Eigenvalue asymptotics of Müller minimizers for atoms and molecules

Abstract: We study the spectral properties of minimizers of the Müller functional for atoms and molecules with $N$ electrons and total nuclear charge $Z$. We prove that under some suitable assumptions on $Z$ and $N$, the $k$-th eigenvalue of a Müller minimizer $γ<em>*$ behaves as $A</em>* k<sup>{-8/3}$ when $k\to \infty$, with a constant $A_<em>&gt;0$ determined explicitly by the density of $γ_</em>$. In particular, in the atomic case $V=Z|x|<sup>{-1}$ our assumption holds if $Z$ is sufficiently large and $N\le Z- C_0 Z<sup>{1/3}$. While our proof is inspired by Sobolev's work on the asymptotic behavior of the one-particle density matrix of Schrödinger ground states, the analysis in Müller theory requires several new ingredients concerning both the singular behavior of the integral kernel of the minimizers near the diagonal and the decay properties at infinity.

• The paper establishes that the k-th eigenvalue decays as A_*·k^(-8/3), mirroring decay laws in Schrödinger theory for high-index spectral tails.
• It introduces novel analytic techniques including kernel regularization and optimal function space embeddings to derive precise Schatten class and Besov space estimates.
• The study demonstrates that the convexity of Müller theory ensures unique density minimizers with exponential decay, guiding improved numerical approaches in quantum chemistry.

Eigenvalue Asymptotics of Müller Minimizers for Atoms and Molecules

Introduction

The paper "Eigenvalue asymptotics of Müller minimizers for atoms and molecules" (2604.18386) addresses spectral properties of the minimizers for the Müller functional, a density-matrix-based energy functional central in computational quantum chemistry. The authors establish sharp asymptotic decay rates for the eigenvalues of the one-particle density matrix minimizer $\gamma_*$ describing $N$-electron atoms and molecules under certain constraints on the nuclear charge $Z$ and electron number $N$. The analysis draws inspiration from recent results on the asymptotics of the one-particle density matrix in Schrödinger theory, specifically the work of Sobolev on eigenvalue decay laws, but introduces new techniques adapted to the convex setting and the singularity structure particular to Müller theory.

Müller Functional and Minimization Framework

The Müller functional modifies the Hartree-Fock approach, yielding a convex energy landscape by replacing the standard exchange term with a term dependent on the operator square root, $X(\gamma^{1/2})$. For a system with $N$ electrons and $K$ nuclei,

$$\mathcal{E}^{\rm M}(\gamma) = (-\tfrac{1}{2}\Delta \gamma) - (V\gamma) + D(\rho_\gamma, \rho_\gamma) - X(\gamma^{1/2}),$$

with $V$ the Coulomb potential, $D$ the classical electrostatic energy, and $N$0 the exchange functional. The minimization is constrained by $N$1 and $N$2, defining the ground state energy $N$3.

Müller minimizers possess distinct spectral features: unlike Hartree-Fock minimizers, any such $N$4 has infinitely many nonzero eigenvalues, reflecting fundamental quantum behavior and unique convexity-induced features such as density uniqueness. The existence of minimizers is secured for $N$5 (total nuclear charge), and the paper concerns the regime where the minimal eigenvalue structure is tractable and physically meaningful.

Main Results: Eigenvalue Asymptotics

The central theorem establishes that the $N$6-th eigenvalue $N$7 of a Müller minimizer $N$8 exhibits the decay

$N$9

where $Z$0 is a positive constant given explicitly by an integral involving the density $Z$1:

$Z$2

This exponent matches the decay established in Schrödinger-type systems, confirming a deep structural parallel between quantum mechanical and density-matrix-functional descriptions. The result is robust in atomic settings ($Z$3) under the condition that $Z$4 for $Z$5 sufficiently large, with explicit dependence of $Z$6 on the spin dimension.

A noteworthy aspect is that the constant $Z$7 is strictly positive in Müller theory, in contrast to possible vanishing in Schrödinger theory for certain configurations.

Regularity and Non-smooth Kernel Analysis

The proof necessitates precise kernel regularity analysis for $Z$8, as its singular structure near the diagonal and at the nuclei dominates the spectral asymptotics. The authors demonstrate:

• $Z$9 for all $N$0,
• $N$1 exhibits optimal regularity in Besov spaces, $N$2.

Further, using a Jastrow-type factor $N$3 that analytically cancels singularities,

$N$4

possesses increased regularity: $N$5 for $N$6. This refinement is critical because the slow decay in the eigenvalues originates from the precise nature of kernel singularities; in particular, the diagonal $N$7 contributes significantly to the eigenvalue tail.

These results employ sharp analytic techniques, including difference quotients, Hardy and Herbst inequalities, and nontrivial embedding arguments. The regularity statements are optimal; any improvement would contradict the established eigenvalue decay law.

Exponential Decay of Density and Spectral Gap Conditions

Eigenvalue asymptotics rely on exponential localization of the density, $N$8. The authors rigorously prove that if the chemical potential $N$9 associated to $X(\gamma^{1/2})$0 satisfies $X(\gamma^{1/2})$1, then

$X(\gamma^{1/2})$2

for any $X(\gamma^{1/2})$3. Using variational upper bounds for $X(\gamma^{1/2})$4 and spectral properties of associated mean-field operators, they show this gap condition is fulfilled for large $X(\gamma^{1/2})$5 and $X(\gamma^{1/2})$6, employing Newton's theorem and the hydrogenic spectrum.

The analytical challenge is heightened because the Euler-Lagrange equations for $X(\gamma^{1/2})$7 involve chemical potentials potentially lying in the essential spectrum and exchange operators that are unbounded, requiring careful decomposition techniques and trial state constructions.

Operator Norm Estimates and Asymptotic Functionals

The asymptotics for the spectrum are derived using Schatten class estimates for integral operators with singular kernels, specifically leveraging results of Birman and Solomyak for operators acting on localized domains. The key functionals $X(\gamma^{1/2})$8 and $X(\gamma^{1/2})$9 characterize the tail behavior of singular values, enabling tight control over the eigenvalue distributions.

The explicit constant in the main asymptotic formula arises from a homogeneous integral operator norm calculation applied to a diagonalized and regularized kernel using localized cutoff functions. Functional calculus, asymptotic comparison, and monotonicity arguments deliver the limiting constants without ambiguity.

Practical and Theoretical Implications

These findings have major implications in mathematical quantum chemistry and spectral theory:

• Spectral Tail Analysis: Confirmation of $N$0 decay quantifies the extent to which high-index eigenvalues contribute to correlation and exchange effects in large systems, clarifying the limitations of truncated density-matrix representations.
• Convex Density-Matrix Functionals: The convex structure of Müller theory enables uniqueness of densities and sharp spectral control, suggesting advantage in systems where Hartree-Fock minimizers fail or are nonunique.
• Numerical Approximations: Sharp regularity and decay estimates provide foundational guidance for basis truncation schemes, error analysis, and adaptive numerical algorithms addressing large atomic or molecular systems.
• Quantum-Statistical Parallelism: The structural parallel to the Schrödinger eigenvalue asymptotics affirms the deep connectivity between density-matrix functional theoretic formulations and pure quantum mechanical models.
• Besov Optimality: Novel optimality results in Besov spaces signal new directions for future investigations of wavefunction regularity, potentially impacting multi-electron modeling and machine learning approaches to quantum chemistry.

The techniques developed—analytic kernel regularization, optimal function space embeddings, spectral variational bounds—are broadly applicable in both theoretical and computational studies of electronic structure, including more general density-matrix functionals and molecular models with complex geometry.

Conclusion

The paper delivers a rigorous and explicit asymptotic characterization of the eigenvalue spectrum for Müller minimizers in atoms and molecules, confirming that the decay law matches that of Schrödinger theory ($N$1), and providing analytically precise constants determined by the density of the minimizer. The proofs combine advanced analytic, variational, and operator-theoretic tools, yielding optimal regularity results in Sobolev and Besov spaces, exponential decay under spectral gap constraints, and sharp Schatten class estimates. The results strengthen the foundational understanding of density-matrix-functional theory and offer essential information for practical and theoretical advancements in quantum chemistry and spectral analysis.