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%\section{Molecular Clusters}
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The term \textit{cluster} was coined by Cotton in the early 1960s to refer to compounds containing metal-metal bonds such as [Re$_2$Cl$_8$]$^{2-}$
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The term \textit{cluster} was coined by F. A. Cotton in the early 1960s to refer to compounds containing metal-metal bonds such as [Re$_2$Cl$_8$]$^{2-}$
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and [Re$_2$Br$_8$]$^{2-}$.\cite{Cotton1964} He defined metal atom cluster compounds as "\textit{those containing a finite group of metal atoms
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which are held together entirely, mainly, or at least to a significant extent, by bonds directly between the metal atoms even though some non-metal
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atoms may be associated intimately with the cluster}". Subsequently, the study of clusters, also referred to as aggregates, has greatly diversified
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@ -92,8 +92,8 @@ potential energy surface (PES), or energy landscape, can be extremely complex an
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exist. The properties of a given cluster can significantly differ from the properties of the corresponding bulk material. For instance, the magnetic moment
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of small iron particles at room temperature is smaller than that of the bulk.\cite{Kimura1991} However, a gradual transition occurs between the properties
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of the clusters and those of the corresponding bulk as cluster size increases.\cite{Jortner1992} This transition can be rough or continuous depending
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on the considered species and properties. For instance, Landman \textit{et al.} reported that anionic gold clusters favor planar structures
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up to $\sim$13 atoms.\cite{Hakkinen2002} L'Hermite \textit{et al.} also reported that the transition temperature extracted from he
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on the considered species and properties. For instance, U. Landman \textit{et al.} reported that anionic gold clusters favor planar structures
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up to $\sim$13 atoms.\cite{Hakkinen2002} J.-M. L'Hermite \textit{et al.} also reported that the transition temperature extracted from he
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heat capacity curve of protonated water clusters (H$_{2}$O)$_{n}$H$^{+}$ has a strong size dependence as seen in Figure~\ref{T_trans}.\cite{Boulon2014}
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Consequently, the study of clusters allow to bridge the gap between single molecule or atom properties and bulk materials, which can be of help
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to reveal microscopic aspects which are hardly observable in the bulk only.
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@ -140,7 +140,7 @@ in order to determine the lowest energy isomer of a given cluster. For instance,
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conducted to determine the low energy structures of (H$_2$O)$_n$ and (H$_2$O)$_n$H$^+$ aggregates. Among them, we can mention
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the studies performed by D. Wales and co-workers using the basin-hopping algorithm.\cite{Wales1997,Wales1998,Wales1999,James2005}
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In more difficult cases, the probe properties result from the contribution of several isomers which have to be taken into account. When
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considering finite-temperature properties, an ergodic exploration of the PES also need to be performed. For instance, Boulon \textit{et al.}
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considering finite-temperature properties, an ergodic exploration of the PES also need to be performed. For instance, J. Boulon \textit{et al.}
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reported heat capacity curves as a function of temperature of mass selected protonated water clusters and highlighted a stronger steepness
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of the curve of (H$_2$O)$_{21}$H$^+$ as compared to adjacent sizes.\cite{Boulon2014} Theoretical simulations latter provided explanations
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for this peculiar behavior.\cite{Korchagina2017} When considering dissociation of clusters, which can be a non-equilibrium process, theoretical
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@ -190,8 +190,8 @@ It is therefore often necessary to make a choice between computational efficienc
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and accuracy. This balance determines the nature of the questions that can be addressed.
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Furthermore, the aforementioned \textbf{enhanced sampling methods} generally require to
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repeat a large amount of calculations. Therefore, they need to be combined with computationally
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efficient approaches to compute the PES. As presented in Chapter~\ref{chap:comput_method}, the method
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I use within this thesis is the \textbf{self-consistent-charge density functional based tight-binding} (SCC-DFTB) method.
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efficient approaches to compute the PES. As presented in chapter~\ref{chap:comput_method}, the method
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I use within this thesis is the \textbf{self-consistent-charge density-functional based tight-binding} (SCC-DFTB) method.
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\end{itemize}
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%Hydrogen bonding is arguably the most extensively studied among all the noncovalent interactions. Hydrogen bonding governs many chemical
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@ -228,7 +228,7 @@ the existence of charged molecular aggregates in the stratosphere,\cite{Arnold19
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and ammonium/ammonia containing aggregates.\cite{Payzant1973, Berden1996} In the latter case, \textbf{ammonia has been highlighted as an
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important component of atmospheric nucleation} together with water and sulphuric acid.\cite{Kulmala1995, Kirkby2011, Dunne2016} This important
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role of ammonia and ammonium water clusters, and the lake of theoretical studies devoted to these species, motivated a thorough benchmark of
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the SCC-DFTB approach to model these systems which is presented in Chapter~\ref{chap:structure}. In parallel, understanding the \textbf{properties of
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the SCC-DFTB approach to model these systems which is presented in chapter~\ref{chap:structure}. In parallel, understanding the \textbf{properties of
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the proton} and how it can impact the solvation properties of molecules of biological interest is of paramount importance for understanding
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fundamental processes in biology and chemistry. In particular, uracil, one of the nucleobases, plays a key role in the encoding and expression of genetic
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information in living organisms. The study of \textbf{water clusters containing uracil} is therefore a good playground to probe how uracil properties
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@ -313,12 +313,11 @@ simulation time of several hundred picoseconds. This approach has thus been use
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collision-induced dissociation experiments.
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To summarize, the goal of this thesis is to go a step further into the theoretical description of the properties of molecular clusters
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in the view to complement complex experimental measurements. It has focused on two different types of molecular clusters. First,
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we have focused on water clusters containing an impurity, \textit{i.e.} an additional ion or molecule. We have first focused our
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in the view to complement complex experimental measurements. It has focused on two different types of molecular clusters. First, I focused on water clusters containing an impurity, \textit{i.e.} an additional ion or molecule. I have first focused my
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studies on \textbf{ammonium and ammonia water clusters} in order to thoroughly explore their PES to characterize in details
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low-energy isomers for various cluster sizes. We then tackle the study of \textbf{protonated uracil water clusters} through two
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low-energy isomers for various cluster sizes. Then I tackle the study of \textbf{protonated uracil water clusters} through two
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aspects: characterize low-energy isomers and model collision-induced dissociation experiments to probe dissociation mechanism
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in relation with recent experimental measurements. Finally, we address the study of the \textbf{pyrene dimer cation} to explore collision
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in relation with recent experimental measurements. Finally, I address the study of the \textbf{pyrene dimer cation} to explore collision
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trajectories, dissociation mechanism, energy partition, mass spectra, and cross-section.
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To introduce, develop, and conclude on these different subjects, this manuscript is organised as follow:
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@ -404,8 +400,8 @@
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@ -87,8 +87,9 @@ In the case of a system composed of $M$ nuclei and $N$ electrons, the Schr{\"o}d
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\label{TDSE}
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i\hbar\frac{\partial\Psi(\mathbf{R}_\alpha, \mathbf{r}_j, t)}{\partial t}=\hat{H}(\mathbf{R}_\alpha, \mathbf{r}_j, t)\Psi(\mathbf{R}_\alpha, \mathbf{r}_j, t), \quad \alpha=1, 2,..., M; j=1, 2,..., N.
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\end{eqnarray}
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This equation describes the evolution of the wavefunction in space and time, where $i$ is the imaginary unit. $\hbar$=$h/2\pi$ ~(1.054572$\times$ 10$^{-34}$~J$\cdot$s) is the reduced Planck constant and $t$ is the time. \textbf{R$_\alpha$} and \textbf{r}$_j$ refer to the coordinates of nucleus $\alpha$ and electron $j$, respectively. $\hat{H}$ is the so called Hamilton operator corresponding to the total energy of the system.
|
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when the Hamiltonian itself is explicitly independent on time (the total wavefunction still has a time dependency), it is possible to decompose the space variables and the time variable to write the time-independent Schr{\"o}dinger equation:
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The vectorial variables are in bold in this manuscript.
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This equation describes the evolution of the wavefunction in space and time, where $i$ is the imaginary unit. $\hbar$=$h/2\pi$ ~(1.054572$\times$ 10$^{-34}$~J$\cdot$s) is the reduced Planck constant and $t$ is the time. \textbf{R$_\alpha$} and \textbf{r}$_j$ refer to the coordinates of nucleus $\alpha$ and electron $j$, respectively. $\hat{H}$ is the so called Hamiltonian operator corresponding to the total energy of the system.
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When the Hamiltonian itself is explicitly independent on time (the total wavefunction still has a time dependency), it is possible to decompose the space variables and the time variable to write the time-independent Schr{\"o}dinger equation:
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\begin{eqnarray}
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\label{TISE}
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\hat{H}\Psi_k(\mathbf{R}_\alpha, \mathbf{r}_j)=E_k\Psi_k(\mathbf{R}_\alpha, \mathbf{r}_j)
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||||
@ -99,7 +100,8 @@ where $E_k$ is the total energy of the system associated with the eigenstate $\P
|
||||
\label{waveFunc}
|
||||
\Psi(\mathbf{R}_\alpha, \mathbf{r}_j, t)=\sum_{k}c_k\Psi_k(\mathbf{R}_\alpha, \mathbf{r}_j)e^{-iE_kt/\hbar}
|
||||
\end{eqnarray}
|
||||
where $c_k$ is the coefficient at t=0. The eigenstates obtained from equation \ref{TISE} are the stationary states of the system and form a complete basis of
|
||||
where $c_k$ is a coefficient. %at t=0.
|
||||
The eigenstates obtained from equation \ref{TISE} are the stationary states of the system and form a complete basis of
|
||||
orthonormal vectors. The lowest energy eigenstate is called the ground state usually denoted as $\Psi_0$ and $E_0$ is the corresponding energy.
|
||||
|
||||
In a system made up of $M$ nuclei and $N$ electrons, the Hamiltonian operator (in a non-relativistic framework) is written as following:
|
||||
@ -115,8 +117,10 @@ In a system made up of $M$ nuclei and $N$ electrons, the Hamiltonian operator (i
|
||||
\label{operatorH}
|
||||
\end{align}
|
||||
%\end{eqnarray}
|
||||
in which the atomic unit system is used. The energy is thus expressed in Hartree. The mass is in unit of the mass of electron and the length is in Bohr. The vectorial variables are in bold in this present manuscript. $M_\alpha$ refers to the mass of nucleus $\alpha$ (in atomic unit) and $Z_\alpha$ is the atomic number of $a$. $\hat{T}_n$ and $\hat{T}_e$ are the kinetic energy operators of nuclei and electrons, respectively. $\hat{V}_{nn}$, $\hat{V}_{ee}$ and $\hat{V}_{ne}$ denote the potential energy operators of the repulsion between the nuclei, repulsion between the electrons and electrostatic attraction between nuclei and electrons, respectively.
|
||||
\textbf{$\nabla^2$} is the Laplace operator. For nucleus $\alpha$ in Cartesian coordinates in three dimensions, its position vector is \textbf{R}$_\alpha$=(X$_\alpha$, Y$_\alpha$, Z$_\alpha$) and \textbf{$\nabla^2$} is expressed as following:
|
||||
in which the atomic unit system is used. The energy is thus expressed in Hartree. The mass is in unit of the mass of electron and the length is in Bohr. $M_\alpha$ refers to the mass of nucleus $\alpha$ (in atomic unit) and $Z_\alpha$ is the atomic number.
|
||||
%of atom $\alpha$.
|
||||
$\hat{T}_n$ and $\hat{T}_e$ are the kinetic energy operators of nuclei and electrons, respectively. $\hat{V}_{nn}$, $\hat{V}_{ee}$ and $\hat{V}_{ne}$ denote the potential energy operators of the repulsion between the nuclei, repulsion between the electrons and electrostatic attraction between nuclei and electrons, respectively.
|
||||
\textbf{$\nabla^2$} is the Laplace operator. For nucleus $\alpha$ in three dimensions using Cartesian coordinates, its position vector is \textbf{R}$_\alpha$=(X$_\alpha$, Y$_\alpha$, Z$_\alpha$) and \textbf{$\nabla^2$} is expressed as following:
|
||||
\begin{eqnarray}
|
||||
\label{laplace}
|
||||
\mathbf{\nabla}^2 = \frac{\partial^2}{\partial{X}_\alpha^2} +
|
||||
@ -127,7 +131,9 @@ in which the atomic unit system is used. The energy is thus expressed in Hartree
|
||||
|
||||
Electrons are very light particles which can not be described correctly even qualitatively through classical mechanics. If we want to describe the electron
|
||||
distribution in details, quantum mechanics must be applied, \textit{i.e.} solving the Schr{\"o}dinger equation. Usually in atomic and molecular systems,
|
||||
it is very hard, if not impossible, to obtain the exact solutions of Schr{\"o}dinger equation, as this requires to separate variables. It is only possible in the
|
||||
it is very hard, if not impossible, to obtain the exact solutions of Schr{\"o}dinger equation.
|
||||
%as this requires to separate variables.
|
||||
It is only possible in the
|
||||
system containing one nucleus and one electron, \textit{i.e.}, the hydrogen atom or hydrogenic ions. For molecular species, the mathematical complexity
|
||||
to solve the Schr{\"o}dinger equation will increase with the number of degrees of freedom of the system. Thus, it is necessary to resort to approximations
|
||||
in almost all cases.
|
||||
@ -150,7 +156,7 @@ BO approximation consists of expressing the total wavefunction of a molecule as
|
||||
an electronic wavefunction, which enables a separation of the Hamiltonian operator into the fast electronic term and the usually much slower nuclear
|
||||
term, where the coupling between electrons and nuclei is neglected so that the two smaller and non-coupled systems can be solved more efficiently.
|
||||
In mathematical terms, the total wavefunction $\Psi_\mathrm{tot}$ of a molecule can be expressed as an expansion in the complete set of electronic
|
||||
wavefunctions $\psi_k^e$ with the expansion coefficients being functions of the nuclei coordinates $\psi^n$:
|
||||
wavefunctions $\psi_k^e$ with the expansion coefficients being parametric functions of the nuclei coordinates \textbf{R}$_\alpha$:
|
||||
\begin{eqnarray}
|
||||
\label{BO}
|
||||
\Psi_\mathrm{tot}(\mathbf{R}_\alpha, \mathbf{r}_j) = \sum_{k=1}^{\infty}
|
||||
@ -180,26 +186,26 @@ where the eigenvalue $E_k^e$, electronic energy, depends on the chosen positions
|
||||
In the second step, the nuclear kinetic energy ${T}_n$ is reintroduced, and the Schr{\"o}dinger equation for the nuclear motion is:
|
||||
\begin{eqnarray}
|
||||
\label{nucSeq}
|
||||
(\hat{T}_n + E_{k}^e(\mathbf{R}_\alpha) + \langle \Psi_k \vert \mathbf{\nabla}_n^2 \vert \Psi_k \rangle )\psi_k^n(\mathbf{R}_\alpha) = E_\mathrm{tot}\psi_k^n(\mathbf{R}_\alpha)
|
||||
(\hat{T}_n + E_{k}^e(\mathbf{R}_\alpha) + \langle \Psi_k \vert \mathbf{\nabla}_n^2 \vert \Psi_k \rangle )\psi_k^n(\mathbf{R}_\alpha) = E_k^\mathrm{tot}\psi_k^n(\mathbf{R}_\alpha)
|
||||
\end{eqnarray}
|
||||
|
||||
The eigenvalue $E_\mathrm{tot}$ is the total energy of the molecule, which includes the overall rotation translation of the molecule, contributions from electrons,
|
||||
The eigenvalue $E_k^\mathrm{tot}$ is the total energy of the molecule, which includes the overall rotation translation of the molecule, contributions from electrons,
|
||||
and nuclear vibrations. This second step involves a separation of vibrational, translational, and rotational motions.
|
||||
Born and Oppenheimer assumed that the integral $\langle \Psi_k \vert \mathbf{\nabla}_n^2 \vert \Psi_k \rangle$ (diagonal correction) weakly depends on the nuclear coordinates,
|
||||
M. Born and R. Oppenheimer assumed that the integral $\langle \Psi_k \vert \mathbf{\nabla}_n^2 \vert \Psi_k \rangle$ (diagonal correction) weakly depends on the nuclear coordinates,
|
||||
so that it can be ignored. \cite{Born1927} Therefore, the Born-Oppenheimer approximation allows to describe the movement of nuclei in the corresponding potential to an adiabatic electronic state by the following equation:
|
||||
\begin{eqnarray}
|
||||
\label{ESeq}
|
||||
(\hat{T}_n + E_{k}^e(\mathbf{R}_\alpha))\psi_k^n(\mathbf{R}_\alpha) = E_\mathrm{tot}\psi_k^n(\mathbf{R}_\alpha)
|
||||
(\hat{T}_n + E_{k}^e(\mathbf{R}_\alpha))\psi_k^n(\mathbf{R}_\alpha) = E_0^\mathrm{tot}\psi_k^n(\mathbf{R}_\alpha)
|
||||
\end{eqnarray}
|
||||
|
||||
In this thesis, we will assume electrons adapt fast to reach their electronic ground state.
|
||||
In this thesis, I will assume electrons adapt fast to reach their electronic ground state.
|
||||
%I only focus on the electronic ground state.
|
||||
The potential energy $E_0^e$ thus equals the ground state electronic
|
||||
energy $E_{0}$ and the total energy is then $E_0^\mathrm{tot}=T_\mathrm{n} + E_{0}$. The Schr{\"o}dinger equation for $\psi_0^n(\mathbf{R}_\alpha)$ can
|
||||
therefore be written as:
|
||||
\begin{eqnarray}
|
||||
\label{classical}
|
||||
(\hat{T}_n + E_{0})\psi_0^n(\mathbf{R}_\alpha) = E_\mathrm{tot}\psi_0^n(\mathbf{R}_\alpha)
|
||||
(\hat{T}_n + E_{0})\psi_0^n(\mathbf{R}_\alpha) = E_0^\mathrm{tot}\psi_0^n(\mathbf{R}_\alpha)
|
||||
\end{eqnarray}
|
||||
The next step is usually to consider the nuclei can be described classically. One can then consider that they evolve classically \textit{i.e.} following new Newton's equation, on a PES defined by the ground state electronic energy. The calculation of the ground state electronic energy is discussed in section~\ref{electronicEnergy}
|
||||
%<<<<<<< HEAD
|
||||
@ -247,7 +253,7 @@ Experimental spectroscopic investigations help in understanding the electronic s
|
||||
emission and scattering. These measurements can often provide a detailed picture of molecular systems but sometimes they are difficult to interpret.
|
||||
In the last few decades, molecular electronic-structure theory has developed to a stage where it can provide invaluable
|
||||
assistance in the interpretation of experimental measurements of a wide range of important properties of molecules in rotational and vibrational spectroscopies,
|
||||
magnetic-resonance spectroscopies, ultraviolet/visible spectroscopies, and otherss.\cite{Puzzarini2010, Helgaker1999, Pedone2010, Fleming1986, Laane1999, Helgaker2012}
|
||||
magnetic-resonance spectroscopies, ultraviolet/visible spectroscopies, and others.\cite{Puzzarini2010, Helgaker1999, Pedone2010, Fleming1986, Laane1999, Helgaker2012}
|
||||
|
||||
Electronic wavefunction of systems including three or more interacting particles can not be obtained analytically, so approximations must be applied.
|
||||
Many approximations have been proposed to obtain approximate solutions of the exact electronic wavefunction. Each one of them is usually the basis
|
||||
@ -260,14 +266,14 @@ Post-HF theory usually generates more accurate results by considering the electr
|
||||
DFT methods in the Kohn-Sham formulation can be regarded as an improvement over the HF theory as it considers
|
||||
approximated electronic correlation.
|
||||
Density-functional based tight-binding (DFTB) formalism is an approximated DFT method that involves additional approximations.\cite{dftb1, dftb2, Elstner1998, Elstner2014}
|
||||
PES can be explored with \textit{ab initio} methods.\cite{Cremer1993}
|
||||
%PES can be explored with \textit{ab initio} methods.\cite{Cremer1993}
|
||||
Compared with quantum chemical (QM) methods that require considerable computer resources, molecular mechanics (MM) calculations are much cheaper but
|
||||
present severe limitations in the treatment of chemical reactivity.. The possibility to model chemical reactions using \textit{ab initio} MD calculation is also
|
||||
severely hindered by the short simulation time accessible with such methods. QM/MM techniques that combine QM for the reactive region and MM for the
|
||||
present severe limitations in the treatment of chemical reactivity.
|
||||
%The possibility to model chemical reactions using \textit{ab initio} MD calculation is also severely hindered by the short simulation time accessible with such methods.
|
||||
QM/MM techniques that combine QM for the reactive region and MM for the
|
||||
remainder are very promising, especially for large systems.\cite{Singh1986, Gao1996, Mordasini1998}
|
||||
|
||||
The following section focuses on the description of wavefunction based methods, density functional theory method, and the density-functional based
|
||||
tight-binding methods used to solve Schr{\"o}dinger equation and the electronic structure problem.
|
||||
The following section focuses on the description of wavefunction based methods, DFT method, and the DFTB methods used to solve Schr{\"o}dinger equation and the electronic structure problem.
|
||||
|
||||
\subsection{Wavefunction based Methods}
|
||||
|
||||
@ -283,31 +289,30 @@ Hartree proposed that the electronic wavefunction could be approximated by assum
|
||||
\end{eqnarray}
|
||||
|
||||
The HF theory assumes that every electron moves in an average field of all the other electrons and the nuclei in the molecule, which is an example of a mean-field approximation.
|
||||
The HF equations for an individual electron $j$ moving in the mean field \textit{V}$_i^{HF}$, can be expressed:
|
||||
|
||||
The HF equations for an individual electron $j$ moving in the mean field \textit{V}$_i^{HF}$, can be expressed as:
|
||||
\begin{equation}
|
||||
\left(-\frac{1}{2}\mathbf{\nabla}_j^2 + V_\sigma(\mathbf{r}_j) + V_\mathrm{H}(\mathbf{r}_j)\right)\psi^\mathrm{HF}_\sigma (\mathbf{r}_j)-
|
||||
\sum_{k=1}^{N_\sigma} \int d^3 {r'} \frac{\psi_{k\sigma}^{\mathrm{HF*}}(\mathbf {r'})\psi_{k\sigma}^\mathrm{HF}(\mathbf r_j)}{\mid \mathbf r_j-\mathbf{r'}\mid}\psi^\mathrm{HF}_{\sigma} (\mathbf{r'})= E_\sigma^\mathrm{HF} \psi_\sigma^\mathrm{HF}(\mathbf{r}_j)
|
||||
\label{HF-individual}
|
||||
\end{equation}
|
||||
|
||||
where $\nabla_j^2$ depends on the $j$th electron coordinates. $V_\sigma(\textbf{r}_j)$ refers to the external potential. $\sigma$ is the spin. The last term on the left-hand side is the HF exchange potential.
|
||||
Using the electronic density, we can obtain:
|
||||
\begin{equation}
|
||||
V_\mathrm{H}(\mathbf{r}_j) = \int d^3 {r}_j \frac{\rho (\mathbf{r'})}{\mid \mathbf{r}_j - \mathbf{r'} \mid}
|
||||
\label{Hartreepotential}
|
||||
\end{equation}
|
||||
|
||||
$V_{H}(\mathbf{r}_j)$ is the Hartree potential:
|
||||
\begin{equation}
|
||||
\rho(\mathbf{r}_j) =\sum_\sigma\sum_j^{N_\sigma}\rho_\sigma(\mathbf{r}_j)=\sum_\sigma\sum_j^{N_\sigma} \mid \psi_{\sigma}^\mathrm{H}(\mathbf {r}_j)\mid^2
|
||||
\label{HF-idensity}
|
||||
\end{equation}
|
||||
\textbf{r'} refers to the positions of all other electrons except electron $j$.
|
||||
%density means the probability of findind an electron
|
||||
where $\nabla_j^2$ depends on the $j$th electron coordinates.
|
||||
$V_\sigma(\textbf{r}_j)$ refers to the external potential. $\sigma$ is the spin. $V_{H}(\mathbf{r}_j)$ is the Hartree potential
|
||||
|
||||
% Hartree potential is a local potential.
|
||||
%j\sigma refers to a state.
|
||||
The last term on the left-hand side of eq \ref{HF-individual} is the HF exchange potential.
|
||||
%which can be viewed as an operator acting on the ortital $\psi^{\mathbf{HF}_{j\sigma}(\mathbf r_j)$ in such a way that the result of its action at position $(\mathbf{r}_j}$ involves an integration of $\psi^{\mathbf{HF}_{j\sigma} over all \mathbf{r'}.
|
||||
\textbf{r'} refers to the positions of all other electrons except electron $j$.
|
||||
|
||||
%The solutions of the individual Hartree equations depend on the solutions of all the other individual Hartree equations via \textit{V}$_i^{\mathrm{HF}}$. Thus, they have to be solved iteratively.
|
||||
%<<<<<<< HEAD
|
||||
%%First, an initial guess for the individual one-electron wavefunctions is needed. After this, iterate continues until all \psi$_{i}$ stay the same, which means a self-consistent field (SCF) has been achieved.
|
||||
@ -372,28 +377,24 @@ more CSFs. Standard multiconfigurational approaches are the CI, CC, and perturba
|
||||
For a long time, approximations based on the wavefunction were systematically applied to solve the Schr{\"o}dinger equation.
|
||||
However, it is usually impractical to perform a wavefunction based calculation with chemical accuracy for complex or large systems.
|
||||
Density functional theory is based on the electron density rather than the electronic wavefunction.\cite{Kohn1965, kohn1999nobel}
|
||||
Because DFT displays a more favourable scaling of computational resources with respect to system size, DFT is nowadays the most
|
||||
widely used method available in computational chemistry, computational physics, and condensed-matter physics for ground state
|
||||
Because DFT displays a more favourable scaling of computational resources with respect to system size, DFT is nowadays the most widely used method available in computational chemistry, computational physics, and condensed-matter physics for ground state
|
||||
calculation of large and complex systems.
|
||||
|
||||
Although DFT has a history almost as old as the Schr{\"o}dinger equation, the modern form dates back to the paper published by P. Hohenberg and W. Kohn \cite{Hohenberg1964}
|
||||
that introduced the two Hohenberg–Kohn (HK) theorems in 1964 and the extension by M. Levy in 1979.\cite{Levy1979} The theory is usually applied in the form latter suggested by W. Kohn and L. J. Sham in 1965.\cite{Kohn1965}
|
||||
DFT makes it possible to transform the problem of electrons interacting and evolving in a nuclear potential to a problem of
|
||||
independent electrons evolving in an effective potential. The electron density $\rho$(\textbf{r}) corresponds to the number of
|
||||
electrons per unit volume in a given state.
|
||||
The central idea of DFT is to promote $\rho$($\mathbf{r}$) (function which only depends on three spatial coordinates) as the
|
||||
key variable in the determination of the electronic energy of a system. From the electronic wavefunction of the system, $\rho$($\mathbf{r}$)
|
||||
is written as:
|
||||
|
||||
%>>>>>>> 92023a10c3aa8b7dc4ace43987c1d571fb99a738
|
||||
\begin{eqnarray}
|
||||
\label{density}
|
||||
\rho(\mathbf{r})=N\int \Psi^*(\mathbf{r}, \mathbf{r}_2, \mathbf{r}_3, ..., \mathbf{r}_N) \Psi(\mathbf{r}, \mathbf{r}_2, \mathbf{r}_3,..., \mathbf{r}_N) d\mathbf{r}_2 d\mathbf{r}_3...d\mathbf{r}_N
|
||||
\end{eqnarray}
|
||||
The central idea of DFT is to promote $\rho$($\mathbf{r}$) (function which only depends on three spatial coordinates) as the
|
||||
key variable in the determination of the electronic energy of a system.
|
||||
%From the electronic wavefunction of the system, $\rho$($\mathbf{r}$) is written as:
|
||||
|
||||
This idea originates from the the model of the uniform electron gas in the phase space around an atom developed in 1927 by Thomas \cite{Thomas1927} and Fermi \cite{Fermi1928}, which is the predecessor to density functional theory. Nevertheless, the Thomas-Fermi model is unable to correctly describe molecular bonds because it does not take into account the exchange and correlation energies.
|
||||
|
||||
Although DFT has a history almost as old as the Schr{\"o}dinger equation, the modern form dates back to the paper published by P. Hohenberg and W. Kohn \cite{Hohenberg1964}
|
||||
that introduced the two Hohenberg–Kohn (HK) theorems in 1964 and the extension by Levy in 1979.\cite{Levy1979} The theory is usually applied in the form latter suggested
|
||||
by Kohn and Sham in 1965.\cite{Kohn1965}
|
||||
%>>>>>>> 92023a10c3aa8b7dc4ace43987c1d571fb99a738
|
||||
This idea originates from the the model of the uniform electron gas in the phase space around an atom developed in 1927 by L. H. Thomas \cite{Thomas1927} and E. Fermi \cite{Fermi1928}, which is the predecessor to density functional theory. Nevertheless, the Thomas-Fermi model is unable to correctly describe molecular bonds because it does not take into account the exchange and correlation energies.
|
||||
|
||||
The first HK theorem shows that, for a many-electron system in its ground state, the energy is uniquely determined by the electron density $\rho$(\textbf{r}).
|
||||
In other words, the first HK theorem shows that it is not necessary to know the wavefunction of the system to obtain its energy and that the knowledge of the electron
|
||||
@ -410,12 +411,12 @@ and $N$ can be obtained via the normalization of $\rho$(\textbf{r}):
|
||||
\label{N}
|
||||
\int \rho(\mathbf{r}) d\mathbf{r}=N
|
||||
\end{eqnarray}
|
||||
and $N$ and $V_\mathrm{ext}$(\textbf{r}) determine the electronic Hamiltonian. $\rho$(\textbf{r}) determines the energy and all other ground state electronic properties of a system. This is clearly shown in Figure \ref{variables}.
|
||||
$N$ and $V_\mathrm{ext}$(\textbf{r}) determine the electronic Hamiltonian. $\rho$(\textbf{r}) determines the energy and all other ground state electronic properties of a system. This is clearly shown in Figure \ref{variables}.
|
||||
\figuremacro{variables}{Interdependence of basic variables in the Hohenberg-Kohn theorem.} \\
|
||||
|
||||
The second HK theorem is a variational electron density theorem which defines an energy functional for a system.
|
||||
For a given external potential $V_\mathrm{ext}$(\textbf{r}), the ground state energy $E_0$ of the system is the minimum
|
||||
value of the energy functional obtained for the exact ground state electron density $\rho$(\textbf{r}) of the system.
|
||||
The second HK theorem is a variational electron density theorem which defines an energy functional of the electronic density for a system.
|
||||
For a given external potential $V_\mathrm{ext}$(\textbf{r}), the ground state energy $E_0$ of the system is obtained by minimizing this energy function with respect to the electronic density, the corresponding density is exactly the ground state electronic density $\rho$(\textbf{r}).
|
||||
%the minimum value of the energy functional obtained for the exact ground state electron density $\rho$(\textbf{r}) of the system.
|
||||
%%
|
||||
From the HK theorems, we can write the functional of the total energy of the system as a sum of the kinetic energy of th
|
||||
electrons $T_\mathrm{e}$[$\rho$(\textbf{r})], and the electronic interaction energy $E_\mathrm{ee}$[$\rho$(\textbf{r})]:
|
||||
@ -445,8 +446,7 @@ $T_\mathrm{no}$[$\rho$(\textbf{r})] is the kinetic energy of a system of non-int
|
||||
T_\mathrm{no}[\rho(\mathbf{r})]=\sum_{i}^{N} \left \langle \Psi_i \left|-\frac{1}{2}\nabla^2 \right | \Psi_i \right \rangle
|
||||
\end{eqnarray}
|
||||
|
||||
$E_H$[$\rho$(\textbf{r})] represents the Hartree energy which corresponds to the interaction energy of a classical charge distribution
|
||||
of density $\rho$(\textbf{r}):
|
||||
$E_H$[$\rho$(\textbf{r})] represents the Hartree energy which corresponds to the interaction energy of a classical charge distribution of density $\rho$(\textbf{r}):
|
||||
|
||||
\begin{eqnarray}
|
||||
\label{Ehartree}
|
||||
@ -456,34 +456,33 @@ E_\mathrm{H}[\rho(\textbf{r})]=\frac{1}{2} \int \int \frac{\rho(\mathbf{r})\rho(
|
||||
$V_\mathrm{ext}$(\textbf{r}) is the external potential.
|
||||
The remaining energy components are assembled in the exchange-correlation energy $E_\mathrm{xc}$[$\rho$(\textbf{r})] functional containing
|
||||
the difference between the kinetic energy of the real system $T$[$\rho$(\textbf{r})] and that of the non-interacting system $T_n$[$\rho$(\textbf{r})]
|
||||
and the non-classical part of $E_\mathrm{ee}$[$\rho$].
|
||||
and the non-classical part of $E_\mathrm{ee}$[$\rho$], adding the difference between $V_\mathrm{ee} [\rho(\mathbf{r})$ and $E_\mathrm H[\rho(\textbf{r})]$.
|
||||
The $E_\mathrm{xc}$[$\rho$(\textbf{r})] functional can thus be expressed as:
|
||||
\begin{eqnarray}
|
||||
\label{Exc}
|
||||
E_\mathrm{xc}[\rho(\mathbf{r})]=(T[\rho(\mathbf{r})]-T_\mathrm{no}[\rho(\mathbf{r})])+(V_\mathrm{ee}[\rho(\mathbf{r})]-E_\mathrm H[\rho(\textbf{r})])
|
||||
\end{eqnarray}
|
||||
|
||||
To minimize the energy $E$[$\rho$(\textbf{r})] with respect to $\rho$(\textbf{r}) applying by the variational principle
|
||||
In practice, to minimize the energy $E$[$\rho$(\textbf{r})] with respect to $\rho$(\textbf{r}) applying by the variational principle
|
||||
while considering the constraints of orbital orthogonality, one performs an optimization under the constraints using Lagrange multipliers.
|
||||
For the optimal $\rho$(\textbf{r}), the energy $E$[$\rho$(\textbf{r})] does not change upon the variation of $\rho$($\mathbf{r}$), providing that
|
||||
$\rho$(\textbf{r}) obey to equation~\ref{N}:
|
||||
|
||||
\begin{eqnarray}
|
||||
\label{Delta}
|
||||
\delta(E[\rho(\mathbf{r})]-\mathcal{L}\rho(\mathbf{r}))=0
|
||||
\end{eqnarray}
|
||||
where $\mathcal{L}$ is a corresponding Lagrange multiplier.
|
||||
Combining eqs \ref{E_KS}, \ref{Tn}, \ref{Ehartree}, and \ref{Delta}, the Euler equation can be written as:
|
||||
|
||||
\begin{eqnarray}
|
||||
\label{Lagrange}
|
||||
\mathcal{L}=V_\mathrm{eff}[\rho(\mathbf{r})]+ \frac{\delta T_n[\rho(\mathbf{r})]}{\delta \rho(\mathbf{r})}
|
||||
\end{eqnarray}
|
||||
where the effective potential $V_\mathrm{eff}$[$\rho$($\mathbf{r}$)] is introduced:
|
||||
%%For the optimal $\rho$(\textbf{r}), the energy $E$[$\rho$(\textbf{r})] does not change upon the variation of $\rho$($\mathbf{r}$), providing that $\rho$(\textbf{r}) obey to equation~\ref{N}:
|
||||
%\begin{eqnarray}
|
||||
%\label{Delta}
|
||||
%\delta(E[\rho(\mathbf{r})]-\mathcal{L}\rho(\mathbf{r}))=0
|
||||
%\end{eqnarray}
|
||||
%where $\mathcal{L}$ is a corresponding Lagrange multiplier.
|
||||
%Combining eqs \ref{E_KS}, \ref{Tn}, \ref{Ehartree}, and \ref{Delta}, the Euler equation can be written as:
|
||||
|
||||
%\begin{eqnarray}
|
||||
%\label{Lagrange}
|
||||
%\mathcal{L}=V_\mathrm{eff}[\rho(\mathbf{r})]+ \frac{\delta T_n[\rho(\mathbf{r})]}{\delta \rho(\mathbf{r})}
|
||||
%\end{eqnarray}
|
||||
Combining eqs \ref{E_KS}, \ref{Tn}, and \ref{Ehartree}, the effective potential $V_\mathrm{eff}$[$\rho$($\mathbf{r}$)] can be introduced as:
|
||||
\begin{align}
|
||||
V_\mathrm{eff}[\rho(\mathbf{r})]&=V_\mathrm{ext}[\rho(\mathbf{r})]+V_\mathrm{H}[\rho(\mathbf{r})]+V_\mathrm{xc}[\rho(\mathbf{r})] \nonumber \\
|
||||
&=V_\mathrm{ext}[\rho(\mathbf{r})]+\int\frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf {r'}|}d\mathbf{r'} +\frac{\delta E_\mathrm{xc}[\rho(\mathbf{r})]}{\delta \rho(\mathbf{r})}
|
||||
&=V_\mathrm{ext}[\rho(\mathbf{r})]+\int\frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf {r'}|}d\mathbf{r'} +\frac{\delta E_\mathrm{xc}[\rho(\mathbf{r})]}{\delta \rho(\mathbf{r})} \\
|
||||
&=\frac{\delta E_\mathrm{eff}[\rho(\mathbf{r})]}{\delta\rho(\mathbf{r})}
|
||||
\label{Veff}
|
||||
\end{align}
|
||||
where $V_{H}$[$\rho$(\textbf{r})] refers to the Hartree potential and $V_{xc}$[$\rho$(\textbf{r})] is the exchange-correlation potential.
|
||||
@ -527,11 +526,10 @@ The computational cost for the incorporation of electron correlation is the one
|
||||
|
||||
DFT is in principle correct if one knows the “exact” exchange-correlation functional. However, despite a lot of work aimed at determining
|
||||
this exact functional, its form is still unknown and a systematic strategy for improvement is not available. Therefore, it is always necessar
|
||||
to use an approximate functional that is characterized by more or less important artefacts in effective calculations. Over the last decades,
|
||||
many exchange-correlation functionals have been proposed. Although they are different, it is possible to classify them into families according
|
||||
to use an approximate functional that is characterized by more or less important artefacts in effective calculations. Over the last decades, many exchange-correlation functionals have been proposed. Although they are different, it is possible to classify them into families according
|
||||
to some common characters. \textbf{Local Density Approximation} (LDA), \textbf{Generalized Gradient Approximation} (GGA), \textbf{meta-GGA},
|
||||
and \textbf{hydrid functionals} (containing to some extent a contribution of HF exact-exchange) are some of the most widely used approximations.
|
||||
\textbf{LDA} is the first approximation of $E_{xc}$[$\rho$(\textbf{r})] proposed by Kohn and Sham,\cite{Kohn1965} which is based on the description
|
||||
\textbf{LDA} is the first approximation of $E_{xc}$[$\rho$(\textbf{r})] proposed by W. Kohn and L. J. Sham,\cite{Kohn1965} which is based on the description
|
||||
of the homogeneous electron gas. For atomic or molecular systems, when densities vary rapidly in space, the assumption of a uniform electron
|
||||
density is not correct and the LDA approximation is not applicable any more. For example, binding energies in molecules are usually overestimated
|
||||
by the LDA approximation. To solve this problem, a new family of functionals, \textbf{GGA functional}, which includes a contribution of the electron
|
||||
@ -565,7 +563,7 @@ provides a computational tool with a remarkable quality and computationally less
|
||||
|
||||
DFT is computationally too expensive for systems with more than hundreds of atoms especially when one needs to perform global optimization
|
||||
or to perform molecular dynamics (MD) simulations with sufficient statistical sampling of the initial conditions or to perform fairly long trajectories.
|
||||
It is therefore necessary to further simplify the method in order to reduce the computational cost. To do so, Foulkes and Haydock showed that
|
||||
It is therefore necessary to further simplify the method in order to reduce the computational cost. To do so, W. Foulkes and R. Haydock showed that
|
||||
tight-binding models can be derived from the DFT.\cite{Foulkes1989} Later, DFTB was proposed by D. Porezag $et ~ al.$.\cite{Porezag1995}
|
||||
|
||||
Non-self-consistent DFTB scheme is suitable to study systems in which polyatomic electronic density is well described by a sum of atom-like densities.
|
||||
@ -841,7 +839,7 @@ located on the same atom. This differs from DFT that explicitly considers atomic
|
||||
As presented above, DFTB was initially developed with Mulliken charges; However, other definitions of atomic charges are possible such as
|
||||
Natural Bond Order (NBO)\cite{Bader1985, Bader1990, Glendening1998, Glendening2012} and Electrostatic Potential Fitting (EPF) charges.\cite{Singh1984, Besler1990}
|
||||
EPF has a fairly good representation of the electrostatic term of a molecule dominated by the Van der Waals interactions. CM3 (Class IV / Charge Model 3)
|
||||
charges were proposed by Li $et al.$ in 1998 and they have been considered in DFTB. They give good results for the description of the electric dipole
|
||||
charges were proposed by J. Li $et al.$ in 1998 and they have been considered in DFTB. They give good results for the description of the electric dipole
|
||||
and the electrostatic potential, partial atomic charges in molecules, and Coulombic intermolecular potential of polycyclic aromatic hydrocarbon clusters.\cite{Li1998, Kalinowski2004, Kelly2005, Rapacioli2009corr}
|
||||
The CM3 charges are defined as:
|
||||
\begin{align}
|
||||
@ -909,7 +907,7 @@ V(R_{\alpha\beta}) = D_{eq}\left(1-e^{-a(R_{\alpha\beta}-R_{eq})} \right)^2
|
||||
where $D_{eq}$ is the depth of the Morse potential well. Parameter $a$ determines the width of the potential, the smaller $a$ the larger the well. The force constant of the bond can be found via the Taylor expansion of $V(R_{\alpha\beta})$ around $R_{\alpha\beta}=R_{eq}$ to the second derivative of the potential energy function, from which it can obtain
|
||||
$a=(k_{eq}/{2D_{eq}})^{\frac{1}{2}}$ in which $k_{eq}$ is the force constant of the minimum well.
|
||||
|
||||
Lennard-Jones potential as known as LJ potential or 12-6 potential is a pair potential, which is proposed by John Lennard-Jones in 1924.\cite{Jones1924, Lennard1924} It models soft repulsive and attractive interactions, therefore, the LJ potential describes electronically neutral atoms or molecules.
|
||||
Lennard-Jones potential as known as LJ potential or 12-6 potential is a pair potential, which is proposed by J. Lennard-Jones in 1924.\cite{Jones1924, Lennard1924} It models soft repulsive and attractive interactions, therefore, the LJ potential describes electronically neutral atoms or molecules.
|
||||
Because of its simple mathematical form, it is one of the most widely used intermolecular potentials especially to describe the interaction within noble gas molecules. The total energy can be written as the sum of the interaction energy of all atomic pairs, which is defined as follows:\cite{Lennard1931}
|
||||
\begin{align}
|
||||
V_\mathrm{LJ}(R_{\alpha\beta}) = 4\varepsilon_0 \left[ \left(\frac{\sigma}{R_{\alpha\beta}}\right)^{12}- \left(\frac{\sigma}{R_{\alpha\beta}}\right)^6
|
||||
@ -1014,8 +1012,7 @@ for a more efficient sampling of the PES. The Monte Carlo, classical MD and PTMD
|
||||
|
||||
The term Monte Carlo denotes a class of algorithmic methods that aims at a probabilistic description, relying on the use of
|
||||
random numbers. The name Monte Carlo alludes to the games of chance taking place at the Monte Carlo casino. The
|
||||
Monte Carlo method was introduced in 1947 by Metropolis,\cite{Metropolis1987} and was first published in 1949 by
|
||||
Metropolis in collaboration with Ulam.\cite{Metropolis1949} Monte Carlo methods have been widely applied in computational
|
||||
Monte Carlo method was introduced in 1947 by N. Metropolis,\cite{Metropolis1987} and was first published in 1949 by N. Metropolis in collaboration with S. Ulam.\cite{Metropolis1949} Monte Carlo methods have been widely applied in computational
|
||||
physics, computational statistics, biomedicine, machine learning, industrial engineering, economics and finance, and other fields.\cite{Kroese2014}
|
||||
\textbf{Monte Carlo methods can} generally be roughly \textbf{divided into two categories}. The first category is applied to problems that have
|
||||
inherent randomness, and the computing power of the computer can directly simulate this random process. For example,
|
||||
@ -1070,7 +1067,7 @@ point in $\Omega$:
|
||||
\label{thermalAverage}
|
||||
\end{align}
|
||||
in which $Z$ is the partition function, and $k_B$ is the Boltzmann constant. $T$ refers to the temperature, and $H(\textbf X)$ denotes the Hamiltonian of the system.
|
||||
To illustrate the general application of Monte Carlo techniques, we take here the standard example of the one-dimensional integral $I$ over integration space $\Omega$:
|
||||
To illustrate the general application of Monte Carlo techniques, here the standard example of the one-dimensional integral $I$ over integration space $\Omega$ is taken:
|
||||
\begin{align}
|
||||
I=\int_\Omega f(\mathbf x)d\mathbf x
|
||||
\label{SingleIntegral}
|
||||
@ -1153,7 +1150,7 @@ In the case of symmetrical movements, $\eta (\mathbf x_i \rightarrow \mathbf x_j
|
||||
%=======
|
||||
MD is a powerful tool for analyzing the physical movements of atoms and molecules of many-body systems.
|
||||
MD was originally developed following the earlier successes of Monte Carlo simulations. The first work about
|
||||
MD was published in 1957 by Alder \textit{et al.} which focused on the integration of classical equations of
|
||||
MD was published in 1957 by B. Alder \textit{et al.} which focused on the integration of classical equations of
|
||||
motion for a system of hard spheres.\cite{Alder1957} Before long, radiation damage at low and moderate
|
||||
energies were studied using MD in 1960 and MD was also applied to simulate liquid argon in 1964.\cite{Gibson1960, Rahman1964}
|
||||
MD experienced an extremely rapid development in the years that followed. MD simulations have been applied
|
||||
@ -1188,7 +1185,7 @@ The reduction from a full quantum description of all particles, electrons and nu
|
||||
main approximations. The first one is the BO approximation as described in section~\ref{BO_approx} which allows to
|
||||
treat separately electrons and nuclei. The second one treats the nuclei (much heavier than electrons) as point charge particles
|
||||
that follow classical Newtonian dynamics.
|
||||
In this thesis, \textbf{classical molecular dynamics was used} to perform simulations and we use the term MD to denote
|
||||
In this thesis, \textbf{classical molecular dynamics was used} to perform simulations and use the term MD to denote
|
||||
classical molecular dynamics only.
|
||||
|
||||
\textbf{Principles.}
|
||||
@ -1339,16 +1336,15 @@ bottoms of the wells can not be explored comprehensively. Therefore, it is not p
|
||||
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}.
|
||||
In \textbf{biased methods}, the dynamics of the system is influenced by a external factor, usually a non-physical
|
||||
orce, which makes it possible to push the system outside of the wells even at low $T$.\cite{Torrie1977, Hansmann1993, Marchi1999, Bartels2000, Darve2001}
|
||||
In \textbf{biased methods}, the dynamics of the system is influenced by a external factor, usually a non-physical force, which makes it possible to push the system outside of the wells even at low $T$.\cite{Torrie1977, Hansmann1993, Marchi1999, Bartels2000, Darve2001}
|
||||
For instance, Metadynamics is a biased method.\cite{Laio2002, Iannuzzi2003, Barducci2011}
|
||||
In \textbf{non-biased methods}, the dynamics of the system is not modified directly. Examples are simulated annealing,\cite{Kirkpatrick1983, Van1987}
|
||||
and multi-replica approaches such as the \textbf{parallel-tempering molecular dynamics} approach, which has been used in this thesis.
|
||||
%For simulated annealing method, a MD or Monte Carlo simulation is performed in the canonical ensemble by slowly decreasing the temperature of the thermostat in order to find a stable structure. The system is then heated again to be out from the basin that was just explored and then it is cooled again to find a new isomer. This mechanism should be done a lot of times in order to get the most stable structure.
|
||||
|
||||
The replica exchange approach also termed parallel-tempering was originally devised by Swendsen $et~ al.$ in 1986 \cite{Swendsen1986}
|
||||
The replica exchange approach also termed parallel-tempering was originally devised by R. H. Swendsen $et~ al.$ in 1986 \cite{Swendsen1986}
|
||||
then extended by Geyer and coauthor in 1991 \cite{Geyer1991} and was further developed by Hukushima and Nemoto,\cite{Hukushima1996}
|
||||
Falcioni and Deem,\cite{Falcioni1999} Earl and coworker.\cite{Earl2005} Sugita and Okamoto formulated a MD version of parallel tempering
|
||||
M. Falcioni and M. W. Deem,\cite{Falcioni1999} D. J. Earl and coworker.\cite{Earl2005} Y. Sugita and Y. Okamoto formulated a MD version of parallel tempering
|
||||
to enhance conformational sampling.\cite{Sugita1999} PTMD is a method which aims at enhancing the ergodicity of MD simulations.
|
||||
The principle of PTMD is shown in Figure~\ref{ptmd_s}.
|
||||
|
||||
|
||||
@ -7,53 +7,53 @@
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@ -1,4 +1,4 @@
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Conclusions% this file is c\emph{•}alled up by thesis.tex
|
||||
%Conclusions% this file is c\emph{•}alled up by thesis.tex
|
||||
% content in this file will be fed into the main document
|
||||
|
||||
%%\chapter{Aims of the project} % top level followed by section, subsection
|
||||
@ -20,9 +20,9 @@ combination of the \textbf{self-consistent-charge density functional based tight
|
||||
energy surfaces (PES) and the \textbf{parallel-tempering molecular dynamics} (PTMD) approach for their exploration. All low-energy isomers
|
||||
reported in this chapter are discussed in terms of structure, relative energy and binding energy which are compared to the literature
|
||||
when available. Calculations at higher level of theory are also performed to refine the results obtained at the SCC-DFTB level or to
|
||||
validate the results it provides. In particular, in this chapter, we propose an improve set of parameters to describe sp$^3$ nitrogen
|
||||
validate the results it provides. In particular, in this chapter, an improve set of parameters is proposed to describe sp$^3$ nitrogen
|
||||
containing compounds at the SCC-DFTB level. Our results are also used to complement collision-induced dissociation experiments
|
||||
performed by S. Zamith and J.-M. l'Hermite at the \textit{Laboratoire Collisions Agr\'egats R\'eactivit\'e} (LCAR).
|
||||
performed by S. Zamith and J.-M. L'Hermite at the \textit{Laboratoire Collisions Agr\'egats R\'eactivit\'e} (LCAR).
|
||||
|
||||
%The background of the systems studied in this thesis and theoretical methods related to the projects of this thesis have been introduced in chapter \ref{chap:general_intro} and chapter \ref{chap:comput_method}. Then the two kinds of projects investigated in my Phd will be shown in the following chapter \ref{chap:structure} and chpater \ref{chap:collision}.
|
||||
|
||||
@ -37,20 +37,16 @@ of the method are presented in section~\ref{sec:DFTB} of chapter~\ref{chap:compu
|
||||
%DFTB is an approximated DFT scheme whose computational efficiency depends on the use of parameterized integrals.\cite{Elstner2014, Elstner1998, dftb1, dftb2}
|
||||
The mio-set for the Slater-Koster tables of integrals was used.\cite{Elstner1998} However, it has been shown that these integrals do not properly described
|
||||
sp$^3$ hybridized nitrogen, in particular, proton affinity.\cite{Gaus2013para} Consequently, in order to avoid spurious deprotonation of the sp$^3$ hybridized
|
||||
nitrogen in NH$_4^+$ and to correctly reproduce binding energies calculated at the MP2/Def2TZVP level, we propose to modify the original mio-set for
|
||||
Slater-Koster tables of N-H integrals by applying them a multiplying factor. Several of them were tested and we present here the results
|
||||
we obtained for two of them: 1.16 and 1.28. For the sp$^2$ nitrogen of uracil, the original integrals of the mio-set were used. To improve
|
||||
nitrogen in NH$_4^+$ and to correctly reproduce binding energies calculated at the MP2/Def2TZVP level, I propose to modify the original mio-set for
|
||||
Slater-Koster tables of N-H integrals by applying them a multiplying factor. Several of them were tested and I present here the results obtained for two of them: 1.16 and 1.28. For the sp$^2$ nitrogen of uracil, the original integrals of the mio-set were used. To improve
|
||||
description of the intermolecular interactions, the original Mulliken charges were replaced by the CM3 charges,\cite{Li1998, Thompson2003, Rapacioli2009corr}
|
||||
(see equation~\ref{CM3} in section~\ref{sec:DFTB}) and an empirical correction term (see equation~\ref{dispersionE} in section~\ref{sec:DFTB}) was used
|
||||
to describe dispersion interactions.\cite{Rapacioli2009corr, Elstner2001, Zhechkov2005} Simon \textit{et al.} developed a SCC-DFTB potential that leads
|
||||
to geometries, frequencies, and relative energies close to the corresponding experimental and CCSD(T)/aug-cc-pVTZ results.\cite{Simon2012, Odutola1980}
|
||||
The corresponding $D_{OH}$ parameter, \textit{i.e.} 0.129, is retained in the studies presented in thus chapter. $D_{NH}$ is tested in the study of
|
||||
ammonium/ammonia water clusters and two values were retained and thoroughly tested: 0.12 and 0.14. D$_{NO}$ is set to zero.
|
||||
to describe dispersion interactions.\cite{Rapacioli2009corr, Elstner2001, Zhechkov2005} A. Simon \textit{et al.} developed a SCC-DFTB potential that leads to geometries, frequencies, and relative energies close to the corresponding experimental and CCSD(T)/aug-cc-pVTZ results.\cite{Simon2012, Odutola1980} The corresponding $D_{OH}$ parameter, \textit{i.e.} 0.129, is retained in the studies presented in thus chapter. $D_{NH}$ is tested in the study of ammonium/ammonia water clusters and two values were retained and thoroughly tested: 0.12 and 0.14. D$_{NO}$ is set to zero.
|
||||
|
||||
\subsection{SCC-DFTB Exploration of PES}
|
||||
|
||||
To determine the lowest-energy isomers of (H$_2$O)$_{1-10,20}${NH$_4$}$^+$, (H$_2$O)$_{1-10}$NH$_3$
|
||||
and (H$_2$O)$_{1-7,11,12}$UH$^+$ clusters, we thoroughly explore their PES using PTMD \cite{Sugita1999, Sugita2000, Earl2005}
|
||||
and \newline (H$_2$O)$_{1-7,11,12}$UH$^+$ clusters, their PES were thoroughly explored using PTMD \cite{Sugita1999, Sugita2000, Earl2005}
|
||||
simulations combined with a SCC-DFTB \cite{Elstner1998} description of the energies and gradients. I describe
|
||||
below the \textbf{detailed parameters used for all the simulations conducted within this chapter}.
|
||||
|
||||
@ -59,12 +55,12 @@ For (H$_2$O)$_{1-3}${NH$_4$}$^+$ and (H$_2$O)$_{1-3}${NH$_3$} clusters, 16 repl
|
||||
with a linear distribution of temperatures with a 15~K step ranging from 10 to 250 K. 40 replicas with a 6~K step ranging from 10 to 250~K were
|
||||
considered for (H$_2$O)$_{4-10,20}${NH$_4$}$^+$ and (H$_2$O)$_{4-10}${NH$_3$} species. All trajectories were 5~ns long and a time step
|
||||
of 0.5 fs was used to integrate the equations of motion. A Nos{\'e}-Hoover chain of 5 thermostats was employed for all the simulations to achieve
|
||||
simulatons in the canonical ensemble.\cite{Nose1984M, Hoover1985} Thermostat frequencies were fixed at 400 cm$^{-1}$.
|
||||
simulations in the canonical ensemble.\cite{Nose1984M, Hoover1985} Thermostat frequencies were fixed at 400 cm$^{-1}$.
|
||||
To identify low-energy isomers of (H$_2$O)$_{1-3}${NH$_4$}$^+$ and (H$_2$O)$_{1-3}${NH$_3$} clusters, 303 geometries were periodically
|
||||
selected from each replicas and further optimized at the SCC-DFTB level, which produced 4848 optimized geometries per cluster. For
|
||||
(H$_2$O)$_{4-10,20}${NH$_4$}$^+$ and (H$_2$O)$_{4-10}${NH$_3$} clusters, 500 geometries were periodically selected from each
|
||||
replicas leading to 20000 optimized geometries per cluster. For (H$_2$O)$_{20}${NH$_4$}$^+$, the initial structure used for the global
|
||||
optimization process was the lowest-energy structure reported by Douady \textit{et al.}.\cite{Douady2009} The five lowest-energy
|
||||
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.
|
||||
|
||||
@ -73,7 +69,7 @@ details on MP2/Def2TZVP calculations.
|
||||
A reasonable time interval for the PT exchanges was 2.5 ps. A Nos{\'e}-Hoover chain of five thermostats with frequencies of 800 cm$^{-1}$ was
|
||||
applied to achieve an exploration in the canonical ensemble.\cite{Nose1984M, Hoover1985} To avoid any spurious influence of the initial
|
||||
geometry on the PES exploration, three distinct PTMD simulations were carried out with distinct initial proton location: on the uracil in two cases
|
||||
and on a water molecule in the other one. In the former cases, we used two isomers u178 and u138 of UH$^+$ shown in Figure~\ref{uracil_s} as
|
||||
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
|
||||
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$^+$,
|
||||
@ -91,8 +87,7 @@ 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 combination
|
||||
with 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 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}
|
||||
|
||||
@ -102,7 +97,7 @@ Following SCC-DFTB optimizations, the five lowest-energy isomers of (H$_2$O)$_{1
|
||||
In section~\ref{sec:ammoniumwater}, relatives energies with respect to the lowest-energy isomer of each
|
||||
cluster are reported. Impact of zero-point vibrational energy (ZPVE) corrections on relative
|
||||
energies were evaluated at MP2/Def2TZVP level. To evaluate the strength of water-ammonium and water-ammonia
|
||||
interactions and to assess the accuracy of the SCC-DFTB method, we also report binding energies.
|
||||
interactions and to assess the accuracy of the SCC-DFTB method, binding energies are also reported.
|
||||
Two distinct approaches were used to calculate binding energies. The first one considers only the binding
|
||||
energy between the water cluster as a whole and the impurity, {NH$_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
|
||||
@ -149,23 +144,19 @@ basicity makes it a potential proton sink that can form a ionic center for nucle
|
||||
also found that even a small amount of atmospherically relevant ammonia can increase the nucleation rate of sulphuric acid particles by several orders of magnitude.\cite{Kirkby2011}
|
||||
The significance of ammonium and ammonia water clusters have thus motivated a large amount of experimental and theoretical studies during the past decades.\cite{Perkins1984, Herbine1985, Stockman1992, Hulthe1997, Wang1998, Chang1998, Jiang1999, Hvelplund2010, Douady2009, Douady2008, Morrell2010, Bacelo2002, Galashev2013}
|
||||
|
||||
As a few examples, in 1984, (H$_2$O)$_{2}${NH$_4$}$^+$ was identified using mass spectrometry by Perkin \textit{et al.}\cite{Perkins1984} In 1997, Stenhagen
|
||||
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} Hvelplund \textit{et al.} later reported a combined experimental and theoretical study devoted to protonated mixed ammonia/water
|
||||
structure.\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,
|
||||
Skurski1998, Jiang1999, Donaldson1999, Sadlej1999, Hvelplund2010, Bacelo2002, Galashev2013} Among them, Novoa \textit{et al.} studied the (H$_2$O)$_4$NH$_3$
|
||||
aggregate and found the existence of a minimum in its potential energy surface corresponding to a (H$_2$O)$_{3}$···{NH$_4$}$^+$···OH$^-$ structure, resulting from
|
||||
one proton transfer from a water molecule to the ammonia molecule.\cite{Lee1996} Bacelo later reported a number of low-energy minima for (H$_2$O)$_{3-4}$NH$_3$
|
||||
clusters obtained from \textit{ab initio} calculation and a Monte Carlo exploration of the potential energy surface (PES).\cite{Bacelo2002} More recently, Douady \textit{et al.}
|
||||
performed a global optimization of (H$_2$O)$_{n}${NH$_4$}$^+$ (n = 1-24) clusters again using a Monte Carlo procedure in combination with a Kozack and Jordan
|
||||
empirical force field.\cite{Douady2008, Kozack1992polar}
|
||||
Skurski1998, Jiang1999, Donaldson1999, Sadlej1999, Hvelplund2010, Bacelo2002, Galashev2013} Among them, J. Novoa \textit{et al.} studied the (H$_2$O)$_4$NH$_3$
|
||||
aggregate and found the existence of a minimum in its potential energy surface corresponding to a (H$_2$O)$_{3}$···{NH$_4$}$^+$···OH$^-$ structure, resulting from one proton transfer from a water molecule to the ammonia molecule.\cite{Lee1996} D. Bacelo later reported a number of low-energy minima for (H$_2$O)$_{3-4}$NH$_3$ clusters obtained from \textit{ab initio} calculation and a Monte Carlo exploration of the PES.\cite{Bacelo2002} More recently, J. Douady \textit{et al.} performed a global optimization of (H$_2$O)$_{n}${NH$_4$}$^+$ ($n$ = 1-24) clusters again using a Monte Carlo procedure in combination with a Kozack and Jordan empirical force field.\cite{Douady2008, Kozack1992polar}
|
||||
In this study, the finite temperature properties as well as vibrational signature of several clusters thus highlighting the key contribution of simulations in understanding such species. Morrell and Shields also studied the
|
||||
(H$_2$O)$_{n}${NH$_4$}$^+$ (n = 1-10) aggregates \textit{via} a mixed molecular dynamics and quantum mechanics methodology to calculate energies and free energies
|
||||
(H$_2$O)$_{n}${NH$_4$}$^+$ ($n$ = 1-10) aggregates \textit{via} a mixed molecular dynamics and quantum mechanics methodology to calculate energies and free energies
|
||||
of formations which were in good agreement with previous experimental and theoretical results.\cite{Morrell2010}
|
||||
More recently, Pei \textit{et al.} determined that (H$_2$O)$_{n}$NH$_4^+$ clusters start to adopt a closed-cage geometry at $n$=8.\cite{Pei2015}
|
||||
Finally, Walters and collaborators determined the geometry of (H$_2$O)$_{16}$NH$_3$ and (H$_2$O)$_{16}$NH$_4^+$ at the HF/6-31G(d) level,
|
||||
More recently, S. Pei \textit{et al.} determined that (H$_2$O)$_{n}$NH$_4^+$ clusters start to adopt a closed-cage geometry at $n$=8.\cite{Pei2015}
|
||||
Finally, W. Walters and collaborators determined the geometry of (H$_2$O)$_{16}$NH$_3$ and (H$_2$O)$_{16}$NH$_4^+$ at the HF/6-31G(d) level,
|
||||
and observed strong hydrogen bonding between water and the lone pair of NH$_3$ and bewteen NH$_4^+$ and the four adjacent water molecules.\cite{Walters2018}
|
||||
|
||||
As for the study of other molecular clusters, the range of applicability of theoretical simulations to describe ammonium and ammonia water clusters is dictated
|
||||
@ -173,16 +164,16 @@ by the balance between accuracy, transferability and computational efficiency. W
|
||||
application to large species is more difficult, in particular when an exhaustive exploration of the PES is required. In contrast, force-field
|
||||
potentials are computationally extremely efficient and can be coupled to global optimization methods but their transferability is limited.
|
||||
The SCC-DFTB approach can be seen as an intermediate approach which combines the strengths of both \textit{ab-initio} and force-field methods.
|
||||
Indeed, it can be as accurate as DFT while computationally more efficient and is more transferable than force fields (see Chapter~2)
|
||||
In the recent years, SCC-DFTB has been successfully applied to the study of various molecular clusters: pure, protonated, and de-protonated water
|
||||
clusters,\cite{Choi2010, Choi2013, Korchagina2017, Simon2019} water clusters on polycyclic aromatic hydrocarbons,\cite{Simon2012, Simon2013water}
|
||||
Indeed, it can be as accurate as DFT while computationally more efficient and is more transferable than force fields (see chapter \ref{chap:comput_method})
|
||||
In recent years, SCC-DFTB has been successfully applied to the study of various molecular clusters: pure, protonated, and de-protonated water
|
||||
clusters,\cite{Choi2010, Choi2013, Korchagina2017, Simon2019} water clusters on PAHs,\cite{Simon2012, Simon2013water}
|
||||
sulfate-containing water clusters,\cite{Korchagina2016} water clusters in an argon matrix,\cite{Simon2017formation} whether it is for global optimization or
|
||||
for the study of finite-temperature properties. However, in its original formulation, SCC-DFTB does not provide good results for the description of ammonia
|
||||
and ammonium as nitrogen hybridization seems to be a problem for minimal basis-set methods like SCC-DFTB.\cite{Winget2003} Elstner and coworkers found
|
||||
and ammonium as nitrogen hybridization seems to be a problem for minimal basis-set methods like SCC-DFTB.\cite{Winget2003} M. Elstner and coworkers found
|
||||
consistent errors (about 14.0 kcal.mol$^{-1}$) for deprotonation energies of sp$^3$ hybridized nitrogen containing systems, whereas sp$^1$ and sp$^2$ systems
|
||||
display much smaller errors.\cite{Gaus2013para}
|
||||
|
||||
In the present section, we first propose an improvement of the SCC-DFTB scheme to describe ammonium and ammonia water clusters by modifying
|
||||
In this section, I first propose an improvement of the SCC-DFTB scheme to describe ammonium and ammonia water clusters by modifying
|
||||
both Hamiltonian and overlap N-H integrals and introducing optimized atomic charges.\cite{Thompson2003, Rapacioli2009} By combining this
|
||||
improved SCC-DFTB scheme with PTMD simulations, global optimization of the (H$_2$O)$_{1-10}${NH$_4$}$^+$ and (H$_2$O)$_{1-10}${NH$_3$}
|
||||
clusters is then performed which allows to report a number of low-energy isomers for these species. Among them, a selected number of structures
|
||||
@ -196,7 +187,7 @@ A full paper devoted to this work is in preparation.
|
||||
|
||||
\subsubsection{Dissociation Curves and SCC-DFTB Potential}
|
||||
|
||||
In order to define the best SCC-DFTB parameter to model ammonia and ammonium water clusters, we have tested various sets of corrections.
|
||||
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
|
||||
@ -279,10 +270,10 @@ From en energetic point of view, it is interesting to first look at the relative
|
||||
(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
|
||||
Chang and co-workers, 3-a is more stable than 3-b.\cite{Wang1998, Jiang1999} The aauhors also complemented their
|
||||
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
|
||||
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, Spiegelman and co-workers, conducted a
|
||||
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}
|
||||
All these results show that for the specific question of lowest-energy isomer of (H$_2$O)$_{3}${NH$_4$}$^+$, SCC-DFTB
|
||||
has an accuracy close to other \textit{ab initio} methods which confirms its applicability.
|
||||
@ -352,10 +343,10 @@ isomers of clusters (H$_2$O)$_{4-10}${NH$_4$}$^+$ and (H$_2$O)$_{4-10}$NH$_3$ ar
|
||||
The five lowest-energy isomers of (H$_2$O)$_{4}${NH$_4$}$^+$ are depicted in Figure~\ref{fig:nh4-4-6w}. 4-a is the lowest-energy isomer obtained
|
||||
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 Chang and co-workers.\cite{Chang1998, Wang1998} Isomer 4-a diplays four hydrogen bonds around the ionic
|
||||
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}
|
||||
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}$. Isomer 4-c was not reported in Chang’s study,\cite{Jiang1999} and the corresponding
|
||||
are slightly higher by$\sim$2.0 kcal.mol$^{-1}$. Isomer 4-c was not reported in H. Chang’s study,\cite{Jiang1999} and the corresponding
|
||||
energy ordering of the five lowest-energy isomers was the same as ours which certainly results from the use of a different basis set.
|
||||
|
||||
\begin{figure}[h!]
|
||||
@ -418,92 +409,92 @@ energy ordering of the five lowest-energy isomers was the same as ours which cer
|
||||
|
||||
The relative binding energy of SCC-DFTB method to MP2/Def2TZVP method with BSSE correction for isomers 4-a to 4-e are listed in Table \ref{reBindE}. When the four water molecules are considered as a whole part to calculate the binding energy, the relative binding energy of isomers 4-a to 4-e are -1.67, 0.00, 0.77, 0.77 and -4.04 kcal·mol\textsuperscript{-1}. As shown in Table \ref{reBindE}, for isomers 4-a to 4-e, when the four water molecules are separately considered using the geometry in the cluster to calculate the binding energy, the biggest absolute value of the relative binding energy is 0.87 kcal·mol\textsuperscript{-1}. This shows the results of SCC-DFTB are in good agreement with those of MP2/Def2TZVP with BSSE correction for (H$_2$O)$_{4}${NH$_4$}$^+$. From the relative binding energy of (H$_2$O)$_{4}${NH$_4$}$^+$, it indicates that all the water molecules considered as a whole part or separately has an effect on the relative binding energy for the cluster (H$_2$O)$_{4}${NH$_4$}$^+$ and the overall $\Delta E_{bind.}^{whole}$ are bigger than $\Delta E_{bind.}^{sep.}$.
|
||||
|
||||
For cluster (H$_2$O)$_{5}${NH$_4$}$^+$, the first five low-energy isomers are illustrated in Figure \ref{fig:nh4-4-6w}. The isomer 5-a is the most stable one, which is consistent with Spiegelman’s result using the global MC optimization and Shields’s results obtained with a mixed molecular dynamics/quantum mechanics moldel.\cite{Douady2008, Morrell2010} The energy order of 5-a to 5-e at SCC-DFTB level is consistent with that at MP2/Def2TZVP level with ZPVE correction. 5-a, 5-d and 5-e have a complete solvation shell while one dangling N-H bond is exposed in 5-b and 5-c. For the first five low-energy isomers, the energy order of our results are not exactly the same with Chang’s calculation results at MP2/6-31+G(d)level with ZPVE correction.\cite{Jiang1999} In Chang’s results, 5-d is the first low-energy isomer and 5-a is the second low-energy isomer. They didn’t find isomers 5-b and 5-c. From the comparison, it implies the combination of SCC-DFTB and PTMD is good enough to find the low-energy isomer and the basis set can affect the energy order when using the MP2 approach.
|
||||
For cluster (H$_2$O)$_{5}${NH$_4$}$^+$, the five low-energy isomers are illustrated in Figure \ref{fig:nh4-4-6w}. The isomer 5-a is the most stable one, which is consistent with F. Spiegelman’s result using the global Monte Carlo optimization and G. Shields’s results obtained with a mixed molecular dynamics/quantum mechanics moldel.\cite{Douady2008, Morrell2010} The energy order of 5-a to 5-e at SCC-DFTB level is consistent with that at MP2/Def2TZVP level with ZPVE correction. 5-a, 5-d and 5-e have a complete solvation shell while one dangling N-H bond is exposed in 5-b and 5-c. For the five low-energy isomers, the energy order of our results are not exactly the same with H. Chang’s calculation results at MP2/6-31+G(d)level with ZPVE correction.\cite{Jiang1999} In H. Chang’s results, 5-d is the low-energy isomer and 5-a is the second low-energy isomer. They didn’t find isomers 5-b and 5-c. From the comparison, it implies the combination of SCC-DFTB and PTMD is good enough to find the low-energy isomer and the basis set can affect the energy order when using the MP2 approach.
|
||||
|
||||
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 first 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$_4$}$^+$.
|
||||
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 first 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 MC 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 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 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.
|
||||
|
||||
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.
|
||||
|
||||
For cluster (H$_2$O)$_{7}${NH$_4$}$^+$, the first 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 Spiegelman’s study, 7-a is also the first low-energy isomer using the MC 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 H2O molecules at MP2/aug-cc-pVDZ level in 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 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.
|
||||
|
||||
\begin{figure}[h!]
|
||||
\includegraphics[width=1.0\linewidth]{nh4-7-10w.png}
|
||||
\centering
|
||||
\caption{The first five low-energy isomers of clusters (H$_2$O)$_{7-10}${NH$_4$}$^+$ and the associated relative energies (in kcal·mol\textsuperscript{-1}) at MP2/Def2TZVP level with (bold) and without ZPVE correction and SCC-DFTB level (italic).}
|
||||
\caption{The five low-energy isomers of clusters (H$_2$O)$_{7-10}${NH$_4$}$^+$ and the associated relative energies (in kcal·mol\textsuperscript{-1}) at MP2/Def2TZVP level with (bold) and without ZPVE correction and SCC-DFTB level (italic).}
|
||||
\label{fig:nh4-7-10w}
|
||||
\end{figure}
|
||||
|
||||
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 first 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 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 Spiegelman’s results obtained using MC 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 first 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 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 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$_4$}$^+$.
|
||||
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.
|
||||
|
||||
For cluster (H$_2$O)$_{10}${NH$_4$}$^+$, 10-a to 10-e are the first 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 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 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 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$_4$}$^+$.
|
||||
|
||||
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 first 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 Novoa et al\cite{Lee1996} 4$^\prime$-d is also the first low-energy isomer in 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 first five low-energy isomers in 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 0.20 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$_3$}.
|
||||
|
||||
\begin{figure}[h!]
|
||||
\includegraphics[width=1.0\linewidth]{nh3-4-7w.png}
|
||||
\centering
|
||||
\caption{The first five low-energy isomers of cluster (H$_2$O)$_{4-7}${NH$_3$} and the associated relative energies (in kcal·mol\textsuperscript{-1}) at MP2/Def2TZVP level with (bold) and without ZPVE correction and SCC-DFTB level (italic).}
|
||||
\caption{The five low-energy isomers of cluster (H$_2$O)$_{4-7}${NH$_3$} and the associated relative energies (in kcal·mol\textsuperscript{-1}) at MP2/Def2TZVP level with (bold) and without ZPVE correction and SCC-DFTB level (italic).}
|
||||
\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$_3$}.
|
||||
|
||||
For cluster (H$_2$O)$_{5}${NH$_3$}, 5$^\prime$-a to 5$^\prime$-e are the first 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 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.
|
||||
|
||||
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 first 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 we 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 0.05 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$_3$}.
|
||||
|
||||
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 first 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 first 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 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$_3$}.
|
||||
|
||||
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 first 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 0.93 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$_3$}.
|
||||
|
||||
\begin{figure}[h!]
|
||||
\includegraphics[width=1.0\linewidth]{nh3-8-10w.png}
|
||||
\centering
|
||||
\caption{The first five low-energy isomers of clusters (H$_2$O)$_{8-10}${NH$_3$} and the associated relative energies (in kcal·mol\textsuperscript{-1}) at MP2/Def2TZVP level with (bold) and without ZPVE correction and SCC-DFTB level (italic).}
|
||||
\caption{The five low-energy isomers of clusters (H$_2$O)$_{8-10}${NH$_3$} and the associated relative energies (in kcal·mol\textsuperscript{-1}) at MP2/Def2TZVP level with (bold) and without ZPVE correction and SCC-DFTB level (italic).}
|
||||
\label{fig:nh3-8-10w}
|
||||
\end{figure}
|
||||
|
||||
The smallest and the biggest values of $\Delta E_{bind.}^{whole}$ of isomers 8$^\prime$-a to 8$^\prime$-e are -0.1 and -1.28 kcal·mol\textsuperscript{-1}, respectively while the smallest absolute value of $\Delta E_{bind.}^{sep.}$ is 3.04 kcal·mol\textsuperscript{-1} shown in Table \ref{reBindE}. The binding energies calculated with SCC-DFTB agree better with those obtained at MP2/Def2TZVP level when all the water molecules are considered as a whole part in cluster (H$_2$O)$_{8}${NH$_3$} than the ones when water molecules calculated separately.
|
||||
|
||||
For cluster (H$_2$O)$_{9}${NH$_3$}, 9$^\prime$-a to 9$^\prime$-e are the first five low-lying energy structures displayed in Figure \ref{fig:nh3-8-10w}. 9$^\prime$-a with a “chair” structure is the first low-energy structure at SCC-DFTB level. 9$^\prime$-b, 9$^\prime$-c and 9$^\prime$-d in which the nine water molecules have the similar configuration are the second, third and fourth isomers. In 9$^\prime$-b and 9$^\prime$-c, the NH$_3$ has three exposed N-H bonds and the energies of 9$^\prime$-b and 9-c are very close at both SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels. The NH$_3$ has two exposed N-H bonds in 9$^\prime$-d. 9$^\prime$-e is the fifth low-energy isomer in the SCC-DFTB calculation results but it is the first low-energy isomer in the calculation results of MP2/Def2TZVP with ZPVE correction. 9$^\prime$-e has a pentagonal prism structure and all the water molecules in it are three-coordinated. The relative energy for each isomer between SCC-DFTB level and MP2/Def2TZVP with ZPVE correction level is less than 1.23 kcal·mol\textsuperscript{-1}. This shows our SCC-DFTB calculation results are consistent with the calculation results of MP2/Def2TZVP with ZPVE correction for low-energy isomers optimization of cluster (H$_2$O)$_{9}${NH$_3$}.
|
||||
For cluster (H$_2$O)$_{9}${NH$_3$}, 9$^\prime$-a to 9$^\prime$-e are the five low-lying energy structures displayed in Figure \ref{fig:nh3-8-10w}. 9$^\prime$-a with a “chair” structure is the first low-energy structure at SCC-DFTB level. 9$^\prime$-b, 9$^\prime$-c and 9$^\prime$-d in which the nine water molecules have the similar configuration are the second, third and fourth isomers. In 9$^\prime$-b and 9$^\prime$-c, the NH$_3$ has three exposed N-H bonds and the energies of 9$^\prime$-b and 9-c are very close at both SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels. The NH$_3$ has two exposed N-H bonds in 9$^\prime$-d. 9$^\prime$-e is the fifth low-energy isomer in the SCC-DFTB calculation results but it is the first low-energy isomer in the calculation results of MP2/Def2TZVP with ZPVE correction. 9$^\prime$-e has a pentagonal prism structure and all the water molecules in it are three-coordinated. The relative energy for each isomer between SCC-DFTB level and MP2/Def2TZVP with ZPVE correction level is less than 1.23 kcal·mol\textsuperscript{-1}. This shows our SCC-DFTB calculation results are consistent with the calculation results of MP2/Def2TZVP with ZPVE correction for low-energy isomers optimization of cluster (H$_2$O)$_{9}${NH$_3$}.
|
||||
|
||||
The relative binding energies of isomers 9$^\prime$-a to 9$^\prime$-e are shown in Table \ref{reBindE}. The absolute values of $\Delta E_{bind.}^{whole}$ are less than 1.09 kcal·mol\textsuperscript{-1} while the smallest absolute value of $\Delta E_{bind.}^{sep.}$ is 2.57 kcal·mol\textsuperscript{-1}. The binding energies calculated with SCC-DFTB agree well with those acquired at MP2/Def2TZVP level when all the water molecules are considered as a whole part for cluster (H$_2$O)$_{9}${NH$_3$}.
|
||||
|
||||
For cluster (H$_2$O)$_{10}${NH$_3$}, 10$^\prime$-a to 10$^\prime$-e are the first five low-energy structures illustrated in Figure \ref{fig:nh3-8-10w}. The energy order for the first five low-energy structures is the same at SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels. 10$^\prime$-a and 10$^\prime$-b are the first and second low-energy isomer in which the ten water molecules constitute the pentagonal prism. The energy differences of 10$^\prime$-a and 10$^\prime$-b are only 0.27 and 0.58 kcal·mol\textsuperscript{-1} at SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels. 10$^\prime$-c and 10$^\prime$-d are the third and fourth low-energy isomers in which eight water molecules constitute a cube and the energy differences between 10$^\prime$-c and 10$^\prime$-d are very small calculated with SCC-DFTB or MP2/Def2TZVP with ZPVE correction. 10$^\prime$-e is the fifth low-energy structure in which eight water molecules also constitute a cube but its energy is obviously higher than those of 10$^\prime$-c and 10$^\prime$-d. The calculation results of SCC-DFTB are consistent with those of MP2/Def2TZ for the optimization of the low-energy isomers of cluster (H$_2$O)$_{10}${NH$_3$}. According to the structures of the first five low-energy isomers of clusters (H$_2$O)$_{1-10}${NH$_3$}, in most cases, the NH$_3$ usually contains two or three exposed N-H bonds.
|
||||
For cluster (H$_2$O)$_{10}${NH$_3$}, 10$^\prime$-a to 10$^\prime$-e are the five low-energy structures illustrated in Figure \ref{fig:nh3-8-10w}. The energy order for the five low-energy structures is the same at SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels. 10$^\prime$-a and 10$^\prime$-b are the first and second low-energy isomer in which the ten water molecules constitute the pentagonal prism. The energy differences of 10$^\prime$-a and 10$^\prime$-b are only 0.27 and 0.58 kcal·mol\textsuperscript{-1} at SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels. 10$^\prime$-c and 10$^\prime$-d are the third and fourth low-energy isomers in which eight water molecules constitute a cube and the energy differences between 10$^\prime$-c and 10$^\prime$-d are very small calculated with SCC-DFTB or MP2/Def2TZVP with ZPVE correction. 10$^\prime$-e is the fifth low-energy structure in which eight water molecules also constitute a cube but its energy is obviously higher than those of 10$^\prime$-c and 10$^\prime$-d. The calculation results of SCC-DFTB are consistent with those of MP2/Def2TZ for the optimization of the low-energy isomers of cluster (H$_2$O)$_{10}${NH$_3$}. According to the structures of the five low-energy isomers of clusters (H$_2$O)$_{1-10}${NH$_3$}, in most cases, the NH$_3$ usually contains two or three exposed N-H bonds.
|
||||
|
||||
The smallest and biggest values of $\Delta E_{bind.}^{whole}$ of isomers 10$^\prime$-a to 10$^\prime$-e are -0.03 and -1.10 kcal·mol\textsuperscript{-1} while the smallest absolute value of $\Delta E_{bind.}^{sep.}$ is 4.80 kcal·mol\textsuperscript{-1} shown in Table \ref{reBindE}. The values of $\Delta E_{bind.}^{whole}$ implies that SCC-DFTB agree very well with MP2/Def2TZVP for cluster (H$_2$O)$_{10}$NH$_3$ when all the water molecules are regarded as a whole part.
|
||||
|
||||
\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 Spiegelman’s group.\cite{Douady2009}
|
||||
We 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.
|
||||
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.
|
||||
|
||||
\begin{figure}[h!]
|
||||
\includegraphics[width=0.6\linewidth]{nh3-nh4-20w.jpeg}
|
||||
\centering
|
||||
\caption{The first 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{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.}
|
||||
\label{fig:nh3-nh4-20w}
|
||||
\end{figure}
|
||||
|
||||
@ -511,7 +502,7 @@ We also calculated the canonical heat capacities of cluster (H$_2$O)$_{20}${NH$_
|
||||
|
||||
The low-energy isomers reported for (H$_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 we propose are in much better agreement with reference calculations than the original SCC-DFTB parameters.
|
||||
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
|
||||
@ -536,9 +527,9 @@ The gas phase study needs to be extended towards more realistic biomolecular sys
|
||||
The nucleobases in DNA and RNA play a significant role in the encoding and expression of genetic information in living systems while water is a natural medium of many reactions in living organisms. The study of the interaction between nucleobase molecules and aqueous environment has attracted a lot of interests among biologists and chemists. Exploring the clusters composed of nucleobase molecules with water is a good workbench to observe how the properties of nucleobase molecules change when going from isolated gas-phase to hydrated species.
|
||||
|
||||
The radiation can cause damages on RNA and DNA molecules, which is proficiently applied in radiotherapy for cancer treatment. The major drawback in radiotherapy is the unselective damage in both healthy and tumor cells, which has a big side effect. This makes it particularly important to explore the radiation fragments.
|
||||
Uracil (U), C$_4$H$_4$N$_2$O$_2$, is one of the four nucleobases of RNA, has been paid attention concerning radiation damage. Protonated uracil UH$^+$ can be generated by radiation damages.\cite{Wincel2009}
|
||||
The reasons for such degradation can be due to the interaction with slow electrons, as shown by the work of Boudaiffa \textit{et al.} \cite{Boudaiffa2000}
|
||||
Several studies have been devoted to the effect of hydration on the electron affinity of DNA nucleobases. \cite{Smyth2011, Siefermann2011, Alizadeh2013} For instance, Rasmussen \textit{et al.} found that a water molecule is more likely to interact with a charged species than with a neutral one though the study of hydration effects on the lowest triplet states of cytosine, uracil, and thymine by including one or two water molecules explicitly, \cite{Rasmussen2010}
|
||||
Uracil , C$_4$H$_4$N$_2$O$_2$, is one of the four nucleobases of RNA, has been paid attention concerning radiation damage. Protonated uracil UH$^+$ can be generated by radiation damages.\cite{Wincel2009}
|
||||
The reasons for such degradation can be due to the interaction with slow electrons, as shown by the work of B. Boudaiffa \textit{et al.} \cite{Boudaiffa2000}
|
||||
Several studies have been devoted to the effect of hydration on the electron affinity of DNA nucleobases. \cite{Smyth2011, Siefermann2011, Alizadeh2013} For instance, A. Rasmussen \textit{et al.} found that a water molecule is more likely to interact with a charged species than with a neutral one though the study of hydration effects on the lowest triplet states of cytosine, uracil, and thymine by including one or two water molecules explicitly, \cite{Rasmussen2010}
|
||||
However, a lot of work is still needed to be performed to understand the role of aqueous environment on charged nucleobases of DNA and RNA.
|
||||
|
||||
Collision experiments is a useful tool that can be applied to understand the reactivity of molecules and provide access to structural information.\cite{Coates2018}
|
||||
@ -547,47 +538,46 @@ however, there are only few studies available concerning the effect of hydration
|
||||
hydrated protonated uracil shows that the most stable tautomeric form of the neutral uracil (diketo) differs from the most stable one for bare
|
||||
protonated uracil (keto-enol).\cite{Bakker2008}
|
||||
%Some theoretical studies reported the structures taken by neutral microhydrated uracil, containing up to 15 water molecules.\cite{Shishkin2000, Gadre2000, Gaigeot2001, Danilov2006, Bacchus2015}
|
||||
However, fragmentation studies of such species under CID conditions have not been performed. \textbf{S. Zamith and J.-M. l'Hermite conducted such CID
|
||||
However, fragmentation studies of such species under CID conditions have not been performed. \textbf{S. Zamith and J.-M. L'Hermite conducted such CID
|
||||
experiments on protonated uracil water species (H$_2$O)$_{1-15}$UH$^+$ during my thesis and I collaborated with them in order to provide a theoretical
|
||||
support to their measurements}.
|
||||
%The dissociation of mass selected (H$_2$O)$_{1-15}$UH$^+$ clusters is induced by collisions with target rare gas atoms (Ne, Ar) or water molecules (H$_2$O, D$_2$O) at a controlled center of mass collision energy 7.2 eV which is chosen high enough so that a large number of fragmentation channels are explored. After the collisions, the remaining charged species, which have lost one or several neutral subunits, are detected. We find essentially the same results whatever the nature of target atoms of molecules is. The resulting inter-molecular dissociation patterns of the (H$_2$O)$_{1-15}$UH$^+$ clusters show that below n = 5 only water molecules are evaporated whereas, for n $\geq$ 5, a new fragmentation channel appears that corresponds to the loss of neutral uracil.
|
||||
%The dissociation of mass selected (H$_2$O)$_{1-15}$UH$^+$ clusters is induced by collisions with target rare gas atoms (Ne, Ar) or water molecules (H$_2$O, D$_2$O) at a controlled center of mass collision energy 7.2 eV which is chosen high enough so that a large number of fragmentation channels are explored. After the collisions, the remaining charged species, which have lost one or several neutral subunits, are detected. We find essentially the same results whatever the nature of target atoms of molecules is. The resulting inter-molecular dissociation patterns of the (H$_2$O)$_{1-15}$UH$^+$ clusters show that below $n$ = 5 only water molecules are evaporated whereas, for n $\geq$ 5, a new fragmentation channel appears that corresponds to the loss of neutral uracil.
|
||||
|
||||
Theoretical studies have already been devoted to mixed uracil-water clusters and intended to describe the lowest energy structures. However,
|
||||
only neutral species ((H$_2$O)$_{n}$U) were considered.\cite{Shishkin2000, Gadre2000, Van2001diffu, Gaigeot2001, Danilov2006, Bacchus2015}
|
||||
Those studies showed that for sizes up to with $n$ = 3, the water molecules arrange in monomers or dimers in the plane of the uracil molecule
|
||||
\cite{Gadre2000, Van2001diffu, Gaigeot2001, Danilov2006, Bacchus2015} with no trimer formation. But for $n$ \textgreater~3, very different structures
|
||||
were predicted depending on the considered study. For instance, Ghomi predicted that for $n$ = 7,\cite{Gaigeot2001} water molecules arrang
|
||||
in dimers and trimers in the plane of the uracil molecule, whereas for n = 11, water molecules form locked chains.\cite{Shishkin2000} 3D configurations were also proposed. For instance, all water molecules lie above the uracil plane for $n$ = 4, 5 reported by Calvo \textit{et al.}.\cite{Bacchus2015} Similarly, for $n$ = 11, Danilov \textit{et al.} also obtained a
|
||||
structure that consists of a water cluster above the uracil molecule.\cite{Danilov2006} Such structures are predicted to start with 4 water molecules
|
||||
reported by Calvo and collaborator \cite{Bacchus2015} or with 6 water molecules (though 5 have not been calculated) reported by Gadre \textit{et al.}.\cite{Gadre2000}
|
||||
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}
|
||||
Those studies may suggest that for few water molecules (up to two), the proton should be located on the uracil molecule, whereas when a large number
|
||||
of water molecules surround the uracil, the charge is expected to be located on the water molecules. Of course, the excess proton is expected to strongly
|
||||
influence the structure of the lowest energy isomers of each species, as observed for pure water clusters, so the size at which the proton is transferred
|
||||
from uracil to water cannot be deduced from the aforementioned studies. Moreover, all those theoretical studies do not lead to the same low-energy
|
||||
structures as highlighted by Danilov and Calvo.\cite{Danilov2006, Bacchus2015} Consequently, although it is instructive from a qualitative point
|
||||
of view, the analysis of the experimental data by S. Zamith and J.-M. l'Hermite cannot be based on those studies. We have therefore undertaken a
|
||||
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
|
||||
complete the experiments by S. Zamith and J.-M. L'Hermite at the \textit{Laboratoire Collisions Agr\'egats R\'eactivit\'e }(LCAR). This work has
|
||||
been published in 2019 in the \textit{The Journal of Chemical Physics}.\cite{Braud2019}
|
||||
|
||||
\subsection{Results and Discussion}
|
||||
|
||||
In the following section, section~\ref{exp_ur}, I present in details the results obtained from the CID experiments of S. Zamith and J.-M. l'Hermite
|
||||
In the following section, section~\ref{exp_ur}, I present in details the results obtained from the CID experiments of S. Zamith and J.-M. L'Hermite
|
||||
and the main concepts used to interpret the data. The following section, section~\ref{calcul_ur}, is devoted to the theoretical determination of
|
||||
the low-energy isomers of the (H$_2$O)$_{1-7,11,12}$UH$^+$ clusters. A more detailed presentation of CID experiments is also provided in
|
||||
section~\ref{exp_cid} of chapter~4, where these details are important to explicitly model CID experiments.
|
||||
|
||||
\subsubsection{Experimental Results} \label{exp_ur}
|
||||
|
||||
\textbf{Time of flight of mass spectrum(TOFMS).}
|
||||
\textbf{Time-of-flight of mass spectrum.}
|
||||
A typical fragmentation mass spectrum obtained by colliding (H$_2$O)$_{7}$UH$^+$ with neon at a center of mass collision energy of 7.2 eV is shown in Figure \ref{mass7w}. The more intense peak on the right comes from the parent cluster (H$_2$O)$_{7}$UH$^+$, the next 7 peaks at the left of the parent peak correspond to the loss of 1-7 water molecules of parent cluster, and the next 5 peaks to the left results from the evaporation of the uracil molecule and several water molecules from parent cluster. This mass spectrum is obtained at the highest pressure explored in the present experiments. This is still true for the largest size investigated here, namely, (H$_2$O)$_{15}$UH$^+$. From the result of the fragmentation mass spectrum displayed in Figure \ref{mass7w}, it indicates multiple collisions are possible, which allows the evaporation of all water molecules. Moreover, the intensity of evaporation of water molecules is bigger than the one of evaporation of U.
|
||||
In our study, we are interested in two specific channels. Channel 1 corresponds to the loss of only neutral water molecules, whereas channel 2 corresponds to the loss of neutral uracil and one or several water molecules,
|
||||
In the study, mainly focus on two specific channels. Channel 1 corresponds to the loss of only neutral water molecules, whereas channel 2 corresponds to the loss of neutral uracil and one or several water molecules,
|
||||
\begin{align}
|
||||
\mathrm{Channel~1, ~(H_2O)_nUH^+} & \rightarrow ~ \mathrm{(H_2O)_{n-x}UH^+ + xH_2O } \\
|
||||
\mathrm{Channel~2, ~(H_2O)_nUH^+} & \rightarrow ~ \mathrm{ (H_2O)_{n-x}H^+ + xH_2O + U}
|
||||
\end{align}
|
||||
|
||||
\figuremacro{mass7w}{Time of flight of mass spectrum obtained by colliding (H$_2$O)$_{7}$UH$^+$ with Ne
|
||||
\figuremacro{mass7w}{Time-of-flight of mass spectrum obtained by colliding (H$_2$O)$_{7}$UH$^+$ with Ne
|
||||
at 7.2 eV center of mass collision energy (93.5 eV in the laboratory frame).}
|
||||
|
||||
\textbf{Fragmentation cross section.}
|
||||
@ -603,33 +593,32 @@ The main differences between the curves in Figure \ref{fragcrosssec} can be rat
|
||||
fragmentation cross sections increase with the size and seem to tend toward the geometrical one.
|
||||
The cross sections measured for clusters containing uracil colliding with water molecules (black squares) are of the same magnitude as the ones previously obtained for deuterated pure water clusters (green full circles) at a similar collision energy.\cite{Zamith2012} For clusters containing uracil, fragmentation cross sections are systematically larger than the one for pure water clusters by an amount of the same magnitude as the one predicted by the geometrical cross sections. For instance, the difference between red squares and blue stars, and the difference between red full line and blue dashed line has the same magnitude.
|
||||
|
||||
The fragmentation cross sections obtained by Dalleska
|
||||
and coworkers \cite{Dalleska1993} for protonated water clusters are within our error bars for n = 5, 6 and about a factor of 2 lower for n = 3, 4. However their cross section is notably lower for (H$_2$O)$_2$H$^+$ as compared to our measurement for (H$_2$O)UH$^+$. This difference may be explained by the fact that UH$^+$ forms a weaker bond with water than H$_2$OH$^+$ does. Indeed the dissociation energy D[H$_2$OH$^+$–H$_2$O] is 1.35 eV \cite{Dalleska1993, Hansen2009} whereas the value for D[UH$^+$–H$_2$O] is estimated between 0.54 \cite{Wincel2009} and 0.73 eV. \cite{Bakker2008} The same behavior is observed for n = 3, and the dissociation energy D[(H$_2$O)$_2$H$^+$–H$_2$O] = 0.86 eV \cite{Dalleska1993, Hansen2009} is greater than the dissociation energy D[U(H$_2$O)H$^+$–H$_2$O] = 0.49 eV.\cite{Wincel2009} Hence the dissociation of water molecules is more favored in the protonated uracil cluster than in the pure water clusters.
|
||||
The fragmentation cross sections obtained by N. Dalleska and coworkers \cite{Dalleska1993} for protonated water clusters are within our error bars for $n$ = 5, 6 and about a factor of 2 lower for $n$ = 3, 4. However their cross section is notably lower for (H$_2$O)$_2$H$^+$ as compared to our measurement for (H$_2$O)UH$^+$. This difference may be explained by the fact that UH$^+$ forms a weaker bond with water than H$_2$OH$^+$ does. Indeed the dissociation energy D[H$_2$OH$^+$–H$_2$O] is 1.35 eV \cite{Dalleska1993, Hansen2009} whereas the value for D[UH$^+$–H$_2$O] is estimated between 0.54 \cite{Wincel2009} and 0.73 eV. \cite{Bakker2008} The same behavior is observed for $n$ = 3, and the dissociation energy D[(H$_2$O)$_2$H$^+$–H$_2$O] = 0.86 eV \cite{Dalleska1993, Hansen2009} is greater than the dissociation energy D[U(H$_2$O)H$^+$–H$_2$O] = 0.49 eV.\cite{Wincel2009} Hence the dissociation of water molecules is more favored in the protonated uracil cluster than in the pure water clusters.
|
||||
|
||||
\figuremacro{fragcrosssec}{Fragmentation cross sections of clusters (H$_2$O)$_{n-1}$UH$^+$ at a collision energy of 7.2 eV plotted as a function of the total number n of molecules in the clusters. The experimental results and geometrical cross sections are shown for collision with H$_2$O and Ne. The results from Dalleska et al.\cite{Dalleska1993} using Xe as target atoms on pure protonated water clusters (H$_2$O)$_{2-6}$H$^+$ and from Zamith \textit{et al.} \cite{Zamith2012} using water as target molecules on deuterated water clusters (D$_2$O)$_{5,10}$H$^+$ are also shown. The geometrical collision cross sections of water clusters in collision with Xe atoms and water molecules are also plotted. Error bars represent one standard deviation.}
|
||||
\figuremacro{fragcrosssec}{Fragmentation cross sections of clusters (H$_2$O)$_{n-1}$UH$^+$ at a collision energy of 7.2 eV plotted as a function of the total number n of molecules in the clusters. The experimental results and geometrical cross sections are shown for collision with H$_2$O and Ne. The results from N. Dalleska et al.\cite{Dalleska1993} using Xe as target atoms on pure protonated water clusters (H$_2$O)$_{2-6}$H$^+$ and from S. Zamith \textit{et al.} \cite{Zamith2012} using water as target molecules on deuterated water clusters (D$_2$O)$_{5,10}$H$^+$ are also shown. The geometrical collision cross sections of water clusters in collision with Xe atoms and water molecules are also plotted. Error bars represent one standard deviation.}
|
||||
|
||||
\textbf{Intermolecular fragmentation.}
|
||||
Figure \ref{Uloss} displays the percentage of the fragments that have lost a neutral uracil molecule over all the fragments, plotted as a function of the number of water molecules in the parent cluster (H$_2$O)$_{n}$UH$^+$. It shows that for the cluster (H$_2$O)$_{n}$UH$^+$ with a small number of water molecules, almost no neutral uracil is evaporated. From n = 5 and more clearly from n = 6, the loss of neutral uracil molecule increases up to about 20\% for (H$_2$O)$_{9}$UH$^+$.
|
||||
Figure \ref{Uloss} displays the percentage of the fragments that have lost a neutral uracil molecule over all the fragments, plotted as a function of the number of water molecules in the parent cluster (H$_2$O)$_{n}$UH$^+$. It shows that for the cluster (H$_2$O)$_{n}$UH$^+$ with a small number of water molecules, almost no neutral uracil is evaporated. From $n$ = 5 and more clearly from $n$ = 6, the loss of neutral uracil molecule increases up to about 20\% for (H$_2$O)$_{9}$UH$^+$.
|
||||
|
||||
\figuremacro{Uloss}{Proportion of neutral uracil molecule loss plotted as a function of the number of water molecules n in the parent cluster (H$_2$O)$_{n}$UH$^+$. Results obtained for collisions with Ne atoms at 7.2 eV center of mass collision energy.}
|
||||
|
||||
The fragmentation can arise from two distinct mechanisms (direct and statistical fragmentation processes) depending on the life time of the collision complex. On the one hand, if the fragmentation occurs in a very short time after collision, the dissociation is impulsive (direct). In this case, we thus assume that the nature of the collision products is partly determined by the nature of the lowest-energy isomers of parent clusters and especially by the location of the excess proton in the structure. In other words, the lowest-energy isomer of the parent cluster obviously plays a major role in determining the fragmentation channels. On the other hand, in the case of long-lived collision complexes, collision energy is transferred to the parent cluster and is redistributed among all degrees of freedom. This is a slow process, and the structures involved during the fragmentation are no longer the lowest-energy isomer, \textit{i.e.}, the structure of the cluster can undergo structural reorganizations before evaporation. Furthermore, the excess proton can also diffuse in the structure and for instance, recombine with the uracil. Then the role of the initial structure of the parent clusters is strongly reduced in determining the fragmentation channels.
|
||||
The fragmentation can arise from two distinct mechanisms (direct and statistical fragmentation processes) depending on the life time of the collision complex. On the one hand, if the fragmentation occurs in a very short time after collision, the dissociation is impulsive (direct). In this case, I thus assume that the nature of the collision products is partly determined by the nature of the lowest-energy isomers of parent clusters and especially by the location of the excess proton in the structure. In other words, the lowest-energy isomer of the parent cluster obviously plays a major role in determining the fragmentation channels. On the other hand, in the case of long-lived collision complexes, collision energy is transferred to the parent cluster and is redistributed among all degrees of freedom. This is a slow process, and the structures involved during the fragmentation are no longer the lowest-energy isomer, \textit{i.e.}, the structure of the cluster can undergo structural reorganizations before evaporation. Furthermore, the excess proton can also diffuse in the structure and for instance, recombine with the uracil. Then the role of the initial structure of the parent clusters is strongly reduced in determining the fragmentation channels.
|
||||
|
||||
In Figure \ref{Uloss}, we focus on the loss of the neutral uracil molecule in the detected fragments since it indicates where the proton lies after collision, namely, on the uracil or on a water cluster. A transition in the nature of fragmentation product is clearly seen from n = 5-6. To account for this transition, we consider that evaporation originates from a direct fragmentation process. A short discussion about the implications of possible structural rearrangement prior to dissociation, which occurs in a statistical process, will be provided in section~\ref{calcul_ur}.
|
||||
In Figure \ref{Uloss}, I focus on the loss of the neutral uracil molecule in the detected fragments since it indicates where the proton lies after collision, namely, on the uracil or on a water cluster. A transition in the nature of fragmentation product is clearly seen from $n$ = 5-6. To account for this transition, the evaporation originates from a direct fragmentation process is considered. A short discussion about the implications of possible structural rearrangement prior to dissociation, which occurs in a statistical process, will be provided in section~\ref{calcul_ur}.
|
||||
|
||||
The relative proton affinities of each component of the
|
||||
mixed clusters gives a first estimate of which molecule, uracil or water, is more likely to carry the positive charge prior to collisions. Experimentally, the gas phase proton affinity of uracil is bracketed to 9 $\pm$ 0.12 eV.\cite{Kurinovich2002} For the proton affinity of water molecule, an experimental value is reported at 7.31 eV \cite{Magnera1991} and a theoretical one at 7.5 eV.\cite{Cheng1998} In the work of Cheng, it shows that the proton affinity of water clusters increases with their size.\cite{Cheng1998} The proton affinities extracted from the different studies for the uracil molecule and for water clusters as a function of the number of water molecules are displayed in Figure~\ref{protonAffinity}.
|
||||
mixed clusters gives a first estimate of which molecule, uracil or water, is more likely to carry the positive charge prior to collisions. Experimentally, the gas phase proton affinity of uracil is bracketed to 9 $\pm$ 0.12 eV.\cite{Kurinovich2002} For the proton affinity of water molecule, an experimental value is reported at 7.31 eV \cite{Magnera1991} and a theoretical one at 7.5 eV.\cite{Cheng1998} In the work of H. Cheng, it shows that the proton affinity of water clusters increases with their size.\cite{Cheng1998} The proton affinities extracted from the different studies for the uracil molecule and for water clusters as a function of the number of water molecules are displayed in Figure~\ref{protonAffinity}.
|
||||
|
||||
\figuremacro{protonAffinity}{The proton affinities of water clusters as a function of the number of water
|
||||
molecules n, which are taken from the work of Magnera (black circles) \cite{Magnera1991} and from the work of Cheng (blue squares).\cite{Cheng1998} The value of the proton affinity of uracil (red dotted dashed line) is also plotted.\cite{Kurinovich2002}}
|
||||
molecules n, which are taken from the work of T. Magnera (black circles) \cite{Magnera1991} and from the work of Cheng (blue squares).\cite{Cheng1998} The value of the proton affinity of uracil (red dotted dashed line) is also plotted.\cite{Kurinovich2002}}
|
||||
|
||||
It clearly shows that the proton affinity of uracil, PA[U], is larger than the one of water monomer PA[H$_2$O]. Thus, for the mono-hydrated uracil, from the energetic point of view, the proton is on the uracil molecule and the only observed fragments are indeed protonated uracil molecules. Moreover, an experimental work \cite{Bakker2008} confirms that there is no proton transfer from the uracil to the water molecule in mono-hydrated clusters. Proton affinity of the uracil molecule is also larger than that of the water dimer, or even the trimer: PA[U] $>$ PA[(H$_2O$)$_n$], n = 2 or 3 depending on the considered data for water. This is still consistent with our experimental observation of no neutral uracil molecule loss for n = 2 and 3. However from the PA values, one would predict that the appearance of neutral uracil should occur for n $\approx$ 3-4. For instance, for n = 4, assuming a statistical fragmentation for which the energies of final products are expected to be of relevance, the channel U + (H$_2$O)$_4$H$^+$ is energetically favorable. If one now assumes a direct dissociation, where the parent protonation state remains unchanged, one also expects that neutral uracil evaporates. However, experimentally, for n = 4, no neutral uracil evaporation is observed. The loss of neutral uracil starts at n = 5 and becomes significant only at n = 6.
|
||||
It clearly shows that the proton affinity of uracil, PA[U], is larger than the one of water monomer PA[H$_2$O]. Thus, for the mono-hydrated uracil, from the energetic point of view, the proton is on the uracil molecule and the only observed fragments are indeed protonated uracil molecules. Moreover, an experimental work \cite{Bakker2008} confirms that there is no proton transfer from the uracil to the water molecule in mono-hydrated clusters. Proton affinity of the uracil molecule is also larger than that of the water dimer, or even the trimer: PA[U] $>$ PA[(H$_2O$)$_n$], $n$ = 2 or 3 depending on the considered data for water. This is still consistent with our experimental observation of no neutral uracil molecule loss for $n$ = 2 and 3. However from the PA values, one would predict that the appearance of neutral uracil should occur for n $\approx$ 3-4. For instance, for $n$ = 4, assuming a statistical fragmentation for which the energies of final products are expected to be of relevance, the channel U + (H$_2$O)$_4$H$^+$ is energetically favorable. If one now assumes a direct dissociation, where the parent protonation state remains unchanged, one also expects that neutral uracil evaporates. However, experimentally, for $n$ = 4, no neutral uracil evaporation is observed. The loss of neutral uracil starts at $n$ = 5 and becomes significant only at $n$ = 6.
|
||||
|
||||
This analysis based on PA is however quite crude. Indeed, it assumes that the protonated uracil cluster would be composed of a uracil molecule attached to an intact water cluster. However, one expects that the hydration of uracil may be more complicated than this simple picture. Therefore, the uracil hydration is explored theoretically in the next Section, Section~\ref{calcul_ur}, in order to determine the proton location more realistically.
|
||||
|
||||
\subsubsection{Calculated Structures of Protonated Uracil Water Clusters} \label{calcul_ur}
|
||||
|
||||
As discussed in section~\ref{sec:ammoniumwater}, we have proposed a modified set of NH parameters to describe sp$^3$ nitrogen atoms. For,
|
||||
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
|
||||
(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
|
||||
@ -672,33 +661,33 @@ have used $D_\textrm{NH}$ = 0.12 in the following.
|
||||
%\figuremacro{a-b}{The structure of isomer a and b of cluster (H$_2$O)U.}
|
||||
|
||||
The lowest-energy isomers determined theoretically for
|
||||
hydrated uracil protonated clusters (H$_2$O)$_{1-7, 11, 12}$UH$^+$ are shown in Figures \ref{1a-f}-\ref{12a-f}. In the experiments, clusters are produced at a temperature of about 25 K, so only a very few isomers are likely to be populated. Indeed, the clusters are produced in the canonical ensemble at the temperature $T_\mathrm c \approx$ 25 K, so only isomers for which the Boltzmann factor exp(-$\Delta E k_\mathrm{B} T_\mathrm{c}$) is larger than 10$^{-7}$ are considered here. In this formula, $\Delta E$ represents the relative energy of a considered isomer with respect to the lowest-energy one. Thus for each isomer, only the first six lowest-energy structures of U(H$_2$O)$_{1-7, 11, 12}$UH$^+$ obtained from the PES exploration will be discussed.
|
||||
hydrated uracil protonated clusters (H$_2$O)$_{1-7, 11, 12}$UH$^+$ are shown in Figures \ref{1a-f}-\ref{12a-f}. In the experiments, clusters are produced at a temperature of about 25 K, so only a very few isomers are likely to be populated. Indeed, the clusters are produced in the canonical ensemble at the temperature $T_\mathrm c \approx$ 25 K, so only isomers for which the Boltzmann factor exp(-$\Delta E k_\mathrm{B} T_\mathrm{c}$) is larger than 10$^{-7}$ are considered here. In this formula, $\Delta E$ represents the relative energy of a considered isomer with respect to the lowest-energy one. Thus for each isomer, only the six lowest-energy structures of U(H$_2$O)$_{1-7, 11, 12}$UH$^+$ obtained from the PES exploration will be discussed.
|
||||
|
||||
Figure \ref{1a-f} displays the six lowest-energy isomers obtained for (H$_2$O)UH$^+$. Two (1a and 1b) of them contain the u138-like isomer of U (each one with a different orientation of the hydroxyl hydrogen). Three of them (1c, 1d, and 1e) contain the u178 isomer and 1f contains the u137\cite{Wolken2000} isomer with a reverse orientation of the hydroxyl hydrogen. From those isomers, different sites are possible for the water molecule attachment which leads to variety of isomers even for such small size system. To the best of our knowledge, (H$_2$O)UH$^+$ is the most studied protonated uracil water cluster and our results are consistent with previous
|
||||
published studies. Indeed, Pedersen and co-workers \cite{Pedersen2014} conducted ultraviolet action spectroscopy on (H$_2$O)UH$^+$ and discussed their measurements in the light of theoretical calculations performed on two isomers: ur138w8 (1a in the present study) and ur178w7 (1c).\cite{Pedersen2014} Their energy ordering at 0 K is the same whatever the computational method they used: B3LYP/6-311++G(3df,2p), M06-2X/6-311++G(3df,2p), MP2/6-311++G(3df,2p), CCSD(T)/6-311++G(3df,2p), and CCSD(T)/augcc-pVTZ and is similar to what we obtain. Similarly, Bakker and co-workers\cite{Bakker2008} considered three isomers: U(DK)H$^+_W$ (1a), U(KE)H$^+_{Wa}$ (1c), and U(KE)H$^+_{Wb}$ (1e) at the B3LYP/6-311++G(3df,2p) level of theory and obtained the same energy ordering as we do. Our methodology has thus allowed us to retrieve those isomers and to locate two new low-energy structures (1b and 1d). 1f is too high in energy to be considered in low-temperature experiments that are in the same range of relative energies but have never been discussed. To ensure that they are not artificially favored in our computational method, calculations were also performed at the B3LYP/6-311++G(3df,2p) level of theory. The results are presented in Figure \ref{1a-f-b3lyp}, which are consistent with the MP2/Def2TZVP ones. This makes us confident in the ability of the present methodology to locate meaningful low energy structures. Importantly, no isomer with the proton on the water molecule was obtained, neither at the DFTB or MP2 levels.
|
||||
published studies. Indeed, S. Pedersen and co-workers \cite{Pedersen2014} conducted ultraviolet action spectroscopy on (H$_2$O)UH$^+$ and discussed their measurements in the light of theoretical calculations performed on two isomers: ur138w8 (1a in the present study) and ur178w7 (1c).\cite{Pedersen2014} Their energy ordering at 0 K is the same whatever the computational method they used: B3LYP/6-311++G(3df,2p), M06-2X/6-311++G(3df,2p), MP2/6-311++G(3df,2p), CCSD(T)/6-311++G(3df,2p), and CCSD(T)/augcc-pVTZ and is similar to what I obtained. Similarly, J. Bakker and co-workers\cite{Bakker2008} considered three isomers: U(DK)H$^+_W$ (1a), U(KE)H$^+_{Wa}$ (1c), and U(KE)H$^+_{Wb}$ (1e) at the B3LYP/6-311++G(3df,2p) level of theory and obtained the same energy ordering as I did. Our methodology has thus allowed us to retrieve those isomers and to locate two new low-energy structures (1b and 1d). 1f is too high in energy to be considered in low-temperature experiments that are in the same range of relative energies but have never been discussed. To ensure that they are not artificially favored in our computational method, calculations were also performed at the B3LYP/6-311++G(3df,2p) level of theory. The results are presented in Figure \ref{1a-f-b3lyp}, which are consistent with the MP2/Def2TZVP ones. This makes us confident in the ability of the present methodology to locate meaningful low energy structures. Importantly, no isomer with the proton on the water molecule was obtained, neither at the DFTB or MP2 levels.
|
||||
|
||||
\figuremacrob{1a-f}{Lowest-energy structures of (H$_2$O)UH$^+$ obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.}
|
||||
|
||||
\figuremacrob{1a-f-b3lyp}{Lowest-energy structures of (H$_2$O)UH$^+$ obtained at the B3LYP/6-311++G(3df,2p) level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. The corresponding values with ZPVE corrections are provided in brackets. Important hydrogen-bond distances are indicated in bold and are given in \AA.}
|
||||
|
||||
Figures \ref{2a-f} and \ref{3a-f} display the first six lowest-energy isomers obtained for (H$_2$O)$_2$UH$^+$ and (H$_2$O)$_3$UH$^+$, respectively. For (H$_2$O)$_2$UH$^+$, the lowest energy structure, 2a contains the u138 isomer of uracil. 2b, 2d, and 2e contain u178 and 2c contains u138 with reverse orientation of the hydroxyl hydrogen. 2f contains u178 with reverse orientation of the hydroxyl hydrogen. This demonstrates that, similarly to (H$_2$O)UH$^+$, a diversity of uracil isomers are present in the low-energy structures of (H$_2$O)$_2$UH$^+$ which makes an exhaustive exploration of its PES more difficult. The same behavior is observed for (H$_2$O)$_3$UH$^+$. The configuration of u138 does not allow for the formation of a water dimer which leads to two unbound water molecules in 2a. By contrast, a water-water hydrogen bond is observed for 2b and 2c. The existence of a water dimer was not encountered in the low-energy isomers of the unprotonated (H$_2$O)$_2$U species due to the absence of the hydroxyl group on U. It is worth pointing out that 2a, 2b, 2c, and 2d are very close in energy which makes their exact energy ordering difficult to determine. However, no isomer displaying an unprotonated uracil in the low-energy isomers of (H$_2$O)$_2$UH$^+$ was located. The lowest-energy structure of (H$_2$O)$_3$UH$^+$, 3a, is characterized by two water-water hydrogen bond that forms a linear water trimer. Higher energy isomers display only one (3b, 3d, and 3e) or zero (3c and 3f) water-water bond (see Figure \ref{3a-f}). Similarly to (H$_2$O)$_2$UH$^+$, no isomer displaying an unprotonated uracil was located for (H$_2$O)$_3$UH$^+$.
|
||||
Figures \ref{2a-f} and \ref{3a-f} display the six lowest-energy isomers obtained for (H$_2$O)$_2$UH$^+$ and (H$_2$O)$_3$UH$^+$, respectively. For (H$_2$O)$_2$UH$^+$, the lowest energy structure, 2a contains the u138 isomer of uracil. 2b, 2d, and 2e contain u178 and 2c contains u138 with reverse orientation of the hydroxyl hydrogen. 2f contains u178 with reverse orientation of the hydroxyl hydrogen. This demonstrates that, similarly to (H$_2$O)UH$^+$, a diversity of uracil isomers are present in the low-energy structures of (H$_2$O)$_2$UH$^+$ which makes an exhaustive exploration of its PES more difficult. The same behavior is observed for (H$_2$O)$_3$UH$^+$. The configuration of u138 does not allow for the formation of a water dimer which leads to two unbound water molecules in 2a. By contrast, a water-water hydrogen bond is observed for 2b and 2c. The existence of a water dimer was not encountered in the low-energy isomers of the unprotonated (H$_2$O)$_2$U species due to the absence of the hydroxyl group on U. It is worth pointing out that 2a, 2b, 2c, and 2d are very close in energy which makes their exact energy ordering difficult to determine. However, no isomer displaying an unprotonated uracil in the low-energy isomers of (H$_2$O)$_2$UH$^+$ was located. The lowest-energy structure of (H$_2$O)$_3$UH$^+$, 3a, is characterized by two water-water hydrogen bond that forms a linear water trimer. Higher energy isomers display only one (3b, 3d, and 3e) or zero (3c and 3f) water-water bond (see Figure \ref{3a-f}). Similarly to (H$_2$O)$_2$UH$^+$, no isomer displaying an unprotonated uracil was located for (H$_2$O)$_3$UH$^+$.
|
||||
|
||||
\figuremacrob{2a-f}{Lowest-energy structures of (H$_2$O)$_2$UH$^+$ obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.}
|
||||
|
||||
\figuremacrob{3a-f}{(H$_2$O)$_3$UH$^+$ lowest-energy structures obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.}
|
||||
|
||||
The first six lowest-energy isomers obtained for (H$_2$O)$_4$UH$^+$ and (H$_2$O)$_5$UH$^+$ are displayed in Figures \ref{4a-f} and \ref{5a-f}, which constitute a transition in the behavior of the proton. Indeed, in (H$_2$O)$_4$UH$^+$, two kind of low-lying energy structures appear: (i) structures composed of UH$^+$, one water trimer, and one isolated water molecule (4b, 4d, 4e, and 4f); (ii) structures composed of U and a protonated water tetramer (4a and 4c). In the latter case, the hydronium ion is always bounded to an uracil oxygen atom. The U–H$_2$OH$^+$ bond is always rather strong as compared to U–H$_2$O bonds as highlighted by the corresponding short oxygen-hydrogen distance. Furthermore, speaking of distances, the difference between the U–H$_2$OH$^+$ and UH$^+$–H$_2$O forms is rather fuzzy and might be sensitive to computational parameters and also to quantum fluctuations of the hydrogen. This suggests that collision with (H$_2$O)$_4$UH$^+$ is more likely to induce evaporation of H$_2$O rather than H$_2$OH$^+$ or a protonated water cluster. The picture is significantly different in (H$_2$O)$_5$UH$^+$ where the lowest-energy structure displays a hydronium ion separated by one water molecule from U. Such structures do not appear in (H$_2$O)$_4$UH$^+$ due to the limited number of water molecules available to separate H$_2$OH$^+$ from U. Such separation suggests that, if considering a direct dissociation process, evaporation of neutral uracil can now occurs in agreement with the experimental observations (see discussion above). One can see that 5b, which is only 0.3 kcal.mol$^{-1}$ higher in energy than 5a, still displays a U–H$_2$OH$^+$ link. This is in line with the low amount of neutral uracil that is evaporated in the experiment (see Figure \ref{Uloss}).
|
||||
The six lowest-energy isomers obtained for (H$_2$O)$_4$UH$^+$ and (H$_2$O)$_5$UH$^+$ are displayed in Figures \ref{4a-f} and \ref{5a-f}, which constitute a transition in the behavior of the proton. Indeed, in (H$_2$O)$_4$UH$^+$, two kind of low-lying energy structures appear: (i) structures composed of UH$^+$, one water trimer, and one isolated water molecule (4b, 4d, 4e, and 4f); (ii) structures composed of U and a protonated water tetramer (4a and 4c). In the latter case, the hydronium ion is always bounded to an uracil oxygen atom. The U–H$_2$OH$^+$ bond is always rather strong as compared to U–H$_2$O bonds as highlighted by the corresponding short oxygen-hydrogen distance. Furthermore, speaking of distances, the difference between the U–H$_2$OH$^+$ and UH$^+$–H$_2$O forms is rather fuzzy and might be sensitive to computational parameters and also to quantum fluctuations of the hydrogen. This suggests that collision with (H$_2$O)$_4$UH$^+$ is more likely to induce evaporation of H$_2$O rather than H$_2$OH$^+$ or a protonated water cluster. The picture is significantly different in (H$_2$O)$_5$UH$^+$ where the lowest-energy structure displays a hydronium ion separated by one water molecule from U. Such structures do not appear in (H$_2$O)$_4$UH$^+$ due to the limited number of water molecules available to separate H$_2$OH$^+$ from U. Such separation suggests that, if considering a direct dissociation process, evaporation of neutral uracil can now occurs in agreement with the experimental observations (see discussion above). One can see that 5b, which is only 0.3 kcal.mol$^{-1}$ higher in energy than 5a, still displays a U–H$_2$OH$^+$ link. This is in line with the low amount of neutral uracil that is evaporated in the experiment (see Figure \ref{Uloss}).
|
||||
|
||||
\figuremacrob{4a-f}{Lowest-energy structures of (H$_2$O)$_4$UH$^+$ obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.}
|
||||
|
||||
\figuremacrob{5a-f}{Lowest-energy structures of (H$_2$O)$_5$UH$^+$ obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.}
|
||||
|
||||
Figures \ref{6a-f} and \ref{7a-f} display the first six lowest-energy isomers obtained for (H$_2$O)$_6$UH$^+$ and (H$_2$O)$_7$UH$^+$. Similarly to (H$_2$O)$_5$UH$^+$, the first lowest-energy structure, 6a and 7a, we located for both species (H$_2$O)$_6$UH$^+$ and (H$_2$O)$_7$UH$^+$ has the excess proton on a water molecule that is separated by one water molecule from the uracil. This appears to be common to the clusters with at least 5 water molecules. This is also observed for higher-energy isomers (6c, 6d, 7c, 7e, and 7f). Other characteristics of the proton are also observed: proton in a similar Zundel form \cite{Zundel1968} bounded to the uracil (6b, 6e, and 7d) or H$_2$OH$^+$ still bounded to uracil (6f and 7b).
|
||||
Figures \ref{6a-f} and \ref{7a-f} display the six lowest-energy isomers obtained for (H$_2$O)$_6$UH$^+$ and (H$_2$O)$_7$UH$^+$. Similarly to (H$_2$O)$_5$UH$^+$, the first lowest-energy structure, 6a and 7a, for both species (H$_2$O)$_6$UH$^+$ and (H$_2$O)$_7$UH$^+$ have the excess proton on a water molecule that is separated by one water molecule from the uracil. This appears to be common to the clusters with at least 5 water molecules. This is also observed for higher-energy isomers (6c, 6d, 7c, 7e, and 7f). Other characteristics of the proton are also observed: proton in a similar Zundel form \cite{Zundel1968} bounded to the uracil (6b, 6e, and 7d) or H$_2$OH$^+$ still bounded to uracil (6f and 7b).
|
||||
|
||||
Finally, due to the neutral uracil loss proportion starts to decrease from $n$=9 (see Figure \ref{Uloss}), which attracted us to perform the optimization of big cluster (H$_2$O)$_{11, 12}$UH$^+$ as examples to explore why it has this change. The first six low-lying energy isomers obtained for cluster (H$_2$O)$_{11, 12}$UH$^+$ are shown in Figures \ref{11a-f} and \ref{12a-f}.
|
||||
Finally, due to the neutral uracil loss proportion starts to decrease from $n$=9 (see Figure \ref{Uloss}), which attracted us to perform the optimization of big cluster (H$_2$O)$_{11, 12}$UH$^+$ as examples to explore why it has this change. The six low-lying energy isomers obtained for cluster (H$_2$O)$_{11, 12}$UH$^+$ are shown in Figures \ref{11a-f} and \ref{12a-f}.
|
||||
|
||||
In all isomers (11a to 11f) of cluster (H$_2$O)$_{11}$UH$^+$, the excess is on the water cluster and was separated by water molecule from uracil.
|
||||
For 12a, 12b, 12c, and 12d, it is obvious that the excess proton is not directly bounded to the uracil. The uracil in 12a and 12d belongs to the di-keto form (there is a hydrogen atom on each nitrogen of uracil), and the excess proton was separated by one water molecule from uracil, additionally, the uracil is surrounded by the water cluster, all of these may lead the excess proton to go to the near oxygen atom of uracil. For 12b, the excess proton is on the water cluster and is very far from the uracil. For 12c, the excess proton was separately by one water molecule from uracil. For isomers 12e and 12f, the excess proton is between the uracil and a water molecule. The uracil is surrounded by the water cluster in 12e but it is not in 12f. Of course, for (H$_2$O)$_{11}$UH$^+$, (H$_2$O)$_{12}$UH$^+$, (H$_2$O)$_6$UH$^+$ and (H$_2$O)$_7$UH$^+$ and also (H$_2$O)$_4$UH$^+$ and (H$_2$O)$_5$UH$^+$, the amount of low-energy isomers is expected to be very large and we do not intended to find them all. Furthermore, due to the limited number of MP2 geometry optimization we performed, there is few chances that we located the global energy minima for (H$_2$O)$_6$UH$^+$, (H$_2$O)$_7$UH$^+$, (H$_2$O)$_{11}$UH$^+$ and (H$_2$O)$_{12}$UH$^+$. However, the general picture we are able to draw from the present discussed structures fully supports the experimental results: from (H$_2$O)$_5$UH$^+$, it exists low-energy structures populated at very low temperature in which the excess proton is not directly bound to the uracil molecule. Upon fragmentation, this allows the proton to remain bounded to the water molecules.
|
||||
For 12a, 12b, 12c, and 12d, it is obvious that the excess proton is not directly bounded to the uracil. The uracil in 12a and 12d belongs to the di-keto form (there is a hydrogen atom on each nitrogen of uracil), and the excess proton was separated by one water molecule from uracil, additionally, the uracil is surrounded by the water cluster, all of these may lead the excess proton to go to the near oxygen atom of uracil. For 12b, the excess proton is on the water cluster and is very far from the uracil. For 12c, the excess proton was separately by one water molecule from uracil. For isomers 12e and 12f, the excess proton is between the uracil and a water molecule. The uracil is surrounded by the water cluster in 12e but it is not in 12f. Of course, for (H$_2$O)$_{11}$UH$^+$, (H$_2$O)$_{12}$UH$^+$, (H$_2$O)$_6$UH$^+$ and (H$_2$O)$_7$UH$^+$ and also (H$_2$O)$_4$UH$^+$ and (H$_2$O)$_5$UH$^+$, the amount of low-energy isomers is expected to be very large and do not intended to find them all. Furthermore, due to the limited number of MP2 geometry optimization I performed, there are few chances that I located the global energy minima for (H$_2$O)$_6$UH$^+$, (H$_2$O)$_7$UH$^+$, (H$_2$O)$_{11}$UH$^+$ and (H$_2$O)$_{12}$UH$^+$. However, the general picture I am able to draw from the present discussed structures fully supports the experimental results: from (H$_2$O)$_5$UH$^+$, it exists low-energy structures populated at very low temperature in which the excess proton is not directly bound to the uracil molecule. Upon fragmentation, this allows the proton to remain bounded to the water molecules.
|
||||
|
||||
\figuremacrob{6a-f}{Lowest-energy structures of (H$_2$O)$_6$UH$^+$ obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.}
|
||||
|
||||
@ -708,21 +697,21 @@ For 12a, 12b, 12c, and 12d, it is obvious that the excess proton is not directly
|
||||
|
||||
\figuremacrob{12a-f}{Lowest-energy structures of (H$_2$O)$_{12}$UH$^+$ obtained at the MP2/Def2TZVP level of theory. Relative ($E_\textrm{rel}$) and binding energies ($E_\textrm{bind}$) are given in kcal.mol$^{-1}$. Important hydrogen-bond distances are indicated in bold and are given in \AA.}
|
||||
|
||||
All the aforementioned low-lying energy structures are relevant to describe the \newline (H$_2$O)$_{1-7, 11, 12}$UH$^+$ species at low temperature and to understand the relation between the parent cluster size and the amount of evaporated neutral uracil in the case of direct dissociation. However, as already stated, one has to keep in mind that upon collision statistical dissociation can also occur. In that case, structural rearrangements are expected to occur which are important to understand each individual mass spectra of the (H$_2$O)$_{1-15}$UH$^+$ clusters and the origin of each collision product. For instance, the fragment UH$^+$ is detected for all cluster sizes in experiment. This means that for the largest sizes, for which we have shown from the calculation that the proton is located away from the uracil, proton transfer does occur prior to dissociation. One possible scenario is that after collision, water molecules sequentially evaporates. When the number of water molecules is small enough, the proton affinity of uracil gets larger than the one of the remaining attached water cluster. Proton transfer is then likely and therefore protonated uracil can be obtained at the end.
|
||||
All the aforementioned low-lying energy structures are relevant to describe the \newline (H$_2$O)$_{1-7, 11, 12}$UH$^+$ species at low temperature and to understand the relation between the parent cluster size and the amount of evaporated neutral uracil in the case of direct dissociation. However, as already stated, one has to keep in mind that upon collision statistical dissociation can also occur. In that case, structural rearrangements are expected to occur which are important to understand each individual mass spectra of the (H$_2$O)$_{1-15}$UH$^+$ clusters and the origin of each collision product. For instance, the fragment UH$^+$ is detected for all cluster sizes in experiment. This means that for the largest sizes, for which I have shown from the calculation that the proton is located away from the uracil, proton transfer does occur prior to dissociation. One possible scenario is that after collision, water molecules sequentially evaporates. When the number of water molecules is small enough, the proton affinity of uracil gets larger than the one of the remaining attached water cluster. Proton transfer is then likely and therefore protonated uracil can be obtained at the end.
|
||||
|
||||
If one turns to the neutral uracil evaporation channel, it appears that the smaller clusters H$_2$OH$^+$ and (H$_2$O)$_2$H$^+$ are not present in the time of flight mass spectra. This absence might have two origins. First, the dissociation energies of the protonated water monomers and dimers are substantially higher than larger sizes, and they are therefore less prone to evaporation. Second, as already mentioned, for such small sizes, the proton affinity of uracil gets larger than the one of the water dimer or trimer and proton transfer to the uracil is likely to occur.
|
||||
If one turns to the neutral uracil evaporation channel, it appears that the smaller clusters H$_2$OH$^+$ and (H$_2$O)$_2$H$^+$ are not present in the time-of-flight mass spectra. This absence might have two origins. First, the dissociation energies of the protonated water monomers and dimers are substantially higher than larger sizes, and they are therefore less prone to evaporation. Second, as already mentioned, for such small sizes, the proton affinity of uracil gets larger than the one of the water dimer or trimer and proton transfer to the uracil is likely to occur.
|
||||
|
||||
In order to confirm the above scenarios, simulations
|
||||
and/or evaporation rate calculation would have to be conducted to describe the fragmentation channels in details. MD simulations of protonated uracil have already been performed by Spezia and co-workers to understand the collision-induced dissociation.\cite{Molina2015, Molina2016} Although, in the present case, the initial position of the excess proton appears as a key parameter to explain the evaporation of neutral uracil, such MD simulations could be additionally conducted to provide a clearer picture on the various evaporation pathways, which will be shown in section \ref{sec:collisionwUH}.
|
||||
and/or evaporation rate calculation would have to be conducted to describe the fragmentation channels in details. MD simulations of protonated uracil have already been performed by R. Spezia and co-workers to understand the collision-induced dissociation.\cite{Molina2015, Molina2016} Although, in the present case, the initial position of the excess proton appears as a key parameter to explain the evaporation of neutral uracil, such MD simulations could be additionally conducted to provide a clearer picture on the various evaporation pathways, which will be shown in section \ref{sec:collisionwUH}.
|
||||
|
||||
\subsection{Conclusions on (H$_2$O)$_{n}$UH$^+$ clusters}
|
||||
|
||||
The work in this section presents the collision-induced dissociation of hydrated protonated uracil (H$_2$O)$_{1-15}$UH$^+$ clusters and their experimental
|
||||
absolute fragmentation cross sections. The experiments demonstrate that the evaporation channels evolve with size: Below n = 5, the observed charged fragments
|
||||
always contain the uracil molecule, whereas from n = 5, the loss of a neutral uracil molecule becomes significant. To understand this transition, I conducted an
|
||||
absolute fragmentation cross sections. The experiments demonstrate that the evaporation channels evolve with size: Below $n$ = 5, the observed charged fragments
|
||||
always contain the uracil molecule, whereas from $n$ = 5, the loss of a neutral uracil molecule becomes significant. To understand this transition, I conducted an
|
||||
exhaustive exploration of the potential energy surface of (H$_2$O)$_{1-7, 11, 12}$UH$^+$ clusters combining a rough exploration at the SCC-DFTB level with
|
||||
fine geometry optimizations at the MP2 level of theory. Those calculations show that below n = 5, the excess proton is either on the uracil or on a water molecule
|
||||
directly bound to uracil, \textit{i.e.}, forming a strongly bound U–H$_2$OH$^+$ complex. From n = 5 and above, clusters contain enough water molecules to allow
|
||||
fine geometry optimizations at the MP2 level of theory. Those calculations show that below $n$ = 5, the excess proton is either on the uracil or on a water molecule
|
||||
directly bound to uracil, \textit{i.e.}, forming a strongly bound U–H$_2$OH$^+$ complex. From $n$ = 5 and above, clusters contain enough water molecules to allow
|
||||
for a net separation between uracil and the excess proton: The latter is often found bound to a water molecule which is separated from uracil by at least one other
|
||||
water molecule. Upon direct dissociation, the excess proton and the uracil can thus belong to different fragments. This study demonstrates that combination of
|
||||
collision-induced dissociation experiments and theoretical calculation allows to probe the solvation and protonation properties of organic molecules such as nucleobases.
|
||||
|
||||
@ -3,18 +3,18 @@
|
||||
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||||
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|
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\FN@pp@footnotehinttrue
|
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\@setckpt{4/collision}{
|
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@ -1,11 +1,15 @@
|
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This is BibTeX, Version 0.99d (TeX Live 2019)
|
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Capacity: max_strings=100000, hash_size=100000, hash_prime=85009
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The top-level auxiliary file: collision.aux
|
||||
White space in argument---line 6 of file collision.aux
|
||||
: \citation{Holm2010,M.
|
||||
: Gatchell2014,Gatchell2017}
|
||||
I'm skipping whatever remains of this command
|
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I found no \bibdata command---while reading file collision.aux
|
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I found no \bibstyle command---while reading file collision.aux
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and the built_in function-call counts, 0 in all, are:
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= -- 0
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> -- 0
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@ -44,4 +48,4 @@ warning$ -- 0
|
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while$ -- 0
|
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width$ -- 0
|
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write$ -- 0
|
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(There were 2 error messages)
|
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(There were 3 error messages)
|
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|
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File diff suppressed because it is too large
Load Diff
@ -30,12 +30,12 @@ The stability of cluster can be investigated from dissociation experiments. Clus
|
||||
For instance, the sodium cluster ions and lithium cluster cation were dissociated with a pulsed UV laser source.\cite{Brechignac1989, Brechignac1994}
|
||||
Gaseous hydrated trivalent metal ions were dissociated using blackbody infrared radiative dissociation (BIRD).\cite{Wong2004, Bush2008}
|
||||
The collision between cluster and high or low energetic particles at different pressure also have been investigated.
|
||||
Collisions between the high energetic projectile ions (such as 3 keV Ar$^+$, 22.5 keV He$^{2+}$) and neutral targets were investigated by Gatchell and Holm.\cite{Holm2010, Gatchell2014, Gatchell2017}
|
||||
Collisions between the high energetic projectile ions (such as 3 keV Ar$^+$, 22.5 keV He$^{2+}$) and neutral targets were investigated by Gatchell and A. Holm.\cite{Holm2010, M. Gatchell2014, Gatchell2017}
|
||||
Collisions between clusters and projectile have been also explored at low collision energy, which allows for the derivation of dissociation energies and the thermal evaporation and stability of clusters. \cite{Boering1992, Wells2005, Zamith2019thermal}
|
||||
|
||||
By colliding a molecule, or a molecular aggregate, with a non-reactive rare gas atom (neon, argon) or a small molecule such as H$_2$O or N$_2$, it is possible to monitor the parent ions and collision products by use, for instance, of tandem mass spectrometry (MS/MS).\cite{Ma1997, Chowdhury2009} The resulting mass spectra provide a wealth of information about the structure of the parent and product ions from which one can infer, for instance, dissociation mechanisms \cite{Nelson1994, Molina2015} or bond and hydration enthalpies \cite{Carl2007}.
|
||||
|
||||
The overall process of collisional activation followed by dissociation/fragmentation is commonly referred to as the collision‐induced dissociation (CID) that is also named collisionally activated dissociation (CAD). CID is a mass spectrometry technique to induce dissociation/fragmentation of selected ions in the gas phase, which is one of standard methods for the determination of dissociation/fragmentation pathways. \cite{Sleno2004ion, Wells2005}
|
||||
The overall process of collisional activation followed by dissociation/fragmentation is commonly referred to as the collision-induced dissociation (CID) that is also named collisionally activated dissociation (CAD). CID is a mass spectrometry technique to induce dissociation/fragmentation of selected ions in the gas phase, which is one of standard methods for the determination of dissociation/fragmentation pathways. \cite{Sleno2004ion, Wells2005}
|
||||
The CID technique consists of accelerating a given ion into a collision gas thereby the ion getting energy and inducing fragmentation. The produced ionic fragments are then mass analyzed, yielding essentially a mass spectrum.\cite{Cody1982}
|
||||
The CID technique has been applied in different context.
|
||||
Higher-energy C-trap dissociation is a CID technique specific to the orbitrap mass spectrometer in which dissociation/fragmentation occurs outside the trap \cite{Olsen2007higher, Hart2011}
|
||||
@ -48,7 +48,7 @@ CID has been applied to a variety of systems, in particular hydrated atomic ions
|
||||
|
||||
Theoretical and experimental studies devoted to fragmentation of hydrated molecular aggregates are scarce, \cite{Li1992, Bobbert2002, Liu2006, Bakker2008, Markush2016, Castrovilli2017} although CID has been applied to water clusters containing an atomic ion \cite{Carl2013, Hofstetter2013, Coates2018} and on charged water
|
||||
clusters \cite{Dawson1982, Bakker2008, Mcquinn2009, Zamith2012}. This is a real lack as understanding hydration of molecules and biomolecules is of paramount importance to get insights into their structure, stability, dynamics and reactivity in aqueous medium. In that respect, CID investigations could play an important role in understanding those properties in a
|
||||
environment free from long-range solvent effects but also for different hydration degrees or protonation states. This can be evidenced by the experimental study of Liu \textit{et al.} on the fragmentation of the singly-charged adenosine 5'-monophosphate (AMP$^-$)
|
||||
environment free from long-range solvent effects but also for different hydration degrees or protonation states. This can be evidenced by the experimental study of B. Liu \textit{et al.} on the fragmentation of the singly-charged adenosine 5'-monophosphate (AMP$^-$)
|
||||
which shows two different fragmentation channel depending on the solvation state of AMP$^-$.\cite{Liu2006} However, to the best of our knowledge, no modelling was performed to complement these experiments except for a few static calculations.\cite{Carl2013, Hofstetter2013, Coates2018}.
|
||||
%colliding with a high or low energetic particles at high or low pressure,\cite{Kambara1977, Kruckeberg1999, Spasov2000, Holm2010, Gatchell2014, Gatchell2017, Zamith2019thermal} and so on has been explored in experiment.
|
||||
%magnetic field and electric field are usually used in the experiment setup for the dissociation of clusters.
|
||||
@ -57,7 +57,7 @@ which shows two different fragmentation channel depending on the solvation state
|
||||
|
||||
%I can not find literature which describe only electric field or magnetic field can lead to dissociation of clusters. You know some ?}
|
||||
|
||||
Threshold collision-induced dissociation (TCID) method has also been used, for instance to study the fragmentation patterns and to measure the dissociation energies of clusters.\cite{Spasov2000, Armentrout2008} Zamith \textit{et al.} did a CID study of the mass-selected protonated uracil water clusters with water molecules and noble gases, respectively.\cite{Braud2019}
|
||||
Threshold collision-induced dissociation (TCID) method has also been used, for instance to study the fragmentation patterns and to measure the dissociation energies of clusters.\cite{Spasov2000, Armentrout2008} S. Zamith \textit{et al.} did a CID study of the mass-selected protonated uracil water clusters with water molecules and noble gases, respectively.\cite{Braud2019}
|
||||
In addition, they also reported the TCID study of pyrene cluster cations. \cite{Zamith2020threshold}
|
||||
For these two projects, the single collision event is the predominant process.
|
||||
In this chapter, MD simulations based on a quantum chemical formalism are able to model such complex dissociation mechanism to provide an atomic-scale description for these collisions to explain and complete these experiments.
|
||||
@ -84,7 +84,7 @@ In this method, ion guides are not used. Therefore, it needs to simulate the ful
|
||||
%\subsection{Experimental setup}
|
||||
\subsection{Experimental Setup} \label{EXPsetup}
|
||||
|
||||
The experimental setup of Laboratoire Collisions Agr{\'e}gats R{\'e}activi{\'e} (LCAR) by Zamith {\textit et al.} for the collision of protonated uracil water clusters or pyrene dimer cation with noble gas atoms is set up as follows:
|
||||
The experimental setup of Laboratoire Collisions Agr{\'e}gats R{\'e}activi{\'e} (LCAR) by S. Zamith {\textit et al.} for the collision of protonated uracil water clusters or pyrene dimer cation with noble gas atoms is set up as follows:
|
||||
|
||||
\figuremacrob{experiment-setup}{Schematic view of the experimental setup. (a) Cluster gas aggregation source. (b) Thermalization chamber. (c) First Wiley–McLaren acceleration stage. (d) Massfilter. (e) Energy focusing. (f) Deceleration. (g) Collision cell. (h) Second Wiley–McLaren acceleration stage. (i) Reflectron. (j) Micro-channel plate detector.}
|
||||
%%%%%%%%%%
|
||||
@ -120,7 +120,7 @@ The preparation for the collisional trajectories for the collision of protonated
|
||||
The argon atom is initially positioned at x=10, y=$b$ and z=0~\AA{}, with $b$ being the impact parameter. At each center of mass collision energy $E_{col}$, a series of 300 collision trajectories were conducted (the center of mass of the aggregate was kept at position (0, 0, 0)) for each of the 13 $b$ values which are evenly distributed (interval being 0.5 \AA) between 0 and R+0.5 \AA. R refers to the radius of Py$_2^+$.
|
||||
600 collision trajectories were performed per isomer of protonated uracil water clusters.
|
||||
Trajectory calculations have been performed with a time step of 0.5~fs and a total duration of 15 ~ps and 3~ps for the collision of argon with protonated uracil water clusters and Py$_2^+$, respectively.
|
||||
For the collision of Py$_2^+$ with argon, we have checked that for high collision energies such as 20 and 25 eV, a time step of 0.1 fs does not change significantly our numerical results, which will be shown in section \ref{sec:MDanalysis}.
|
||||
For the collision of Py$_2^+$ with argon, I have checked that for high collision energies such as 20 and 25 eV, a time step of 0.1 fs does not change significantly our numerical results, which will be shown in section \ref{sec:MDanalysis}.
|
||||
|
||||
It should be noted that the quaternion was used for rotation process in the generation of initial inputs.
|
||||
%In some cases it may be advisable to use the rigid-body approximation.
|
||||
@ -154,14 +154,14 @@ Mean values are computed using the same approach, followed by a division by $\pi
|
||||
|
||||
\subsection{Introduction}
|
||||
|
||||
Motivated by the recent CID experiments conducted by Braud \textit{et al.} consisting in (H$_2$O)$_{1-15}$UH$^+$ clusters colliding with an impacting atom or molecule M (M = H$_2$O, D$_2$O, neon, and argon) at a constant center of mass collision energy of 7.2~eV,\cite{Braud2019} the dynamical simulations of the collision between the protonated uracil water clusters (H$_2$O)$_{1-7, 11, 12}$UH$^+$ and an argon atom were performed.
|
||||
Motivated by the recent CID experiments conducted by I. Braud \textit{et al.} consisting in (H$_2$O)$_{1-15}$UH$^+$ clusters colliding with an impacting atom or molecule M (M = H$_2$O, D$_2$O, neon, and argon) at a constant center of mass collision energy of 7.2~eV,\cite{Braud2019} the dynamical simulations of the collision between the protonated uracil water clusters (H$_2$O)$_{1-7, 11, 12}$UH$^+$ and an argon atom were performed.
|
||||
%First, it appears important to understand the interaction of DNA or RNA basis with water seeing
|
||||
%their relevance in living organisms. They can also be subject to radiation damages which is still a medical challenge and thus
|
||||
%needs to be further investigated. In that context, a number of studies have been conducted on molecules deriving from uracil
|
||||
%or on uracil with only a few water molecules. \cite{Rasmussen2010,Imhoff2007,Abdoul2000,Champeaux2010,Delaunay2014,
|
||||
%Bacchus2009,Kossoski2015} \cite{Maclot2011, Domaracka2012, Markush2016, Castrovilli2017}. Second, a very recent
|
||||
The low collision energy (7.2~eV) only leads to intermolecular bond breaking, without any electronic excitation, rather than intramolecular bond breaking. The branching ratios for different charged fragments were determined in experiment, which allows to deduce the fragmentation cross section for all
|
||||
(H$_2$O)$_{1-15}$UH$^+$ species and the location of the excess proton after collision: on a uracil containing cluster or on a pure water cluster. This allows to determine the proportion of neutral uracil loss (corresponding to cases where the excess proton is located on pure water clusters) as a function of the number $n$ of water molecules. A sharp increase of neutral uracil loss was observed for $n$ = 5-6 (2.8\% and 25.0\% for n = 4 and 7, respectively).
|
||||
(H$_2$O)$_{1-15}$UH$^+$ species and the location of the excess proton after collision: on a uracil containing cluster or on a pure water cluster. This allows to determine the proportion of neutral uracil loss (corresponding to cases where the excess proton is located on pure water clusters) as a function of the number $n$ of water molecules. A sharp increase of neutral uracil loss was observed for $n$ = 5-6 (2.8\% and 25.0\% for $n$ = 4 and 7, respectively).
|
||||
%
|
||||
Those experiment were complemented by theoretical calculations that aim at characterizing the lowest-energy isomers of (H$_2$O)$_{n}$UH$^+$ ($n$ = 1-7, 11, 12) clusters (see section \ref{structureUH}), which
|
||||
%They show that (i) For $n$ = 1-2, the uracil is protonated; (ii) For $n$ = 3-4, the excess proton is still on the uracil but has a tendency to be displaced towards adjacent water molecules; (iii) When $n$ is larger than 4, the excess proton is transferred to the water molecules.
|
||||
@ -302,7 +302,7 @@ From Figure~\ref{proporEachFrag-7a12a-zoom}, it is clear that for both aggregate
|
||||
(H$_2$O)$_{10}$UH$^+$ and (H$_2$O)$_{11}$UH$^+$, and the increase of proportion of (H$_2$O)$_{6}$UH$^+$,
|
||||
(H$_2$O)$_{7}$UH$^+$ and (H$_2$O)$_{8}$UH$^+$ indicate sequential dissociation after collision is occurring. It is worth noting that, in contrast to 7a, the proportions of the main fragments of 12a do not tend to be a constant at the end of the simulations.
|
||||
This implies that, for this large aggregate, structural rearrangements are more likely to occur prior to complete dissociation. Proportions of the main fragments of clusters 7d and 12c shown in Figure \ref{proporEachFrag-7d12c-zoom} display similar behavior as for 7a and 12a.
|
||||
As a first conclusion, Figure~\ref{proporEachFrag-7a12a-zoom} suggests that clusters with 7 water molecules experience a direct dissociation mechanism as was hypothesised by Braud \textit{et al.}.\cite{Braud2019} A similar conclusion can be drawn for smaller cluster sizes as supported by Figures \ref{proporEachFrag-1a2a}-\ref{proporEachFrag-5a6a-zoom}. In contrast, cluster with 11 (see Figure \ref{proporEachFrag-11a-zoom}) and 12 water molecules shows a behavior compatible with a certain amount of statistical dissociation, namely a long-time evolution that allows structural rearrangements. These important observations can now be refined by looking at more detailed properties.
|
||||
As a first conclusion, Figure~\ref{proporEachFrag-7a12a-zoom} suggests that clusters with 7 water molecules experience a direct dissociation mechanism as was hypothesised by I. Braud \textit{et al.}.\cite{Braud2019} A similar conclusion can be drawn for smaller cluster sizes as supported by Figures \ref{proporEachFrag-1a2a}-\ref{proporEachFrag-5a6a-zoom}. In contrast, cluster with 11 (see Figure \ref{proporEachFrag-11a-zoom}) and 12 water molecules shows a behavior compatible with a certain amount of statistical dissociation, namely a long-time evolution that allows structural rearrangements. These important observations can now be refined by looking at more detailed properties.
|
||||
|
||||
\figuremacrob{proporEachFrag-1a2a}{Time-dependent proportions of the main fragments obtained from the dissociation of the lowest-energy isomers of (H$_2$O)$_1$UH$^+$ (left) and (H$_2$O)$_{2}$UH$^+$ (right).} \\
|
||||
|
||||
@ -321,7 +321,7 @@ As a first conclusion, Figure~\ref{proporEachFrag-7a12a-zoom} suggests that clus
|
||||
|
||||
In order to get more insights in the fragmentation, molecular dynamics trajectories were analysed in terms of proportion of neutral uracil loss ($P_{NUL}$) defined in section \ref{convergence} and total fragmentation
|
||||
cross sections ($\sigma_{frag}$) defined in section~\ref{trajecanylysis}. These two properties are also accessible from experiments. Another property extracted from the MD simulations, but not accessible from experiment,
|
||||
is the proportion of protonated uracil ($P_{PU}$) which is equal to the ratio of the number of simulations leading to a protonated uracil molecule over the number of simulations leading to a fragment containing the uracil and the excess proton. In order to correlate the outcome of the collision and the structure of the aggregate undergoing the collision, all considered low-energy isomers are characterized by there relative energies ($E_{rel.}$) and the location of the excess proton (LEP). For the latter, three distinct configurations were considered: The excess proton is bounded to the uracil molecule (noted U-H); The excess proton is bounded to a water molecule that is adjacent to an oxygen atom of the uracil molecule (noted W-H-U); The excess proton is bounded to a water molecule that is separated by at least one other water molecule from the uracil molecule (noted W-H). All these data are gathered in Table~\ref{tab:full} and we first discuss the behavior of the small species (H$_2$O)$_{1-7}$UH$^+$.
|
||||
is the proportion of protonated uracil ($P_{PU}$) which is equal to the ratio of the number of simulations leading to a protonated uracil molecule over the number of simulations leading to a fragment containing the uracil and the excess proton. In order to correlate the outcome of the collision and the structure of the aggregate undergoing the collision, all considered low-energy isomers are characterized by there relative energies ($E_{rel.}$) and the location of the excess proton (LEP). For the latter, three distinct configurations were considered: The excess proton is bounded to the uracil molecule (noted U-H); The excess proton is bounded to a water molecule that is adjacent to an oxygen atom of the uracil molecule (noted W-H-U); The excess proton is bounded to a water molecule that is separated by at least one other water molecule from the uracil molecule (noted W-H). All these data are gathered in Table~\ref{tab:full} and will first discuss the behavior of the small species (H$_2$O)$_{1-7}$UH$^+$.
|
||||
|
||||
\begin{table*}
|
||||
\begin{center}
|
||||
@ -375,7 +375,7 @@ $\sigma_{{frag}_{exp}}$ are the experimental values for $P_{NUL}$ and $\sigma_{f
|
||||
\end{table*}
|
||||
|
||||
Various information can be inferred from these properties. Firstly, one observes a general increase of $\sigma_{frag}$ as a function of cluster size with values ranging from 25.9~\AA$^2$~for isomer 1b to 60.2~\AA$^2$~for isomer 12a. Interestingly, only slight variations of $\sigma_{frag}$ are observed for different isomers of the same aggregate. In contrast, $P_{NUL}$ is much more sensitive to the nature of the considered isomers, in particular when these isomers display different LEP values. For instance, $P_{NUL}$ is 46.6 ~\% for 5a (W-H) while it is only 0.1~\% for 5d (U-H). More interestingly, there seems to exist a strong correlation between $P_{NUL}$ and LEP.
|
||||
Indeed, $P_{NUL}$ values below 1.0~\% are characterized by an excess proton initially bounded to uracil (U-H type). This suggests that when uracil is protonated, probability for deprotonation after collision is very low and thus $P_{NUL}$ is close to 0\%. $P_{NUL}$ values between 9.7 and 29.4~\% are obtained from W-H-U configurations while larger $P_{NUL}$ values, above 31.1~\%, arise from W-H configurations in clusters (H$_2$O)$_{5-7}$UH$^+$. This demonstrates that, from the excess proton point of view, the outcome of the collision is highly sensitive to the nature of the isomer undergoing the collision as hypothesised by Braud \textit{ al.} \cite{Braud2019} This important finding can be of help to determine which isomer, or set of isomers, is likely to undergo collision by comparing experimental and theoretical $P_{NUL}$ as this is not necessarily the lowest-energy isomer as discussed below.
|
||||
Indeed, $P_{NUL}$ values below 1.0~\% are characterized by an excess proton initially bounded to uracil (U-H type). This suggests that when uracil is protonated, probability for deprotonation after collision is very low and thus $P_{NUL}$ is close to 0\%. $P_{NUL}$ values between 9.7 and 29.4~\% are obtained from W-H-U configurations while larger $P_{NUL}$ values, above 31.1~\%, arise from W-H configurations in clusters (H$_2$O)$_{5-7}$UH$^+$. This demonstrates that, from the excess proton point of view, the outcome of the collision is highly sensitive to the nature of the isomer undergoing the collision as hypothesised by I. Braud \textit{ et al.} \cite{Braud2019} This important finding can be of help to determine which isomer, or set of isomers, is likely to undergo collision by comparing experimental and theoretical $P_{NUL}$ as this is not necessarily the lowest-energy isomer as discussed below.
|
||||
|
||||
For (H$_2$O)$_{1-2}$UH$^+$, the theoretical and experimental $P_{NUL}$ values, close to zero, are in good agreement regardless of the considered isomer. For (H$_2$O)$_3$UH$^+$, the experimental $P_{NUL}$ is 1.7~\% which is well reproduced by both isomers 3a
|
||||
and 3b although 3b is the one closer to the experimental value, 0.0~\% against 5.7~\% for 3a. This was expected as they are very close in energy, only 0.3 kcal.mol$^{-1}$ difference, and in structure, as displayed in Figure~\ref{fig-1a-3b}, both being of U-H type structure. Consequently, in the experiment, each one of them could be at the origin of the experimental signal. (H$_2$O)$_4$UH$^+$ behaves differently. The two low-energy isomers, 4a and 4b, display very different $P_{NUL}$ values, 29.4 and 2.6~\%, respectively. The experimental value is 2.8~\% which suggests that 4b, although slightly higher in energy by 0.9 kcal.mol$^{-1}$, is the isomer prevailing during the collision process. The difference in behavior can be explained by the U-H configuration of 4b, in which the excess proton is bounded to the uracil, whereas in 4a, it is bounded to a water molecule adjacent to uracil (see Figure~\ref{fig-4a-5d}). The case of (H$_2$O)$_5$UH$^+$ is more complex as this is the first species displaying the three types of LEP configuration among its four lowest-energy isomers as can be seen on Figure~\ref{fig-4a-5d}. This implies very different $P_{NUL}$ values: 46.6~\% for 5a, 28.5 and 27.1~\% for 5b and 5c, respectively, while it is only 0.1~\% for 5d. The experimental $P_{NUL}$ value for (H$_2$O)$_5$UH$^+$ is still relatively low, 7.5~\%, which suggests that a U-H type structure prevails during the collision process. Although 5d is 2.4~kcal.mol$^{-1}$
|
||||
@ -406,8 +406,8 @@ small aggregates such as (H$_2$O)$_{5-7}$UH$^+$ ((H$_2$O)$_{1-4}$UH$^+$ do not d
|
||||
\figuremacro{fig-7a-7d}{Selected low-energy configurations of (H$_2$O)$_{7}$UH$^+$. Relative energies at the MP2/Def2TZVP level are in kcal.mol$^{-1}$.}
|
||||
|
||||
The clusters discussed above are characterized by complex potential energy surfaces characterized by several low-energy isomers, with relative energies that can be lower than 1~kcal.mol$^{-1}$, and which get more complex as the number of water molecules
|
||||
increases. Consequently, the exact energetic ordering between the low-energy isomers can not be precisely known as this is below chemical accuracy and we thus can not claim here to have found the lowest-energy structure of each aggregate, or the isomer undergoing the collision. Nevertheless, what we show is that $P_{NUL}$ is mainly determined by the initial position
|
||||
of the proton in the isomer undergoing the collision. Consequently, for the collision energy and the range of cluster size we have considered, the structure of the aggregate undergoing the collision plays a key role in determining the dissociation process and collision
|
||||
increases. Consequently, the exact energetic ordering between the low-energy isomers can not be precisely known as this is below chemical accuracy and thus can not claim here to have found the lowest-energy structure of each aggregate, or the isomer undergoing the collision. Nevertheless, what I show is that $P_{NUL}$ is mainly determined by the initial position
|
||||
of the proton in the isomer undergoing the collision. Consequently, for the collision energy and the range of cluster size I have considered, the structure of the aggregate undergoing the collision plays a key role in determining the dissociation process and collision
|
||||
outcomes much more than energetics. This is consistent with the analysis of the time-dependent proportion of fragments which suggests a direct dissociation mechanism.
|
||||
This is further highlighted on Figure~\ref{neutralUloss-Ne-Ar}, which presents the experimental $P_{NUL}$ for collision with Ar and Ne, respectively as a function of $n$ and the corresponding theoretical values obtained from the lowest-energy isomers as well as from the isomers for which $P_{NUL}$ matches best to the experimental data. As can be seen, a very good agreement can be obtained with the experimental data by considering a specific set of isomers. Interestingly, if a similar plot is drawn for $\sigma_{frag}$ considering the same isomers
|
||||
(see Figure~\ref{cross-section-Ne-Ar}), a good agreement with the experimental data and much better than $\sigma_{geo}$ (calculated from formula \ref{cross-section-geo})is also obtained with the two sets of isomers which confirms the weaker dependence upon isomer of $\sigma_{frag}$.
|
||||
@ -420,7 +420,7 @@ Theory 1 (green line) is obtained from the isomers which $P_{NUL}$ matches best
|
||||
|
||||
\subsection{Behaviour at Larger Sizes, the Cases of (H$_2$O)$_{11, 12}$UH$^+$} \label{large}
|
||||
|
||||
In the experiments conducted by Braud \textit{et al.},\cite{Braud2019} $P_{NUL}$ starts to decrease at $n$=8. This decrease is not consistent with the above argument of a direct dissociation mechanism and larger species more likely characterized by W-H and W-H-U type structures. This apparent discrepancy motivated us to extend the present study to a larger cluster, namely (H$_2$O)$_{11, 12}$UH$^+$. For (H$_2$O)$_{12}$UH$^+$, the only available experimental data is for collisions with Ne instead of Ar, although for the same center of
|
||||
In the experiments conducted by I. Braud \textit{et al.},\cite{Braud2019} $P_{NUL}$ starts to decrease at $n$=8. This decrease is not consistent with the above argument of a direct dissociation mechanism and larger species more likely characterized by W-H and W-H-U type structures. This apparent discrepancy motivated us to extend the present study to a larger cluster, namely (H$_2$O)$_{11, 12}$UH$^+$. For (H$_2$O)$_{12}$UH$^+$, the only available experimental data is for collisions with Ne instead of Ar, although for the same center of
|
||||
mass collision energy.
|
||||
As shown in Figure~\ref{neutralUloss-Ne-Ar}, experimental $P_{NUL}$ values for Ne or Ar, although not equal, display similar trend. In the following,
|
||||
I thus discuss the experimental data of (H$_2$O)$_{12}$UH$^+$ colliding with Ne. For cluster (H$_2$O)$_{1-7, 11}$UH$^+$, keep discussing the experimental data from colliding with Ar.
|
||||
@ -436,7 +436,7 @@ assuming that the dissociation mechanism in (H$_2$O)$_{12}$UH$^+$ involves some
|
||||
of structural rearrangement that softens the impact of the isomer undergoing the collision. Indeed,
|
||||
as (H$_2$O)$_{12}$UH$^+$ has more degrees of freedom, it can more easily accommodate the kinetic energy transferred by the colliding atom prior to dissociation which thus takes place on a longer time scale. This excess of internal energy thus fosters structural rearrangements, in particular proton transfers toward the uracil, explaining the smaller $P_{NUL}$ value for
|
||||
(H$_2$O)$_{12}$UH$^+$. This is in full agreement with the conclusions obtained in section~\ref{time}
|
||||
from Figures~\ref{proporEachFrag-11a-zoom},\ref{proporEachFrag-7a12a-zoom} and \ref{proporEachFrag-7d12c-zoom}. To further support this conclusion, we conducted 200 MD simulations in the micro-canonical ensemble in which the whole kinetic energy of Ar was randomly distributed in all the vibrational modes of isomer 12c by drawing initial velocities in a 1185~K Boltzmann
|
||||
from Figures~\ref{proporEachFrag-11a-zoom},\ref{proporEachFrag-7a12a-zoom} and \ref{proporEachFrag-7d12c-zoom}. To further support this conclusion, I conducted 200 MD simulations in the micro-canonical ensemble in which the whole kinetic energy of Ar was randomly distributed in all the vibrational modes of isomer 12c by drawing initial velocities in a 1185~K Boltzmann
|
||||
distribution. Among them, 166 simulations display dissociation with one or two water molecules dissociating from the main cluster. No neutral uracil loss is observed. To conclude, although the present simulations are too short to assert that (H$_2$O)$_{12}$UH$^+$ undergoes a statistical dissociation mechanism, they clearly show that a direct mechanism is not sufficient to account for the theoretical and experimental results.
|
||||
Consequently, structural rearrangements are very likely to occur prior to dissociation and the experimental
|
||||
results for $P_{NUL}$ and $\sigma_{frag}$ values can not result from a single (H$_2$O)$_{12}$UH$^+$ isomer.
|
||||
@ -530,7 +530,7 @@ level can provide a wealth of information about collision-induced mechanism in m
|
||||
|
||||
PAH clusters have been investigated in several scientific fields.
|
||||
In combustion science, the role of PAH clusters in combustion processes is still under debate, in particular they might or not be the intermediate systems in the growth of soot particles. \cite{Chung2011, Saggese2015, Eaves2015, Mao2017, Wang2018}
|
||||
In atmospheric and environmental science, PAHs are known as the pollutants, which is harmful to human health. For instance, the carcinogenic PAHs associated with particulate matter in air pollution has showed clear evidence of genotoxic effects, such as DNA adducts, chromosome aberrations. \cite{Kyrtopoulos2001, Farmer2003}
|
||||
In atmospheric and environmental science, PAHs are known as the pollutants, which is harmful to human health. For instance, the carcinogenic PAHs associated with particulate matter in air pollution has showed clear evidence of genotoxic effects, such as DNA adduct, chromosome aberrations. \cite{Kyrtopoulos2001, Farmer2003}
|
||||
In new energy resources field, for the understanding of the properties of organic crystal or the design of new organic solar cell devices, PAH stacks are investigated as the prototypes.\cite{Aumaitre2019}
|
||||
In astrophysics, PAHs species are believed to be ubiquitous and abundant in the interstellar medium because of their compact and stable structure. \cite{Tielens2008} The PAH clusters are important contributors to the diffuse interstellar bands and UV-visible absorption bands. PAH clusters have been proposed to be the origin of a series of infrared emission bands, which are ubiquitous in the Universe. \cite{Leger1984,Allamandola1985} The broadening of these bands in regions protected from the star's UV flux suggests the following scenario: PAHs are trapped in clusters in UV-protected regions and photo-evaporated by star's UV photons in the so-called photodissociated region. \cite{Rapacioli2005, Berne2008} For all these topics, it is necessary to make a better understanding of the fundamental properties of PAH clusters. The crucial quantities are the stability, molecular growth processes, dissociation energies and their evolution with PAH charge, species, cluster size.
|
||||
|
||||
@ -544,14 +544,14 @@ The range of collision energies considered experimentally is quite large, rangin
|
||||
The quantitative data from experiments of PAH clusters are still rather limited, which motivates the modeling studies of them. In the calculation of PAH clusters, the size of the systems limits the use of {\it ab initio} wave function methods to the investigation of properties of the smallest clusters, namely dimers \cite{Piacenza2005, Birer2015}, whereas larger clusters can be addressed either at the DFT level or with more semi-empirical schemes \cite{Zhao2008truhlar, Rapacioli2009corr, Mao2017, Bowal2019}. Many of these studies, focused on structural properties, evidence a stacking growth process in agreement with experimental results. In addition, IR properties were also reported at the DFT level \cite{Ricca2013}. Most of the theoretical studies involve neutral clusters, mostly due to the fact that treating charge resonance process in ions is a challenging task for DFT based methods \cite{Grafenstein2009}. The singly charged PAH clusters are more stable than their neutral counterparts due to charge resonance stabilization.\cite{Rapacioli2009} Cationic PAH clusters are expected to be abundant in the photo-dissociation regions because the ionization energy of the PAH cluster is lower than that of the isolated PAH, which leads to the efficient formation of cationic PAH clusters. In addition, the ionized PAH clusters are easier to control, so it is more important to study them. It should be mentioned the recent studies computing ionisation potentials \cite{Joblin2017} as well as structural \cite{Dontot2019} and spectral (electronic \cite{Dontot2016} and vibrational \cite{Dontot2020}) properties of cations, performed with an original model combining DFTB \cite{Porezag1995,
|
||||
Seifert1996, Elstner1998, Spiegelman2020} with a configuration interaction scheme\cite{Rapacioli2011}.
|
||||
|
||||
With respect to these studies, very few is known about the dynamical aspects of PAH clusters carrying internal energy. High energy collisions of PAH clusters with energetic ions have been simulated by Gatchel {\textit et al.} \cite{Gatchell2016, Gatchell2016knockout} at the semi-empirical and DFTB levels.
|
||||
Recently experiments at lower collision energies were performed by Zamith \textit{et al.} from LCAR \cite{Zamith2020threshold} (the principle of this experiment and the experimental setup were shown in sections \ref{principleTCID} and \ref{EXPsetup}), which were analysed by treating statistically the dissociation after collision energy deposition. Namely, the dissociation rate of pyrene clusters has been computed using phase space theory (PST)\cite{Zamith2019thermal}. A fair agreement with experimental results was obtained concerning the collision energy dependence of the dissociation cross section. However, the employed model failed at reproducing in details the shape of the peaks in the time-of-flight (TOF) spectra. In this section, it is aimed at extending the description of such low energy collision processes (less than several tens of eV) combining a dynamical simulations to describe the fast processes in addition to the statistical theory to address dissociation at longer timescales. With this approach, (i) good agreement between simulated and experimental mass spectra will be shown, thus validating the model, (ii) dissociation cross sections as a function of the collision energy is derived, (iii) the kinetic energy partition between dissociative and non-dissociative modes will be discussed and (iv) the energy transfer efficiency between intra and intermolecular modes will also be discussed.
|
||||
With respect to these studies, very few is known about the dynamical aspects of PAH clusters carrying internal energy. High energy collisions of PAH clusters with energetic ions have been simulated by M. Gatchel {\textit et al.} \cite{Gatchell2016, Gatchell2016knockout} at the semi-empirical and DFTB levels.
|
||||
Recently experiments at lower collision energies were performed by S. Zamith \textit{et al.} \cite{Zamith2020threshold} (the principle of this experiment and the experimental setup were shown in sections \ref{principleTCID} and \ref{EXPsetup}), which were analysed by treating statistically the dissociation after collision energy deposition. Namely, the dissociation rate of pyrene clusters has been computed using phase space theory (PST)\cite{Zamith2019thermal}. A fair agreement with experimental results was obtained concerning the collision energy dependence of the dissociation cross section. However, the employed model failed at reproducing in details the shape of the peaks in the time-of-flight (TOF) spectra. In this section, it is aimed at extending the description of such low energy collision processes (less than several tens of eV) combining a dynamical simulations to describe the fast processes in addition to the statistical theory to address dissociation at longer timescales. With this approach, (i) good agreement between simulated and experimental mass spectra will be shown, thus validating the model, (ii) dissociation cross sections as a function of the collision energy is derived, (iii) the kinetic energy partition between dissociative and non-dissociative modes will be discussed and (iv) the energy transfer efficiency between intra and intermolecular modes will also be discussed.
|
||||
|
||||
% \ref{sec:compapp},
|
||||
|
||||
\subsection{Calculation of Energies}
|
||||
|
||||
In the analysis, we will discuss the kinetic energy contributors, applying the following decomposition of the total kinetic energy $E^k_{tot}$ of the dimer:
|
||||
In the analysis, I will discuss the kinetic energy contributors, applying the following decomposition of the total kinetic energy $E^k_{tot}$ of the dimer:
|
||||
\begin{align}
|
||||
\label{Eparti}
|
||||
E^k_{tot}&=E^k_{Ar} + E^k_{td} + E^k_{Py^1} + E^k_{Py^2} + E^k_{Re} \nonumber \\
|
||||
@ -603,12 +603,12 @@ $\Vec{\omega}$ is the angular velocity.
|
||||
$\Vec{r}_{i}^n$ and $\Vec{r}_{_{CM}}(Py^n)$ denote the coordinates of atom $i$ and center of mass of dimer of monomer $n$, respectively.
|
||||
$\Vec{r}_{i}$ and $\Vec{r}_{_{CM}}(Py_2)$ and denote the coordinates of atom $i$ and center of mass of dimer, respectively.
|
||||
|
||||
From the endpoint of the simulation, we can also compute the total energy transferred towards internal rovibrational modes of the pyrene dimer as:
|
||||
From the endpoint of the simulation, the total energy transferred towards internal rovibrational modes of the pyrene dimer cal also be computed as:
|
||||
\begin{equation}
|
||||
\Delta E_{int}^{Py_2} = E^{k,0}_{Ar} - E^k_{Ar} - E^k_{td}
|
||||
\end{equation}
|
||||
where $E^{k,0}_{Ar}$ is the initial argon kinetic energy whereas $E^k_{Ar}$ and $E^k_{td}$ correspond to kinetic energies at the end of the MD simulation.
|
||||
In the case of dissociated dimers at the end of the simulations, we can also deduce the energy deposited in the rovibrational modes of the monomers as:
|
||||
In the case of dissociated dimers at the end of the simulations, the energy deposited in the rovibrational modes of the monomers can be deduced as:
|
||||
\begin{equation}
|
||||
\Delta E_{int}^{Py^1+Py^2} = E^{k,0}_{Ar} - E^k_{Ar} - E^k_{td} - E^k_{Re}
|
||||
\end{equation}
|
||||
@ -618,21 +618,21 @@ In the case of dissociated dimers at the end of the simulations, we can also ded
|
||||
|
||||
The experimental TOFMS are reproduced by simulating the ion trajectories through the experimental setup in the presence of the electric fields. These are calculated by solving numerically the Laplace equation. Equations of motion are integrated using the fourth order Runge-Kutta method with adaptive step size. The occurrence of collision or dissociation is decided at each time step of the ion trajectory based on the collision and dissociation probabilities.
|
||||
|
||||
In the work of Zamith {\textit et al.}, \cite{Zamith2020threshold} the energy transfer was treated upon collision by using the Line of Center model (LOC) \cite{Levine1987}. In the LOC model, the transferred energy is the kinetic energy along the line of centers. Evaporation rates were then estimated using PST, in which only statistical dissociation to be possible after energy deposition in the cluster by collision was conside. Although this approach, which will be referred to as PST in the following, has been proved to be able to satisfactorily reproduce CID cross section experiments,\cite{Zamith2020threshold} it fails to reproduce in details the shape and position of the fragment peaks in the TOFMS, as will be shown in section \ref{sec:MS}.
|
||||
In the work of S. Zamith {\textit et al.}, \cite{Zamith2020threshold} the energy transfer was treated upon collision by using the Line of Center model (LOC) \cite{Levine1987}. In the LOC model, the transferred energy is the kinetic energy along the line of centers. Evaporation rates were then estimated using PST, in which only statistical dissociation to be possible after energy deposition in the cluster by collision was conside. Although this approach, which will be referred to as PST in the following, has been proved to be able to satisfactorily reproduce CID cross section experiments,\cite{Zamith2020threshold} it fails to reproduce in details the shape and position of the fragment peaks in the TOFMS, as will be shown in section \ref{sec:MS}.
|
||||
|
||||
In order to better reproduce the position and peak shapes, the MD and PST methods were combined. The outputs of the MD simulations were used to treat the collisions in the ion trajectories. At each time step the probability for a collision is evaluated. The principle of MD+PST is displayed in Figure \ref{MDPST}.
|
||||
|
||||
\figuremacrob{MDPST}{Principle of MD+PST.}
|
||||
|
||||
One MD trajectory (with proper weighting of the $b$ values) was randomly picked from all outputs of MD simulations at a given collision energy. Then two cases have to be considered. First, if the dissociation occurred during the picked MD calculation (short time dissociation), then we use the MD final velocities of the fragments to further calculate the ion trajectories. On the other hand, if the pyrene dimer is still intact at the end of the picked MD calculation, then we update the dimer velocity and use the collision energy transfer $\Delta E_{int}^{Py_2}$ deduced from the MD calculation to increase the internal energy of the cluster. The dissociation rate resulting from this new internal energy is then evaluated using PST.
|
||||
One MD trajectory (with proper weighting of the $b$ values) was randomly picked from all outputs of MD simulations at a given collision energy. Then two cases have to be considered. First, if the dissociation occurred during the picked MD calculation (short time dissociation), then the MD final velocities of the fragments are used to further calculate the ion trajectories. On the other hand, if the pyrene dimer is still intact at the end of the picked MD calculation, then the dimer velocity is updated and use the collision energy transfer $\Delta E_{int}^{Py_2}$ deduced from the MD calculation to increase the internal energy of the cluster. The dissociation rate resulting from this new internal energy is then evaluated using PST.
|
||||
In the latter case, if dissociation occurs (long time dissociation), the relative velocities of the fragment are evaluated using the PST outcome. The whole process of MD+PST is performed many times for different trajectories to ensure the reliability of the final obtained data. For each time, the TOFMS of Py$^+$ or Py$_2^+$ is updated.
|
||||
%Then we can obtain the update TOFMS histogram of the monomer cation Py$^+$ from the short and long time dissociation.
|
||||
|
||||
Here we emphasize that, due to the short time scale of the MD calculations (3 ps), only direct dissociation can be captured by the MD simulations. Therefore, one has to evaluate the probability of dissociation at longer time scales after the energy deposition by collision. This is done here by considering that at longer time scales, dissociation occurs statistically and is treated by using PST.
|
||||
Here I emphasize that, due to the short time scale of the MD calculations (3 ps), only direct dissociation can be captured by the MD simulations. Therefore, one has to evaluate the probability of dissociation at longer time scales after the energy deposition by collision. This is done here by considering that at longer time scales, dissociation occurs statistically and is treated by using PST.
|
||||
|
||||
\subsection{\label{sec:results}Results and Discussion}
|
||||
|
||||
In the following, we will discuss the dissociation at short, experimental and infinite timescales. The first two ones correspond to dissociation occurring during the MD simulation only or with the MD+PST model. The dissociation at infinite time accounts for all MD trajectories where the amount of energy transferred to the internal dimer rovibrational modes $\Delta E_{int}^{Py_2}$ is larger than the dissociation energy of 1.08~eV (value from references \cite{Dontot2019, Zamith2020threshold}). It can be regarded as the dissociation occurring after an infinite time neglecting any cooling processes, such as thermal collisions or photon emissions.
|
||||
In the following, I will discuss the dissociation at short, experimental and infinite timescales. The first two ones correspond to dissociation occurring during the MD simulation only or with the MD+PST model. The dissociation at infinite time accounts for all MD trajectories where the amount of energy transferred to the internal dimer rovibrational modes $\Delta E_{int}^{Py_2}$ is larger than the dissociation energy of 1.08~eV (value from references \cite{Dontot2019, Zamith2020threshold}). It can be regarded as the dissociation occurring after an infinite time neglecting any cooling processes, such as thermal collisions or photon emissions.
|
||||
|
||||
\subsubsection{\label{sec:MS}TOFMS Comparison}
|
||||
|
||||
@ -649,7 +649,7 @@ In Figure~\ref{expTOF}(b), the experimental result is compared to the PST and MD
|
||||
|
||||
\textbf{Description of selected trajectories}
|
||||
|
||||
A first qualitative description of the collision processes can be obtained from the analysis of some arbitrarily selected MD trajectories. Figure \ref{collisions} (top and bottom) reports some snapshots extracted from two trajectories with the same collision energy (17.5~eV) and impact parameter (3.5~\AA). Only the top one leads to the Py$_2^+$ dissociation. During the results collection, we extract the final snapshot for each trajectory and consider that the dimer is dissociated if the distance between the two monomers molecular mass centers is larger than 10~\AA.
|
||||
A first qualitative description of the collision processes can be obtained from the analysis of some arbitrarily selected MD trajectories. Figure \ref{collisions} (top and bottom) reports some snapshots extracted from two trajectories with the same collision energy (17.5~eV) and impact parameter (3.5~\AA). Only the top one leads to the Py$_2^+$ dissociation. During the results collection, the final snapshot for each trajectory is extracted and the dimer is dissociated is considered if the distance between the two monomers molecular mass centers is larger than 10~\AA.
|
||||
Figures \ref{collisions}-1/1* represent the system after its preliminary thermalization, when the argon atom introduced in the simulation with its initial velocity.
|
||||
Figures \ref{collisions}-2/2* and 3/3* represent the beginning and end of the collision. From these points, the two trajectories show different behaviors.
|
||||
For the top trajectory in Figure \ref{collisions}, snapshot 5 corresponds to the step where the two pyrene monomers start to go away from each other. After this, the intermolecular distance continues to increase further in snapshot 6. For the bottom trajectory in Figure \ref{collisions}, Figures \ref{collisions}-5* and 6* correspond to the middle and ending snapshots of the simulation, respectively. The snapshots 4*,~5* and 6* show the process of energy redistribution within the clusters. In particular, the soft modes associated to global deformation of the molecular planes appear to be excited.
|
||||
@ -669,10 +669,10 @@ All these effects are intrinsically taken into account in the MD simulations on
|
||||
|
||||
\figuremacro{distriPerc-Etf-175eV-d-bin03}{Distribution of transferred energy in rovibrational modes $\Delta E_{int}^{Py_2}$ for trajectories leading to dissociation at the end of MD (center of mass collision energy of 17.5~eV). The dashed line shows the distribution of transferred energy used in the LOC model.}
|
||||
|
||||
Finally, we note that the pyrene monomers remained intact (no fragmentation) up to collision energies of 25~eV.
|
||||
Finally, I note that the pyrene monomers remained intact (no fragmentation) up to collision energies of 25~eV.
|
||||
The snapshots of a fragmentation trajectory at collision energy of 27.5~eV are shown in Figure \ref{fragmentation}.
|
||||
It can be seen that the pyrene molecule impacted by the argon undergoes an opening of an aromatic cycle and the loss of two hydrogen atoms, leaving as a H$_2$ molecule.
|
||||
As the study of monomer's fragmentation is beyond the scope of the present paper, we will focus on trajectories with collision energies below this fragmentation threshold energy in the following.
|
||||
As the study of monomer's fragmentation is beyond the scope of the present paper, I will focus on trajectories with collision energies below this fragmentation threshold energy in the following.
|
||||
|
||||
\figuremacrob{fragmentation}{Snapshots for molecular dynamics trajectory with impact parameter of 0.5~\AA{} and a collision energy of 27.5 eV leading to intramolecular fragmentation.}
|
||||
|
||||
@ -688,7 +688,7 @@ The purple curve corresponds to dissociation at infinite timescales. Figure \ref
|
||||
It can be seen that, for low collision energies, the MD and MD+PST cross sections are very close, indicating that most of the dissociations occur on the short timescales. On the opposite, at high collision energies, a non-negligible fraction of the dimers, which are not dissociated at the end of the MD simulation, carry enough energy to evaporate on the experimental timescales.
|
||||
At the experimental center of mass collision energy of 17.5~eV, the MD+PST cross section (about 70~\AA$^2$) is slightly above the pure MD dissociation ratio, which indicates that the dissociation at long timescales represents a small fraction of the dissociated pyrene dimers as already seen from the TOF spectra analysis (see Figure \ref{expTOF}).
|
||||
|
||||
We have also plotted in Figure~\ref{cross-section} the model cross section $\sigma_\infty$ that successfully reproduced the threshold collision-induced dissociation experimental results \cite{Zamith2020threshold}. This model cross section is obtained by considering that the collision energy transfer is given by the LOC model and the expression for the cross section is given by:
|
||||
I have also plotted in Figure~\ref{cross-section} the model cross section $\sigma_\infty$ that successfully reproduced the threshold collision-induced dissociation experimental results \cite{Zamith2020threshold}. This model cross section is obtained by considering that the collision energy transfer is given by the LOC model and the expression for the cross section is given by:
|
||||
\begin{equation}
|
||||
\sigma_{LOC}(E_{col}) = \sigma_0 (E_{col} - D)/(E_{col}).
|
||||
\end{equation}
|
||||
@ -705,7 +705,7 @@ The dissociation cross sections for MD timescales with time step being 0.1 fs at
|
||||
The mean value obtained for the transferred energy after removing the translation kinetic energy of the dimer, namely $\Delta E_{int}^{Py_2}$, is plotted in Figure \ref{transferredE-Ar-300} as a function of the collision energy. Although this quantity evolves almost linearly with the collision energy, the curves are different when one considers only the trajectories leading to dissociation or non-dissociation.
|
||||
For trajectories where the dimer does not dissociate, $\Delta E_{int-ud}^{Py_2}$ remains small for all collision energies below 20~eV and shows a very slight increase for collision energies larger than 20~eV.
|
||||
For trajectories leading to dissociation, $E_{int-d}^{Py_2}$ grows almost linearly, but above 10-15~eV most of the absorbed energy is actually used to heat the individual monomers (the green curve) whereas the energy given in the dissociative mode (difference between the blue and green curves) remains almost constant.
|
||||
We note that, despite the trends of the mean energy values derived from all simulations or restricted to the undissociated cases are interesting, their absolute values have small meaning as they depend on the arbitrarily chosen $b_{max}$ value, {\it i.e.} increasing $b_{max}$ would result in more undissociated trajectories with less and less energy transfer. On the opposite, absolute values of mean energies for the dissociation trajectories are relevant, as increasing the $b_{max}$ value would not result in new dissociation trajectories.
|
||||
I note that, despite the trends of the mean energy values derived from all simulations or restricted to the undissociated cases are interesting, their absolute values have small meaning as they depend on the arbitrarily chosen $b_{max}$ value, {\it i.e.} increasing $b_{max}$ would result in more undissociated trajectories with less and less energy transfer. On the opposite, absolute values of mean energies for the dissociation trajectories are relevant, as increasing the $b_{max}$ value would not result in new dissociation trajectories.
|
||||
|
||||
For MD simulations with time step being 0.1 fs at collision energy of 20 and 25 ~eV, the corresponding energies in Figure~\ref{transferredE-Ar-300} are close to the ones of time step being 0.5 fs, which indicates a time step of 0.1 fs used in the MD simulation does not change significantly the corresponding deposition of the total transferred energy.
|
||||
|
||||
@ -822,7 +822,7 @@ This is actually in line with the fact that the part of the absorbed collision e
|
||||
|
||||
\figuremacrob{E-time-abcdef}{Instantaneous kinetic energies as a function of time for intra and intermolecular modes of the pyrene dimer at a collision energy of 22.5 eV. Impact parameters $b$ are (a) 2, (b) 3, (c) 0, (d) 2.5, (e) 2, and (f) 2 \AA{}. In cases (a) and (b), dissociation takes place whereas in the other cases the dimer remains undissociated at the end of the MD simulation.}
|
||||
|
||||
In this section, we address how the energy is shared inside the dimer after the collision. In particular, we look at the efficiency of energy transfer between the intramolecular modes of each unit and the intermolecular modes. The amount of deposited energy as well as its partition between the intramolecular modes of each molecule and the intermolecular modes is strongly dependent on the collision condition: the impact parameter, the orientation of the dimer, whether a head on collision occurs with one of the dimer atoms (and its nature, carbon or hydrogen). This results in very different evolutions of the subsequent energy flows for which precise values concerning timescales can hardly be derived. Nevertheless, the analysis of the trajectories allows to identify some characteristic behaviors. In order to estimate the thermalization process efficiency, the instantaneous intra and intermolecular kinetic temperatures are evaluated using the following formula:
|
||||
In this section, I address how the energy is shared inside the dimer after the collision. In particular, I look at the efficiency of energy transfer between the intramolecular modes of each unit and the intermolecular modes. The amount of deposited energy as well as its partition between the intramolecular modes of each molecule and the intermolecular modes is strongly dependent on the collision condition: the impact parameter, the orientation of the dimer, whether a head on collision occurs with one of the dimer atoms (and its nature, carbon or hydrogen). This results in very different evolutions of the subsequent energy flows for which precise values concerning timescales can hardly be derived. Nevertheless, the analysis of the trajectories allows to identify some characteristic behaviors. In order to estimate the thermalization process efficiency, the instantaneous intra and intermolecular kinetic temperatures are evaluated using the following formula:
|
||||
\begin{align}
|
||||
\label{kineticT}
|
||||
T^k=2 \frac{<E^k>}{nk_b}
|
||||
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||||
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: @book{Kaplan2006
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: @article{Karplus2002
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: @book{Allen2017
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: @article{Raghavachari1989
|
||||
: ,
|
||||
I'm skipping whatever remains of this entry
|
||||
Repeated entry---line 5155 of file 6_backmatter/references.bib
|
||||
Repeated entry---line 5160 of file 6_backmatter/references.bib
|
||||
: @article{Purvis1982
|
||||
: ,
|
||||
I'm skipping whatever remains of this entry
|
||||
Repeated entry---line 5185 of file 6_backmatter/references.bib
|
||||
Repeated entry---line 5190 of file 6_backmatter/references.bib
|
||||
: @article{Head1994
|
||||
: ,
|
||||
I'm skipping whatever remains of this entry
|
||||
Repeated entry---line 5221 of file 6_backmatter/references.bib
|
||||
Repeated entry---line 5226 of file 6_backmatter/references.bib
|
||||
: @article{Vcivzek1966
|
||||
: ,
|
||||
I'm skipping whatever remains of this entry
|
||||
Repeated entry---line 5351 of file 6_backmatter/references.bib
|
||||
Repeated entry---line 5356 of file 6_backmatter/references.bib
|
||||
: @book{Jensen2017
|
||||
: ,
|
||||
I'm skipping whatever remains of this entry
|
||||
Repeated entry---line 5431 of file 6_backmatter/references.bib
|
||||
Repeated entry---line 5435 of file 6_backmatter/references.bib
|
||||
: @article{Elstner1998
|
||||
: ,
|
||||
I'm skipping whatever remains of this entry
|
||||
Warning--can't use both volume and number fields in Castleman1978
|
||||
Warning--empty journal in Shields2010
|
||||
Warning--empty year in Shields2010
|
||||
Warning--empty journal in Finlayson1986
|
||||
Warning--empty journal in Fleming1986
|
||||
Warning--empty journal in Slater1968
|
||||
Warning--empty year in Magnasco2009
|
||||
Warning--empty year in Oliveira2009
|
||||
Warning--empty year in Unger1993
|
||||
Warning--empty title in GaussianCode
|
||||
Warning--empty journal in GaussianCode
|
||||
Warning--empty year in GaussianCode
|
||||
You've used 484 entries,
|
||||
You've used 483 entries,
|
||||
2776 wiz_defined-function locations,
|
||||
3607 strings with 98136 characters,
|
||||
and the built_in function-call counts, 223751 in all, are:
|
||||
= -- 23367
|
||||
> -- 9299
|
||||
3553 strings with 95663 characters,
|
||||
and the built_in function-call counts, 222660 in all, are:
|
||||
= -- 23248
|
||||
> -- 9241
|
||||
< -- 22
|
||||
+ -- 3712
|
||||
- -- 3172
|
||||
* -- 17596
|
||||
:= -- 34021
|
||||
add.period$ -- 1458
|
||||
call.type$ -- 484
|
||||
change.case$ -- 2539
|
||||
+ -- 3689
|
||||
- -- 3150
|
||||
* -- 17482
|
||||
:= -- 33857
|
||||
add.period$ -- 1454
|
||||
call.type$ -- 483
|
||||
change.case$ -- 2527
|
||||
chr.to.int$ -- 0
|
||||
cite$ -- 492
|
||||
duplicate$ -- 6870
|
||||
empty$ -- 18932
|
||||
format.name$ -- 3172
|
||||
if$ -- 51345
|
||||
cite$ -- 495
|
||||
duplicate$ -- 6853
|
||||
empty$ -- 18849
|
||||
format.name$ -- 3150
|
||||
if$ -- 51104
|
||||
int.to.chr$ -- 0
|
||||
int.to.str$ -- 484
|
||||
missing$ -- 506
|
||||
newline$ -- 2420
|
||||
num.names$ -- 968
|
||||
pop$ -- 3754
|
||||
int.to.str$ -- 483
|
||||
missing$ -- 505
|
||||
newline$ -- 2414
|
||||
num.names$ -- 966
|
||||
pop$ -- 3757
|
||||
preamble$ -- 1
|
||||
purify$ -- 2538
|
||||
purify$ -- 2526
|
||||
quote$ -- 0
|
||||
skip$ -- 11354
|
||||
skip$ -- 11313
|
||||
stack$ -- 0
|
||||
substring$ -- 13809
|
||||
swap$ -- 1980
|
||||
substring$ -- 13691
|
||||
swap$ -- 1972
|
||||
text.length$ -- 22
|
||||
text.prefix$ -- 0
|
||||
top$ -- 0
|
||||
type$ -- 1882
|
||||
warning$ -- 8
|
||||
while$ -- 1758
|
||||
width$ -- 487
|
||||
write$ -- 5299
|
||||
(There were 18 error messages)
|
||||
type$ -- 1878
|
||||
warning$ -- 12
|
||||
while$ -- 1750
|
||||
width$ -- 486
|
||||
write$ -- 5280
|
||||
(There were 22 error messages)
|
||||
|
||||
@ -202,7 +202,6 @@
|
||||
\backcite {dftb2}{{19}{2.3}{section.2.3}}
|
||||
\backcite {Elstner1998}{{19}{2.3}{section.2.3}}
|
||||
\backcite {Elstner2014}{{19}{2.3}{section.2.3}}
|
||||
\backcite {Cremer1993}{{19}{2.3}{section.2.3}}
|
||||
\backcite {Singh1986}{{19}{2.3}{section.2.3}}
|
||||
\backcite {Gao1996}{{19}{2.3}{section.2.3}}
|
||||
\backcite {Mordasini1998}{{19}{2.3}{section.2.3}}
|
||||
@ -215,7 +214,7 @@
|
||||
\backcite {Helgaker2014}{{20}{2.3.1}{equation.2.3.15}}
|
||||
\backcite {Roothaan1951}{{20}{2.3.1}{equation.2.3.15}}
|
||||
\backcite {Blinder2018}{{20}{2.3.1}{equation.2.3.15}}
|
||||
\backcite {koch1999}{{21}{2.3.1}{equation.2.3.15}}
|
||||
\backcite {koch1999}{{20}{2.3.1}{equation.2.3.15}}
|
||||
\backcite {Moskowitz1965}{{21}{2.3.1}{equation.2.3.15}}
|
||||
\backcite {Lowdin1995}{{21}{2.3.1}{equation.2.3.15}}
|
||||
\backcite {Jensen2017}{{21}{2.3.1}{equation.2.3.15}}
|
||||
@ -235,145 +234,145 @@
|
||||
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|
||||
\backcite {Magnasco2009}{{21}{2.3.1}{equation.2.3.15}}
|
||||
\backcite {Dacosta2011}{{21}{2.3.1}{equation.2.3.15}}
|
||||
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|
||||
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|
||||
\backcite {Hohenberg1964}{{22}{2.3.2}{subsection.2.3.2}}
|
||||
\backcite {Levy1979}{{22}{2.3.2}{subsection.2.3.2}}
|
||||
\backcite {Kohn1965}{{22}{2.3.2}{subsection.2.3.2}}
|
||||
\backcite {kohn1999nobel}{{22}{2.3.2}{subsection.2.3.2}}
|
||||
\backcite {Thomas1927}{{22}{2.3.2}{equation.2.3.16}}
|
||||
\backcite {Fermi1928}{{22}{2.3.2}{equation.2.3.16}}
|
||||
\backcite {Hohenberg1964}{{22}{2.3.2}{equation.2.3.16}}
|
||||
\backcite {Levy1979}{{22}{2.3.2}{equation.2.3.16}}
|
||||
\backcite {Kohn1965}{{22}{2.3.2}{equation.2.3.16}}
|
||||
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|
||||
\backcite {Kohn1965}{{26}{2.3.2}{equation.2.3.28}}
|
||||
\backcite {Kohn1965}{{26}{2.3.2}{equation.2.3.28}}
|
||||
\backcite {Herman1969}{{26}{2.3.2}{equation.2.3.28}}
|
||||
\backcite {Perdew1996}{{26}{2.3.2}{equation.2.3.28}}
|
||||
\backcite {Kohn1965}{{25}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Kohn1965}{{25}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Herman1969}{{25}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Perdew1996}{{25}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Becke1993}{{25}{2.3.2}{equation.2.3.27}}
|
||||
\backcite {Becke1993}{{26}{2.3.2}{equation.2.3.28}}
|
||||
\backcite {Becke1993}{{26}{2.3.2}{equation.2.3.29}}
|
||||
\backcite {Becke1988}{{26}{2.3.2}{equation.2.3.29}}
|
||||
\backcite {Lee1988}{{26}{2.3.2}{equation.2.3.29}}
|
||||
\backcite {Stephens1994}{{26}{2.3.2}{equation.2.3.29}}
|
||||
\backcite {Adamo1999}{{26}{2.3.2}{equation.2.3.29}}
|
||||
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|
||||
\backcite {Krukau2006}{{26}{2.3.2}{equation.2.3.29}}
|
||||
\backcite {Zhao2008}{{26}{2.3.2}{equation.2.3.29}}
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
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|
||||
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|
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|
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|
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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@ -69,6 +69,8 @@
|
||||
\nompageref{viii}
|
||||
\item [{QM}]\begingroup quantum mechanics \nomeqref {0}
|
||||
\nompageref{viii}
|
||||
\item [{RRKM}]\begingroup Rice-Ramsperger-Kassel-Marcus \nomeqref {0}
|
||||
\nompageref{viii}
|
||||
\item [{SCC-DFTB}]\begingroup self-consistent-charge density-functional based tight-binding \nomeqref {0}
|
||||
\nompageref{viii}
|
||||
\item [{SCF}]\begingroup self-consistent field \nomeqref {0}
|
||||
|
||||
Binary file not shown.
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@ -8,59 +8,59 @@
|
||||
\contentsline {subsection}{\numberline {2.3.1}Wavefunction based Methods}{19}{subsection.2.3.1}%
|
||||
\contentsline {subsection}{\numberline {2.3.2}Density Functional Theory}{21}{subsection.2.3.2}%
|
||||
\contentsline {subsection}{\numberline {2.3.3}Density Functional based Tight-Binding Theory}{26}{subsection.2.3.3}%
|
||||
\contentsline {subsection}{\numberline {2.3.4}Force Field Methods}{34}{subsection.2.3.4}%
|
||||
\contentsline {section}{\numberline {2.4}Exploration of PES}{36}{section.2.4}%
|
||||
\contentsline {subsection}{\numberline {2.4.1}Monte Carlo Simulations}{37}{subsection.2.4.1}%
|
||||
\contentsline {subsection}{\numberline {2.3.4}Force Field Methods}{33}{subsection.2.3.4}%
|
||||
\contentsline {section}{\numberline {2.4}Exploration of PES}{35}{section.2.4}%
|
||||
\contentsline {subsection}{\numberline {2.4.1}Monte Carlo Simulations}{36}{subsection.2.4.1}%
|
||||
\contentsline {subsection}{\numberline {2.4.2}Classical Molecular Dynamics}{40}{subsection.2.4.2}%
|
||||
\contentsline {subsection}{\numberline {2.4.3}Parallel-Tempering Molecular Dynamics}{45}{subsection.2.4.3}%
|
||||
\contentsline {subsection}{\numberline {2.4.4}Global Optimization}{47}{subsection.2.4.4}%
|
||||
\contentsline {chapter}{\numberline {3}Exploration of Structural and Energetic Properties}{51}{chapter.3}%
|
||||
\contentsline {section}{\numberline {3.1}Computational Details}{52}{section.3.1}%
|
||||
\contentsline {subsection}{\numberline {3.1.1}SCC-DFTB Potential}{52}{subsection.3.1.1}%
|
||||
\contentsline {subsection}{\numberline {3.1.2}SCC-DFTB Exploration of PES}{52}{subsection.3.1.2}%
|
||||
\contentsline {subsection}{\numberline {3.1.3}MP2 Geometry Optimizations, Relative and Binding Energies}{53}{subsection.3.1.3}%
|
||||
\contentsline {subsection}{\numberline {3.1.4}Structure Classification}{54}{subsection.3.1.4}%
|
||||
\contentsline {section}{\numberline {3.2}Structural and Energetic Properties of Ammonium/Ammonia including Water Clusters}{55}{section.3.2}%
|
||||
\contentsline {subsection}{\numberline {3.2.1}General introduction}{55}{subsection.3.2.1}%
|
||||
\contentsline {subsection}{\numberline {3.2.2}Results and Discussion}{57}{subsection.3.2.2}%
|
||||
\contentsline {subsubsection}{\numberline {3.2.2.1}Dissociation Curves and SCC-DFTB Potential}{57}{subsubsection.3.2.2.1}%
|
||||
\contentsline {subsubsection}{\numberline {3.2.2.2}Small Species: (H$_2$O)$_{1-3}${NH$_4$}$^+$ and (H$_2$O)$_{1-3}${NH$_3$}}{60}{subsubsection.3.2.2.2}%
|
||||
\contentsline {subsubsection}{\numberline {3.2.2.3}Properties of (H$_2$O)$_{4-10}${NH$_4$}$^+$ Clusters}{63}{subsubsection.3.2.2.3}%
|
||||
\contentsline {subsubsection}{\numberline {3.2.2.4}Properties of (H$_2$O)$_{4-10}${NH$_3$} Clusters}{70}{subsubsection.3.2.2.4}%
|
||||
\contentsline {subsubsection}{\numberline {3.2.2.5}Properties of (H$_2$O)$_{20}${NH$_4$}$^+$ Cluster}{75}{subsubsection.3.2.2.5}%
|
||||
\contentsline {subsection}{\numberline {3.2.3}Conclusions for Ammonium/Ammonia Including Water Clusters}{76}{subsection.3.2.3}%
|
||||
\contentsline {section}{\numberline {3.3}Structural and Energetic Properties of Protonated Uracil Water Clusters}{77}{section.3.3}%
|
||||
\contentsline {subsection}{\numberline {3.3.1}General introduction}{77}{subsection.3.3.1}%
|
||||
\contentsline {subsection}{\numberline {3.3.2}Results and Discussion}{79}{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}{85}{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}{163}{chapter*.82}%
|
||||
\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 {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}%
|
||||
\contentsline {subsection}{\numberline {3.1.3}MP2 Geometry Optimizations, Relative and Binding Energies}{51}{subsection.3.1.3}%
|
||||
\contentsline {subsection}{\numberline {3.1.4}Structure Classification}{52}{subsection.3.1.4}%
|
||||
\contentsline {section}{\numberline {3.2}Structural and Energetic Properties of Ammonium/Ammonia including Water Clusters}{53}{section.3.2}%
|
||||
\contentsline {subsection}{\numberline {3.2.1}General introduction}{53}{subsection.3.2.1}%
|
||||
\contentsline {subsection}{\numberline {3.2.2}Results and Discussion}{55}{subsection.3.2.2}%
|
||||
\contentsline {subsubsection}{\numberline {3.2.2.1}Dissociation Curves and SCC-DFTB Potential}{55}{subsubsection.3.2.2.1}%
|
||||
\contentsline {subsubsection}{\numberline {3.2.2.2}Small Species: (H$_2$O)$_{1-3}${NH$_4$}$^+$ and (H$_2$O)$_{1-3}${NH$_3$}}{58}{subsubsection.3.2.2.2}%
|
||||
\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}{161}{chapter*.82}%
|
||||
|
||||
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