Integer electron-count cuts in variational second-order reduced-density-matrix theory. II. A convergent hierarchy of rigorous relaxations for real-space pair-density representability
Abstract
Variational second-order reduced-density-matrix calculations can dissociate molecules into fractionally charged fragments; Part I largely removed this failure with integer electron-count conditions on fixed partitions. Building on the realizability theory of point processes, we place these conditions in a convergent hierarchy of rigorous relaxations, defined by finite occupation linear programs (LPs) on spatial partitions, for real-space pair densities at fixed one-particle density. Every level contains all representable pair densities, the levels tighten under nested refinement, and their intersection characterizes pair densities admitting a symmetric N-particle probability-measure extension. In a fixed orbital basis the levels give monotonically increasing lower bounds to the full configuration-interaction energy, which they need not reach, even for the Coulomb repulsion. The spin-summed real-space cuts of Part I are its two-cell members. The LP optimum is also a measure: at fixed one-particle density it equals the L1 distance of the coarse-grained candidate from representability and increases under refinement to that of the candidate, with a vanishing coarse-graining error; its primal law constructs a representable approximation, and its dual a violated valid inequality, used as a cut in molecular calculations. The count conditions are not implied by the usual positivity conditions: moment data that satisfy all three-particle positivity conditions, including T1 and T2, give through Gaussian orbitals a smooth pair density that violates an integer count. For atoms at fractional electron number, ensemble relaxations with the integer-count condition lie closer to the exact piecewise-linear energy than the zero-variance fixed-N form.
Related
- Variational Polaron Theory for Ground States of Strongly Coupled Light-Matter and Electron-Phonon Systems
- Automated generation and execution of large-scale semidefinite programs for variational two-electron reduced density matrix theory
- Integer electron-count cuts in variational second-order reduced-density-matrix theory. I. Molecular dissociation with fixed real-space and orbital partitions
- A constructive solution to the n-representability problem
- V-Representability Beyond Singular Times in Time-Dependent Density Functional Theory: The Hubbard Dimer Case
BibTeX
@misc{nakata2026integer,
title = {{Integer electron-count cuts in variational second-order reduced-density-matrix theory. II. A convergent hierarchy of rigorous relaxations for real-space pair-density representability}},
author = {Maho Nakata},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.11707},
eprint = {2611.11707},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.11707},
}
RIS
TY - UNPB
TI - Integer electron-count cuts in variational second-order reduced-density-matrix theory. II. A convergent hierarchy of rigorous relaxations for real-space pair-density representability
AU - Maho Nakata
PY - 2026
DA - 2026/11/09
PB - arXiv
DO - 10.48550/arxiv.2611.11707
UR - https://arxiv.org/abs/2611.11707
AB - Variational second-order reduced-density-matrix calculations can dissociate molecules into fractionally charged fragments; Part I largely removed this failure with integer electron-count conditions on fixed partitions. Building on the realizability theory of point processes, we place these conditions in a convergent hierarchy of rigorous relaxations, defined by finite occupation linear programs (LPs) on spatial partitions, for real-space pair densities at fixed one-particle density. Every level contains all representable pair densities, the levels tighten under nested refinement, and their intersection characterizes pair densities admitting a symmetric N-particle probability-measure extension. In a fixed orbital basis the levels give monotonically increasing lower bounds to the full configuration-interaction energy, which they need not reach, even for the Coulomb repulsion. The spin-summed real-space cuts of Part I are its two-cell members. The LP optimum is also a measure: at fixed one-particle density it equals the L^1 distance of the coarse-grained candidate from representability and increases under refinement to that of the candidate, with a vanishing coarse-graining error; its primal law constructs a representable approximation, and its dual a violated valid inequality, used as a cut in molecular calculations. The count conditions are not implied by the usual positivity conditions: moment data that satisfy all three-particle positivity conditions, including T_1 and T_2, give through Gaussian orbitals a smooth pair density that violates an integer count. For atoms at fractional electron number, ensemble relaxations with the integer-count condition lie closer to the exact piecewise-linear energy than the zero-variance fixed-N form.
ER -