Exact SO(3) Averages and Directional Analysis of Molecular Hyperpolarizabilities: Closed Form β/γ Invariant Lab...

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 (B₁,B₃; N₀, N₂, N₄). 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 M₆/M₈ invariant formulation, for the THS channels ⟨|γ_ZZZZ |²⟩ and ⟨|γ_ZXXX|²⟩ as exact rational combinations of N₀,N₂,N₄ 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 M₆/M₈ 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 Qβ² (n), Qγ² (n) and introduce charge-transfer normalized effective hyperpolarizabilities β_eff^CT and γ_eff^CT, which scale invariant RMS amplitudes by the S₀ → S₁ 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.