This report concerns the remarkable fine structure reported recently in the optical absorption spectrum of the ubiquitous icosahedral Au144(SR)60 cluster compounds when measured under cryogenic conditions. The theoretical explanation of the spectrum relied upon an I-symmetrized variant of the conventional Pd145-type structure-model; real-time TDDFT calculations revealed that, in contradistinction to the prior state of knowledge, the spectrum is profoundly structured and rich in quantum-state information.1 Reported herein is an investigation of the sensitivity of the theoretical electronic absorption spectra of this compound to variations in the structure. Both I-symmetric as well as asymmetric structure-models are considered; having the same core structure and connectivity, these differ in the mutual configurations about the pyramidal S atoms, which produce significant structure differences penetrating into the gold core. As R-groups, both methyl (R=CH3) and hydrogen (R=H) are considered. The effects on the structure and spectra of local optimizations employing different exchange-correlation (xc-) functionals are also considered. The results may be summarized as follows: all computed spectra show a rich fine-structure when computed at a similar level of resolution (∼0.16 eV, transform limited); the I-symmetric structure with R=CH3 has more pronounced features than the asymmetric structure with the same rest group. This is consistent with the high degree of symmetry-imposed degeneracy in the electronic states of the former. These spectral differences between the I-symmetric and the unsymmetrical models are reduced when the CH3 R group is replaced by the smaller R=H. Many other systematic differences are noted. In particular, we show explicitly the differences caused by changing the R group, the exchange-correlation functional in the geometry relaxation, and the charge state. The present study contains clear indications as to what factors need to be well-controlled in order to achieve good agreement with available experiment. They will be useful for understanding ligand effects on the optical characteristics of thiolate-protected (and other) noble-metal clusters in this interesting size-range where the plasmon (LSPR) emerges.