Quantum Chemistry for Spectroscopy { A Tale of 1 Three Spins (S = 0, 2, and 1) by Bryan Matthew Wong Submitted to the Department of Chemistry in partial ful¯llment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2007 °c Massachusetts Institute of Technology 2007. All rights reserved. Author.............................................................. Department of Chemistry May 18, 2007 Certi¯ed by. Robert W. Field Haslam and Dewey Professor of Chemistry Thesis Supervisor Accepted by . Robert W. Field Chairman, Department Committee on Graduate Students This doctoral thesis has been examined by a Committee of the Department of Chemistry that included Professor Jianshu Cao (Chairperson) Professor Troy Van Voorhis Professor Robert W. Field (Thesis Supervisor) 2 Quantum Chemistry for Spectroscopy { A Tale of Three 1 Spins (S = 0, 2, and 1) by Bryan Matthew Wong Submitted to the Department of Chemistry on May 18, 2007, in partial ful¯llment of the requirements for the degree of Doctor of Philosophy Abstract Three special topics in the ¯eld of molecular spectrocopy are investigated using a variety of computational techniques. First, large-amplitude vibrational motions on ground-state singlet (S0) potential energy surfaces are analyzed for both the acety- lene/vinylidene and the HCN/HNC isomerization systems. Electronic properties such as electric dipole moments and nuclear quadrupole coupling constants are used as diagnostic markers of progress along the isomerization path. Second, the topic of electronically excited triplet states and their relevance to doorway-mediated intersys- tem crossing for acetylene is considered. A new diabatic characterization of the third triplet electronic state, T3, enables a vibrational analysis of data obtained from current and past experiments. The last part of this thesis reviews the techniques and ideas of electron-molecule collisions relevant to Rydberg states of diatomic molecules. Pre- viously developed and current methods of treating the excited Rydberg electron are evaluated and extended. Each of these three topics in molecular spectroscopy is stud- ied using ab initio approaches coupled with experimental observations or chemically intuitive models. The unique combination of quantum chemistry and spectroscopy stimulates further developments in both theory and experiment. Thesis Supervisor: Robert W. Field Title: Haslam and Dewey Professor of Chemistry 3 4 Acknowledgments It is a pleasure to thank those people who have helped me with various aspects of this thesis. First and foremost, I want to express my gratitude to Prof. Robert W. Field for devoting an enormous investment of time and patience to teaching me about the spectroscopy of small molecules. I would like to thank Bob for constantly reminding me that there is often simplicity behind every complexity. His invaluable view and intuition has improved this work in many ways. In addition, several colleagues have provided support in both professional and personal capacities. Adam H. Steeves deserves recognition for writing most of the J. Phys. Chem. B manuscript on large-amplitude motions of S0 acetylene. Additionally, Dr. Hans A. Bechtel was always available to explain and provide the HCN/HNC nuclear quadrupole coupling constants measured from his experiments. I also owe both Adam and Hans a non-research related acknowledgement for helping me \¯nd Aerosmith" (on two separate occasions) for my computer. I acknowledge Dr. Ryan L. Thom and Prof. John F. Stanton for teaching me about excited states of triplet acetylene. I would also like to thank John for several helpful lessons (in the great state of Texas) on obtaining diabatic parameters from quantum chemistry calculations. Dr. Serhan N. Altunata and Dr. Stephen L. Coy deserve credit for introducing me to electron-molecule scattering techniques. I thank Serhan for explaining the methodological subtleties of scattering theory to me whenever I required his help. I am grateful to Kyle L. Bittinger for extremely quali¯ed help with writing programs in perl. An advantage of working in the Field group is the variety of science that I learned, and I would like to thank all the other group members at M.I.T. during my stay. I would like to express particular thanks to Dr. Kate L. Bechtel and Samuel H. Lipo® for their support during and after my thesis defense. Sam organized the celebration after my thesis defense, and Kate wore the M.I.T. beaver mascot costume during my entire thesis defense (it gets very hot in that furry costume). Their encouragement is greatly appreciated. My last few years at M.I.T. were particularly di±cult, and it is appropriate to 5 express my gratitude to those who prayed for me and gave me invaluable advice. Dr. Sumathy Raman, Dr. Oleg A. Mazyar, and Dr. Andrei A. Golosov all helped me ¯g- ure out what I should do during these years and what my future plans should be after my time at M.I.T. I am grateful that they also remain close scienti¯c collaborators. Finally, the completion of my education at M.I.T. would not have been possible with- out my family. Although they did not help me with the scienti¯c details of singlet, triplet, or Rydberg states, they did something far better. They constantly prayed for me and supported me, even when I did not have con¯dence in myself. Extra special thanks goes to Mom, Dad, and my sister, Bonnie. 6 To my family: Mom, Dad, and Bonnie 7 8 Contents 1 Introduction 23 1.1 Motivation . 23 1.2 Outline . 24 2 One-Dimensional Molecular Hamiltonians 29 2.1 Introduction . 29 2.2 Hamiltonian . 31 2.3 Eckart Reduced Inertias . 35 2.4 Pitzer Reduced Inertias . 37 2.5 Examples and Applications . 39 2.6 Conclusion . 46 3 Electronic Signatures of Large-Amplitude Motion in S0 Acetylene 49 3.1 Introduction . 49 3.2 Hamiltonian . 52 3.3 Ab Initio Calculations . 55 3.4 Dipole Moments in the Unsymmetrized Local Mode Basis . 60 3.5 Local Bending in the Polyad Model . 62 3.6 Results . 62 3.7 Evolution of the Dipole Moment . 64 3.8 Assignment of Large-Amplitude Local Bender States . 67 3.9 Conclusions . 70 9 4 The Hyper¯ne Structure of HCN and HNC 71 4.1 Introduction . 71 4.2 Quadrupole Coupling Constants of Nuclei . 73 4.3 Application to the HCN ­ HNC Isomerization System . 75 4.4 Results for HCN and HNC . 78 4.5 Conclusion . 83 5 Valence-Excited States of Triplet Acetylene 85 5.1 Introduction . 85 5.2 Previous Studies on Triplet Acetylene . 87 5.3 A New Ab Initio Study of the T3 Surface . 90 5.4 A Brief Digression on Adiabatic and Diabatic Representations . 92 5.5 The T2/T3 Vibronic Model . 96 5.6 S1/T3 Vibrational Overlap Integrals . 99 5.7 Results for T3 Acetylene . 105 5.8 Conclusion . 109 6 Computational Techniques for Electron-Molecule Scattering 111 6.1 Introduction . 111 6.2 Variational Derivation of the K matrix . 115 6.3 Numerical Evaluation of Integrals . 120 6.4 Representing the Continuum Functions . 122 6.5 Additional Computational Details . 124 1 + 6.6 Test Calculations on the 1sσg4pσu §u State of H2 .......... 126 6.7 Conclusion . 128 7 Developments in Electron Scattering for Polar Molecules 131 7.1 Introduction . 131 7.2 The Electron Scattering Equations . 133 7.3 A Partial Di®erential Equation Approach . 135 7.4 Boundary Conditions Adapted for Long Range Dipoles . 139 10 7.5 The Finite Element Approach . 145 7.6 Extracting the K matrix . 150 7.7 Conclusion . 151 8 Conclusion 153 8.1 Summary . 153 8.2 Future Directions . 154 A Expansion of the Internal Coordinate Hamiltonian 157 B Computer Codes for Calculating Vibrational Overlap Integrals 165 B.1 overlap integral.m . 165 B.2 make overlap table.m . 169 B.3 load acetylene data.m . 171 B.4 b matrix acetylene.m . 176 C Analytical Expressions for One- and Two-Electron Integrals in K matrix Calculations 179 C.1 General Expansion of Cartesian Gaussian Orbitals . 179 C.2 Overlap Integrals . 181 C.3 Kinetic Energy Integrals . 183 C.4 Nuclear Attraction Integrals . 183 C.5 Electron Repulsion Integrals . 185 11 12 List of Figures 2-1 Eckart e®ective inertias (Eq. (2.12)) for six molecules displaying in- ternal rotation compared with those calculated from Pitzer's formulae (Eq. (2.19)). 42 2-2 The lowest 2,500 eigenvalues for the torsional motion of 1,2-dichloroethane. The eigenvalues obtained from the instantaneous Eckart inertias are considerably larger than those obtained from a constant-valued Pitzer inertia at the trans geometry. 43 2-3 (a) Relaxed torsional potential and energies for 1,2-dichloroethane ob- tained at the MP2(full)/6-31G(d) level of theory. Each of the tor- sional eigenvalues is associated with symmetric (red-colored) and anti- symmetric (green-colored) torsional states. (b) Torsionally averaged Eckart inertias, hIEckarti, for the lowest 150 torsional states of 1,2- dichloroethane. The broken line indicates the numerical value of IPitzer = 16.37 amu Aº2 calculated at the trans global minimum. 44 2-4 Reduced moment of inertia calculated by the Powell dogleg algorithm for the HCC bending motion of acetylene. The e®ective inertia in- creases rapidly near 110± when the rightmost hydrogen moves in con- cert with the leftmost hydrogen to form vinylidene. 45 13 3-1 (a) One-dimensional relaxed potential for the acetylene-vinylidene iso- merization as a function of the HCC valence bend angle. A transi- tion state structure is labeled by the number 2, and local minima are denoted by numbers 3 and 4. (b) Spatial positions of the acetylene coordinates color-coded to match their corresponding location on the one-dimensional potential. The rightmost hydrogen retraces its path near the transition state but moves in concert with the leftmost hydro- gen in the red-colored post transition state 2 regions.