| \chapter{Tutorial} |
|
|
| \section{Adsorption isotherm of N$_2$ in a metal-organic framework (MOF), Henry coefficients, enthalpy of adsorption} |
|
|
|
|
| \begin{figure}[H] |
| \centering |
| \subfloat[]{\includegraphics[width=7.5cm]{./Tutorial/MIL-47.jpg}} |
| \subfloat[]{\includegraphics[width=7.5cm]{./Tutorial/MIL-47-2x2x4.jpg}} |
| \caption{(left) The MIL-47 unit cell, $6.8179\times 16.1430 \times 13.9390$ \AA, $\alpha=\beta=\gamma=90^\circ$, (right) the $4\times2\times2$ supercell.} |
| \label{Tutorial-MIL-47.jpg} |
| \end{figure} |
|
|
| We are going to look at adsorption properties of methane in MIL-47 \cite{Barthelet2002}. The MIL-47 structure is shown in Fig. \ref{Tutorial-MIL-47.jpg}. |
| The first step is to select the size of the system. We are going to choose a VDW cutoff of 12 \AA. This implies that all \emph{perpendicular} lengths |
| of the unit cell should be larger than twice the cutoff, i.e.\ 24 \AA. So, as a \emph{minimum}, we should use at least a $4\times2\times2$ unit cells. |
| This requirement follows from the "minimum-image convention" where the interactions are computed with only the closest periodic image. |
|
|
| \begin{table}[t] |
| \begin{tabularx}{\linewidth}{X|X|X|X|X} |
| type & V$^{0+}$ & V$^{2+}$ & V$^{4+}$ & DFT \\ |
| \hline |
| V & 2.67677 & 1.45833 & 1.83592 & 1.68 \\ |
| O1 & -0.986909 & -0.527963 & -0.661157 & -0.6 \\ |
| O2 & -0.712381 & -0.439958 & -0.516643 & -0.52 \\ |
| O3 & -0.693279 & -0.427135 & -0.501819 & -0.52 \\ |
| C1 & 0.00680379 & -0.0146643 & -0.0110838 & -0.15 \\ |
| H1 & 0.0434488 & 0.0574796 & 0.055858 & 0.12 \\ |
| C2 & 0.0116383 & -0.0118276 & -0.00782077 & -0.15 \\ |
| H2 & 0.0444475 & 0.0586772 & 0.0570134 & 0.12 \\ |
| C3 & -0.150672 & -0.0720208 & -0.083311 & 0.0 \\ |
| C4 & 0.605064 & 0.384265 & 0.420426 & 0.56 \\ |
| \end{tabularx} |
| \caption{Obtaining charges for MIL-47: charge equilibration vs. DFT-derived charges.} |
| \label{Tutorial-charges} |
| \end{table} |
|
|
| The positions of the atoms are usually known from experiments, or alternatively can be obtained from QM optimizations. |
| Using the atomic information of the framework only, we can compute the frameworks-mass as 14787.6 g mol$^{-1}$, |
| and we can already compute a few interesting properties. |
| The first is a measurement of how void the structure is. |
|
|
| \begin{center} |
| \shadowbox{ |
| \begin{minipage}[c]{15cm} |
| Exercise 1: go to the sub-directory '\verb+1-Helium-void-fraction+'. Compute the helium void-fraction |
| (look for `\verb+Rosenbluth factor new: 0.608 [-]+' in the output file). |
| \end{minipage}} |
| \end{center} |
|
|
| About 61\% of the structure is empty! |
|
|
| \begin{center} |
| \shadowbox{ |
| \begin{minipage}[c]{15cm} |
| Exercise 2: go to the sub-directory '\verb+1-Helium-void-fraction+'. Add a line '\verb+HeliumVoidFraction 0.61+' below '\verb+Framework 0+', |
| use zero cycles, and check the structural properties (i.e.\ accesible pore volume and loading conversion factors) of the system.' |
| \end{minipage}} |
| \end{center} |
|
|
| We see that the available pore volume is 0.60977519 cm$^3$ g$^{-1}$. This density is important to know, because, using the liquid density, |
| it gives a first approximation of the "maximum" number of molecules in the pore. |
|
|
| Two other very useful properties are the accessible surface and pore-size distribution. |
| \begin{center} |
| \shadowbox{ |
| \begin{minipage}[c]{15cm} |
| Exercise 3: go to the sub-directory '\verb+3-Surface-area+'. Run and compute the surface area. |
| \end{minipage}} |
| \end{center} |
|
|
| The nitrogen surface area of MIL-47 is about 1650 m$^2$ g$^{-1}$. This is much larger than most zeolites, but smaller than most large pore MOFs |
| which can go up to an incredible 5000-7000 m$^2$ g$^{-1}$. |
|
|
| \begin{center} |
| \shadowbox{ |
| \begin{minipage}[c]{15cm} |
| Exercise 4: go to the sub-directory '\verb+4-Pore-size-distribution+'. Run and compute the pore-size-distribution. |
| Plot the output-file in '\verb+PoreSizeDistributionHistogram+' in gnuplot using column 1 vs.\ 3 |
| (\verb+plot 'PoreSizeDistributionHistogram' us 1:3 with lines+) to see what the typical pore sizes are. |
| \end{minipage}} |
| \end{center} |
|
|
| In general, the individual framework atoms are charged, but the overall framework should be charge neutral (or compensated by cations when the |
| framework itself has a net-charge). |
| For the charges there is ambiguity, since charge is not an ab-initio observable. That means that different methods to obtain charges |
| give different answers. For adsorption however, we are not really interested in the charges themselves but rather of their influence on the |
| electrostatic potential energy field \emph{inside} the cavities. The CHELPG-type methods aim to do just that: they optimize the classical |
| point charges on the framework work atoms to match the PES (potential energy surface) computed with ab-initio methods. |
| For crystals, the REPEAT method is a very nice variant taking the periodicity into account \cite{Campana2009}. However, such |
| computations can take several hours (or even days). A fast alternative is "charge-equilibration" \cite{Rappe1991,Wilmer2011b,Wilmer2012}. |
|
|
| \begin{center} |
| \shadowbox{ |
| \begin{minipage}[c]{15cm} |
| Exercise 5: go to the sub-directory '\verb+5-ChargeEquilibration+'. Compute the charges using charge-equilibration |
| for various oxidation states of vanadium (edit the '\verb+pseudo_atoms.def+'-file). The output-charges are written to |
| '\verb+Movies/System_0/Framework_0_final_1_1_1.cif+'. |
| \end{minipage}} |
| \end{center} |
|
|
| In Table \ref{Tutorial-charges} we summarize the results: charge-equilibration can give a good approximation in a matter of seconds (or minutes) |
| provided the charge-expansion is performed around the appropriate oxidation state \cite{Wilmer2012}. |
|
|
|
|
| Next we are going to choose N$_2$ as the adsorbate molecule. Since this is a small molecule, it is probably fine to use the small $4\times2\times2$ supercell. |
| For much larger molecules finite-size effects occur and a larger system should be used. For example, a long chain-molecule could even interacts with |
| \emph{itself} if the system was small, which obviously leads to erroneous results. |
| In order to compute an adsorption isotherm we need the framework positions and charges, a force field for the adsorbate and interactions with the framework. |
| Here we will use a generic force field based on DREIDING and UFF \cite{MayoOlafsonGoddard1990,RappeCasewitColwellGoddardSkiff1992}. |
|
|
| An adsorption isotherm describes the adsorption at a fixed (chosen) temperature as a function of pressure. The first question is to examine the appropriate |
| pressure range for adsorption. |
|
|
| \begin{center} |
| \shadowbox{ |
| \begin{minipage}[c]{15cm} |
| Exercise 6: go to the sub-directory '\verb+6-Adsorption+'. Use a few thousand cycles and determine at what pressure |
| adsorption starts to occur (pressures units are Pascal). Do this for 600K (and if time permits 600K). |
| \end{minipage}} |
| \end{center} |
|
|
| \begin{center} |
| \shadowbox{ |
| \begin{minipage}[c]{15cm} |
| Exercise 7: go to the sub-directory '\verb+6-Adsorption+'. Use 10000 cycles initialization, a few ten-thousand cycles for production |
| and compute 5 to 10 points from the starting pressure to 1000 bar. |
| Put the data in a file and plot the loading vs. pressure in normal scale and in log-scale. |
| \end{minipage}} |
| \end{center} |
|
|
| \begin{verbatim} |
| SimulationType MonteCarlo |
| NumberOfCycles 10000 |
| NumberOfInitializationCycles 10000 |
| PrintEvery 100 |
| RestartFile no |
| |
| Forcefield ExampleMOFsForceField |
| UseChargesFromCIFFile yes |
| |
| Framework 0 |
| FrameworkName MIL-47 |
| UnitCells 4 2 2 |
| HeliumVoidFraction 0.61 |
| ExternalTemperature 300.0 |
| ExternalPressure 100000.0 |
| |
| Component 0 MoleculeName N2 |
| MoleculeDefinition ExampleDefinitions |
| TranslationProbability 0.5 |
| RotationProbability 0.5 |
| ReinsertionProbability 0.5 |
| SwapProbability 1.0 |
| CreateNumberOfMolecules 0 |
| \end{verbatim} |
|
|
| By examing the isotherm, the slope of the curve can be related to the Henry's coefficient. This property can also be conveniently computed |
| by Widom's insertion using a single probe adsorbate and is directly related to the excess chemical |
| potential and the free energy \cite{FrenkelSmit2002}. The Henry coefficient can be computed by |
| \begin{equation} |
| K_H = \frac{1}{R T \rho_f}\frac{\left\langle W \right\rangle}{\left\langle W^{IG}\right\rangle} |
| \end{equation} |
| where $\rho_f$ is the density of the framework, and $\left\langle W \right\rangle$ is the Rosenbluth weight. This weight is in general |
| the Rosenbluth weight when configurational bias is used, and reduces to the Boltzmann factor $\left\langle \exp\left(-\beta\Delta U\right) \right\rangle$ |
| without biasing. The ideal Rosenbluth weight $\left\langle W^{IG}\right\rangle$ is the value for a single molecule in the ideal gas phase and serves |
| as the reference state. |
| \begin{center} |
| \shadowbox{ |
| \begin{minipage}[c]{15cm} |
| Exercise 8: go to the sub-directory '\verb+7-Henry coefficient+'. Compute the Henry coefficient and compare it to the value you |
| obtain from the isotherm at low loading. |
| \end{minipage}} |
| \end{center} |
|
|
| Similarly, the limit of the enthalpy of adsorption can be computed from the limit of using a single adsorbate in the NVT-ensemble |
| The affinity of a molecule with the framework can be expressed as the binding energy, or more general, as the enthalpy of adsorption |
| at infinite dilution $\Delta H$: |
| \begin{equation} |
| \Delta H = \Delta U - RT = \left\langle U_{hg}\right\rangle - \left\langle U_{h}\right\rangle - \left\langle U_{g}\right\rangle - RT |
| \label{Eq: enthalpy of adsorption at zero loading} |
| \end{equation} |
| where $\Delta U$ is the internal energy, and $\left\langle U_{hg}\right\rangle$, $\left\langle U_{h}\right\rangle$, and $\left\langle U_{g}\right\rangle$ |
| are the average energy of the guest molecule inside the host-framework, the average energy of the host-framework, and |
| the average energy of the guest-molecule, respectively. |
| In simulations a common approximation is to assume the framework is rigid, and in this case |
| the enthalpy of adsorption at infinite dilution can be understood to be the difference in internal energy of a single molecule outside |
| and inside the confinement of the host framework. In the limit of zero temperature, the enthalpy of adsorption becomes the binding energy. |
| Note: for rigid molecules $\left\langle U_{g}\right\rangle = 0$. |
| \begin{center} |
| \shadowbox{ |
| \begin{minipage}[c]{15cm} |
| Exercise 9: go to the sub-directory '\verb+8-Heat of adsorption+'. Compute the limit of the enthalphy of adsorption at zero loading. |
| Compare this value to the values from the fluctuation formula computed during the isotherm. |
| \end{minipage}} |
| \end{center} |
| Infinite dilution enthalpy of adsorption $\Delta H$ is related to the Henry's coefficient $K_H$ as |
| \begin{equation} |
| \Delta H = - \frac{\partial \ln K_H}{\partial \beta} |
| \label{Tutorial heat-relation} |
| \end{equation} |
| where $\beta=1/\left(k_BT\right)$ is the inverse temperature, and $k_B$ the Boltzmann's constant. The Henry's coefficient is the slope of the |
| isotherm at zero pressure/loading. |
| \begin{center} |
| \shadowbox{ |
| \begin{minipage}[c]{15cm} |
| Exercise 10 (optional): check relation Eq.\ \ref{Tutorial heat-relation} with Henry's coefficient simulations as a function of temperature. |
| \end{minipage}} |
| \end{center} |
|
|
| \section{NPT density of super-critical CO$_2$, RDF, diffusion} |
|
|
| The density of CO2 at 400 and 100 bar is about 161.53 kg m$^{-3}$, at 500 bar 745.45 kg m$^{-3}$ and at 1000 bar the density is |
| 932.81 kg m$^{-3}$ (NIST chemical database). In this example the density is |
| computed using Monte Carlo in the NPT-ensemble. Given the pressure $P$, the temperature $T$, and the amount of molecules $N$, |
| the density is computed. |
|
|
| \begin{verbatim} |
| SimulationType MonteCarlo |
| NumberOfCycles 50000 |
| NumberOfInitializationCycles 10000 |
| PrintEvery 1000 |
| RestartFile no |
| |
| Forcefield ExampleZeolitesForceField |
| |
| Box 0 |
| BoxLengths 30 30 30 |
| ExternalTemperature 400.0 |
| ExternalPressure 10e5 |
| ComputeMolecularPressure yes |
| |
| VolumeChangeProbability 0.05 |
| |
| Component 0 MoleculeName CO2 |
| MoleculeDefinition ExampleDefinitions |
| TranslationProbability 0.5 |
| RotationProbability 0.5 |
| RegrowProbability 0.5 |
| CreateNumberOfMolecules 256 |
| \end{verbatim} |
|
|
| \begin{center} |
| \shadowbox{ |
| \begin{minipage}[c]{15cm} |
| Exercise 1: go to the sub-directory '\verb+FluidCO2/MC_NPT+'. Verify the three densities listed from NIST experimental data. |
| \end{minipage}} |
| \end{center} |
|
|
| Next we are going to compute two important fluid properties that give inside in the structure of the fluid: the radial |
| distribution function (RDF) and the self-diffusion. |
|
|
| \begin{center} |
| \shadowbox{ |
| \begin{minipage}[c]{15cm} |
| Exercise 2: go to the sub-directory '\verb+FluidCO2/MC_NPT+'. Set the volume to the average of the previous step |
| and switch off the volume move, e.g.\ remove '\verb+VolumeChangeProbability 0.05+'. |
| Also replace '\verb+ComputeMolecularPressure yes+' by\\ |
| ComputeRDF yes\\ |
| WriteRDFEvery 1000\\ |
| Plot and analyze the output rdf's that can be found in the directory '\verb+RadialDistributionFunctions+'. Analyze what the peaks mean. |
| \end{minipage}} |
| \end{center} |
|
|
| Finally, we are going to compute a dynamic property. Therefore, we change '\verb+SimulationType MonteCarlo+' to |
| '\verb+SimulationType MolecularDynamics+' and we are going to use Molecular Dynamics. |
|
|
| \begin{verbatim} |
| SimulationType MolecularDynamics |
| NumberOfCycles 1000000 |
| NumberOfInitializationCycles 1000 |
| NumberOfEquilibrationCycles 10000 |
| PrintEvery 10000 |
| PrintPropertiesEvery 10000 |
| |
| Ensemble NVT |
| TimeStep 0.0005 |
| |
| Forcefield ExampleZeolitesForceField |
| |
| Box 0 |
| BoxLengths 25 25 25 |
| ExternalTemperature 400.0 |
| ComputeMSD yes |
| PrintMSDEvery 5000 |
| |
| Component 0 MoleculeName CO2 |
| MoleculeDefinition ExampleDefinitions |
| TranslationProbability 1.0 |
| ReinsertionProbability 1.0 |
| CreateNumberOfMolecules 100 |
| |
| \end{verbatim} |
|
|
| \begin{center} |
| \shadowbox{ |
| \begin{minipage}[c]{15cm} |
| Exercise 3: Compute the diffusion via the mean-square displacement. Using gnuplot, plot the file |
| '\verb+MSDOrderN/System_0/msd_self_methane_0.dat+'. Use the slope to extract the diffusion coefficient. |
| \end{minipage}} |
| \end{center} |
|
|
| \section{Reaction-ensemble of ammonia} |
|
|
| We are going to study the ammonia synthesis reaction |
| \begin{equation} |
| N_2 + 3H_2 \leftrightarrow 2NH_3 |
| \end{equation} |
| Ammonia ranks second among synthesis chemicals in amount produced, and there has been a great deal of experimental and |
| theoretical research into the ammonia synthesis reaction over the past 100 years. Thus there is an abundance |
| of experimental reference data on this reaction, allowing an accurate check of simulation models \cite{Turner2001}. |
|
|
| One of the most commonly used approaches in molecular simulation is to simulate |
| reaction equilibria in the reaction ensemble (RxMC). In this approach, the chemical reaction is |
| carried out by a Monte Carlo (MC) trial move. Beside thermalization (translation, rotation, etc), |
| trial moves are carried out in which reactants are removed and reaction products are inserted |
| in the system, in such a way that an equilibrium distribution of reactants and reaction products |
| is obtained. The mechanism and the transition state of the reaction are not considered as |
| this approach is purely thermodynamic. As a result, the efficiency of this simulation technique |
| is not affected by the height of the activation energy barrier of the reaction as reaction kinetics |
| are not considered. The RxMC method |
| requires the ideal gas partition functions of all reactant and reaction product molecules, |
| a list of all possible chemical reactions in the system, and an appropriate force field |
| accurately describing interactions between molecules. |
|
|
| \begin{figure}[t] |
| \centering |
| \subfloat[]{\includegraphics[width=7.5cm]{./Tutorial/Ammonia-system.jpg}} |
| \subfloat[]{\includegraphics[width=7.5cm]{./Tutorial/Ammonia-fractional.jpg}} |
| \caption{(left) the N$_2$-3H$_3$-2NH$_3$ system, (right) the fractional molecules involved in the reaction.} |
| \label{Tutorial-reaction-system} |
| \end{figure} |
|
|
| Figure \ref{Tutorial-reaction-system} shows a snapshot of the N$_2$-3H$_3$-2NH$_3$ system. To efficiently |
| perform the reaction we use the reaction-ensemble using continuous fractional component MC. The reaction is performed |
| along a $\lambda$-parameter from 0 to 1, where 0 denotes the full N$_2$-3H$_3$ reactant state for the fractional components and |
| 1 the full product state 2NH$_3$. Using fractional molecules for each component the reaction can be |
| performed gradually. In addition to the usual thermalization moves we have a $\lambda$-move that attempts to |
| change $\lambda$ with three possible outcomes: |
| \begin{enumerate} |
| \item{$\lambda$ remains between 0 and 1.} |
| \item{$\lambda$ goes beyond 1.}\\ |
| We have formed real 2NH$_3$ molecules and choose new fractional molecule (randomly) with a value $\lambda-1$. |
| \item{$\lambda$ goes below 0.}\\ |
| We have formed real N$_2$-3H$_3$ molecules and choose new fractional molecule (randomly) with a value $\lambda+1$. |
| \end{enumerate} |
| The $\lambda$-moves are switched on by the '\verb+ProbabilityCFCRXMCLambdaChangeMove+' input-parameter. We |
| also perform volume moves to impose the pressure using '\verb+VolumeChangeProbability+' option. |
| The example input below defines the box, the 3 components, and the reaction using |
| \begin{verbatim} |
| Reaction 1 3 0 0 0 2 |
| \end{verbatim} |
| which list the stoichiometry of the reactants and the product. So, 1 of component 0 and 3 of component 1 forms 2 |
| molecules of component 2. |
|
|
|
|
| \begin{verbatim} |
| SimulationType MC |
| NumberOfCycles 15000 |
| NumberOfInitializationCycles 10000 |
| NumberOfEquilibrationCycles 20000 |
| RestartFile no |
| PrintEvery 500 |
| |
| ChargeMethod Ewald |
| Forcefield Local |
| CutOffVDW 9.0 |
| CutOffCoulomb 9.0 |
| EwaldPrecision 1e-5 |
| |
| |
| Box 0 |
| BoxLengths 38 38 38 |
| ExternalTemperature 573.0 |
| ExternalPressure 3e7 |
| |
| |
| Reaction 1 3 0 0 0 2 |
| |
| ProbabilityCFCRXMCLambdaChangeMove 1.0 |
| VolumeChangeProbability 0.1 |
| |
| Component 0 MoleculeName N2 |
| MoleculeDefinition Local |
| LnPartitionFunction 208.188 |
| TranslationProbability 40.0 |
| RotationProbability 53.9 |
| ReinsertionProbability 5.0 |
| ExtraFrameworkMolecule no |
| CreateNumberOfMolecules 13 |
| |
| Component 1 MoleculeName H2 |
| MoleculeDefinition Local |
| LnPartitionFunction 93.9084 |
| TranslationProbability 40.0 |
| RotationProbability 53.9 |
| ReinsertionProbability 5.0 |
| ExtraFrameworkMolecule no |
| CreateNumberOfMolecules 39 |
| |
| Component 2 MoleculeName NH3 |
| MoleculeDefinition Local |
| LnPartitionFunction 253.69 |
| TranslationProbability 40.0 |
| RotationProbability 53.9 |
| ReinsertionProbability 5.0 |
| ExtraFrameworkMolecule no |
| CreateNumberOfMolecules 134 |
| \end{verbatim} |
|
|
|
|
| \begin{figure}[H] |
| \centering |
| \subfloat[Experiments]{ |
| \begin{tabular}{l|l|l|l|l} |
| P [bar] & 573K & 673K & 773K & 873K\\ |
| \hline |
| 100 & 0.53 & 0.25 & 0.10 & 0.05\\ |
| 200 & 0.67 & 0.39 & 0.18 & 0.09\\ |
| 300 & 0.75 & 0.48 & 0.25 & 0.13\\ |
| 400 & 0.80 & 0.55 & 0.32 & 0.16\\ |
| 500 & 0.84 & 0.61 & 0.37 & 0.20\\ |
| 600 & 0.87 & 0.66 & 0.42 & 0.24\\ |
| 700 & 0.89 & 0.70 & 0.47 & 0.27\\ |
| 800 & 0.91 & 0.74 & 0.51 & 0.31\\ |
| 900 & 0.93 & 0.77 & 0.55 & 0.34\\ |
| 1000 & 0.94 & 0.80 & 0.58 & 0.37\\ |
| \hline |
| \end{tabular}} |
| \hspace{2cm} |
| \subfloat[Simulations]{ |
| \begin{tabular}{l|l|l|l|l} |
| P [bar] & 573K & 673K & 773K & 873K\\ |
| \hline |
| 100 & 0.56 & 0.27 & 0.12 & 0.05\\ |
| 200 & 0.69 & 0.41 & 0.20 & 0.09\\ |
| 300 & 0.78 & 0.49 & 0.26 & 0.14\\ |
| 400 & 0.82 & 0.57 & 0.32 & 0.17\\ |
| 500 & 0.86 & 0.62 & 0.37 & 0.20\\ |
| 600 & 0.88 & 0.66 & 0.42 & 0.24\\ |
| 700 & 0.90 & 0.69 & 0.45 & 0.27\\ |
| 800 & 0.91 & 0.73 & 0.50 & 0.30\\ |
| 900 & 0.93 & 0.77 & 0.53 & 0.33\\ |
| 1000 & 0.94 & 0.79 & 0.56 & 0.35\\ |
| \hline |
| \end{tabular}} |
| \caption{Mol-fractions of the NH$_3$ in the ammonio bulk phase reaction of N$_2$ and H$_2$ |
| computed from simulation compared to experiments over a wide range of temperatures |
| and pressures.} |
| \end{figure} |
|
|
|
|
| \begin{table}[H] |
| \centering |
| \begin{tabular}{l|l|l|l|l} |
| T [K] & N$_2$ & H$_2$ & NH$_3$ & Eq. constant K$_p$\\ |
| \hline |
| 573 & 2.60E+90 & 6.08E+40 & 1.50E+110 & 0.006327104\\ |
| 673 & 6.89E+77 & 1.28E+35 & 5.42E+94 & 0.000244159\\ |
| 773 & 3.44E+68 & 8.28E+30 & 2.12E+83 & 2.06653E-05\\ |
| 873 & 2.42E+61 & 5.08E+27 & 3.65E+74 & 2.97405E-06\\ |
| \hline |
| \end{tabular} |
| \caption{Input partition function in units of \AA$^3$ and the equilibrium constant K$_p$. |
| The partition functions are computed based on the vibrational and rotational constants reported in the book by McQuarrie \cite{McQuarrie2000}.} |
| \label{Tutorial reaction-input} |
| \end{table} |
|
|
|
|
|
|
| \begin{center} |
| \shadowbox{ |
| \begin{minipage}[c]{15cm} |
| Exercise 1: go to the sub-directory '\verb+Tutorial/ReactionEnsembleAmmonia+'. |
| Using the input-parameters of Table \ref{Tutorial reaction-input} reproduce the simulation results. |
| \end{minipage}} |
| \end{center} |
|
|
| \bibliographystyle{Tutorial/jpc} |
| \bibliography{Tutorial/biblio} |
|
|
