Computational Strategies for Spectroscopy: from Small Molecules to Nano Systems

Computational spectroscopy is a rapidly evolving field that is becoming a versatile and widespread tool for the assignment of experimental spectra and their interpretation as related to chemical physical effects. This book is devoted to the most significant methodological contributions in the field, and to the computation of IR, UV-VIS, NMR and EPR spectral parameters with reference to the underlying vibronic and environmental effects. Each section starts with a chapter written by an experimental spectroscopist dealing with present challenges in the different fields; comprehensive coverage of conventional and advanced spectroscopic techniques is provided by means of dedicated chapters written by experts. Computational chemists, analytical chemists and spectroscopists, physicists, materials scientists, and graduate students will benefit from this thorough resource.

• Language: English
• Publisher: Wiley
• Release date: Nov 1, 2011
• ISBN: 9781118008713

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Contributors

Hans Ågren, Theoretical Chemistry, School of Biotechnology, Royal Institute of Technology, SE10044 Stockholm, Sweden

Vincenzo Barone, Scuola Normale Superiore, Piazza dei Cavalieri 7, I56126 Pisa, Italy

Maurizzio Becucci, LENS and Dipartimento di Chimica Ugo Schiff, Polo Scientifico e Tecnologico, Università degli Studi di Firenze, Via N. Carrara 1, I50019 Sesto Fiorentino, Florence, Italy

Malgorzata Biczysko, Scuola Normale Superiore, Piazza dei Cavalieri 7, I56126 Pisa, Italy and Dipartimento di Chimica Paolo Corradini, Università di Napoli Federico II, Complesso Univ. Monte S. Angelo, Via Cintia, I80126 Naples, Italy

Julien Bloino, Scuola Normale Superior, Piazza dei Cavalieri 7, I56126 Pisa, Italy and Dipartimento di Chimica Paolo Corradini, Università di Napoli Federico II, Complesso Univ. Monte S. Angelo, Via Cintia, I80126 Naples, Italy

Giuseppe Brancato, Italian Institute of Technology, IIT@NEST Center for Nanotechnology Innovation, Piazza San Silvestro 12, I56125 Pisa, Italy

Marina Brustolon, Dipartimento di Scienze Chimiche, Università degli Studi di Padova, Via Marzolo 1, 35131 Padova, Italy

Chiara Cappelli, Dipartimento di Chimica e Chimica Industriale, Università di Pisa, Via Risorgimento 35, I56126 Pisa, Italy

Vincenzo Carravetta, CNR—Consiglio Nazionale delle Ricerche, Istituto per i Processi Chimico-Fisici (IPCF), Via G. Moruzzi, I56124 Pisa, Italy

Sonia Coriani, Dipartimento di Scienze Chimiche, Università degli Studi di Trieste Via L. Giorgieri 1, I34127 Trieste, Italy and Centre for Theoretical and Computational Chemistry (CTCC), University of Oslo, Blindern, N0315 Oslo, Norway

Orlando Crescenzi, Dipartimento di Chimica Paolo Corradini, Università di Napoli Federico II, Complesso Univ. Monte S. Angelo, Via Cintia, I80126 Naples, Italy

Wolfgang Domcke, Department of Chemistry, Technische Universität München, D-85747 Garching, Germany

Sabine Van Doorslaer, Department of Physics, University of Antwerp, Universiteitsplein 1 (N 2.16), B-2610 Antwerp, Belgium

Dassia Egorova, Institute of Physical Chemistry, Universität Kiel, D-24098 Kiel, Germany

Jack H. Freed, Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853

Maxim F. Gelin, Department of Chemistry, Technische Universität München, D-85747 Garching, Germany

Roberto Improta, CNR—Consiglio Nazionale delle Ricerche, Istituto Biostrutture e Bioimmagini, Via Mezzocannone 16I, I80134 Naples, Italy

Alessandro Lami, CNR—Consiglio Nazionale delle Ricerche, Istituto di Chimica dei Composti OrganoMetallici, UOS di Pisa, Via G. Moruzzi, I56124 Pisa, Italy

Ermelinda S. M. Maçôas, Centro de Química-Física Molecular (CQFM) and Institute of Nanoscience and Nanotechnology (IN), Instituto Superior Técnico, 1049-001 Lisbon, Portugal

Alfonso Pedone, Scuola Normale Superiore, Piazza dei Cavalieri 7, I56126 Pisa, Italy

Giangaetano Pietraperzia, LENS and Dipartimento di Chimica Ugo Schiff, Polo Scientifico e Tecnologico, Università degli Studi di Firenze, Via N. Carrara 1, I50019 Sesto Fiorentino, Florence, Italy

Antonino Polimeno, Dipartimento di Scienze Chimiche, Università degli Studi di Padova, Via Marzolo 1, I35131 Padova, Italy

Cristina Puzzarini, Dipartimento di Chimica G. Ciamician, Università degli Studi di Bologna, Via Selmi 2, I40126 Bologna, Italy

Nadia Rega, Dipartimento di Chimica Paolo Corradini, Università di Napoli Federico II, Complesso Univ. Monte S. Angelo, Via Cintia, I80126 Naples, Italy

Antonio Rizzo, CNR—Consiglio Nazionale delle Ricerche, Istituto per i Processi Chimico-Fisici (IPCF), Via G. Moruzzi, I56124 Pisa, Italy

Kenneth Ruud, Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Troms , N9037 Troms , Norway

Fabrizio Santoro, CNR—Consiglio Nazionale delle Ricerche, Istituto di Chimica dei Composti OrganoMetallici, UOS di Pisa, Via G. Moruzzi, I56124 Pisa, Italy

Fabio Trani, Scuola Normale Superiore, Piazza dei Cavalieri 7, I56126 Pisa, Italy

Preface

Within the plethora of modern experimental techniques, vibrational, electronic, and resonance spectroscopies are uniquely suitable to probe the static and dynamic properties of molecular systems under realistic environmental conditions and in a noninvasive fashion. Indeed, the impact of spectroscopic techniques in practical applications is huge, ranging from astrophysics to drug design and biomedical studies, from the field of cultural heritage to characterizations of materials and processes of technological interest, and so on. However, the development of more and more sophisticated experimental techniques poses correspondingly stringent requirements on the quality of the models employed to interpret spectroscopic results and on the accuracy of the underlying chemical–physical descriptions. As a matter of fact, spectra do not provide direct access to molecular structure and dynamics, and interpretation of the indirect information that can be inferred from analysis of the experimental data is seldom straightforward. Typically these complications arise from the fact that spectroscopic properties depend on the subtle interplay of several different effects, whose specific roles are not easy to separate and evaluate.

In such a complex scenario, theoretical studies can be extremely helpful, essentially at three different levels: (i) supporting and complementing the experimental results to determine structural, electronic, and dynamical features of target molecule(s) starting from spectral properties; (ii) dissecting and quantifying the role of different effects in determining the spectroscopic properties of a given molecular/supramolecular system; and (iii) predicting electronic, molecular, and spectroscopic properties for novel/modified systems. For this reason, computational spectroscopy is rapidly evolving from a highly specialized research field into a versatile and fundamental tool for the assignment of experimental spectra and their interpretation in terms of basic physical–chemical processes.

The predictive and interpretative ability of computational chemistry experiment can be clearly demonstrated by state-of-the-art quantum mechanical approaches to spectroscopy, which at present yield results comparable to the most accurate experimental measurements. In this book several examples of such highly accurate studies with particular reference to rotational spectroscopy or electronic transitions for small molecular systems showing complex nonadiabatic interactions will be presented. However, the highly accurate approaches available for small molecular systems are not the main scope of the present work, as they are not transferable directly to the study of large, complex molecular systems. Clearly, the definition of efficient computational approaches aimed at spectroscopic studies of macrosystems is in general a nontrivial task, and the basic requirement is that such effective models need to reflect a correct physical picture. Then, as will be presented, appropriate schemes can be introduced even for challenging cases, retaining the reliability of more demanding computational approaches for molecular systems of, for example, drug design, materials science, and nanotechnology.

The main aim of the book is the presentation and analysis of several examples illustrating the current status of computational spectroscopy approaches applicable to medium-to-large molecular system in the gas phase and in more complex environments. Particular attention is devoted to theoretical models able to provide data as close as possible to the results directly available from experiment in order to avoid ambiguities in the interpretation of the latter. Additionally, the main focus is on approaches easily accessible to nonspecialists, possibly through integrated computational strategies available in standard computational packages.

In fact, one of the objectives of the book is to introduce nonexpert readers to modern computational spectroscopy approaches. In this respect, the essential basic background of the described theoretical models is provided, but for the extended description of concepts related to theory of molecular spectra readers are referred to the widely available specialized volumes. Similarly, although computational spectroscopy studies rely on quantum mechanical computations, only necessary aspects of quantum theory related directly to spectroscopy will be presented. Additionally, we have chosen to analyze only those physical–chemical effects which are important for molecular systems containing atoms from the first three rows of the periodic table, while we will not discuss in detail effects and computational models specifically related to transition metals or heavier elements. Particular attention has been devoted to the description of computational tools which can be effectively applied to the analysis and understanding of complex spectroscopy data. In this respect, several illustrative examples are provided along with discussions about the most appropriate computational models for specific problems.

The book has been set as a joint effort of members of an Italian network devoted to applications of computational approaches to molecular and supramolecular problems (http://m3village.sns.it) along with some international collaborators and is organized as follows. After the preface by the editor, short chapters (authored by Brustolon, Van Doorslaer, Maçôas, Becucci, and Pietraperzia) summarize the point of view of experimental spectroscopists about the status and many interesting perspectives in the field. Then, different topics of computational spectroscopy are examined starting with a section devoted to transitions between electronic and spin states within a static framework. This part starts with a chapter (by Improta) describing electronic spectroscopy in the ultraviolet (UV)–visible region with particular attention to environmental effects; the following chapters deal with response function theory applied to linear and nonlinear optical spectroscopy (by Rizzo, Coriani, and Ruud), X-ray spectroscopy (by Carravetta and Agren), magnetic resonance spectroscopies (by Pedone and Crescenzi), and photoluminescence in nanocrystals (by Trani). Then, time-independent approaches to nuclear motions, with special reference to rotational, vibrational, and electronic spectroscopies, are analyzed by Puzzarini Cappelli, Biczysko, Bloino, and Santoro. The last section is devoted to time-dependent approaches and includes a contribution by Domcke and co-workers concerning the computation of ultrafast time- and frequency-resolved spectroscopic signals; followed by two chapters devoted to quantum, semiclassical, and classical dynamical approaches authored by Lami, Santoro, Rega, and Brancato; and closed by a chapter by Freed and Polimeno devoted to the application of stochastic techniques to slow spectroscopies like nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR).

Computational spectroscopy is a rapidly evolving field that is producing versatile and widespread tools for the assignment of experimental spectra and their interpretation in terms of basic chemical–physical effects. We hope that the topics covered in the book and related to the computation of infrared (IR), UV–visible, NMR, and EPR spectral parameters, with reference to the underlying vibronic and environmental effects, will allow computational chemists, analytical chemists and spectroscopists, physicists, materials scientists, and graduate students to benefit from this thorough resource.

Vincenzo Barone

Note: Color versions of selected figures are available at ftp://ftp.wiley.com/public/sci_tech_med/computational_strategies.

Introduction to Electron Paramagnetic Resonance

Marina Brustolon

Dipartimento di Scienze Chimiche Universita degli Studi di Padova, Padova, Italy

Sabine Van Doorslaer

Department of Physics, University of AntwerpAntwerpBelgium

Ever since its first observation in 1944, electron paramagnetic resonance (EPR) has offered a unique tool to investigate paramagnetic systems. In the first four decades of EPR, continuous-wave (cw) EPR at X-band frequencies (9.5 GHz) was the main technique. Although a lot of information can sometimes already be determined from these cw-EPR experiments, in many cases these spectra consist of very uninformative single lines. The introduction of the cw electron nuclear double resonance (cw-ENDOR) technique in 1956 [1] offered a first way of obtaining more detailed information about the interactions of the unpaired electrons with the surrounding magnetic nuclei (and hence about the electronic state of the system). However, a revolutionary new area of EPR started in the 1980s with the development of pulsed EPR spectrometers and more recently with the introduction of high-frequency (HF) EPR. The EPR toolbox has now moved from the X-band cw-EPR analysis to a large number of different EPR methods, each of them resulting in particular information. Combined multi frequency cw and pulsed EPR techniques allow characterizing in detail the dynamics and the electronic and geometric structure of long-living and transient paramagnetic species.

In the following the potential of some of these methods will be outlined further. At the same time we will point out the increasing importance of a synergic development of theoretical models and computational tools for a full interpretation of the new EPR experiments.

EPR: Dynamics and Spin Relaxation

The analysis of cw-EPR spectral profiles for radicals in solution to obtain information on their intermolecular and intramolecular dynamics has been a customary procedure since chemists began to use this spectroscopy in the 1960s. For motions sufficiently fast to average the magnetic anisotropies, the spectral profile can be simulated as a collection of Lorentzian lines at the resonance frequencies, with linewidths proportional to the spin–spin relaxation rates. This type of simulation is based on the Redfield theory, in which the random magnetic interactions are considered as a time-dependent perturbation of the static spin Hamiltonian. The linewidths are obtained as functions of the magnetic anisotropies (g and hyperfine tensors for radicals) and of the correlation times of the motions [2]. The border between fast and slow motions is defined by the comparison between the correlation times and the inverse of magnetic anisotropies. Therefore, in a multifrequency EPR approach (MF-EPR), each experiment requires an adjustment of the time scale on the basis of the frequency of the spectrometer, since the Zeeman anisotropy depends on the external magnetic field. The use of MF-EPR therefore facilitates the study of complex dynamics. HF-EFR is a fast time-scale technique freezing out the slower motions and giving lineshapes affected by the faster motions only, whereas EPR at lower frequencies is sensitive to slower motions. The combined approach thus permits the separation of different types of motion [e.g., 3].

Today the spectral profiles can be simulated for any motional regime by a numerical integration of the stochastic Liouville equation, as discussed in Chapter 12 and in the references therein. The noticeable improvement in the techniques of calculation of the magnetic parameters and their dependence on the solvent, and of the minimum energy conformation of the molecules, have opened the possibility of an integrated computational approach. Since it gives calculated spectral profiles completely determined by the molecular and physical properties of the radical and of the solvent at a given temperature, this method is a step forward in the direction of a sound interpretation of complex spectra.

EPR spectral profiles of paramagnetic probes in solid phases are determined by the anisotropic distribution of spin packets due to the nonaveraged magnetic anisotropies. The inhomogeneously broadened bands are scarcely informative, and the so-called hyperfine methods (see below) are used in these cases to extract information on the hyperfine parameters.

A viable method to extract further information on the residual motions of paramagnetic probes in solids is echo-detected EPR (ED-EPR), where the spectral profile versus the magnetic field is given by the integrated electron spin echo (ESE). The echo experiment probes directly the homogeneous linewidth at each spectral position affected by the motional process, bypassing the inhomogeneous broadening [4]. This method has been applied conveniently when residual motions of paramagnetic probes are present in glasses [e.g., 5], in biological systems [6], and in complex solids. Multifrequency ED-EPR has been applied to study nitroxide spin probes in a supramolecular compound [7]. In another example, ED-EPR has been used for the detection of a photoexcited dye triplet in glassy solution and in an inclusion channel compound; the different types of tumbling motions in the two environments give ED–time resolved EPR spectra with strongly different spectral profiles [8].

Finally, pulsed EPR experiments in solids allow determination of spin–lattice relaxation and phase memory times (T1, TM), whose dependence on temperature and nature of the environment can be analyzed to give information on the collective relaxation phenomena due to nuclear spin diffusion, electron–electron dipolar interaction, and instantaneous diffusion [9]. Note that the electron–electron dipolar interaction is also exploited in the DEER (PELDOR) technique for measuring the distance between paramagnetic centers. Instantaneous diffusion can give information on the microconcentration of radicals produced by high-energy irradiation in solids [e.g., 10].

The studies on spin dynamics and spin relaxation properties of paramagnetic species in solids are particularly interesting today in the perspective of a development of spintronics. In this respect the relaxation properties of the relatively simple, well-known and studied organic radicals in solids can help in understanding more complex behaviors [e.g., 11].

Probing Electronic Structure Through Hyperfine Spectroscopy

In paramagnetic systems, the unpaired electron(s) can interact with the surrounding magnetic nuclei (hyperfine interaction) [2, 12]. This interaction reflects the spin density distribution (and hence the electronic energy), and the electron spin–nuclear spin distances. Furthermore, for a nuclear spin larger than , the nuclear quadrupole interaction reflects the electric field gradient experienced by the nucleus [12, 13]. The measurement of these hyperfine and nuclear quadrupole interactions thus completes the electronic information obtained from determining the electron Zeeman interaction and (for ) the zero-field interaction and, hence, allows in principle for a full description of the electronic structure of the paramagnetic species at hand. The majority of the hyperfine and nuclear quadrapole interactions unfortunately remain unresolved in the field-swept EPR spectra, explaining the need of other EPR approaches, the so-called hyperfine spectroscopy techniques, to obtain this information.

A large variety of hyperfine spectroscopy methods exist that allow the detection of hyperfine and nuclear quadrupole interactions: electron spin-echo envelope modulation (ESEEM), ENDOR, and ELDOR-detected NMR (electron--electron double-resonance detected nuclear magnetic resonance) [13]. Although there are cases in which ESEEM and ENDOR perform equally well, ESEEM-like methods tend to be favorable for the detection of small nuclear frequencies (<10 MHz), whereas ENDOR and ELDOR-detected NMR methods are more appropriate for the larger nuclear transition frequencies. At high microwave frequencies, the use of ELDOR-detected over ENDOR techniques may be advantageous [14].

All ESEEM pulse sequences have in common that they consist only of microwave pulses with one microwave frequency and that they generate an electron spin echo, whose intensity is detected as a function of one or two interpulse distances. The detected modulation of this echo intensity reflects the nuclear interactions. Details on the physical origin of these nuclear modulations can be found in the literature [13]. Although a large variety of ESEEM methods are available, the HYSCORE (hyperfine sublevel correlation) spectroscopy method [15] has become the most favored technique. This four-pulse ESEEM method leads to a two-dimensional (2D) time-domain signal. In the frequency-domain spectrum obtained after 2D Fourier transformation of this signal the cross peaks directly link the nuclear frequencies of different electron spin manifolds. In this way, spectral interpretation is largely facilitated. While HYSCORE experiments are easy to perform at lower microwave frequencies (2–35 GHz), high-frequency HYSCORE is hard to realize, needing high-power microwave pulses, which are at present only attainable in prototype spectrometers [16, 17].

This limitation is not there for pulsed ENDOR methods, which can be used at all microwave frequencies. In ENDOR, the sample is irradiated with a combination of microwaves and radio waves. Continuous-wave ENDOR was already introduced in 1956 by Feher [1] and for a long time remained an important tool to determine the hyperfine and nuclear quadrupole interactions. However, nowadays this technique is largely replaced by the pulsed counterparts, which are more versatile. Two of the most commonly used ENDOR pulse sequences are Davies ENDOR [18] and Mims ENDOR [19]. In these techniques a combination of microwave pulses and a π radio frequency (RF) pulse with variable RF is used. A first set of microwave pulses creates electron polarization. When the RF matches one of the nuclear transitions, the populations of the different energy levels will be affected. This will change the electron polarization that is read out by a last sequence of microwave pulses, usually via electron spin echo detection as a function of the radio frequency. In this way, the nuclear frequencies can be directly detected.

In some cases, it may be beneficial to use ELDOR-detected NMR [20] instead of ENDOR to obtain the larger nuclear transitions. This technique bears similarities with the above-mentioned ENDOR techniques, but instead of using an RF pulse, a microwave pulse with variable microwave frequency is used to affect the electron populations. The nuclear transition frequencies follow from monitoring the polarization changes as a function of the difference between the two microwave frequencies.

In practice, the EPR spectroscopist will thus be using a variety of multifrequency cw-EPR and hyperfine spectroscopy techniques to determine the different spin Hamiltonian parameters. This has led to new insights in biological systems as well as in material sciences and it would be impossible to give an exhaustive account of all these applications. Here, we merely give a few examples to give the reader an idea about the wide applicability of the hyperfine techniques. ENDOR and HYSCORE spectroscopy allowed for instance for the direct detection of a hydrogen ligand in the [NiFe] center of the regulatory H2-sensing hydrogenase of Ralstonia eutropha in its reduced state [21], a ligand about which a lot had been speculated, but no hard proof could be provided by other (spectroscopic) means. Similarly, hyperfine spectroscopies were found to be key in the identification of a stable Rh-coordinated aminyl radical metal complex obtained after one-electron oxidation of an amide complex [22] and in the detection of a Co (III)-bound phenoxyl radical formed during the activation of a chiral cobalt salen catalyst [23]. Pulsed EPR and ENDOR of N@C60 in polycrystalline C60 allowed the detection of the freezing of cage rotation and the symmetry lowering induced by a phase transition [24]. High-field EPR and ENDOR even allowed, in a unique way, the probing of the wave function of shallow Li and Na donors in ZnO nanoparticles [25].

One of the big challenges is the translation of the different spin Hamiltonian parameters in a structural model. Indeed, the g values, zero-field splittings, and hyperfine and nuclear quadrupole tensors depend in a complicated way on the geometric structure and electronic ground state. Although some simplified approaches, such as the point dipolar approximation to extract distance information from the dipolar part of the hyperfine interaction [26], are readily used, in most cases these approaches are not sufficient to determine all information from the experimental parameters. Therefore, EPR analyses are increasingly combined with quantum chemical computation, whereby density functional theory (DFT) plays an extremely important role. While DFT reproduces the EPR parameters of organic radicals very well [27, 28], the computations of these parameters for paramagnetic transition metal complexes is often still problematic [29–31]. Especially challenging is the computation of zero-field parameters of high-spin systems and the g and metal hyperfine values of many transition metal systems. Nevertheless, it is by now clear that quantum chemical methods are pivotal for the correct interpretation of the EPR findings and that an improvement in the quality of these computations will have a huge impact on the application potential of hyperfine spectroscopy.

Interelectron Spin Distances: Probing Nanometer Distances with EPR

As mentioned earlier, dipolar electron spin–spin interactions can be probed with EPR techniques. If the interspin distance is smaller than 2 nm, considerable broadening of the cw-EPR spectrum occurs that can be interpreted in terms of the spin–spin distance [32]. Recent advances in pulsed EPR, most importantly the introduction of the four-pulse DEER (or four-pulse PELDOR) method [33] and the double quantum coherence techniques [34], have extended the electron interspin distance range accessible by EPR from 1 to 8 nm. However, only few natural systems contain two paramagnetic sites, limiting the applications. Wayne Hubbel and co-workers [35, 36] overcame this limitation in their pioneering work on site-directed spin labeling (SDSL) of proteins. This method allows attaching paramagnetic spin labels in proteins at specific points. In this way, any diamagnetic protein can be turned into an EPR-active molecule. By careful selection of the spin label insertion points and subsequent distance determination, a lot of structural information can be obtained about the protein. The spin labels are usually nitroxides, with the (1-oxyl-2,2,5,5-tetramethylpyrroline-3-methyl) methanethiosulfonate label (MTSSL) [37] being the most popular. Similarly, routes to spin label DNA or RNA have been introduced. Spin label EPR is now a booming area, and the number of systems that can be addressed by this technique is growing continuously. We refer the reader to a number of excellent reviews on this matter [38–43]. It is important to mention that the spin label orientation typically will show a certain amount of flexibility around the tether attaching the label to the protein. This spread in orientations will influence the distance determination. Hence, spin label EPR results need to be combined with molecular dynamics computations to link the experimentally observed distances to the molecule's structure [44].

Time-Resolved EPR: Detection of Short-Living Paramagnetic States

Time-resolved EPR (TR-EPR) allows the investigation of shortly living excited states of molecules with lifetimes down to hundreds of nanoseconds. It is used to study paramagnetic states produced by photophysical as well as photochemical events. The spin populations shortly after a laser pulse are far from equilibrium, and the system is spin polarized. The acquisition of the EPR spectrum after a short delay via cw-EPR or pulsed EPR then allows the recording of spin-polarized lines, in emission or enhanced absorption, with a much better signal-to-noise (S/N) ratio than for equilibrium spin populations. The first observation of spin polarization was made for radical pairs produced in solution by photolysis, the CIDEP (chemically induced dynamic electron polarization) effect [45]. Radicals produced in pairs are spin correlated (SCRP); for example, they remember whether they are deriving by dissociation from a molecule in a singlet or in an excited triplet state. Studies on the effect of spins in chemical reactions have led to a new branch of chemistry, spin chemistry, involving specific studies on reaction yield detected magnetic resonance (RYDMR), magnetic field effects (MFE), in chemistry and so on [46]. Short-lived radical pairs (RP) are formed also during the primary energy conversion steps of natural photosynthesis [e.g., [47], 48] and in donor–acceptor pairs, or donor–bridge–acceptor triads via electron or charge transfer in materials mimicking natural photosynthesis [49]. The search is for materials allowing intramolecular electron transfer events ultimately resulting in a charge transfer (CT) excited state with a relatively long lifetime [50].

Photoexcited triplet states of many organic molecules have been characterized also by TR-EPR, obtaining information on the electron distribution in the triplet (via the zero-field splitting parameters), on the photophysical path leading to the triplet (as the spin populations depend on the previous history), and on the evolution of the excited state in time [51]. The spin evolution of systems of higher multiplicity, such as an organic excited triplet on a dye and spin exchanging with the unpaired electron on a stable radical, has also been studied [52]. This has been performed also on a series of fullerenes linked to a stable nitroxide radical [e.g., 53].

Time-resolved EPR observations have played a pivotal role in obtaining information about the properties and evolution of spin states in natural biological events driven by light and in complex materials. This information is of relevance not only in relation to solar energy conversion but also in relation to perspectives such as molecular electronics, molecular photonics, and molecular spintronics, where the spin manipulation by light is a very challenging topic. As dipolar electron–electron interactions and spin exchange interactions are fundamental parameters for the interpretation of these experiments, any further improvement in the ability in calculating zero field splitting (ZFS) parameters and spin exchange energy is very important. Reliable calculations of ZFS parameters are also of relevance for PELDOR experiments mentioned in the previous part, since the distance measured via this method between pairs of paramagnetic species is based on the value of their electron–electron dipolar interaction [54, 55].

EPR Imaging

EPR is not only a spectroscopic tool but can also be used for imaging. This branch of EPR focuses on the imaging and location of paramagnetic species in different samples. One of the most interesting recent developments in this field is the electron spin resonance microscopy aimed at the observation of paramagnetic species in small (solid-state) samples with (sub-)micrometer-scale resolution [56]. In case of in vivo imaging, the strong absorption of the microwaves by the water results in large-dimension constraints. As a result, L-band (1–1.5 GHz) EPR spectrometers are used for studies of animals no larger than mice, whereas EPR spectrometers working at RFs (200–300 MHz) allow examination of rats [57]. Despite this limitation, EPR imaging is finding growing medical applications, such as in the monitoring of drug delivery processes [57], skin and hair research [58, 59], and cardiac research [60]. Further information can be found elsewhere [57, 58, 60].

Conclusion

In the last decades, EPR has become a very versatile research field, with different subdisciplines. Recent developments, including the introduction of high-field EPR, different pulsed EPR methodologies, spin-labeling techniques, and miniaturization, have enormously increased the number of problems that can be addressed with EPR. At the same time, there is a strong need of new and dedicated theoretical models and calculation tools in order to extract the maximum of information from the obtained data.

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Challenge of Optical Spectroscopies

Ermelinda M. S. MaÇôas

Centro de Química-Física Molecular (CQFM) and Institute of Nanoscience and Nanotechnology (IN), Instituto Superior Técnico, Lisbon, Portugal

Introduction

Optical spectroscopy reveals information about the perturbation of the oscillating charged particles of atoms and molecules (nuclei and electrons) induced by the oscillating electric field of light. The extent of the perturbation depends on the structure of the material giving insight into key structure–reactivity relationships. The interaction of the electric field with matter can lead to a series of events that are better understood considering either the wavelike (refraction and reflection) or particlelike (absorption, emission, scattering) nature of radiation and matter.

A light wave can be refracted (bent) and reflected when crossing the boundary between media of different density (different index of refraction). Refraction is a very important phenomenon when dealing with high-power, short light pulses. The magnitude of the wave vector (k) defining the direction of propagation of a light wave depends on the index of refraction (n) of the medium (k = 2π/λ = ωn/c), which is frequency and intensity dependent. This dependence results in the time delay, duration broadening, and frequency variation in time (chirp) experienced by short pulses propagating through a transparent medium [1].

Resonant light–matter interaction occurs when the photon energy (or frequency of the oscillating field) matches the energy difference (transition frequency) between rotational, vibrational, or electronic states of the system leading to absorption or emission of a photon. The excess energy introduced in a molecular system by interaction with the electric field can be relaxed by different photochemical and photophysical processes (Figure 1). The energy can be redistributed among the vibrational states of the system and further relaxed by intermolecular energy transfer. It can lead to transitions between states of the same spin (internal conversion, IC) or states of different spin (intersystem crossing, IST) as well as reactions in the excited state (R). It can also be deactivated radiatively by singlet–singlet emission (fluorescence, F) and triplet–singlet emission (phosphorescence, P) [2]. Light scattering is a nonresonant phenomenon that is best understood by considering the periodic oscillating charge induced within the system due to perturbation of the electron cloud by the oscillating electric field. The induced oscillating dipole moment is by definition a source of radiation, thereby resulting in emission of light. The light is emitted at either the same or different frequencies from the incident light in the so-called elastic and inelastic scattering interaction, respectively. Raman spectroscopy probes the inelastic scattering due to molecular rotations and vibrations.

Figure. 1 State energy diagram representing possible photochemical and photophysical processes triggered by absorption of a photon: A, absorption; VR, vibrational relaxation; F, fluorescence; IC, internal conversion; P, phosphorescence; IST, intersystem crossing; R, reaction.

The molecular response to a perturbation induced by an incident radiation field can be generalized by the polynomial expansion of the macroscopic electric polarizability vector (P) as follows:

(1)

equation

where ε0 is the electric permittivity, E is the complex electric field function of the incident light wave, and χ(n) are the (n + 1)-rank tensors of the macroscopic susceptibility. For weak radiation fields interacting with the material, the induced perturbation scales linearly with the magnitude of the electric field due to the vanishingly small contributions from higher order terms than the first order. When two or more independent radiation fields interact with the medium or intense radiation fields are used, the higher order terms must be taken into account. In this situation, the response of the material becomes nonlinear. The imaginary parts of the odd terms are used to express the linear, two-photon and three-photon absorption cross sections. The even terms are at the origin of phenomena like second-harmonic generation, sum-and-difference frequency mixing, optical rectification, and linear electro-optical effects. They are relevant for surfaces and interfaces but vanish in isotropic centrosymmetric media. Depending on the particular orientation and delay between independent radiation fields acting on the material, the third-order term can be used to express many of the nonlinear optical spectroscopies commonly used in photophysic and dynamic studies, like coherent Raman, pump–probe, transient grating, and photon echo spectroscopies.

The optical susceptibility χ(n) is a property of the medium that relates to its microscopic structure. It can be measured experimentally, but a fundamental understanding of the underlying complex structure–optical properties relationship necessarily relies on the development of theoretical models able to predict the nth-order nonlinear optical effects. Even though optical susceptibility can be best evaluated by a full quantum mechanical treatment, for practical purposes, interpretation of the experimental results is based on some sort of simplified models. The following sections in this chapter are a brief illustration of the application of linear and third-order spectroscopies to study the photophysics and dynamics of molecular systems pointing out the interplay between experiments and computational modeling. The focus will be mainly on third-order spectroscopies that have largely superseded the more conventional linear spectroscopy approach due to the great detail of information that can be extracted from spectroscopic methods based on nonlinear interaction of the electric field with matter.

Steady-State Linear Spectroscopy

In steady-state linear optical spectroscopies, resonance frequencies, amplitudes, and lineshapes are measured that provide valuable information about structure, reactivity, and photophysics. Interpretation of the observed spectral features is greatly aided by computational simulations. Vibrational spectra of small- to medium- size molecules and aggregates are routinely interpreted based on computed harmonic vibrational transitions and oscillator strength for optimized ground-state geometries, which are readily available from different computational programs at various levels of theoretical approximation. Spectroscopic investigation of the conformational landscape and reaction pathways in prototypical molecules and supramolecular assemblies relevant in biochemistry (e.g., biomolecular building blocks, physiologically active compounds, models for ligand–receptor interactions) are extensively supported by ab initio and density functional theory (DFT) calculations of ground-state potential energy surface, allowing us to understand at a fundamental level intra- and intermolecular interactions like hydrogen-bonding effects, isomerization, and specific solute–solvent interactions [3–5]. Due to the high computational cost associated with anharmonic calculations, vibrational spectra of medium- to large-size molecules are typically done under the harmonic approximation. In conformationally flexible systems characterized by highly anharmonic large-amplitude modes, this treatment has significant limitations. This is usually the case in carboxylic acids and amino acid groups so ubiquitous in all sorts of chemical and biological processes (e.g., molecular recognition, self-assembly, protein folding, formation of micellar carriers for drug delivery, and smart polymers) [6–11]. In order to achieve spectroscopic accuracy in these types of systems, different strategies are being explored to introduce realistic anharmonic potential. Efforts have been made in finding suitable computationally tractable solutions of the vibrational Schrödinger equation, including anharmonic potentials defined at different levels of approximation, in introducing anharmonic corrections via a perturbation approach based on the harmonic oscillator states, and in the parallelization of the calculations of both the anharmonic potential and the vibrational eigenvalues and eigenvectors [12–14].

Computation of excited-state energies, structures, and electronic properties has also become an indispensable tool in the interpretation of electronic spectra and excited-state dynamics. Electron density based time-dependent DFT (TD-DFT) methods have become very popular in the simulation of excited-state properties due to their ability to give impressive quantitative results at a moderate computational cost for relevant medium- to large-size molecule, as opposed to the more time-consuming correlated wavefunction-based methods, such as multireference configuration interaction (MRCI), multiconfigurational self-consistent field (MCSCF), complete active-space SCF (CASCF), and complete active-space perturbation theory (CASPT) [15]. Every week several reports are published demonstrating the usefulness of the TD-DFT methods in increasingly complex systems. TD-DFT has been systematically applied in photophysical studies addressing metal complexes and dyes as potential sensitizers in solar cells [16, 17], common fluorescent probes used in spectroscopy and microscopy [18, 19], nucleobases present in RNA and in DNA [20], conjugated polymers with applications in photonic devides [21], and many other complex systems. The recent simulations of the electronic spectra of carbonyl-protected gold cluster are but one example of how far this method can go [22]. Implementation of TD-DFT in electronic structure calculation programs was a great driving force for routine application of these methods in photochemistry and photophysical studies. In addition, the set of programs nowadays available to visualize molecular orbitals can give a very intuitive picture of the electronic transitions. TD-DFT also has its own limitations that one should be aware of in order to extract reliable information. The choice of a functional and basis set should be critically accessed against known experimental data about the particular system under study. The failure of common functionals to describe long-range dispersion forces of significant importance to supramolecular systems and the limited accuracy in describing delocalized excited states (charge transfer state, valence states of molecules with extended π systems, Rydberg states, and doubly excited states) are known shortfalls in the performance of TD-DFT [15, 23, 24]. Even within the same molecule it is not unreasonable to question the validity of a particular choice of functional to address different electronic states. Nevertheless, development of corrected exchange-correlation functional is still a very active area of research.

The apparent structureless bandshape of broad electronic transitions observed in ultraviolet (UV)–visible absorption or emission spectra hides a complicated set of vibronic transitions that can only be unraveled by excited-state electronic structure calculation methods [18, 25]. Even though the formulation exists to include vibrational couplings and anharmonic corrections in computing electronic transitions that would allow for a quantitative comparison with experimental data, their implementation has so far not been trivial [18, 25]. However, this picture is most likely about to change as an integrated set of computational tools easy to handle by nonspecialists and able to run on standard personal computers has been recently reported [26].

For molecules in the condensed phase, including environment effects in the simulations of electronic and vibrational spectra is of utmost importance. The environment can stabilize preferentially specific molecular conformations and excited states either due to polarity effects or via specific interactions. Indeed, information about the environmental polarity can be retrieved from solvatochromic effects observed in electronic absorption spectra due to differences in the electric dipole moment of the states involved in the electronic transition [27]. Several approaches are available to account for environmental effects, from the simplest model representing the solvent by a continuum characterized by a dielectric constant to the supramoleular approach where the solute plus solvent molecules introduced as near neighbors are treated quantum mechanically within an external potential created by the solvent [28, 29].

Steady-State Third-Order spectroscopy

In the last two decades many creative technological applications of nonlinear absorption have been demonstrated stimulated by the development of suitable powerful pulsed lasers operating at near-IR wavelengths. Frequency up-conversion lasing; optical power limiting, stabilization, and reshaping; optical 3D microfabrication; drug delivery; high-density optical data storage; and laser scanning multiphoton excited fluorescence microscopy are among the areas where exploration of different events triggered by multiphoton absorption has led to significant technological developments [30]. Multiphoton absorption is here understood as the simultaneous absorption of more than one photon by the same molecule (Figure 2). It provides a mean to access a given excited state by absorption of lower energy photons when compared to the corresponding one-photon-induced transition. The need for high-power pulsed lasers stems from the fact that multiphoton absorption cross sections, σn, are considerably lower than linear absorption cross sections (e.g., σ1 10−19−10−16 cm² , σ2 10−51−10−46 cm⁴ s, and σ3 10−78−10−80 cm⁸ s²). The high-order dependence of the multiphoton absorption probability on the light intensity is responsible for an intrinsic confinement of the absorption to a smaller volume when compared to the linear absorption. Elimination of undesirable out-of-focus absorption when conjugated with a greater depth of penetration of the longer wavelengths used in multiphoton excitation constitutes the fundamental advantages of nonlinear absorption over the corresponding linear process. Under diffraction-limited focusing conditions, the absorption can be confined to a nanometric volume by high-numerical-aperture objectives enabling the activation of chemical or physical processes with high 3D spatial resolution.

Fig. 2 Schematic representation of simultaneous two-photon absorption followed by vibrational energy relaxation and emission of higher energy photon. The quadratic dependence of the two-photon absorption cross section on the excitation intensity leads to a confinement of the absorption process to a smaller volume when compared with linear absorption.

Due to the ease of implementation, connected with the widespread availability of high-peak-power pulsed lasers tunable in the range of 700–1000 nm (Ti–sapphire oscillators), two-photon absorption of molecular and supramolecular systems with linear absorption in the range of 350–500 nm has been by far the most extensively explored process. The complexity of optimized two-photon absorption chromophores range from simple dipolar molecules to hyperbranched dendritic structures, polymers, conjugated polymer dots, semiconductor, and up-conversion nanoparticles. In cases where the optimized system to perform a given task (e.g., delivery of a given drug, singlet oxygen release) is not a particularly good two-photon absorption chromophore, ingenious architectures have been devised where good two-photon absorption chromophores are used as antenna that later will channel the excitation energy into the desired reaction center [30, 31].

The two-photon absorption cross section is expressed in terms of the third-order macroscopic susceptibility (χ(3) in expression 1), which is an orientational average of the microscopic polarizabilities induced by the effective local field [32]. The microscopic polarizabilities can be expanded in terms of the transition dipole moments associated with many-body excited states, so that the two-photon absorption cross section is proportional to the sum over the transition dipole moments between the ground- and excited-state wavefuncions [33]. The general requirements for successful two-photon absorption systems are quite well understood [30]. These systems will preferably have either a permanent ground-state dipole or resonant charge transfer excited states. Thus, strong electron-donating (D) and electron-withdrawing (A) groups bridged by polarizable π-conjugated systems are needed. Many different geometries have been explored in order to establish the relevant structure–activity relationships, from simple dipolar molecules (D–π–A), quadrupolar (D–π–A–π–D, A–π–D–π–A), and octupolar [D(–π–A)3, A(–π–D)3] systems to dendrimers, cyclic oligomers, and polymers. Systematic studies have brought insight into how factors like the nature of the D and A groups, type and extent of the bridge, branching, symmetry, and solvation affect the two-photon absorption cross section. Nowadays, there are records of supramolecular systems with extremely high two-photon absorption cross sections (200,000 GM, 1 GM = 10−50 cm⁴ s, more than three orders of magnitude higher than the two-photon active molecules known 10 years ago) [34]. Much of this achievement is due to an extremely huge effort of molecular engineering in the quest to maximize two-photon absorption cross sections. Nevertheless, as the complexity of the structures increases, simple structure–activity relationships became less clear as the factors affecting the two-photon activity became strongly entangled.

The guidance that computational estimations can bring to the molecular engineering strategies designed to tackle specific applications is precious. Just as in the case of linear absorption, wave function-based correlated ab initio methods are currently of little use in the treatment of nonlinear optical properties of molecules of practical interest. Two-photon absorption properties have been most successfully modeled by TD-DFT extensions for the calculation of molecular nonlinear optical properties based on the residues of the quadratic response function [35], and on the quasi-particle formalism of the TD-DFT equations for arbitrary frequency-dependent nonlinear optical polarizability [36, 37]. Fortunately, for organic molecules only a limited number of states are optically active in the UV–visible spectral region. These are usually states with strong transition dipole moments associated with a charge transfer nature of the excited state and the delocalized π-electron systems. Indeed, the complete sum-of-all-states expression providing expansions of the molecular polarizability into dipolar contributions from different many-body excited states can be simplified under certain assumptions to an actual sum over few states. These simplified expressions have been applied with great success to understand the trends in two-photon absorption activity of dipolar and quadrupolar molecules [33]. Branched molecules with several chromophores are usually simulated by a Frenkel exciton model [33]. Nevertheless, it has become clear that electrostatic interactions alone cannot provide an accurate description of the two-photon absorption process in these systems and electronic correlation as to be explicitly introduced [33, 38].

Despite the remarkable progresses made in the last decade, it should be noted that while many of the identified potential applications require high-energy excited states that can efficiently transfer the excitation energy into desired reaction centers, the high two-photon absorption cross sections reported so far are usually connected with low-energy excited states. Materials that can be excited by simultaneous absorption of two-photons at higher energies are needed. In addition, for two-photon active material to be of practical use, optimization of secondary properties key to specific applications must also be carefully considered (e.g, solubility, cytotoxicity). Future developments can be envisaged in the exploration of different types of systems (charged and open-shell molecules) and different types of processes connected with two-photon absorption (e.g., circular dichroism). Simplified theoretical models that include electronic correlation are needed to give clear indication as the way to proceed avoiding the up-to-now trial-and-error search for improved chromophores.

Transient Third-Order Spectroscopy

The linewidth characteristic of steady-state spectroscopic features contains contributions from coherent dephasing, inhomogeneous dephasing, and relaxation. Nonlinear transient spectroscopy is able to isolate these contributions giving precious insight into the evolution of coherences in the excited state, lifetimes, and solvation dynamics. All of those are crucial to understand the dynamics of ultrafast events such as bond breaking/bond formation, photoisomerization, electron and proton transfer, exciton relaxation, and the dynamics of energy relaxation in general. The widespread availability of powerful femtosecond laser pulses together with the development of tools to effectively shape and control these pulses (tuning of central frequencies, bandwidth, pulsewidth, phase and polarization direction) has led to major developments in ultrafast spectroscopic techniques in the last two decades.

Time-correlated single-photon counting (TCSPC) and fluorescence up-conversion are the most popular techniques for real-time measurements of fluorescence emission [39]. In TCSPC, the arrival time of single photons emitted by a sample excited by a train of ultrafast pulses (femtoseconds to picoseconds) is recorded. After accumulation of enough single-photon events, a histogram of detected times allows for reconstruction of the waveform of the emitted signal [40]. Even when using the fastest avalanche photodiodes as detector, time resolutions are limited to 20–50 ps. In the less sensitive fluorescence up-conversion technique, the fluorescence lifetime can be recorded with a resolution that is only limited by the duration of the laser pulse (typically less than 100 fs). An ultrafast laser beam is split so that part of it is used for excitation of the sample and part of it is used for optical gating of the emission signal. The gate pulse is sum–frcquency mixed with the photons emitted by the sample in a nonlinear crystal with an appropriate phase-matching geometry. The up-converted signal, which is separated in space from both the gate pulse and the fluorescence pulse, can be measured by a slow detector. The fluorescence up-conversion signal is measured as a function of a time delay between the excitation event and the optical gate [39]. Just like most of the transient spectroscopy methods, the fluorescence up-conversion technique is a variation of the general pump–probe method [41, 42]. In pump–probe methods a strong pump beam interacts with the sample and a weaker beam arriving at a variable delay time is used to probe the time-dependent changes induced by the pump pulse (Figure 3). The frequencies of the two pulses and the time delays between them are the controllable variables of the setup. There are many variations of the pump–probe scheme depending on which processes are being monitored (absorption and emission) and on which type of wavelength is used for the pump and probe beams (UV–vis, UV–IR, vis–IR, IR–IR, etc). Other popular third-order transient spectroscopics used to interrogate coherent dynamics are transient grating (TG), coherent anti-Stokes–Raman (CARS), and photon echo (PE) spectroscopies [43]. All of those are based on noncollinear four-wave mixing arrangements differing in the pulse sequence and the number of controllable wavelengths (Figure 3). Several time-resolved wave-mixing experiments can be performed with minimum modifications on the setup. The type of response observed is determined by the phase-matching geometry based on momentum conservation of the involved photons. In TG spectroscopy two time-coincident pulses cross at the sample with some angle [44]. The two beams create a spatially varying interference pattern along the transverse direction that interacts with the sample. The optical grating can be written by changes in absorption (excited-state grating), subsequent relaxation can lead to a periodic temperature profile (thermal grating), and non-resonant scattering can create a spatial modulation of the refractive index. A third pulse approaching the grating at an angle that satisfies the Bragg condition is diffracted off the grating probing the time evolution of the system. In CARS spectroscopy the two-time coincident pulses have different colors and the energy difference matches a vibrational frequency of the sample creating a vibrational coherence [45]. The probe measures the degree of vibrational coherence remaining after a certain time delay. PE spectroscopy is the optical analogue of spin echo spectroscopy in NMR [46]. It is the method of excellence to disentangle homogeneous and inhomogeneous line broadening. A laser pulse creates a coherence between ground (g) and excited state (e), which is allowed to evolve freely with a frequency for a given time interval (τ). A second pulse switches the coherence to evolve at a frequency −ωeg restoring the initial phase after a time 2τ. The emitted coherent signal recorded as a function of the evolution time bares only the homogeneous relaxation dynamics. In the three-pulse version of PE, also known as stimulated photon echo, the second pulse converts the ωeg coherence into a ground- or excited-state population. After a time delay T, a third pulse comes that will initiate the rephasing process. The output coherence finally emits after a time t < τ. The polarization echo is measured as a function of T and gives information about population changes in the ground and excited states. Phase information can also be collected if the echo pulse is made to interfere with the phase-locked pulse, called the local oscillator.

Fig. 3 Folded BOXCARS geometry applied in several transient nonlinear optical spectroscopies. In pump–probe spectroscopy, one of the three beams is blocked and the intensity of one of the incoming beams is monitored as a function of the time delay between the remaining two beams (e.g., beam 3 is blocked and beam 2 is monitored as a function of its delay with respect to beam 1, phase-matching condition would be ). Beams 4 and 5 are photon echo signals generated from beams 1 and 2. Beams 6 and 7 can be stimulated photon echo or transient grating signals generated from beams 1, 2, and 3. In transient grating two of the beams are time coincident. In coherent anti-Stokes–Raman spectroscopy, beams 1 and 3 are time coincident and carry the same frequency; the difference between this frequency and that of beam 2 (so-called Stokes beam) matches a vibrational frequency of the system; and beam 6 will correspond to the anti-Stokes emission.

Generally, a combination of the described techniques is used to interrogate in great detail the dynamics of the systems. The femtosecond time resolution provided by this set of spectroscopic techniques is the key for observation of atomic-scale dynamics [47]. Molecular rotations and vibrations can be observed in real time rather than in energy spectra. Following directly the motion of the system along the potential energy surface while it moves from the reactant well trough the transition state into the product well became possible with femtosecond spectroscopies. Dissociation of the NaI molecule is a paradigmatic example of chemistry probed in real time [47]. With dephasing times slower than the nuclear motion, the wavepacket dynamics and quantum beats could be investigated by femtosecond spectroscopy. As expected for reactive systems with slow dephasing, the wavepacket repeatedly sampled the reactive curve crossing, each time crossing with some probability into the product side. The kinetics of product formation showed a stepwise rising curve with a frequency corresponding to the period of the wavepacket motion [48].

The theoretical treatment of transient nonlinear spectroscopic features is based on the density matrix and the perturbative solution of the quantum mechanical Liouville equations describing the temporal evolution of the system under the influence of the incident electromagnetic fields as well as the intrinsic relaxation processes [32, 49]. Generally, the signals observed are understood in terms of a diagrammatic representation on double-sided Feynman diagrams of the time evolution of the density matrix operator and its transformations by interaction with a particular time order and geometric arrangement (phase-matching conditions in Figure 3) of the electric fields involved [50]. The corresponding mathematical expressions can then be written in a very intuitive way following some general rules that translate the relevant diagrammatic representations [32]. Frequently, a reduction of the high dimensionality of the problem is needed in order to obtain computationally tractable expressions. Simulations at diverse levels of theoretical approximation, from classical mechanics to mixed quantum mechanical/molecular mechanical (QM/MM) models to full quantum mechanics, are applied depending on the nature of the problem at hands [32, 47, 51]. The computation of electronic energies as a function of molecular geometry can be used to obtain adiabatic transition energies and identify state crossings important in ultrafast nonradiative decay probed by transient spectroscopies. Such calculations at a high level of accuracy have located conical intersections pointed out to be responsible for the ultrafast energy relaxation observed in DNA base pairs providing the mechanism for protection of UV-induced phodamage in DNA [52, 53]. Much of our present understanding of the rich photophysics and photochemistry observed in transition metal complexes, with relevance to their application in solar cells, organic light-emitting diodes (LEDs), and photoswitches, results from a combined effort of transient spectroscopic studies and TD-DFT simulations [54]. Modeling of transient spectroscopy signals has to account also for transition frequency fluctuations due to solute–solvent interactions. The environment plays a key role in the excited-state dynamics. Besides providing a bath for energy relaxation [55], environment fluctuations can aid intramolecular energy redistribution by providing frequency components that equal the energy mismatch between two