Analysis, Calculation and Understanding of Molecular Vibrations: decomposition of empirical spectra, simulation targeted to a frequency band, and theoretical framework of oscillatory motion as fat manifolds

The idea of molecular dynamics seems intuitively simple,that atoms should
move through space on complex but well defined trajectories that reflect the
chemical interactions between them.Unfortunately,atoms are quantum ob-
jects therefore to define even the concept of movement,for example as the
change of a well-defined position versus a smoothly evolving time coordinate,
is problematic.
In the present thesis I address molecular motion beginning from the funda-
mentals,presenting initially a software tool for analysis of empirically obtained
spectra such that the optical (or acoustic)properties of a material can be con-
nected to basic statements about its inherent vibrational resonances.Moving
on from this empirically-grounded signal processing approach I develop a
framework for selectively interrogating dynamics when confined to a specific
window of frequency space,providing a toolkit to answer the question“what
molecular motion connects to what observed peaks in a spectrum”.
The final mathematical approach offers a novel perspective on molecular
motion derived from the tradition of Riemannian geometry,framing molecular
trajectories as geodesics on a Riemannian manifold shaped by the Jacobi met-
ric.In this setting,vibrational motion emerges as curvature-driven structure,
captured without recourse to predefined coordinates or harmonic approxima-
tions.The Riemannian approach offers a novel,efficient,and highly robust
integration algorithm which can be used to produce the geodesics of complex
systems in a way that is naturally adapted to interrogate vibrational spectra,
even in the presence of relativistic and quantum effects.