First and Second Order Expansions for Origin Independent Vibronic Calculations of Electronic Chiroptical Spectra Beyond the Franck–Condon Approximation

We investigate the origin dependence in the computation of vibrationally resolved electronic circular dichroism (ECD) and circularly polarized luminescence (CPL) spectral shapes with particular attention to non-Condon approximations. To that end, we adopt the length and velocity representations for the electric transition dipole moment (TDM), expressing both electric and magnetic TDMs as Taylor expansions in nuclear coordinates, including the constant (Franck-Condon, FC), non-Condon linear (Herzberg-Teller, HT) and, in some cases, second-order terms. Our analysis evidence that HT spectra, in the standard formulation, i.e., including the full first-order expansions of both TDMs, are not origin invariant, even in the velocity gauge. This inconsistency arises because the corresponding expression for the rotatory strength, which, unlike the individual TDMs, directly encodes the measurable chiroptical response, involves an incomplete second-order expansion. We show that origin invariance in the velocity gauge is restored when the product terms between the electric and magnetic TDM expansions are selected to yield a complete either first- or second-order expansion of the rotatory strength. In particular, excluding the cross-linear terms in the HT expression results in a consistent linear expansion, which delivers origin invariant rotatory strengths and ECD/CPL lineshapes. We refer to this formulation as the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> approach. Similarly, including the additional terms required for a complete second-order expansion, i.e., those combining constant and quadratic TDM terms, defines the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> approach, which also produces origin-invariant rotatory strengths. However, the ECD/CPL spectra lineshapes computed at this level still depend weakly on the gauge origin because the intensity of each single vibronic transition does. Nonetheless, the fact that the integrated ECD/CPL <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> lineshapes remain origin invariant effectively mitigates origin-related variations in the spectra. Simulations of ECD and CPL lineshapes for three representative systems, spanning different non-Condon strengths and sensitivities to the gauge origin, confirm these predictions. All calculations are performed at the TDDFT level, using analytical first-order derivatives and numerical second-order derivatives of the TDMs with respect to nuclear coordinates. The analytical time-correlation functions necessary to run <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> vibronic computations, in a time-dependent framework, were derived in harmonic approximation, providing a general and robust route toward origin-independent vibronic simulations of chiroptical spectra.