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\author{Linjie Zheng}
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\affiliation[Universit\'e de Toulouse (UPS) and CNRS]{Laboratoire de Chimie et Physique Quantiques LCPQ/IRSAMC, Universit\'e de Toulouse (UPS) and CNRS, 118 Route de Narbonne, F-31062 Toulouse, France}
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\author{Jérôme Cuny}
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\affiliation[Universit\'e de Toulouse (UPS) and CNRS]{Laboratoire de Chimie et Physique Quantiques LCPQ/IRSAMC, Universit\'e de Toulouse (UPS) and CNRS, 118 Route de Narbonne, F-31062 Toulouse, France}
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\email{jerome.cuny@irsamc.ups-tlse.fr}
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\title[An \textsf{achemso} demo]
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{Structure and Stability of {NH$_4$}$^+$(H$_2$O)$_n$ and NH$_3$(H$_2$O)$_n$ Clusters: A SCC-DFTB Study }
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%\abbreviations{PES,ISM,COSAC,FT-IR,MRCI+Q,CCD,CCSD(T),RCCSD(T),UCCSD(T),MP2, MBPT, BSSE, ZPE, DFT, SCC-DFTB, CM3, MD, MDPT}
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\keywords{SCC-DFTB, PTMD, Ammonium, Ammonia, Water clusters}
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\begin{abstract}
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The global optimization to obtain the low-energy isomers of {NH$_4$}$^+$(H$_2$O)$_{n=1-10, 20}$ and NH$_3$(H$_2$O)$_{1-10}$ clusters is investigated using the self-consistent-charge density-functional based tight-binding (SCC-DFTB) approach in combination with the parallel-tempering molecular dynamics (PTMD) simulations. Different integral parameters and N-H bond parameters are tried in SCC-DFTB method for calculating the low-energy isomers of ammonium ane ammmonia water clusters. MP2 method is also used to optimize the low-energy isomers of {NH$_4$}$^+$(H$_2$O)$_{1-10}$ and NH$_3$(H$_2$O)$_{1-10}$ clusters to be sure the best parameters in SCC-DFTB level. Comparing with the caloric curves of the low-energy isomer {NH$_4$}$^+$(H$_2$O)$_{20}$ obtained using DFT calculation in the literature, the SCC-DFTB method can provide a quite good result. From the present study, we get the benchmark on small ammonium and ammonia water clusters, which shows the ability of SCC-DFTB to depict the potential energy surfaces of this kind of complexes. Furthermore, SCC-DFTB make it possible to simulate the large ammonium and ammonia containing water clusters.
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\section{Introduction}
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Small scale water cluster systems can provide valuable insights into the structure characterization of bulk water and different phenomena associated with water. For instance, if one ionic or neutral molecule is introduced into a water cluster, it becomes possible to study the solvation in water and the combination with water. Water clusters play an important role in diverse areas, such as the atmospheric science, chemistry, and biology.\cite{Keesee1989, Gilligan2000, Sennikov2005, Cabellos2016, Orabi2013, Bommer2016} Water clusters are able to absorb amounts of radiant energy, thus lessening the greenhouse effect.\cite{Galashev2010} The physical and chemical properties of water clusters are closely related to the critical stages in natural nucleation, growth of water-containing droplets and particles processes in atmosphere chemistry.\cite{Kulmala2004} Ammonia is an important substance found in the atmosphere, it can be as the ionic center of nucleation because of its relatively high basicity making it a potential proton sink.\cite{Perkins1984, Arnold1997} Ammonia can lead to the formation of aerosols in the atmosphere. Aerosols shape the global climate system and affect the weather. In addition, both ammonium and ammonia can lead to the pollution of the surface water system of earth. Due to the significance of ammonium and ammonia water clusters, they have been widely researched in both experiment and theory during the past decades.\cite{Stockman1992, Herbine1985, Hvelplund2010, Douady2009, Douady2008, Morrell2010, Bacelo2002, Wang1998}
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In 1984, {NH$_4$}$^+$(H$_2$O)$_{2}$ was identified using the first mass spectrometric measurements at ground level by Perkin et al.\cite{Perkins1984} In 1997, Stenhagen and coworkers studied positively charged water clusters {H$_3$O}$^+$(H$_2$O)$_{20}$ and {NH$_4$}$^+$(H$_2$O)$_{20}$ by a slight modification of the electrospray ionization method and they found that the two types clusters have the similar structures.\cite{Hulthe1997} Uggerud et al. reported the protonated mixed ammonia water clusters mainly exist in three forms, {(NH$_3$)$_4$}H$^+$(H$_2$O)$_{n}$, {(NH$_3$)$_5$}H$^+$(H$_2$O)$_{n}$, {(NH$_3$)$_6$}H$^+$(H$_2$O)$_{n}$ (n=1-25) through the mass spectrometric experiments in 2010 and their research supports the idea that small protonated mixed clusters of water and ammonia contain a central core {NH$_4$}$^+$.\cite{Hvelplund2010} The NH$_3$(H$_2$O) complex has been experimentally investigated via radio frequency and microwave spectra by Herbine et al., and via microwave and tunable far-infrared spectroscopy by Stockman and co-workers.\cite{Herbine1985, Stockman1992} A number of theoretical calculations about ammonium and ammonia water clusters also have been extensively studied in recent years.\cite{Chang1998, Jiang1999, Hvelplund2010, Bacelo2002, Galashev2013} Spiegelman gave the lowest-energy structures of {NH$_4$}$^+$(H$_2$O)$_{n}$ and investigated combined effects of size and temperature on the stable {NH$_4$}$^+$(H$_2$O)$_{n}$ clusters using an empirical potential and DFT method.\cite{Douady2008} Shields’s group studied the ammonium with 1 to 10 water molecules systems via a mixed molecular dynamics and quantum mechanics model and structures and thermodynamic values they got were in good agreement with previous experimental and theoretical results.\cite{Morrell2010} The structural, energetic, and spectroscopic properties of NH$_3$(H$_2$O) complex has been studied by Sadlej and co-workers.\cite{Sadlej1999} In addition, NH$_3$(H$_2$O) complex was also researched with an ab initio method by Skurski.\cite{Skurski1998} In 1996, Novoa et al. have already reported the existence of a minimum energy structure in the potential energy surface of NH$_3$(H$_2$O)$_4$ cluster, which is the result of one proton transfer from a water molecule to the ammonia molecule.\cite{Lee1996} Donaldson found a global minimum for NH$_3$(H$_2$O) and a cyclic structure for NH$_3$(H$_2$O)$_2$ through an experimental and theoretical investigation of the structure, kinetics, and thermodynamics of the formation of NH$_3$(H$_2$O)$_{1-2}$ clusters in the year of 1999.\cite{Donaldson1999} Bacelo reported a number of low-lying minima on the potential hypersurfaces structures of NH$_3$(H$_2$O)$_{1-4}$ clusters, which were identified by ab initio Monte Carlo simulated annealing in 2002.\cite{Bacelo2002}
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DFTB approach is derived from a second order Taylor series expansion of the density functional theory (DFT) total energy expression around a reference density. DFTB approach is two or three orders of magnitude faster than DFT method, and can provide accurate results, which makes it possible to calculate much larger systems.\cite{Elstner1998, Elstner2014} However, the usual DFTB approach can not calculate the ammonium water clusters accurately, because nitrogen hybridization seems to pose a problem for minimal basis set methods like DFTB methods.\cite{Winget2003} Elstner and coworkers found consistent errors (about 14.0 kcal·mol\textsuperscript{-1}) for deprotonation energies of sp$^3$ hybridized nitrogen containing systems, whereas sp1 and sp$^2$ systems are with smaller errors.\cite{Gaus2013para} They tried to solve this problem by giving a special parametrization N-H-mod, where they shift the repulsive potential by 14.0 kcal·mol\textsuperscript{-1} but this limits the transferability of DFTB, which doesn’t solve the problem fundamentally.
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In the present work, we improve the SCC-DFTB scheme by correcting integral parameter and bond parameter between hydrogen and nitrogen atoms. The structures of {NH$_4$}$^+$(H$_2$O)$_{n=1-10,20}$ and {NH$_3$}(H$_2$O)$_{1-10}$ clusters are investigated via combining the improved SCC-DFTB scheme with the molecular dynamics parallel-tempering (MDPT) simulations, which allows to explore their potential energy surfaces (PES) and periodic gradient driven quenches to obtain low-energy isomers. Moreover, the candidate structures calculated at the SCC-DFTB level are reoptimised with the MP2/Def2TZVP method to confirm the lowest-energy isomers. The labels \textquotedblleft n-x\textquotedblright and \textquotedblleft n$^\prime$-x\textquotedblright are utilized to distinguish all the obtained isomers of {NH$_4$}$^+$(H$_2$O)$_{n}$ and {NH$_3$}(H$_2$O)$_{n}$ clusters at SCC-DFTB level, and \textquotedblleft n-x$^*$\textquotedblright and \textquotedblleft n$^\prime$-x$^*$\textquotedblright represent the obtained isomers of {NH$_4$}$^+$(H$_2$O)$_{n}$ and {NH$_3$}(H$_2$O)$_{n}$ clusters at MP2/ Def2TZVP level, in which n and n$^\prime$ are in terms of the number water molecules in ammonium and ammonia water clusters respectively, and x denotes the alphabet corresponding to each isomer in given \textquotedblleft n-x\textquotedblright or \textquotedblleft n$^\prime$-x\textquotedblright series. In the following, the isomers are characterized by their relative energies, E$_{rel}$(DFTB) representing the relative energy optimized at the SCC-DFTB level, E$_{rel}$(MP2) in terms of the relative energy optimized at the MP2/Def2TZVP level.
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\section{Computational Methods} \label{Comput_meth}
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\textbf{Description of PES.}
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To describe the PES of different clusters studied in this work, the improved SCC-DFTB scheme is performed in deMonNano code.\cite{demonnanoCode} In this method, the electronic energy is written as
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\begin{equation} \label{E}
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\begin{aligned}
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E_{SCC-DFTB} & =\sum_{i}^{occ}\langle \Psi_i\mid\widehat{H}^0\mid\Psi_i\rangle + \sum_{\alpha\beta}U_{\alpha\beta} + \frac{1}{2}\sum_{\alpha\beta}\Delta q_\alpha\Delta q_\beta\gamma_{\alpha\beta} \\
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&\quad -\sum_{\alpha\beta}f_{damp}(R_{\alpha\beta}) \dfrac{C_6^{\alpha\beta}}{R_{\alpha\beta}^6} \end{aligned}
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where the first term is a band energy defined from parameterized integrals and the second term is a sum of repulsive interaction over all pairs of atoms. In the present work, we employ the mio-set with improved parameter for Slater-Koster tables of integrals.\cite{Elstner1998} The third term in equation \ref{E} donates the second order correction term expressed as a function of the atomic charge fluctuations $\Delta$q and the last term is an empirical correction which describes the dispersion interactions.\cite{Elstner2001, Zhechkov2005, Rapacioli2009} Rapacioli and co-workers made an improvement for the description of electrostatic interactions in molecular cluster that can be obtained by replacing the original Mulliken charges with the class IV/charge model 3 (CM3) charges,\cite{Rapacioli2009,Thompson2003, Winget2002} defined as the following equation \ref{charge}
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\begin{equation} \label{charge}
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q_{\alpha}^{CM_3} & =q_{\alpha}^{Mull} &+ &\sum_{\alpha^\prime\neq\alpha}^{atoms}(D_{Z_\alpha Z_{\alpha^\prime}}B_{KK^\prime} + C_{Z_\alpha Z_{\alpha^\prime}}{B_{KK^\prime}}^2)
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where B$_{KK^\prime}$ in terms of the Mayer’s bond order,\cite{Mayer1983} C$_{Z_\alpha Z_{\alpha^\prime}}$ and D$_{Z_\alpha Z_{\alpha^\prime}}$ are the empirical parameters to determine. In the research of water clusters, Simon et al. developed a SCC-DFTB potential that geometries, frequency, and energy are close to the corresponding experimental values and the results of expensive CCSD(T)/aug-cc-pVTZ calculations.\cite{Simon2012, Odutola1980} The parameters of the CM3 charges used in this work, i.e. D$_{OH}$ is 0.129, is retained in the present work, while D$_{NH}$ is tested in order to reproduce the values of the electrostatic potential (ESP) derived from the atomic charges at the level of MP2/Def2TZVP. This appropriate value of D$_{NH}$ is 0.14. The D$_{NO}$ parameter is set to zero. In order to avoid deprotonation of the sp3 hybridized nitrogen and reproduce the binding energy calculated at the level of MP2/Def2TZVP, we developed the parameters in Slater-Koster integral tables, namely, 1.28 times the original parameters of nitrogen and hydrogen atoms.
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\textbf{Exploration of the PES}
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In the present study, we use finite temperature to explore the PES combing periodic gradient quenching to perform the global optimization to obtain the minima. The finite temperature exploration was carried out by MDPT algorithm,\cite{Sugita1999, Sugita2000, Earl2005} which was implemented in deMonNano code.\cite{Oliveira2015, Korchagina2016} A linear distribution of temperature with a 15 K step ranged from 10 to 250 K with 16 replicas for {NH$_4$}$^+$(H$_2$O)$_{1-3}$ and {NH$_3$}(H$_2$O)$_{1-3}$ clusters was used. And 40 replicas with a 6 K step ranged from 10 to 250 K for {NH$_4$}$^+$(H$_2$O)$_{n=4-10,20}$ and {NH$_3$}(H$_2$O)$_{4-10}$ clusters was performed. The trajectories were 5 ns for each cluster, and a time step of 0.5 fs was used to integrate the motion equations in the MDPT simulations. The Nosé-Hoover chain thermostat was employed for all the simulations performed in this study. The thermostat frequency and the number of thermostats in the chain were fixed at 400 cm-1 and 5, respectively to achieve an exploration in the canonical ensemble.\cite{Nose1984, Hoover1985}
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To acquire the low-lying energy isomers, the MDPT algorithm combining SCC-DFTB description the energies and gradients was conducted to explore the PES. 303 geometries were periodically selected from each replica and optimized using the improved SCC-DFTB method, which produced 4848 optimized geometries for {NH$_4$}$^+$(H$_2$O)$_{1-3}$ and {NH$_3$}(H$_2$O)$_{1-3}$ clusters. 500 geometries were periodically selected from each replica and optimized with the improved SCC-DFTB method leading to 20000 optimized geometries for {NH$_4$}$^+$(H$_2$O)$_{n=4-10,20}$ and {NH$_3$}(H$_2$O)$_{4-10}$ clusters. Following, the first five lowest-energy isomers selected from 4848 or 20000 geometries were optimized using the MP2/Def2TZVP method to obtain the lowest-energy isomer of {NH$_4$}$^+$(H$_2$O)$_{n=1-10,20}$ and {NH$_3$}(H$_2$O)$_{1-10}$ clusters. In addition, to evaluate the strength of water-ammonium and -ammonia interaction and to further assess the accuracy of the SCC-DFTB method, we also calculated the binding energy of the selected isomers. The detailed results and discussion on the structure and binding energy of small aggregates (n = 1-3) are shown in the Supporting Information (SI), and the most relevant features are discussed in the following part. To confirm the accuracy, the binding energies and geometries obtained at the SCC-DFTB level were compared with those carried out at MP2/Def2TZVP level and corresponding results in the literature. Finally, the influence of basis set superposition errors (BSSE) was tested by using the counterpoise method of Boys and Bernardi.\cite{Boys2002} In our calculations, both all water molecules in the ammonium and ammonia water clusters considered separately and as a whole were calculated when the counterpoise method was used. All MP2 calculations were performed in the Gaussian 09 package.\cite{GaussianCode}
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\section{Results and Discussion} \label{resul_disc}
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SCC-DFTB Performances on Ammonium and Ammonia Water Clusters
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To validate the accuracy of SCC-DFTB approach on ammonium and ammonia water clusters, the low-energy isomers obtained at SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels are compared.
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\textbf{SCC-DFTB Performances on clusters {NH$_4$}$^+$(H$_2$O)$_{1-3}$ and {NH$_3$}(H$_2$O)$_{1-3}$.} Here we only show the first low-energy isomer of clusters {NH$_4$}$^+$(H$_2$O)$_{1-2}$ and {NH$_3$}(H$_2$O)$_{1-3}$ and show the first two low-energy isomers of clusters {NH$_4$}$^+$(H$_2$O)$_{3}$. As displayed in Figure \ref{fig:nh3-nh4-1w} and \ref{fig:nh3-nh4-2-3w}, the configurations of 1-a, 1$^\prime$-a, 2-a, 2$^\prime$-a, 3-a and 3$^\prime$-a obtained at SCC-DFTB level are similar to those of 1-a$^*$, 1$^\prime$-a$^*$, 2-a$^*$, 2$^\prime$-a$^*$, 3-a$^*$ and 3$^\prime$-a$^*$ obtained at MP2/Def2TZVP level and the corresponding bond lengths at SCC-DFTB and MP2/Def2TZVP levels are very close. From the results of our calculation, the energy of isomer 3-b is 2.12 kcal·mol\textsuperscript{-1} higher than that of 3-a at SCC-DFTB level, and the energy of isomer 3-b$^*$ relative to 3-a$^*$ at MP2/Def2TZVP level is -0.30 kcal·mol\textsuperscript{-1} while it is 1.21 kcal·mol\textsuperscript{-1} at MP2/Def2TZVP level corrected with zero-point vibrational energy (ZPVE) correction. From the experiment results of Chang’s group, 3-a$^*$ is more stable than 3-b$^*$ and from their theoretical results at B3LYP/6-31+G(d) level, the energy of 3-a$^*$ is higher than that of 3-b$^*$ but at MP2/6-31+G(d) level corrected with ZPVE, the energy of 3-a$^*$ is lower than that of 3-b$^*$, which has a reordering between isomers due to the ZPVE correction.\cite{Wang1998, Jiang1999} According to results of Spiegelman’s group, 3-a$^*$ is the lowest-energy isomer which were obtained using global Monte Carlo (MC) optimizations with the intermolecular polarizable potential.\cite{Douady2008} All these results show the accuracy of SCC-DFTB method is between B3LYP/6-31+G(d), MP2/6-31+G(d) methods and the MP2/6-31+G(d) method corrected by ZPVE, which indicates the combining of SCC-DFTB and MDPT is good enough to find the lowest-energy isomer. In additional, other isomers of clusters {NH$_4$}$^+$(H$_2$O)$_{3}$ and {NH$_3$}(H$_2$O)$_{3}$ at SCC-DFTB level are in good agreement with those calculated at the MP2/Def2TZVP level shown in Figure S1 in the SI.
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\begin{figure}[H]
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\includegraphics[width=0.4\linewidth]{figure_ammoni/nh3-nh4-1w.jpeg}
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\centering
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\caption{The geometries of 1-a, 1$^\prime$-a are optimized at SCC-DFTB level and 1-a$^*$, 1$^\prime$-a$^*$ are optimized at MP2/Def2TZVP level. The energies are in kcal·mol\textsuperscript{-1} and the selected bond lengths are in Å.}
|
|
||||||
\label{fig:nh3-nh4-1w}
|
|
||||||
\end{figure}
|
|
||||||
|
|
||||||
|
|
||||||
\begin{figure}[H]
|
|
||||||
\includegraphics[width=0.9\linewidth]{figure_ammoni/nh3-nh4-2-3w.jpeg}
|
|
||||||
\centering
|
|
||||||
\caption{The geometries of n-a, n$^\prime$-a (n=2-3) and 3-b are optimized at SCC-DFTB level and n-a$^*$, n$^\prime$-a$^*$ (n=2-3) and 3-b$^*$ are optimized at MP2/Def2TZVP level. The energies are in kcal·mol\textsuperscript{-1} and the selected bond lengths are in Å.}
|
|
||||||
\label{fig:nh3-nh4-2-3w}
|
|
||||||
\end{figure}
|
|
||||||
For the binding energy calculation of {NH$_4$}$^+$ and NH$_3$ containing water clusters, two methods were used. One is to consider all the water molecules in the cluster as a whole part to calculate the binding energy and the other one is to calculate the energy of water molecule separately using their geometries within the clusters. The binding energies with the two methods of clusters {NH$_4$}$^+$(H$_2$O)$_{1-3}$ and NH$_3$(H$_2$O)$_{1-3}$ at SCC-DFTB and MP2/Def2TZVP levels with the BSSE correction are listed in Table \ref{reBindE}.
|
|
||||||
|
|
||||||
\begin{table} \label{reBindE}
|
|
||||||
\begin{center}
|
|
||||||
\caption{The relative binding energies (E$_{bind}$(SCC-DFTB)-E$_{bind}$(MP2/Def2TZVP)) of the low-energy isomers for clusters {NH$_4$}$^+$(H$_2$O)$_{1-10}$ and {NH$_3$}(H$_2$O)$_{1-10}$. E$_{rela-whole}$ refers to the relative binding energy when all the water molecules are considered as a whole part. E$_{rela-sepa}$ refers to the relative binding energy when the water molecules are separately considered using the geometry in the cluster.}
|
|
||||||
|
|
||||||
\label{reBindE}
|
|
||||||
\begin{tabular}{c|c|c|c|c|c}
|
|
||||||
\textbf{clusters} & \textbf{E$_{rela-whole}$} & \textbf{E$_{rela-sepa}$} & \textbf{clusters} & \textbf{E$_{rela-whole}$} & \textbf{E$_{rela-sepa}$} \\
|
|
||||||
\hline
|
|
||||||
1-a & 1.21 & 1.21 & 1$^\prime$-a & -1.17 & -1.17 \\
|
|
||||||
2-a & 0.82 & 0.91 & 2$^\prime$-a & 0.57 & 0.28 \\
|
|
||||||
3-a & -0.25 & 0.11 & 3$^\prime$-a & 0.91 & 0.01 \\
|
|
||||||
3-b & 1.21 & -0.15 & - & - & - \\
|
|
||||||
4-a & -1.67 & -0.87 & 4$^\prime$-a & -1.11 & -1.76 \\
|
|
||||||
4-b & 0.00 & 0.61 & 4$^\prime$-b & -0.29 & -1.62 \\
|
|
||||||
4-c & 0.77 & 0.44 & 4$^\prime$-c & -0.29 & -1.38 \\
|
|
||||||
4-d & 0.77 & 0.42 & 4$^\prime$-d & 1.08 & -0.49 \\
|
|
||||||
4-e & -4.04 & 0.69 & 4$^\prime$-e & 1.02 & -1.07 \\
|
|
||||||
5-a & -1.62 & 0.56 & 5$^\prime$-a & 0.82 & -1.78 \\
|
|
||||||
5-b & 0.72 & 0.48 & 5$^\prime$-b & -0.23 & -2.26 \\
|
|
||||||
5-c & 0.69 & 0.55 & 5$^\prime$-c & -0.34 & -2.50 \\
|
|
||||||
5-d & -1.08 & -0.78 & 5$^\prime$-d & -0.59 & -1.84 \\
|
|
||||||
5-e & -2.08 & 0.88 & 5$^\prime$-e & -0.38 & -2.60 \\
|
|
||||||
6-a & -1.71 & -0.38 & 6$^\prime$-a & -0.27 & -3.05 \\
|
|
||||||
6-b & -1.14 & -0.76 & 6$^\prime$-b & -0.31 & -3.55 \\
|
|
||||||
6-c & -2.06 & 0.27 & 6$^\prime$-c & -1.11 & -4.67 \\
|
|
||||||
6-d & -2.90 & -1.06 & 6$^\prime$-d & -0.05 & -4.44 \\
|
|
||||||
6-e & -1.18 & -0.60 & 6$^\prime$-e & 0.55 & -1.96 \\
|
|
||||||
7-a & -2.95 & -0.39 & 7$^\prime$-a & 1.09 & -2.02 \\
|
|
||||||
7-b & -2.92 & -0.38 & 7$^\prime$-b & -0.02 & -4.07 \\
|
|
||||||
7-c & -2.17 & 0.09 & 7$^\prime$-c & -0.40 & -4.15 \\
|
|
||||||
7-d & -1.28 & -1.35 & 7$^\prime$-d & -0.14 & -3.10 \\
|
|
||||||
7-e & -3.22 & -2.27 & 7$^\prime$-e & -1.11 & -4.32 \\
|
|
||||||
8-a & -2.20 & -1.63 & 8$^\prime$-a & -1.12 & -4.41 \\
|
|
||||||
8-b & -1.61 & -2.01 & 8$^\prime$-b & -0.10 & -3.04 \\
|
|
||||||
8-c & -3.71 & -1.17 & 8$^\prime$-c & -0.41 & -4.46 \\
|
|
||||||
8-d & -2.43 & -0.36 & 8$^\prime$-d & 0.20 & -3.68 \\
|
|
||||||
8-e & -0.55 & 0.35 & 8$^\prime$-e & -1.28 & -4.75 \\
|
|
||||||
9-a & -2.02 & -1.39 & 9$^\prime$-a & -0.15 & -4.47 \\
|
|
||||||
9-b & 0.51 & -0.84 & 9$^\prime$-b & -1.01 & -4.45 \\
|
|
||||||
9-c & -3.31 & -0.85 & 9$^\prime$-c & -1.04 & -4.42 \\
|
|
||||||
9-d & -1.58 & -1.78 & 9$^\prime$-d & -1.09 & -5.14 \\
|
|
||||||
9-e & -2.39 & -0.91 & 9$^\prime$-e & 0.41 & -2.57 \\
|
|
||||||
10-a & -2.64 & -1.94 & 10$^\prime$-a & -0.03 & -4.80 \\
|
|
||||||
10-b & -5.79 & -4.35 & 10$^\prime$-b & 0.13 & -5.61 \\
|
|
||||||
10-c & -1.26 & -2.36 & 10$^\prime$-c & -0.62 & -6.50 \\
|
|
||||||
10-d & -1.98 & -1.42 & 10$^\prime$-d & -1.10 & -6.30 \\
|
|
||||||
10-e & -7.17 & -1.54 & 10$^\prime$-e & 0.23 & -8.36 \\
|
|
||||||
\end{tabular}
|
|
||||||
\end{center}
|
|
||||||
\end{table}
|
|
||||||
|
|
||||||
As listed in Table \ref{reBindE}, the relative binding energies (E$_{bind}$(SCC-DFTB)-E$_{bind}$(MP2/Def2TZVP)) of {NH$_4$}$^+$(H$_2$O) and {NH$_3$}(H$_2$O) are 1.21 and -1.17 kcal·mol\textsuperscript{-1}, separately, which indicates the results of SCC-DFTB agree with those of MP2/Def2TZVP with BSSE correction on {NH$_4$}$^+$(H$_2$O) and NH$_3$(H$_2$O).
|
|
||||||
|
|
||||||
When the water molecules in the cluster are considered as a whole part to calculate the binding energy, the relative binding energies (E$_{bind}$(SCC-DFTB)-E$_{bind}$(MP2/Def2TZVP)) of 2-a, 2$^\prime$-a, 3-a, 3$^\prime$-a and 3-b are 0.82, 0.57, -0.25, 0.91 and 1.21 kcal·mol\textsuperscript{-1}, respectively shown in Table \ref{reBindE}. When the water molecules are separately considered using the geometry in the cluster to calculate the binding energy, the relative binding energies of 2-a, 2$^\prime$-a, 3-a, 3$^\prime$-a and 3-b are 0.91, 0.28, 0.11, 0.01 and -0.15 kcal·mol\textsuperscript{-1}, respectively shown in Table \ref{reBindE}. This demonstrates the SCC-DFTB is in good agreement with MP2/Def2TZVP for {NH$_4$}$^+$(H$_2$O)$_{2-3}$ and {NH$_3$}(H$_2$O)$_{2-3}$. The relative binding energies of other isomers of clusters {NH$_4$}$^+$(H$_2$O)$_{n=2-3}$ and {NH$_3$}(H$_2$O)$_{2-3}$ are shown in Table S1 in the SI and the results of SCC-DFTB also agrees well with those of MP2/Def2TZVP with BSSE correction. In addition, these results show when there are two or three water molecules in the cluster, considering the water molecules as a whole part or not does not affect a lot on the relative binding energy.
|
|
||||||
|
|
||||||
It is worth mentioning that the difference of binding energy between {NH$_4$}$^+$(H$_2$O) and NH$_3$(H$_2$O) was expected owing to a stronger electrostatic contribution of {NH$_4$}$^+$ to the binding energy.
|
|
||||||
|
|
||||||
\textbf{Properties of clusters {NH$_4$}$^+$(H$_2$O)$_{n=4-10,20}$ and NH$_3$(H$_2$O)$_{4-10}$.}
|
|
||||||
Clusters {NH$_4$}$^+$(H$_2$O)$_{n=4-10,20}$ have been studied by molecular dynamic simulations, global optimization with Monte Carlo method, DFT and MP2 methods and so on, but these methods cost expensively.\cite{Wang1998, Jiang1999, Douady2008, Lee2004, Douady2009, Morrell2010} However, to the best of our knowledge, no theoretical calculation about {NH$_3$}(H$_2$O)$_{5-10}$ clusters have been studied. The inexpensive computational cost of SCC-DFTB approach and its good performances on small clusters provide an appealing opportunity for people to explore the PES of both large ammonium and ammonia water clusters. In the following, the first five low-energy isomers of clusters {NH$_4$}$^+$(H$_2$O)$_{n=4-10,20}$ and NH$_3$(H$_2$O)$_{4-10}$ will be shown and discussed in detail.
|
|
||||||
|
|
||||||
\textbf{Properties of clusters {NH$_4$}$^+$(H$_2$O)$_{n=4-10,20}$.}
|
|
||||||
|
|
||||||
For cluster {NH$_4$}$^+$(H$_2$O)$_{4}$, the first five low-energy isomers are shown in Figure \ref{fig:nh4-4-6w}. The energy of structure 4-a is the minima of the global optimization with SCC-DFTB method and also the lowest-energy configuration optimized at MP2/Def2TZVP level with ZPVE correction in the five isomers. This geometry of 4-a is consistent with the previous computational results reported by several groups\cite{Wang1998, Jiang1999, Douady2008, Lee2004, Pickard2005} and experimental studies by Chang’s group.\cite{Chang1998, Wang1998} Isomer 4-a presents itself as the most stable structure of the listed five isomers of cluster {NH$_4$}$^+$(H$_2$O)$_{4}$, forming four hydrogen bonds around the ion symmetrically. In other words, no free N-H bonds in isomer 4-a. Other isomers of comparable stability are displayed in Figure 2, where 4-b is only 0.20 kcal·mol\textsuperscript{-1} higher compared to 4-a. The energy differences among 4-c, 4-d, and 4-e are within 2.00 kcal·mol\textsuperscript{-1} at MP2/Def2TZVP level with ZPVE correction. The energy order of 4-a to 4-e at SCC-DFTB level is consistent with that at MP2/Def2TZVP level with ZPVE correction. It’s worth noting that isomer 4-c was not found in Chang’s study,\cite{Jiang1999} and the energy order for the first five low-energy isomers are not completely the same with our results because of the difference of the basis set.
|
|
||||||
|
|
||||||
\begin{figure}[H]
|
|
||||||
\includegraphics[width=1.0\linewidth]{figure_ammoni/nh4-4-6w.jpeg}
|
|
||||||
\centering
|
|
||||||
\caption{The first five low-energy isomers of cluster {NH$_4$}$^+$(H$_2$O)$_{4-6}$ and the associated relative energies (in kcal·mol\textsuperscript{-1}) at SCC-DFTB level (italic) and MP2/Def2TZVP level with and without ZPVE correction (bold).}
|
|
||||||
\label{fig:nh4-4-6w}
|
|
||||||
\end{figure}
|
|
||||||
|
|
||||||
|
|
||||||
The relative binding energy of SCC-DFTB method to MP2/Def2TZVP method with BSSE correction (E$_{bind}$(SCC-DFTB)-E$_{bind}$(MP2/Def2TZVP)) 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 {NH$_4$}$^+$(H$_2$O)$_{4}$. From the relative binding energy of {NH$_4$}$^+$(H$_2$O)$_{4}$, it indicates that all the water molecules considered as a whole part or not has an effect on the relative binding energy for the cluster {NH$_4$}$^+$(H$_2$O)$_{4}$ and when all the water molecules are considered as a whole, the difference between SCC-DFTB and MP2/Def2TZVP with BSSE correction are smaller than that when the water molecules are considered separately.
|
|
||||||
|
|
||||||
For cluster {NH$_4$}$^+$(H$_2$O)$_{5}$, 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 whereas a 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 MDPT 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 energy of water molecules. As listed in Table \ref{reBindE}, for isomers 5-a to 5-e, the relative binding energy E$_{rela-whole}$ are -1.62, 0.72, 0.69, -1.08 and -2.08 kcal·mol\textsuperscript{-1} and the E$_{rela-sepa}$ are -0.56, 0.48, 0.55, -0.78 and 0.88 kcal·mol\textsuperscript{-1}, respectively. The E$_{rela-whole}$ is bigger than corresponding E$_{rela-sepa}$, which indicates it is better to calculate the binding energy with considering the water molecules separately. The E$_{rela-sepa}$ is less than 1.00 kcal·mol\textsuperscript{-1} for the first five low-energy isomers of cluster {NH$_4$}$^+$(H$_2$O)$_{5}$, so the SCC-DFTB method is good enough compared to MP2/Def2TZVP with BSSE correction for cluster {NH$_4$}$^+$(H$_2$O)$_{5}$.
|
|
||||||
|
|
||||||
For cluster {NH$_4$}$^+$(H$_2$O)$_{6}$, 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 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 {NH$_4$}$^+$(H$_2$O)$_{6}$ 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-e 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-e. It shows SCC-DFTB is good to find the low-energy isomers of cluster {NH$_4$}$^+$(H$_2$O)$_{6}$ compared to MP2 and B3LYP methods.
|
|
||||||
|
|
||||||
As shown in Table \ref{reBindE}, for isomers 6-a to 6-e, the relative binding energy E$_{rela-whole}$ are -1.71, -1.14, -2.06, -2.90 and -1.18 kcal·mol\textsuperscript{-1} and the E$_{rela-sepa}$ 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 E$_{rela-whole}$ is bigger than corresponding E$_{rela-sepa}$ because it can not calculate interaction energy of water molecules perfectly using SCC-DFTB when all the water molecules are considered as a whole part.
|
|
||||||
|
|
||||||
For cluster {NH$_4$}$^+$(H$_2$O)$_{7}$, 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 H2O 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} In 7-c is the first low-energy isomer at MP2/Def2TZVP with ZPVE correction level including three three-coordinated water molecules. 7-c is 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 displayed in Figure 2, 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 accessible than 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 {NH$_4$}$^+$(H$_2$O)$_{7}$ verify the accuracy of SCC-DFTB approach.
|
|
||||||
|
|
||||||
\begin{figure}[H]
|
|
||||||
\includegraphics[width=1.0\linewidth]{figure_ammoni/nh4-7-10w.jpeg}
|
|
||||||
\centering
|
|
||||||
\caption{The first five low-energy isomers of cluster {NH$_4$}$^+$(H$_2$O)$_{7-10}$ and the associated relative energies (in kcal·mol\textsuperscript{-1}) at SCC-DFTB level (italic) and MP2/Def2TZVP level with and without ZPVE correction (bold).}
|
|
||||||
\label{fig:nh4-7-10w}
|
|
||||||
\end{figure}
|
|
||||||
|
|
||||||
As shown in Table \ref{reBindE}, for isomers 7-a to 7-e, the relative binding energy E$_{rela-whole}$ are -2.95, -2.92, -2.17, -1.28 and -3.22 kcal·mol\textsuperscript{-1} and the E$_{rela-sepa}$ are -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 levels when water molecules are calculated separately. When all the water molecules are considered as a whole part, the results of SCC-DFTB are not as good as those of the MP2 with BSSE method.
|
|
||||||
|
|
||||||
For cluster {NH$_4$}$^+$(H$_2$O)$_{8}$, 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 H2O 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 H2O molecules in its cage frame. The energies of isomers 8-a to 8-e are very close at both 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 {NH$_4$}$^+$(H$_2$O)$_{8}$.
|
|
||||||
|
|
||||||
As shown in Table \ref{reBindE}, for isomers 8-a to 8-e, the relative binding energy E$_{rela-whole}$ are -2.20, -1.61, -3.71, -2.43 and -0.55 kcal·mol\textsuperscript{-1} and the biggest E$_{rela-sepa}$ is -2.01 kcal·mol\textsuperscript{-1}, respectively. It shows the binding energies at SCC-DFTB level agree well with those at MP2/Def2TZVP with BSSE correction levels when water molecules are calculated separately. 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 {NH$_4$}$^+$(H$_2$O)$_{9}$, the first five low-energy structures of {NH$_4$}$^+$(H$_2$O)$_{9}$ are illustrated in Figure \ref{fig:nh4-7-10w}. 9-a with seven three-coordinated H2O 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 levels 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 are also close. The energy difference of isomers 8-a to 8-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 very 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 {NH$_4$}$^+$(H$_2$O)$_{9}$.
|
|
||||||
|
|
||||||
As shown in Table \ref{reBindE}, for isomers 9-a to 9-e, the relative binding energy E$_{rela-whole}$ are -2.20, -1.61, -3.71, -2.43 and -0.55 kcal·mol\textsuperscript{-1} and the relative binding energy E$_{rela-sepa}$ is -1.39, -0.84, -0.85, -1.78, and -0.91 kcal·mol\textsuperscript{-1}, respectively and the corresponding E$_{rela-whole}$ are bigger than those of E$_{rela-sepa}$. It shows the binding energies at SCC-DFTB level agree well with those at MP2/Def2TZVP with BSSE correction levels when water molecules are calculated separately. 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.
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||||||
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|
||||||
For cluster {NH$_4$}$^+$(H$_2$O)$_{10}$, 10-a to 10-e are the first five low-energy isomers which has a has a complete solvation shell in the ion core {NH$_4$}$^+$ 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, while it is 0.31 kcal·mol\textsuperscript{-1} lower than that of 10-a at MP2/Def2TZVP with ZPVE correction level. The energy of isomers 8-a to 8-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 with ZPVE correction for cluster {NH$_4$}$^+$(H$_2$O)$_{10}$.
|
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||||||
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|
||||||
As shown in Table \ref{reBindE}, for isomers 10-a to 10-e, the relative binding energies E$_{rela-whole}$ and E$_{rela-sepa}$ is not as small as the corresponding results of clusters {NH$_4$}$^+$(H$_2$O)$_{1-9}$, which implies the error of the relative binding energy increases with the number of water molecules in the cluster. However, the whole results of E$_{rela-whole}$ are still bigger than those of E$_{rela-sepa}$ for isomers 10-a to 10-e.
|
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||||||
|
|
||||||
textbf{Description of the heat capacity curves}
|
|
||||||
For cluster {NH$_4$}$^+$(H$_2$O)$_{20}$, the lowest-energy structure 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 obtained using exchange Monte Carlo simulations by Spiegelman’s group.\cite{Douady2009}
|
|
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We also calculated the canonical heat capacities of cluster {NH$_4$}$^+$(H$_2$O)$_{20}$ using the combination of SCC-DFTB and PTMD depicted in Figure \ref{fig:nh3-nh4-20w}.
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|
||||||
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||||||
\begin{figure}[H]
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||||||
\includegraphics[width=0.6\linewidth]{figure_ammoni/nh3-nh4-20w.jpeg}
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||||||
\centering
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||||||
\caption{The first five low-energy isomers of cluster {NH$_4$}$^{+}$(H$_2$O)$_{20}$ (a) and {NH$_3$}(H$_2$O)$_{20}$ (b) at SCC-DFTB level.}
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||||||
\label{fig:nh3-nh4-20w}
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|
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\end{figure}
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||||||
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|
|
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\textbf{Properties of {NH$_3$}(H$_2$O)$_{4-10}$ water clusters.}
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|
||||||
For cluster {NH$_3$}(H$_2$O)$_{4}$, 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 and 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 and the four oxygens 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 in 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 {NH$_3$}(H$_2$O)$_{4}$.
|
|
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|
||||||
\begin{figure}[H]
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|
||||||
\includegraphics[width=1.0\linewidth]{figure_ammoni/nh3-4-7w.jpeg}
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\centering
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\caption{The first five low-energy isomers of cluster {NH$_3$}$^+$(H$_2$O)$_{4-7}$ and the associated relative energies (in kcal·mol\textsuperscript{-1}) at SCC-DFTB level (italic) and MP2/Def2TZVP level with and without ZPVE correction (bold).}
|
|
||||||
\label{fig:nh3-4-7w}
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|
||||||
\end{figure}
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||||||
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|
||||||
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 E$_{rela-whole}$ for 4$^\prime$-a to 4$^\prime$-e are smaller than the corresponding values of E$_{rela-sepa}$. The E$_{rela-sepa}$ of 4$^\prime$-d is smaller than those of other isomers, it has a nearly planar pentagonal structure that only contains three O-H hydrogen bonds among the four water molecules while other isomers contain four O-H 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 E$_{rela-sepa}$ of 4$^\prime$-d is smaller than those of other isomers. In general, both relative binding energies E$_{rela-sepa}$ and E$_{rela-sepa}$ are not big that indicates SCC-DFTB performs well compared to the MP2 method with BSSE correction for calculating the binding energy of cluster {NH$_3$}(H$_2$O)$_{4}$.
|
|
||||||
|
|
||||||
For cluster {NH$_3$}(H$_2$O)$_{5}$, 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. 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 structures of 5$^\prime$-a and 5$^\prime$-e are almost the same, but the energy of 5$^\prime$-e is 1.51 kcal·mol\textsuperscript{-1} higher than 5$^\prime$-a at MP2/Def2TZVP with ZPVE correction level, which implies the O-H hydrogen bond types have an influence on the stability of the isomers. The results show the SCC-DFTB approach agrees well those of MP2/Def2TZVP with ZPVE correction method to find the low-energy isomers for cluster {NH$_3$}(H$_2$O)$_{5}$.
|
|
||||||
|
|
||||||
The relative binding energies of isomers 5$^\prime$-a to 5$^\prime$-e are shown in Table \ref{reBindE}. The values of E$_{rela-whole}$ are less than 0.82 kcal·mol\textsuperscript{-1} for 5$^\prime$-a to 5$^\prime$-e. The values of E$_{rela-sepa}$ are bigger than the corresponding values of E$_{rela-whole}$ that indicates SCC-DFTB agree better with E$_{rela-whole}$ than with E$_{rela-sepa}$ for calculating the binding energy of cluster {NH$_3$}(H$_2$O)$_{5}$.
|
|
||||||
|
|
||||||
For cluster {NH$_3$}(H$_2$O)$_{6}$, 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 in 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 and 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 good enough to find the low-energy isomers of cluster {NH$_3$}(H$_2$O)$_{6}$.
|
|
||||||
|
|
||||||
The relative binding energies of isomers 6$^\prime$-a to 6$^\prime$-e are shown in Table \ref{reBindE} that the smallest and the biggest values of E$_{rela-whole}$ are -0.05 and -1.11 kcal·mol\textsuperscript{-1}, respectively. The smallest absolute value of E$_{rela-sepa}$ is 1.96 kcal·mol\textsuperscript{-1}. The binding energies calculated with SCC-DFTB agree well with those of E$_{rela-whole}$ in cluster {NH$_3$}(H$_2$O)$_{6}$.
|
|
||||||
|
|
||||||
For cluster {NH$_3$}(H$_2$O)$_{7}$, the first five low-energy isomers 7$^\prime$-a to 7$^\prime$-e are shown in Figure \ref{fig:nh3-4-7w}. 7$^\prime$-a with a cubic structure is the first low-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-energy isomers at SCC-DFTB level and their energy are close. 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 that we used agree well with those of MP2/Def2TZVP with ZPVE correction for the first five low-energy isomers of cluster {NH$_3$}(H$_2$O)$_{7}$.
|
|
||||||
|
|
||||||
The relative binding energies of isomers 7$^\prime$-a to 7$^\prime$-e that the smallest and the biggest values of E$_{rela-whole}$ are -0.02 and -1.11 kcal·mol\textsuperscript{-1}, respectively and the smallest absolute value of E$_{rela-sepa}$ is 2.02 kcal·mol\textsuperscript{-1} shown in Table \ref{reBindE}. The binding energies calculated with SCC-DFTB agree well with those of E$_{rela-whole}$ in cluster {NH$_3$}(H$_2$O)$_{7}$.
|
|
||||||
|
|
||||||
For cluster {NH$_3$}(H$_2$O)$_{8}$, 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-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 very small at both SCC-DFTB and MP2/Def2TZVP with ZPVE correction levels. From Figure 6, the fifth low-energy isomer 8$^\prime$-e includes less number of hydrogen bonds than other isomers. The energy of 8$^\prime$-e has a clearly increase than other isomers. The results show our SCC-DFTB method is good enough to obtain the low-energy isomers of cluster {NH$_3$}(H$_2$O)$_{8}$.
|
|
||||||
|
|
||||||
\begin{figure}[H]
|
|
||||||
\includegraphics[width=1.0\linewidth]{figure_ammoni/nh3-8-10w.jpeg}
|
|
||||||
\centering
|
|
||||||
\caption{The first five low-energy isomers of cluster {NH$_3$}(H$_2$O)$_{8-10}$ and the associated relative energies (in kcal·mol\textsuperscript{-1}) at SCC-DFTB level (italic) and MP2/Def2TZVP level with and without ZPVE correction (bold).}
|
|
||||||
\label{fig:nh3-8-10w}
|
|
||||||
\end{figure}
|
|
||||||
|
|
||||||
The relative binding energies of isomers 8$^\prime$-a to 8$^\prime$-e that the smallest and the biggest values of E$_{rela-whole}$ are -0.1 and -1.28 kcal·mol\textsuperscript{-1}, respectively while the smallest absolute value of E$_{rela-sepa}$ is 3.04 kcal·mol\textsuperscript{-1} shown in Table \ref{reBindE}. The binding energies calculated with SCC-DFTB agree well with those of E$_{rela-whole}$ that all the water molecules are considered as a whole part in cluster {NH$_3$}(H$_2$O)$_{8}$.
|
|
||||||
|
|
||||||
For cluster {NH$_3$}(H$_2$O)$_{9}$, 9$^\prime$-a to 9$^\prime$-e are the first five low-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-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-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. In general the energy differences of 9$^\prime$-a to 9$^\prime$-e at both MP2/Def2TZVP with ZPVE correction levels are not less than 1.39 kcal·mol\textsuperscript{-1}, and the relative energy for each isomer between SCC-DFTB level and MP2/Def2TZVP with ZPVE correction level is less than 1.39 kcal·mol\textsuperscript{-1}. This shows our SCC-DFTB calculation results agree well with the calculation results of MP2/Def2TZVP with ZPVE correction for low-energy isomers optimization of cluster {NH$_3$}(H$_2$O)$_{9}$.
|
|
||||||
|
|
||||||
The relative binding energies of isomers 9$^\prime$-a to 9$^\prime$-e are shown in Table \ref{reBindE}. The absolute values of E$_{rela-whole}$ are less than 1.09 kcal·mol\textsuperscript{-1} while the smallest absolute value of E$_{rela-sepa}$ is 2.57 kcal·mol\textsuperscript{-1}. The binding energies calculated with SCC-DFTB agree well with those of E$_{rela-whole}$ that all the water molecules are considered as a whole part in cluster N{NH$_3$}(H$_2$O)$_{9}$.
|
|
||||||
|
|
||||||
For cluster {NH$_3$}(H$_2$O)$_{10}$, 10$^\prime$-a to 10$^\prime$-e are the first five low-energy structures displayed 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 energies of 10$^\prime$-a and 10$^\prime$-b are very close at both 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 that eight water molecules also constitute a cube but its energy is obviously higher than that 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 {NH$_3$}(H$_2$O)$_{10}$.
|
|
||||||
|
|
||||||
The relative binding energies of isomers 10$^\prime$-a to 10$^\prime$-e that the smallest and biggest values of E$_{rela-whole}$ are -0.03 and -1.10 kcal·mol\textsuperscript{-1} while the smallest absolute value of E$_{rela-sepa}$ is 4.80 kcal·mol\textsuperscript{-1} shown in Table \ref{reBindE}. The binding energies calculated with SCC-DFTB agree well with those of E$_{rela-whole}$ that all the water molecules are considered as a whole part in cluster {NH$_3$}(H$_2$O)$_{10}$.
|
|
||||||
|
|
||||||
For the first five low-energy isomers of clusters {NH$_3$}(H$_2$O)$_{1-10}$, in most cases, the NH$_3$ usually has two or three exposed N-H bonds.
|
|
||||||
|
|
||||||
\begin{figure}[H]
|
|
||||||
\includegraphics[width=1.0\linewidth]{figure_ammoni/E-distance-nh4-w-bsse.eps}
|
|
||||||
\centering
|
|
||||||
\caption{The binding energy according to the distane to distance of N and O atoms in cluster {NH$_4$}$^+$(H$_2$O).}
|
|
||||||
\label{fig:E-distance-nh4-w-bsse.eps}
|
|
||||||
\end{figure}
|
|
||||||
|
|
||||||
|
|
||||||
\begin{figure}[H]
|
|
||||||
\includegraphics[width=1.0\linewidth]{figure_ammoni/E-distance-nh3-w-bsse.eps}
|
|
||||||
\centering
|
|
||||||
\caption{The binding energy according to the distane to distance of N and O atoms in cluster NH$_3$(H$_2$O).}
|
|
||||||
\label{fig:E-distance-nh3-w-bsse.eps}
|
|
||||||
\end{figure}
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
||||||
\section{Conclusions} \label{Concl}
|
|
||||||
aaaa
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|
||||||
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|
||||||
%% The "Acknowledgement" section can be given in all manuscript
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|
||||||
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|
||||||
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|
||||||
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|
||||||
|
|
||||||
\begin{acknowledgement}
|
|
||||||
The authors acknowledge the supercomputing facility of CALMIP for generous allocation
|
|
||||||
of computer resources (projects P1320 and P0059). The authors declare that there has
|
|
||||||
been no significant financial support for this work.
|
|
||||||
\end{acknowledgement}
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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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%\bibliographystyle{achemso}
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\bibliography{biblio}
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\end{document}
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