Understanding the structure of complex multidimensional wave functions. A case study of excited vibrational states of ammonia

Generalization of an earlier reduced-density-matrix-based vibrational assignment algorithm is given, applicable for systems exhibiting both large-amplitude motions, including tunneling, and degenerate vibrational modes. The algorithm developed is used to study the structure of the excited vibrational wave functions of the ammonia molecule, ¹⁴NH₃. Characterization of the complex dynamics of systems with several degenerate vibrations requires reconsidering the traditional degenerate-mode description given by vibrational angular momentum quantum numbers and switching to a symmetry-based approach that directly predicts state degeneracy and uncovers relations between degenerate modes. Out of the 600 distinct vibrational eigenstates of ammonia obtained by a full-dimensional variational computation, the developed methodology allows for the assignment of about 500 with meaningful labels. This study confirms that vibrationally excited states truly have modal character recognizable up to very high energies even for the non-trivial case of ammonia, a molecule which exhibits a tunneling motion and has two two-dimensional normal modes. The modal characteristics of the excited states and the interplay of the vibrational modes can be easily visualized by the reduced-density matrices, giving an insight into the complex modal behavior directed by symmetry.