Development and Application of Diffusion Monte Carlo Methods to Study Molecular Vibrations

Abstract

Understanding the vibrational structure of molecules that exhibit large-amplitude motion remains a central challenge in theoretical spectroscopy. In such systems, the vibrational wave function is often highly delocalized, the harmonic approximation breaks down, and strong coupling among vibrational coordinates complicates both the calculation and the interpretation of the results. These challenges are especially pronounced in fluxional molecular ions such as C$_2$H$_5^+$ and CH$_5^+$, where anharmonicity, isotopic substitution, and hydrogen scrambling strongly influence the observed spectra. This dissertation uses diffusion Monte Carlo (DMC) to address these challenges, while also extending the methodology so that it can provide broader access to excited-state properties and spectroscopic observables. To begin, the theoretical foundations of diffusion Monte Carlo and several of its extensions are reviewed, including guided DMC, fixed-node approaches for excited states, rigid-body formulations, and treatments involving multiple potential energy surfaces. Particular attention is given to the theory and its connection to the algorithm. The first application focuses on the ethyl cation, H$^+$(C$_2$H$_4$), and its deuterated analogues. DMC calculations show that the ground-state wave function is localized near the nonclassical bridged structure where the proton is equidistant from the two carbon atoms, although the motion of the bridging proton remains large in amplitude. Fixed-node DMC calculations of excited states are used to examine effects of isotopic substitution on the proton-transfer and CH stretching vibrations, clarifying how deuteration changes both the vibrational frequencies and the extent of delocalization associated with the bridging proton. The second part of this work develops approaches for evaluating infrared intensities within the guided DMC framework. Trial-wave-function and descendant-weighting approaches are developed for calculating transition dipole matrix elements and are applied to model systems, water, H$_3$O$_2^-$, and H$_5$O$_2^+$. These calculations show that trial-wave-function approaches provide useful lower-cost approximations while keeping the accuracy of the descendant-weighting approach and help reveal how vibrational couplings are reflected in transition intensities. The third part presents a guided DMC formulation in internal coordinates for ground and excited vibrational states. This approach extends DMC to coordinate representations that more directly reflect the underlying nuclear motion and is designed for cases in which the coordinate dependence of the effective masses can be treated locally. Applications to water show good agreement with Cartesian-coordinate DMC and converged variational calculations, while also demonstrating that the method can be applied to excited states whose nodal surfaces are not determined by symmetry. The final application addresses the CH stretching spectra of CH$_5^+$ and CH$_4$D$^+$, systems whose flat potential energy surfaces and highly delocalized ground-state wave functions have made their spectra difficult to interpret. Structures sampled from the DMC ground-state probability density are used together with reduced spectral models to describe the CH stretching region. These calculations reproduce the main features of the reported spectra and suggest that isomerization of CH$_5^+$ is more restricted in helium droplets than in the gas phase. Taken together, the results presented in this dissertation demonstrate that diffusion Monte Carlo can serve not only as a tool for evaluating vibrational energies and wave functions, but also as a flexible framework for probing excited states, transition intensities, and the spectroscopic consequences of fluxional motion. By combining methodological development with applications to challenging molecular ions, this work expands the scope of DMC for vibrational spectroscopy and provides new insight into how anharmonicity, delocalization, and isotopic substitution shape molecular structure and spectra.