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Generating Vibronic Coupling Models and Simulating Photoelectron Spectra

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Generating Vibronic Coupling Models and Simulating Photoelectron Spectra Generating Vibronic Coupling Models and Simulating Photoelectron Spectra Submitted as partial completion of the requirements for CHEM 494 By: Julia Endicott 20311750 02-04-2012 Supervisor: Marcel Nooijen Julia Endicott Acknowledgements This project would not have been possible without the support and guidance of Dr. Marcel Nooijen. I would like to thank Dr. Nooijen for the great opportunity he has given me to gain experience doing research in this field. ii Julia Endicott Summary To create accurate spectra the effects of coupling between electronic states must be included. To include this coupling the Born-Oppenheimer approximation must not be assumed. The coupling constants are found by calculating a potential matrix and then doing a Taylor series expansion up to the quartic constant. With these vibronic coupling constants potential energy surfaces are generated for ionized states and photoelectron spectra are simulated. To generate vibronic coupling models, a series of calculations were done using the ACES2 program and the VIBRON program created in Dr. Nooijen's group. These calculations were done for twenty different small molecules to show that the process of simulating spectra that include vibronic effects can be straightforward. The resultant spectra showed improved agreement with experimental spectra when compared to the spectra generated using the Franck-Condon approach. iii Julia Endicott Table of Contents Introduction ..............................................................................................................................1 Theoretical Background: The Born-Oppenheimer Approximation.......................3 Theoretical Background: Vibronic Coupling..................................................................5 Procedures.................................................................................................................................7 Results and Discussion ....................................................................................................... 13 Conclusions............................................................................................................................. 25 Further Work......................................................................................................................... 25 References .............................................................................................................................. 26 iv Julia Endicott List of Figures Figure 1: Molecules modeled.............................................................................................. 2 Figure 2: Example of a conical intersection between three states in methane.................... 5 Figure 3: Potential energy surfaces along normal modes of formaldehyde...................... 14 Figure 4: PES of furan normal mode 15 generated before fitting to scan points.............. 16 Figure 5: PES of furan normal mode 15 after fitting to scan points................................. 16 Figure 6: Comparison of S(q) and q..................................................................................17 Figure 7: PES for furan normal mode 15 using the S(q) substitution................................17 Figure 6: Comparison of photoelectron spectrum of formaldehyde generated using the S(q) substitution and the normal polynomial expansion........................................... 18 Figure 7: Simulated photoelectron spectrum of formaldehyde......................................... 20 Figure 8: Experimental photoelectron spectrum of formaldehyde.6 ................................. 20 Figure 9: Simulated photoelectron spectrum for chloroethylene...................................... 21 Figure 10: Experimental photoelectron spectrum for chloroethylene.7 ............................ 21 Figure 11: Experimental photoelectron spectra of methane.8 ........................................... 22 Figure 12: Simulated photoelectron spectrum of methane. .............................................. 22 Figure 13: PESs for ionized states of methane along three normal modes....................... 22 Figure 14: Photoelectron spectrum of furan simulated using the TDH method............... 24 Figure 15: Photoelectron spectrum of furan simulated using the FC approach................ 24 Figure 16: Experimental photoelectron spectrum of furan. 8............................................ 24 v Julia Endicott Introduction Spectroscopy is a technique widely used by experimentalists to probe molecular properties using electromagnetic radiation such as light. Understanding spectra requires an understanding of the quantum mechanics taking place on the molecular level. This is where theoretical chemists become important. If spectra can be accurately simulated using computational methods then computational tools can be used to predict spectra for a given molecule. The most commonly used method for simulating spectra involves the Born- Oppenheimer and Franck-Condon approximations because they give a relatively simple approach which explains a lot of the features of spectroscopy. However it is also well known that in the case of degenerate or nearly degenerate states the approximation does not accurately predict results. The Born-Oppenheimer approximation assumes that since the relative velocity of the nuclei is much less than the velocity of the electrons the problem can be treated in stages. First the electronic eigenvalues are found at a fixed geometry then the nuclear coordinates are optimized on a particular electronic potential energy surface (PES). A more technical description of the Born-Oppenheimer approximation is given in the next section. The Franck-Condon approximation uses harmonic oscillators to model PESs. Using the same idea that nuclei move much slower than electrons, electrons are assumed to be excited from a region near the ground state geometry. Therefore the most intense peaks in the Franck-Condon spectra are those where the excited state wavefunction overlaps most with the ground state wavefunction. The purpose of this project is to create models which include the effect of coupling between the motions of electrons and nuclei through the vibrational normal 1 Julia Endicott modes of the molecule. This is called the vibronic approach and is explained in detail in the theory section. With this project it is shown that through straightforward calculations in ACES2 and VIBRON vibronic models can be generated for many different molecules. The molecules vibronic models were calculated for are shown in Figure 1. The following report contains two sections on theoretical background, in the first the Schrödinger equation is derived using the Born-Oppenheimer approximation and in the second the mathematical procedure for finding the vibronic coupling constants is described. In the procedure section the details of the calculations used to generate the coupling constants are explained. Next in the results section the vibronic models generated and the methods for doing simulations of spectra are discussed. Finally examples of simulated spectra are compared to experimental spectra. The differences between spectra which include vibronic coupling and those generated using the Franck- Condon approach are discussed. Figure 1: Molecules modeled. 2 Julia Endicott Theoretical Background: The Born-Oppenheimer Approximation The purpose of this project is to create vibronic coupling models but in order to understand the importance of this the Born-Oppenheimer approximation must first be explored in more detail. Using the Born-Oppenheimer approximation means each electronic state is treated independently of all others. Given the Hamiltonian: 1 where: 2 It is clear that the electronic Hamiltonian depends on the nuclear coordinates . The wavefunction is defined as 4 where is an eigenstate of the electronic Hamiltonian and is the nuclear contribution. The Born-Oppenheimer approximation can be derived by expanding the Hamiltonian acting on the wavefunction ⎛ 1 ⎞ ⎛ ⎞ Hˆ r, R 2 Hˆ r, R R Ψ( ) = ⎜ −∑ ∇α + el ⎟ ⎜ ∑ϕλ ( )χλ ( )⎟ ⎝ α 2mα ⎠ ⎝ λ ⎠ 5 ⎡ 2 2 ⎤ 1 χλ (R)∇αϕλ (r, R) + ϕλ (r, R)∇α χλ (R) = ∑ ∑⎢ ⎥ α 2mα λ ⎢+∇ ϕ r, R ⋅∇ χ R + E R ϕ r, R χ R ⎥ ⎣ α λ ( ) α λ ( ) λ ( ) λ ( ) λ ( )⎦ and then integrating against 3 Julia Endicott ⎡ 1 ⎤ = φ* r, R ∇2φ r, R dr χ R ∑⎢∫ µ ( )∑ α λ ( ) ⎥ λ ( ) (A) λ ⎣ α 2mα ⎦ ⎛ 1 ⎞ 2 R +δ µλ ⎜ ∑ ∇α ⎟ χλ ( ) ⎝ α 2mα ⎠ (B) 6 1 + φ* r, R ∇ φ r, R dr ⋅ ∇ χ R (C) ∑∫ µ ( ) α λ ( ) α λ ( ) α mα (D) +δ µλ Eλ (R)χλ (R) = Eδ χ R µλ λ ( ) If the Born-Oppenheimer approximation is used the terms A and C are neglected because ∇ φ r, R for a particular geometry is zero if there is no coupling between the α λ ( ) electronic wavefunction and the nuclear coordinates. The final result is the Schrödinger equation within the Born-Oppenheimer approximation ⎡Tˆ + V R ⎤ χ R = E χ R 7 ⎣ N λ ( )⎦ λ,n ( ) n λ,n ( ) where λ labels the electronic states and n labels the rotational-vibrational levels. Therefore, the Schrödinger equation can be solved at a fixed geometry to find the electronic wavefunction ϕ r, R and the potential energy surface (PES)V R . In the λ ( ) λ ( ) Born-Oppenheimer approximation it is assumed that one point on the PES is enough to describe the state. The Born-Oppenheimer method is commonly used in many areas of chemistry, however is not accurate when there are two electronic states that are close together in energy or when the character of a state changes rapidly. Examples of situations where vibronic coupling is important
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