# Extending the reach of algorithms for the calculation of molecular vibronic spectra

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Theoretical spectroscopy is an important field of chemistry that can help extract useful information about the properties of a molecule from experimental spectral data. Ab initio calculations of molecular spectra can be performed and compared against experimental data to determine the validity of various calculated molecular properties. Unfortunately, the computational cost of these spectral simulations rises quickly with the number of atoms in the molecule of interest. As a result, current techniques for simulating molecular spectra are often limited for use with only the smallest of molecules. The main purpose of this work is to develop new computational tools in an effort to extend the reach of current state-of-the-art spectral simulation algorithms and allow for the spectroscopic study of larger molecules than is currently feasible. The calculation of vibronic spectra requires the solution of the time-independent Schrödinger equation to obtain the vibronic energy levels of a molecule and their corresponding transition intensities. When the Born-Oppenheimer approximation is applicable for the solution of the time-independent Schrödinger equation, the vibrational energy levels of a molecule can be easily determined analytically, if the harmonic approximation is used. What remains, then, for a spectral simulation, is the calculation of the transition intensities associated with each energy level. Under the harmonic approximation, the transition intensities (also known as Franck-Condon factors) can be calculated via a set of recurrence equations developed by Doktorov, Malkin, and Manko. The implementation of these recurrence equations, though, can be computationally intensive for medium-to-large molecules, especially for finite-temperature simulations. In this work, I present a new algorithm for the calculation of Franck-Condon factors via the Doktorov recurrence equations that achieves significantly better computational performance than existing implementations, with speedups of roughly thirty times on a single processor. When the Born-Oppenheimer approximation is not applicable, vibronic coupling effects must be accounted for to achieve an accurate spectral simulation. A common approach for treating vibronic coupling effects is to solve the time-independent Schrödinger equation using a model Hamiltonian developed by Köppel, Domcke, and Cederbaum (KDC). Using the KDC approach, the problem of solving the Schrödinger equation becomes a problem of solving for the eigenstates of a large, sparse matrix. The computational difficulty of this problem is then dependent on the size of the matrix. Unfortunately, the size of the matrix at hand grows exponentially with the number of vibrational modes of the molecule of interest, and matrix dimensions can easily reach upwards of one billion and beyond. In an attempt to make tractable problems involving very large matrices, I present in this work a distributed-memory parallelization strategy for the KDC approach. The resulting parallel algorithm achieves impressive parallel scalability and has been used to study several previously intractable spectroscopic problems. I conclude this work by presenting ab initio calculations and spectral simulations for the molecule trans-1,3-butadiene. The spectroscopy of butadiene has been studied by expermentalists and theorist alike, but a complete KDC spectral model has yet to be achieved due in part to the large size of the resulting matrices. Using the newly-developed parallel algorithm described above, I am able to present spectra from simulations using the most complete KDC model for butadiene to date and discuss what these results may tell us about butadiene’s electronic structure.