Methods to simulate photo-excited properties of aromatic molecules by Marcus Peter Taylor A thesis submitted to the University of Birmingham for the examination of DOCTOR OF PHILOSOPHY School of Chemistry University of Birmingham 2018 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. Abstract The study of ultrafast dynamics is crucial for understanding the photophysics of many chemical and fundamental processes. Theoretical models have become vi- tal for providing the rationale for experimental observations. These models require solving the time-dependent Schr¨odingerequation (TDSE). An efficient way for solv- ing the TDSE is the MCTDH method, an algorithm for efficient propagation of wavepackets. A pre-requisite for performing quantum dynamics is the construction of potential energy surfaces (PES), which often require expensive electronic struc- ture calculations, and are further complicated by non-adiabatic features such as conical intersections. In this thesis, the photo-excited properties of a range of aromatic molecules are studied, including chromophores of fluorescent proteins, the most renowned being the green fluorescent protein (GFP). Much focus is therefore given to the photo- chemistry of GFP and its corresponding chromophore, HBDI. The stability of the HBDI anion and its photoelectron spectrum was understood through considering its molecular components, imidazolide and phenoxide, as well as the totally symmetric analogues bis-imidazoloxy and bis-phenoxy. A vibronic coupling Hamiltonian was developed for each molecule to calculate the photoelec- tron spectrum. Good agreement with experimental data was achieved, indicating the fitted surfaces provide accurate PES. From the results, the contribution of the phenoxide and imidazolide ring to the photophysics of HBDI was determined. A simple linear vibronic coupling Hamiltonian has been developed to calculate the photoelectron spectra for the two lowest bands of phenol. The obtained spectra were in good agreement with experiment, however an alternative assignment of the vibrational fine structure is proposed. The existence of a conical intersection be- tween D0 and D1 was found and its effect on the photodynamics determined from 0.0 diabatic state populations. The ion surfaces, along with those for the excited states, are necessary for modelling time-resolved photoelectron spectra to complement ex- perimental data. To ensure accuracy of the S1 and S2 surfaces, the absorption spectrum of phenol was also calculated. Two alternative methods for calculating PESs are then investigated. The first uses gradients and second derivatives from Hessian calculations to express the poten- tials for all normal modes as displaced harmonic potentials. This methodology was applied to calculating the absorption and emission spectra for a range of anthracene derivatives and linear polyacenes. The success and limitations of the model are eval- uated. The second is the recently developed DD-vMCG method, where Gaussian wavepackets are propagated on PES calculated “on-the-fly”. Using this approach, the excited state proton transfer in GFP is studied and the mechanism elucidated. Despite the dynamics not being fully completed, preliminary results provide details on the mechanism, in support of previous studies. 2 \No fair! You changed the outcome by measuring it!" Professor Hubert J. Farnsworth The Luck of the Fryrish Futurama, Season 3, Episode 36, Fox Television, March 11, 2001 Acknowledgements A PhD thesis is not easily accomplished by an individual reliant only on himself. It requires a dedicated supervisor, a supportive group, a loving family and a group of crazy, yet wonderful friends. These last four years have been hard graft, but have also presented me with numerous opportunities to develop as a scientist, but also much more. I am lucky to have been to Paris several times for workshops and visited many cities across the country for conferences. It's funny to think that at one point I was perfectly content with only three years of undergraduate chemistry and here I am now having done many more years. Firstly, a massive thank you to my supervisor, Graham, for his guidance, knowl- edge and enthusiasm. Your generosity with your time, as well as many a pint of beer, has always kept me encouraged and determined over the last few years. The Worth group, which despite undergoing a few personnel changes, have always been friendly, willing to help and have made for some great group outings, conferences and pub \group meetings". One faces many challenges during a PhD, although one that is more unusual is your group and supervisor moving down to London. Luckily, I was taken in by the Johnston Group and all of office 204D, to whom I am very grateful for adopting me and providing an ample supply of laughs and serious discussions about everything, but science. In return, I enlightened you with lots of long group presentations and brilliantly crafted witticisms! Thank you to my surrogate supervisors at Birming- ham, Roy and Sarah. Roy for his patience whilst I explain what I've been doing and Sarah for always taking the time to talk to me. You have both provided a lot of positive and constructive advice and feedback. As for Hydra and the other computers residing in the server room - I hate you all! Many, many hours have been spent managing, fixing and occasionally hitting you. 0.0 I am relieved to finally relinquish all administrative, network and security duties! Good luck to you Chris! For all their love and support, I am deeply indebted to Mum and Dad. I know that the majority of this thesis will seem like inane jargon to you, but hopefully you will enjoy the figures and those \weird, funny looking things," I call molecules! It is with great sadness that Grandma did not see the completion of this thesis, as I know she would have been so proud and would have thoroughly enjoyed perusing through. Your kindness was deeply appreciated. Thank you to all the friends, both new and old, who have helped me through the last few years and kept me sane (or least matched my sanity!). The importance of living somewhere you can relax and feel at ease cannot be overstated, nor can living with friends you can laugh with, rant at and build the occasional igloo. For this reason, I am especially grateful to Phil and Alex. From the walks to and from University, to many late night games, it has always been great fun. Being on a first floor flat, was never going to prevent us growing windowsill fruit and herbs. Thank you also Rupert for being there to listen and talk whenever times were tough. During those moments of self-doubt, you encouraged me to believe in myself and carry on. Your continued support is greatly appreciated. 2 This thesis and the work described in it are entirely my own, except where I have acknowledged either help from a named person or a refer- ence is given to a published source or a thesis. Text taken from another source will be enclosed in quotation marks and a reference will be given. May 26, 2019 List of Publications M. P. Taylor and G. A. Worth, \Vibronic coupling model to calculate the photoelectron spectum of phenol", Chemical Physics, 515, 719 (2018) O. J. Daubney, M. P. Taylor, C. Soto and G. A. Worth, \Expanding the Spec- troscopic Toolbox - Efficient Emission Prediction of Small Polyaromatic Hy- drocarbons", Manuscript in Preparation. G. Christopoulou, J. Woodhouse, M. P. Taylor, G. A. Worth and H. H. Fielding \TR-PES of phenol", Manuscript in Preparation. Contents List of Figures xxii List of Tables xxxi Glossary xxxv 1 Introduction 1 2 Theory 20 2.1 The nuclear Schr¨odingerequation . 20 2.2 Born-Oppenheimer and Adiabatic approximations . 22 2.3 The Diabatic Representation . 25 2.4 Conical Intersections . 27 2.5 Photochemistry . 30 2.5.1 Group Theory and Symmetry . 30 2.5.2 Franck-Condon Principle . 31 2.5.3 Selection Rules . 33 2.5.4 Vibrational Structure . 36 2.6 Vibronic Coupling Hamiltonian . 39 3 Quantum Calculations 42 3.1 Many-electron wavefunction . 42 3.2 Hartree Fock . 45 Contents i Contents 3.3 Configuration Interaction . 49 3.4 Møller-Plesset Perturbation Theory . 51 3.5 Coupled Cluster Theory . 54 3.5.1 Equation of Motion - Coupled Cluster . 59 3.6 Multi-Configurational Self Consistent Field . 60 3.7 Complete Active Space . 62 3.8 Basis Set . 64 3.9 Density Functional Theory (DFT) . 68 3.9.1 Pair Density . 70 3.9.2 Hohenberg-Kohn Theorems . 72 3.9.3 Kohn-Sham Theory . 73 3.9.4 Approximations for the exchange-correlation functional . 77 3.10 Time Dependent Density Functional Theory . 81 3.10.1 Runge-Gross Theorem . 81 3.10.2 Time-Dependent Kohn Sham Theory . 83 3.10.3 Linear Response TDDFT . 84 3.11 Multi-Configuration Time-Dependent Hartree . 85 3.11.1 Time Dependent Hartree . 85 3.11.2 Multi-Configuration Time-Dependent Hartree . 86 3.11.3 Discrete Variable Representation . 88 3.11.4 Memory and Efficiency . 90 3.11.5 Analysis and Spectra . 90 3.12 Direct Dynamics - Variational Multi-configurational Gaussian . 93 3.12.1 Singularities and Matrix Inversion . 95 Contents ii Contents 3.12.2 Direct Dynamics . 97 4 Components of the HBDI chromophore 100 4.1 Introduction . 100 4.2 The model Hamiltonian .