A practical method for the use of curvilinear coordinates in calculations of normal-mode-projected displacements and Duschinsky rotation matrices for large molecules

While use of curvilinear coordinates such as bond lengths and bond angles is common in accurate spectroscopic and/or scattering calculations for triatomic and other small molecules, their use for large molecules is uncommon and restricted. For large molecules, normal-mode analysis is feasible but gives sensible results only if the dynamical or spectroscopic process being considered involves changes in angular coordinates, including ring deformations, which are so small that the motion can be approximated by its tangential component. We describe an approximate method by which curvilinear normal-mode-projected displacements and hence Franck–Condon factors, reorganization energies, and vibronic coupling constants, as well as Duschinsky (Dushinsky, Duschinskii) rotation matrices, can be evaluated for large systems. Three illustrative examples are provided: (i) to understand the nature of the first excited state of water, illustrating properties of large-amplitude bending motions; (ii) to understand the nature of the “boat” relaxation of the first excited state of pyridine, illustrating properties of large-amplitude torsional motions; and (iii) to understand the coupling of vibrational modes to the oxidation of bacteriochlorophyll-a, a paradigm with many applications to both chemical and biological electron transfer, illustrating properties of macrocyclic deformations. The method is interfaced to a wide variety of computational chemistry computer programs.