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arXiv:2606.09763

RPA as a Hessian Closure: Effective Functionals and Source-Variable Duality Across DFT, LR-TDDFT, 1RDMFT, and MBPT

Nan Sheng

Published Jun 8, 2026• 0 citations
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Abstract

We present a variational formulation of the random phase approximation (RPA) that places density functional theory (DFT), linear-response time-dependent density functional theory (LR-TDDFT), one-body reduced density matrix functional theory (1RDMFT), and Green's function many-body perturbation theory (MBPT) into a common source-variable hierarchy. The central claim is that a broad class of RPA constructions can be organized, independently of any one problem-specific formula, diagrammatic resummation, or small-amplitude equation of motion, as closure approximations to the exact Hessian of an effective functional. In this language, exact linear response is governed by the Hessian of the corresponding effective functional, while RPA is obtained by retaining a reference contribution together with an explicit interaction kernel and discarding the irreducible remainder. The hierarchy has two independent enrichments of the density-level description. One may enlarge the static local density to a time-dependent density, giving the dynamical density channel of LR-TDDFT, or enlarge it to an equal-time bilocal one-body reduced density matrix, giving the static bilocal channel of 1RDMFT. The Green's function level combines both enrichments, since the one-particle Green's function is bilocal in both space and time. This picture clarifies the relation between DFT, LR-TDDFT, 1RDMFT, and MBPT through exact forward reductions and source restrictions, while emphasizing that the corresponding RPA closures need not commute under projection. The hierarchy also distinguishes the local branch-wise Legendre geometry common to all four levels from the stronger global convex duality that may emerge only in source sectors possessing additional positivity and global regularity.

DOI
10.48550/arxiv.2606.09763

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We derive a renormalized many-body perturbation theory (MBPT) starting from the random phase approximation (RPA). This RPA-renormalized perturbation theory extends the scope of single-reference MBPT methods to small-gap systems without significantly increasing the computational cost. The leading correction to RPA, termed the approximate exchange kernel (AXK), substantially improves upon RPA atomization energies and ionization potentials without affecting other properties such as barrier heights where RPA is already accurate. Thus, AXK is more balanced than second-order screened exchange [A. Grüneis et al., J. Chem. Phys. 131, 154115 (2009)], which tends to overcorrect RPA for systems with stronger static correlation. Similarly, AXK avoids the divergence of second-order Møller-Plesset (MP2) theory for small gap systems and delivers a much more consistent performance than MP2 across the periodic table at comparable cost. RPA+AXK thus is an accurate, non-empirical, and robust tool to assess and improve semi-local density functional theory for a wide range of systems previously inaccessible to first-principles electronic structure calculations.
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