QUANTUM CHEMISTRY PROJECT 2: THE FRANCK CONDON PRINCIPLE

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1 Chemistry 460 Fall 2017 Dr. Jean M. Standard October 4, 2017 OUTLINE QUANTUM CHEMISTRY PROJECT 2: THE FRANCK CONDON PRINCIPLE This project deals with the Franck-Condon Principle, electronic transitions in diatomic molecules, and computation of Franck-Condon factors. In the first part of this project, you will treat the diatomic molecule using the harmonic oscillator approximation as a model for the potential energy curves. In the rest of the project, you will treat the diatomic molecule as a more realistic anharmonic oscillator. DUE DATE The project is due on MONDAY, OCTOBER 23, GRADING Project 2 is worth 50 points and consists of three parts. Part A is worth 20 points, Part B is worth 15 points, and Part C is worth 15 points.

2 2 PART A (20 points): A MODEL FOR ELECTRONIC TRANSITIONS IN DIATOMIC MOLECULES In this part of the project, you will construct a model for electronic transitions in diatomic molecules. Although a more realistic treatment of diatomic molecules is to consider them as anharmonic oscillators, a reasonable starting point is to consider the vibrational motion of the molecules to be represented by harmonic oscillators. In electronic spectroscopy, a molecule absorbs radiation and undergoes a transition from a lower energy electronic state (usually the ground state) to an excited electronic state. During this transition, the vibrational energy of the molecule may also change. This leads to the possibility of many different transitions, all involving a transition from the ground electronic state to the excited electronic state, but with different vibrational levels. Not all of the possible transitions have the same intensity. The Franck-Condon Principle states that the probability I of making a transition from a vibrational level (given by quantum number n) of the ground electronic state to a vibrational level (given by quantum number m) of the excited electronic state is proportional to the square of the overlap integral, S nm. The overlap integral is defined as S nm = ψ g* n (x) ψ e m (x) dx, (1) where x is the bond displacement relative to the equilibrium geometry of the ground state. In Eq. (1), g denotes the ground electronic state and e denotes the excited electronic state. The intensity of a transition in a UV-Vis spectrum is therefore I FCF = S nm 2. (2) The square of the overlap integral is called the Franck-Condon factor (FCF). The intensity of the transition is proportional to this Franck-Condon factor. Even though diatomic molecules behave as anharmonic oscillators, a good first approximation is to represent them as harmonic oscillators. Thus, the potentials for the two electronic states may be defined as and V g (x) = 1 2 k g x 2, (3) V e (x) = ε k e (x b)2. (4) These potentials are shown in Figure 1. The lower (ground state) potential is V g (x) and the upper (excited state) potential is V e (x). The force constants for the ground and excited electronic states are k g and k e, and b is the difference between the locations of the potential minima. The energy of the potential minimum of state e is and amount ε greater than the energy of the potential minimum of state g. The variable x represents the displacement from the equilibrium position of state g.

3 3 Figure 1. Harmonic oscillator models for ground and excited electronic states of diatomic molecules. Since the potentials for the ground and excited electronic states are harmonic oscillators, the wavefunctions are known. For the ground state, the wavefunctions are ψ ng (x) ( ) n = 2 n! )1/ 4 α g x 2 / 2 Hn ( + e 'π * 1/ 2 & α g ( ) αg x, n = 0, 1, 2,.... (5) The functions H n are Hermite polynomials with # µ k g &1/ 2 αg = % 2 (. $! ' (6) Similarly, for the excited electronic state, the wavefunctions are ψ me (x) ( m ) = 2 m! )1/ 4 α (x b )2 / 2 Hm ( e+ e e 'π * 1/ 2 & α ( ) α e (x b), m = 0, 1, 2,..., (7) where # µ k &1/ 2 α e = % 2e (. $! ' (8) In these equations, µ is the reduced mass of the diatomic molecule, µ = where m1 and m2 are the masses of the two atoms. m1 m2, m1 + m2 (9)

4 4 Notice that the excited state wavefunctions are similar to the ground state wavefunctions, but are shifted along the x- axis by the amount b. Now that we have defined the harmonic oscillator model, we will examine the Franck-Condon factors for electronic transitions from the ground to excited electronic states. Follow the instructions below to investigate what happens to the intensities of the transitions when you vary the displacement b and the force constant of the excited state. 1. Construction of Ground Electronic State Wavefunctions Use Microsoft Excel to construct the first four vibrational wavefunctions (n = 0-3) of the ground electronic state, Eq. (5). Assume that the diatomic molecule has a reduced mass µ given by atomic units (a.u.) and a force constant k g equal to 1.0 a.u. Remember that in atomic units,! = 1. In order to construct the wavefunctions, you will have to obtain the forms of the Hermite polynomials. Plot the four wavefunctions as a function of the displacement x with a range in x from 1.0 to 1.0 a.u. Use a stepsize of 0.01 a.u. for the x coordinate. You can check the shapes of the wavefunctions by comparing to figures in your textbook. Turn these graphs in with your project. 2. Construction of Excited Electronic State Wavefunctions Use Microsoft Excel to construct the first four vibrational wavefunctions (m = 0-3) of the excited electronic state, Eq. (7). Again, assume that the diatomic molecule has a reduced mass µ given by a.u. The force constant for the excited state may be different than that of the ground state; however, for this part of the assignment, you will assume the excited state force constant is the same as the ground state force constant, 1.0 a.u.; you will change it later. To construct these excited state wavefunctions, you should include the displacement b. We will initially assume that b=0; however, later you also will change this value. Construct the four excited state wavefunctions with a range in x from 1.0 to 1.0 a.u. These wavefunctions should be exactly the same as the ground state wavefunctions (you do not have to turn in any graphs of them since they are the same as the ground state ones). 3. Calculating Franck-Condon Factors In order to get the intensities of the transitions (to predict what the most intense lines in the UV-Vis spectrum would be), the Franck-Condon factors must be calculated. For the first four vibrational levels of the ground state (n = 0-3) and the first four vibrational levels of the excited state (m = 0-3), there are 16 possible transitions from ground to excited state levels. Using Microsoft Excel, calculate the 16 overlap integrals S nm. Because of symmetry, you only actually have to calculate 10 overlaps, since S nm = S mn. To calculate one value of S nm, in a new column in your Excel workbook, calculate the product ψ g n (x) ψ e m (x) for each value of x. Then, the overlap integral over all space may be approximated by summing up these values and multiplying by the step size, S nm = n n ψ g* n (x) ψ e m (x) dx ψ g n (x i ) ψ e m (x i ) δ = δ ψ g n (x i ) ψ e m (x i ), (10) i=1 i=1 where δ is the step size in x (0.01 a.u.) and we have assumed that the harmonic oscillator wavefunctions are real. Though the integral theoretically ranges from negative to positive infinity, you may integrate from x = 1.0 to 1.0 a.u. since the wavefunctions are nearly zero outside this range. Square the values of the overlaps S nm to obtain the Franck Condon factors (FCFs). Tabulate the 16 FCF values. Indicate which are the most intense transitions. The values that you obtained for the FCFs in this particular case should be very simple. Explain why.

5 5 4. Calculating Franck-Condon Factors for a Shifted Excited State Potential Now, see what happens when the excited state potential is shifted. In this case, set b = 0.2 a.u. in your workbook for the excited state wavefunctions. Note that this changes the wavefunctions of the excited state, but the wavefunctions of the ground state remain the same. Again, for the first four vibrational levels of the ground state (n = 0-3) and the first four vibrational levels of the excited state (m = 0-3), there are 16 possible transitions from ground to excited state levels. Using Microsoft Excel and the same method described in step 3, calculate the 16 overlap integrals S nm in this case. You may again integrate from x = 1.0 to 1.0 a.u. since the wavefunctions are nearly zero outside this range. Square the overlaps S nm to obtain the Franck Condon factors. Tabulate these 16 values. How are they different from the ones that you calculated in step 3? Which are the most intense transitions? In order to help you see why the results are the way they are, a plot of the two wavefunctions involved in the overlap S nm is helpful. For the example overlap integrals S 30, S 31, S 32, and S 33, plot the wavefunctions for the ground and excited states on the same graph. You should have one graph for each overlap (four total graphs for this part). Include these plots with your project. Use these plots to help rationalize and explain the results that you obtained for these and the rest of the overlap integrals. You may find the literature article from the Journal of Chemical Education useful in discussing the results from this step and from step 5 [J. M. Standard and B. K. Clark, J. Chem. Ed. 1999, 76, ]. 5. Calculating Franck-Condon Factors for a Different Excited State Force Constant This time you will observe what happens when the force constant of the excited state potential is reduced. In this case, you should set the displacement b = 0.0 a.u and the excited state force constant k e = 0.25 a.u. All other parameters stay the same. Note that this again changes the wavefunctions of the excited state, but the wavefunctions of the ground state remain the same. As you did in step 4, calculate the 16 overlap integrals S nm for this case. Square the overlaps S nm to obtain the Franck Condon factors. Tabulate these 16 values. How are they different from the ones that you calculated in step 3? Which are the most intense transitions? For the selected overlap integrals S 30, S 31, S 32, and S 33, plot the wavefunctions for the ground and excited states on the same graphs. You should have one graph for each overlap (four total graphs for this part). Include these graphs with your project. From these graphs, rationalize the results that you obtained for the magnitudes of the overlap integrals in this case.

6 6 PART B (15 points): ELECTRONIC SPECTROSCOPY OF DIATOMIC POTASSIUM This part of the project investigates a real diatomic molecular system, K 2. The potential energy functions for two different electronic states will be explored and the vibrational energy levels for each of the states will be calculated. 1. Equilibrium Geometries and Dissociation Energies of the K 2 Electronic States The ground electronic state of K 2 is denoted by the symbol X. The second excited electronic state of K 2 is denoted by the symbol B. To create graphs of the potential energy curves of these states, download the file named "k2pot.xls" from the course web site (this file is available on the course web site under Handouts for Wednesday, October 4). This file contains data for both potential energy curves (on separate worksheets) that may be copied into Microsoft Excel and plotted. The first column of each worksheet gives the K 2 bond length in atomic units and the second column gives the potential energy in cm 1. Plot both potential energy curves on the same graph and include the graph with your project. From the graph, estimate the equilibrium bond length for each electronic state. Using the two values of the equilibrium bond lengths, compute the displacement b in a.u. for K 2. Since b is not zero, we might expect the Franck-Condon factors for K 2 to be qualitatively similar to those obtained in Part A, Step 4. Determine graphically the energy difference between the minima of the two potential energy curves. This value is called T e, and it also corresponds in the harmonic approximation to ε in Eq. (4). Also using the graph, calculate the dissociation energy D e of the X and B states. The dissociation energy is the difference between the dissociation limit at large distance and the minimum of the potential energy curve. Tabulate the values of the equilibrium bond lengths and dissociation energies for each state of K 2 along with the values of b and T e and include in your report. 2. Force Constants of the K 2 Electronic States Using the potential energy curves of the X and B states obtained in step 1, fit the low energy portion of each curve to a parabola centered at the equilibrium distance. That is, if the equilibrium distance is r ex, then a harmonic approximation to the potential energy of the K 2 X state, V X r ( ), has the form V X ( r) = 1 2 k g ( r r ex ) 2, (11) where r is the bond distance and k g is the force constant. Since the B state is shifted up in energy by an amount T e, the potential energy of the K 2 B state, V B r ( ), must have the form V B ( r) = T e k e ( r r eb) 2, (12) where r eb is the equilibrium distance of the B state and k e is the force constant. Use your own discretion in deciding the range of the potential energy to use in fitting to a parabola, but explain how you did it. Once you have fit the potential curves to parabolas, report the force constants that you obtained in units of N/m. The requisite conversion factors are 1 atomic unit of length = 1 bohr = Å = m. 1 atomic unit of energy = 1 hartree = cm 1 = Joule.

7 7 3. Harmonic Frequencies of the K 2 Electronic States Compute the reduced mass µ of K 2 in kg (not kg/mol). Using the force constants determined in step 2, calculate the harmonic vibrational frequencies ν 0 of the X and B states of K 2 using the definition ν 0 = 1 2π 1/ 2 $ k ' & ). (13) % µ ( Convert into units of cm 1 and report the results for the harmonic frequencies using the relation ν 0 = ν 0 c, (14) where ν 0 is the harmonic frequency in units of s 1 and ν 0 is the harmonic frequency in units of cm Vibrational Energy Levels of the K 2 Electronic States The next step in the project is to use a method similar to the Numerov algorithm that you employed in Project 1 to compute the first four vibrational energy levels and wavefunctions of the X and B states of K 2. The algorithm is called the Cooley-Cashion-Zare (CCZ) method. This algorithm will be employed using computer programs on the Linux workstations in SLB 260. Communication with these workstations is through a software package available for Mac OS X called XQuartz and accessed from the Macintosh computers in JH 216. You will first need to use the harmonic frequencies obtained in step 3 to input guesses for the appropriate energy levels. For the X state, the harmonic approximation to the vibrational energy levels is E nx = ν 0X ( n + 1 2), (15) where E nx is the vibrational energy of the X state in cm 1, ν 0X is the harmonic frequency in cm 1 of the X state, and n is the quantum number. For the B state, the harmonic approximation to the vibrational energies is E mb = ν 0B ( m + 1 2), (16) where E mb is the vibrational energy of the B state in cm 1, ν 0B is the harmonic frequency in cm 1 of the B state, and m is the quantum number. Once you have the guesses for the first four vibrational energies of the X and B states using the harmonic approximations, along with the reduced mass of K 2 in amu (g/mol), log on to any of the Macintosh computers in JH 216 using your ULID and password. To start the X11 application, click on the "X" icon in the Dock (or find its icon in the Applications folder). A single white window should appear this is called an "xterm" window or simply a terminal window. To log on to the Linux computer, type the following command in the xterm window: ssh Y che460@host.che.ilstu.edu Here, "ssh" stands for secure shell and this application provides a secure connection to the remote computer, host is the name of one of the Linux computers (the choices are frodo, samwise, gandalf, aragorn, legolas, and gimli). It does not matter which one you use, but if multiple people are working on the assignment at once, it is helpful to select different computers for each person so that the program is not in use simultaneously. Finally, enter the password when prompted; it is "erwin". Once you have done this, you are connected to the Linux computer.

8 8 To run the CCZ program for the X state of K 2, type "k2xccz" in the terminal window. The number of integration points is The minimum and maximum coordinate values (xmin and xmax) should be entered as 1.5 and 10 Å. For the energy guess, enter the value of the energy in cm 1 of the n=0 vibrational level of the X state. Record the converged energy that is listed on the screen. Repeat this process for levels n=1, 2, and 3. Record the energies in cm 1. To determine the first four vibrational energy levels of the B state, type "k2bccz" in the terminal window. The number of integration points is 2001, and the minimum and maximum coordinate values (xmin and xmax) again should be entered as 1.5 and 10 Å. For the energy guess, enter the value of the energy in cm 1 of the m=0 vibrational level of the B state. Record the converged energy that is listed on the screen. Repeat this process for the B state levels m=1, 2, and 3. Record the energies in cm 1. Tabulate the n=0-3 X state and m=0-3 B state vibrational energy levels that you obtained from these calculations, and include a table of these results in your report. PART C (15 points) CALCULATION OF FRANCK-CONDON FACTORS FOR K 2 X B STATE TRANSITIONS Not only were the vibrational energy levels calculated by the computer programs in the previous section, but the wavefunctions also were determined. In this part of the project, you will transfer the wavefunction data files to a Macintosh (and then to your own computer) and use Microsoft Excel to compute the Franck-Condon factors. 1. Vibrational Wavefunctions of the K 2 X State Log on to one of the Macintosh computers in JH 216 as you did in the previous section and start XQuartz. If you are still connected to the Linux computer via the ssh command from the previous section, type "exit" to disconnect. Then, in the X11 terminal window, type the command: sftp che460@host.che.ilstu.edu Here host is the name of the Linux workstation that you used in the previous section (frodo, samwise, etc.). You will then be prompted to enter the password (erwin). To retrieve the wavefunction data files, type the command: mget psi* Here, "psi*" is a generic name that refers to all the wavefunction files. You will then be prompted about whether you actually want to retrieve each file of that type (answer "y" for yes each time). There should be eight wavefuction files in total. The files containing the vibrational wavefunctions of the X state are named "psixvnnj00", where NN is the vibrational level (i.e., 00 for n=0, 01 for n=1, etc.). The files containing the vibrational wavefunctions of the B state are named "psibvmmj00", where MM is the vibrational level (i.e., 00 for m=0, 01 for m=1, etc.). Once you have all the wavefunction files, you may type "quit" and close the XQuartz program. You can then get the wavefunction data files off the Macintosh computer using either a thumb drive or by ing the files to yourself. They are located in the home folder (if you can't find them, see Dr. Standard for assistance). Once you have the wavefunction files, import them into a single Excel workbook. You should then be able to create graphs of all of the wavefunctions. Include graphs of the four vibrational wavefunctions of the X state and the four vibrational wavefunctions of the B state in your project report.

9 9 2. Calculation of Franck-Condon Factors for K 2 Transitions Use Microsoft Excel to calculate the 16 overlap integrals S nm for the K 2 molecule. Use the same approach that you used in Part A of this project, but make sure you use the appropriate step size for the K 2 wavefunctions. Though the overlap integral theoretically ranges from negative to positive infinity, it is safe to use the integration range of x = 1.5 to 10.0 Å since the K 2 vibrational wavefunctions are nearly zero outside this range. Unlike the overlaps that you calculated in Part A, though, this time there is no symmetry, so you have to calculate all 16 overlaps. Square the overlaps S nm to obtain the Franck Condon factors. Tabulate these 16 values. Indicate the most intense transitions. Compare and contrast the Franck-Condon factors for K 2 with those calculated for the harmonic oscillator model in Part A, steps 4 and Comparison with Literature Please refer to the literature paper by B. K. Clark and coworkers [B. K. Clark, J. M. Standard, Z. J. Smolinski, D. P. Ripp, and J. R. Fleming, Chem. Phys. 1996, 213, ]. The Franck-Condon factors give information about the most intense UV-Vis transitions for K 2. These are shown in Figure 9, and are listed in Table 4 of the paper. From the literature results, what is the most intense transition overall? Give the quantum numbers of the ground and excited vibrational levels of the most intense transition. Why is the n=0 m=0 transition not the most intense transition? From the literature, for n=0 what are the most intense transitions? Give the quantum numbers for the excited state corresponding to the two most intense transitions. What about for n=1, 2, and 3? Give the most intense transitions in these cases. Compare the results that you have with the literature study. Do your results agree? 4. Bose-Einstein Condensates One of the reasons that the spectroscopy of diatomic alkali molecules is of current interest is related to their use in the formation of Bose-Einstein condensates. Give a brief definition of a Bose-Einstein condensate. What were the first types of atoms used to form a Bose- Einstein condensate and in what year was a Bose-Einstein condensate first experimentally realized? What sorts of temperatures are required in order for formation of a Bose-Einstein condensate?