Interaction#

Interaction evaluates the four-center matrix element after the two transition fields have been transformed to momentum space. For transition fields \(\rho_{13}\) and \(\rho_{42}\), QUARTIC2D evaluates the radial-kernel reduction of

\[U_{1234}=\iint d^2\mathbf r\,d^2\mathbf r'\, \rho_{13}(\mathbf r)\,U(|\mathbf r-\mathbf r'|)\, \rho_{42}^*(\mathbf r').\]

The ordinary constructor evaluates the requested method once. Use Interaction.converge_parameters() when the interaction itself requires a recorded refinement study.

class quartic2d.Interaction(deltas, field1, field2, U_q, *, method='gl4', interpolator='cubic', n=512, bias=-0.5, subdivisions=1, N=1024, h=None)#

Evaluate four-center matrix elements from transformed transition fields.

For transition fields

\[\rho_{13}(\mathbf r)=\phi_1^*(\mathbf r)\phi_3(\mathbf r), \qquad \rho_{42}(\mathbf r)=\phi_4^*(\mathbf r)\phi_2(\mathbf r),\]

Interaction evaluates the radial-kernel reduction of

\[U_{1234}(\boldsymbol\delta)= \iint d^2\mathbf s\,d^2\mathbf t\, \rho_{13}(\mathbf s) U(|\mathbf s-\mathbf t+\boldsymbol\delta|) \rho_{42}^*(\mathbf t).\]

The two fields are supplied as HarmonicTransform objects. The ordinary constructor performs one evaluation using explicit numerical parameters. converge_parameters() is the recommended opt-in path when the assembled interaction requires a recorded refinement study.

Parameters:

  • deltas (array_like, shape (D, 2)) – Cartesian displacement vectors \(\boldsymbol\delta\).

  • field1 (HarmonicTransform) – Momentum-space harmonic representations of the two transition fields.

  • field2 (HarmonicTransform) – Momentum-space harmonic representations of the two transition fields.

  • U_q (callable) – Scalar radial interaction kernel in momentum space. The callable is evaluated at non-negative momentum magnitudes.

  • method ({'fftlog', 'trapezoid', 'simpson', 'gl4', 'gl8', 'ogata'}, default='gl4') – Numerical method used for the final radial interaction integrals. The methods have different convergence controls; see converge_parameters() and the numerical-method guide rather than interpreting the method name as an accuracy ranking.

  • interpolator ({'linear', 'cubic', 'pchip'}, default='cubic') – Interpolator used for the sampled momentum-space harmonics.

  • n (int, default=512) – FFTLog sequence length. Used only for method='fftlog'.

  • bias (float, default=-0.5) – FFTLog power-law bias. Used only for method='fftlog'.

  • subdivisions (int, default=1) – Number of equal finite-rule subdivisions per original momentum interval. Used by trapezoid, simpson, gl4, and gl8.

  • N (int, default=1024) – Ogata node count. Used only for method='ogata'.

  • h (float or None, optional) – Ogata resolution parameter. None selects the package default for the requested N.

Phi_mm#

Angular prefactors for every harmonic pair and displacement.

Type:

ndarray

H_mm#

Radial interaction integrals for every harmonic pair and displacement.

Type:

ndarray

V_mm#

Harmonic-pair contributions to the final matrix element.

Type:

ndarray

V#

Total interaction for each displacement.

Type:

ndarray, shape (D,)

convergence#

Calibration record attached when the object is constructed through InteractionConvergenceResult.interaction().

Type:

InteractionConvergenceResult or None

See also

Interaction.converge_parameters

Recommended interaction-level calibration workflow.

InteractionConvergenceResult

Reusable selected parameters and convergence history.

HarmonicTransform

Momentum-space representation of one transition field.

Typical workflow#

interaction = Interaction(deltas, field_13, field_42, U_q)
values = interaction.V

For quantitative production work:

result = Interaction.converge_parameters(
    deltas,
    field_13,
    field_42,
    U_q,
    rtol=1e-4,
)
if not result.converged:
    raise RuntimeError("interaction convergence search did not converge")
fig, axes = result.plot_convergence()
interaction = result.interaction(deltas, field_13, field_42, U_q)
interaction.plot_convergence()

The returned InteractionConvergenceResult stores the selected parameters and the method-specific search history. It can be serialized, reused to construct an interaction, and inspected with the convergence plots.

Advanced mutation and low-level convergence#

set_method(method, *[, n, bias, ...])

Select the interaction quadrature and recompute V.

set_interpolator(interpolator)

Select the momentum-space interpolator and recompute V.

converge(*[, rtol, atol, subdivisions, ...])

Refine the active interaction method at fixed transformed fields.