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Exact SO(3) Averages and Directional Analysis of Molecular Hyperpolarizabilities: Closed Form β/γ Invariant Lab Channel Metrics with Certified Sampling
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DOI：10.1021/acs.jctc.6c00320.png)
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Abstract
En 中文
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 ⟨|βZZZ |2⟩ and ⟨|βZXX |2⟩ and, within a fully Cartesian M6/M8 invariant formulation, for the THS channels ⟨|γZZZZ |2⟩ and ⟨|γZXXX|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 βeffCT and γeffCT, 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.
Keywords:
Chromophores
Optical properties
Polarizability
Polarization
Journal
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