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thesis/2_Introduction/introduction.tex
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@ -1306,7 +1306,7 @@ he dynamics from the direction of time. The other property is the conservation
%Hamiltonian function
total energy over time.
Because of the discretization of trajectories, this conservation can not be insured. A stable integration algorithm must
impose this conservation for long enough time steps ($\delta t$) to allow for sufficiently long simulation times. The VV algorithm
impose this conservation for long enough time steps ($\delta t$) to allow for sufficiently long simulation time. The VV algorithm
is able to do this due to its sufficient numerical stability.
According to the conservation of energy, the natural ensemble corresponding to such dynamics is the microcanonical ensemble
($N, V, E$).\cite{Ray1981, Ray1999} $N$ is the number of particles. $V$ denotes the volume and $E$ is the energy of the system
@ -1328,7 +1328,7 @@ barriers are too high to be crossed. In this case, if $E_b$ refers to the energy
the simulation, one can consider that $E_b >> k_BT$ where $k_B$ is the Boltzmann constant. At intermediate temperature,
the possibility of crossing the energy barriers during a simulation increases, but this can not guarantee the PES to
be explored exhaustively. For high temperatures, one has a high probability to cross the energy barriers whereas the
bottoms of the wells can not be explored comprehensively. Therefore, it is not possible to both cross the energy barriers
bottoms of the wells can not be explored comprehensively. Therefore, it is very difficult to both cross the energy barriers
and thoroughly explore the bottom of the wells using a unique MD simulation at a given temperature.
Many methods have been proposed to solve this question and are referred to as \textbf{enhanced sampling methods}
They are classified into two groups: \textbf{biased methods} and \textbf{non-biased methods}.
@ -1363,12 +1363,8 @@ $N$ replicas ($C_i, i = 1, 2, ..., N$) of the same system are simulated in paral
The time evolution of each replica is independent with each other but exchanges of configurations between adjacent
replicas $C_i$ and $C_j$, where $T_i < T_j$ and $i = j - 1$ are permitted at regular time intervals.
The choice of the extreme temperatures $T_1$ and $T_N$ is very important for the algorithm to be optimal.
The lowest temperature ($T_1$) should be the one at which usual simulations are blocked and the highest
temperature ($T_N$) should be chosen so that all significant energy barriers can be overcome during the
simulation. Moreover, the temperatures between $T_1$ and $T_N$ must be chosen to lead to sufficient overlap
between the density of states of the adjacent replicas. Indeed, if this overlap is too small, the probability of
exchange is very low, which makes the PTMD simulations inefficient and leads to a bad exploration of the PES.
In contrast, if the overlap is too large, a large amount of redundant information will be produced, which will
The lowest temperature ($T_1$) should be the one at which usual simulations are blocked in basins and the highest temperature ($T_N$) should be chosen so that all significant energy barriers can be overcome during the simulation. Moreover, the temperatures between $T_1$ and $T_N$ must be chosen to lead to sufficient overlap between the density of states of the adjacent replicas. Indeed, if this overlap is too small, the probability of exchange is very low, which makes the PTMD simulations inefficient and leads to a bad exploration of the PES.
In contrast, if the overlap is too large, a significant amount of redundant information will be produced, which will
cost unnecessary computational resources. Configurations between two neighbouring replicas at different
$T$ are exchanged based on the Metropolis–Hastings criterion with probability:
\begin{align}
@ -1400,26 +1396,26 @@ all particles can be renormalized as follows:
%=======
%>>>>>>> 92023a10c3aa8b7dc4ace43987c1d571fb99a738
\textbf{Global Optimization} refers to the determination of the lowest energy point on a PES, \textit{i.e.} the global minimum. As this latter usually
\textbf{Global optimization} refers to the determination of the lowest energy point on a PES, \textit{i.e.} the global minimum. As this latter usually
includes a large number of stationary points, it is not straightforward to find the global minimum. Local optimization methods do not
make it possible to cross the energy barriers between local minima. Therefore, a global optimization scheme such as MD or Monte Carlo
simulations is needed to perform a more exhaustive exploration of the PES to get to the lowest energy minimum.
There exits a vast amount of methods to perform global optimization and each one has its strength and weaknesses.
For instance, the Basin-Hopping method is a particular useful global optimization technique in high-dimensional landscapes
There exits a vast amount of methods to perform global optimization and each one has its strength and weaknesses. The ergodicity problem appears in all of these global optimization methods. In principle, one can only be sure of having found the real global minimum after an infinite number of iterations.
The Basin-Hopping method is a particular useful global optimization technique in high-dimensional landscapes
that iterates by performing a random perturbation of coordinates, making a local optimization, and rejecting or accepting
new coordinates based on a minimized function value.\cite{Wales1997, Wales1999}
Genetic algorithms are also among the most used methods to find a global minimum.\cite{Hartke1993, Unger1993, Sivanandam2008, Toledo2014}
A genetic algorithm is inspired by the process of natural selection. Genetic algorithms are usually applied to generate high-quality solutions
of optimization. The ergodicity problem appears in all of these global optimization methods. In principle, one can only be sure of having
found the real global minimum after an infinite number of iterations.
of optimization.
In order to avoid ergodicity problems, it is interesting to combine global and local optimization methods. A very popular combination
is the simulated annealing method combined with local optimizations. The PES of molecular aggregates display many degrees of freedom
and contains a large number of low-energy isomers. The PTMD algorithm coupled with a great number of local optimizations is a good
In order to avoid ergodicity problems, an interesting tool is to combine global and local optimization methods. A very popular combination
is the simulated annealing method combined with local optimizations.
%The PES of molecular aggregates display many degrees of freedom and contains a large number of low-energy isomers.
The PTMD algorithm coupled with a great number of local optimizations is a good
choice to search for low-energy structures in this kind of system. Local optimizations are performed many times from initial conditions
structures which are extracted from all PTMD trajectories, whether it be low or high temperature, in order to maximize sampling. \textbf{This
approach, in combination with SCC-DFTB, has been conducted along this thesis to perform global Optimization.}
approach, in combination with SCC-DFTB, has been conducted along this thesis to perform global optimization.}

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thesis/3/structure_stability.aux
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@ -7,10 +7,10 @@
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thesis/3/structure_stability.tex
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@ -10,14 +10,13 @@
\graphicspath{{3/figures/EPS/}{3/figures/}}
\fi
\chapter{Exploration of Structural and Energetic Properties} \label{chap:structure}
\chapter{Investigation 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
combination of the \textbf{self-consistent-charge density functional based tight-binding} (SCC-DFTB) method for the efficient calculation of the PES and the \textbf{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, an improve set of parameters is proposed to describe sp$^3$ nitrogen
@ -60,17 +59,17 @@ To identify low-energy isomers of (H$_2$O)$_{1-3}${NH$_4$}$^+$ and (H$_2$O)$_{1-
selected from each replicas and further optimized at the SCC-DFTB level, which produced 4848 optimized geometries per cluster. For
(H$_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 J. Douady \textit{et al.}.\cite{Douady2009} The five lowest-energy
optimization process was the lowest-energy structure reported by J. 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.
40 replicas with temperatures ranging 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, I 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
the initial geometries.\cite{Wolken2000, Pedersen2014} 600 geometries per temperature were periodically 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$^+$,
@ -87,21 +86,19 @@ level. See below for the details on MP2/Def2TZVP calculations.
\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 combinationwith an all electron Def2TZVP basis-set.\cite{Weigend2005, Weigend2006} All calculations used a a tight criteria for geometry
Some low-energy isomers obtained at the SCC-DFTB level were further optimized at the MP2 level of theory in combinationwith an all electron Def2TZVP basis-set.\cite{Weigend2005, Weigend2006} All calculations used 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
In section~\ref{sec:ammoniumwater}, relative energy with respect to the lowest-energy isomer of each
cluster will be shown. 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, binding energies are also reported.
interactions and to assess the accuracy of the SCC-DFTB method, binding energies are also calculated.
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))
energy between the water cluster as a whole and the impurity, {NH$_4$}$^+$ or NH$_3$ that the corresponding binding energy is denoted as $E_{bind.}^{whole}$, while the second one considers the binding energy between all the molecules of the cluster corresponding to $E_{bind.}^{sep.}$. 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
@ -115,17 +112,12 @@ of clusters (H$_2$O)$_{2-7, 11, 12}$UH$^+$ are discussed in section~\ref{structu
\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
To classify 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 reported (H$_2$O)$_{n}${NH$_4$}$^+$ and (H$_2$O)$_{n}${NH$_3$} isomers, respectively, obtained at the SCC-DFTB level. 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.
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}
@ -145,8 +137,7 @@ also found that even a small amount of atmospherically relevant ammonia can incr
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 M. D. 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} P. Hvelplund \textit{et al.} later reported a combined experimental and theoretical study devoted to protonated mixed ammonia/water
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 similar structures.\cite{Hulthe1997} P. 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,
@ -180,7 +171,7 @@ clusters is then performed which allows to report a number of low-energy isomers
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}
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}
@ -190,9 +181,9 @@ A full paper devoted to this work is in preparation.
In order to define the best SCC-DFTB parameter to model ammonia and ammonium water clusters, I 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
corrections have provided satisfactory results, 1.16/0.12 and 1.28/0.14. 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 1.28/0.14 and SCC-DFTB
1.16/0.12 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.
@ -200,7 +191,7 @@ ammonium/ammonia was shifted along the N--O vector, all other geometrical parame
\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.}
original SCC-DFTB (plain red), SCC-DFTB (1.28/0.14) (dotted red) and SCC-DFTB (1.16/0.12) (dashed red) levels of theory.}
\label{fig:E_nh4}
\end{figure}
@ -208,22 +199,22 @@ ammonium/ammonia was shifted along the N--O vector, all other geometrical parame
\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.}
original SCC-DFTB (plain red), SCC-DFTB (1.28/0.14) (dotted red) and SCC-DFTB (1.16/0.12) (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 -25.57 and -21,07~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 0.47~kcal.mol$^{-1}$. The SCC-DFTB 0.14/1.28 curve is also very close with a difference in binding energy only 0.16~kcal.mol$^{-1}$ higher. It is
binding energies vary between -25.57 and -21,07~kcal.mol$^{-1}$ at the original SCC-DFTB and SCC-DFTB 1.28/0.14 levels, respectively. The binding
energy obtained at the SCC-DFTB 1.16/0.12 level is the closest to that obtained at MP2/Def2TZVP level with BSSE correction with a binding energy difference of
only 0.47~kcal.mol$^{-1}$. The SCC-DFTB 1.28/0.14 curve is also very close with a difference in binding energy only 0.16~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 (2.64~\AA) than with corrections (2.73~\AA). 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 -3.82 to -7,39~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 0.12/1.16 level is the closest to that obtained at MP2/Def2TZVP level with BSSE correction with a binding energy
difference of only 0.01~kcal.mol$^{-1}$. The SCC-DFTB 0.14/1.28 curve is also rather close with a difference in binding energy only 1.3~kcal.mol$^{-1}$
vary from -3.82 to -7.39~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 1.16/0.12 level is the closest to that obtained at MP2/Def2TZVP level with BSSE correction with a binding energy
difference of only 0.01~kcal.mol$^{-1}$. The SCC-DFTB 1.28/0.14 curve is also rather close with a difference in binding energy of only 1.3~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 well reproduced by the corrected potentials. In contrast to (H$_2$O){NH$_4$}$^+$, the shape of the curves for (H$_2$O){NH$_3$} obtained
at the SCC-DFTB level differs significantly from those obtained at MP2 level. Vibrational frequencies calculated at the SCC-DFTB level for this systems
@ -232,7 +223,7 @@ are therefore expected to be inacurate. It is worth mentioning that the large di
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
(H$_2$O){NH$_4$}$^+$ dimer. Figure~\ref{dimers} compares the structure obtained from geometry optimization at the SCC-DFTB 1.28/0.14
and original SCC-DFTB levels. The N-H covalent bond involved in the hydrogen bond is longer with the original potential while the N---O
distance is smaller by 0.14~\AA. 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
@ -244,34 +235,34 @@ realistic molecular dynamics simulation as it leads to a spurious deprotonation.
\begin{figure}[h!]
\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.}
\caption{Structure of (H$_2$O){NH$_4$}$^+$ obtained from geometry optimization at the SCC-DFTB 1.28/0.14 (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 0.12/1.16 better describe both (H$_2$O){NH$_3$} and
Figures~\ref{fig:E_nh4} and ~\ref{fig:E_nh3} show that SCC-DFTB 1.16/0.12 better describe both (H$_2$O){NH$_3$} and
(H$_2$O){NH$_4$}$^+$ dissociation curves. Furthermore, as (H$_2$O){NH$_3$} is characterized by a much lower
binding energy than (H$_2$O){NH$_4$}$^+$, an error of the order of $\sim$1.0~kcal.mol$^{-1}$ is more likely to play
a significant role for ammonia than ammonium containing species. All the following discussion therefore involve the
SCC-DFTB 0.12/1.16 potential.
SCC-DFTB 1.16/0.12 potential.
\subsubsection{Small Species: (H$_2$O)$_{1-3}${NH$_4$}$^+$ and (H$_2$O)$_{1-3}${NH$_3$}}
As a first test case for the application of the SCC-DFTB 0.14/1.28 potential is the study of small ammonium and ammonia water clusters:
As a first test case for the application of the SCC-DFTB 1.28/0.14 potential is the study of small ammonium and ammonia water clusters:
(H$_2$O)$_{1-3}${NH$_4$}$^+$ and (H$_2$O)$_{1-3}${NH$_3$}. Due to the limited number of low-energy isomers for these species, we
only consider the lowest-energy isomer of (H$_2$O)$_{1-2}${NH$_4$}$^+$ and (H$_2$O)$_{1-3}${NH$_3$} and the two lowest-energy
isomers for (H$_2$O)$_{3}${NH$_4$}$^+$.
As displayed in Figure~\ref{fig:nh3-nh4-1w} and\ref{fig:nh3-nh4-2-3w}, the reported low-energy isomers 1-a, 1$^\prime$-a, 2-a, 2$^\prime$-a, 3-a, 3-b, and
3$^\prime$ display a structure very similar to those obtained at the MP2/Def2TZVP level (1-a$^*$, 1$^\prime$-a$^*$, 2-a$^*$, 2$^\prime$-a$^*$, 3-a$^*$,
3$^\prime$ display structures very similar to those obtained at the MP2/Def2TZVP level (1-a$^*$, 1$^\prime$-a$^*$, 2-a$^*$, 2$^\prime$-a$^*$, 3-a$^*$,
3-b$^*$ and 3$^\prime$-a$^*$). Indeed, although differences in bond lengths are observed, they are rather small.
In terms of energetics,
From en energetic point of view, it is interesting to first look at the relative energy between the two reported isomers of
From an energetic point of view, it is interesting to first look at the relative energy between the two reported isomers of
(H$_2$O)$_{3}${NH$_4$}$^+$. Isomer 3-b is 2.12~kcal·mol\textsuperscript{-1} higher than 3-a at the SCC-DFTB level.
At the MP2/Def2TZVP level, 3-b is 0.30~kcal·mol\textsuperscript{-1} lower than 3-a when ZPVE is not considered while
it is 1.21 kcal·mol\textsuperscript{-1} higher when it is considered. In comparison, in the experimental results by
H. Chang and co-workers, 3-a is more stable than 3-b.\cite{Wang1998, Jiang1999} The authors also complemented their
measurements by theoretical calculations that show that at the B3LYP/6-31+G(d) level, 3-a is higher than 3-b but. In
measurements by theoretical calculations which show that at the B3LYP/6-31+G(d) level, 3-a is higher than 3-b. In
contrast, at the MP2/6-31+G(d) level corrected with ZPVE, the energy of 3-a is lower than that of 3-b while it is inverted
if ZPVE is taken into account.\cite{Wang1998, Jiang1999} Additionally, F. Spiegelman and co-workers, conducted a
global Monte Carlo optimizations with an intermolecular polarizable potential that lead to 3-a as lowest-energy isomer.\cite{Douady2008}
@ -326,8 +317,8 @@ has an accuracy close to other \textit{ab initio} methods which confirms its app
As listed in Table~\ref{reBindE-small}, the relative binding energies $\Delta E_{bind.}^{whole}$ or $\Delta E_{bind.}^{sep.}$ of
(H$_2$O){NH$_4$}$^+$ and (H$_2$O){NH$_3$} are 1.21 and -1.17 kcal·mol\textsuperscript{-1}, respectively, which again
highlights that SCC-DFTB is in agreement with MP2/Def2TZVP. For (H$_2$O){NH$_3$}, the negative value
show that MP2/Def2TZVP binding energy is smaller than the SCC-DFTB value. This is inverse to what is shown in Figure~\ref{fig:E_nh3}
and results from structural reorganization after optimization. All other values of Table~\ref{reBindE-small} are equal of smaller than these
shows that MP2/Def2TZVP binding energy is smaller than the SCC-DFTB value. This is inverse to what is shown in Figure~\ref{fig:E_nh3}
and results from structural reorganization after optimization. All other values of Table~\ref{reBindE-small} are equal or smaller than these
values, whether considering $\Delta E_{bind.}^{whole}$ or $\Delta E_{bind.}^{sep.}$, which again demonstrates that the presently proposed
SCC-DFTB potential provides results in line with reference MP2/Def2TZVP calculations.
@ -344,7 +335,7 @@ The five lowest-energy isomers of (H$_2$O)$_{4}${NH$_4$}$^+$ are depicted in Fig
from the global SCC-DFTB optimization and also the lowest-energy configuration after optimization at MP2/Def2TZVP level with ZPVE
corrections. This result is consistent with previous computational studies\cite{Wang1998, Jiang1999, Douady2008, Lee2004, Pickard2005} and
the experimental studies by H. Chang and co-workers.\cite{Chang1998, Wang1998} Isomer 4-a displays four hydrogen bonds around the ionic
center which lead to no dangling N-H bonds. Other isomers of comparable stability are displayed in Figure~\ref{fig:nh4-4-6w}
center which lead to no dangling N-H bonds. Other isomers of comparable stability are displayed in Figure~\ref{fig:nh4-4-6w}.
The energy ordering of 4-a to 4-e at SCC-DFTB level is consistent with that at MP2/Def2TZVP level with ZPVE correction, although they
are slightly higher by$\sim$2.0 kcal.mol$^{-1}$. Isomer 4-c was not reported in H. Chang’s study,\cite{Jiang1999} and the corresponding
energy ordering of the five lowest-energy isomers was the same as ours which certainly results from the use of a different basis set.
@ -413,11 +404,11 @@ For cluster (H$_2$O)$_{5}${NH$_4$}$^+$, the five low-energy isomers are illustra
When all the water molecules are considered as a whole part, the obtained binding energy has a deviation due to the interaction of water molecules. As listed in Table \ref{reBindE}, for isomers 5-a to 5-e, the relative binding energy $\Delta E_{bind.}^{whole}$ are -1.62, 0.72, 0.69, -1.08 and -2.08 kcal·mol\textsuperscript{-1} and $\Delta E_{bind.}^{sep.}$ are -0.56, 0.48, 0.55, -0.78 and 0.88 kcal·mol\textsuperscript{-1}, respectively. The $\Delta E_{bind.}^{whole}$ is bigger than corresponding $\Delta E_{bind.}^{sep.}$, which indicates it is better to calculate the binding energy with considering the water molecules separately. The $\Delta E_{bind.}^{sep.}$ is less than 1.00 kcal·mol\textsuperscript{-1} for the five low-energy isomers of cluster (H$_2$O)$_{5}${NH$_4$}$^+$, so the SCC-DFTB method is good enough compared to MP2/Def2TZVP with BSSE correction for cluster (H$_2$O)$_{5}${NH$_4$}$^+$.
For cluster (H$_2$O)$_{6}${NH$_4$}$^+$, no N-H bond is exposed in the five low-energy isomers displayed in Figure \ref{fig:nh4-4-6w}. 6-a is the first low-energy isomer at SCC-DFTB level, which is a symmetric double-ring species connected together by eight hydrogen bonds making it a robust structure. 6-a is also the first low-energy isomer obtained using the Monte Carlo optimizations with the intermolecular polarizable potential.\cite{Douady2008} 6-d is the first low-energy isomer at MP2/Def2TZVP level with ZPVE correction but it is only 0.22 kcal·mol\textsuperscript{-1} lower than 6-a. In Shields’s results, 6-d is also the first low-energy isomer at MP2/aug-cc-pVDZ level.\cite{Morrell2010} In H. Chang’s study, 6-b with a three-coordinated H2O molecule is the first low-energy isomer for cluster (H$_2$O)$_{6}${NH$_4$}$^+$ at B3LYP/6-31+G(d) level.\cite{Wang1998} 6-b is also the first low-energy isomer at B3LYP/6-31++G(d,p) level including the harmonic ZPE contribution.\cite{Douady2008} The energy of 6-b is only 0.14 kcal·mol\textsuperscript{-1} higher than that of 6-a at MP2/Def2TZVP level with ZPVE correction. The energies of 6-a, 6-b and 6-d are very close at both SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels, which implies it is easy to have a transformation among 6-a, 6-b and 6-d. It shows SCC-DFTB is good to find the low-energy isomers of cluster (H$_2$O)$_{6}${NH$_4$}$^+$ compared to MP2 and B3LYP methods.
For cluster (H$_2$O)$_{6}${NH$_4$}$^+$, no N-H bond is exposed in the five low-energy isomers displayed in Figure \ref{fig:nh4-4-6w}. 6-a is the first low-energy isomer at SCC-DFTB level, which is a symmetric double-ring species connected together by eight hydrogen bonds making it a robust structure. 6-a is also the first low-energy isomer obtained using the Monte Carlo optimizations with the intermolecular polarizable potential.\cite{Douady2008} 6-d is the first low-energy isomer at MP2/Def2TZVP level with ZPVE correction but it is only 0.22 kcal·mol\textsuperscript{-1} lower than 6-a. In Shields’s results, 6-d is also the first low-energy isomer at MP2/aug-cc-pVDZ level.\cite{Morrell2010} In H. Chang’s study, 6-b with a three-coordinated H$_2$O molecule is the first low-energy isomer for cluster (H$_2$O)$_{6}${NH$_4$}$^+$ at B3LYP/6-31+G(d) level.\cite{Wang1998} 6-b is also the first low-energy isomer at B3LYP/6-31++G(d,p) level including the harmonic ZPE contribution.\cite{Douady2008} The energy of 6-b is only 0.14 kcal·mol\textsuperscript{-1} higher than that of 6-a at MP2/Def2TZVP level with ZPVE correction. The energies of 6-a, 6-b and 6-d are very close at both SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels, which implies it is easy to have a transformation among 6-a, 6-b and 6-d. It shows SCC-DFTB is good to find the low-energy isomers of cluster (H$_2$O)$_{6}${NH$_4$}$^+$ compared to MP2 and B3LYP methods.
As shown in Table \ref{reBindE}, for isomers 6-a to 6-e, the relative binding energy $\Delta E_{bind.}^{whole}$ are -1.71, -1.14, -2.06, -2.90 and -1.18 kcal·mol\textsuperscript{-1} and the $\Delta E_{bind.}^{sep.}$ are -0.38, -0.76, 0.27, -1.06 and -0.60 kcal·mol\textsuperscript{-1}, respectively. It indicates the binding energy are very close at SCC-DFTB and MP2/Def2TZVP with BSSE correction levels when water molecules are calculated separately. The $\Delta E_{bind.}^{whole}$ is bigger than corresponding $\Delta E_{bind.}^{sep.}$ because of the interaction of water molecules when all the water molecules are considered as a whole part.
As shown in Table \ref{reBindE}, for isomers 6-a to 6-e, the relative binding energy $\Delta E_{bind.}^{whole}$ are -1.71, -1.14, -2.06, -2.90 and -1.18 kcal·mol\textsuperscript{-1} and the $\Delta E_{bind.}^{sep.}$ are -0.38, -0.76, 0.27, -1.06 and -0.60 kcal·mol\textsuperscript{-1}, respectively. It indicates the binding energies are very close at SCC-DFTB and MP2/Def2TZVP with BSSE correction levels when water molecules are calculated separately. The $\Delta E_{bind.}^{whole}$ is bigger than corresponding $\Delta E_{bind.}^{sep.}$ because of the interaction of water molecules when all the water molecules are considered as a whole part.
For cluster (H$_2$O)$_{7}${NH$_4$}$^+$, the five low-energy isomers are shown in Figure \ref{fig:nh4-7-10w}. The ion core {NH$_4$}$^+$ has a complete solvation shell in isomers 7-a to 7-e. 7-a and 7-b with three three-coordinated H$_2$O molecules are the first low-energy isomers at SCC-DFTB level. In F. Spiegelman’s study, 7-a is also the first low-energy isomer using the Monte Carlo optimizations with the intermolecular polarizable potential.\cite{Douady2008} 7-c is the first low-energy isomer at MP2/Def2TZVP with ZPVE correction level including three three-coordinated water molecules. 7-c is also the first low-energy isomer at B3LYP/6-31++G(d,p) level including the harmonic ZPE contribution.\cite{Douady2008} 7-e is the first low-energy isomer with three three-coordinated H$_2$O molecules at MP2/aug-cc-pVDZ level in G. Shields’s study.\cite{Morrell2010} As illustrated in Figure \ref{fig:nh4-7-10w}, the energy difference between 7-a, 7-c and 7-e at SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels are less than 0.61 kcal·mol\textsuperscript{-1} so it is possible that the first low-energy iosmer is different when different method are applied. The energy of 7-a and 7-b are the same at both SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels and their structures are similar, which indicates it is easy for them to transform to each other. The results for cluster (H$_2$O)$_{7}${NH$_4$}$^+$ verify the accuracy of SCC-DFTB approach.
For cluster (H$_2$O)$_{7}${NH$_4$}$^+$, the five low-energy isomers are shown in Figure \ref{fig:nh4-7-10w}. The ion core {NH$_4$}$^+$ has a complete solvation shell in isomers 7-a to 7-e. 7-a and 7-b with three three-coordinated H$_2$O molecules are the first low-energy isomers at SCC-DFTB level. In F. Spiegelman’s study, 7-a is also the first low-energy isomer using the Monte Carlo optimizations with the intermolecular polarizable potential.\cite{Douady2008} 7-c is the first low-energy isomer at MP2/Def2TZVP with ZPVE correction level including three three-coordinated water molecules. 7-c is also the first low-energy isomer at B3LYP/6-31++G(d,p) level including the harmonic ZPE contribution.\cite{Douady2008} 7-e is the first low-energy isomer with three three-coordinated H$_2$O molecules at MP2/aug-cc-pVDZ level in G. Shields’s study.\cite{Morrell2010} As illustrated in Figure \ref{fig:nh4-7-10w}, the energy difference between 7-a, 7-c and 7-e at SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels are less than 0.61 kcal·mol\textsuperscript{-1} so it is possible that the first low-energy isomer is different when different method are applied. The energy of 7-a and 7-b are the same at both SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels and their structures are similar, which indicates it is easy for them to transform to each other. The results for cluster (H$_2$O)$_{7}${NH$_4$}$^+$ verify the accuracy of SCC-DFTB approach.
\begin{figure}[h!]
\includegraphics[width=1.0\linewidth]{nh4-7-10w.png}
@ -428,22 +419,22 @@ For cluster (H$_2$O)$_{7}${NH$_4$}$^+$, the five low-energy isomers are shown in
As shown in Table \ref{reBindE}, for isomers 7-a to 7-e, the relative binding energy $\Delta E_{bind.}^{whole}$ are -2.95, -2.92, -2.17, -1.28 and -3.22 kcal·mol\textsuperscript{-1} and the $\Delta E_{bind.}^{sep.}$are only -0.39, -0.38, 0.09, -1.35 and -2.27 kcal·mol\textsuperscript{-1}, respectively. It indicates the binding energies of 7-a to 7-e at SCC-DFTB agree well especially for 7-a to 7-d with those at MP2/Def2TZVP with BSSE correction level when water molecules are calculated separately. When all the water molecules are regarded as a whole part, the results of SCC-DFTB are not as good as those of the MP2 with BSSE method.
For cluster (H$_2$O)$_{8}${NH$_4$}$^+$, 8-a to 8-e are the five low-energy isomers displayed in Figure \ref{fig:nh4-7-10w}. In 8-a to 8-d, the ion core {NH$_4$}$^+$ has a complete solvation shell. 8-a is the first low-energy isomer in our calculation at SCC-DFTB level. In F. Spiegelman’s study, 8-b is the first low-energy isomer at B3LYP/6-31++G(d,p) level including the harmonic ZPE contribution.\cite{Douady2008} The structures of 8-a and 8-b are very similar and the energy differences are only 0.09 and 0.18 kcal·mol\textsuperscript{-1} at SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels, respectively. 8-d with seven three-coordinated H$_2$O molecules in the cube frame is the first low-energy isomer in our calculation at MP2/Def2TZVP with ZPVE correction level, which is consistent with F. Spiegelman’s results obtained using Monte Carlo optimizations.\cite{Douady2008} In 8-e, {NH$_4$}$^+$ has an exposed N-H bond and it also has seven three-coordinated H$_2$O molecules in its cage frame. The energies of isomers 8-a to 8-e are very close calculated using SCC-DFTB and MP2 methods, so it’s possible that the energy order will change when different methods or basis sets are applied. The results certificate the SCC-DFTB is good enough to find the low-energy isomers for cluster (H$_2$O)$_{8}${NH$_4$}$^+$.
For cluster (H$_2$O)$_{8}${NH$_4$}$^+$, 8-a to 8-e are the five low-energy isomers displayed in Figure \ref{fig:nh4-7-10w}. In 8-a to 8-d, the ion core {NH$_4$}$^+$ has a complete solvation shell. 8-a is the first low-energy isomer in our calculation at SCC-DFTB level. In F. Spiegelman’s study, 8-b is the first low-energy isomer at B3LYP/6-31++G(d,p) level including the harmonic ZPE contribution.\cite{Douady2008} The structures of 8-a and 8-b are very similar and the energy differences are only 0.09 and 0.18 kcal·mol\textsuperscript{-1} at SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels, respectively. 8-d with seven three-coordinated H$_2$O molecules in the cube frame is the first low-energy isomer in our calculation at MP2/Def2TZVP with ZPVE correction level, which is consistent with F. Spiegelman’s results obtained using Monte Carlo optimizations.\cite{Douady2008} In 8-e, {NH$_4$}$^+$ has an exposed N-H bond and it also has seven three-coordinated H$_2$O molecules in its cage frame. The energies of isomers 8-a to 8-e are very close calculated using SCC-DFTB and MP2 methods, so it’s possible that the energy order will change when different methods or basis sets are applied. The results certificate the SCC-DFTB is good enough to find the low-energy isomers for cluster (H$_2$O)$_{8}${NH$_4$}$^+$.
As shown in Table \ref{reBindE}, for isomers 8-a to 8-e, the relative binding energy $\Delta E_{bind.}^{whole}$are -2.20, -1.61, -3.71, -2.43 and -0.55 kcal·mol\textsuperscript{-1}, respectively and the biggest $\Delta E_{bind.}^{sep.}$ is -2.01 kcal·mol\textsuperscript{-1}. It shows the binding energies at SCC-DFTB level agree well with those at MP2/Def2TZVP with BSSE correction level when water molecules are calculated separately. From these results, when all the water molecules are considered as a whole part, the results of SCC-DFTB didn’t agree well with those of the MP2 with BSSE correction method.
For cluster (H$_2$O)$_{9}${NH$_4$}$^+$, the five low-energy structures of (H$_2$O)$_{9}${NH$_4$}$^+$ are illustrated in Figure \ref{fig:nh4-7-10w}. 9-a with seven three-coordinated H$_2$O molecules in the cage frame is the first low-energy isomer at SCC-DFTB level. 9-a is also the first low-energy structure at B3LYP/6-31++G(d,p) level including the harmonic ZPE contribution in F. Spiegelman’s study.\cite{Douady2008} 9-b with one N-H bond exposed in {NH$_4$}$^+$ is the second low-energy isomer whose energy is only 0.22 kcal·mol\textsuperscript{-1} higher than that of 9-a in the results of SCC-DFTB calculation. 9-b is the first low-energy isomer at MP2/Def2TZVP with ZPVE correction level in our calculation and it is also the first low-energy isomer at B3LYP/6-31++G(d,p) level in F. Spiegelman’s study.\cite{Douady2008} 9-c, 9-d and 9-e have a complete solvation shell. All the water molecules are connected together in the structure of 9-c. The structures of 9-a and 9-e are very similar and their energy difference is only 0.11 kcal·mol\textsuperscript{-1} at MP2/Def2TZVP with ZPVE correction level. The energy difference of isomers 9-a to 9-e is less than 0.51 kcal·mol\textsuperscript{-1} at SCC-DFTB and less than 0.86 kcal·mol\textsuperscript{-1} at MP2/Def2TZVP with ZPVE correction, so it’s easy for them to transform to each other making it possible for the variation of the energy order. The results certificate the SCC-DFTB is good enough to find the low-energy isomers for cluster (H$_2$O)$_{9}${NH$_4$}$^+$.
As shown in Table \ref{reBindE}, for isomers 9-a to 9-e, the relative binding energy $\Delta E_{bind.}^{whole}$ are -2.20, -1.61, -3.71, -2.43 and -0.55 kcal·mol\textsuperscript{-1} and the relative binding energy $\Delta E_{bind.}^{sep.}$ are -1.39, -0.84, -0.85, -1.78, and -0.91 kcal·mol\textsuperscript{-1}, respectively.
It is obvious that the absolute values of $\Delta E_{bind.}^{whole}$ are bigger than the corresponding $\Delta E_{bind.}^{sep.}$. It shows the binding energies at SCC-DFTB level agree well with those at MP2/Def2TZVP with BSSE correction level when water molecules are calculated separately. According to the results, When all the water molecules are considered as a whole part, the results of SCC-DFTB didn’t agree well with those of the MP2 with BSSE correction method.
It is obvious that the absolute values of $\Delta E_{bind.}^{whole}$ are bigger than the corresponding $\Delta E_{bind.}^{sep.}$. It shows the binding energies at SCC-DFTB level agree well with those at MP2/Def2TZVP with BSSE correction level when water molecules are calculated separately. According to the results, when all the water molecules are considered as a whole part, the results of SCC-DFTB didn’t agree well with those of the MP2 with BSSE correction method.
For cluster (H$_2$O)$_{10}${NH$_4$}$^+$, 10-a to 10-e are the five low-energy isomers in which the ion core {NH$_4$}$^+$ has a complete solvation shell shown in Figure \ref{fig:nh4-7-10w}. 10-a with eight three-coordinated H2O molecules in its big cage structure is the first low-energy isomer calculated using the SCC-DFTB approach. 10-a is also the first low-energy structure at B3LYP/6-31++G(d,p) level including the harmonic ZPE contribution in F. Spiegelman’s study.\cite{Douady2008} In 10-b and 10-e, there is a four-coordinated H$_2$O molecule in their cage structures. 10-d is the first low-energy structure in our calculation results using MP2/Def2TZVP with ZPVE correction, which is also the first low-energy isomer at B3LYP/6-31++G(d,p) level in F. Spiegelman’s study.\cite{Douady2008} The energy of 10-b is only 0.17 kcal·mol\textsuperscript{-1} higher than that of 10-a at SCC-DFTB level, and it is only 0.31 kcal·mol\textsuperscript{-1} lower than that of 10-a at MP2/Def2TZVP with ZPVE correction level. The energy of isomers 10-a to 10-e are very close at both SCC-DFTB and MP2/Def2TZVP levels, which indicates the results with SCC-DFTB agree well with those using MP2/Def2TZVP method for cluster (H$_2$O)$_{10}${NH$_4$}$^+$.
For cluster (H$_2$O)$_{10}${NH$_4$}$^+$, 10-a to 10-e are the five low-energy isomers in which the ion core {NH$_4$}$^+$ has a complete solvation shell shown in Figure \ref{fig:nh4-7-10w}. 10-a with eight three-coordinated H$_2$O molecules in its big cage structure is the first low-energy isomer calculated using the SCC-DFTB approach. 10-a is also the first low-energy structure at B3LYP/6-31++G(d,p) level including the harmonic ZPE contribution in F. Spiegelman’s study.\cite{Douady2008} In 10-b and 10-e, there is a four-coordinated H$_2$O molecule in their cage structures. 10-d is the first low-energy structure in our calculation results using MP2/Def2TZVP with ZPVE correction, which is also the first low-energy isomer at B3LYP/6-31++G(d,p) level in F. Spiegelman’s study.\cite{Douady2008} The energy of 10-b is only 0.17 kcal·mol\textsuperscript{-1} higher than that of 10-a at SCC-DFTB level, and it is only 0.31 kcal·mol\textsuperscript{-1} lower than that of 10-a at MP2/Def2TZVP with ZPVE correction level. The energy of isomers 10-a to 10-e are very close at both SCC-DFTB and MP2/Def2TZVP levels, which indicates the results with SCC-DFTB agree well with those using MP2/Def2TZVP method for cluster (H$_2$O)$_{10}${NH$_4$}$^+$.
As shown in Table \ref{reBindE}, for isomers 10-a to 10-e, the relative binding energies $\Delta E_{bind.}^{whole}$ and $\Delta E_{bind.}^{sep.}$ are not as small as the corresponding ones of clusters (H$_2$O)$_{1-9}${NH$_4$}$^+$, which implies the error of the relative binding energy increases with the number of water molecules in the cluster. The whole results of $\Delta E_{bind.}^{whole}$ are still bigger than those of $\Delta E_{bind.}^{sep.}$ for isomers 10-a to 10-e.
\subsubsection{Properties of (H$_2$O)$_{4-10}${NH$_3$} Clusters}
For cluster (H$_2$O)$_{4}${NH$_3$}, the five low-energy structures 4$^\prime$-a to 4$^\prime$-e are displayed in Figure \ref{fig:nh3-4-7w}. 4$^\prime$-a with three N-H bonds exposed is the first low-energy isomer at SCC-DFTB level. 4$^\prime$-b with two N-H bonds exposed is the second low-energy isomer at SCC-DFTB level but it is the first low-energy isomer at MP2/Def2TZVP with ZPVE correction level. The energy differences between 4$^\prime$-a to 4$^\prime$-b are only 0.20 and 0.07 kcal·mol\textsuperscript{-1} at SCC-DFTB and MP2/Def2TZVP with ZPVE correction level, respectively. The energy difference of isomers 4$^\prime$-a to 4$^\prime$-e is less than 0.75 kcal·mol\textsuperscript{-1} at MP2/Def2TZVP with ZPVE correction, so it’s possible for the variation of the energy order when different methods or basis sets are used. 4$^\prime$-d with a nearly planar pentagonal structure with nitrogen atom and the four oxygen atoms at the apexes is the first low-energy isomer at MP2/6-31+G(d,p) studied by J. Novoa et al\cite{Lee1996} 4$^\prime$-d is also the first low-energy isomer in D. Bacelo’s study using QCISD(T) for a single-point energy calculation based on the MP2/6-311++G(d,p) results.\cite{Bacelo2002} In addition, 4$^\prime$-a to 4$^\prime$-e are also the five low-energy isomers in D. Bacelo’s study even the energy order is different.\cite{Bacelo2002} The results show the SCC- DFTB is good enough to find the low-energy isomers isomers for cluster (H$_2$O)$_{4}${NH$_3$}.
For cluster (H$_2$O)$_{4}${NH$_3$}, the five low-energy structures 4$^\prime$-a to 4$^\prime$-e are displayed in Figure \ref{fig:nh3-4-7w}. 4$^\prime$-a with three N-H bonds exposed is the first low-energy isomer at SCC-DFTB level. 4$^\prime$-b with two N-H bonds exposed is the second low-energy isomer at SCC-DFTB level but it is the first low-energy isomer at MP2/Def2TZVP with ZPVE correction level. The energy differences between 4$^\prime$-a to 4$^\prime$-b are only 0.07 and 0.2 kcal·mol\textsuperscript{-1} at MP2/Def2TZVP with ZPVE correction level and SCC-DFTB level, respectively. The energy difference of isomers 4$^\prime$-a to 4$^\prime$-e is less than 0.75 kcal·mol\textsuperscript{-1} at MP2/Def2TZVP with ZPVE correction, so it’s possible for the variation of the energy order when different methods or basis sets are used. 4$^\prime$-d with a nearly planar pentagonal structure with nitrogen atom and the four oxygen atoms at the apexes is the first low-energy isomer at MP2/6-31+G(d,p) studied by J. Novoa et al\cite{Lee1996} 4$^\prime$-d is also the first low-energy isomer in D. Bacelo’s study using QCISD(T) for a single-point energy calculation based on the MP2/6-311++G(d,p) results.\cite{Bacelo2002} In addition, 4$^\prime$-a to 4$^\prime$-e are also the five low-energy isomers in D. Bacelo’s study even the energy order is different.\cite{Bacelo2002} The results show the SCC- DFTB is good enough to find the low-energy isomers isomers for cluster (H$_2$O)$_{4}${NH$_3$}.
\begin{figure}[h!]
\includegraphics[width=1.0\linewidth]{nh3-4-7w.png}
@ -452,21 +443,21 @@ For cluster (H$_2$O)$_{4}${NH$_3$}, the five low-energy structures 4$^\prime$-a
\label{fig:nh3-4-7w}
\end{figure}
The relative binding energies of isomers 4$^\prime$-a to 4$^\prime$-e are shown in Table \ref{reBindE}. Except 4$^\prime$-d, the values of $\Delta E_{bind.}^{whole}$ for 4$^\prime$-a to 4$^\prime$-e are smaller than the corresponding values of $\Delta E_{bind.}^{sep.}$. The $\Delta E_{bind.}^{sep.}$ of 4$^\prime$-d is smaller than those of other isomers. 4$^\prime$-d has a nearly planar pentagonal structure that only contains three O-H···O hydrogen bonds among the four water molecules while other isomers contain four O-H···O hydrogen bonds among the four water molecules. So the intermolecular interaction of the four water molecules in 4$^\prime$-d is not as strong as it in other isomers, this may explain the $\Delta E_{bind.}^{sep.}$ of 4$^\prime$-d is smaller than those of other isomers. In general, both relative binding energies $\Delta E_{bind.}^{sep.}$ and $\Delta E_{bind.}^{sep.}$ are not big that indicates SCC-DFTB performs well compared to the MP2 method with BSSE correction for calculating the binding energy of cluster (H$_2$O)$_{4}${NH$_3$}.
The relative binding energies of isomers 4$^\prime$-a to 4$^\prime$-e are shown in Table \ref{reBindE}. Except 4$^\prime$-d, the values of $\Delta E_{bind.}^{whole}$ for 4$^\prime$-a to 4$^\prime$-e are smaller than the corresponding values of $\Delta E_{bind.}^{sep.}$. The $\Delta E_{bind.}^{sep.}$ of 4$^\prime$-d is smaller than those of other isomers. 4$^\prime$-d has a nearly planar pentagonal structure that only contains three O-H···O hydrogen bonds among the four water molecules while other isomers contain four O-H···O hydrogen bonds among the four water molecules. So the intermolecular interaction of the four water molecules in 4$^\prime$-d is not as strong as it is in other isomers, this may explain the $\Delta E_{bind.}^{sep.}$ of 4$^\prime$-d is smaller than those of other isomers. In general, both relative binding energies $\Delta E_{bind.}^{sep.}$ and $\Delta E_{bind.}^{sep.}$ are not big that indicates SCC-DFTB performs well compared to the MP2 method with BSSE correction for calculating the binding energy of cluster (H$_2$O)$_{4}${NH$_3$}.
For cluster (H$_2$O)$_{5}${NH$_3$}, 5$^\prime$-a to 5$^\prime$-e are the five low-energy isomers shown in Figure \ref{fig:nh3-4-7w}. 5$^\prime$-a with four three-coordinated water molecules is the first low-energy structure at both SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels. 5$^\prime$-b and 5$^\prime$-c are the second and third isomers at SCC-DFTB level and they are the third and second isomers at MP2/Def2TZVP level with ZPVE. The energy difference between 5$^\prime$-b and 5$^\prime$-c is only 0.05 and 0.44 kcal·mol\textsuperscript{-1} at SCC-DFTB level and MP2/Def2TZVP with ZPVE correction level, respectively. In addition, the structures of 5$^\prime$-b and 5$^\prime$-c are very similar so it is possible for them to transform to each other. 5$^\prime$-d with two three-coordinated water molecules is the fourth low-energy structure at both SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels. 5$^\prime$-e with four three-coordinated water molecules is the fifth low-energy structure at both SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels. The frames of 5$^\prime$-a and 5$^\prime$-e are almost the same but the water molecule who offers the hydrogen or oxygen to form the O-H···O hydrogen bonds has a small difference. The energy of 5$^\prime$-e is 1.51 kcal·mol\textsuperscript{-1} higher than that of 5$^\prime$-a at MP2/Def2TZVP with ZPVE correction level, which implies the intermolecular connection mode has an influence on the stability of the isomers. The results show the SCC-DFTB approach performs well to find the low-energy isomers for cluster (H$_2$O)$_{5}${NH$_3$} compared with MP2/Def2TZVP with ZPVE correction method.
For cluster (H$_2$O)$_{5}${NH$_3$}, 5$^\prime$-a to 5$^\prime$-e are the five low-energy isomers shown in Figure \ref{fig:nh3-4-7w}. 5$^\prime$-a with four three-coordinated water molecules is the first low-energy structure at both MP2/Def2TZVP with ZPVE correction and SCC-DFTB levels. 5$^\prime$-b and 5$^\prime$-c are the second and third isomers at SCC-DFTB level and they are the third and second isomers at MP2/Def2TZVP level with ZPVE. The energy difference between 5$^\prime$-b and 5$^\prime$-c is only 0.44 and 0.05 kcal·mol\textsuperscript{-1} at MP2/Def2TZVP with ZPVE correction level and SCC-DFTB level, respectively. In addition, the structures of 5$^\prime$-b and 5$^\prime$-c are very similar so it is possible for them to transform to each other. 5$^\prime$-d with two three-coordinated water molecules is the fourth low-energy structure at both MP2/Def2TZVP with ZPVE correction and SCC-DFTB levels. 5$^\prime$-e with four three-coordinated water molecules is the fifth low-energy structure at both MP2/Def2TZVP with ZPVE correction and SCC-DFTB levels. The frames of 5$^\prime$-a and 5$^\prime$-e are almost the same but the water molecule who offers the hydrogen or oxygen to form the O-H···O hydrogen bonds has a small difference. The energy of 5$^\prime$-e is 1.51 kcal·mol\textsuperscript{-1} higher than that of 5$^\prime$-a at MP2/Def2TZVP with ZPVE correction level, which implies the intermolecular connection mode has an influence on the stability of the isomers. The results show the SCC-DFTB approach performs well to find the low-energy isomers for cluster (H$_2$O)$_{5}${NH$_3$} compared with MP2/Def2TZVP with ZPVE correction method.
The relative binding energies of isomers 5$^\prime$-a to 5$^\prime$-e are shown in Table \ref{reBindE}. The values of $\Delta E_{bind.}^{whole}$ are less than 0.82 kcal·mol\textsuperscript{-1} for 5$^\prime$-a to 5$^\prime$-e. The values of $\Delta E_{bind.}^{sep.}$ are bigger than the corresponding values of $\Delta E_{bind.}^{whole}$. It indicates SCC-DFTB agrees better with MP2/Def2TZVP $\Delta E_{bind.}^{whole}$ when all the water molecules are regarded as a whole part than considering separately for calculating the binding energy of cluster (H$_2$O)$_{5}${NH$_3$}.
For cluster (H$_2$O)$_{6}${NH$_3$}, the five low-energy structures 6$^\prime$-a to 6$^\prime$-e are displayed in Figure \ref{fig:nh3-4-7w}. 6$^\prime$-a is the first low-energy structure at SCC-DFTB level. All water molecules in 6$^\prime$-a are three-coordinated. 6$^\prime$-b is the second low-energy isomer at SCC-DFTB level and it’s only 0.05 and 0.42 kcal·mol\textsuperscript{-1} higher than the ones of 6$^\prime$-a at SCC-DFTB level and MP2/Def2TZVP with ZPVE correction level, respectively. 6$^\prime$-c to 6$^\prime$-d are the third and fourth low-energy isomers in which the six water molecules form a triangular prism structure and there are one and two four-coordinated water molecules in 6$^\prime$-c to 6$^\prime$-d, respectively. 6$^\prime$-e is the fifth low-energy structure at SCC-DFTB level but it’s the first low-energy isomer at MP2/Def2TZVP with ZPVE correction level. The energy of 6$^\prime$-a to 6$^\prime$-e are very close at both SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels that it is difficult to keep the energy order when different methods or basis sets are applied. This also shown the SCC-DFTB method used is efficient to find the low-energy isomers of cluster (H$_2$O)$_{6}${NH$_3$}.
For cluster (H$_2$O)$_{6}${NH$_3$}, the five low-energy structures 6$^\prime$-a to 6$^\prime$-e are displayed in Figure \ref{fig:nh3-4-7w}. 6$^\prime$-a is the first low-energy structure at SCC-DFTB level. All water molecules in 6$^\prime$-a are three-coordinated. 6$^\prime$-b is the second low-energy isomer at SCC-DFTB level and it’s only 0.42 and 0.05 kcal·mol\textsuperscript{-1} higher than the ones of 6$^\prime$-a at MP2/Def2TZVP with ZPVE correction level and SCC-DFTB level, respectively. 6$^\prime$-c to 6$^\prime$-d are the third and fourth low-energy isomers in which the six water molecules form a triangular prism structure and there are one and two four-coordinated water molecules in 6$^\prime$-c to 6$^\prime$-d, respectively. 6$^\prime$-e is the fifth low-energy structure at SCC-DFTB level but it’s the first low-energy isomer at MP2/Def2TZVP with ZPVE correction level. The energy of 6$^\prime$-a to 6$^\prime$-e are very close at both both MP2/Def2TZVP with ZPVE correction and SCC-DFTB levels that it is difficult to keep the energy order when different methods or basis sets are applied. This also shows the SCC-DFTB method used is efficient to find the low-energy isomers of cluster (H$_2$O)$_{6}${NH$_3$}.
The relative binding energies of isomers 6$^\prime$-a to 6$^\prime$-e are listed in Table \ref{reBindE}. The smallest and the biggest values of $\Delta E_{bind.}^{whole}$ are -0.05 and -1.11 kcal·mol\textsuperscript{-1}, respectively. The smallest absolute value of $\Delta E_{bind.}^{sep.}$ is 1.96 kcal·mol\textsuperscript{-1}. The binding energies calculated with SCC-DFTB agree well with those calculated at MP2/Def2TZVP level for cluster (H$_2$O)$_{6}${NH$_3$} when all the water molecules are considered as a whole part.
For cluster (H$_2$O)$_{7}${NH$_3$}, the five low-energy isomers 7$^\prime$-a to 7$^\prime$-e are illustrated in Figure \ref{fig:nh3-4-7w}. 7$^\prime$-a with a cubic structure is the first low-lying energy structure at both SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels. 7$^\prime$-b is the second low-energy structure at both SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels. 7$^\prime$-b has a similar structure with 7$^\prime$-a but the NH$_3$ in it has two exposed N-H bonds. 7$^\prime$-c and 7$^\prime$-d have similar structures and they are the third and fourth low-lying energy isomers at SCC-DFTB level and their energy difference is only 0.74 kcal·mol\textsuperscript{-1}. 7$^\prime$-e with three exposed N-H bonds is the fifth low-energy isomer at both SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels. The results of SCC-DFTB method agree well with those of MP2/Def2TZVP with ZPVE correction for the five low-energy isomers of cluster (H$_2$O)$_{7}${NH$_3$}.
For cluster (H$_2$O)$_{7}${NH$_3$}, the five low-energy isomers 7$^\prime$-a to 7$^\prime$-e are illustrated in Figure \ref{fig:nh3-4-7w}. 7$^\prime$-a with a cubic structure is the first low-lying energy structure at both MP2/Def2TZVP with ZPVE correction and SCC-DFTB levels. 7$^\prime$-b is the second low-energy structure at both MP2/Def2TZVP with ZPVE correction and SCC-DFTB levels. 7$^\prime$-b has a similar structure with 7$^\prime$-a but the NH$_3$ in it has two exposed N-H bonds. 7$^\prime$-c and 7$^\prime$-d have similar structures and they are the third and fourth low-lying energy isomers at SCC-DFTB level and their energy difference is only 0.74 kcal·mol\textsuperscript{-1}. 7$^\prime$-e with three exposed N-H bonds is the fifth low-energy isomer at both MP2/Def2TZVP with ZPVE correction and SCC-DFTB levels. The results of SCC-DFTB method agree well with those of MP2/Def2TZVP with ZPVE correction for the five low-energy isomers of cluster (H$_2$O)$_{7}${NH$_3$}.
The smallest and the biggest values of $\Delta E_{bind.}^{whole}$ of isomers 7$^\prime$-a to 7$^\prime$-e are -0.02 and -1.11 kcal·mol\textsuperscript{-1}, respectively and the smallest absolute value of $\Delta E_{bind.}^{sep.}$ is 2.02 kcal·mol\textsuperscript{-1} shown in Table \ref{reBindE}. The binding energies calculated with SCC-DFTB agree well with those obtained using MP2/Def2TZVP for cluster (H$_2$O)$_{7}${NH$_3$} when all the water molecules are considered as a whole part.
For cluster (H$_2$O)$_{8}${NH$_3$}, 8$^\prime$-a to 8$^\prime$-e are the five low-energy structures shown in Figure \ref{fig:nh3-8-10w}. 8$^\prime$-a in which eight water molecules constitute a cube is the first low-lying energy structure in SCC-DFTB calculation results. 8$^\prime$-b also with a water-cube structure is the second low-energy structure at SCC-DFTB level and it is the first low-energy isomer at MP2/Def2TZVP with ZPVE correction level. The energy differences between 8$^\prime$-a and 8$^\prime$-b are only 0.93 an 0.30 kcal·mol\textsuperscript{-1} at SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels. From Figure \ref{fig:nh3-8-10w}, the fifth low-energy isomer 8$^\prime$-e includes less number of hydrogen bonds than other isomers and its energy has a clearly increase compared to other isomers. The results show the SCC-DFTB method performs well to obtain the low-energy isomers of cluster (H$_2$O)$_{8}${NH$_3$}.
For cluster (H$_2$O)$_{8}${NH$_3$}, 8$^\prime$-a to 8$^\prime$-e are the five low-energy structures shown in Figure \ref{fig:nh3-8-10w}. 8$^\prime$-a in which eight water molecules constitute a cube is the first low-lying energy structure in SCC-DFTB calculation results. 8$^\prime$-b also with a water-cube structure is the second low-energy structure at SCC-DFTB level and it is the first low-energy isomer at MP2/Def2TZVP with ZPVE correction level. The energy differences between 8$^\prime$-a and 8$^\prime$-b are only 0.30 and 0.93 kcal·mol\textsuperscript{-1} at MP2/Def2TZVP with ZPVE correction level and SCC-DFTB level. From Figure \ref{fig:nh3-8-10w}, the fifth low-energy isomer 8$^\prime$-e includes less hydrogen bonds than other isomers and its energy has a clearly increase compared to other isomers. The results show the SCC-DFTB method performs well to obtain the low-energy isomers of cluster (H$_2$O)$_{8}${NH$_3$}.
\begin{figure}[h!]
\includegraphics[width=1.0\linewidth]{nh3-8-10w.png}
@ -487,35 +478,90 @@ The smallest and biggest values of $\Delta E_{bind.}^{whole}$ of isomers 10$^\pr
\subsubsection{Properties of (H$_2$O)$_{20}${NH$_4$}$^+$ Cluster}
For cluster (H$_2$O)$_{20}${NH$_4$}$^+$, the lowest-energy structure shown in Figure \ref{fig:nh3-nh4-20w} (a) was obtained with the combination of SCC-DFTB and PTMD which is consistent with that of previous study.\cite{Kazimirski2003, Douady2009, Bandow2006}
Microcanonical and canonical caloric curves were obtained using exchange Monte Carlo simulations by F. Spiegelman’s group.\cite{Douady2009}
I also calculated the canonical heat capacities of cluster (H$_2$O)$_{20}${NH$_4$}$^+$ using the combination of SCC-DFTB and PTMD depicted in Figure.
The lowest-energy isomer of (H$_2$O)$_{20}${NH$_4$}$^+$ was previously reported by Douady \textit{et al.}\cite{Douady2008,Douady2009}
as well as by other studies.\cite{Kazimirski2003, Bandow2006} Douady \textit{et al.} conducted a Monte Carlo simulation in combination with
the Kozack-Jordan polarizable potential.\cite{Kozack1992polar,Kozack1992empiri} This isomer displays a closed-shell structure, similar to the
well-know structure of (H$_2$O)$_{21}${H}$^+$, with the {NH$_4$}$^+$ ion fully solvated at the center of the aggregate. The structure of
(H$_2$O)$_{20}${NH$_4$}$^+$ is depicted in Figure~\ref{fig:nh3-nh4-20w}. Starting from the coordinates Douady \textit{et al.} kindly
sent to us, our PTMD exploration of the PES does not lead any lower-energy isomer. We thus consider this isomer to be also the lowest-energy
isomer at the SCC-DFTB level.
\begin{figure}[h!]
\includegraphics[width=0.6\linewidth]{nh3-nh4-20w.jpeg}
\includegraphics[width=0.4\linewidth]{nh4-20.png}
\centering
\caption{The five low-energy isomers of cluster (H$_2$O)$_{20}${NH$_4$}$^{+}$ (a) and (H$_2$O)$_{20}${NH$_3$} (b) at SCC-DFTB level.}
\caption{Lowest-energy isomer of (H$_2$O)$_{20}${NH$_4$}$^{+}$.}
\label{fig:nh3-nh4-20w}
\end{figure}
Douady \textit{et al.} further computed the heat capacity of (H$_2$O)$_{20}${NH$_4$}$^+$ as a function of the
temperature.\cite{Douady2009} In order to further demonstrate the accuracy of SCC-DFTB we have conducted a
similar calculation. The modelling of heat capacity as a function of the temperature can be achieved in different
ways. The simplest one consists in performing MC or MD simulations at different temperatures and extracting
for each one the heat capacity from a direct calculation of the variance of the potential energy. This approach
is somewhat statistically inefficient and thus alternative approaches have been proposed. Indeed, to reduce
the statistical noise and to extrapolate heat capacities at temperatures not explicitly simulated, one can benefit
from the fact that in MD or MC simulations a given configuration may be visited at different temperatures.
Labastie and Whetten,\cite{Labastie1990} proposed a method to take advantage of these overlaps to calculate
heat capacity curves. It is referred to as the multiple histogram method. It uses probability densities, extracted
from a 10~ns PTMD simulation, of finding an energy at a given temperature leading to a set of histograms for
each simulated temperature. The entropy and partition functions are extracted from these histograms which
can then give access to internal energy at any given temperature. The heat capacity at temperature $T$ is finally
calculated as:
\begin{equation} \label{heatCapacity}
C(T) = \frac{N_{dof}k_B}{2} + \frac{\langle V^{2} \rangle - \langle V \rangle ^{2}}{k_{B}T^{2}}
\end{equation}
where $k_{B}$ is the Boltzmann constant, $N_{dof}$ the number of degrees of freedom, $\langle V \rangle$ and
$\langle V^{2} \rangle$ the internal energy and square of internal energy at temperature $T$, respectively.
The first term of equation~\ref{heatCapacity} is the classical limit of the heat capacity at T = 0~K.
Figure~\ref{fheat_c} displays the heat capacity curve we obtained. It is very similar to the one obtained by Douady
and co-workers. It is flat up to $\sim$150~K with a sharp increase starting at $\sim$165~K. This behaviour
also exits in (H$_2$O)$_{21}${H}$^+$,\cite{Douady2009,Korchagina2017} and was interpreted by a weak
density of low-energy isomers above the global minimum, \textit{i.e.} a particularly stable lowest-energy structure.\cite{Korchagina2017}
Douady\textit{et al.} observed a slightly higher transition temperature of (H$_2$O)$_{20}${NH$_4$}$^+$ as
compared to (H$_2$O)$_{21}${H}$^+$. This is also true at the SCC-DFTB level as the transition
temperature of (H$_2$O)$_{21}${H}$^+$ was evaluated at $\sim$140~K.\cite{Korchagina2017}
This confirms that SCC-DFTB with the improved set of N---H parameters, besides properly describing
structures and binding energies, also lead to correct thermodynamical properties.
\begin{figure}[h!]
\includegraphics[width=0.5\linewidth]{capacity-curve-new.eps}
\centering
\caption{Canonical heat capacity as a function of the temperature of (H$_2$O)$_{20}${NH$_4$}$^{+}$.}
\label{fheat_c}
\end{figure}
\subsection{Conclusions for Ammonium/Ammonia Including Water Clusters}
The low-energy isomers reported for (H$_2$O)$_{1-10, 20}${NH$_4$}$^+$ and (H$_2$O)$_{1-10, 20}$NH$_3$ clusters are obtained with a combination of
SCC-DFTB (0.12/1.16) and PTMD. Binding energies as a function of the N---O distance in (H$_2$O){NH$_4$}$^+$ and (H$_2$O)NH$_3$ demonstrate
that the improve parameters I proposed are in much better agreement with reference calculations than the original SCC-DFTB parameters.
The low-energy isomers of clusters (H$_2$O)$_{1-10}${NH$_4$}$^+$ and (H$_2$O)$_{1-10}$NH$_3$ at the SCC-DFTB (0.12/1.16) level
agree well with those at MP2/Def2TZVP level and the corresponding results in the literature. The SCC-DFTB binding energies also agree well
with those calculated with MP2/Def2TZVP method with BSSE correction. This demonstrate that SCC-DFTB (0.12/1.16) approach is good
enough to model ammonium and ammonia containing water clusters.
In this paper, we have introduced a modification to the N-H set of parameters by modifying both the original mio-set of N---H integrals
and the evaluation of the charges. The proposed new set of parameters solve the spurious deprotonation observed for {NH$_4$}$^+$
using the original set of parameters. We first demonstrate that this new set of parameters leads also to improved description of the
dissociation curves of both (H$_2$O){NH$_4$}$^+$ and (H$_2$O)NH$_3$ dimers as compared to MP2/Def2TZVP method with BSSE
correction reference calculations.
By combining this new potential, SCC-DFTB (0.12/1.16), to PTMD simulations, a number of low-energy isomers are reported for
(H$_2$O)$_{1-10}${NH$_4$}$^+$ and (H$_2$O)$_{1-10}$NH$_3$ clusters. Further geometry optimizations at the MP2/Def2TZVP
lead to structures very similar to those reported at the SCC-DFTB level and also reported in the literature. The SCC-DFTB binding
energies also agree well with those calculated with the MP2/Def2TZVP approch including with BSSE correction. This demonstrates
that SCC-DFTB (0.12/1.16) approach is well suited to the description of ammonium and ammonia containing water clusters.
Among the five lowest-energy structures of (H$_2$O)$_{4}${NH$_4$}$^+$, four of them display a dangling N-H bond. Among the five lowest-energy
structures of (H$_2$O)$_{5}${NH$_4$}$^+$, only two structures display a dangling N-H bond. Among the five lowest-energy isomers of (H$_2$O)$_{6-10}${NH$_4$}$^+$,
all the structures, except 8-e, display a ion core {NH$_4$}$^+$ that has a complete solvation shell but it is not located at the center of the water cluster.
In the most stable structures of (H$_2$O)$_{20}${NH$_4$}$^+$, the ion core {NH$_4$}$^+$ has a complete solvation shell and it is in the center of the
water cluster. In contrast, in the low-energy isomers of (H$_2$O)$_{1-10}$NH$_3$ clusters, NH$_3$ is never fully solvated by the water cluster.
Among the five lowest-energy structures of (H$_2$O)$_{4}${NH$_4$}$^+$, four of them display a dangling N-H bond. Among the five
lowest-energy structures of (H$_2$O)$_{5}${NH$_4$}$^+$, only two structures display a dangling N-H bond. Among the five
lowest-energy isomers of (H$_2$O)$_{6-10}${NH$_4$}$^+$, all the structures, except 8-e, display a ion core {NH$_4$}$^+$
that has a complete solvation shell but it is not located at the center of the water cluster. In the most stable structures of
(H$_2$O)$_{20}${NH$_4$}$^+$, reported in a previous study, the ion core {NH$_4$}$^+$ has a complete solvation shell
and it is located in the center of the water cluster. (H$_2$O)$_{1-10}$NH$_3$ clusters display significantly different structures.
Indeed, NH$_3$ is never fully solvated by the water molecules whatever the cluster size. It either participates to the surface
hydrogen bond network in a few cases, or acts as a surface molecule, only bonded to the water molecules by a unique
hydrogen bond.
The present study demonstrate thet ability of SCC-DFTB to model small size ammonium and ammonia containing water clusters, which is less
expensive than \textit{ab initio} methods. It is possible for SCC-DFTB to describe the larger scaled ammonium and ammonia containing water clusters.
Application of the presently proposed potential to the calculation of the heat capacity curve of (H$_2$O)$_{4}${NH$_4$}$^+$
further demonstrates the quality of the potential as the SCC-DFTB curve is close to the previously reported curve. The
present study therefore demonstrates the ability of SCC-DFTB to model small size ammonium and ammonia containing
water clusters. Due to the low computational cost of SCC-DFTB as compared to \textit{ab initio} methods, one can envisage
its application to describe the larger size ammonium and ammonia containing water clusters. One can also envisage the
study of water clusters containing a mixture of nitrogen and sulphur compounds, for instance, ammonium and sulfate ions.
These species, their conjugated basis and acid in combination with dimethylamine and water molecules represent the basis
for nucleation of atmospheric particles and SCC-DFTB could play a major in the theoretical description of these species.
\section{Structural and Energetic Properties of Protonated Uracil Water Clusters} \label{structureUH}
@ -548,13 +594,13 @@ only neutral species ((H$_2$O)$_{n}$U) were considered.\cite{Shishkin2000, Gadre
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, M. Ghomi predicted that for $n$ = 7,\cite{Gaigeot2001} water molecules arrange
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 F. Calvo \textit{et al.}.\cite{Bacchus2015} Similarly, for $n$ = 11, V. 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 F. Calvo and collaborator \cite{Bacchus2015} or with 6 water molecules (though 5 have not been calculated) reported by S. Gadre \textit{et al.}.\cite{Gadre2000}
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 F. Calvo \textit{et al}.\cite{Bacchus2015} Similarly, for $n$ = 11, V. 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 F. Calvo and collaborator \cite{Bacchus2015} or with 6 water molecules (though 5 have not been calculated) reported by S. 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 V. Danilov and F. Calvo.\cite{Danilov2006, Bacchus2015} Consequently, although it is instructive from a qualitative point
structures as highlighted by V. Danilov and F. 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. I 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
@ -619,10 +665,10 @@ This analysis based on PA is however quite crude. Indeed, it assumes that the pr
\subsubsection{Calculated Structures of Protonated Uracil Water Clusters} \label{calcul_ur}
As discussed in section~\ref{sec:ammoniumwater}, I 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
sp$^2$ nitrogen atoms there is no need to modify 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} presents the binding energies of the two
(H$_2$O)U isomers represented in Figure~\ref{uracil_i} 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
$D_\textrm{NH}$ = 0.14 lead to reasonable binding energies. So, to be consistent with the work performed in the previous section, we
have used $D_\textrm{NH}$ = 0.12 in the following.
\begin{figure}[h!]

596
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\citation{Kyrtopoulos2001,Farmer2003}
\citation{Aumaitre2019}
@ -240,35 +240,35 @@
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@article{Labastie1990,
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Journal = {Phys. Rev. Lett.},
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Pages = {1567--1570},
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Volume = {65},
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@article{Bandow2006,
title={Larger water clusters with edges and corners on their way to ice: Structural trends elucidated with an improved parallel evolutionary algorithm},
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Year = {1976}}
@article{Stegmaier2011,
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title={A Bronze Matryoshka: The Discrete Intermetalloid Cluster [Sn@Cu$_{12}$@Sn$_{20}$]$_{12}^-$ in the Ternary Phases A$_{12}$Cu$_{12}$Sn$_{21}$ (A= Na, K)},
author={Stegmaier, Saskia and Fässler, Thomas F},
journal={J. Am. Chem. Soc.},
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@ -60,7 +60,7 @@
\bibitem{Stegmaier2011}
{\sc Saskia Stegmaier and Thomas~F Fässler}.
\newblock {\bf A Bronze Matryoshka: The Discrete Intermetalloid Cluster
[Sn@Cu$_{12}$@Sn$_{20}$]$_{12}^-$ -in the Ternary Phases
[Sn@Cu$_{12}$@Sn$_{20}$]$_{12}^-$ in the Ternary Phases
A$_{12}$Cu$_{12}$Sn$_{21}$ (A= Na, K)}.
\newblock {\em J. Am. Chem. Soc.}, {\bf 133}(49):19758--19768, 2011.
@ -2367,6 +2367,17 @@
algorithm}.
\newblock {\em J. Phys. Chem. A}, {\bf 110}(17):5809--5822, 2006.
\bibitem{Kozack1992empiri}
{\sc RE~Kozack and PC~Jordan}.
\newblock {\bf Empirical models for the hydration of protons}.
\newblock {\em J. Chem. Phys.}, {\bf 96}(4):3131--3136, 1992.
\bibitem{Labastie1990}
{\sc Pierre Labastie and Robert~L Whetten}.
\newblock {\bf {Statistical Thermodynamics of the Cluster Solid-Liquid
Transition}}.
\newblock {\em Phys. Rev. Lett.}, {\bf 65}(13):1567--1570, 1990.
\bibitem{Maclot2011}
{\sc Sylvain Maclot, Michael Capron, R{\'e}mi Maisonny, Arkadiusz {\L}awicki,
Alain M{\'e}ry, Jimmy Rangama, Jean-Yves Chesnel, Sadia Bari, Ronnie

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@ -12,99 +12,100 @@ A level-1 auxiliary file: 4/collision.aux
A level-1 auxiliary file: 5/general_conclusion.aux
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Warning--I didn't find a database entry for "Korchagina2017theor"
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@ -117,45 +118,45 @@ Warning--empty year in Unger1993
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@ -448,9 +448,9 @@
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@ -472,7 +472,7 @@
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@ -484,12 +484,12 @@
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\BOOKMARK [2][]{subsection.2.4.2}{2.4.2 Classical Molecular Dynamics}{section.2.4}% 13
\BOOKMARK [2][]{subsection.2.4.3}{2.4.3 Parallel-Tempering Molecular Dynamics}{section.2.4}% 14
\BOOKMARK [2][]{subsection.2.4.4}{2.4.4 Global Optimization}{section.2.4}% 15
\BOOKMARK [0][]{chapter.3}{3 Exploration of Structural and Energetic Properties}{}% 16
\BOOKMARK [0][]{chapter.3}{3 Investigation of Structural and Energetic Properties}{}% 16
\BOOKMARK [1][]{section.3.1}{3.1 Computational Details}{chapter.3}% 17
\BOOKMARK [2][]{subsection.3.1.1}{3.1.1 SCC-DFTB Potential}{section.3.1}% 18
\BOOKMARK [2][]{subsection.3.1.2}{3.1.2 SCC-DFTB Exploration of PES}{section.3.1}% 19
@ -62,4 +62,4 @@
\BOOKMARK [0][]{chapter.5}{5 General Conclusions and Perspectives}{}% 62
\BOOKMARK [1][]{section.5.1}{5.1 General Conclusions}{chapter.5}% 63
\BOOKMARK [1][]{section.5.2}{5.2 Perspectives}{chapter.5}% 64
\BOOKMARK [0][]{chapter*.82}{References}{}% 65
\BOOKMARK [0][]{chapter*.83}{References}{}% 65

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@ -14,7 +14,7 @@
\contentsline {subsection}{\numberline {2.4.2}Classical Molecular Dynamics}{39}{subsection.2.4.2}
\contentsline {subsection}{\numberline {2.4.3}Parallel-Tempering Molecular Dynamics}{44}{subsection.2.4.3}
\contentsline {subsection}{\numberline {2.4.4}Global Optimization}{46}{subsection.2.4.4}
\contentsline {chapter}{\numberline {3}Exploration of Structural and Energetic Properties}{49}{chapter.3}
\contentsline {chapter}{\numberline {3}Investigation of Structural and Energetic Properties}{49}{chapter.3}
\contentsline {section}{\numberline {3.1}Computational Details}{50}{section.3.1}
\contentsline {subsection}{\numberline {3.1.1}SCC-DFTB Potential}{50}{subsection.3.1.1}
\contentsline {subsection}{\numberline {3.1.2}SCC-DFTB Exploration of PES}{50}{subsection.3.1.2}
@ -28,39 +28,39 @@
\contentsline {subsubsection}{\numberline {3.2.2.3}Properties of (H$_2$O)$_{4-10}${NH$_4$}$^+$ Clusters}{61}{subsubsection.3.2.2.3}
\contentsline {subsubsection}{\numberline {3.2.2.4}Properties of (H$_2$O)$_{4-10}${NH$_3$} Clusters}{68}{subsubsection.3.2.2.4}
\contentsline {subsubsection}{\numberline {3.2.2.5}Properties of (H$_2$O)$_{20}${NH$_4$}$^+$ Cluster}{73}{subsubsection.3.2.2.5}
\contentsline {subsection}{\numberline {3.2.3}Conclusions for Ammonium/Ammonia Including Water Clusters}{74}{subsection.3.2.3}
\contentsline {section}{\numberline {3.3}Structural and Energetic Properties of Protonated Uracil Water Clusters}{75}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}General introduction}{75}{subsection.3.3.1}
\contentsline {subsection}{\numberline {3.3.2}Results and Discussion}{77}{subsection.3.3.2}
\contentsline {subsubsection}{\numberline {3.3.2.1}Experimental Results}{77}{subsubsection.3.3.2.1}
\contentsline {subsubsection}{\numberline {3.3.2.2}Calculated Structures of Protonated Uracil Water Clusters}{83}{subsubsection.3.3.2.2}
\contentsline {subsection}{\numberline {3.3.3}Conclusions on (H$_2$O)$_{n}$UH$^+$ clusters}{92}{subsection.3.3.3}
\contentsline {chapter}{\numberline {4}Dynamical Simulation of Collision-Induced Dissociation}{97}{chapter.4}
\contentsline {section}{\numberline {4.1}Experimental Methods}{97}{section.4.1}
\contentsline {subsection}{\numberline {4.1.1}Principle of TCID}{99}{subsection.4.1.1}
\contentsline {subsection}{\numberline {4.1.2}Experimental Setup}{100}{subsection.4.1.2}
\contentsline {section}{\numberline {4.2}Computational Details}{102}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}SCC-DFTB Potential}{102}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}Collision Trajectories}{103}{subsection.4.2.2}
\contentsline {subsection}{\numberline {4.2.3}Trajectory Analysis}{104}{subsection.4.2.3}
\contentsline {section}{\numberline {4.3}Dynamical Simulation of Collision-Induced Dissociation of Protonated Uracil Water Clusters}{105}{section.4.3}
\contentsline {subsection}{\numberline {4.3.1}Introduction}{105}{subsection.4.3.1}
\contentsline {subsection}{\numberline {4.3.2}Results and Discussion}{106}{subsection.4.3.2}
\contentsline {subsubsection}{\numberline {4.3.2.1}Statistical Convergence}{106}{subsubsection.4.3.2.1}
\contentsline {subsection}{\numberline {4.3.3}Time-Dependent Proportion of Fragments}{109}{subsection.4.3.3}
\contentsline {subsection}{\numberline {4.3.4}Proportion of Neutral Uracil Loss and Total Fragmentation Cross Sections for Small Clusters}{112}{subsection.4.3.4}
\contentsline {subsection}{\numberline {4.3.5}Behaviour at Larger Sizes, the Cases of (H$_2$O)$_{11, 12}$UH$^+$}{122}{subsection.4.3.5}
\contentsline {subsection}{\numberline {4.3.6}Mass Spectra of Fragments with Excess Proton}{126}{subsection.4.3.6}
\contentsline {subsection}{\numberline {4.3.7}Conclusions about CID of (H$_2$O)$_{n}$UH$^+$}{129}{subsection.4.3.7}
\contentsline {section}{\numberline {4.4}Dynamical Simulation of Collision-Induced Dissociation for Pyrene Dimer Cation}{131}{section.4.4}
\contentsline {subsection}{\numberline {4.4.1}Introduction}{131}{subsection.4.4.1}
\contentsline {subsection}{\numberline {4.4.2}Calculation of Energies}{133}{subsection.4.4.2}
\contentsline {subsection}{\numberline {4.4.3}Simulation of the Experimental TOFMS}{135}{subsection.4.4.3}
\contentsline {subsection}{\numberline {4.4.4}Results and Discussion}{137}{subsection.4.4.4}
\contentsline {subsubsection}{\numberline {4.4.4.1}TOFMS Comparison}{137}{subsubsection.4.4.4.1}
\contentsline {subsubsection}{\numberline {4.4.4.2}Molecular Dynamics Analysis}{138}{subsubsection.4.4.4.2}
\contentsline {subsection}{\numberline {4.4.5}Conclusions about CID of Py$_2^+$}{154}{subsection.4.4.5}
\contentsline {chapter}{\numberline {5}General Conclusions and Perspectives}{157}{chapter.5}
\contentsline {section}{\numberline {5.1}General Conclusions}{157}{section.5.1}
\contentsline {section}{\numberline {5.2}Perspectives}{160}{section.5.2}
\contentsline {chapter}{References}{163}{chapter*.82}
\contentsline {subsection}{\numberline {3.2.3}Conclusions for Ammonium/Ammonia Including Water Clusters}{75}{subsection.3.2.3}
\contentsline {section}{\numberline {3.3}Structural and Energetic Properties of Protonated Uracil Water Clusters}{76}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}General introduction}{76}{subsection.3.3.1}
\contentsline {subsection}{\numberline {3.3.2}Results and Discussion}{78}{subsection.3.3.2}
\contentsline {subsubsection}{\numberline {3.3.2.1}Experimental Results}{79}{subsubsection.3.3.2.1}
\contentsline {subsubsection}{\numberline {3.3.2.2}Calculated Structures of Protonated Uracil Water Clusters}{84}{subsubsection.3.3.2.2}
\contentsline {subsection}{\numberline {3.3.3}Conclusions on (H$_2$O)$_{n}$UH$^+$ clusters}{94}{subsection.3.3.3}
\contentsline {chapter}{\numberline {4}Dynamical Simulation of Collision-Induced Dissociation}{99}{chapter.4}
\contentsline {section}{\numberline {4.1}Experimental Methods}{99}{section.4.1}
\contentsline {subsection}{\numberline {4.1.1}Principle of TCID}{101}{subsection.4.1.1}
\contentsline {subsection}{\numberline {4.1.2}Experimental Setup}{102}{subsection.4.1.2}
\contentsline {section}{\numberline {4.2}Computational Details}{104}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}SCC-DFTB Potential}{104}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}Collision Trajectories}{105}{subsection.4.2.2}
\contentsline {subsection}{\numberline {4.2.3}Trajectory Analysis}{106}{subsection.4.2.3}
\contentsline {section}{\numberline {4.3}Dynamical Simulation of Collision-Induced Dissociation of Protonated Uracil Water Clusters}{107}{section.4.3}
\contentsline {subsection}{\numberline {4.3.1}Introduction}{107}{subsection.4.3.1}
\contentsline {subsection}{\numberline {4.3.2}Results and Discussion}{108}{subsection.4.3.2}
\contentsline {subsubsection}{\numberline {4.3.2.1}Statistical Convergence}{108}{subsubsection.4.3.2.1}
\contentsline {subsection}{\numberline {4.3.3}Time-Dependent Proportion of Fragments}{111}{subsection.4.3.3}
\contentsline {subsection}{\numberline {4.3.4}Proportion of Neutral Uracil Loss and Total Fragmentation Cross Sections for Small Clusters}{114}{subsection.4.3.4}
\contentsline {subsection}{\numberline {4.3.5}Behaviour at Larger Sizes, the Cases of (H$_2$O)$_{11, 12}$UH$^+$}{124}{subsection.4.3.5}
\contentsline {subsection}{\numberline {4.3.6}Mass Spectra of Fragments with Excess Proton}{128}{subsection.4.3.6}
\contentsline {subsection}{\numberline {4.3.7}Conclusions about CID of (H$_2$O)$_{n}$UH$^+$}{131}{subsection.4.3.7}
\contentsline {section}{\numberline {4.4}Dynamical Simulation of Collision-Induced Dissociation for Pyrene Dimer Cation}{133}{section.4.4}
\contentsline {subsection}{\numberline {4.4.1}Introduction}{133}{subsection.4.4.1}
\contentsline {subsection}{\numberline {4.4.2}Calculation of Energies}{135}{subsection.4.4.2}
\contentsline {subsection}{\numberline {4.4.3}Simulation of the Experimental TOFMS}{137}{subsection.4.4.3}
\contentsline {subsection}{\numberline {4.4.4}Results and Discussion}{139}{subsection.4.4.4}
\contentsline {subsubsection}{\numberline {4.4.4.1}TOFMS Comparison}{139}{subsubsection.4.4.4.1}
\contentsline {subsubsection}{\numberline {4.4.4.2}Molecular Dynamics Analysis}{140}{subsubsection.4.4.4.2}
\contentsline {subsection}{\numberline {4.4.5}Conclusions about CID of Py$_2^+$}{156}{subsection.4.4.5}
\contentsline {chapter}{\numberline {5}General Conclusions and Perspectives}{159}{chapter.5}
\contentsline {section}{\numberline {5.1}General Conclusions}{159}{section.5.1}
\contentsline {section}{\numberline {5.2}Perspectives}{162}{section.5.2}
\contentsline {chapter}{References}{165}{chapter*.83}