6
Conclusions and Perspectives
“Where shall I start, please, Your Majesty?” he asked. “Begin at the beginning,” the king said gravely, “and go on till you come to the end: then stop.”
L. Carroll
The major modelistic result of this work is the development of a novel methodology to study the photophysics of large nonadiabatic molecular systems, where the Potential Energy Surfaces of interest are described at the harmonic (quadratic in the coordinates) level, also with the inclusion of the Duschinsky effect (Sec. 2.6.4).
In this context, the method was elaborated to define, in a rigorous manner, a partition of the coordinates into blocks, so to originate a hierarchical sequence of effective reduced-dimensionality
Hamiltonians, able to reproduce the short-time dynamics of the system up to a certain time, which
grows when a larger number of effective coordinates is included in the dynamics (Chapter 3). Theoretical and numerical evidence has been given about the property of the hierarchy to nicely describe the short-time dynamics of a Franck-Condon wave packet promoted on the coupled PES, only by the first members of the sequence of blocks.
The developed model allows to describe the effect of frequency changes and Duschinsky rotations (neglected in the Linear Vibronic Coupling model extensively used so far) on the electronic spectra lineshapes, providing an effective route to upgrade the current models for the calculation of electronic spectra in nonadiabatic systems (multiple coupled states), to those nowadays routinely adopted for Born-Oppenheimer (single-state) adiabatic systems. In these latter, such effects, connected to quadratic couplings, are known to play a relevant role, most of all if compared to the precision nowadays achievable by state-of-the-art electronic methods.[31]
Even more relevant is the fact that the proposed methodology also allows to investigate the role of the quadratic terms in the Hamiltonian on the dynamics around Conical Intersections. In fact, having the possibility of tuning the relative energies between the coupled states, and the positions
94 conclusions and perspectives
and energies of the CoIs, the quadratic terms are expected to play a significant role when the states lie close in energy already in the FC region, so that the CoIs are not excessively peaked.
Shifting to numerical applications, in single-state systems (Sec. 5.5) the method allows to reproduce, with few effective coordinates, the exact spectra obtained with time-independent approaches, only available when nonadiabatic effects are negligible.
Then, the methodology was applied to the study of the quantum dynamics and the electronic spectra of the coupled ππ∗/nπ∗ states in gas-phase thymine. The computational scheme shows very good convergence properties, with respect to the number of effective modes included in the dynamics. The results indicate that the nπ∗state is populated on the ultrafast time scale <50 fs and therefore it takes part to the subsequent decay to the ground electronic state. The subps time constant measured in time-resolved experiments (Sec. 1) can thus be assigned to such process. Moreover, it was shown that the absence of a structure related to the nπ∗ state in the spectral envelope is not necessarily indicative of an absence of population and dynamics on that state.
The quadratic terms in the Hamiltonian modify both the electronic absorption spectrum (Sec. 5.7), mainly red-shifting it by≈ 0.2 eV and broadening the vibrational peaks, and the electronic population dynamics (Sec. 5.6.1), increasing the extent of the ππ∗−→nπ∗ population transfer by ≈15%.
Finally, in the present example, the effect of intrastate quadratic couplings (Duschinsky effect) on the dynamics is comparable in magnitude and takes place on the same time scale needed at which the members beyond the first block of the hierarchy for the LVC model are activated. When
such blocks are needed to describe the dynamics of interest, the Duschinsky terms should be taken into account as well.
In the last part of the work, the effect of the coordinates cut away in the reduced hierarchical modes was evaluated by mixed quantum-classical dynamics. Essentially the modes beyond the first nine coordinates of the hierarchy have a minor effect on the dynamics and concur in broadening the UV spectra.
6.1 Next applications: first results
As mentioned several times in this manuscript, the hierarchy of Quadratic Vibronic Coupling Hamiltonians described in this work is a novel methodology, so the results discussed here are only the first applications. Before concluding this work, some preliminar results about two quite different model systems will be briefly sketched, whithout giving too many computational details.
6.1 next applications: first results 95
6.1.1 An anharmonic potential
The model of effective modes described in Chapter 3, allows to define, in a rigorous way, a hierarchy of reduced-dimensionality effective Hamiltonians, for a system of coupled states, once the Hessians of the electronic diabatic states (at a specific geometry) are known. For two coupled states, the first member of the hierarchy is a three-dimensional Hamiltonian.
However, the generalized Lanczos algorithm illustrated in Sec. 3.2 is valid for an arbitrary number of coordinates for the first block. This means that anharmonicities can be treated as well, if these are restricted to few specific collective coordinate.
For example, in many reactions, the ’reaction coordinate’ is a specific combination of normal modes to go from the geometries of the reactants to those of the products. A possible model for the Potential Energy Surface of such system consists in quadratic terms and anharmonic terms only related to the reaction coordinate.
For what concerns the thymine molecule, as anticipated in Sec. 5.1, the true minimum of the
ππ∗ state features a non-planar geometry, as sketched in Fig. 6.1. This minimum can be reached moving along to non-totalsymmetric coordinates, represented by the normal modes of Sπ, at the S2-min geometry, which have imaginary frequencies, denoted Q(1π) and Q(2π).
-Q(π)j ππ∗
nπ∗
Figure 6.1: Schematic sketch of the double well of the ππ∗ state of thymine, along the two
non-totalsymmetric coordinates with negative frequencies
Therefore, to a first approximation, we can concentrate all the main anharmonicities of the Sπ
PES in the first member of the hierarchy, by including the ’effective modes’ Q(1π)and Q(2π) in the first member of the hierarchy, and adding to the intra-state Sπ Hamiltonian the anharmonic terms,
γ1Q (π) 1 4 , γ2Q2(π) 4 , γ12Q (π) 1 2 Q2(π)2 . (6.1)
With this scheme in mind, a hierarchy of Hamiltonian has been generated, starting from five modes (three of them are, of course, the gradients at the FC point) and simultaneously block
6.1 next applications: first results 97
6.1.2 An excitonic coupling problem
Donor-acceptor system where the excitonic coupling may occur have a considerable applicative interest and are suitable for being studied by means of quantum dynamics, in order to understand the vibronic mechanism by which an electronic excitation moves from one site to another.
As an example, we report here the results of quantum dynamical spectral simulations of a model system to simulate an anthracene dimer (the ’real’ measured system is depicted in Fig. 6.4),[67]) where the two monomers lie on planes formig an angle of 135° (this angle is actually important since it affects the absorption spectrum).
In this case the system is constituted by a couple of equal chromophores, and we have the same normal modes (66 in total) for each monomer. For this system we have two electronic states, |1iand|2iwhich are degenerate by definition, corresponding to excitations localized on the first and the second monomer, respectively. Treating the PESs at the Adiabatic Hessian level, it has been considered that for each of these two states only the normal modes of corresponding excited monomer have a nonzero gradient at the FC point. The coupling has been simply modelled with a constant, given by half the energy difference between the adiabatic states of the process at the FC geometry, computed at the TD-PBE0/svp level.
This simple model does not take into account the relative orientation between the monomers, which affects the spectrum. In fact when the excitonic transfer occurs, the spectrum becomes sensitive to the angle β between the transition dipole moments of the two monomers. In particular, it can be proven that the effect of the relative orientation can be recovered within the model adopted here, by considering an initial excitation on the state|1iand calculating the spectrum as the Fourier transform of the autocorrelation function[40]
A(t) = hΦ, 0|Φ, ti +cos βhΦ, 0|1ih2|Φ, ti. (6.2) Thus, the hierarchical method has been applied to define, for each monomer, the first effective coordinate and the other ones, by tridiagonalization of the Hessian of the excited states.
In Fig. 6.3, the convergence results for the absorption spectra are reported,[68] for the case where the excitonic coupling is absent (single monomer) and that where a donor → acceptor transfer occurs (dimer). For this molecule, the hierarchy works particularly well, since using 4 (for the single monomer) or 6 (for the dimer) effective modes, we get the converged spectra.
Some differences arise between the converged spectra (Fig. 6.4), that can be ascribed to the excitonic coupling. In essence the peaks in the dimer are broader and less defined. Even some
6.2 some ideas for future work 99
6.2 Some ideas for future work
• Inclusion of the quadratic terms in the diabatic couplings. In many quantum dynamical calculations reported so far in literature such terms were not included in the Hamiltonian. Their role on the dynamics around a conical intersection and the way how to include them in hierarchical models still need to be partially addressed. As mentioned in Chapter 3, the determination of such parameters is now at hand at the TD-DFT level.[41]
• Improvement and development of methods for wave packet propagation. Efficient codes for quantum dynamics have been developed, starting for the 90s, at the Theoretical Chemistry Group of the University of Heidelberg.[69] However, all the codes used in this work have been developed at the Theoretical and Computational Chemistry Lab of the ICCOM-CNR in recent times, so they are currently under development. Future work will involve the improvement and the development of methods for quantum dynamics, and the optimization of the codes utilizing the MCTDH method (Sec. 4.4) and its multi-layer version.[70] Moreover, alternative variational approaches, such as the multiconfigurational Gaussian wave packet method[46], need an investigation, since these methods, which describe the evolving wave packet in terms of moving Gaussians, open the route to on-the-fly quantum dynamical simulations, where the knowledge of the PESs is not required priori.
Finally, another interesting task is to describe a chromofore in a complex enviroment, e. g. a protein, using, for example, a molecular mechanics force field: the whole system could be treated with mixed quantum-classical model outlined in this work.
• Inclusion of solvent effects. The solvent can alter the dynamics with either ’static’ effects, i. e. modifying the Potential Energy Surfaces, and with ’dynamical’ effects, for example when solute-solvent hydrogen bonds occurs. Both the effects can be taken into account with the same machinery developed in this work, essentially with two additional calculations: i) the construction of ’solvated’ PESs, for example using an electrostatic continuum solvation model; ii) the inclusion, in the system under study, of some solvent molecules which may be involved in the vibrational dynamics, e. g. hydrogen-bonding water molecules. Besides, the statistical behaviour of the solvent could be taken into account with the inclusion of a properly defined classical ’solvent coordinate’.[71]
• Simulation of time-resolved spectra. Many theoretical developments in this field have been made recently.[5][9] In practice, it is not always simple to define the computational scheme to
100 conclusions and perspectives
reproduce the time-resolved signals (for example, in transient absorption experiments, it is not always clear what state is populated after the pump pulse). When the electronic states involved in the experiments are known (e. g. in fluorescence upconversion experiments), the procedure to compute the two-dimensional signals requires several quantum dynamical calculations.[72][9] Therefore, even for this task, the computational method illustrated here is expected to be useful.
Bibliography
[1] A. Griesbeck, M. Oelgem ¨oller and F. Ghetti, CRC Handbook of Organic Photochemistry and
Photobiology, (CRC Press, 2012)
[2] C. E. Crespo-Hernandez, B. Cohen, P. M. Hare, and B. Kohler, Chem. Rev. 104, 1977 (2004) [3] Conical Intersections: Electronic Structure, Dynamics and Spectroscopy, edited by W. Domcke, D.
R. Yarkony and H. K ¨oppel, (World Scientific, Singapore, 2004)
[4] H. K ¨oppel, W. Domcke, and L. S. Cederbaum, Adv. Chem. Phys. 57, 59 (1984)
[5] S. Mukamel, Principles of Nonlinear Optical Spectroscopy, Oxford University Press, New York, 1999
[6] M. Cho, Two-dimensional optical spectroscopy, CRC Press, 2009 [7] J.C. Tully, J. Chem. Phys. 93, 1061, (1990)
[8] G. Stock and W. Domcke, in Conical Intersections: Electronic Structure, Dynamics and Spectroscopy, edited by W. Domcke, D. R. Yarkony and H. K ¨oppel, chapter 17, p. 739 (World Scientific, Singapore, 2004)
[9] M. F. Gelin, W. Domcke, and D. Egorova, in Computational Strategies for Spectroscopy: from
Small Molecules to Nano Systems, edited by V. Barone, chapter 9, p. 447 (Wiley, Chichester, UK,
2011)
[10] D. Picconi, V. Barone, A. Lami, F. Santoro and R. Improta, ChemPhysChem 12, 1957 (2011) [11] L. S. Cederbaum, E. Gindensperger, and I. Burghardt, Phys. Rev. Lett. 94, 113003 (2005) [12] E. Gindensperger, I. Burghardt, and L. S. Cederbaum, J. Chem. Phys. 124, 144103 (2006) [13] E. Gindensperger, I. Burghardt, and L. S. Cederbaum, J. Chem. Phys. 124, 144104 (2006) [14] E. Gindensperger, H. K ¨oppel, and L. S. Cederbaum, J. Chem. Phys. 126, 034106 (2007) [15] D. Gust, T. A. Moore and A. L. Moore, Acc. Chem. Res. 42, 1890 (2009)
102 bibliography
[16] W. J. Schreier, T. E. Schrader, F. O. Koller, P. Gilch, C. E. Crespo-Hern`andez, V. N. Swami-nathan, T. Carell, W. Zinth and B. Kohler, Science 315, 625 (2005)
[17] F. Santoro, V. Barone and R. Improta, Proc. Nat. Acad. Sci. USA 104, 9931 (2007) [18] C. E. Crespo-Hern`andez, B. Cohen and B. Kohler, Nature, 436, 1141 (2005) [19] F. Santoro, V. Barone and R. Improta J. Am. Chem. Soc., 131, 15232 (2009)
[20] A. L. Sobolewski, W. Domcke and C. H¨attig, Proc. Nat. Acad. Sci. USA 102, 17903 (2005) [21] T. Schultz, E. Samoylova, W. Radloff, I. V. Hertel, A. L. Sobolewski and W. Domcke, Science
306, 1765 (2004)
[22] E. Samoylova, H. Lippert, S. Ullrich, I. V. Vertel, W. Radloff and T. Schultz, J. Am. Chem. Soc.
127, 1782 (2005)
[23] D. Onidas, D. Markovitsi, S. Marguet, A. Sharonov and T. Gustavsson, J. Phys. Chem. B 106, 11367(2002)
[24] P. M. Hare, C. E. Crespo-Hern`andez and B. Kohler, Proc. Nat. Acad. Sci. 104, 435 (2007) [25] Y. Mercier, F. Santoro, M. Reguero and R. Improta, J. Phys. Chem B, 112, 10769 (2008)
[26] L. S. Cederbaum, in Conical Intersections: Electronic Structure, Dynamics and Spectroscopy, edited by W. Domcke, D. R. Yarkony and H. K ¨oppel, chapter 1, p. 3 (World Scientific, Singapore, 2004)
[27] F. Santoro, R. Improta, A. Lami, J. Bloino, and V. Barone, J. Chem. Phys. 126, 084509 (2007) [28] F. Santoro and V. Barone, Int. J. Quant. Chem. 110, 476 (2010)
[29] A. Lami and F. Santoro, in Computational Strategies for Spectroscopy: from Small Molecules to
Nano Systems, edited by V. Barone, chapter 10, p.475 (Wiley, Chichester, UK, 2011)
[30] H. C. Longuet-Higgins and G. Herzberg, Proc. R. Soc. Lond. A 344, 147 (1975)
[31] M. Biczysko, J. Bloino, F. Santoro, and V. Barone, in Computational Strategies for Spectroscopy:
from Small Molecules to Nanosystems, edited by V. Barone, chap. 8, p. 361 (John Wiley & Sons,
2011)
bibliography 103
[33] H. K ¨oppel, in Conical Intersections: Electronic Structure, Dynamics and Spectroscopy, edited by W. Domcke, D. R. Yarkony and H. K ¨oppel, chapter 4, p. 175 (World Scientific, Singapore, 2004) [34] B. Lasorne, M. A. Robb and G. A. Worth, Phys. Chem. Chem. Phys., 9, 3210 (2007)
[35] ´A. Vib ´ok, A. Csehi, E. Gindensperger, H. K ¨oppel, and G. J. Hal´asz, J. Phys. Chem. A 116, 2659, (2012)
[36] D. Picconi, A. Lami and F. Santoro, J. Chem. Phys. 136, 244104 (2012)
[37] J. J. Sakurai, Modern Quantum Mechanics. Revised Edition, (Addison-Wesley Publishing Com-pany, Inc. 1994)
[38] C. Lanczos, Res. Nat. Bur. Stand. 45, 225 (1950)
[39] T. J. Park and J. C. Light, J. Chem. Phys. 85, 5870 (1986)
[40] R. Improta, F. Santoro, V. Barone and A. Lami, J. Phys. Chem. A 113, 15346 (2009)
[41] S. Coriani, T. Kjærgaard, P.Jørgensen, K. Ruud, J. Huh, R. Berger, J. Chem. Theory Comput. 6, 1028(2010)
[42] A. L. Fetter and J. D. Walecka, Quantum theory of many-particle systems, (McGraw-Hill, New York, 1971).
[43] P. Kramer, M. Saraceno, Geometry of the Time-Dependent Variational Principle, Springer, Berlin (1981)
[44] T. J. Park and J. C. Light, J. Chem. Phys. 85, 5870 (1986)
[45] M. H. Beck, A. J¨ackle, G. A. Worth and H.-D. Meyer, Phys. Rep. 324, 1 (2000)
[46] H.-D. Meyer, F. Gatti and G. A. Worth Eds., Multidimensional Quantum Dynamics. MCTDH
theory and applications, (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2009)
[47] P. A. M. Dirac, Proc. Cambridge Philos. Soc., 26, 376 (1930)
[48] G. C. Schatz and M. A. Ratner, Quantum Mechanics in Chemistry, Dover Publications, Inc., Mineola, New York (2002)
104 bibliography
[50] J. J. Szymczak, M. Barbatti, J. T. Soo Hoo, J. A. Adkins, T. L. Windus, D. Nachtigallova and H. Lischka, J. Phys. Chem. A, 111, 8500 (2007)
[51] K. A. Kistler and S. Matsika, J. Phys. Chem. A, 113, 12396 (2009)
[52] J. M. Olsen, K. Aidas, K. V. Mikkelsen and J. Kongsted, J. Chem. Theory Comput., 6, 249 (2010) [53] Gaussian 09, Revision A.2, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, ¨O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009. [54] W. H. Press, S. A. Teulosky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in Fortran 77.
The art of Scientific Computing, Cambridge University Press (1992) Chapter 16, p. 718
[55] F. Santoro, FCClasses: a Fortran 77 code: visit http://village.pi.iccom.cnr.it/en/Software [56] V. Barone, J. Bloino, M. Biczysko, and F. Santoro, J. Chem. Theory and Comp. 5, 540 (2009) [57] D. Asturiol, B. Lasorne, G. A. Worth, M. A. Robb and L. Blancafort, Phys. Chem. Chem. Phys,
12, 4949 (2010)
[58] S. Perun, A. L. Sobolewski and W. Domcke, J. Phys. Chem. A, 113, 12686 (2009)
[59] H. R. Hudock, B. G. Levine, A. L. Thompson, H. Satzger, D. Townsend, N. Gador, S. Ullrich, A. Stolow and T. J. Martinez, J. Phys. Chem. A, 111, 8500 (2007)
[60] F. Santoro, R. Improta and V. Barone, Theor. Chem. Acc., 123, 273 (2009) [61] L. B. Clark, G. G. Peschel and I. Tinoco Jr., J. Phys. Chem., 69, 3615 (1965)
[62] T. Gustavsson, N. Sarkar, ´A. B´any´asz, Dimitra Markovitsi and R. Improta, Photochem. Photobiol.,
bibliography 105
[63] A. J. Jerri, The Gibbs phenomenon in Fourier Analysis, Splines and Wavelet Approximation, (Kluwer, Dordrecht, 1998)
[64] A. Raab, G. A. Worth, H.-D. Meyer, and L. S. Cederbaum, J. Chem. Phys. 110, 2 (1999) [65] H. K ¨oppel, Chem. Phys. 77, 359 (1983)
[66] M. D ¨oscher and K ¨oppel, Chem. Phys. 225, 93 (1997)
[67] N. Harada, Y. Takuma and H. Uda, J. Am. Chem. Soc., 100, 13 (1978)
[68] D. Padula, D. Picconi, G. Pescitelli, A. Lami, L. Di Bari and F. Santoro, in preparation [69] H.-D. Meyer, WIREs Comput. Mol. Sci., 2, 351 (2012)
[70] H. Wang and M. Thoss, J. Chem. Phys., 119, 1289 (2003) [71] I. Burghardt and B. Bagchi, Chem. Phys. 329, 343 (2006)
Acknowledgements
It is necessary, at the end of this work, to dedicate some thoughts to the people who have contributed to its production and to those who have accompanied me during my activities in Pisa.
First of all, my deepest gratitude is to my supervisor, Dr. Fabrizio Santoro. He has been supportive and has given me the possibility to explore on my own and pursue many projects, being at the same time a guide, indispensable for not losing the thread of my research.
My co-supervisor , Dr. Chiara Cappelli, has always helpful in listening and giving advice. I am indebted to her for the discussions that helped me sort out the organization and the technical details of my work.
I am also grateful to Dr. Giovanni Granucci for his insightful comments and constructive criticisms, which helped me to focus my ideas.
Very important thanks go to Dr. Alessandro Lami, at ICCOM-CNR, who is the main developer of the numerical domestic codes used in this work and the researcher of the group who had the first ideas about the model introduced in this work. Sandro is one of the smartest people I know and he extensively helped me to introduce my developments in the codes and to interpret the results of the simulations.
I would thank also Prof. Vincenzo Barone, who has been always extremely helpful and open to discuss with me (not only about scientific subjects).
Besides, I am clearly thankful to a lot of familiar, friends and other people, who have been very close to me in these years. I report them quite randomly here (surely I will forgot someone). First of all the family, my parents, my twin brother, his girlfriend, my grandparents, my dogs, uncles and cousins. Then friends and others: those of the CNR, Michele, Daniele, Ilaria, Francisco and Alberto (flawless in his informatic assistance); the flatmates Alessandro (the most talented physicist I know), Federico, Mattia Carlo, Rainaldo and Ascanio and the probable future flatmate in Munich Alessandro; the friends in Pisa Ilario, il Cera, Filippo, Franco, Anastasia, Beatrice, Giada, Francesca, Delio, Paolo, Mauro, Roberto, Lorenzo, Valeria, Tilde, Marialaura; those from Tuscania, the Gesk, Dario, Fausto Maria, Federico called ’Stendardi’, Gabriele; those from SNS, Lin, Antonio, Michael Churchill, ilBolzo, the Alf, the Soba, Gabbo, Suzzi, Simo Surdi, Claudia, Fedecaspa, Mariolone,
108 bibliography
Darius Russian, Kerrison, Ugo, Filippo Sala, and many others; the volleyball team Matteo Vezza, Rigo, Tap, Gianmarco, Ilenia, Ida, Francesca, Ursic, Lorenzo; then some other random people and things: the bricklayers who arrived here from Timpano, Le Piagge, the pizza delivery boys, Sylvester Stallone, L’Orzo Bruno, the ice-cream, Benny Lava, Maccio Capatonda, the Higgs boson, Rainaldo House, the ’Pandino’ without headlamps, Bologna, Francesco Guccini, Dylan Dog, Tex, Diabolik and the others, the Winter School in Helsinki (at -20°C), the guys of Munich, the national soccer team, Mario Balotelli, the thieves who stolen two bikes in a night, the owner of the flat and the cats in general.
Affectionate thanks go to Chiara, who bears me for more than two years, and who is always an important and tender support.
Finally, I would like to thank all those people I met, even only briefly, during my studies and research experiences, with whom I often exchanged just an idea or a smile. In some sense, they all have concurred to my formation, stimulating me toward new challenges and suggesting, directly or indirectly, the way to face them.
Pisa, July 2012
