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arXiv:2301.02848 · Annales Henri Poincaré

Boundedness of the Fifth Derivative for the One-Particle Coulombic Density Matrix at the Diagonal

Peter Hearnshaw

Published Feb 18, 2025• 1 citation
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Abstract

Boundedness is demonstrated for the fifth derivative of the one-particle reduced density matrix for non-relativistic Coulombic wavefunctions in the vicinity of the diagonal. To prove this result, improved pointwise bounds are obtained for cluster derivatives of wavefunctions involving multiple clusters.
DOI
10.1007/s00023-025-01548-0

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We obtain bounds for all derivatives of the non-relativistic Coulombic one-particle density matrix γ(x,y)γ(x, y)γ(x,y) near the diagonal x=yx = yx=y.
An explicit expression relating the magnitude of the fifth-order off-diagonal cusp in the real part of the one-electron reduced density matrix to the “on-top” two-electron density is derived in a rigorous manner from the behavior of the underlying electronic wavefunction at the electron–electron coalescence. The implications of the presence of this cusp upon electronic structure calculations of quantum chemistry and solid-state physics, including the limits imposed upon their accuracy, are elucidated. In particular, the power-law decay of the occupation numbers of the natural orbitals is demonstrated for 1S states of systems composed of arbitrary even numbers of electrons. The practical importance of analogous off-diagonal cusps in many-electron reduced density matrices is briefly discussed.
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