Fourier and coordinate conventions

Real-space angular expansion

PETAL2D uses

\[ \rho(r,\theta)=\sum_{m\in\mathbb Z}\rho_m(r)e^{im\theta}, \]

with

\[ \rho_m(r)=\frac{1}{2\pi}\int_0^{2\pi} \rho(r,\theta)e^{-im\theta}\,d\theta. \]

The integer \(m\) is the signed angular-harmonic index.

Transition-field Fourier transform

QUARTIC2D uses

\[ \rho(\mathbf r)= \int\frac{d^2\mathbf q}{2\pi} \rho(\mathbf q)e^{i\mathbf q\cdot\mathbf r}, \]

and

\[ \rho(\mathbf q)= \frac{1}{2\pi}\int d^2\mathbf r\, \rho(\mathbf r)e^{-i\mathbf q\cdot\mathbf r}. \]

With the Hankel definition

\[ F_m(q)=\int_0^\infty r\,dr\,\rho_m(r)J_m(qr), \]

the transformed field is

\[ \rho(q,\phi_q) =\sum_m(-i)^m e^{im\phi_q}F_m(q). \]

Kernel Fourier transform

The radial interaction kernel is defined by

\[ U(\mathbf R)= \int\frac{d^2\mathbf q}{(2\pi)^2} U(q)e^{i\mathbf q\cdot\mathbf R}. \]

The callable supplied to Interaction is this \(U(q)\).

Displacement sign

For local transition-field expansion centers \(\mathbf R_1\) and \(\mathbf R_2\), define

\[ \boldsymbol\delta=\mathbf R_1-\mathbf R_2. \]

With the local-coordinate convention derived in Four-center reduction, QUARTIC2D uses

\[ e^{-i\mathbf q\cdot\boldsymbol\delta} \]

in the momentum-space interaction. The input deltas array therefore fixes both the displacement magnitude and its polar angle.

Negative Hankel order

For integer \(k\ge0\),

\[ J_{-k}(x)=(-1)^kJ_k(x). \]

HankelTransform preserves the sign convention for negative harmonic indices. Interaction uses nonnegative translation Bessel orders internally and includes the corresponding parity in Phi_mm.

Units

QUARTIC2D does not define a unit system. If the real-space coordinates have dimensions of length, q has dimensions of inverse length. The dimensions of the final matrix element follow from the normalization of the transition fields and the supplied \(U(q)\).

For reproducible calculations, record the coordinate unit, orbital normalization, kernel convention, and any dielectric or screening parameters together with the numerical tolerances.