DEVELOPMENT OF ALGORITHMS FOR CLASSICAL AND SEMICLASSICAL DYNAMICS
Driven by questions regarding the significance of atom-atom correlations, this research introduces novel concepts and algorithms aimed at discerning whether or not a molecule is composed of a collection of approximately independent vibrating fragments. Specifically, from a time-dependent perspective, the uncorrelated degrees of freedom must obey Liouville theorem or, more generally, conserve the symplectic form of the Hamiltonian flow. We present a practical approach to identify such vibrating fragments utilizing data collected from molecular dynamics simulations. Our method is based on a heuristic evolutionary algorithm that we call the Probability-Graph Evolutionary Algorithm (PGEA). We demonstrate its usefulness with applications on both model systems and ab initio potentials. We implement these analyses for semiclassical theories, in which the symplectic monodromy matrix naturally occurs. When the calculation of the monodromy matrix becomes prohibitively expensive, we show how to use a neural gas algorithm to tessellate the phase space traversed by trajectories and approximate the local curvature of the potential. We further investigate the correlation among atoms with an experimentalist mindset: We want to artificially “decouple” pairs of atoms and see how the sys tem as a whole rearranges. In practice, we run simulations of artificially modified potentials (called the pair-decoupled potential), in which selected pairs of atoms (or degrees of freedom) do not perceive each other’s displacement. When this artificial disturbance takes place, it affects the molecule as a whole, resulting in a loss (or gain) in energy. A measure of the average energy lost allows to understand if the decoupled pairs can or cannot be seen as belonging to approximately independent groups. To evolve a system under the pair-decoupled potential we provide a practical numerical integration rule, which is based on a local harmonic approximation of the potential, yet it preserves the symplectic form of the Hamiltonian flow and is exact in the limit of quadratic potentials. The SEF algorithm is devised to show that the hypotheses of uncorrelated motion between degrees of freedom may lead to very surprising effects. Application of SEF to the vibrational spectroscopy of the Salicylic Acid molecule shows that for the molecule to remain overall planar in time the hydroxyl and carboxyl groups must either correlate over time, or remain completely uncorrelated. Partial decoupling of selected atoms lead to an unphysical twist of the carboxyl group. From the pair-decoupling concept we propose the Dynamical Coupling Index (DCI). DCI is a new molecular descriptor derived as the energy loss caused by a short-time atom-pair-decoupling perturbation, averaged over a statistical ensemble. We measure the DCI patterns of many small sized organic molecules and investigate the importance of all the atom-atom interactions. This analysis gives a quantitative picture of the fundamental groups composing each molecule from the perspective of dynamical correlation. The DCI picture of rigid molecules, such as benzene, naphthalene and uracil is equivalent to the traditional ball-and-stick picture (after disregarding the weakest DCI couplings). On the other hand, molecules with flexible and rotating fragments have very peculiar DCI pictures, where the coupling of flexible fragment with other pieces of the molecule may be more important then the coupling between covalently bonded atoms.
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