Generating Vibronic 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.

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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 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.

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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

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List of Figures

Figure 1: Molecules modeled...... 2 Figure 2: Example of a 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

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Introduction 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 .

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 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.

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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 are conical intersections and avoided crossings, where two potential energy surfaces approach one another.

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Figure 2: Example of a conical intersection between three states in methane.

Complicated surfaces where the Born-Oppenheimer approximation is not adequate are common in polyatomics especially amongst higher energy states. By including vibronic coupling effects better models can be generated and therefore simulations of spectra will be more accurate.

Theoretical Background: Vibronic Coupling The derivation of the Born-Oppenheimer approximation uses the assumption that

  the φ r, R are eigenstates of the electronic Hamiltonian at a particular set of nuclear λ ( )  coordinates R , these are called the adiabatic states. In order to include vibronic coupling the two terms neglected when using the Born-Oppenheimer approximation must be calculated. However, these terms would be very hard to calculate using the adiabatic states. Since several states are included in the vibronic calculation the adiabatic states do not have to be used as the basis. Instead a combination of states, called the diabatic states,

5 Julia Endicott can be used. The diabatic states are selected to make the term C in 6 as close to zero as possible. The derivation is the same except term D in 6 becomes

       ϕ * r, R Hˆ ϕ r, R dr χ R = E χ R 8 ∫ µ ( ) el λ ( ) λ ( ) µλ λ ( )

With the diabatic states a potential energy matrix, Eµλ , is produced rather than just a potential energy surface. To get the diabatic states a select number of the adiabatic states are calculated and then a linear combination of these states is used to form the diabatic states.      9 ϕa (r, R) = ∑ϕλ (r, R)Uλa (R) λ   where U R is a unitary matrix. To find U R the overlap between the adiabatic λa ( ) λa ( ) states at the ground state geometry and at a geometry slightly displaced along the normal mode qi is found.

    S = ϕ * r, R(q ) ϕ r, R(q + dq 10 µλ ∫ µ ( 0 ) λ ( 0 i ) The unitary matrix is found such that

11

This cannot be done perfectly but it can be done to sufficient accuracy. This optimization essentially means that the diabatic states which change as little as possible with changes in geometry are selected. From this optimization the potential matrix is generated

12

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To get the vibronic coupling constants a Taylor series expansion of the potential energy matrix is done. For example the Taylor series expansion for a system with two electronic states a and b and m normal modes is

m 1 m V (q) = δ E Ei q + Eij q q … 13 ab ab ab ∑ ab i 2 ∑ ab i j i i, j Numerical derivatives are used to get the coefficients. For example the first coefficient is

E q + dq − E (q − dq ) i ab ( 0 i ) ab 0 i 14 E ab = 2dqi

Higher order coefficients can be found using similar steps, usually including up to the quartic coefficients. The coefficients are called the vibronic coupling constants and a vibronic model is a matrix of these constants for each normal mode. An example of the number of constants which this entails can be seen by looking at the file in Appendix 1.

For an extensive review of vibronic coupling theory please see reference 1. The diabatization scheme outlined here was first discussed by Nooijen in reference 2.

Procedures The process of making vibronic coupling models involves several computational steps.

1. ACES2: Geometry optimization and frequency calculation of the ground state.

2. ACES2: IP-EOMCC/STEOM single point calculations.

3. ACES2: Preparation of data for vibronic model.

4. VIBRON: Construct the vibronic model and PESs.

5. VIBRON: Spectra Simulations.

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The steps in the calculation of a vibronic model using formaldehyde (see structure

Fig. 1 top left) will be described as an example. Geometry optimizations are very commonly done and there are many different programs capable of doing this. For this project the optimizations were done in ACESII at the CCSD level with the TZ2P basis set. The frequency calculation was done at the same time as the optimization. From this calculation the optimized geometry, the normal mode frequencies, and the orbital eigenvalues are found. There is also an important file which must be specifically requested, this file contains information about the normal mode coordinates needed in later steps of the calculation and is called the normal_fdif file. The only input is the Z- matrix which gives the program a starting point from which to optimize the geometries.

The next step is the Ionization Potential EOMCC or the STEOM calculation.

When ionized states are of interest the IP-EOMCC calculation is used and the STEOM calculation is used for excited states. Both of these calculations are used to determine which states need to be included in the vibronic model. In these calculations the states to be included are selected by setting a cut-off for the upper and lower bounds for the energy of the states. The cut-offs are placed at the points where there is a jump in orbital eigenvalue energy, to ensure that the number of states in the window remains constant during the calculations. These parameters are much easier to determine for the IP case

8 Julia Endicott than the EE case because the upper limit is always zero and there is usually a clear jump in energies at the lower limit. The input file shown below shows how the cut-offs are implemented in an IP-EOMCC calculation.

The STEOM calculation is more complicated because of course there are many more excited states than ionized states. The lower limit is the same as in the IP case but the upper limit must be chosen more carefully and sometimes several attempts are necessary before an acceptable cut-off is found. The number of states in each symmetry representation must also be indicated in the input file as shown below.

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From this calculation we can check that the given cut-off is acceptable and use the output to determine the input for the next calculation. The acceptability of a given cut-off depends on the % singles and the % active character of each state. When % singles is above 85% and % active is above 95% the cut-offs are acceptable and the calculation of the model can continue. This step is crucial to getting good results and finding a cut-off that works can be a difficult task. More information on using the STEOM program can be found on the course webpage for Introduction to with Marcel

Nooijen.

The next calculation is the actual calculation of the vibronic model. We use the same limits that we tested using the IP-EOMCC/STEOM calculations but for the excited states we need to add more restrictions. We create another window which includes only the excited states of interest. The lower limit is a bit below the energy of the lowest excited state and the upper limit is chosen to include a few of the excited states but cuts- off at a point where there is a jump in energy. We also need to indicate the symmetry of the excited states, which will be included in the calculation. This is not totally straightforward because the symmetry of the molecule will change as it is distorted during the calculation. To allow for the collapse of the symmetry we must have enough states in the first symmetry block to account for all the states in the window, and enough states in each higher symmetry block to account for all the states in still higher symmetry blocks. In the IP case the limits are set to 0 and 200 since the number of ionized states is determined by the ip_low threshold. There is no symmetry requirement for the IP calculation. Below there is an example of an excitation energy vibron calculation on the

10 Julia Endicott left and an ionization potential vibron calculation on the right. The normal_fdif file mention earlier is also a required input for this calculation.

After this step we do one more calculation to construct the vibronic model from the prep_vib outputs. This calculation is what gives us the cp.auto file shown in

Appendix 1. The cp.auto file contains the coupling constants and is used to calculate potential energy surfaces and to simulate spectra.

To check the validity of our vibronic model plots of the PESs can be compared to scans of the PESs. A scan calculates the single point energy of each state at a series of displacements along a particular normal mode. The examples shown below are for normal mode 5 in hydrogen peroxide. There are two steps to a scan calculation. First a small scan is done to determine the symmetry of the electronic states in the displaced geometry. The input for the small scan is almost identical to the prep_vib input but the

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PREP_VIBRON=ON, and GRID_VIBRON=6 keywords are replaced by the

SCAN_NORMAL=ON keyword. A second file called scan_info is required, this file contains information on the number of normal modes and electronic states that are being scanned, and on the number steps and the step size for the scan. Again the normal_fdif file must also be included.

The small scan gives the symmetries of the displaced geometries so that they can be included in the large scan. The large scan input file shown below shows a scan for a normal mode where at the displaced geometry all five electronic states are in the first symmetry block. The scan_info file must be changed to give more scan points, 10 steps in each direction with a stepsize of 0.5 gives good results.

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The large scan gives an output file which contains all the points calculated for each electronic state at a series of displacements along the normal mode. By plotting the scans and the PESs together the fit of the model to the true value can be ascertained. The use of scans for correcting models which give unbound potentials is discussed in the next section. To fix the model we do a fit calculation which adjusts the cp.auto file so that the coupling constants give a polynomial that fits the points seen in the scan.

Results and Discussion The first step in determining whether the vibronic model has given good results is to look at the PESs. The program used to generate the PESs is called VIBRON and was developed in the Nooijen group. The vibronic model gives us a potential energy matrix for the molecule, by diagonalizing the matrix we can generate the adiabatic PESs. The surfaces can give insight into what is happening to the molecule as it vibrates. Some surfaces show interesting features such as avoided crossings or conical intersections which would not be as accurately described if the vibronic coupling were not taken into account.

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Figure 3: Potential energy surfaces along normal modes of formaldehyde.

An example of the PESs generated from the vibronic model calculated for formaldehyde are shown in Figure 3. The first three graphs are of the symmetric modes and the last three are the asymmetric modes. The first normal mode shows a possible conical intersection between the third and fourth states. The PESs of formaldehyde are well behaved and therefore we can trust that the model is correct. If we wanted to double check the curves or if we thought that there was something wrong with the model we

14 Julia Endicott could do a scan calculation. Sometimes the vibronic model does not give suitable PESs for certain normal modes, an example of this is when the potentials go negative at large displacements, see Figure 4. This problem prevents the simulation of spectra because if the potential is unbounded the calculation of the spectrum will not converge.

When unbounded potentials are seen the VIBRON program can be used to fit the polynomial of the vibronic model to the points calculated in the scan and then the fitted coupling constants are used to simulate the spectra. The unbound potential problem occurs because of a negative quartic constant in the Taylor series expansion which dominates at large q and causes the function to tend towards negative infinity. The original calculation of the model only uses small displacements to determine the equation of the polynomial for the potential, which can cause the quartic constant to be negative when it shouldn’t be. If this is the case then fitting to the larger displacements scan points will give a positive quartic constant and bounded potentials. It is also possible that fitting to the scan points will give a negative quartic constant because the true potential is best described that way. If the fit still gives a negative quartic constant then the fitting procedure will not solve the problem of unbounded potentials.

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Figure 4: PES of furan normal mode 15 generated before Figure 5: PES of furan normal mode 15 after fitting to fitting to the scan points. scan points.

For ionized states fitting to scans is usually an effective way to solve the problem of unbound potentials, but it is a tedious process which involves checking each normal mode and doing scans and fits for all modes with unbound potentials. Each molecule has

3N-6 normal modes which means for a molecule like furan there are 21 normal modes, doing scans for all of these modes slows the process of simulating spectra considerably.

In order to make the calculations more routine a method which can be applied to all normal modes at once is preferable.

For excited states the PESs are more complicated and often have the issue of unbound potentials. Unlike the ionized states the potentials for the excited states are often unbounded even after they have been fit to the scan points. It is the nature of the polynomial expansion which is causing the problem so in order to fix it a substitution which allows the polynomial to follow the scans at small displacements but will not go to infinity is needed.

To try to solve the problem of unbound potentials more efficiently a substitution was made for q in the Taylor series expansion of the potential. The function to be used

16 Julia Endicott needed to follow q around the 0 displacement point, which is most important for simulating spectra but, go to a constant value at large displacements. The function used was

1/3 ⎛ 1 3 ⎞ S(q) = ⎜ arctan λq ⎟ 15 ⎝ λ ( )⎠

Figure 6: Comparison of S(q) and q. Figure 7: PES for furan normal mode 15 using the S(q) substitution.

In Figure 5 the behaviour of the S(q) substitution can be seen to match our desired criteria for a substitution. The result of using the S(q) substitution to fix the unbound potentials in furan is shown on the right of Figure 5. The PES follows the scans for small displacements and at large displacements it goes to a constant. This substitution makes the simulation of photoelectron spectra a more routine process and still provides accurate spectra, see Figure 8.

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Figure 8: Comparison of photoelectron spectrum of formaldehyde generated using the S(q) substitution and the normal polynomial expansion

Once the model has been checked by looking at the PESs, and fitting them to the scans if necessary or using the S(q) substutution, the photoelectron spectrum can be generated. There are several different methods which can be used to generate spectra.

The focus of this work was on generating the vibronic coupling models not on the process of simulating spectra so the details of the methods will be omitted. The method used for the smaller molecules is coded in VIBRON and uses a time-independent method involving a Lanczos diagonalization scheme to simulate the vibronic spectra. This method is very expensive because it scales as ~10N, where N is the number of normal modes. VIBRON can also calculate the Franck-Condon spectra even for large molecules.

To get spectra which include vibronic coupling for large molecules the Multi

Configuration Time Dependent Hartree (MCTDH) method can be used. This method propagates the wave packet over the PES using the time dependent Schrödinger equation.

The MCTDH method is less computationally expensive than time independent programs like VIBRON and a protocol for producing spectra in MCTDH given the vibronic model

18 Julia Endicott calculated in VIBRON has been developed by Yao Li, another undergraduate student in the Nooijen group.

As discussed the spectrum can be calculated using the vibronic model and using the Franck-Condon model. By comparing the two spectra to an experimental spectrum it is possible to see the importance of vibronic coupling. In Figure 7 the spectrum of formaldehyde shows the difference between calculations which include vibronic coupling and those that do not. The lowest energy state shows little difference between the two types of calculation which means that state does not experience much vibronic coupling.

However, the effect of vibronic coupling is clear for the highest electronic state ( at 17-18 eV) where the vibronic simulation shows a broad peak while the Franck-Condon simulation shows a series of sharp peaks. Comparing to the experimental spectrum shows that the vibronic simulation is more accurate.

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Figure 9: Simulated photoelectron spectrum of formaldehyde.

Figure 10: Experimental photoelectron spectrum of formaldehyde.6

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The photoelectron spectrum of chloroethylene seen in Figure 8 is another example of the improved accuracy of the spectrum simulated using vibronic coupling over the

Franck-Condon approach. It is important to note that the Franck-Condon spectrum does give good results for molecules like formaldehyde and chloroethylene. The location of the peaks lines up well with experiment for most peaks and at lower energies the two approaches give almost the same result. However, the improved accuracy can be seen from the lack of extra fine structure for the higher states in the vibronic model, and the capturing of features such as the long tail seen around 14 eV in the vibronic and experimental spectra.

Figure 11: Simulated photoelectron spectrum for chloroethylene.

Figure 12: Experimental photoelectron spectrum for 21 chloroethylene.7 Julia Endicott

Figure 13: Simulated photoelectron spectrum of methane.

Figure 15: PESs for ionized states of methane along three normal modes.

Figure 14: Experimental photoelectron spectra of 8 methane.

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Methane is a good example of a case where vibronic coupling is very important.

The Td symmetry in methane means all of the ionized states are degenerate at the optimized geometry. In Figure 9 the PES for normal mode 1 shows a single curve because this is the symmetric stretching mode in the molecule and so the symmetry of the molecule is not affected by distorting along this mode. In normal mode 2 the effect of breaking the Td symmetry of the molecule is seen. The electronic states exhibit the Jahn-

Teller effect because two states are more stable at displaced geometries. Normal mode 8 is an example of a mode that breaks some of the symmetry of the molecule while maintaining at least some of the symmetry because two states are still degenerate in this case. Comparing the experimental spectrum to the two calculated spectra shows how important vibronic coupling is in this case. The Franck-Condon spectrum has a lot of fine structure while the experiment shows the spectrum should have one broad peak. The vibronic model captures the broadening of the spectrum because it is able to properly describe the crossings of the degenerate states. Here we see the effect of the inaccuracy of the Born-Oppenheimer approximation when there are degenerate states.

All of the spectra shown above were generated using the VIBRON program.

VIBRON is only able to simulate the full photoelectron spectrum for molecules as large as chloroethylene. As discussed earlier it is possible to use the models developed in

VIBRON to simulate spectra using MCTDH. In Figure 10 we show a spectra calculated using the TDH method, which is similar to the MCTDH method but only includes one single particle function rather than several. TDH is faster than MCTDH but the spectrum shows small negative intensities due to convergence issues.

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Figure 16: Photoelectron spectrum of furan simulated using the TDH method.

Figure 17: Photoelectron spectrum of furan simulated using the FC approach.

Figure 18: Experimental photoelectron spectrum of furan.8

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Conclusions The method for simulating photoelectron spectra described above has been shown to give spectra which compare well to experiment and give improved results when compared to spectra calculated using the Franck-Condon approximation. Further more, the vibronic models can be calculated through a fairly routine set of calculations. The issue of unbound potentials has been solved in two ways for ionized states, by fitting to scanned single point energies and by substituting S(q) for q in the Taylor series expansion. It is hoped that the S(q) substitution will help in the creation of models for excited states which have unbounded potentials. Spectra have been simulated for the smaller molecules and it has been shown that the MCTDH method can be used to simulate the spectra for larger molecules.

Further Work The most immediate extension of this work is to create models for the excited states of the molecules studied here and to use those models in the simulation of UV visible absorption spectra. We would like the process to be as straightforward for excited states as it is for ionized states. In the more distant future it is hoped that a method similar to this can be used to create accurate models of the PESs in transition metals.

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References

1. Koppel, H.; Domcke, W.; Cederbaum, L. S. Multimode Molecular-Dynamics Beyond the Born-Oppenheimer Approximation. Advances in Chemical Physics 1984, 57, 59-246. 2. M. Nooijen, "First Principles simulation of the absorption spectrum of ketene", Int. J. Quantum Chem. 95, 768-783 (2003). 3. H. Chang, From Electronic Structure Theory to Molecular Spectroscopy. B.A. Thesis. Princeton University, U.S. (2003). 4. Nooijen, M.; Bartlett, R. J. A new method for excited states: Similarity transformed equation-of-motion coupled-cluster theory. Journal of Chemical Physics 1997, 106, 6441-6448. 5. A. Hazra, H. H. Chang, M. Nooijen, First principles simulation of the UV absorption spectrum of ethylene using the vertical Franck- Condon approach, J. Chem. Phys. 121, 2125-2136, 2004. 6. Niu, B. Shirley, D. A. Bai, Y. Daymo, E. “High-resolution He Iα photoelectron spectroscopy of H2CO and D2CO using supersonic molecular beams” Chem. Phys. Lett. 201, 212-216 (1993). 7. Locht, R.; Leyh, B.; Hottmann, K.; Baumgartel, H. “The He(I), threshold photoelectron and constant ion state spectroscopy of vinylchloride”, Chem. Phys. 220, 217-232 (1997). 8. Turner, D. W., “Molecular photoelectron spectroscopy”, Phil. Trans. Roy. Soc. Lond. A. 268, 7-31 (1970).

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Appendix: Example cp.auto file for formaldehyde

Vibronic Coupling elements for case HEFF_IP2 ------

Vibronic grid in ACESII : 6 Total number of vibrational irreps 4 Number of modes per symmetry 3 0 1 2 [1] 3 A1 1562.50 1808.21 2979.22 [3] 1 B2 1199.28 [4] 2 B1 1300.00 3058.88

Gradients parent energy along normal modes ------

Normal mode 1 Gradient :-.5935731906E-07 Normal mode 2 Gradient :0.9015982982E-05 Normal mode 3 Gradient :-.7823018677E-06 Normal mode 4 Gradient :0.0000000000E+00 Normal mode 5 Gradient :0.7105427358E-10 Normal mode 6 Gradient :0.1136868377E-10

Diagonal Hessian parent energy ------

Normal mode 1 Frequency : 1562.48 cm-1 0.193724 eV Normal mode 2 Frequency : 1808.19 cm-1 0.224188 eV Normal mode 3 Frequency : 2979.20 cm-1 0.369376 eV Normal mode 4 Frequency : 1199.28 cm-1 0.148692 eV Normal mode 5 Frequency : 1300.00 cm-1 0.161181 eV Normal mode 6 Frequency : 3058.87 cm-1 0.379254 eV

Reference Hamiltonian 15.983761 0.000000 0.000000 0.000000 0.000000 17.379321 0.000000 0.000000 0.000000 0.000000 10.653803 0.000000 0.000000 0.000000 0.000000 14.504730

Gradients of heff along normal modes ------

Normal mode : 1 Frequency : 1562.50 -0.117668 0.000000 0.000000 0.000000 0.000000 0.417238 -0.049255 0.000000 0.000000 -0.049255 -0.072118 0.000000 0.000000 0.000000 0.000000 0.107009

Normal mode : 2 Frequency : 1808.21 0.323218 0.000000 0.000000 0.000000 0.000000 0.022881 0.409333 0.000000 0.000000 0.409333 -0.006233 0.000000 0.000000 0.000000 0.000000 0.485266

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Normal mode : 3 Frequency : 2979.22 -0.006768 0.000000 0.000000 0.000000 0.000000 0.461602 -0.295391 0.000000 0.000000 -0.295391 0.051206 0.000000 0.000000 0.000000 0.000000 -0.064124

Normal mode : 4 Frequency : 1199.28 0.000000 0.000000 0.000000 -0.197230 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.197230 0.000000 0.000000 0.000000

Normal mode : 5 Frequency : 1300.00 0.000000 0.144927 -0.509812 0.000000 0.144927 0.000000 0.000000 0.000000 -0.509812 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

Normal mode : 6 Frequency : 3058.88 0.000000 0.194289 -0.150823 0.000000 0.194289 0.000000 0.000000 0.000000 -0.150823 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

Diagonal second order corrections of heff ------

Normal mode : 1 Frequency : 1562.50 -0.006026 0.000000 0.000000 0.000000 0.000000 -0.076383 0.068022 0.000000 0.000000 0.068022 -0.052585 0.000000 0.000000 0.000000 0.000000 0.008969

Normal mode : 2 Frequency : 1808.21 0.004681 0.000000 0.000000 0.000000 0.000000 -0.002598 0.057123 0.000000 0.000000 0.057123 0.001092 0.000000 0.000000 0.000000 0.000000 0.057345

Normal mode : 3 Frequency : 2979.22 -0.006514 0.000000 0.000000 0.000000 0.000000 0.039747 -0.020905 0.000000 0.000000 -0.020905 0.021025 0.000000 0.000000 0.000000 0.000000 0.011798

Normal mode : 4 Frequency : 1199.28 -0.020207 0.000000 0.000000 0.000000 0.000000 -0.166654 0.106606 0.000000 0.000000 0.106606 -0.029971 0.000000 0.000000 0.000000 0.000000 0.062416

Normal mode : 5 Frequency : 1300.00 -0.078473 0.000000 0.000000 0.000000 0.000000 -0.093904 0.084135 0.000000

28 Julia Endicott

0.000000 0.084135 0.005922 0.000000 0.000000 0.000000 0.000000 -0.000829

Normal mode : 6 Frequency : 3058.88 0.031442 0.000000 0.000000 0.000000 0.000000 -0.148410 0.062684 0.000000 0.000000 0.062684 0.009879 0.000000 0.000000 0.000000 0.000000 0.020466

Off-diagonal second order corrections heff ------

[1] 2 1808.21 1 1562.50 0.014727 0.000000 0.000000 0.000000 0.000000 0.054029 -0.026522 0.000000 0.000000 -0.026522 0.016052 0.000000 0.000000 0.000000 0.000000 0.010809

[1] 3 2979.22 1 1562.50 -0.012657 0.000000 0.000000 0.000000 0.000000 0.058367 -0.015115 0.000000 0.000000 -0.015115 0.001254 0.000000 0.000000 0.000000 0.000000 -0.000906

[1] 3 2979.22 2 1808.21 -0.011314 0.000000 0.000000 0.000000 0.000000 -0.036891 0.002113 0.000000 0.000000 0.002113 -0.028815 0.000000 0.000000 0.000000 0.000000 -0.023630

[4] 6 3058.88 5 1300.00 0.025082 0.000000 0.000000 0.000000 0.000000 -0.080861 0.028639 0.000000 0.000000 0.028639 0.010515 0.000000 0.000000 0.000000 0.000000 0.008696

Diagonal Cubic corrections of heff ------

Normal mode : 1 Frequency : 1562.50 0.002221 0.000000 0.000000 0.000000 0.000000 -0.015969 0.006465 0.000000 0.000000 0.006465 -0.000829 0.000000 0.000000 0.000000 0.000000 0.001593

Normal mode : 2 Frequency : 1808.21 0.069461 0.000000 0.000000 0.000000 0.000000 0.073323 0.003310 0.000000 0.000000 0.003310 0.071466 0.000000 0.000000 0.000000 0.000000 0.075148

Normal mode : 3 Frequency : 2979.22 0.176413 0.000000 0.000000 0.000000 0.000000 0.167882 -0.001221 0.000000

29 Julia Endicott

0.000000 -0.001221 0.171912 0.000000 0.000000 0.000000 0.000000 0.169400

Normal mode : 4 Frequency : 1199.28 0.000000 0.000000 0.000000 0.013400 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.013400 0.000000 0.000000 0.000000

Normal mode : 5 Frequency : 1300.00 0.000000 0.006312 0.013584 0.000000 0.006312 0.000000 0.000000 0.000000 0.013584 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

Normal mode : 6 Frequency : 3058.88 0.000000 -0.092968 0.042792 0.000000 -0.092968 0.000000 0.000000 0.000000 0.042792 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

Diagonal quartic corrections of heff ------

Normal mode : 1 Frequency : 1562.50 0.000500 0.000000 0.000000 0.000000 0.000000 0.008523 -0.004395 0.000000 0.000000 -0.004395 0.004122 0.000000 0.000000 0.000000 0.000000 0.000536

Normal mode : 2 Frequency : 1808.21 0.019301 0.000000 0.000000 0.000000 0.000000 0.020266 -0.001930 0.000000 0.000000 -0.001930 0.020074 0.000000 0.000000 0.000000 0.000000 0.019882

Normal mode : 3 Frequency : 2979.22 0.062647 0.000000 0.000000 0.000000 0.000000 0.064417 0.006613 0.000000 0.000000 0.006613 0.065207 0.000000 0.000000 0.000000 0.000000 0.065418

Normal mode : 4 Frequency : 1199.28 0.018408 0.000000 0.000000 0.000000 0.000000 0.050873 -0.015797 0.000000 0.000000 -0.015797 0.027176 0.000000 0.000000 0.000000 0.000000 0.013170

Normal mode : 5 Frequency : 1300.00 0.009701 0.000000 0.000000 0.000000 0.000000 0.011580 -0.001494 0.000000 0.000000 -0.001494 0.015274 0.000000 0.000000 0.000000 0.000000 0.011209

Normal mode : 6 Frequency : 3058.88

30 Julia Endicott

0.069878 0.000000 0.000000 0.000000 0.000000 0.130049 -0.018212 0.000000 0.000000 -0.018212 0.075545 0.000000 0.000000 0.000000 0.000000 0.073663

Reference Transition Moments 0.1000E+00 0.1000E+00 0.1000E+00 0.1000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Gradients of tmom along normal modes ------

Normal mode : 1 Frequency : 1562.50 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Normal mode : 2 Frequency : 1808.21 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Normal mode : 3 Frequency : 2979.22 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Normal mode : 4 Frequency : 1199.28 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Normal mode : 5 Frequency : 1300.00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Normal mode : 6 Frequency : 3058.88 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Reference Magnetic Transition Moments 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Gradients of tmag along normal modes ------

Normal mode : 1 Frequency : 1562.50 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

31 Julia Endicott

0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Normal mode : 2 Frequency : 1808.21 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Normal mode : 3 Frequency : 2979.22 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Normal mode : 4 Frequency : 1199.28 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Normal mode : 5 Frequency : 1300.00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Normal mode : 6 Frequency : 3058.88 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

32