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Overview
Over the past few decades, experimental excited state chemistry has moved into the femtochemistry era, where time resolution is short enough to resolve nuclear dynamics. Recently, the time resolution has moved into the attosecond domain, where electronic motion can be resolved as well. Theoretical chemistry is becoming an essential partner in such experimental investigations; not only for the interpretation of the results, but also to suggest new experiments.
This book provides an integrated approach. The three main facets of excited-state theoretical chemistry; namely, mechanism, which focuses on the shape of the potential surface along the reaction path, multi-state electronic structure methods, and non-adiabatic dynamics, have been brought together into one volume. Theoretical Chemistry for Electronic Excited States is aimed at both theorists and experimentalists, involved in theoretical chemistry, in electronic structure computations and in molecular dynamics. The book will provide both with the knowledge and understanding to discover ways to work together more closely through its unified approach.
Product Details
| ISBN-13: | 9781788014359 |
|---|---|
| Publisher: | Royal Society of Chemistry |
| Publication date: | 03/02/2018 |
| Series: | ISSN |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 226 |
| File size: | 16 MB |
| Note: | This product may take a few minutes to download. |
Read an Excerpt
CHAPTER 1
Introduction and Motivation
There are two main threads associated with the theoretical chemistry of the excited state. On the one hand, we have to understand the shapes of potential energy surfaces that are associated with the nonadiabatic event that occurs when the reaction path passes from one state to another. This is associated with a conical intersection. The other thread is associated with methods for computing such potential energy surfaces and possibly studying the dynamics associated with nuclear motion.
The shapes of these potential surfaces result from the fact that the force field of an excited state, i.e. the strength and position of the various bonds, is different from that of the ground state. We will show that the shapes of potential energy surfaces are intimately connected with a theory that can be used to predict their shape, but without doing actual computations. This is valence bond (VB) theory. So in this chapter we briefly introduce the subject of VB theory and how it controls the shapes of potential energy surfaces.
Electronic structure methods for computing potential energy surfaces and studying the dynamics associated with nuclear motion are huge fields. Our discussion must be limited. In this book our objective is not to discuss the various methods associated with electronic structure techniques or dynamics. Rather, we wish to elucidate the general conceptual principles that lie behind these methods. Our objective is to suggest how the reader can make informed decisions about which methods may be most appropriate for the problem to hand. Thus we believe that we can present the important aspects of the relevant electronic structure methods from a unified point of view using the partitioned eigenvalue problem and the perturbation theory that stems from this partitioning. So our purpose in this chapter is just to give the most basic algebraic development.
1.1 The Chemical Nature of Electronic Excited States
Any discussion of electronically excited states usually starts with a Jablonski diagram, as shown in Figure 1.1a. In this case we are discussing electronically excited states that do not involve ionization. This figure presents the electronic states of a molecule much like a diatomic molecule, i.e. as a set of electronic and vibrational energy levels. When one considers nuclear motion, e.g. a chemical reaction, then the electronic energy levels evolve on a potential energy curve as the geometry changes, along the reaction coordinate in Figure 1.1b. The vibrational energy levels shown in Figure 1.1a are replaced by the classical idea of a ball rolling on the potential curve in Figure 1.1b.
In Figure 1.1b we distinguish an adiabatic trajectory or reaction path and a nonadiabatic trajectory or path. The nonadiabatic path, e.g. FC (Franck–Condon) -> CoIn -> P', that moves from one potential surface to another via a conical intersection. The adiabatic path, e.g. FC (Franck–Condon) -> R* -> TS* -> p*, remains on a single potential curve.
In the Jablonski diagram, we distinguish excited states by their spin multiplicity, e.g. singlet excited states S1 and triplet excited states T1, and their associated vibrational manifolds, together with the radiationless processes that interconnect these manifolds such as internal conversion (IC) and intersystem crossing (ISC) as well as relaxation in the vibrational manifold (internal vibrational relaxation, IVR). In addition we have processes involving absorption of radiation (A) and emission, fluorescence (F) or phosphorescence (P). Once we allow nuclear motion then the vibrational energy levels can be represented, classically, in the continuous form as a potential curve, as shown in Figure 1.1b. We then imagine reactivity as a "ball", or mass point, moving on the potential curves according to the classical equations of motion. In this picture, a radiationless process occurs at a topological feature (at a specific geometry) associated with the curve crossing, a conical intersection (CoIn) in Figure 1.1b. Otherwise, various topological features on the potential curve have their usual meaning, e.g. transition state (TS) and various minima, e.g. P for product, etc.
The potential energy curves in Figure 1.1b will have been obtained (in practice) from an electronic structure computation. However, there is as yet no chemical information, i.e. the nature of the geometrical change as we progress along the reaction coordinate. Thus we do not understand why the potential energy curves behave the way they do as the reaction coordinate changes. We need to understand something about the chemical nature (bonding characteristics) of electronically excited states. For example, the curve connecting the FC (Frank Condon point) and P0 (product), through the surface crossing appears to be continuous. This would imply that the bonding characteristics are changing only in a gradual continuous fashion. Thus, a molecule excited to the point FC, simply relaxes from FC to P0, without changing bonding characteristics. However, the bonding situation in the excited state (S1) at the point FC is different from for the ground state (S0). As a consequence, the bond lengths must adjust and the energy goes down along the reaction coordinate. A potential curve, or a surface in higher dimensions, that changes in this way, corresponds to a quasi-diabatic state in which the electronic structure/configuration does not change along the reaction co-ordinate. For the moment, we wish to inquire about the nature of the bonding that makes the curves in Figure 1.1b behave the way they do.
The Jablonski diagram (Figure 1.1a) also contains radiative processes such as F (fluorescence), A (absorption), and P (phosphorescence). The relationship between the corresponding energy level diagram (Figure 1.2a) and a potential curve (Figure 1.2b) can also be developed. The well-known rationalization of emission and absorption behaviour in the Jablonski diagram in terms of potential surfaces is illustrated for fluorescence in Figure 1.2. Figure 1.2a is a simplified form of the Jablonski diagram with the vibrational energy levels omitted for clarity. Then in Figure 1.2b we show the fluorescence process as absorption, followed by intramolecular vibrational relaxation, followed by vertical decay into the vibrational energy levels of the ground state. The surface crossing in Figure 1.1b can also be treated using the density of vibrational states via the Fermi golden rule method, but we will not develop this point and the reader is directed to standard textbooks for a discussion.
Thus in the following discussions we will focus mainly on excited state reactivity and dynamics on a potential surface of the form shown in Figure 1.1b. Of course when we come to consider laser chemistry then we need to consider the details of the interaction of light with molecules.
The concepts of VB theory provide the link that explains why the potential energy surfaces or curves for ground and excited states behave in a different way. We will illustrate this simple idea with an example, as shown in Figure 1.3. Figure 1.3 illustrates the photophysics and photochemistry of DMABN ((dimethylamino)benzonitrile), which has been studied theoretically in our group. We are interested in two excited states of DMABN: S1, a locally excited state (LE) in which only the benzene chromophore has been excited, and S2, a charge transfer (CT) state in which an electron has been transferred from the nitrogen lone pair to the benzene ring in the photo-excitation process. (For a recent study with dynamics see the work of Martinez et al.) We can represent the three states with valence bond pictures (ground state I, LE: II and CT: IIIao-4IIIb, as indicated in Figure 1.3). The ground state corresponds to the well-known Kekulé structure I of benzene while the excited state LE is the anti-Kekulé structure II, i.e. the negative combination of the two locally bonded Kekulé structures. The CT state has a negative charge on the benzene ring with two VB forms, IIIa and IIIb, the quinoid and anti-quinoid structures of the benzene radical anion. Along the appropriate geometrical coordinate, the S2 LE state changes continuously to the S1 LE and similarly for the CT state with a crossing along the reaction coordinate. Thus the subscripts on Si refer to the state ordering (1 or 2), while the notation LE or CT, refers to the chemical nature (VB) of the state, i.e. the diabatic state. Understanding the chemical nature of the electronic state in terms of its bonding pattern enables one to predict or at least rationalize the behaviour as the geometry changes.
The main component of the reaction coordinate is shown in Figure 1.4a as a skeletal quinoid anti-quinoid deformation of the benzene ring. Since the LE state of benzene is totally symmetric, the energy goes up along this coordinate (S1 LE -> S1 CT), because of the non-symmetric distortion benzene ring. The energy of the CT state goes down, since the CT state has a quinoid equilibrium structure, with the resulting curve crossing. Thus, in Figure 1.3 we illustrate the main ideas that connect the chemical concepts via VB theory to the shape of the potential surface. While sophisticated electronic structure computations may be required to obtain the potential surface to a high level of accuracy, extracting VB information from such computations proves to be useful both in designing electronic structure computations themselves and in rationalizing both experimental and theoretical results a posteriori.
In Figure 1.3 we have shown a real crossing of the two quasi-diabatic states, along which the VB states LE and CT do not mix, along the coordinate given in Figure 1.4a. We have also shown an avoided crossing (dashed curve) with a transition state maximum. This curve is displaced along the coordinate shown in Figure 1.4b. This is the coordinate that lifts the degeneracy at the real crossing and takes one to an avoided crossing. The two coordinates shown in Figure 1.4 have a precise mathematical definition, in the same way that a transition vector corresponding to an imaginary frequency does.
The point that we wish to emphasise at this stage is that the photophysics of DMABN is easily rationalized with such a simple figure based upon the most elementary ideas of VB theory, discussed in more detail elsewhere. The experimental aspects, see for example the discussion by Zacharise or Martinez, relate to the observation of dual fluorescence, or not, from the S1 CT and or S1 LE minima shown in Figure 1.3 and whether the S1 CT minimum is populated via an adiabatic path from the S1 LE minima or via a nonadiabatic path from the S2 CT state in the Franck–Condon region. These issues in turn depend on the position, and stability, of the S2 CT -> S1 CT potential curve in Figure 1.3.
In Figure 1.3 we show a "real" crossing of two potential energy curves together with an "avoided" crossing. These correspond to slightly different slices through the potential energy surface, displaced along the co-ordinate shown in Figure 1.4b. This idea can be more clearly explained with a three-dimensional picture, as shown in Figure 1.5b. On the left-hand side of Figure 1.5a we show a reaction path through an avoided crossing, similar to the dashed curve in Figure 1.3. On the right-hand side we show a conical intersection in three dimensions. For an introductory article on conical intersections in photochemistry see Robb, which is available as a free download.
The real crossing of the two quasi-diabatic states (dotted lines) shown in Figure 1.3, along which the VB states LE and CT do not mix, corresponds to the trajectory via the apex of the double cone shown in Figure 1.5b. The avoided crossing, on the other hand, corresponds to the slice through the cone shown in Figure 1.5b, also shown as the avoided crossing in Figure 1.5a. In Figure 1.5a, the slow radiationless decay that would occur at the intermediate M* on the excited state would be governed by the Fermi golden rule dynamics referred to previously, while passage through the conical intersection in Figure 1.5b occurs without impediment. We shall return to a more detailed discussion of the dynamics through a conical intersection shortly. The point to appreciate at this stage is the contrast between the two-dimensional entities, projections or slices, associated with an avoided crossing and at a real crossing, as they are shown in Figure 1.3, and as they are illustrated in three dimensions in Figure 1.5b. The two coordinates in the case of DMABN shown in Figure 1.4a and b correspond to the space that contains the double cone and Figure 1.5b.
1.2 Chemical Reactivity in Electronic Excited States
We would now like to discuss a few more examples of the way in which the reactivity in electronically excited states can be rationalized and understood using simple VB concepts and how these can rationalize the occurrence of features such as conical intersections. A more extensive discussion can be found in a review by Robb.
In general, each electronically excited state can be represented as a valence bond isomer or combination of VB isomers. For example, in benzene the ground state is the familiar sum of the two localized hexatriene-like Kekulé structures, while the first excited state is the difference between the two localized Kekulé structures. Each of these VB isomers has different equilibrium bond lengths corresponding to different shapes of the corresponding potential energy surfaces. After vertical excitation, the geometry then relaxes according to forces arising from the particular VB structures associated with that potential energy surface. Thus each diabatic potential energy surface can be understood as arising from the different VB force fields associated with the different bonding arrangements for the particular excited state. By force field we mean an equilibrium value of an internal degree of freedom together with a force constant. We now continue in a qualitative fashion, returning to a more mathematical presentation later.
The classic textbook excited state chemistry example, the 2 + 2 cycloaddition of two ethylene molecules (Figure 1.6a). is a simple but useful starting point. We shall consider the face-to-face approach (Figure 1.6b) where the new σ bonds are formed synchronously, as well as a bi-radical approach (Figure 1.6d), where one σ bond is formed first to yield a diradical intermediate. The coordinate that connects the two approaches is a trapezoidal distortion coordinate, as shown in Figure 1.6c. The schematic potential energy surface in the space of these two coordinates (Figure 1.6a and b) for the ground and excited states is shown in Figure 1.7. For our purposes, we imagine that the starting point of the excited state cycloaddition, the Franck–Condon geometry, corresponds to two isolated ethylene molecules, and the product is cyclobutane in a square planar geometry.
(Continues…)
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Copyright © 2018 Michael A. Robb.
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