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\chapter{Exploration of Structural and Energetic Properties}  \label{chap:structure}

This \textbf{third chapter} of my thesis merges two independent studies dealing with the determination of the low-energy isomers of
ammonium/ammonia water clusters, (H$_2$O)$_{n}${NH$_4$}$^+$ and (H$_2$O)$_{n}$NH$_3$, and protonated uracil water clusters, (H$_2$O)$_{n}$UH$^+$.
As highlighted in the general introduction of this thesis and in chapter~\ref{chap:comput_method}, performing global optimization of
molecular clusters is not straightforward. The two studies presented in this chapter thus share a main common methodology which is the
combination of the \textbf{self-consistent-charge density functional based tight-binding} (SCC-DFTB) method for the efficient calculation of the potential
energy surfaces (PES) and the \textbf{parallel-tempering molecular dynamics} (PTMD) approach for their exploration. All low-energy isomers 
reported in this chapter are discussed in terms of structure, relative energy and binding energy which are compared to the literature
when available. Calculations at higher level of theory are also performed to refine the results obtained at the SCC-DFTB level or to
validate the results it provides. In particular, in this chapter, we propose an improve set of parameters to describe sp$^3$ nitrogen
containing compounds at the SCC-DFTB level. Our results are also used to complement collision-induced dissociation experiments
performed by S. Zamith and J.-M. l'Hermite at the \textit{Laboratoire Collisions Agr\'egats R\'eactivit\'e} (LCAR).

%The background of the systems studied in this thesis and theoretical methods related to the projects of this thesis have been introduced in chapter \ref{chap:general_intro} and chapter \ref{chap:comput_method}. Then the two kinds of projects investigated in my Phd will be shown in the following chapter \ref{chap:structure} and chpater \ref{chap:collision}.
 
%For the topics about ammonium/ammonia water clusters and protonated uracil water clusters, both of the two  studies are about the structural and energetic properties. For instance, some lowest-energy isomers were obtained and the relative binding energies of the lowest-energy isomers were calculated. So the work of these two topics are put in the same chapter. These two studies demonstrates the accuracy of SCC-DFTB to describe ammonium/ammonia water clusters and protonated uracil water clusters. The calculation methods for the topics about ammonium/ammonia water clusters and protonated uracil water clusters are almost the same, therefore, the methods of these two topics will be described together. But their introduction, results, and conclusion will be shown independently (in different subsections). For the topic about ammonium/ammonia water clusters, it is a pure theoretical calculation while for the topic about protonated uracil water clusters, it is collaborated with the experiment and some experimental results will be displayed.

\section{Computational Details} \label{sec:structure-methods}

\subsection{SCC-DFTB Potential}

SCC-DFTB electronic structure calculations presented in this chapter were all performed with the deMonNano code.\cite{deMonNano2009} The details
of the method are presented in section~\ref{sec:DFTB} of chapter~\ref{chap:comput_method}.
%DFTB is an approximated DFT scheme whose computational efficiency depends on the use of parameterized integrals.\cite{Elstner2014, Elstner1998, dftb1, dftb2}
The mio-set for the Slater-Koster tables of integrals was used.\cite{Elstner1998} However, it has been shown that these integrals do not properly described
sp$^3$ hybridized nitrogen, in particular, proton affinity.\cite{Gaus2013para} Consequently, in order to avoid spurious deprotonation of the sp$^3$ hybridized
nitrogen in NH$_4^+$ and to correctly reproduce binding energies calculated at the MP2/Def2TZVP level, we propose to modify the original mio-set for
Slater-Koster tables of N-H integrals by applying  them a multiplying factor. Several of them were tested and we present here the results
we obtained for two of them: 1.16 and 1.28. For the sp$^2$ nitrogen of uracil, the original integrals of the mio-set were used. To improve
description of the intermolecular interactions, the original Mulliken charges were replaced by the CM3 charges,\cite{Li1998, Thompson2003, Rapacioli2009corr}
(see equation~\ref{CM3} in section~\ref{sec:DFTB}) and an empirical correction term (see equation~\ref{dispersionE} in section~\ref{sec:DFTB}) was used
to describe dispersion interactions.\cite{Rapacioli2009corr, Elstner2001, Zhechkov2005}  Simon \textit{et al.} developed a SCC-DFTB potential that leads
to geometries, frequencies, and relative energies close to the corresponding experimental and CCSD(T)/aug-cc-pVTZ results.\cite{Simon2012, Odutola1980}
The corresponding $D_{OH}$ parameter, \textit{i.e.} 0.129, is retained in the studies presented in thus chapter. $D_{NH}$ is tested in the study of
ammonium/ammonia water clusters and two values were retained and thoroughly tested: 0.12 and 0.14. D$_{NO}$ is set to zero.

\subsection{SCC-DFTB Exploration of PES}

To determine the lowest-energy isomers of (H$_2$O)$_{1-10,20}${NH$_4$}$^+$, (H$_2$O)$_{1-10}$NH$_3$
and (H$_2$O)$_{1-7,11,12}$UH$^+$ clusters, we thoroughly explore their PES using PTMD \cite{Sugita1999, Sugita2000, Earl2005}
simulations combined with a SCC-DFTB \cite{Elstner1998} description of the energies and gradients. I describe
below the \textbf{detailed parameters used for all the simulations conducted within this chapter}.

\textbf{Detailed parameters for PTMD simulations of (H$_2$O)$_{1-10,20}${NH$_4$}$^+$ and (H$_2$O)$_{1-10}$NH$_3$ clusters} are as follows.
For (H$_2$O)$_{1-3}${NH$_4$}$^+$  and (H$_2$O)$_{1-3}${NH$_3$} clusters, 16 replicas were used 
with a linear distribution of temperatures with a 15~K step ranging from 10 to 250 K. 40 replicas with a 6~K step ranging from 10 to 250~K were
considered for (H$_2$O)$_{4-10,20}${NH$_4$}$^+$ and (H$_2$O)$_{4-10}${NH$_3$}  species. All trajectories were 5~ns long and a time step
of 0.5 fs was used to integrate the equations of motion. A Nos{\'e}-Hoover chain of 5 thermostats was employed for all the simulations  to achieve
simulatons in the canonical ensemble.\cite{Nose1984M, Hoover1985} Thermostat frequencies were fixed at 400 cm$^{-1}$.
To identify low-energy isomers of (H$_2$O)$_{1-3}${NH$_4$}$^+$ and (H$_2$O)$_{1-3}${NH$_3$} clusters, 303 geometries were periodically
selected from each replicas and further optimized at the SCC-DFTB level, which produced 4848 optimized geometries per cluster. For
(H$_2$O)$_{4-10,20}${NH$_4$}$^+$ and (H$_2$O)$_{4-10}${NH$_3$} clusters, 500 geometries were periodically selected from each
replicas leading to 20000 optimized geometries per cluster. For (H$_2$O)$_{20}${NH$_4$}$^+$, the initial structure used for the global
optimization process was the lowest-energy structure reported by Douady \textit{et al.}.\cite{Douady2009} The five lowest-energy
isomers among the 4848 or 20000 optimized geometries were further optimized using the MP2/Def2TZVP method. See below for the
details on MP2/Def2TZVP calculations.

\textbf{Detailed parameters for PTMD simulations of (H$_2$O)$_{1-7,11,12}$UH$^+$ clusters} are as follows.
40 replicas with temperatures rnaging linearly from 50 to 350 K were used. Each trajectory was 4 ns long, and the integration time step was 0.5 fs.
A reasonable time interval for the PT exchanges was 2.5 ps. A Nos{\'e}-Hoover chain of five thermostats with frequencies of 800 cm$^{-1}$ was
applied to achieve an exploration in the canonical ensemble.\cite{Nose1984M, Hoover1985} To avoid any spurious influence of the initial
geometry on the PES exploration, three distinct PTMD simulations were carried out with distinct initial proton location: on the uracil in two cases
and on a water molecule in the other one. In the former cases, we used two isomers u178 and u138 of UH$^+$ shown in Figure~\ref{uracil_s} as
the initial geometries.\cite{Wolken2000, Pedersen2014} 600 geometries per temperature were linearly selected along each PTMD simulation
for subsequent geometry optimization leading to 72000 structures optimized at SCC-DFTB level. These structures were sorted in ascending
energy order and checked for redundancy. 9, 23, 46, 31, 38, 45, 63, 20, and 29 structures were then selected for (H$_2$O)UH$^+$,
(H$_2$O)$_{2}$UH$^+$, (H$_2$O)$_{3}$UH$^+$, (H$_2$O)$_{4}$UH$^+$, (H$_2$O)$_{5}$UH$^+$, (H$_2$O)$_{6}$UH$^+$,
(H$_2$O)$_{7}$UH$^+$, (H$_2$O)$_{11}$UH$^+$ and (H$_2$O)$_{12}$UH$^+$ respectively, to perform geometry optimizations at the  MP2/Def2TZVP
level. See below for the details on MP2/Def2TZVP calculations.

%\figuremacro{uracil}{Structures of the two protonated uracil isomers, u178 (keto-enol form) and u138 (di-keto form), used as initial conditions in the PTMD simulations.}
\begin{figure}[h!]
\includegraphics[width=0.5\linewidth]{a-b.pdf}
\centering
\caption{Structures of the two protonated uracil isomers, u178 (keto-enol form) and u138 (di-keto form), used as initial conditions in the PTMD simulations.}
\label{uracil_s} 
\end{figure}
 
\subsection{MP2 Geometry Optimizations, Relative and Binding Energies}

Some low-energy isomers obtained at the SCC-DFTB level were further optimized at the MP2 level of theory in combination
with an all electron Def2TZVP basis-set.\cite{Weigend2005, Weigend2006} All calculations used a a tight criteria for geometry
convergence and an ultrafine grid for the numerical integration. All MP2 calculations were performed with the Gaussian 09
package.\cite{GaussianCode} 

\textbf{Detailed parameters for (H$_2$O)$_{1-10,20}${NH$_4$}$^+$ and (H$_2$O)$_{1-10}$NH$_3$ clusters.}
Following SCC-DFTB optimizations, the five lowest-energy isomers of (H$_2$O)$_{1-10}${NH$_4$}$^+$ and
(H$_2$O)$_{1-10}${NH$_3$} clusters  were further optimized at the MP2/Def2TZVP level of theory.
In section~\ref{sec:ammoniumwater}, relatives energies with respect to the lowest-energy isomer of each
cluster are reported. Impact of zero-point vibrational energy (ZPVE) corrections on relative
energies were evaluated at MP2/Def2TZVP level. To evaluate the strength of water-ammonium and water-ammonia
interactions and to assess the accuracy of the SCC-DFTB method, we also report binding energies.
Two distinct approaches were used to calculate binding energies. The first one considers only the binding
energy between the water cluster as a whole and the impurity, {NH$_4$}$^+$ or NH$_3$, while the second one
considers the binding energy between all the molecules of the cluster. In both cases, the geometry of the molecules
is the one found in the optimized cluster. Using these two methods, relative binding energies (E$_{bind.}$(SCC-DFTB)-E$_{bind.}$(MP2/Def2TZVP))
$\Delta E_{bind.}^{whole}$ and $\Delta E_{bind.}^{sep.}$ were obtained. For all binding energies of
(H$_2$O)$_{1-10}${NH$_4$}$^+$ and (H$_2$O)$_{1-10}${NH$_3$} clusters calculated at MP2/Def2TZVP level,
basis set superposition errors (BSSE) correction was considered by using the counterpoise method of Boys and
Bernardi.\cite{Boys2002}  

\textbf{Detailed parameters for (H$_2$O)$_{1-7, 11, 12}$UH$^+$ clusters.}
Following SCC-DFTB optimizations, the six lowest-energy isomers of 
(H$_2$O)$_{1-7, 11, 12}$UH$^+$ clusters were further optimized at the MP2/Def2TZVP level of theory.
The binding and relative energies calculated at MP2/Def2TZVP level without BSSE correction
of clusters (H$_2$O)$_{2-7, 11, 12}$UH$^+$ are discussed in section~\ref{structureUH}.

\subsection{Structure Classification}

For clusters (H$_2$O)$_{1-10}${NH$_4$}$^+$ and (H$_2$O)$_{1-10}${NH$_3$},  
\textquotedblleft n-x\textquotedblright and \textquotedblleft n$^\prime$-x\textquotedblright labels are used to distinguish between the
(H$_2$O)$_{n}${NH$_4$}$^+$ and (H$_2$O)$_{n}${NH$_3$} reported isomers, respectively, obtained at the SCC-DFTB level. For comparison,
\textquotedblleft n-x\textquotedblright and \textquotedblleft n$^\prime$-x\textquotedblright  isomers are also optimized at the MP2/Def2TZVP
level. In that case, the resulting structures are referred to as \textquotedblleft n-x$^*$\textquotedblright and
\textquotedblleft n$^\prime$-x$^*$\textquotedblright~ to distinguish them more easily although they display the same general topology as
\textquotedblleft n-x\textquotedblright and \textquotedblleft n$^\prime$-x\textquotedblright isomers. 
In these notations, n and n$^\prime$ denote the number of water molecules in the ammonium and ammonia water clusters, respectively.
x is an alphabetic character going from a to e that differentiates between the five low-energy isomers reported for each cluster in ascending
energy order, \textit{i.e.} a designates the lowest-energy isomer.


%\section{\label{sec:ammoniumwater}The structure and energetics properties study of ammonium or ammonia including water clusters}
\section{Structural and Energetic Properties of Ammonium/Ammonia including Water Clusters}   \label{sec:ammoniumwater}

\subsection{General introduction}
%Ionic clusters (typically made up of a core ion surrounded by one or more solvating molecules) are known to be involved in the chemistry of the upper and mid atmosphere [6].《Ion rearrangement at the beginning of cluster formation: isomerization pathways and dissociation kinetics for the ionized dimethylamine dimer》

Water clusters play an important role in various areas such as atmospheric and astrochemical science, chemistry and biology.\cite{Keesee1989, Gilligan2000,
Sennikov2005, Cabellos2016, Orabi2013, Bommer2016, Rodgers2003, Van2004, Gibb2004, Tielens2005, Parise2005, Boogert2015, Dulieu2010, Michoulier2018}
They are involved into the critical stages of nucleation and growth of water-containing droplets in the atmosphere thus contributing to the physical and
chemical properties of this medium.\cite{Kulmala2004}
%For instance, water clusters are able to absorb a significant amount of radiant energy thus decreasing the greenhouse effect.\cite{Galashev2010}
In many cases, the presence of chemical impurities interacting with water aggregates strongly affect their properties.
For instance, ammonia is an important compound commonly found in the atmosphere and  which displays a key role in aerosol chemistry.\cite{Ziereis1986} Its high
basicity makes it a potential proton sink that can form a ionic center for nucleation.\cite{Perkins1984, Arnold1997} E. Dunne and co-workers also reported that most nucleation occurring in the atmosphere involves ammonia or biogenic organic compounds, in addition to sulfuric acid.\cite{Dunne2016} J. Kirkby \textit{et al.} 
also found that even a small amount of atmospherically relevant ammonia can increase the nucleation rate of sulphuric acid particles by several orders of magnitude.\cite{Kirkby2011}
The significance of ammonium and ammonia water clusters have thus motivated a large amount of experimental and theoretical studies during the past decades.\cite{Perkins1984, Herbine1985, Stockman1992, Hulthe1997, Wang1998, Chang1998, Jiang1999, Hvelplund2010, Douady2009, Douady2008, Morrell2010, Bacelo2002, Galashev2013} 

As a few examples, in 1984, (H$_2$O)$_{2}${NH$_4$}$^+$ was identified using mass spectrometry by Perkin \textit{et al.}\cite{Perkins1984} In 1997, Stenhagen
and co-workers studied the {(H$_2$O)$_{20}$H$_3$O}$^+$ and (H$_2$O)$_{20}${NH$_4$}$^+$ clusters and found that both species display a similar
structure.\cite{Hulthe1997} Hvelplund \textit{et al.} later reported a combined experimental and theoretical study devoted to protonated mixed ammonia/water
which highlighted the idea that small protonated mixed clusters of water and ammonia contain a central {NH$_4$}$^+$ core.\cite{Hvelplund2010}
%The (H$_2$O)NH$_3$ complex has been experimentally investigated via radio frequency and microwave spectra by Herbine \textit{et al.}, and via microwave and tunable far-infrared spectroscopy by Stockman and co-workers.\cite{Herbine1985, Stockman1992}
Theoretical calculations devoted to ammonium and ammonia water clusters have also been extensively conducted.\cite{Lee1996, Chang1998,
Skurski1998, Jiang1999, Donaldson1999, Sadlej1999, Hvelplund2010, Bacelo2002, Galashev2013} Among them, Novoa \textit{et al.} studied the (H$_2$O)$_4$NH$_3$
aggregate and found the existence of a minimum in its potential energy surface corresponding to a (H$_2$O)$_{3}$···{NH$_4$}$^+$···OH$^-$ structure, resulting from
one proton transfer from a water molecule to the ammonia molecule.\cite{Lee1996} Bacelo later reported a number of low-energy minima for (H$_2$O)$_{3-4}$NH$_3$
clusters obtained from \textit{ab initio} calculation and a Monte Carlo exploration of the potential energy surface (PES).\cite{Bacelo2002} More recently, Douady \textit{et al.}
performed a global optimization of (H$_2$O)$_{n}${NH$_4$}$^+$ (n = 1-24) clusters again using a Monte Carlo procedure  in combination with a Kozack and Jordan
empirical force field.\cite{Douady2008, Kozack1992polar}
In this study, the finite temperature properties as well as vibrational signature of several clusters thus highlighting the key contribution of simulations in understanding such species. Morrell and Shields also studied the
(H$_2$O)$_{n}${NH$_4$}$^+$ (n = 1-10) aggregates \textit{via} a mixed molecular dynamics and quantum mechanics methodology to calculate energies and free energies
of formations which were in good agreement with previous experimental and theoretical results.\cite{Morrell2010} 
More recently, Pei \textit{et al.} determined that  (H$_2$O)$_{n}$NH$_4^+$ clusters start to adopt a closed-cage geometry at $n$=8.\cite{Pei2015}
Finally, Walters and collaborators determined the geometry of (H$_2$O)$_{16}$NH$_3$ and (H$_2$O)$_{16}$NH$_4^+$ at the HF/6-31G(d) level,
and observed strong hydrogen bonding between water and the lone pair of NH$_3$ and bewteen NH$_4^+$ and the four adjacent water molecules.\cite{Walters2018}

As for the study of other molecular clusters, the range of applicability of theoretical simulations to describe ammonium and ammonia water clusters is dictated 
by the balance between accuracy, transferability and computational efficiency. While \textit{ab-initio} methods can accurately model small aggregates, their
application to large species is more difficult, in particular when an exhaustive exploration of the PES is required. In contrast, force-field
potentials are computationally extremely efficient and can be coupled to global optimization methods but their transferability is limited.
The SCC-DFTB approach can be seen as an intermediate approach which combines the strengths of both \textit{ab-initio} and force-field methods. 
Indeed, it can be as accurate as DFT while computationally more efficient and is more transferable than force fields (see Chapter~2)
 In the recent years, SCC-DFTB has been successfully applied to the study of various molecular clusters: pure, protonated, and de-protonated water
clusters,\cite{Choi2010, Choi2013, Korchagina2017, Simon2019} water clusters on polycyclic aromatic hydrocarbons,\cite{Simon2012, Simon2013water}
sulfate-containing water clusters,\cite{Korchagina2016} water clusters in an argon matrix,\cite{Simon2017formation} whether it is for global optimization or
for the study of finite-temperature properties. However, in its original formulation, SCC-DFTB does not provide good results for the description of ammonia
and ammonium as nitrogen hybridization seems to be a problem for minimal basis-set methods like SCC-DFTB.\cite{Winget2003} Elstner and coworkers found
consistent errors (about 14.0 kcal.mol$^{-1}$) for deprotonation energies of sp$^3$ hybridized nitrogen containing systems, whereas sp$^1$ and sp$^2$ systems
display much smaller errors.\cite{Gaus2013para}

In the present section, we first propose an improvement of the SCC-DFTB scheme to describe ammonium and ammonia water clusters by modifying
both Hamiltonian and overlap N-H integrals and introducing optimized atomic charges.\cite{Thompson2003, Rapacioli2009} By combining this
improved SCC-DFTB scheme with PTMD simulations, global optimization of the (H$_2$O)$_{1-10}${NH$_4$}$^+$ and (H$_2$O)$_{1-10}${NH$_3$}
clusters is then performed which allows to report a number of low-energy isomers for these species. Among them, a selected number of structures
are further optimized at the MP2/Def2TZVP level of theory to confirm they are low-energy structures of the PES and to rationalize the difference in
relative energy between both methods. A detailed description of the reported low-energy isomers is then provided as well as comparisons with the
literature. The heat capacity curve of (H$_2$O)$_{20}${NH$_4$}$^+$ is also obtained at the SCC-DFTB level and compared to previously published
simulations. Some conclusions are finally presented.  A very small part of this work has been published in 2019 in a review in\textit{Molecular Simulation}.\cite{Simon2019}
A full paper devoted to this work is in preparation.

\subsection{Results and Discussion}

\subsubsection{Dissociation Curves and SCC-DFTB Potential}

In order to define the best SCC-DFTB parameter to model ammonia and ammonium water clusters, we have tested various sets of corrections.
Each correction involves two modifications of the potential, the first one is the CM3 charge parameter D$_{NH}$ and the second one is the
multiplying factor, noted $xNH$, applied to the NH integrals in the Slater-Koster tables. So a given set is noted D$_{NH}$/$xNH$. Two sets of
corrections have provided satisfactory results, 0.12/1.16 and 0.14/1.28. Figure~\ref{fig:E_nh4} and ~\ref{fig:E_nh3} present dissociation
curves obtained at the  MP2/Def2TZVP, MP2/Def2TZVP with BSSE correction, original SCC-DFTB,  SCC-DFTB 0.14/1.28 and SCC-DFTB
0.12/1.16 levels of theory. These curves are obtained using the same set of geometries regardless of the method applied to calculate
the binding energies. They are obtained from the MP2/Def2TZVP optimized structures in which the distance between the water and the
ammonium/ammonia was shifted along the N--O vector, all other geometrical parameters being kept fixed.

\begin{figure}[h!]
  \includegraphics[width=0.6\linewidth]{E-distance-nh4-w.png}
     \centering
    \caption{Binding energies of (H$_2$O){NH$_4$}$^+$ as a function of the  N---O distance at MP2/Def2TZVP (plain black), MP2/Def2TZVP with BSSE correction (dotted black),
    original SCC-DFTB (plain red),  SCC-DFTB (0.14/1.28) (dotted red) and SCC-DFTB (0.12/1.16) (dashed red)  levels of theory.}
    \label{fig:E_nh4} 
 \end{figure}

\begin{figure}[h!]
  \includegraphics[width=0.6\linewidth]{E-distance-nh3-w.png}
     \centering
    \caption{Binding energies of (H$_2$O){NH$_3$} as a function of the  N---O distance at MP2/Def2TZVP (plain black), MP2/Def2TZVP with BSSE correction (dotted black),
    original SCC-DFTB (plain red),  SCC-DFTB (0.14/1.28) (dotted red) and SCC-DFTB (0.12/1.16) (dashed red)  levels of theory.}
    \label{fig:E_nh3} 
 \end{figure}

From Figure~\ref{fig:E_nh4}, the five curves display the same trends with a minimum located at almost the same N---O distance. At the curve minimum,
binding energies  vary between XX and XX~kcal.mol$^{-1}$ at the original SCC-DFTB and SCC-DFTB 0.14/1.28 levels, respectively. The binding energy
obtained at the SCC-DFTB 0.12/1.16 level is the closest to that obtained at MP2/Def2TZVP level with BSSE correction with a binding energy difference of
only XX~kcal.mol$^{-1}$. The SCC-DFTB 0.14/1.28 curve is also very close  with a difference in binding energy only XX~kcal.mol$^{-1}$ higher. It is
worth mentioning that both sets of corrections lead to improved results as compared to the original SCC-DFTB parameters which leads to a too low
binding energy as compared to MP2/Def2TZVP level with BSSE correction. Also the position of the minimum is more shifted at the original SCC-DFTB
level than with corrections. So from structural and energetic point of views, both sets of corrections are satisfactory.

From Figure~\ref{fig:E_nh3}, the five curves display significant differences. This effect is accentuated by smaller binding energy values:  they
vary from XX to XX~kcal.mol$^{-1}$ at the original SCC-DFTB and MP2/Def2TZVP levels, respectively, at the minimum of the curves. The binding
energy obtained at the SCC-DFTB 0.14/1.28 level is the closest to that obtained at MP2/Def2TZVP level with BSSE correction with a binding energy
difference of only XX~kcal.mol$^{-1}$. The SCC-DFTB 0.12/1.16 curve is also rather close  with a difference in binding energy only XX~kcal.mol$^{-1}$
higher. Here also, both sets of corrections lead to improved results as compared to the original SCC-DFTB parameters. The position of the minimum
is also  better reproduced by the the corrected potentials than by the original SCC-DFTB. It is worth mentioning that the rather difference in binding
energy between (H$_2$O){NH$_4$}$^+$ and (H$_2$O)NH$_3$ was expected owing to a stronger electrostatic contribution of {NH$_4$}$^+$ to the
binding energy.

Another very important point when comparing the original SCC-DFTB potential and the corrected potentials, is the structure obtained for the 
(H$_2$O){NH$_4$}$^+$ dimer. Figure~\ref{dimers} compares the structure obtained from geometry optimization at the SCC-DFTB 0.14/1.28
and original SCC-DFTB levels. The N-H covalent bond involved in the hydrogen bond is significantly longer with the original potential while
the N---O distance is XXX. This is reminiscent of the too low proton affinity of {NH$_4$}$^+$ predicted by the original SCC-DFTB potential.
This discrepancy has been previously highlighted in other studies,\cite{Winget2003,Gaus2013para} and makes this potential unusable in any
realistic molecular dynamics simulation as it leads to a spurious deprotonation. Both sets of corrections are free of this error.  \red{An additional
proof of this assertion is presented in Figure~\ref{XX} that displays the energy barrier for the proton transfer at constant N---O distance obtained
from different methods.}

%\begin{figure}[H]
\begin{figure}
  \includegraphics[width=0.3\linewidth]{dimers.png}
     \centering
    \caption{Structure of (H$_2$O){NH$_4$}$^+$ obtained from geometry optimization at the SCC-DFTB 0.14/1.28 (right) and original SCC-DFTB (left) levels.}
    \label{dimers} 
 \end{figure}
 
Figures~\ref{fig:E_nh4} and ~\ref{fig:E_nh3} show that SCC-DFTB 0.14/1.28 better describe (H$_2$O){NH$_3$} dissociation curve while SCC-DFTB 0.12/1.16
better describe (H$_2$O){NH$_4$}$^+$. As one looks for a unique potential to describe both ammonium and ammonia water clusters, we choose  SCC-DFTB
0.14/1.28 for the present study. Indeed, as (H$_2$O){NH$_3$} is characterized by a much lower binding energy than (H$_2$O){NH$_4$}$^+$, an error of
the of order of XX~kcal.mol$^{-1}$ is more likely to play a significant role for ammonia than ammonium containing species. All the following discussion therefore
involve the SCC-DFTB 0.14/1.28 potential.


\subsubsection{Small Species: (H$_2$O)$_{1-3}${NH$_4$}$^+$ and (H$_2$O)$_{1-3}${NH$_3$}}



\subsubsection{Properties of (H$_2$O)$_{4-10}${NH$_4$}$^+$ Clusters}



\subsection{Conclusions for Ammonium/Ammonia Including Water Clusters}




\section{Structural and Energetic Properties  of Protonated Uracil Water Clusters} \label{structureUH}

\subsection{General introduction}

Gas phase investigations of molecules help to understand  the intrinsic properties of molecules that are free from the effects of solvents. 
The gas phase study needs to be extended towards more realistic biomolecular systems, to reveal how the intrinsic molecular properties are affected by the surrounding medium when the biomolecules are in a natural environment.\cite{Maclot2011, Domaracka2012, Markush2016, Castrovilli2017} The hydration study of biomolecules is of paramount importance to get insights into their behavior in aqueous medium, especially the effects on their structure, stability and dynamics. 

The nucleobases in DNA and RNA play a significant role in the encoding and expression of genetic information in living systems while water is a natural medium of many reactions in living organisms. The study of the interaction between nucleobase molecules and aqueous environment has attracted a lot of interests among biologists and chemists. Exploring the clusters composed of nucleobase molecules with water is a good workbench to observe how the properties of nucleobase molecules change when going from isolated gas-phase to hydrated species.

The radiation can cause damages on RNA and DNA molecules, which is proficiently applied in radiotherapy for cancer treatment. The major drawback in radiotherapy is the unselective damage in both healthy and tumor cells, which has a big side effect. This makes it particularly important to explore the radiation fragments. 
Uracil (U), C$_4$H$_4$N$_2$O$_2$, is one of the four nucleobases of RNA, has been paid attention concerning radiation damage. Protonated uracil UH$^+$ can be generated by radiation damages.\cite{Wincel2009} 
The reasons for such degradation can be due to the interaction with slow electrons, as shown by the work of Boudaiffa \textit{et al.} \cite{Boudaiffa2000}
Several studies have been devoted to the effect of hydration on the electron affinity of DNA nucleobases. \cite{Smyth2011, Siefermann2011, Alizadeh2013} For instance, Rasmussen \textit{et al.} found that a water molecule is more likely to interact with a charged species than with a neutral one though the study of hydration effects on the lowest triplet states of cytosine, uracil, and thymine by including one or two water molecules explicitly, \cite{Rasmussen2010} 
However, a lot of work is still needed to be performed to understand the role of aqueous environment on charged nucleobases of DNA and RNA.

Collision experiments is a useful tool that can be applied to understand the reactivity of molecules and provide access to structural information.\cite{Coates2018} 
Fragmentation of the bare protonated U has already been performed under collision-induced dissociation (CID) with tandem mass spectrometry,\cite{Nelson1994, Sadr2014, Molina2016}
however, there are only few studies available concerning the effect of hydration on such process. Infrared photodissociation spectroscopy of singly
hydrated protonated uracil shows that the most stable tautomeric form of the neutral uracil (diketo) differs from the most stable one for bare
protonated uracil (keto-enol).\cite{Bakker2008}
%Some theoretical studies reported the structures taken by neutral microhydrated uracil, containing up to 15 water molecules.\cite{Shishkin2000, Gadre2000, Gaigeot2001, Danilov2006, Bacchus2015} 
However, fragmentation studies of such species under CID conditions have not been performed. \textbf{S. Zamith and J.-M. l'Hermite conducted such CID
experiments on protonated uracil water species (H$_2$O)$_{1-15}$UH$^+$ during my thesis and I collaborated with them in order to provide a theoretical
support to their measurements}.
%The dissociation of mass selected (H$_2$O)$_{1-15}$UH$^+$ clusters is induced by collisions with target rare gas atoms (Ne, Ar) or water molecules (H$_2$O, D$_2$O) at a controlled center of mass collision energy 7.2 eV which is chosen high enough so that a large number of fragmentation channels are explored. After the collisions, the remaining charged species, which have lost one or several neutral subunits, are detected. We find essentially the same results whatever the nature of target atoms of molecules is. The resulting inter-molecular dissociation patterns of the (H$_2$O)$_{1-15}$UH$^+$ clusters show that below n = 5 only water molecules are evaporated whereas, for n $\geq$ 5, a new fragmentation channel appears that corresponds to the loss of neutral uracil. 

Theoretical studies have already been devoted to mixed uracil-water clusters and intended to describe the lowest energy structures. However,
only neutral species ((H$_2$O)$_{n}$U) were considered.\cite{Shishkin2000, Gadre2000,  Van2001diffu, Gaigeot2001, Danilov2006, Bacchus2015}
Those studies showed that for sizes up to with $n$ = 3, the water molecules arrange in monomers or dimers in the plane of the uracil molecule 
\cite{Gadre2000, Van2001diffu, Gaigeot2001, Danilov2006, Bacchus2015} with no trimer formation. But for $n$ \textgreater~3, very different structures 
were predicted depending on the considered study. For instance, Ghomi predicted that for $n$ = 7,\cite{Gaigeot2001}  water molecules arrang
 in dimers and trimers in the plane of the uracil molecule, whereas for n = 11, water molecules form locked chains.\cite{Shishkin2000} 3D configurations were also proposed. For instance, all water molecules lie above the uracil plane for $n$ = 4, 5 reported by Calvo \textit{et al.}.\cite{Bacchus2015} Similarly, for $n$ = 11, Danilov \textit{et al.} also obtained a
 structure that consists of a water cluster above the uracil molecule.\cite{Danilov2006} Such structures are predicted to start with 4 water molecules
 reported by Calvo and collaborator \cite{Bacchus2015} or with 6 water molecules (though 5 have not been calculated) reported by Gadre \textit{et al.}.\cite{Gadre2000}
 Those studies may suggest that for few water molecules (up to two), the proton should be located on the uracil molecule, whereas when a large number
 of water molecules surround the uracil, the charge is expected to be located on the water molecules. Of course, the excess proton is expected to strongly
 influence the structure of the lowest energy isomers of each species, as observed for pure water clusters, so the size at which the proton is transferred
 from uracil to water cannot be deduced from the aforementioned studies. Moreover, all those theoretical studies do not lead to the same low-energy
 structures as highlighted by Danilov and Calvo.\cite{Danilov2006, Bacchus2015} Consequently, although it is instructive from a qualitative point
 of view, the analysis of the  experimental data by S. Zamith and J.-M. l'Hermite  cannot be based on those studies. We have therefore undertaken a
 theoretical simulation of  hydrated protonated uracil clusters (H$_2$O)$_{1-7, 11, 12}$UH$^+$  to determine their lowest-energy structures to
 complete the experiments by S. Zamith and J.-M. l'Hermite at the  \textit{Laboratoire Collisions Agr\'egats R\'eactivit\'e }(LCAR). This work has
 been published in 2019 in the  \textit{The Journal of Chemical Physics}.\cite{Braud2019}

\subsection{Results and Discussion}

In the following section, section~\ref{exp_ur}, I present in details the results obtained from the CID experiments of S. Zamith and J.-M. l'Hermite
and the main concepts used to interpret the data. The following section, section~\ref{calcul_ur}, is devoted to the theoretical determination of
the low-energy isomers of  the (H$_2$O)$_{1-7,11,12}$UH$^+$ clusters. A more detailed presentation of CID experiments is also provided in 
section~\ref{exp_cid} of chapter~4, where these details are important to explicitly model CID experiments.

\subsubsection{Experimental Results} \label{exp_ur}

\textbf{Time of flight of mass spectrum(TOFMS) .}
A typical fragmentation mass spectrum obtained by colliding (H$_2$O)$_{7}$UH$^+$ with neon at a center of mass collision energy of 7.2 eV is shown in Figure \ref{mass7w}. The more intense peak on the right comes from the parent cluster (H$_2$O)$_{7}$UH$^+$, the next 7 peaks at the left of the parent peak correspond to the loss of 1-7 water molecules of parent cluster, and the next 5 peaks to the left results from the evaporation of the uracil molecule and several water molecules from parent cluster. This mass spectrum is obtained at the highest pressure explored in the present experiments. This is still true for the largest size investigated here, namely, (H$_2$O)$_{15}$UH$^+$. From the result of the fragmentation mass spectrum displayed in Figure \ref{mass7w}, it indicates multiple collisions are possible, which allows the evaporation of all water molecules. Moreover, the intensity of evaporation of water molecules is bigger than the one of evaporation of U. 
In our study, we are interested in two specific channels. Channel 1 corresponds to the loss of only neutral water molecules, whereas channel 2 corresponds to the loss of neutral uracil and one or several water molecules,
\begin{align}
\mathrm{Channel~1, ~(H_2O)_nUH^+} & \rightarrow ~ \mathrm{(H_2O)_{n-x}UH^+ + xH_2O }  \\
\mathrm{Channel~2, ~(H_2O)_nUH^+} & \rightarrow ~ \mathrm{ (H_2O)_{n-x}H^+ + xH_2O + U}
\end{align}

\figuremacro{mass7w}{Time of flight of mass spectrum obtained by colliding (H$_2$O)$_{7}$UH$^+$ with Ne
at 7.2 eV center of mass collision energy (93.5 eV in the laboratory frame).}

\textbf{Fragmentation cross section.}
The total fragmentation cross sections of clusters
(H$_2$O)$_{n-1}$UH$^+$, pure water clusters \newline (H$_2$O)$_{n=2-6}$H$^+$,\cite{Dalleska1993} and deuterated water clusters (D$_2$O)$_{n=5, 10}$H$^+$ \cite{Zamith2012} are plotted in Figure \ref{fragcrosssec} as a function of the cluster size n. Here n stands for the total number of molecules when the cluster includes uracil molecule. Different target atoms and molecules were used in these experiments: Water molecules or neon atoms in our experiments, xenon atoms in Dalleska’s experiments. These experimental data are compared to the geometrical (\textit{i.e.}, hard sphere) cross sections given by:
\begin{align}
\label{cross-section-geo}
\sigma_{geo} = \pi \left(\left[n_w\times r_w^3 + n_Ur_U^3\right]^{1/3} + r_T \right)^2
\end{align}
where $n_w$ is the number of water molecules, and $n_U$ is the number of uracil molecules ($n_U$ = 0 or 1 in the present study). $r_w$, $r_U$, and $r_T$ refer to the molecular radii of water, uracil, and the target atom or molecule, respectively. The molecular radii are deduced from macroscopic densities that gives $r_U$ = 3.2 \AA \cite{Myers2007} and $r_w$ = 1.98 \AA. The radii of rare gas target atoms are taken as their Van der Waals radii $r_{Ne}$ = 1.54 \AA and $r_{Xe}$ = 2.16 \AA.

The main differences between the curves in Figure  \ref{fragcrosssec} can be rationalized as follows: The larger the size of the target atom (or molecule) is, the bigger the fragmentation cross section will be. The experimental fragmentation cross sections of clusters (H$_2$O)$_{n-1}$H$^+$ colliding with water molecules are larger than the values obtained for collisions with Ne atoms. In the same vein, for a given number of molecules in the cluster, the cross section is larger for clusters containing uracil. The overall trend of all curves in Figure \ref{fragcrosssec} is the same: The
fragmentation cross sections increase with the size and seem to tend toward the geometrical one. 
The cross sections measured for clusters containing uracil colliding with water molecules (black squares) are of the same magnitude as the ones previously obtained for deuterated pure water clusters (green full circles) at a similar collision energy.\cite{Zamith2012} For clusters containing uracil, fragmentation cross sections are systematically larger than the one for pure water clusters by an amount of the same magnitude as the one predicted by the geometrical cross sections. For instance, the difference between red squares and blue stars, and the difference between red full line and blue dashed line has the same magnitude.

The fragmentation cross sections obtained by Dalleska
and coworkers \cite{Dalleska1993} for protonated water clusters are within our error bars for n = 5, 6 and about a factor of 2 lower for n = 3, 4. However their cross section is notably lower for (H$_2$O)$_2$H$^+$ as compared to our measurement for (H$_2$O)UH$^+$. This difference may be explained by the fact that UH$^+$ forms a weaker bond with water than H$_2$OH$^+$ does. Indeed the dissociation energy D[H$_2$OH$^+$–H$_2$O] is 1.35 eV \cite{Dalleska1993, Hansen2009} whereas the value for D[UH$^+$–H$_2$O] is estimated between 0.54 \cite{Wincel2009} and 0.73 eV. \cite{Bakker2008} The same behavior is observed for n = 3, and the dissociation energy D[(H$_2$O)$_2$H$^+$–H$_2$O] = 0.86 eV \cite{Dalleska1993, Hansen2009} is greater than the dissociation energy D[U(H$_2$O)H$^+$–H$_2$O] = 0.49 eV.\cite{Wincel2009} Hence the dissociation of water molecules is more favored in the protonated uracil cluster than in the pure water clusters.

\figuremacro{fragcrosssec}{Fragmentation cross sections of  clusters (H$_2$O)$_{n-1}$UH$^+$ at a collision energy of 7.2 eV plotted as a function of the total number n of molecules in the clusters. The experimental results and geometrical cross sections are shown for collision with H$_2$O and Ne. The results from Dalleska et al.\cite{Dalleska1993} using Xe as target atoms on pure protonated water clusters (H$_2$O)$_{2-6}$H$^+$ and from Zamith \textit{et al.} \cite{Zamith2012} using water as target molecules on deuterated water clusters (D$_2$O)$_{n=5,10}$H$^+$ are also shown. The geometrical collision cross sections of water clusters in collision with Xe atoms and water molecules are also plotted. Error bars represent one standard deviation.}

\textbf{Intermolecular fragmentation.}
Figure \ref{Uloss} displays the percentage of the fragments that have lost a neutral uracil molecule over all the fragments, plotted as a function of the number of water molecules in the parent cluster (H$_2$O)$_{n}$UH$^+$. It shows that for the cluster (H$_2$O)$_{n}$UH$^+$ with a small number of water molecules, almost no neutral uracil is evaporated. From n = 5 and more clearly from n = 6, the loss of neutral uracil molecule increases up to about 20\% for (H$_2$O)$_{9}$UH$^+$. 

\figuremacro{Uloss}{Proportion of neutral uracil molecule loss plotted as a function of the number of water molecules n in the parent cluster (H$_2$O)$_{n}$UH$^+$. Results obtained for collisions with Ne atoms at 7.2 eV center of mass collision energy.}

The fragmentation can arise from two distinct mechanisms (direct and statistical fragmentation processes) depending on the life time of the collision complex. On the one hand, if the fragmentation occurs in a very short time after collision, the dissociation is impulsive (direct). In this case, we thus assume that the nature of the collision products is partly determined by the nature of the lowest-energy isomers of parent clusters and especially by the location of the excess proton in the structure. In other words, the lowest-energy isomer of the parent cluster obviously plays a major role in determining the fragmentation channels. On the other hand, in the case of long-lived collision complexes, collision energy is transferred to the parent cluster and is redistributed among all degrees of freedom. This is a slow process, and the structures involved during the fragmentation are no longer the lowest-energy isomer, \textit{i.e.}, the structure of the cluster can undergo structural reorganizations before evaporation. Furthermore, the excess proton can also diffuse in the structure and for instance, recombine with the uracil. Then the role of the initial structure of the parent clusters is strongly reduced in determining the fragmentation channels.

In Figure \ref{Uloss}, we focus on the loss of the neutral uracil molecule in the detected fragments since it indicates where the proton lies after collision, namely, on the uracil or on a water cluster. A transition in the nature of fragmentation product is clearly seen from n = 5-6. To account for this transition, we consider that evaporation originates from a direct fragmentation process. A short discussion about the implications of possible structural rearrangement prior to dissociation, which occurs in a statistical process, will be provided in section~\ref{calcul_ur}. 

The relative proton affinities of each component of the
mixed clusters gives a first estimate of which molecule, uracil or water, is more likely to carry the positive charge prior to collisions. Experimentally, the gas phase proton affinity of uracil is bracketed to 9 $\pm$ 0.12 eV.\cite{Kurinovich2002} For the proton affinity of water molecule, an experimental value is reported at 7.31 eV \cite{Magnera1991} and a theoretical one at 7.5 eV.\cite{Cheng1998} In the work of Cheng, it shows that the proton affinity of water clusters increases with their size.\cite{Cheng1998} The proton affinities extracted from the different studies for the uracil molecule and for water clusters as a function of the number of water molecules are displayed in Figure~\ref{protonAffinity}.  

\figuremacro{protonAffinity}{The proton affinities of water clusters as a function of the number of water
molecules n, which are taken from the work of Magnera  (black circles) \cite{Magnera1991} and from the work of Cheng (blue squares).\cite{Cheng1998}  The value of the proton affinity of uracil (red dotted dashed line) is also plotted.\cite{Kurinovich2002}}

It clearly shows that the proton affinity of uracil, PA[U], is larger than the one of water monomer PA[H$_2$O]. Thus, for the mono-hydrated uracil, from the energetic point of view, the proton is on the uracil molecule and the only observed fragments are indeed protonated uracil molecules. Moreover, an experimental work \cite{Bakker2008} confirms that there is no proton transfer from the uracil to the water molecule in mono-hydrated clusters. Proton affinity of the uracil molecule is also larger than that of the water dimer, or even the trimer: PA[U] $>$ PA[(H$_2O$)$_n$], n = 2 or 3 depending on the considered data for water. This is still consistent with our experimental observation of no neutral uracil molecule loss for n = 2 and 3. However from the PA values, one would predict that the appearance of neutral uracil should occur for n $\approx$ 3-4. For instance, for n = 4, assuming a statistical fragmentation for which the energies of final products are expected to be of relevance, the channel U + (H$_2$O)$_4$H$^+$ is energetically favorable. If one now assumes a direct dissociation, where the parent protonation state remains unchanged, one also expects that neutral uracil evaporates. However, experimentally, for n = 4, no neutral uracil evaporation is observed. The loss of neutral uracil starts at n = 5 and becomes significant only at n = 6.

This analysis based on PA is however quite crude. Indeed, it assumes that the protonated uracil cluster would be composed of a uracil molecule attached to an intact water cluster. However, one expects that the hydration of uracil may be more complicated than this simple picture. Therefore, the uracil hydration is explored theoretically in the next Section, Section~\ref{calcul_ur}, in order to determine the proton location more realistically.

\subsubsection{Calculated Structures of Protonated Uracil Water Clusters} \label{calcul_ur}

As discussed in section~\ref{sec:ammoniumwater}, we have proposed a modified set of NH parameters to describe sp$^3$ nitrogen atoms. For,
sp$^2$ nitrogen atoms there is no need to modified the integral parameters as SCC-DFTB describe them rather correctly. Consequently, only the 
$D_{NH}$ parameter needs to be defined for the present calculations. Table~\ref{tab:DNH} present the binding energy of the two
(H$_2$O)U isomers represented in Figure~\ref{uracil_s} at MP2/Def2TZVP and SCC-DFTB levels of theory. Both $D_\textrm{NH}$ = 0.12 and
$D_\textrm{NH}$ = 0.14 lead to consistent binding energies. So, to be consistent with the work performed in the previous section, we
have used  $D_\textrm{NH}$ = 0.14 in the following.

 \begin{table}
  %\footnotesize
  \begin{center}
  %\begin{adjustbox}{width=1\textwidth}
 % \small
    \caption{Binding energy of two (H$_2$O)U isomers at MP2/Def2TZVP and SCC-DFTB levels of theory.}      
    \label{tab:DNH}
    \resizebox{1.0\textwidth}{!}{
    \begin{tabular}{|c|c|c|c|c|c|c|c|}
    \hline
      \textbf{isomer} & \boldm{$E_{bind_{MP2}}$} & \boldm{$E_{bind_{DFTB}}$}
      & \boldm{$E_{Re}$} &\boldm{$E_{bind_{DFTB}}$}
       & \boldm{$E_{Re}$}
      &\boldm{$E_{bind_{DFTB}}$}
      & \boldm{$E_{Re}$} \\
             &     &  \boldm{$D_{{NH}_{0.0}}$} & \boldm{$D_{{NH}_{0.0}}$} & \boldm{$D_{{NH}_{0.12}}$} & \boldm{$D_{{NH}_{0.12}}$} & \boldm{$D_{{NH}_{0.14}}$} &\boldm{$D_{{NH}_{0.14}}$} \\ 
     \hline
     \bf{a} & -8.3  &  -8.6  & -0.3 &  -9.8  &-1.5 & -10.0 &-1.7 \\
      \hline
     \bf{b} & -6.9  &  -6.6  & 0.3 &  -6.9  & 0.0 & -6.9  & 0.0 \\
      \hline
    \end{tabular}}
  %  \end{adjustbox}
  \end{center}
\end{table} 

%\figuremacro{a-b}{The structure of isomer a and b of cluster (H$_2$O)U.}

The lowest-energy isomers determined theoretically for
hydrated uracil protonated clusters (H$_2$O)$_{n=1-7, 11, 12}$UH$^+$ are shown in Figures \ref{1a-f}-\ref{12a-f}. In the experiments, clusters are produced at a temperature of about 25 K, so only a very few isomers are likely to be populated. Indeed, the clusters are produced in the canonical ensemble at the temperature $T_\mathrm c \approx$ 25 K, so only isomers for which the Boltzmann factor exp(-$\Delta E k_\mathrm{B} T_\mathrm{c}$) is larger than 10$^{-7}$ are considered here. In this formula, $\Delta E$ represents the relative energy of a considered isomer with respect to the lowest-energy one. Thus for each isomer, only the first six lowest-energy structures of U(H$_2$O)$_{n=1-7, 11, 12}$UH$^+$ obtained from the PES exploration will be discussed. 

Figure \ref{1a-f} displays the six lowest-energy isomers obtained for (H$_2$O)UH$^+$. Two (1a and 1b) of them contain the u138-like isomer of U (each one with a different orientation of the hydroxyl hydrogen). Three of them (1c, 1d, and 1e) contain the u178 isomer and 1f contains the u137\cite{Wolken2000} isomer with a reverse orientation of the hydroxyl hydrogen. From those isomers, different sites are possible for the water molecule attachment which leads to variety of isomers even for such small size system. To the best of our knowledge, (H$_2$O)UH$^+$ is the most studied protonated uracil water cluster and our results are consistent with previous
published studies. Indeed, Pedersen and co-workers \cite{Pedersen2014} conducted ultraviolet action spectroscopy on (H$_2$O)UH$^+$ and discussed their measurements in the light of theoretical calculations performed on two isomers: ur138w8 (1a in the present study) and ur178w7 (1c).\cite{Pedersen2014} Their energy ordering at 0 K is the same whatever the computational method they used: B3LYP/6-311++G(3df,2p), M06-2X/6-311++G(3df,2p), MP2/6-311++G(3df,2p), CCSD(T)/6-311++G(3df,2p), and CCSD(T)/augcc-pVTZ and is similar to what we obtain. Similarly, Bakker and co-workers\cite{Bakker2008} considered three isomers: U(DK)H$^+_W$ (1a), U(KE)H$^+_{Wa}$ (1c), and U(KE)H$^+_{Wb}$ (1e) at the B3LYP/6-311++G(3df,2p) level of theory and obtained the same energy ordering as we do. Our methodology has thus allowed us to retrieve those isomers and to locate two new low-energy structures (1b and 1d). 1f is too high in energy to be considered in low-temperature experiments that are in the same range of relative energies but have never been discussed. To ensure that they are not artificially favored in our computational method, calculations were also performed at the B3LYP/6-311++G(3df,2p) level of theory. The results are presented in Figure \ref{1a-f-b3lyp}, which are consistent with the MP2/Def2TZVP ones. This makes us confident in the ability of the present methodology to locate meaningful low energy structures. Importantly, no isomer with the proton on the water molecule was obtained, neither at the DFTB or MP2 levels.

\figuremacrob{1a-f}{Lowest-energy structures of (H$_2$O)UH$^+$ obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.}

\figuremacrob{1a-f-b3lyp}{Lowest-energy structures of (H$_2$O)UH$^+$ obtained at the B3LYP/6-311++G(3df,2p) level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. The corresponding values with ZPVE corrections are provided in brackets. Important hydrogen-bond distances are indicated in bold and are given in \AA.} 

Figures \ref{2a-f} and \ref{3a-f} display the first six lowest-energy isomers obtained for (H$_2$O)$_2$UH$^+$ and (H$_2$O)$_3$UH$^+$, respectively. For (H$_2$O)$_2$UH$^+$, the lowest energy structure, 2a contains the u138 isomer of uracil. 2b, 2d, and 2e contain u178 and 2c contains u138 with reverse orientation of the hydroxyl hydrogen. 2f contains u178 with reverse orientation of the hydroxyl hydrogen. This demonstrates that, similarly to (H$_2$O)UH$^+$, a diversity of uracil isomers are present in the low-energy structures of (H$_2$O)$_2$UH$^+$ which makes an exhaustive exploration of its PES more difficult. The same behavior is observed for (H$_2$O)$_3$UH$^+$. The configuration of u138 does not allow for the formation of a water dimer which leads to two unbound water molecules in 2a. By contrast, a water-water hydrogen bond is observed for 2b and 2c. The existence of a water dimer was not encountered in the low-energy isomers of the unprotonated (H$_2$O)$_2$U species due to the absence of the hydroxyl group on U. It is worth pointing out that 2a, 2b, 2c, and 2d are very close in energy which makes their exact energy ordering difficult to determine. However, no isomer displaying an unprotonated uracil in the low-energy isomers of (H$_2$O)$_2$UH$^+$ was located. The lowest-energy structure of (H$_2$O)$_3$UH$^+$, 3a, is characterized by two water-water hydrogen bond that forms a linear water trimer. Higher energy isomers display only one (3b, 3d, and 3e) or zero (3c and 3f) water-water bond (see Figure \ref{3a-f}). Similarly to (H$_2$O)$_2$UH$^+$, no isomer displaying an unprotonated uracil was located for (H$_2$O)$_3$UH$^+$.

\figuremacrob{2a-f}{Lowest-energy structures of (H$_2$O)$_2$UH$^+$ obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.}

\figuremacrob{3a-f}{(H$_2$O)$_3$UH$^+$ lowest-energy structures obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.}

The first six lowest-energy isomers obtained for (H$_2$O)$_4$UH$^+$ and (H$_2$O)$_5$UH$^+$ are displayed in Figures \ref{4a-f} and \ref{5a-f}, which constitute a transition in the behavior of the proton. Indeed, in (H$_2$O)$_4$UH$^+$, two kind of low-lying energy structures appear: (i) structures composed of UH$^+$, one water trimer, and one isolated water molecule (4b, 4d, 4e, and 4f); (ii) structures composed of U and a protonated water tetramer (4a and 4c). In the latter case, the hydronium ion is always bounded to an uracil oxygen atom. The U–H$_2$OH$^+$ bond is always rather strong as compared to U–H$_2$O bonds as highlighted by the corresponding short oxygen-hydrogen distance. Furthermore, speaking of distances, the difference between the U–H$_2$OH$^+$ and UH$^+$–H$_2$O forms is rather fuzzy and might be sensitive to computational parameters and also to quantum fluctuations of the hydrogen. This suggests that collision with (H$_2$O)$_4$UH$^+$ is more likely to induce evaporation of H$_2$O rather than H$_2$OH$^+$ or a protonated water cluster. The picture is significantly different in (H$_2$O)$_5$UH$^+$ where the lowest-energy structure displays a hydronium ion separated by one water molecule from U. Such structures do not appear in (H$_2$O)$_4$UH$^+$ due to the limited number of water molecules available to separate H$_2$OH$^+$ from U. Such separation suggests that, if considering a direct dissociation process, evaporation of neutral uracil can now occurs in agreement with the experimental observations (see discussion above). One can see that 5b, which is only 0.3 kcal.mol$^{-1}$ higher in energy than 5a, still displays a U–H$_2$OH$^+$ link. This is in line with the low amount of neutral uracil that is evaporated in the experiment (see Figure \ref{Uloss}).

\figuremacrob{4a-f}{Lowest-energy structures of (H$_2$O)$_4$UH$^+$ obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.}

\figuremacrob{5a-f}{Lowest-energy structures of (H$_2$O)$_5$UH$^+$ obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.}

Figures \ref{6a-f} and \ref{7a-f} display the first six lowest-energy isomers obtained for (H$_2$O)$_6$UH$^+$ and (H$_2$O)$_7$UH$^+$. Similarly to (H$_2$O)$_5$UH$^+$, the first lowest-energy structure, 6a and 7a, we located for both species (H$_2$O)$_6$UH$^+$ and (H$_2$O)$_7$UH$^+$ has the excess proton on a water molecule that is separated by one water molecule from the uracil. This appears to be common to the clusters with at least 5 water molecules. This is also observed for higher-energy isomers (6c, 6d, 7c, 7e, and 7f). Other characteristics of the proton are also observed: proton in a similar Zundel form \cite{Zundel1968} bounded to the uracil (6b, 6e, and 7d) or H$_2$OH$^+$ still bounded to uracil (6f and 7b).

Finally, due to the neutral uracil loss proportion starts to decrease from n=9 (see Figure \ref{Uloss}), which attracted us to perform the optimization of big cluster (H$_2$O)$_{11, 12}$UH$^+$ as examples to explore why it has this change. The first six low-lying energy isomers obtained for cluster (H$_2$O)$_{11, 12}$UH$^+$ are shown in Figures \ref{11a-f} and \ref{12a-f}. 

In all isomers (11a to 11f) of cluster (H$_2$O)$_{11}$UH$^+$, the excess is on the water cluster and was separated by water molecule from uracil.  
For 12a, 12b, 12c, and 12d, it is obvious that the excess proton is not directly bounded to the uracil. The uracil in 12a and 12d belongs to the di-keto form (there is a hydrogen atom on each nitrogen of uracil), and the excess proton was separated by one water molecule from uracil, additionally, the uracil is surrounded by the water cluster, all of these may lead the excess proton to go to the near oxygen atom of uracil. For 12b, the excess proton is on the water cluster and is very far from the uracil. For 12c, the excess proton was separately by one water molecule from uracil. For isomers 12e and 12f, the excess proton is between the uracil and a water molecule. The uracil is surrounded by the water cluster in 12e but it is not in 12f. Of course, for (H$_2$O)$_{11}$UH$^+$, (H$_2$O)$_{12}$UH$^+$, (H$_2$O)$_6$UH$^+$ and (H$_2$O)$_7$UH$^+$ and also (H$_2$O)$_4$UH$^+$ and (H$_2$O)$_5$UH$^+$, the amount of low-energy isomers is expected to be very large and we do not intended to find them all. Furthermore, due to the limited number of MP2 geometry optimization we performed, there is few chances that we located the global energy minima for (H$_2$O)$_6$UH$^+$, (H$_2$O)$_7$UH$^+$, (H$_2$O)$_{11}$UH$^+$ and (H$_2$O)$_{12}$UH$^+$. However, the general picture we are able to draw from the present discussed structures fully supports the experimental results: from (H$_2$O)$_5$UH$^+$, it exists low-energy structures populated at very low temperature in which the excess proton is not directly bound to the uracil molecule. Upon fragmentation, this allows the proton to remain bounded to the water molecules.

\figuremacrob{6a-f}{Lowest-energy structures of (H$_2$O)$_6$UH$^+$ obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.}

\figuremacrob{7a-f}{Lowest-energy structures of (H$_2$O)$_7$UH$^+$ obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.}

\figuremacrob{11a-f}{Lowest-energy structures of (H$_2$O)$_{11}$UH$^+$ obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.} \\

\figuremacrob{12a-f}{Lowest-energy structures of (H$_2$O)$_{12}$UH$^+$ obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.}

All the aforementioned low-lying energy structures are relevant to describe the \newline (H$_2$O)$_{n=1-7, 11, 12}$UH$^+$ species at low temperature and to understand the relation between the parent cluster size and the amount of evaporated neutral uracil in the case of direct dissociation. However, as already stated, one has to keep in mind that upon collision statistical dissociation can also occur. In that case, structural rearrangements are expected to occur which are important to understand each individual mass spectra of the (H$_2$O)$_{n=1-15}$UH$^+$ clusters and the origin of each collision product. For instance, the fragment UH$^+$ is detected for all cluster sizes in experiment. This means that for the largest sizes, for which we have shown from the calculation that the proton is located away from the uracil, proton transfer does occur prior to dissociation. One possible scenario is that after collision, water molecules sequentially evaporates. When the number of water molecules is small enough, the proton affinity of uracil gets larger than the one of the remaining attached water cluster. Proton transfer is then likely and therefore protonated uracil can be obtained at the end.
 
If one turns to the neutral uracil evaporation channel, it appears that the smaller clusters H$_2$OH$^+$ and (H$_2$O)$_2$H$^+$ are not present in the time of flight mass spectra. This absence might have two origins. First, the dissociation energies of the protonated water monomers and dimers are substantially higher than larger sizes, and they are therefore less prone to evaporation. Second, as already mentioned, for such small sizes, the proton affinity of uracil gets larger than the one of the water dimer or trimer and proton transfer to the uracil is likely to occur. 

In order to confirm the above scenarios, simulations
and/or evaporation rate calculation would have to be conducted to describe the fragmentation channels in details. MD simulations of protonated uracil have already been performed by Spezia and co-workers to understand the collision-induced dissociation.\cite{Molina2015, Molina2016} Although, in the present case, the initial position of the excess proton appears as a key parameter to explain the evaporation of neutral uracil, such MD simulations could be additionally conducted to provide a clearer picture on the various evaporation pathways, which will be shown in section \ref{sec:collisionwUH}.

\subsection{Conclusions on (H$_2$O)$_{n}$UH$^+$ clusters}

The work in this section presents the collision-induced dissociation of hydrated protonated uracil (H$_2$O)$_{n=1-15}$UH$^+$ clusters and their experimental
absolute fragmentation cross sections. The experiments demonstrate that the evaporation channels evolve with size: Below n = 5, the observed charged fragments
always contain the uracil molecule, whereas from n = 5, the loss of a neutral uracil molecule becomes significant. To understand this transition, I conducted an
exhaustive exploration of the potential energy surface of (H$_2$O)$_{n=1-7, 11, 12}$UH$^+$ clusters combining a rough exploration at the SCC-DFTB level with
fine geometry optimizations at the MP2 level of theory. Those calculations show that below n = 5, the excess proton is either on the uracil or on a water molecule
directly bound to uracil, \textit{i.e.}, forming a strongly bound U–H$_2$OH$^+$ complex. From n = 5 and above, clusters contain enough water molecules to allow
for a net separation between uracil and the excess proton: The latter is often found bound to a water molecule which is separated from uracil by at least one other
water molecule. Upon direct dissociation, the excess proton and the uracil can thus belong to different fragments. This study demonstrates that combination of
collision-induced dissociation experiments and theoretical calculation allows to probe the solvation and protonation properties of organic molecules such as nucleobases.
This is a step toward a better understanding of the role of water in the chemistry of in vivo DNA and RNA bases. However, the knowledge of the lowest-energy isomers
of the species involved in CID experiments is not enough to understand all the collision process. To get a deeper understanding of the collision mechanism, an explicit
modelling of the collision is needed?. This question is addressed in the next chapter of this thesis. 


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