Kamal Ziadi
We present a unified implementation ready computational framework for rotation-averaged, frame-invariant evaluation of the first- (β) and second-order (γ) molecular hyperpolarizabilities in isotropic media. Starting directly from ab initio Cartesian tensors as delivered by standard electronic-structure codes, these tensors are projected onto Frobenius-orthogonal irreducible components (β: J = 1, 3; γ: L = 0, 2, 4), defining isotropic invariants ( B 1, B 3; N 0, N 2, N 4 ). Using isotropic tensor algebra, we derive closed-form expressions for the HRS channels ⟨|β ZZZ | 2 ⟩ and ⟨|β ZXX | 2 ⟩ and, within a fully Cartesian M 6 /M 8 invariant formulation, for the THS channels ⟨|γ ZZZZ | 2 ⟩ and ⟨|γ ZXXX | 2 ⟩ as exact rational combinations of N 0, N 2, N 4 consistent with established THS tensor analysis, the mixed ZXXX channel is insensitive to the scalar ( L = 0) sector. These formulas are directly useable for HRS and THS, yielding third-harmonic scattering intensities and depolarization ratios directly from invariant norms without invoking symmetry assumptions. To ensure robustness of implementation, we employ three mutually consistent routes: (i) analytic SO(3) contractions with M 6 / M 8 tensors as the reference solution; (ii) deterministic Euler-angle quadrature; and (iii) uniform-quaternion Monte Carlo with variance-based stopping criteria used exclusively as independent verification and quality control. Beyond scalar averages, we compute directional power maps Qβ 2 ( n ), Qγ 2 ( n ) and introduce charge-transfer normalized effective hyperpolarizabilities β eff CT and γ eff CT, which scale invariant RMS amplitudes by the S 0 → S 1 charge-transfer length to quantify the nonlinear response per CT length, and D–π–A chromophores confirm cross-validated β/γ metrics for HRS/THS. Windows-friendly, turnkey code is provided to facilitate reproducible use by computational practitioners.