Direct and exchange couplings in a two-orbital localized basis

Multiorbital effective Hamiltonians require more than an onsite or intersite density-density repulsion. Distinct orbital products generate exchange and other off-diagonal four-center terms, and sign changes in those products are part of the physical matrix element rather than a numerical pathology.

A two-orbital basis already contains this distinction. Consider the normalized orbitals

\[ \phi_s=\frac{e^{-r^2/2}}{\sqrt{\pi}}, \qquad \phi_{p_x}=\sqrt{\frac{2}{\pi}}\,x e^{-r^2/2}. \]

For the direct channel, the transition fields are

\[ \rho_{ss}=|\phi_s|^2, \qquad \rho_{pp}=|\phi_{p_x}|^2, \]

whereas exchange uses

\[ \rho_{sp}=\phi_s^*\phi_{p_x}. \]

The latter changes sign across the \(p_x\) node. The interaction kernel is the regular two-dimensional Yukawa form

\[ U(q)=\frac{2\pi}{\sqrt{q^2+\kappa^2}}, \qquad \kappa=0.35. \]

Angular content

The analytic angular structure is

\[ \rho_{ss}: m=0, \qquad \rho_{pp}: m=0,\pm2, \qquad \rho_{sp}: m=\pm1. \]

The executable example constructs all three decompositions and records the measured PETAL2D reconstruction errors reported below.

For an interactive calculation, PETAL2D and QUARTIC2D provide the relevant inspection helpers directly:

dec_sp.plot_harmonics_with_hist("Exchange transition field")
hcal_sp.plot_convergence()
field_sp.plot_harmonics()

Self-convergence diagnostics for the exchange transition-field transform.

Direct and exchange interactions

After separately calibrating the three harmonic transforms, calibrate the two interaction channels on the production displacement domain:

direct_cal = Interaction.converge_parameters(
    calibration_vectors,
    field_ss,
    field_pp,
    yukawa,
    rtol=1.0e-4,
    atol=1.0e-12,
    method="gl4",
    verbose=False,
)
exchange_cal = Interaction.converge_parameters(
    calibration_vectors,
    field_sp,
    field_sp,
    yukawa,
    rtol=1.0e-4,
    atol=1.0e-12,
    method="gl4",
    verbose=False,
)

exchange_cal.plot_convergence()

Self-convergence diagnostics for the exchange interaction calibration.

Direct and exchange matrix elements from the same localized orbital pair. The upper plot compares independent reference curves with QUARTIC2D values; the lower plot shows the exchange-to-direct ratio. The sign change of the exchange channel follows from the nodal transition field and is a genuine four-center effect rather than a numerical artifact.

PETAL2D
rho_ss: modes=[0], reconstruction error=1.377e-14%
rho_pp: modes=[0, 2, -2], reconstruction error=3.233e-14%
rho_sp: modes=[1, -1], reconstruction error=2.852e-14%

Automatic convergence
all harmonic searches satisfied: True
direct interaction satisfied: True
exchange interaction satisfied: True
delta=0.0  direct=+0.667798  exchange=+0.299776
delta=1.0  direct=+0.664678  exchange=+0.053394
delta=2.0  direct=+0.451379  exchange=-0.112760

The complete script is examples/four_center_interaction.py. Field preparation, transform convergence, and interaction convergence are reported separately for the two channels.