Response properties

dftax.ks.properties is a PySCF-runtime-free property layer: dipole, polarizability, Hessian → vibrational frequencies → IR / Raman, and alchemical derivatives. The geometric quantities are finite differences of the analytic Pulay-free forces; the dipole and polarizability are exact (matrix trace and finite field). Everything is validated against PySCF and/or finite difference.

import jax; jax.config.update("jax_enable_x64", True)
import numpy as np
from dftax import dipole, polarizability, ir_spectrum, alchemical_deriv
from dftax.system import Molecule
from dftax.energy.xc import PBE

mol = Molecule.from_xyz("O 0 0 0; H 0.7586 0 0.5043; H 0.7586 0 -0.5043", "sto-3g")

mu = np.asarray(dipole(mol, PBE(), debye=True))
print("dipole:", np.linalg.norm(mu), "Debye")            # ~1.92 (vs PySCF to 6e-7 a.u.)

alpha = np.asarray(polarizability(mol, PBE()))           # field finite difference
print("isotropic α:", np.trace(alpha) / 3, "a.u.")

ir = ir_spectrum(mol, PBE(), n_radial=60, lebedev=194)
for f, a in sorted(zip(np.asarray(ir["frequencies"]), np.asarray(ir["intensities"]))):
    if f > 100:                                          # skip ~0 translations/rotations
        print(f"{f:8.1f} cm^-1   {a:6.2f} km/mol")       # freqs match PySCF to ~1 cm^-1

print("dE/dZ:", np.asarray(alchemical_deriv(mol, PBE())))  # Hellmann-Feynman alchemy

What's available

Function	Quantity	Method
dipole	μ = Σ Z_A R_A − Tr(P r)	exact (dipole integrals)
polarizability	α_ij = ∂μ_i/∂E_j	finite field, or method="analytic" (CPHF)
hessian	∂²E/∂R∂R'	FD of analytic forces
vibrations	harmonic frequencies + normal modes	mass-weighted Hessian, Eckart-projected
ir_spectrum	frequencies + IR intensities	|dμ/dQ|²
raman_spectrum	frequencies + Raman activities	dα/dQ (expensive)
alchemical_deriv	∂E/∂Z_A	Hellmann-Feynman (fixed-density autodiff)

The harmonic analysis projects out translations and rotations (Eckart), so the six external modes come back at ~0 and the vibrational frequencies match PySCF's harmonic_analysis (water/sto-3g: <5 cm⁻¹). For the exact CPHF polarizability and the status of the analytic Hessian, see implicit differentiation.