Example 5: real \(p_x\) and \(d_{x^2-y^2}\) orbitals

Real orbitals illustrate why PETAL2D treats real and complex fields differently. A real field obeys

\[ \rho_{-m}(r)=\rho_m(r)^*, \]

so nonzero harmonics occur in conjugate pairs.

\(p_x\): an \(m=\pm1\) pair

Using

\[ x=r\cos\theta =\frac{r}{2}\left(e^{i\theta}+e^{-i\theta}\right), \]

a localized \(p_x\)-like orbital contains equal \(m=+1\) and \(m=-1\) power. PETAL2D therefore treats \((1,-1)\) as one adaptive selection unit rather than keeping either member alone and creating an artificial complex reconstruction.

Run:

python examples/px_orbital.py

=== p_x-like orbital ===
domain_consistency: 1.00000003
selected_pairs: [(1, -1)]
retained_modes: [1, -1]
reconstruction_error_percent: 2.70394e-14
target_reached: True
     m           power      fraction   r_power_support   r_amplitude_support      r_cutoff  retained
----------------------------------------------------------------------------------------------------
    -1    1.250000e-01  5.000000e-01           4.08711                4.2065        4.2065      True
     1    1.250000e-01  5.000000e-01           4.08711                4.2065        4.2065      True

The real \(p_x\) orbital has equal \(m=\pm1\) power.

Conjugate-pair selection preserves the signed real orbital.

\(d_{x^2-y^2}\): an \(m=\pm2\) pair

Similarly,

\[ x^2-y^2=r^2\cos2\theta =\frac{r^2}{2}\left(e^{i2\theta}+e^{-i2\theta}\right), \]

so a real \(d_{x^2-y^2}\)-like orbital contains equal \(m=\pm2\) power.

python examples/dx2_y2_orbital.py

=== d_(x^2-y^2)-like orbital ===
domain_consistency: 1.00000000
selected_pairs: [(2, -2)]
retained_modes: [2, -2]
reconstruction_error_percent: 4.43516e-14
target_reached: True
     m           power      fraction   r_power_support   r_amplitude_support      r_cutoff  retained
----------------------------------------------------------------------------------------------------
    -2    2.500000e-01  5.000000e-01           4.37494               4.52437       4.52437      True
     2    2.500000e-01  5.000000e-01           4.37494               4.52437       4.52437      True

The \(d_{x^2-y^2}\) orbital has equal \(m=\pm2\) power.

Signed original field and retained-mode reconstruction.