ bohr per $n_{\max}$ step, $3×$ smaller than PK's $+0.15$–$0.22$ bohr/$l_{\max}$), not numerical. GeoVac Paper 19. Part of the GeoVac research corpus (quantum chemistry group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

DOI
10.5281/zenodo.21286727

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Structurally Sparse Molecular Hamiltonians for Near-Term Quantum Simulation

Josh Loutey

We present a library of qubit Hamiltonians for molecular quantum simulation that are structurally sparser than conventional Gaussian-basis alternatives by one to three orders of magnitude. The Hamiltonians are constructed from a spectral graph theory framework (GeoVac) that exploits natural coordinate separations in molecular electronic structure, producing block-diagonal electron repulsion integrals and Gaunt-enforced selection rules that persist through Jordan-Wigner encoding. For the composed architecture (PK-partitioned), the composed LiH Hamiltonian is near parity with minimal-basis STO-3G on Pauli count (334 vs. STO-3G's 276 reduced / 907 raw) while using more qubits; the decisive resource advantages are the $O(Q^{2.5})$ Pauli scaling and a 190$×$ reduction versus cc-pVDZ. The balanced coupled architecture (PK-free) binds LiH with an FCI minimum at a computed $R_{\mathrm{eq}} = 3.227$ bohr (7.0% above the experimental 3.015 bohr) at $n_{\max}=2$ (878 Pauli terms, 30 qubits); a single-point evaluation at the experimental geometry reaches 0.20% energy error at $n_{\max}=3$ (84 qubits) when solved exactly. Chemistry-accuracy scope at $n_{\max}=2$ is row-conditional: first-row hydrides (LiH, BeH$_2$, H$_2$O, $\ldots$) bind with documented potential-energy-surface (PES) topology, while second-row hydrides (NaH and below) exhibit monotone overattraction under every tested configuration — frozen-core balanced, multi-zeta valence, screened cross-center, explicit-core Hartree–Fock — with no interior PES minimum at the qubit-Hamiltonian level. We document this scope boundary empirically in \S(ref); the framework's primary value proposition is structural sparsity with quantitative truncation-convergence rates (Paper 38 state-space GH convergence), not energy-accuracy parity with CCSD(T) on second-row systems. The hamiltonian() registry covers 37 systems (35 composed molecules spanning $Z=1$–56, H through Ba, plus the He and H$_2$ reference atoms), including multi-center systems (CO, N$_2$, F$_2$), mixed frozen-core molecules (NaCl), and the full first transition series as hydrides (ScH through ZnH), with Pauli scaling $O(Q^{2.5})$ confirmed universal across all three rows, compared to $O(Q^{3.9–4.3})$ for Gaussian baselines. The composed Pauli count is exactly linear in qubit count (coefficient $11.10$, exact, for main-group molecules; $9.23$ for $d$-orbital blocks under the pair-diagonal ERI approximation (Sec. (ref))—the exact global-$M_L$ rule makes the $d$-block coefficient higher, not lower), so adding atoms to a molecule increases quantum cost linearly. Second- and third-row molecules use analytical frozen cores ([Ne], [Ar], [Ar]3$d^{10}$); the frozen-core energy enters only through the identity operator and does not affect quantum measurement cost. Isostructural molecules with frozen cores produce identical Pauli counts across periodic table rows. The Hamiltonians are available as a Python library with Qiskit, OpenFermion, and PennyLane export. GeoVac Paper 20. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

•Jul 9, 2026•0 citations

Nuclear Shell Model Hamiltonians on the Hyperspherical Lattice

Josh Loutey

We present structurally sparse qubit Hamiltonians for the nuclear shell model, constructed by applying the GeoVac angular-sparsity framework to the harmonic oscillator (HO) single-particle basis with a Mayer–Jensen spin–orbit term. The HO shell closures $2, 8, 20, 40, 70, 112$ emerge directly from graph state counting, and the physical magic numbers $2, 8, 20, 28, 50, 82, 126$ are recovered with a spin–orbit strength $v_{ls}/\hbar\omega \approx 0.17$ and a Nilsson $\ell(\ell+1)$ correction $d_{\ell\ell}/\hbar\omega \approx 0.02$. We build the first two-species qubit encoding for the deuteron ($1p+1n$, $16$ qubits, $592$ non-identity Pauli terms, $1$-norm $\approx 342$ MeV) using the Minnesota nucleon–nucleon potential and Moshinsky–Talmi brackets, and extend the architecture to helium-4 ($2p+2n$, $16$ qubits, $712$ Pauli terms) with intra-species antisymmetrization and proton–proton Coulomb repulsion. Despite a Hilbert-space dimension $12.25×$ larger than the deuteron, He-4 requires only $1.20×$ more Pauli terms — a direct consequence of the angular selection rules that carry through to multi-particle systems. A composed nuclear–electronic deuterium encoding ($26$ qubits) validates the multi-block architecture and quantifies the $\sim 10^{13}$ coefficient ratio between nuclear, electronic, and hyperfine scales. We state and apply the Fock projection rigidity theorem: the $S^3$ conformal projection that makes electronic Hamiltonians equivalent to a graph Laplacian is unique to the Coulomb potential and does not extend to HO or Woods–Saxon systems. For nuclear Hamiltonians, the GeoVac contribution is therefore the angular machinery and qubit encoding, not the conformal projection. All nuclear physics inputs (Minnesota parameters, spin–orbit strength, $\hbar\omega$) follow the standard shell-model literature; the quantitative binding energies at $N_{\mathrm{shells}}=2$ are far from experiment and should be read as encoding-validation benchmarks, not as a nuclear-structure calculation. GeoVac Paper 23. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

•Jul 9, 2026•0 citations

GeoVac (The Geometric Vacuum): Structurally Sparse Qubit Hamiltonians and the Discrete Spectral Triple

Josh Loutey

Four papers establishing the natural geometry hierarchy for quantum mechanics from discrete graph topology. Paper 7 proves that the discrete graph Laplacian on S^3 is mathematically equivalent to the Schrodinger equation (18/18 symbolic proofs). Paper 11 extends the framework to diatomic molecules via prolate spheroidal coordinates (H2+ at 0.70% error). Paper 12 introduces algebraic two-electron integrals via the Neumann expansion (H2 at 92.4% D_e). Paper 13 implements the hyperspherical adiabatic method for two-electron atoms (He at 0.05% error) and demonstrates ab initio molecular spectroscopy from electron lattice to rovibrational lines.

•Feb 23, 2026•0 citations

Structurally Sparse Qubit Hamiltonians from Spectral Graph Theory

Josh Loutey

We show that qubit Hamiltonians derived from spectral graph theory lattices exhibit structurally favorable Pauli decompositions for quantum simulation. After Jordan–Wigner transformation, the GeoVac lattice Hamiltonian yields $O(Q^{3.15})$ Pauli string terms in the number of qubits $Q$, compared to $O(Q^{4+})$ for conventional Gaussian-basis Hamiltonians (our own log-log fit of the published Pauli counts of Trenev et al. [ref] gives $Q^{4.25}$ for LiH and $Q^{3.92}$ for H$_2$O, spanning $Q^{3.9}$–$Q^{4.3}$). The advantage originates in angular momentum selection rules—enforced by Gaunt integrals at the two-electron integral level—which cause the electron repulsion integral (ERI) tensor density to decay as $\sim 1/M$ (empirically $\approx M^{-0.85}$–$M^{-1}$) with the number of spatial orbitals $M$, while Gaussian ERI tensors remain near-dense at $O(1)$. Beyond term counts, we quantify two operationally relevant cost metrics: the number of qubitwise-commuting (QWC) measurement groups scales as $O(Q^{3.36})$, and the Pauli 1-norm $\lambda = \sum_i |c_i|$—which controls both Trotter step counts and qubitization query complexity—scales as $O(Q^{1.69})$ with $R^2 = 0.997$. The sub-quadratic 1-norm scaling is the key result for fault-tolerant simulation: it implies that Trotter circuit depth grows far more slowly with system size than term counts alone would suggest. All Gaussian baselines are validated against published integral values (Szabo and Ostlund for H$_2$; analytical STO for He). For composed multi-center systems, the pseudopotential approximation of the composed natural geometry framework [ref] yields block-diagonal electron repulsion integrals with a universal Pauli scaling of $O(Q^{2.5})$, independent of molecular topology (exponent spread $0.02$ across three molecules). At equal qubit counts, this represents two or more orders of magnitude fewer Pauli terms than Gaussian baselines (published counts of Trenev et al. [ref], $Q^{3.9}$–$Q^{4.3}$ span fitted by us)—an advantage that grows with system size due to the $\sim 1.5$ exponent gap. These comparisons are at matched qubit counts, not matched accuracy; the composed basis carries classical equilibrium-geometry errors of 5–26% (LiH/BeH$_2$/H$_2$O; Sec. (ref)), making the framework optimal for fixed-geometry quantum simulation rather than geometry optimisation. Across the composed molecular library (35 molecules spanning $Z = 1$–56, H through Ba, including the first transition series ScH–ZnH), the composed Pauli count is exactly linear in the qubit count: $N_{\mathrm{Pauli}} = 11.10 × Q$ (exact, non-identity terms; $d$-orbital blocks give 9.23). These counts are realized under a pair-diagonal angular approximation ($m_a=m_c$, $m_b=m_d$) adopted deliberately for sparsity; the exact global-$M_L$ Coulomb rule (used in the framework's precision-physics paths, and considered but left on the table here) gives $\approx 2.5×$ more terms with the same scaling, under which the $d$-block coefficient is in fact higher, not lower (Sec. (ref)). GeoVac Paper 14. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

•Jul 9, 2026•0 citations

Structural Sparsity for Quantum Simulation: A Synthesis of the GeoVac Quantum-Computing Arc

Josh Loutey

The Geometric Vacuum (GeoVac) framework discretizes electronic structure on natural geometries, producing sparse graph Hamiltonians whose angular content is fixed by Gaunt selection rules. This synthesis assembles the framework's quantum-computing arc—the four Group 4 papers—into a single resource-oriented narrative. The spine is: the qubit encoding and its Pauli/$1$-norm scaling [ref]; the benchmarked resource library across three periodic-table rows [ref]; the $S_N$ representation-theoretic foundation that classifies the atomic building blocks [ref]; and the nuclear shell-model extension that delineates exactly which part of the framework transfers [ref]. The unifying value proposition is structural, not accuracy-based: after the Jordan–Wigner transformation the GeoVac lattice Hamiltonian yields $O(Q^{3.15})$ Pauli terms for atoms and $O(Q^{2.5})$ for composed multi-center molecules (versus $O(Q^{3.9–4.3})$ for Gaussian baselines), with a sub-quadratic Pauli $1$-norm $\lambda = O(Q^{1.69})$ ($R^2=0.997$) that controls Trotter depth and qubitization query cost. Two disciplines run through the arc. First, we separate the robust scaling exponents—unchanged under the choice of electron-repulsion selection rule—from the absolute multipliers, which depend on a stated pair-diagonal angular approximation and are disclosed as such. Second, every comparison is made at matched qubit count, not matched accuracy: the composed basis carries classical equilibrium-geometry errors of $5$–$26%$, so the framework is positioned for fixed-geometry NISQ/VQE simulation (Pauli count, QWC measurement groups), not for fault-tolerant geometry optimisation. The nuclear extension is honest in the same way: its binding energies are encoding-validation benchmarks far from experiment, and the Fock projection rigidity theorem states precisely that the conformal $S^3$ structure does not transfer to the harmonic-oscillator basis—only the angular machinery and the qubit encoding do. GeoVac synthesis document. Part of the GeoVac research corpus (synthesis / reading guide group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

•Jul 9, 2026•0 citations

Entanglement Structure of the Angular Momentum Eigenbasis: Energy-Entanglement Decoupling and Basis-Intrinsic Sparsity

Josh Loutey

We characterize the entanglement structure of multi-electron atoms in the angular momentum eigenbasis—the basis whose orbitals are products of hydrogenic radial functions and spherical harmonics, labeled by quantum numbers $(n, l, m)$. Four computational results are presented. First, energy-entanglement decoupling: the off-diagonal one-body Hamiltonian (graph Laplacian topology, coupling constant $\kappa = -1/16$) contributes $10$–$39%$ of the correlation energy but less than $0.2%$ of the entanglement entropy; the off-diagonal electron repulsion contributes $100%$ of the entanglement. Entanglement entropy scales as $S \sim Z^{-2.56}$ across He-like ions. Second, basis-intrinsic sparsity: among orbital bases for systems with spherical symmetry, the angular momentum eigenbasis is the unique basis producing sparse electron repulsion integrals (ERIs). Any unitary rotation of orbital indices, no matter how small, fills the ERI tensor from $42%$ to $99%$ density in a step-function transition at the identity. The nonzero ERI count is $Z$-independent, confirming that sparsity is purely geometric. Third, entanglement network topology: orbital mutual information maps reveal a hub migration pattern—$1s \to 2s \to 2p$—that tracks the partially filled shell across the first row of the periodic table. Fourth, core-valence decoupling: the core-valence mutual information falls to $\approx 2×10^{-3}$ by $Z = 4$ and below $10^{-3}$ for $Z \geq 5$, quantitatively justifying the composed fiber bundle factorization used in the GeoVac molecular framework. In the composed molecular architecture, per-block entanglement is $R$-independent, with core/bond entropy ratios of $50×$ at matched qubit count. These results establish that the angular momentum eigenbasis decouples one-body energy transport from two-body entanglement, and that the $O(Q^{2.5})$ Pauli scaling of the GeoVac qubit Hamiltonians is a direct consequence of this basis choice. This is Paper 26 in the GeoVac series. GeoVac Paper 26. Part of the GeoVac research corpus (precision observations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

•Jul 9, 2026•0 citations
The composed natural geometry framework (Paper 17) constructs molecular Hamiltonians by partitioning electrons into single-center blocks and coupling them via Phillips–Kleinman (PK) pseudopotentials. This produces structurally sparse qubit Hamiltonians—O(Q2.5)O(Q^{2.5})O(Q2.5) Pauli scaling, 51–1{,}712××× fewer terms than Gaussian baselines (Paper 14)—but PK limits accuracy to 5.3–19.4% equilibrium bond length error. PK cannot be improved: six modification attempts have failed (v2.0.6–23), and the overcounting diverges at higher angular momentum. We report a negative result (Track CB, v2.0.37): replacing PK with explicit cross-block electron-electron integrals without corresponding cross-center nuclear attraction integrals produces an unbalanced Hamiltonian (29% error, worse than PK's 15%). The root cause is identified: two-center one-body integrals ⟨ψnlmA∣VB∣ψn′l′m′A⟩\langle \psi_{nlm}^A | V_B | \psi_{n'l'm'}^A \rangle⟨ψnlmA​∣VB​∣ψn′l′m′A​⟩, where an orbital on center AAA feels the nuclear potential of center BBB, are required for energy balance but absent from the single-center framework. We observe that these missing integrals are precisely the Shibuya–Wulfman integrals of the Coulomb Sturmian literature [ref], which are computed via Fock's momentum-space projection onto S3S^3S3—the same conformal map that underlies the GeoVac framework (Paper 7). The mathematical foundation is therefore already in place. We review the convergence properties of the Shibuya–Wulfman expansion, identify the Coulomb resolution technique [ref] as a potentially faster alternative, and outline the ``coupled composition'' architecture that would result from incorporating two-center integrals. If the angular expansion converges in few terms at molecular bond lengths, the coupled Hamiltonian would be PK-free, structurally sparse, and significantly more accurate than the current composed framework. This paper presents the balanced coupled architecture, documents its mathematical foundations, and reports computational benchmarks across three molecules (LiH, BeH2_22​, H2_22​O). We report computational results from the first three phases. The convergence concern is resolved: the multipole expansion of the cross-center potential terminates exactly at Lmax⁡=2lmax⁡L_{\max} = 2l_{\max}Lmax​=2lmax​ by Gaunt selection rules (33 one-body terms at nmax⁡=2n_{\max}=2nmax​=2). The balanced Hamiltonian has 878 Pauli terms (Conjecture 1 confirmed). Four-electron FCI gives ReqR_{\rm eq}Req​ error of 7.0% and De=0.037D_e = 0.037De​=0.037 Ha—the only bound configuration among all four coupling schemes tested (Conjecture 2 not confirmed in strict sense, but the comparison crosses FCI regimes). At nmax⁡=3n_{\max} = 3nmax​=3 (Q=84Q = 84Q=84), single-point FCI gives 0.20% energy error (down from 1.8%) with analytical integrals (incomplete gamma functions, machine precision), confirming convergence. The ReqR_{\rm eq}Req​ drift is structural to the block decomposition (+0.053+0.053+0.053 bohr per nmax⁡n_{\max}nmax​ step, 3×3×3× smaller than PK's +0.15+0.15+0.15–0.220.220.22 bohr/lmax⁡l_{\max}lmax​), not numerical. GeoVac Paper 19. Part of the GeoVac research corpus (quantum chemistry group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.
We present a library of qubit Hamiltonians for molecular quantum simulation that are structurally sparser than conventional Gaussian-basis alternatives by one to three orders of magnitude. The Hamiltonians are constructed from a spectral graph theory framework (GeoVac) that exploits natural coordinate separations in molecular electronic structure, producing block-diagonal electron repulsion integrals and Gaunt-enforced selection rules that persist through Jordan-Wigner encoding. For the composed architecture (PK-partitioned), the composed LiH Hamiltonian is near parity with minimal-basis STO-3G on Pauli count (334 vs. STO-3G's 276 reduced / 907 raw) while using more qubits; the decisive resource advantages are the O(Q2.5)O(Q^{2.5})O(Q2.5) Pauli scaling and a 190××× reduction versus cc-pVDZ. The balanced coupled architecture (PK-free) binds LiH with an FCI minimum at a computed Req=3.227R_{\mathrm{eq}} = 3.227Req​=3.227 bohr (7.0% above the experimental 3.015 bohr) at nmax⁡=2n_{\max}=2nmax​=2 (878 Pauli terms, 30 qubits); a single-point evaluation at the experimental geometry reaches 0.20% energy error at nmax⁡=3n_{\max}=3nmax​=3 (84 qubits) when solved exactly. Chemistry-accuracy scope at nmax⁡=2n_{\max}=2nmax​=2 is row-conditional: first-row hydrides (LiH, BeH2_22​, H2_22​O, …\ldots…) bind with documented potential-energy-surface (PES) topology, while second-row hydrides (NaH and below) exhibit monotone overattraction under every tested configuration — frozen-core balanced, multi-zeta valence, screened cross-center, explicit-core Hartree–Fock — with no interior PES minimum at the qubit-Hamiltonian level. We document this scope boundary empirically in \S(ref); the framework's primary value proposition is structural sparsity with quantitative truncation-convergence rates (Paper 38 state-space GH convergence), not energy-accuracy parity with CCSD(T) on second-row systems. The hamiltonian() registry covers 37 systems (35 composed molecules spanning Z=1Z=1Z=1–56, H through Ba, plus the He and H2_22​ reference atoms), including multi-center systems (CO, N2_22​, F2_22​), mixed frozen-core molecules (NaCl), and the full first transition series as hydrides (ScH through ZnH), with Pauli scaling O(Q2.5)O(Q^{2.5})O(Q2.5) confirmed universal across all three rows, compared to O(Q3.9–4.3)O(Q^{3.9–4.3})O(Q3.9–4.3) for Gaussian baselines. The composed Pauli count is exactly linear in qubit count (coefficient 11.1011.1011.10, exact, for main-group molecules; 9.239.239.23 for ddd-orbital blocks under the pair-diagonal ERI approximation (Sec. (ref))—the exact global-MLM_LML​ rule makes the ddd-block coefficient higher, not lower), so adding atoms to a molecule increases quantum cost linearly. Second- and third-row molecules use analytical frozen cores ([Ne], [Ar], [Ar]3d10d^{10}d10); the frozen-core energy enters only through the identity operator and does not affect quantum measurement cost. Isostructural molecules with frozen cores produce identical Pauli counts across periodic table rows. The Hamiltonians are available as a Python library with Qiskit, OpenFermion, and PennyLane export. GeoVac Paper 20. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.
We present structurally sparse qubit Hamiltonians for the nuclear shell model, constructed by applying the GeoVac angular-sparsity framework to the harmonic oscillator (HO) single-particle basis with a Mayer–Jensen spin–orbit term. The HO shell closures 2,8,20,40,70,1122, 8, 20, 40, 70, 1122,8,20,40,70,112 emerge directly from graph state counting, and the physical magic numbers 2,8,20,28,50,82,1262, 8, 20, 28, 50, 82, 1262,8,20,28,50,82,126 are recovered with a spin–orbit strength vls/ℏω≈0.17v_{ls}/\hbar\omega \approx 0.17vls​/ℏω≈0.17 and a Nilsson ℓ(ℓ+1)\ell(\ell+1)ℓ(ℓ+1) correction dℓℓ/ℏω≈0.02d_{\ell\ell}/\hbar\omega \approx 0.02dℓℓ​/ℏω≈0.02. We build the first two-species qubit encoding for the deuteron (1p+1n1p+1n1p+1n, 161616 qubits, 592592592 non-identity Pauli terms, 111-norm ≈342\approx 342≈342 MeV) using the Minnesota nucleon–nucleon potential and Moshinsky–Talmi brackets, and extend the architecture to helium-4 (2p+2n2p+2n2p+2n, 161616 qubits, 712712712 Pauli terms) with intra-species antisymmetrization and proton–proton Coulomb repulsion. Despite a Hilbert-space dimension 12.25×12.25×12.25× larger than the deuteron, He-4 requires only 1.20×1.20×1.20× more Pauli terms — a direct consequence of the angular selection rules that carry through to multi-particle systems. A composed nuclear–electronic deuterium encoding (262626 qubits) validates the multi-block architecture and quantifies the ∼1013\sim 10^{13}∼1013 coefficient ratio between nuclear, electronic, and hyperfine scales. We state and apply the Fock projection rigidity theorem: the S3S^3S3 conformal projection that makes electronic Hamiltonians equivalent to a graph Laplacian is unique to the Coulomb potential and does not extend to HO or Woods–Saxon systems. For nuclear Hamiltonians, the GeoVac contribution is therefore the angular machinery and qubit encoding, not the conformal projection. All nuclear physics inputs (Minnesota parameters, spin–orbit strength, ℏω\hbar\omegaℏω) follow the standard shell-model literature; the quantitative binding energies at Nshells=2N_{\mathrm{shells}}=2Nshells​=2 are far from experiment and should be read as encoding-validation benchmarks, not as a nuclear-structure calculation. GeoVac Paper 23. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.
Four papers establishing the natural geometry hierarchy for quantum mechanics from discrete graph topology. Paper 7 proves that the discrete graph Laplacian on S^3 is mathematically equivalent to the Schrodinger equation (18/18 symbolic proofs). Paper 11 extends the framework to diatomic molecules via prolate spheroidal coordinates (H2+ at 0.70% error). Paper 12 introduces algebraic two-electron integrals via the Neumann expansion (H2 at 92.4% D_e). Paper 13 implements the hyperspherical adiabatic method for two-electron atoms (He at 0.05% error) and demonstrates ab initio molecular spectroscopy from electron lattice to rovibrational lines.
We show that qubit Hamiltonians derived from spectral graph theory lattices exhibit structurally favorable Pauli decompositions for quantum simulation. After Jordan–Wigner transformation, the GeoVac lattice Hamiltonian yields O(Q3.15)O(Q^{3.15})O(Q3.15) Pauli string terms in the number of qubits QQQ, compared to O(Q4+)O(Q^{4+})O(Q4+) for conventional Gaussian-basis Hamiltonians (our own log-log fit of the published Pauli counts of Trenev et al. [ref] gives Q4.25Q^{4.25}Q4.25 for LiH and Q3.92Q^{3.92}Q3.92 for H2_22​O, spanning Q3.9Q^{3.9}Q3.9–Q4.3Q^{4.3}Q4.3). The advantage originates in angular momentum selection rules—enforced by Gaunt integrals at the two-electron integral level—which cause the electron repulsion integral (ERI) tensor density to decay as ∼1/M\sim 1/M∼1/M (empirically ≈M−0.85\approx M^{-0.85}≈M−0.85–M−1M^{-1}M−1) with the number of spatial orbitals MMM, while Gaussian ERI tensors remain near-dense at O(1)O(1)O(1). Beyond term counts, we quantify two operationally relevant cost metrics: the number of qubitwise-commuting (QWC) measurement groups scales as O(Q3.36)O(Q^{3.36})O(Q3.36), and the Pauli 1-norm λ=∑i∣ci∣\lambda = \sum_i |c_i|λ=∑i​∣ci​∣—which controls both Trotter step counts and qubitization query complexity—scales as O(Q1.69)O(Q^{1.69})O(Q1.69) with R2=0.997R^2 = 0.997R2=0.997. The sub-quadratic 1-norm scaling is the key result for fault-tolerant simulation: it implies that Trotter circuit depth grows far more slowly with system size than term counts alone would suggest. All Gaussian baselines are validated against published integral values (Szabo and Ostlund for H2_22​; analytical STO for He). For composed multi-center systems, the pseudopotential approximation of the composed natural geometry framework [ref] yields block-diagonal electron repulsion integrals with a universal Pauli scaling of O(Q2.5)O(Q^{2.5})O(Q2.5), independent of molecular topology (exponent spread 0.020.020.02 across three molecules). At equal qubit counts, this represents two or more orders of magnitude fewer Pauli terms than Gaussian baselines (published counts of Trenev et al. [ref], Q3.9Q^{3.9}Q3.9–Q4.3Q^{4.3}Q4.3 span fitted by us)—an advantage that grows with system size due to the ∼1.5\sim 1.5∼1.5 exponent gap. These comparisons are at matched qubit counts, not matched accuracy; the composed basis carries classical equilibrium-geometry errors of 5–26% (LiH/BeH2_22​/H2_22​O; Sec. (ref)), making the framework optimal for fixed-geometry quantum simulation rather than geometry optimisation. Across the composed molecular library (35 molecules spanning Z=1Z = 1Z=1–56, H through Ba, including the first transition series ScH–ZnH), the composed Pauli count is exactly linear in the qubit count: NPauli=11.10×QN_{\mathrm{Pauli}} = 11.10 × QNPauli​=11.10×Q (exact, non-identity terms; ddd-orbital blocks give 9.23). These counts are realized under a pair-diagonal angular approximation (ma=mcm_a=m_cma​=mc​, mb=mdm_b=m_dmb​=md​) adopted deliberately for sparsity; the exact global-MLM_LML​ Coulomb rule (used in the framework's precision-physics paths, and considered but left on the table here) gives ≈2.5×\approx 2.5×≈2.5× more terms with the same scaling, under which the ddd-block coefficient is in fact higher, not lower (Sec. (ref)). GeoVac Paper 14. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.
The Geometric Vacuum (GeoVac) framework discretizes electronic structure on natural geometries, producing sparse graph Hamiltonians whose angular content is fixed by Gaunt selection rules. This synthesis assembles the framework's quantum-computing arc—the four Group 4 papers—into a single resource-oriented narrative. The spine is: the qubit encoding and its Pauli/111-norm scaling [ref]; the benchmarked resource library across three periodic-table rows [ref]; the SNS_NSN​ representation-theoretic foundation that classifies the atomic building blocks [ref]; and the nuclear shell-model extension that delineates exactly which part of the framework transfers [ref]. The unifying value proposition is structural, not accuracy-based: after the Jordan–Wigner transformation the GeoVac lattice Hamiltonian yields O(Q3.15)O(Q^{3.15})O(Q3.15) Pauli terms for atoms and O(Q2.5)O(Q^{2.5})O(Q2.5) for composed multi-center molecules (versus O(Q3.9–4.3)O(Q^{3.9–4.3})O(Q3.9–4.3) for Gaussian baselines), with a sub-quadratic Pauli 111-norm λ=O(Q1.69)\lambda = O(Q^{1.69})λ=O(Q1.69) (R2=0.997R^2=0.997R2=0.997) that controls Trotter depth and qubitization query cost. Two disciplines run through the arc. First, we separate the robust scaling exponents—unchanged under the choice of electron-repulsion selection rule—from the absolute multipliers, which depend on a stated pair-diagonal angular approximation and are disclosed as such. Second, every comparison is made at matched qubit count, not matched accuracy: the composed basis carries classical equilibrium-geometry errors of 555–2626%26, so the framework is positioned for fixed-geometry NISQ/VQE simulation (Pauli count, QWC measurement groups), not for fault-tolerant geometry optimisation. The nuclear extension is honest in the same way: its binding energies are encoding-validation benchmarks far from experiment, and the Fock projection rigidity theorem states precisely that the conformal S3S^3S3 structure does not transfer to the harmonic-oscillator basis—only the angular machinery and the qubit encoding do. GeoVac synthesis document. Part of the GeoVac research corpus (synthesis / reading guide group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.
We characterize the entanglement structure of multi-electron atoms in the angular momentum eigenbasis—the basis whose orbitals are products of hydrogenic radial functions and spherical harmonics, labeled by quantum numbers (n,l,m)(n, l, m)(n,l,m). Four computational results are presented. First, energy-entanglement decoupling: the off-diagonal one-body Hamiltonian (graph Laplacian topology, coupling constant κ=−1/16\kappa = -1/16κ=−1/16) contributes 101010–3939%39 of the correlation energy but less than 0.20.2%0.2 of the entanglement entropy; the off-diagonal electron repulsion contributes 100100%100 of the entanglement. Entanglement entropy scales as S∼Z−2.56S \sim Z^{-2.56}S∼Z−2.56 across He-like ions. Second, basis-intrinsic sparsity: among orbital bases for systems with spherical symmetry, the angular momentum eigenbasis is the unique basis producing sparse electron repulsion integrals (ERIs). Any unitary rotation of orbital indices, no matter how small, fills the ERI tensor from 4242%42 to 9999%99 density in a step-function transition at the identity. The nonzero ERI count is ZZZ-independent, confirming that sparsity is purely geometric. Third, entanglement network topology: orbital mutual information maps reveal a hub migration pattern—1s→2s→2p1s \to 2s \to 2p1s→2s→2p—that tracks the partially filled shell across the first row of the periodic table. Fourth, core-valence decoupling: the core-valence mutual information falls to ≈2×10−3\approx 2×10^{-3}≈2×10−3 by Z=4Z = 4Z=4 and below 10−310^{-3}10−3 for Z≥5Z \geq 5Z≥5, quantitatively justifying the composed fiber bundle factorization used in the GeoVac molecular framework. In the composed molecular architecture, per-block entanglement is RRR-independent, with core/bond entropy ratios of 50×50×50× at matched qubit count. These results establish that the angular momentum eigenbasis decouples one-body energy transport from two-body entanglement, and that the O(Q2.5)O(Q^{2.5})O(Q2.5) Pauli scaling of the GeoVac qubit Hamiltonians is a direct consequence of this basis choice. This is Paper 26 in the GeoVac series. GeoVac Paper 26. Part of the GeoVac research corpus (precision observations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.