Information encoding in spherical DFT

TL;DR

This work extends spherical DFT by showing that, for Coulombic systems, the set of atom-centered spherical densities $\{\bar{\rho}_i(r_i)\}$ intrinsically encodes nuclear coordinates through cusp information, enabling reconstruction of the external potential $v({\bf r})$ without origin tags. Building on Theophilou's local proofs and Nagy's cusp and constrained-search approaches, the authors prove an extended HK-like theorem for spherical DFT using distance geometry and Euclidean distance matrices. They validate the theory with LiF and glycine, demonstrating that interatomic distances can be extracted from spherical densities and used to recover 3D coordinates via multidimensional scaling, even when some peaks are missing due to basis-set smoothing. The results justify using sphericalized atomic densities for potentials and descriptors in atomistic modeling and ML applications, linking foundational DFT results to practical, distance-based representations of molecular structure.

Abstract

Spherical density functional theory (DFT) is a reformulation of the classic theorems of DFT, in which the role of the total density of a many-electron system is replaced by a set of sphericalized densities, constructed by spherically-averaging the total electron density about each atomic nucleus. In Hohenberg-Kohn DFT and its constrained-search generalization, the electron density suffices to reconstruct the spatial locations and atomic numbers of the constituent atoms, and thus the external potential. However, the original proofs of spherical DFT require knowledge of the atomic locations at which each sphericalized density originates, in addition to the set of sphericalized densities themselves. In the present work, we utilize formal results from geometric algebra -- in particular, the subfield of distance geometry -- to show that for Coulombic systems this spatial information is encoded within the ensemble of sphericalized densities themselves, and does not require independent specification. Consequently, the set of sphericalized densities uniquely determines the total external potential of the system, exactly as in Hohenberg-Kohn DFT. This theoretical result is illustrated through numerical examples for LiF and for glycine, the simplest amino acid. In addition to establishing a sound practical foundation for spherical DFT as applied to Coulombic systems, the extended theorem provides a rationale for the use of sphericalized atomic basis densities -- rather than orientation-dependent basis functions -- when designing classical or machine-learned potentials for atomistic simulation.

Figures (12)

• Figure 1: Spherical averaging geometry, illustrating the relation between 3D vectors ${\bf r}$, ${\bf R}$, and ${\bf r'}$.
• Figure 2: Sphericalized densities for the heteronuclear diatomic, LiF, at the equilibrium internuclear separation 1.564 Å (2.956 a.u.) (a) Origin of sphericalized density at Li. The location of the F atom is discernible as a peak in the distribution at the correct internuclear separation (see inset). (b) The second sphericalized density distribution, with F located at the origin. Although the peak for Li is smaller (due to its smaller size) it is still discernible in the inset, again at the correct internuclear separation.
• Figure 3: Optimized geometry of glycine from quantum mechanical calculations. Converged atomic coordinates are given in Table \ref{['tab:opt-coords']}.
• Figure 4: Spherical density distribution as a function of radial distance from the nucleus of atom 6 of glycine (subsequently identified as a hydrogen atom). Although the electron density at the nucleus dominates the distribution, additional peaks are easily identified numerically as local maxima (indicated via red vertical lines), corresponding to the radial distances of other atoms from the center. As the distance from the origin increases, the magnitude of the peaks decreases, but they remain present.
• Figure 5: Spherical density distribution as a function of radial distance from the nucleus of a heavier atom, atom 3 (subsequently identified as a nitrogen atom), of glycine. In a zoomed-out view, the density at the nucleus dominates the distribution. However, peaks are still detectable, and their form can be observed, when zooming in (see subplots). A total of five peaks are present, including the one at the origin. As described in the text, none of the hydrogen atoms of glycine are detectable.

Theorems & Definitions (4)

• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1