PowerSphericalPotentialwCutoff Overview

• PowerSphericalPotentialwCutoff is an isotropic radial pair potential defined by a power-law term modulated with a smooth cutoff to ensure vanishing potential and derivatives beyond a cutoff radius.
• The selection of cutoff functions and their polynomial order (n=0 to n=3) directly controls the smoothness (C^n continuity) of the potential, affecting computational stability and physical accuracy.
• This framework is crucial in energy minimization problems and simulations (e.g., Lennard-Jones systems), optimizing trade-offs between computational efficiency and simulation fidelity.

A PowerSphericalPotentialwCutoff is a class of isotropic, radial pair potentials of the form $V(r) = C\,r^{-m}$ or more generally $V(r)=\epsilon\,(\sigma/r)^m$, defined on Euclidean space (notably on $S^2 \subset \mathbb{R}^3$ for sphere packing problems), and augmented by a smooth cutoff function that ensures the potential and its derivatives vanish beyond a specified outer cutoff radius $r_c$. This formalism is widely used in both mathematical studies of energy-minimizing point configurations and in computational simulations of condensed matter systems, where controlling boundary discontinuities and truncation errors in power-law decaying potentials is essential. The precise choice of cutoff function critically affects the physical properties and numerical stability of simulations, as well as the mathematical tractability of analytical studies.

1. Mathematical Definition and Construction

A general power-spherical potential with cutoff is constructed by multiplying a base power-law potential $V(r)=C\,r^{-m}$ by a compactly supported, smooth cutoff function $f_{c\,n}(r)$: $U_n(r) = V(r)\,f_{c\,n}(r)$ where $f_{c\,n}(r)$ satisfies:

• $f_{c\,n}(r) = 1$ for $r \le r_i$ (inner flat region)
• $V(r)=\epsilon\,(\sigma/r)^m$0 for $V(r)=\epsilon\,(\sigma/r)^m$1 (outer cutoff)
• $V(r)=\epsilon\,(\sigma/r)^m$2 and its derivatives up to order $V(r)=\epsilon\,(\sigma/r)^m$3 vanish at $V(r)=\epsilon\,(\sigma/r)^m$4
• $V(r)=\epsilon\,(\sigma/r)^m$5 and its derivatives up to order $V(r)=\epsilon\,(\sigma/r)^m$6 vanish at $V(r)=\epsilon\,(\sigma/r)^m$7

The cutoff variable $V(r)=\epsilon\,(\sigma/r)^m$8, $V(r)=\epsilon\,(\sigma/r)^m$9, defines the interpolation window. The cutoff functions are polynomials $S^2 \subset \mathbb{R}^3$0 of degree $S^2 \subset \mathbb{R}^3$1 or higher, constructed so that $S^2 \subset \mathbb{R}^3$2 for $S^2 \subset \mathbb{R}^3$3 and with specified vanishing derivatives at endpoints (Müser, 2022).

Explicit forms for $S^2 \subset \mathbb{R}^3$4 are: $S^2 \subset \mathbb{R}^3$5 The combined potential and its derivatives are: $S^2 \subset \mathbb{R}^3$6

$S^2 \subset \mathbb{R}^3$7

2. Cutoff Function Properties and Selection

The choice of the cutoff polynomial order $S^2 \subset \mathbb{R}^3$8 directly determines the smoothness class ($S^2 \subset \mathbb{R}^3$9) of the truncated potential at the cutoff radius $r_c$0. For the combined potential $r_c$1:

• $r_c$2 ensures $r_c$3 continuity (the potential itself is continuous) but the force is discontinuous.
• $r_c$4 enforces $r_c$5 continuity, yielding continuous forces.
• $r_c$6 ensures $r_c$7 (continuous second derivatives), eliminating discontinuities in stress for crystalline systems.
• $r_c$8 yields $r_c$9 continuity for highly demanding applications (e.g., requiring a continuous elastic modulus in crystals).

The inner radius $V(r)=C\,r^{-m}$0 controls the width $V(r)=C\,r^{-m}$1 of the smoothing window. Selection of $V(r)=C\,r^{-m}$2 involves balancing retention of the physically significant first coordination shell against excessive sharpening, which can generate large derivative magnitudes and artifacts.

In practical terms, for Lennard-Jones type ($V(r)=C\,r^{-m}$3) simulations, $V(r)=C\,r^{-m}$4 and $V(r)=C\,r^{-m}$5 are found to minimize RMS energy and force errors while maintaining stable thermodynamic properties. Using $V(r)=C\,r^{-m}$6 or $V(r)=C\,r^{-m}$7 reduces energy and force errors by approximately 25% compared to traditional hard or linear (shifted) cutoffs, without additional evaluations when using precomputed tables for $V(r)=C\,r^{-m}$8 and its derivatives (Müser, 2022).

3. Application to Energy Minimization on the Sphere

Power-spherical potentials with cutoff play a central role in the generalization of the classical Thomson problem (minimum-energy configurations of $V(r)=C\,r^{-m}$9 electrons on the sphere with Coulombic repulsion) to arbitrary power-law potentials $f_{c\,n}(r)$0, also called Riesz potentials (Schwartz, 2015). The potential is used to define the energy

$f_{c\,n}(r)$1

for configurations $f_{c\,n}(r)$2. For $f_{c\,n}(r)$3, the minimization over all possible arrangements has led to the identification of the triangular bi-pyramid (TBP) as the unique global minimizer of $f_{c\,n}(r)$4 for $f_{c\,n}(r)$5, a result established through rigorous analytic interpolation (Tumanov's lemma) and computer-assisted proofs.

Auxiliary polynomials $f_{c\,n}(r)$6 are introduced to construct convex combinations that dominate $f_{c\,n}(r)$7, and the TBP is shown to uniquely minimize these auxiliary energies, ensuring the optimality extends to the power potential itself. The optimality range extends up to a sharp cutoff exponent $f_{c\,n}(r)$8, with a proven lower bound of $f_{c\,n}(r)$9 and a conjectured threshold near $U_n(r) = V(r)\,f_{c\,n}(r)$0 (Schwartz, 2015).

4. Physical and Computational Implications

The properties of cutoff functions substantially impact simulation fidelity in computational materials science and statistical mechanics:

• Discontinuity in the potential or its derivatives at $U_n(r) = V(r)\,f_{c\,n}(r)$1 induces artifacts such as spurious jumps in computed stress, bulk modulus, or energy as a function of density, especially acute for low $U_n(r) = V(r)\,f_{c\,n}(r)$2 values ($U_n(r) = V(r)\,f_{c\,n}(r)$3, $U_n(r) = V(r)\,f_{c\,n}(r)$4).
• Higher-order polynomial cutoffs ($U_n(r) = V(r)\,f_{c\,n}(r)$5, $U_n(r) = V(r)\,f_{c\,n}(r)$6) alleviate these artifacts, yielding smoother thermodynamic properties, reduced systematic energy errors, and, by optimizing the cutoff window, fewer pairwise calculations for a fixed accuracy target.
• Choice of $U_n(r) = V(r)\,f_{c\,n}(r)$7 and the smoothing window directly tunes the trade-off between computational cost and accuracy.

This is exemplified in the Lennard-Jones context, where $U_n(r) = V(r)\,f_{c\,n}(r)$8 or $U_n(r) = V(r)\,f_{c\,n}(r)$9 cutoffs with appropriate $f_{c\,n}(r)$0 achieve continuous equations of state and force errors ∼25% lower than traditional cut-and-shifted schemes (Müser, 2022).

5. Implementation Considerations

Implementation of a PowerSphericalPotentialwCutoff is explicit and algorithmically efficient:

• Once the cutoff polynomials and their coefficients are tabulated, switching smoothness order $f_{c\,n}(r)$1 is a matter of updating coefficient tables in the radial loop of pair interaction computations.
• Evaluation overhead per pair is minimal if polynomial tables are used ($f_{c\,n}(r)$2 extra FLOPs for $f_{c\,n}(r)$3 without tabulation; negligible with tabulation).
• Only the parameters $f_{c\,n}(r)$4, $f_{c\,n}(r)$5, and the cutoff polynomials $f_{c\,n}(r)$6, $f_{c\,n}(r)$7, $f_{c\,n}(r)$8 must be stored and managed at runtime.
• The same construction generalizes to any radial, power-law–type short-range potential and can be adapted to techniques requiring smooth truncations, such as those in Wolf summation for long-range Coulomb interactions (Müser, 2022).

6. Connections to Broader Methodologies and Further Developments

Power-spherical potentials with cutoff are central to the analysis of energy minimization problems in discrete geometry, optimization of atomic configurations, and computational scripting for physical systems. Their smooth cutoff framework has enabled more accurate modeling of finite systems, improved computational efficiency, and rigorous analysis of minimizer uniqueness and stability.

Recent work outlines analytic and computer-assisted approaches to extend the proven cutoff for minimizer optimality in Riesz energy problems—specifically, by tuning auxiliary cutoff polynomials and expanding interpolation frameworks, the threshold for which certain symmetric configurations are globally optimal is being quantitatively sharpened (Schwartz, 2015). This suggests future refinements may close the gap to the conjectured cutoff $f_{c\,n}(r)$9, providing a more complete resolution of longstanding open problems in discrete potential theory.

Cutoff designs are also proving essential in large-scale simulations where balancing cost and simulation fidelity is paramount, particularly for potentials with long-range decay or inhomogeneous systems where naive truncations introduce undesirable artifacts (Müser, 2022).