CO 2 molecule. Morse Potential One of the potentials used to simulate chemical bond is a Morse potential of the following form: O C O

Size: px

Start display at page:

Download "CO 2 molecule. Morse Potential One of the potentials used to simulate chemical bond is a Morse potential of the following form: O C O"

Gwen Ford
5 years ago
Views:

CO 2 molecule The aim of this project is a numerical analysis of adsorption spectra of CO2 molecule simulated by a double Morse potential function.

1 CO 2 molecule The aim of this project is a numerical analysis of adsorption spectra of CO2 molecule simulated by a double Morse potential function. In the project you should achieve following tasks: 1. To plot a surface plot of the potential energy landscape of CO2 molecule. Using the plot to analyze and obtain equilibrium conformation of CO2 molecule. 2. To analyze trajectory of C and O atoms within the molecule using Euler s and Verlet algorithm. 3. Using Fourier analysis to obtain fundamental frequencies of CO2 molecule. 4. To apply harmonic approximation to the problem and obtain resonant frequencies. Using this result to calculate asymmetric stretch wavenumber of the adsorption spectra of the CO2 molecule. roc= xo1 - xc roc= xc xo2 O C O xo1 xc xc Morse Potential One of the potentials used to simulate chemical bond is a Morse potential of the following form:

2 V(r) = d(e 2α(r b) 2e α(r b) ) where r interatomic distance b equilibrium interatomic distance d coupling energy (depth of potential well) α the width of the potential well For C-O bond the parameters have following values: d = 7.65 ev b = Å α = 2.5 1/Å Model Let us consider three atomic O=C=O molecule (mo = 16, mc = 12). Each bond is characterized by the Morse potential: O=C bond: V(r OC1 ) = d(e 2α(r OC1 b) 2e α(r OC1 b) ) Where rco1 = xc xo1 is an interatomic distance between left-hand side oxygen atom and carbon atom. C=O bond V(r OC2 ) = d(e 2α(r OC2 b) 2e α(r OC2 b) ) Where rco2 = xo2 xc is an interatomic distance between right-hand side oxygen atom and carbon atom.

3 The total energy of the system is given by a sum of the aforementioned formulas: V(r OC1, r OC2 ) = V(r OC1 ) + V(r OC2 ) Physical properties of the system F F F F CO1 CO1 C02 C02 Force acting on: a) Left oxygen atom: b) Carbon atom: c) Right oxygen atom: F O2 = F CO2 = dv dr CO 2

Units In the simulation you can use SI units or non-dimensional units. If so you need to calculate what the time step in your simulation is in order to associate it with your wave number.

66e 27kg, a. u. = 1.6e 10J, A = 1e 10m So Or the wave number may be calculated using frequency: Project 1. Plot Morse potential for C-O bond. Using the plot identify equilibrium distance.

4 Units In the simulation you can use SI units or non-dimensional units. If so you need to calculate what the time step in your simulation is in order to associate it with your wave number. The relation between time, energy, length and mass in terms of units is given as: If E = ev, M = a.u., L = A then the time step in the simulation is: ev = 1.66e 27kg, a. u. = 1.6e 10J, A = 1e 10m So Or the wave number may be calculated using frequency: Project 1. Plot Morse potential for C-O bond. Using the plot identify equilibrium distance. Interpret the meaning of the α parameter. 2. Determine what are the conditions necessary for adsorption of light by three atomic molecule? How many normal modes does three atomic molecules possess? Which of them are electromagnetically active? Why? 3. Plot potential energy surface for O=C=O bonds (2D function of (rco1, rco2), hint: use meshgrid and surfc function).

4. The thermal energy at room temperature is kt = 0.025 ev, where k is the Boltzmann constant.

5 4. The thermal energy at room temperature is kt = ev, where k is the Boltzmann constant. Determine the amplitude of the vibrational motion at room temperature by reading off the plot of the Morse potential. What is the amplitude at 300, 600 and 900K? 5. Using formulas from p. 3 derive finite difference scheme and apply Euler s algorithm to compute atom s trajectory. As initial conditions set velocities to zero and initial positions far from equilibrium e.g.: xo1 = b 0.15 xc = xo2 = b Plot the system trajectory (current value of (rco1, rco2) and superimpose on the system s potential energy. Can you determine if the system is the harmonic approximation regime? 7. Save obtained trajectory to xyz formatted file. Visualize trajectory in the VMD program. 8. Plot as a function of time: atom s velocities, potential energy, kinetic energy and total energy. What do you observe? Is the total energy of the system is conserved? Why? 9. Euler s algorithm can easily modified into more accurate version (so called Verlet algorithm) by simply adding acceleration dependent part: Apply Verlet algorithm to analyze CO2 molecule. Plot the energy as a function of time. What has changed? 10. Decrease the atom s amplitudes and observe its trajectory. Set initial conditions so the you will obtain asymmetric and symmetric stretch (picture below). Decide (justify your decision) when (from which values of amplitudes) harmonic approximation is valid. Calculate spring constants k OC and k CO.

Figure 1: System of three coupled harmonic oscillators with fixed boundary conditions. 2 (u i+1 u i ) 2. i=0

Figure 1: System of three coupled harmonic oscillators with fixed boundary conditions. 2 (u i+1 u i ) 2. i=0 Department of Applied Physics, Chalmers, Göteborg Martin Gren, Göran Wahnström 014-11-03 E1 Coupled harmonic oscillators Oscillatory motion is common in physics. Here we will consider coupled harmonic