Development of a General Time-Dependent Approach for the Calculation of Vibronic Spectra of Medium- and Large-Size Molecules

Universita` di Pisa Facolta` di Scienze Matematiche Fisiche e Naturali Corso di Laurea Magistrale in Chimica Development of a general time-dependent approach for the calculation of vibronic spectra of medium- and large-size molecules Alberto Baiardi Relatore Controrelatore Prof. Vincenzo Barone Prof.ssa Benedetta Mennucci Anno Accademico 2013/2014 Contents Chapter 1. Introduction 5 Chapter 2. General theory and time-independent approach to vibronic spectroscopy 11 1. Semiclassical theory of light absorption 11 2. General theory of resonance Raman spectroscopy 16 3. Harmonic theory of molecular vibrations 23 4. Harmonic theory of molecular vibrations in internal coordinates 26 5. Calculation of the transition dipole moment 29 6. Extension to internal coordinates system 32 7. Beyond the harmonic approximation 35 Chapter 3. Time-dependent theory 39 1. General considerations on time-dependent approaches to molecular spectroscopies 39 2. Time-dependent formulation for OP spectroscopies 41 3. The null temperature singularity 51 4. Extension to the RR case 53 5. Simplified formulations 60 6. Band-broadening effects 62 Chapter 4. Implementation 67 1. Implementation of TD-OP and TD-RR 67 2. Bandshape reproduction 72 3. From redundant to non-redundant internal coordinates 76 1 2 Contents Chapter 5. Applications 91 1. Chiroptical properties of quinone camphore derivatives 92 2. Resonance-Raman spectra of open-shell systems 96 3. Resonance-Raman spectra of metal complexes 102 4. Applications of vibronic spectroscopy in internal coordinates 107 Chapter 6. Conclusions 123 Acknowledgement 3 Chapter 1 Introduction Electronic spectroscopy is nowadays among the most powerful tools for the study of chemical systems. Several techniques have been developed, based either on one- or multiphoton-processes. The former include the widely used optical absorption (OPA) and emission (OPE) spectroscopies, as well as their chiroptical counterparts, electronic circular dichroism (ECD) and circularly polarized luminescence (CPL), which have been developed only recently. On the other hand, multiphoton spectroscopies include more advanced experimental techniques, like two-photon absorption (TPA), two-photon circular dichroism (TPCD) and vibrational resonance Raman spectroscopy (RR). In general, the interpretation of spectroscopic data arising from one-photon spectroscopies is not straightforward, in particular for large and complex systems, and requires the combination of experimental methods with computational approaches to reach a com- plete comprehension of the experimental data. However, from a computational point of view, one-photon electronic spectra are usually simulated by using a single peak for each electronic transition, whose height depends on a specific electronic property (like the dipole strength for one-photon absorption spectroscopy). Band-broadening effects are then simulated by superimposing a symmetric broadening function of this peak. How- ever, in those simplified models, the fine vibrational structure arising from the presence of the vibrational levels of each electronic state is completely neglected. Therefore, they cannot be used to simulate both high-resolution electronic spectra and low-resolution ones where an asymmetrical bandshape is present due to the convolution of all the vibronic transitions. 5 6 Introduction Two parallel routes have been followed for the simulation of vibronic spectra, the time-dependent (TI) and the time-dependent (TD) methods. In the time-independent method, usually called also sum-over-state method, the spectrum is obtained as the en- semble of all transitions between the vibrational initial and final states treated indepen- dently from one another, and the intensity of the transition is calculated by computing the so-called Franck-Condon factor for each vibronic transition. This method suffers from different fundamental problems, especially when dealing with large systems. First of all, due to the huge number of transitions to take into account, ad hoc schemes must be employed to select a priori only the transitions, which actually contribute to the spectrum band-shape. Such models are difficult to automatize and apply on any system, since they generally rely on semi-empirical tests. Furthermore, the computational cost of TI calculations grows quickly when temperature effects are taken into account, since vibronic transitions starting from a large number of initial states are possible. The shortcomings of the time-independent approach can be overcome by using a TD model. By working in the time domain, it is possible to express specific experimental observable, which depend on the type of spectroscopy, in terms of the Fourier transform of an appropriate correlation function. The correlation function can be calculated by propagating a wavepacket corresponding to the initial state over the potential energy surface of the excited electronic state. Therefore, once a model for the propagation of the wavepacket has been derived, the only approximated step is the numerical Fourier transform. Even if this goal can be accomplished by using ab-initio quantum dynam- ics methods, under the harmonic approximation the time-evolution of the correlation function can be computed analytically, and therefore the spectrum can be obtained di- rectly by Fourier transform. TD models permit to overcome both the shortcomings of the TI approach presented before. First of all, the sum over all the possible final vibrational levels is carried out in the definition of the correlation function by using a general, complex-valued time variable. As a consequence, no schemes are needed to select a-priori the final states to take into account. Furthermore, temperature effects can be included implicitly in the time-evolution of the wavepacket, and therefore the computational costs are independent on the temperature. Even if the TD theory of vibronic spectroscopy has been formulated for the first time around 1980 in the works of Heller and Mukamel[1, 2], the TD approaches which 7 are usually used to compute vibronic spectra usually rely on approximated models. Therefore, the first goal of this project is the development and the implementation of a TD framework for the calculation of OP spectra, complementary to the TI one already available in our group[3], in which the approximations used to simulate the evolution of the correlation function are as much limited as possible. Let us remark that the TI and TD approaches can be combined to profit from their advantages and to minimize their limits. In fact, in the TD route all the vibrational levels are automatically included, together with temperature effets. On the other hand, the TI route allows to assign single vibronic transitions. Once the TD formalism of OP spectroscopy has been developed, the following goal of this project has been to extend its application also to Resonance Raman spectroscopy. As already proven by Dirac in 1927, the simulation of RR spectra is related to the calculation of the transition polarizability tensor and, as for the case of one-photon spectroscopies, both TI and TD approaches have been proposed to compute it. The generalization of the TI approach to RR spectroscopy is rather straightforward, since also the polarizability tensor can be expressed in terms of Franck-Condon integrals[4]. However, the TI approaches to compute the polarizability tensor present the same draw- backs as for the one-photon case, and the computational costs are even higher, since a vibronic calculation must be performed for each peak of the RR spectrum. Even if, as for the one-photon case, by expressing the polarizability tensor in a time-dependent way it is possible to overcome all those limitations, the TD models which have been proposed in the last years are even more approximated than in the TD-OP case. However, the interpretation of RR spectra is far from being straightforward due to the much higher complexity to derive simple selection rules with respect to vibrational spectroscopies (IR and nRR, for instance). Therefore, a theoretical model able to simulate a RR spectrum, in which the approximations are as much limited as possible, is essential to identify unequivocally the various contributions to the experimental spectrum and to facilitate its interpretation. In order to accomplish this goal, the same TD formalism derived by us for one-photon spectroscopies has been used as a reference to derive a new TD formulation of RR spectroscopy at the same level of accuracy of the OP case. 8 Introduction Both the TD-OP and the TD-RR theories have been implemented inside the well- known computational chemistry program Gaussian, within a pre-existing module ded- icated to vibronic spectroscopy. Particular care has been paid to the computational efficiency of the whole procedure, in order to make it usable also for the study of large- size systems. Furthermore, the TD procedure has been automatized as much as possible, in order to make its use possible also for non-specialist users, and its computational efficiency has been improved thanks to the use of different numerical algorithms and to an efficient parallelization of the code. As mentioned above, the TD formalism is well-suited for the study of vibronic spectra of large-size molecules. In fact, all the virtually infinite vibrational levels of both the initial (when temperature effects are included in the simulation) and the final electronic state are taken automatically into account. However, the extension of the TD procedure to large-size systems requires

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