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.

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 P′ (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 P′, 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.^3–6 

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^8–12  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.⁷  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 photoexcitation process. (For a recent study with dynamics see the work of Martinez et al.¹³ ) We can represent the three states with valence bond pictures (ground state I, LE: II and CT: IIIa <->IIIb, 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 S_i 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 coordinate⁷  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.⁷  The experimental aspects, see for example the discussion by Zacharise¹⁴  or Martinez,¹³  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^16–25  in three dimensions. For an introductory article on conical intersections in photochemistry see Robb,²²  which is available as a free download.²⁶ 

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.

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.²⁷ 

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).^28–31  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.

We can use the two “sheets” of the potential surfaces shown in Figure 1.7 to compare and contrast what might happen in a thermal and a photochemical reaction. We have distinguished two molecular motions X₁ and X₂ in which to plot the surfaces. The variable X₁ is a reaction coordinate corresponding to the approach of the two ethylenes (Figure 1.6a and b). The variable X₂ is a rhomboidal distortion (Figure 1.6c). As we will presently discuss, radiationless decay must involve two such distinguished coordinates. In contrast, a transition state, on the ground state A′, is associated with one distinguished coordinate, X₁ in this case, corresponding to the reaction path. The discussion is similar to that for DMABN except that here we work in three dimensions.

On the ground state surface (Figure 1.7) there are two possible transition states, shown as A′ for the synchronous reaction, where both bonds are formed simultaneously, and C for the asynchronous reaction were one bond is formed first (Figure 1.6d). The Woodward–Hoffman (WH)³⁰  rules predict that the asynchronous reaction, via C, has the lower energy.

Now let us examine a region of the potential energy surface along a line connecting the two transition states A′ and C (X₂). We can see that the ground state energy passes through a very high-energy point E where the ground state and excited states become degenerate. This is known as a conical intersection,^32–34  as first discussed in chemistry by Zimmerman,³⁵  Michl³⁶  and Ruedenberg.³⁷  Recent reviews are available in several places.^16–25  At this point, we notice again (cf. Figures 1.3, 1.4 and 1.5) that a double cone at point E requires the two coordinates X₁ and X₂ to describe it.

Now, notice on the excited state sheet that E is the lowest energy point. The excited state reaction progresses on the upper potential energy surface and is assumed to begin at the FC geometry, corresponding to two separated ethylenes. The reaction would progress along a coordinate leading to a minimum A if the system were constrained to have a rectangular D_2h geometry. Notice the avoided crossing at the points A and A′; first discussed in Figure 1.5.

The excited state potential energy is unstable along the X₂ coordinate, rhomboidal distortion, as shown in Figure 1.6c. Along a reaction path directed towards the point E there is a negative direction of curvature so that A, rather than being a local minimum, is in fact, a transition state along a reaction path leading to the point E. Thus the geometrical changes corresponding to reaction paths on the excited state are quite different from the ground state. The motion which brings the two ethylenes together along the coordinate that preserves rectangular symmetry is a maximum on the ground state involving a transition state at A′; in contrast, it is a local minimum on the excited state at A. However this excited state reaction path is not stable, and a lower energy pathway is available, which involves motion along the rhomboidal distortion coordinate, leading via point E to the ground state asynchronous pathway at point C via a conical intersection at point E. Thus the point E plays two roles:

• (1) It is the lowest energy point on the excited state energy sheet. But it is not a minimum because the gradient of the energy is not zero; rather it is a singularity.

• (2) Point E is a conical intersection point where the energy of the ground and excited states are the same.

The potential energy surfaces shown in Figure 1.7 can be easily rationalized using VB theory. The two VB structures are shown in Figure 1.8. As one can see, in the VB structure I the bonding arrangement of two ethylenes corresponds to the initial part of the ground state potential energy surface, which is repulsive as one brings the two ethylenes together. In contrast, the VB structure II in Figure 1.8 corresponds to the formation of the two new bonds (dashed lines). Of course, it is attractive, and corresponds to the excited state minimum at point A. Trapezoidal distortion of A leads downwards to the conical intersection E. Why does the energy of the upper sheet go down in this way? Well, if we look at the structure of point E we see that we are forming an incipient cross bond that will correlate with the bi-radical structure C.

The cartoon in Figure 1.7 embodies many of the ideas associated with the concept of a chemical mechanism. A mechanism is a sequence of molecular structures through which the reaction passes on its way from reactants to products. The difference for an excited state reaction mechanism is that some of these structures lie on the excited state, viz. structure A, and others lie on the ground state, e.g. structure C. The reaction path thus has two segments, the excited state and the ground state, divided by the conical intersection point E. In a reaction occurring on a single potential energy surface, we are mainly interested in the shapes associated with minima and transition states. In contrast, an excited state reaction is nonadiabatic, so we have a new type of molecular structure, a conical intersection, where the reaction passes from the excited state to the ground state in a radiationless nonadiabatic event.

Before leaving the discussion of Figure 1.7, it is important to mention that this Figure is a “cartoon”. With present-day computational methods, one computes the geometries of points where the gradient is zero, such as minima and transition states. One can also compute the energies and geometries of low-energy conical intersection points.³⁸  The cartoon that one draws in Figure 1.7 is intended to convey the shape of the potential energy surface and the way in which various critical points, minima, etc., are connected, rather than presenting the results of actual computations on a grid.

If the mechanistic information just discussed in Figure 1.7 is to be really useful, then it must be an intrinsic property of the chromophores themselves, i.e. the two ethylene molecules. So let us examine this idea using another 2+2 cycloaddition reaction, the intra-strand thymine dimerization in DNA (see Figure 1.9a), which is recognized as the most common process leading to DNA damage under ultraviolet (UV) irradiation.^39,40  The formation of thymine dimers can disrupt the function of DNA and trigger complex biological responses, including apoptosis, immune suppression and carcinogenesis. In Figure 1.9b we show the geometry corresponding to the point E in Figure 1.7 as well as the computed directions³⁹  X₁ and X₂ for the 2+2 cycloaddition reaction of two thymine molecules. Accordingly, we can regard the cartoon in Figure 1.7 as a “picture” of the mechanism of a general photochemical 2+2 reaction. We will see that such cartoons can be obtained from computation, or inferred from simple ideas of electronic structure mainly derived from VB theory. However, in “real” 2+2 cycloadditions, the real situation is more complicated⁴⁰  because the reaction begins in a different state. The VB states I and II in Figure 1.8 are “dark” states and cannot be reached directly via optical excitation. So the mechanistic picture needs more detail than we have given so far.

Why should we look for such generalities? The answer, as we have just stated, lies in the fact that the cartoon in Figure 1.7 is a “picture” of the “mechanism” of a general photochemical 2+2 reaction. Detailed electronic structure computations³¹  can provide the geometries and energies of the various mechanistic points in Figure 1.7. However, without some sort of mechanistic insight (e.g. Figure 1.7) one cannot generalise the results to different systems. The mechanistic insight comes from “thinking VB”.

The preceding example can be rationalized using VB theory involving four orbitals and four electrons. Another case where a mechanistic analysis is possible using VB theory is the case of three orbitals and three electrons. We now discuss such an example.

The photoinduced ring opening (see the horizontal axis in Figures 1.10 and 1.11) of 1,3-cyclohexadiene (CHD) to cZc-hexatriene (HT)^42–52  is another “classic” photochemical reaction like the 2+2 cycloaddition of two ethylenes. The topic has been reviewed by Deb.⁵³  In fact this problem involves three electronically excited states, as well as Rydberg states. The excitation of CHD generates a zwitteronic excited state (1B₂ in Figure 1.10), the reaction path then crosses to a covalent excited state (2A₁) in Figure 1.10, and then to the ground state (1A₁) by another crossing. In the present discussion we will focus on the second part of the reaction, namely the crossing of the two covalent states. (But of course, the details of the 1B₂/1A₁ crossing may control the ultimate product distribution.) As we shall see this can be understood in terms of the VB model with three orbitals and three electrons. We should also see that in this case we need three geometric variables: X₁X₂ (shown as X_1/2 in Figures 1.10 and 1.11) to describe the space of the nonadiabatic transition at the conical intersection and a “reaction co-ordinate” X₃. We emphasize again that these pictures are cartoons. We focus in this case on three geometric variables; however, the cartoons are extracted from computations that treat all the geometric variables.

The main mechanistic points are illustrated schematically in Figure 1.11. The reaction coordinate is the conrotatory ring-opening motion. This motion is plotted as the horizontal axis in Figure 1.11. The energy profile along this co-ordinate shows an avoided crossing (see Figure 1.5) of the 2A₁/1A₁ states. In the 2+2 cycloaddition of two ethylenes there were two directions X₁ and X₂ that corresponded to the space of the double cone of the conical intersection. In Figure 1.11 we show only one of these two directions, which we denote as X_1/2. It is clear that radiationless decay takes place via a conical intersection only following displacement, from the conrotatory ring-opening path, along this coordinate X_1/2, which is orthogonal to the reaction coordinate X₃. This is in contrast to the excited state molecular motion shown in Figure 1.5, where the reaction path lies in the space X₁X₂ of the conical intersection itself, and the reaction is like “sand in a funnel”. We now discuss the CHD ring opening mechanism, which involves three molecular coordinates, in a little bit more detail.

Nonadiabatic decay during a photochemical reaction was first clarified mechanistically by van der Lugt and Oosterhoff.^54,55  The central idea uses the concept of an avoided crossing, which forms the photochemical funnel,⁴  arising from the interaction of the ground state and an excited state along a common reaction co-ordinate (along X₃ in Figure 1.11). Note also, that, earlier in the progression along the reaction co-ordinate, there is another avoided crossing between the initially photoexcited 1B₂ state, the ionic, HOMO–LUMO state and the doubly excited 2A₁ state, which is the dark covalent state. This latter feature is associated with the forbidden vs. allowed nature in WH theory. The work of Nenov et al.⁵⁶  showed that the reaction path does indeed pass via the avoided crossing between the 2A₁ and 1A₁, but does not extend to the conical intersection that lies along an orthogonal co-ordinate X_1/2 in Figure 1.11.

This example provides the opportunity to introduce some quite general aspects associated with the concept of the mechanism of radiationless decay. As we discussed in the previous paragraph, the central concept is associated with three distinguished molecular coordinates: X₁X₂ associated with the double cone of the conical intersection as shown in Figure 1.5, and the reaction coordinate X₃, as shown orthogonal to X_1/2 in Figure 1.11. While the reaction coordinate can lie in the plane X₁X₂, as shown in Figure 1.5, i.e. the sand in the funnel picture, in general, the reaction coordinate is independent of X₁X₂. In the limiting case, the reaction coordinate X₃ can be completely orthogonal to X₁X₂ over most of its range. In fact, that is what happens in the ring opening of CHD. As we show in Figure 1.11 we have an extended seam^22,57  of intersection lying approximately parallel to the excited state reaction path X₃. This seam in Figure 1.11 was computed via a seam-following minimum energy path method (seam-MEP).⁵⁸  The seam-MEP is just the steepest descent path in the restricted space (orthogonal to X₁X₂) where the two excited states are degenerate. Thus motion orthogonal to the reaction path in the direction of the seam X_1/2 must control the ultrafast decay to the ground state.

We now discuss Figure 1.11 in a little more depth since it illustrates a general mechanistic paradigm. In Figure 1.11 we show (i) the ground state reaction path (CHD, , cZc-HT and a second ), (ii) the excited state reaction path (with labels FC, CHD* and ) and (iii) the S₁ seam-MEP⁵⁸  (with labels CoIn_bu, , , , and ) The complete minimum energy S₀ and S₁ reaction paths (X₃) for the conrotatory ring-opening reaction of CHD cover the region from the closed (CHD) to the open ring structure (HT). Then, displaced along a skeletal deformation co-ordinate X_1/2, some vector in the plane X₁X₂, we see the almost parallel corresponding conrotatory S₁ seam-MEP.⁵⁸  The avoided crossing feature on the excited state branch of reaction path X₃ is thus displaced from the lowest energy point of the conical intersection seam, shown as CoIn_min in Figure 1.11. Note that the seam-MEP has local transition state features (such as and ) and corresponding seam-imaginary frequencies are also given in Figure 1.11. These frequencies were obtained using the second order representation of the seam and correspond to motion along the intersection space, dominated by bond breaking.

In summary, in the WH treatment of photochemistry, as reformulated by van der Lugt,^54,55  the excited state and ground state reaction paths were assumed to be similar, with the “photochemical funnel” occurring at an avoided crossing. In this classic example, computations show that the ground state and excited state reaction paths (i.e. X₃ in Figure 1.11) are indeed very similar. However the conical intersection seam, which provides the locus of radiationless decay, is displaced from the excited state/ground state MEP along the coordinate X_1/2 in Figure 1.11.

We now look at another example where the surface crossing involves an extended seam: cyanine dyes. Cyanine dyes have proven to be a very useful model system for the study of photochemical cis–trans isomerization, for both theory^59,60  and experiment.^61,62  The structure of one such cyanine dye that has been studied experimentally is shown in Figure 1.12. Notice the extended conjugated π system connecting to nitrogen terminal groups (the two adjacent benzene rings have been treated as inert substituents for the present discussion). Notice also the ionic nature of the structure that exists in two valence bond forms with the charge on different N atoms. Different cyanine dyes have different lengths chains of conjugated carbon atoms connecting the ionic nitrogens. The simplest model system would have three carbon atoms and this is the system that has been studied theoretically.^59,60 

In Figure 1.13, we show a cartoon of a potential energy surface obtained as the result of theoretical calculations.⁵⁹  In a similar fashion to Figure 1.10, in Figure 1.13 we plot the energy as a function of two geometrical parameters. In Figure 1.10, we choose the variable torsion X₃, the reaction co-ordinate for double bond isomerization. As a second geometrical variable X_1/2, we choose skeletal deformation, labelled asymmetric stretch. In Figure 1.13 we see again (viz. Figure 1.8) that the two potential surfaces intersect along a “seam” (shown as CI seam in Figure 1.13) rather than a point. Notice, however, that in this example the reaction path meets the seam eventually near the lowest energy point on the potential surface.

The dynamics of photochemical reactivity that occurs on the potential energy surfaces of the form shown in Figure 1.13 can be understood using the idea of a ball that rolls on the potential energy surface, tracing out a trajectory as a function of time. One such trajectory is shown in Figure 1.13. In this trajectory, after photoexcitation to the excited state surface, the system has some initial momentum along a skeletal, wavy line, deformation co-ordinate, as it moves more slowly downhill along a torsional co-ordinate. It is clear from Figure 1.13 that the point at which the system encounters this seam depends upon the distribution of momenta between the fast skeletal deformation vibrations and the slow torsional motion. If there is a lot of energy in the skeletal deformation co-ordinate (X_1/2) initially, then the surface crossing may be encountered early along the torsional reaction co-ordinate and will decay in the region of the reactants. Alternatively, as shown in Figure 1.13, if the amount of energy in the skeletal deformation co-ordinate is smaller, then the system may decay after relaxation near the bottom of the potential well. In fact, this is what is observed experimentally by Dietzek et al.⁶²  They demonstrated that one can effectively “turn off” the cis–trans isomerization if sufficient energy is placed into high-frequency vibrational modes by laser excitation. This is an example of what is known as a coherent or intelligent control. We shall return to this idea subsequently. Quantum dynamics illustrate this idea nicely⁶⁰  and we shall look at this in some detail in the last chapter.

Now that we have briefly introduced the way in which the shape of the potential surface determines reactivity, we turn briefly to the problem of understanding why ground and excited state surfaces may have different shapes. Clearly, understanding this connection between the change in electron distribution on photoexcitation and the subsequent changing of the positions of the nuclei is an essential feature of a photochemical mechanism. We want to understand what the electrons are doing in the ground and excited states and what the nuclei do in response to that electronic change and thus how the potential surface changes.

Let us illustrate this idea with a simple example (Figure 1.14), which shows some potential energy curves for butadiene in its ground (S₀) and excited states (S₁). For simplicity we have ignored the zwitterionic B₁ that is also present and shown for hexatriene in Figure 1.10. The types of questions that we will be continually asking are:

• (1) What are the “shapes” of potential energy surfaces involved? In Figure 1.14 we used S₁ or S₀ to denote the first excited singlet state and ground state, respectively. We shall refer to these states as the adiabatic states.

• (2) Are the states singlet or triplet spin multiplicity?

• (3) What is the nature of the geometrical change that occurs on absorption of light? What are the electrons doing? We shall use the term diabatic states as corresponding to states where the electronic configuration does not change.

Figure 1.14 is an example of the way that we will understand the shapes of potential energy surfaces in terms of simple VB ideas. In the potential energy diagram shown in Figure 1.14 we have plotted the energy of the ground and excited state against the bond length of the middle bond in butadiene. On the extreme left of the figure we show two VB structures for ground and excited state electronic configurations, or, more correctly, diabatic states. Classically, we would draw the VB ground state butadiene with two double bonds, linked by one single bond. One excited state of butadiene corresponds, electronically, to a VB isomer of the ground state with a double bond in the middle of the molecule and two un-paired electrons on the terminal methylenes. Thus we “label” the potential curves with a VB structure, i.e. the diabatic state.

The next issue is to understand why the potential energy curves have the shape they do in terms of these two VB structures. For the ground state, the S₀ ground state minima occurs on the right-hand side of the figure corresponding to a “long”, i.e. single, middle bond length. However, when one populates the excited state, the electronic structure changes and corresponds to the VB structure with a double bond in the middle of the molecule. Thus the minimum of the excited state potential energy curve S₁ is found at a short middle C–C bond length, consistent with the excited state VB structure. Further, when the molecule is vertically excited, at the long middle bond length corresponding to the ground state equilibrium geometry, it will arrive on the excited state at a geometry that is not an equilibrium geometry. The first thing that will happen on the excited state is that the molecule will relax and the nuclear geometry will change, because the electronic structure corresponds to a double bond for the middle bond, and the nuclear geometry is thus in the wrong place. Thus, after arriving on the excited state surface, in butadiene, the central bond length will shorten and the outer bond lengths will increase as the molecule moves to a new position on the excited-state potential energy surface where the central bond length is short.

This idea is quite general; one populates an electronic excited-state, following vertical excitation. Electronically the bonds are in a different place from on the ground state. The position of the atomic nuclei no longer matches the electron distribution and the initial geometry on the excited state is not an equilibrium geometry, and consequently the nuclei will move until a new equilibrium on S₁ is reached, as shown in Figure 1.14. This change in the electron distribution on the excited state is the driving force for photochemical change. Here, in butadiene, the bonds are in the wrong place on the excited state and therefore the geometry of the molecule will relax until the geometry is consistent with the electronic structure.

We now wish to collect the ideas of the previous two subsections into some general discussion about a photochemical reaction, or a general nonadiabatic transformation. The textbook discussion of a photochemical reaction path^3,64  is summarized in Figure 1.15. Starting at the geometry GS₁, an excited state (EX₁) is created at the same geometry. The system then evolves on the excited state surface, on the different reaction paths that are possible, depending upon initial conditions. The reaction path may progress to conical intersections, points CI₁ or CI₂, where decay to the ground state occurs and the ground state reaction path progresses to GS₁ or GS₂. Thus the photochemical reaction path has two branches, one on the excited state and one on the ground state.

It has now been established, by both theoretical computations and complementary experiments, that the point where the excited state reaction path and ground state reaction path are connected is a conical intersection.^16–25 

Conical intersections have been known since the 1930s.^32–34  Zimmerman,³⁵  Michl³⁶  and Ruedenberg³⁷  were among the first to suggest that internal conversion occurring at a conical intersection was the key feature to enable understanding certain photochemical mechanisms. Modern theoretical developments began to occur once the necessary theoretical methods^58,65–70  were developed that enabled the location of minimum energy points on conical intersections (MECI). The location of many such MECI at low energy has demonstrated that such features are an essential part of photochemistry. The historical development of the subject was summarized elegantly by Michl, in the preface of a collective volume on conical intersections.²¹  Organic reactivity involving conical intersections was reviewed by Olivucci et al.¹⁸  in this same volume. We continued this discussion in the second volume of this series²²  where we focused on the extended nature of the conical intersection seam.

As we have just discussed, to rationalise excited state chemical reactivity semi-quantitatively in the region of a conical intersection, three “model” coordinates X₁X₂ and X₃ are required.^41,58  The coordinates X₁ and X₂ describe the “branching space” of the conical intersection (double cone insert in Figure 1.15). Motion along these two directions lifts the degeneracy at the apex of the cone. The third coordinate X₃ denotes the “intersection space”.²²  This coordinate is intended to be representative of all the remaining nuclear coordinates, explicitly excluding the branching space.³⁷  Plotting the energy in the space of X₃ and either X₁ or X₂ (i.e. X₁₂) shows a “seam”, as illustrated in Figure 1.15.

Of course, there are other types of nonadiabatic transition other than photochemical decay, e.g. singlet–triplet transformation as well as ionization. Since electronic states with different spin multiplicity do not mix, motion on the triplet surface will progress through a crossing with the singlet surface without any interaction. We get a true nonadiabatic event, corresponding to crossing from the triplet to the singlet, only if we have spin orbit interaction. In this case, the singlet and the triplet states mix in the passage from singlet to triplet corresponding to a transition state at an avoided crossing. The most important point is that the crossing of the singlet and triplet state is (n−1)-dimensional, i.e. one less than the crossing of two singlet states. Thus one of the characteristic directions of a surface crossing, X₁X₂, disappears. We return to this topic in the next chapter.

Thus the main feature of single excited state reactivity is embodied in Figure 1.15 and focuses on the locus of the nonadiabatic event at a surface crossing, which is referred to as a conical intersection. This is the point where an excited state reaction path, controlled by transition states, minima, etc., is transformed into a ground state reaction path. Of course the ground state reaction path can be studied using conventional methods of quantum chemistry. However, the quantum chemistry for the excited state is more challenging because one must obtain a balanced representation of two states rather than one. Furthermore, the idea that a reaction path a simply flows downhill from reactants to products may not be valid (see Figure 1.15 for example) because in a nonadiabatic event there are three coordinates involved: X₃, the reaction path, and the branching space, X₁X₂, associated with the locus of radiationless decay at the conical intersection. We now introduce some of the main features and constraints associated with electronic structure computations on the electronically excited state.

Electronic structure computations coupled to semi-classical or quantum dynamics.

The question we want to ask in this last introductory section is “how is an electronic excited state computed within the orbital based methods used in ground state chemistry?” Our objective is not so much to go into the detailed quantum chemistry, but rather to have some discussion of excited state computation within the configuration interaction method and the associated algebraic eigenvalue problem. The ideas then occur in context in methods like the CIS (CI singles), TD-DFT (time-dependent density functional theory) and CASSCF (complete active space SCF). Our objective here is only to give a simplified discussion of where potential energy “surfaces” have come from. Further, reactivity corresponds to “rolling a ball” on these surfaces, i.e. dynamics. So we will briefly introduce the formulation of classical dynamics within quantum chemistry.

Quantum chemistry “solves” the Schrödinger equation within the algebraic approximation, which replaces the solution of the Schrödinger equation with expansions and an algebraic eigenvalue problem. The central features are:

• (1) Orbitals are commonly expanded in Gaussian type functions.

• (2) The SCF equations, for one-electron orbitals, are solved within LCAO (linear combination of atomic orbitals, i.e. AO, expansion) as an optimization of the energy with respect to linear variation of coefficients (the weights of the AO in the expansion of an SCF MO).

• (3) Many electron wavefunctions, i.e. the many-electron states in the configuration interaction method are built from orbitals as linear combinations of determinants leading to the algebraic eigenvalue problem.

• (4) Perturbation theory is an approximate solution of the eigenvalue problem by series expansion.

• (5) DFT and related methods can be understood in terms of an “effective” algebraic Hamiltonian in which correlation and other effects are “folded in” to yield “dressed” integrals. By “dressed” integrals we mean that, for example, the two electron repulsion integrals over the operator r$12−1$ are replaced either (i) by an integral over a function rather than r$12−1$ itself, or (ii) the integral is replaced by a series expansion from perturbation theory.

Here we focus on the CI method and discuss how one characterizes an excited state in such a formalism.

CI is a variational method for solving the electronic Schrödinger equation that leads to an algebraic eigenvalue problem. We can write the wavefunction of a state K (with energy E_k) as |Ψ_K〉 as a linear combination of N-electron basis functions:

The expansion functions |Φ_λ〉 can be Slater determinants (SD_λ) or spin functions (e.g. Yamanouchi–Kotani (YK_λ) spin coupled basis eigenfunctions of (Ŝ²Ŝ_z) built from orbitals. Some examples of the determinants that can be built from three-orbitals and three-electrons are shown in Figure 1.16.

The two types of basis functions in use are shown in eqn (1.2a), Slater determinants (SD_λ) and eqn (1.2b), eigenfunctions of spin (Ŝ²), such as the Yamanouchi–Kotani (YK_λ) spin coupled basis

This aspect is a specialized topic in quantum chemistry^71,72  and we shall not discuss it in detail.

This approach (expansion SD_λ or YK_λ) gives rise to an algebraic eigenvalue problem, eqn (1.3) and (1.4), where 〈Φ_λ|Ĥ|Φ_μ〉 are the Hamiltonian matrix elements (eqn (1.4a)) and U and ε are matrices collecting the eigenvectors and eigenvalues (eqn (1.4b))

In eqn (1.3) the Φ_λ are the expansion functions, say determinants, in eqn (1.1) built from the orbitals. The Kth column of U contains the individual eigenvectors {c$λK$}, λ=1 … in eqn (1.1) with energy ε_κ.

Any eigenvalue ε_κ can be shown to be an upper bound to the corresponding exact excited state energy^73,74  only if the trial wave function is orthogonal to all the lower lying states of the same symmetry. Thus, within the CI method the excited states are given, approximately, as the higher roots of the equation system (1.3). As the space of n-electron basis functions is increased we approach the exact energy, for the infinite basis from above, as shown in Figure 1.17. Of course this is only true if the configurations are built from the same set of orbitals.

The computation of the matrix elements H_λμ=〈Φ_λ|Ĥ|Φ_μ〉 on the basis of determinants or spin coupled basis is summarized in eqn (1.5).

The terms 〈a|h|b〉 and [ab|cd] are the integrals over the orbitals while are so-called “symbolic” matrix elements for H_λμ that give the contribution of the integral 〈a|h|b〉 to 〈Φ_λ|Ĥ|Φ_μ〉. The computation of such matrix elements is a “solved problem” that is incorporated in most electronic structure programs. A perspective article contains the historical development.⁷⁵ 

Of course the algebraic eigenvalue problem, eqn (1.2) and (1.3), is different for each position of the nuclei R. Thus we have a potential energy surface ε_k (R) for each state K. In practice one does not compute the potential surface. The gradients, eqn (1.6), and second derivatives are usually sufficient. Thus there are critical points on the surface corresponding to minima where the gradient, eqn (1.6), goes to zero. One can define the second derivatives in a similar fashion:

If we know the gradients and second derivatives (Hessian) then we can easily write down the equation for the classical dynamics on any potential surface:⁷⁶ 

We have written eqn (1.7) in a representation where the hessian is diagonal with eigenvalues λ_i. Thus the dynamics equations are simple when they are written in the space of normal modes i. Then g_i is the gradient component along normal mode i, and q_i is the position, and is the acceleration. The integrated form of eqn (1.7) then has the simple form (for real λ_i)

Of course such equations need to be solved iteratively over the region of the potential surface that is quadratic.

In summary, given the algebraic CI eigenvalue problem, one can evaluate the gradient for state K via eqn (1.5) and the corresponding Hessian eigenvalues (λ_i). Then the time evolution of the system is given by propagation using the standard Newtonian equations of motion. Further, mimima, etc. are defined by the gradient and the Hessian.

The various numerical electronic structure approaches differ in detail leading to various realizations of wavefunctions of a state K (energy E_K) as |Ψ_K〉 as a linear combination of N-electron basis functions. We will discuss these ideas in detail subsequently. For the discussion in Chapter 2 on the shapes of potential surfaces we will use the VB method, which we introduced qualitatively in this chapter.

Many of the approaches used to study the excited state have their origins in perturbation theory. This is easily formulated within the algebraic eigenvalue problem so we end this section with a brief introduction. There are two essential features associated with perturbation theory that are relevant for the discussion of excited states:

1. the partition of a set of states into two subsets: A, the reference or active space, which contains the excited states of interest, and B, the secondary space, and

2. the partition of the Hamiltonian into a zeroth order part H₀ and a higher order part V in the reference space, and a similar partition of W (eqn (1.9)) in the secondary space into E_λ and W−E_λ so that the algebraic eigenvalue equation can be written

where is the Kth column of U in eqn (1.4).

It is straightforward to rewrite eqn (1.9) in terms of an effective Hamiltonian in the A (reference space) space (H^eff(K)) as defined in eqn (1.10a) and (1.10b). One derives this by formally solving for U$KB$ in eqn (1.9) and substituting in the first A partition:

One has a perturbation series by expanding (E_K−W)⁻¹ in eqn (1.10).

It is possible to convert this to an energy independent form⁷⁷ 

Here, the “higher order terms” come from the expansion of E_K shown in eqn (1.12)

The terms in eqn (1.12) correspond to the folded terms of Brandow.⁷⁷  We use the direct sum symbol ⊕ to denote that we are taking the appropriate column or row of Z.

One can then expand the inverse in eqn (1.10) and (1.11) via the formula in eqn (1.13)

to give a perturbation series expansion.

H^eff has some remarkable properties via the solution of eqn (1.10a). Firstly it is energy independent so that all of the eigenvalues associated with the subspace A are obtained by diagonalization of this operator. Secondly it is non-Hermitian. All this really means for us is that the eigenvectors are not orthogonal. This latter point is a practical issue. But we are never going to use this equation numerically, in any case, in our discussions. Rather we will focus on understanding the electron correlation problem for excited states via this Hamiltonian. This is a useful aspect from our point of view. The effective Hamiltonian produces solutions within a reference space that are essentially exact. The effect of the secondary space occurs only in the “dressing” of the Hamiltonian via the second term in eqn (1.11). We will see that this “dressing” has a simple physical origin that enables us to understand the effects of electron correlation on excited states and thus design computations that take this into account efficiently. However from the point of view of physical interpretation we can restrict our attention to the active space. The practical issue is associated with the partition of the CI space into a reference space and a secondary space. In order to do this one requires some physical insight and this is obtained via VB theory, as we shall discuss subsequently.