## Ab initio molecular diffraction

##### Abstract

In 1915, Debye derived his well-known equation for the X-ray scattering from a
sample of randomly orientated gas-phase molecules. He approximated the molecular
scattering by adding the contributions of isolated atomic constituents. This
is known as the Independent Atom Model (IAM). However, it omits the redistribution
of valence electrons due to bonding, and is limited to the electronic ground
state. The main proposition of this thesis is that it is worthwhile going beyond
the IAM when interpreting X-ray scattering data. In part, this is motivated by
the arrival of new X-ray sources called X-ray Free-Electron Lasers (XFELs).
A new method called Ab Initio X-ray Diffraction (AIXRD) is introduced. It calculates
the elastic X-ray molecular scattering factor directly from wave functions
calculated by ab initio electronic structure theory, for instance Hartree-Fock or
multiconfigurational self-consistent field. In this way, the valence electrons are
correctly taken into account, and calculations based on electronically excited wave
functions become possible. The wave functions must be constructed from spatial
orbitals made up of Gaussian-Type Orbitals (GTOs), giving an analytical
solution to the Fourier transform integrals involved, and is key to computationally
efficient and accurate results. This is compared to a fast Fourier transform
(FFT) method, where the electron density is computed on a 3D grid and an FFT
algorithm is used to obtain the elastic X-ray molecular scattering factor.
Inspired by post-crystallography experiments such as serial femtosecond crystallography
and single-particle imaging at XFELs, the AIXRD method is expanded
to allow accurate X-ray diffraction calculations from large molecules such as proteins.
To make the underlying ab initio problem tractable, the molecule is split
into fragments. In other words, the electron density is constructed by a sum of
fragment contributions, as is the corresponding molecular form-factor. In this
way, it is analogous to the IAM approach except that instead of isolated atoms,
there are isolated fragments. A pairwise summation of fragment contributions is
also used to account for fragment-fragment interactions. Various fragment definitions
are compared based on their effect on the X-ray diffraction signal, and
are compared to the IAM method.
Finally, X-ray diffraction from molecules in specific quantum states is calculated,
revealing a distinct quantum fingerprint in the X-ray diffraction, and a comparison
to experiment is made. In particular, the elastic X-ray diffraction is
calculated from gas-phase H2 pumped to various electronic, vibrational, and electronic
states. This is expanded upon for polyatomic molecules using the harmonic
approximation for the vibrational states.