Scope and limitations#

PETAL2D is designed for localized two-dimensional scalar fields around a single expansion center.

Supported problem class#

PETAL2D is designed for

  • two-dimensional real or complex scalar fields

  • localized structure around one meaningful expansion center

  • uniformly spaced Cartesian coordinates for sampled arrays

  • polar angular-Fourier analysis with retained radial profiles.

Not currently provided#

PETAL2D does not currently provide

  • 3D spherical harmonics

  • coupled vector/tensor harmonic decomposition

  • interpolation from nonuniform Cartesian grids

  • automatic crystallographic point-group classification

  • automated multi-center decomposition

  • boundary-condition-aware periodic decomposition of an entire unit cell

  • inference of an unaliased continuum harmonic beyond the angular Nyquist limit

  • a guarantee that every tiny nonzero FFT bin is physical.

Interpretation limits#

Angular-harmonic labels are basis labels about the chosen origin. For a complex wavefunction they coincide with eigenchannels of the planar \(L_z\) operator about that origin, but in a crystal a dominant \(m\) should not automatically be interpreted as an exact isolated-atom orbital quantum number: the lattice can mix angular channels allowed by the local symmetry.

For a density such as \(|\psi|^2\), PETAL2D analyzes the angular morphology of the density. It cannot recover wavefunction phase or angular-momentum information that was removed before the density was supplied.

Likewise, a spectrum compatible with a \(C_n\) rotational selection rule is evidence about angular content around the chosen center. By itself it is not a complete proof of the full point-group symmetry of the underlying physical system.