permittivity, and ћ the reduced Planck constant. The sums run over Nfs excited states and Np nuclear geometries xi. For each of such geometries in the ensemble, transition energies ΔE0n(xi) and oscillator strengths f0n(xi) between the ground (0) and the excited (n) states are computed. Each transition in the ensemble is convoluted with a normalized line shape function centered at ΔE0n(xi) and with width δ. Each xi is a vector collecting the cartesian components of the geometries of each atom. The line shape function may be, for instance, a normalized Gaussian function given by Although δ is an arbitrary parameter, it must be much narrower than the band width, not to interfere in its description. As the average value of band widths is around 0.3 eV, it is a good practice to adopt δ ≤ 0.05 eV. The geometries xi can be generated by any method able to describe the ground state distribution. Two of the most employed are dynamics and Wigner distribution nuclear normal modes. Molar extinction coefficient ε can be obtained from absorption cross section through Because of the dependence of f0n on xi, NEA is a post-Condon approximation, and it can predict dark vibronic bands. NEA for emission spectrum In the case of fluorescence, the differential emission rate is given by . This expression assumes the validity of the Kasha's rule, with emission from the first excited state. NEA for other types of spectrum NEA can be used for many types of steady-state and time-resolved spectrum simulations. Some examples beyond absorption and emission spectra are: two-dimensional differential transmission photoelectron ultrafast Auger X-ray photo-scattering Limitations of NEA By construction, NEA does not include information about the target (final) states. For this reason, any spectral information that depends on these states cannot be described in the framework of NEA. For example, vibronically resolved peaks in the absorption spectrum will not appear in the simulations, only the band envelope around them, because these peaks depend on the wavefunction overlap between the ground and excited state. NEA can be, however, coupled to excited-state dynamics to recover these effects. NEA may be too computationally expensive for large molecules. The spectrum simulation requires the calculation of transition probabilities for hundreds of different nuclear geometries, which may become prohibitive due to the high computational costs. Machine learning methods coupled to NEA have been proposed to reduce these costs. References Theoretical chemistry