Vibronic Coupling Effects in Spectroscopy and Non-adiabatic Transitions in Molecular Photodynamics

Abstract

A brief historic and systematic survey is given of our efforts to elucidate important features of the nuclear motion on interacting potential energy surfaces (PESs). Starting with our early work in 1977, a variety of small to medium-sized polyatomic molecules have been treated by quantum-dynamical methods. As the key topological feature signalling the effects in question, conical intersections of PESs have been established. The associated strong nonadiabatic coupling effects manifest themselves as diffuse (at low resolution) or irregular (at high resolution) spectral structures upon electronic transitions. The concomitant fs electronic population decay governs the photophysical and photochemical properties of these systems. Representative examples with 2–5 strongly coupled electronic states are given, and the quantum nature of the phenomena is emphasized.

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6.1 General Introduction

In this contribution our quantum-dynamical treatment of nuclear motion on coupled potential energy surfaces (PESs) is surveyed. The present general introduction deals with the early history of the field (i.e. from 1977 to 1991), followed by a more systematic classification of types of surface intersections and a phenomenology of the ensuing nuclear dynamics.

6.1.1 Outline of Early History

In 1974 the group of E. Heilbronner in Basel had measured the photoelectron spectrum of the butatriene molecule, which displays a peculiar shape [1]. In the ionization energy range from 9 to 10 eV they observed three electronic bands where only two states of the radical cation and, hence, only two bands are expected. Moreover, the bands show an asymmetrically broadened profile and a rather complex structure (see Fig. 6.1, left panel). The upper and lower energy bands 1 and 2 could be correlated with electronic states of the radical cation, whereas the intensity of the central band 1′ far exceeds any conceivable mechanism for a corresponding electronic feature of the system. Thus it was also termed “mystery band”. In subsequent theoretical work it could be established that the “mystery” band represents a vibronic effect [2]. By setting up a suitable model Hamiltonian for coupling to and through vibrational modes, and based on ab initio values for the system parameters, the band system could be very satisfactorily reproduced, see the central panel in Fig. 6.1. It proved crucial that the model not only describes vibrational excitation (coupling to modes) but also allowed for interaction between the electronic states (coupling through modes). This is made apparent by the right panel where this latter interaction has been suppressed in the calculation and the mystery band is absent [3]. Apparently the interaction leads to a redistribution and build-up of spectral intensity in the centre and high-energy part of the spectrum, which is vital for the irregularities and the appearance of the mystery band of butatriene.

Fig. 6.1

Comparison of the experimental photoelectron spectrum of butatriene with the “full” theoretical result and the one obtained in the Condon approximation with uncoupled PESs (from left to right). The band 1′ in the left panel was termed “mystery band”

Fig. 6.2

Cuts through the two-dimensional model PES of the butatriene radical cation underlying the theoretical treatment of Fig. 6.1. One coordinate (Q _g or Q _u ) is varied as indicated in the panels, and the other fixed at zero.

In later work it was realized that the key feature of the underlying PESs is a conical intersection (CoIn) [3, 4]. This can be understood from Fig. 6.2 which represents cuts (cross sections) through the ionic PESs along the two nuclear coordinates considered in the model. These feature shifts in opposite directions along the totally symmetric mode Q _g (C = C stretching mode) which leads to a crossing of the PES near the minimum of the upper state (the zeros of coordinate and energy scale denote the neutral ground state equilibrium geometry and energy). For displacements along the non-symmetric coordinate Q _u a repulsion of the PES occurs due to the associated symmetry lowering, which enables an interaction between the states to take place (for Q _u  = 0 the electronic states have different symmetry). The repulsion leads to a double minimum PES of the ionic ground state, which in turn is responsible for the low-energy progression in the spectra (between 9 and 9.5 eV). The situation is summarized in the perspective drawing of Fig. 6.3 which illustrates the double minimum of the lower adiabatic PES as well as the point of degeneracy between upper and lower adiabatic PES in the centre of the figure. This degeneracy is lifted in first order of the nuclear displacements and is thus of conical shape. From simple arguments the nonadiabatic coupling elements are seen to diverge at these points, causing a severe failure of the Born–Oppenheimer approximation for energies at and above that of the intersection [3]. This is just borne out by the comparison of the spectra in the central and right panels in Fig. 6.1.

Fig. 6.3

Perspective drawing of the two-dimensional model PES of the butatriene radical cation

What first seemed little more than a theoretical curiosity, of interest only to a few experts, proved later to be of broad relevance, and with ramifications in many different areas of physical chemistry and chemical physics. Nonadiabatic processes following molecular photoexcitation have received broad attention from experiment and theory alike, and CoIns are nowadays recognized as key topological features of molecular PES signalling these effects [5–10]. The above example opened the door to a quantum-dynamical treatment of these phenomena by the Heidelberg group, which has been pursued and widened ever since. In the present contribution I attempt to give a lucid, historic, and also pedagogic overview over the field as it evolved since around 1980, with an emphasis on our line of work. This also fits into the scope of this book, because the quantum nature of the dynamics at CoIns has been a focus of our work over all the years. While nuclear motion is sometimes treated classically or semiclassically in the literature, the consideration of quantum effects is important in at least three different situations, regarding zero point energy effects, tunneling processes, and coupled electronic PESs. General reasoning in this direction has been presented in the introduction to this book and need not be repeated here. Suffice it to say that a semiclassical treatment of nuclear motion on coupled PES (surface hopping, mean path approach, etc.) has been proposed and is further discussed in another chapter of this book. An a priori justified such treatment in the general case, however, remains problematic and I confine myself to citing a recent in-depth discussion of the topic by one of the pioneers in the field [11].

In the following I continue this outline of historic developments, followed by more systematic considerations on different coupling scenarios. As an important early step, the analysis of the photoelectron (PE) spectrum of ethene revealed that strong nonadiabatic coupling effects can be operative also for seemingly distant electronic states, with well-separated spectral bands [12]. Namely, the combination of several active vibrational modes can lead to a CoIn also when the energetic separation of the states in the centre of the Franck–Condon (FC) zone is as large as 2 eV. While nowadays also relevant CoIns with vertical energy gaps of the PES of 4–5 eV are known (and will be mentioned below) this result proved striking at that time, and strongly indicated that only the energetic accessibility of the CoIn matters for the dynamics, thus rendering it much more important than originally thought. Similarly, it was realized that the well-known paradigm of the dynamic Jahn–Teller (JT) effect [13] is to be considered a special case of a CoIn. In JT systems the electronic degeneracy is enforced by symmetry and, by the very nature of the JT theorem, is lifted in first order of the displacements when distorting the system along the coordinate of a JT active mode. From the point of view of structural chemistry, this makes the symmetry lowering “spontaneous” and occurring independently of actual values of the coupling parameters. Regarding the dynamics, the JT-split PES necessarily take a conical shape near the high-symmetry nuclear configuration, rendering the nuclear motion highly nonadiabatic after an electronic transition from a nondegenerate state (which usually takes the system to the point of degeneracy). In early work, the PE spectra of allene and pentatetraene were investigated [14] and the doubly degenerate ground states identified as structurally related to the cases of ethene and butatriene mentioned above [15]. This established a homologous series of molecules (“cumulene radical cations”) with a systematic trend in the PES and dynamics. As a further instructive example we show in Fig. 6.4 the first band of the PE spectrum of benzene [16]. The electronic ground state of the radical cation is doubly degenerate in the D_6h point group, and there are four JT active vibrational modes (also doubly degenerate). The photoionization from the nondegenerate neutral ground state creates the radical cation directly at the point of degeneracy (CoIn) and the vibrational modes excited in the spectral band are supported by both JT-split PESs simultaneously. An adiabatic separation would fail equally badly as already demonstrated in Fig. 6.1. This is emphasized because the vibrational structure is quite regular which can thus not be taken as evidence for the validity of the adiabatic (or synonymously: Born–Oppenheimer) approximation, even in such seemingly simple cases. We further see from Fig. 6.4 that in well-resolved spectra as the present one the quantum nature of the vibrational modes is evident, i.e. the discrete level structure becomes directly visible by inspection.

Fig. 6.4

Comparison of experimental (lowest panel) and theoretical (upper two panels) first photoelectron band of benzene. The upper panel represents the results for the JT active modes as indicated in the panel, the central one displays the full result, obtained by including the totally symmetric C = C stretching mode ν ₂

While these developments were important early indications for the relevance of CoIns and nonadiabatic coupling effects, the seminal contributions by Ruedenberg [17], Robb [5, 6], Yarkony [7, 8] and their groups, and subsequently also Martinez and coworkers [18], provided breakthroughs for the field (mostly in the 1990s) and made it known to a broad range of physical and organic chemists and beyond (see also the important work of Klessinger, Michl and Bonacic–Koutecky [19, 20]). In our group, at about the same time the investigations were becoming broader regarding the phenomena addressed, the introduction of different coupling scenarios and also by considering more than two strongly interacting electronic states. These more systematic aspects shall be addressed in the following sub-section.

6.1.2 Methodology and Phenomena

6.1.2.1 Diabatic Electronic States

The complexity of the theoretical treatment strongly suggests, and often necessitates, to introduce suitable model Hamiltonians. This holds not only in view of the quantum treatment of the nuclear motion but also regarding the underlying electronic structure (ab initio) computations, which are always an essential part of the theoretical methodology. Global PES in more than 6–8 dimensions can hardly be obtained accurately even on present-day computer hardware. The set-up of suitable models is usually combined with the use of a so-called diabatic electronic basis [21–23]. The latter is of special importance since it avoids the singular derivative couplings appearing in the usual adiabatic basis at degeneracies of different molecular PESs (say V ₁(Q) and V ₂(Q)). Denoting the corresponding adiabatic electronic wavefunctions by Ψ ₁(Q) and Ψ ₂(Q), we have the well-known identity (see, for example, [24])

$$\displaystyle{ \langle \varPsi _{1}(Q)\vert \frac{\partial } {\partial Q}\vert \varPsi _{2}(Q)\rangle = \frac{\langle \varPsi _{1}(Q)\vert \frac{\partial H} {\partial Q}\vert \varPsi _{2}(Q)\rangle } {V _{2}(Q) - V _{1}(Q)} }$$

(6.1)

The diabatic wavefunctions are smooth functions of the nuclear coordinates Q also at these degeneracies V ₁(Q) = V ₂(Q), and so are the potential energy matrix elements in this basis. This allows to introduce a Taylor series expansion which may sometimes be truncated at low order, thus defining the linear (LVC) or linear-plus-quadratic (QVC) coupling scheme [3]. The LVC or QVC schemes have been successfully employed by the Heidelberg, Munich and other groups, and can naturally be extended to carry the expansion to third or fourth (or similar) order. More recently, also a systematic approach to go to considerably higher order has been suggested [25].

The LVC and QVC approaches avoid the explicit construction of diabatic states because they result in a parametrized form of the adiabatic PES; this can be used to determine the coupling parameters by comparing the parametrized form of the adiabatic PES with results from electronic structure calculations for these surfaces (diabatization by ansatz [26]). On the other hand, the fixed functional form leads to a model shape of the PES which may not always be flexible enough to reproduce these data well. To overcome this limitation, a modified construction scheme for the diabatic potential matrix has been introduced [27, 28] where the LVC approach is applied to the adiabatic-to-diabatic (ADT) mixing angle “only”. In this form it can be applied to general adiabatic PESs as given from electronic structure calculations and reproduce them without loss of generality. It can be shown to eliminate the singular derivative couplings rigorously (in principle) and has therefore been coined construction scheme of “regularized diabatic states” [28, 29]. Interestingly, the relevant system parameters can again be determined from potential energy data alone, at least in many relevant cases, for example, if a symmetry is present. The scheme has been successfully applied and its properties have been examined for a Jahn–Teller system [27], a seam of symmetry-allowed CoIns [28] and a general CoIn with two relevant degrees of freedom [29]. Nowadays the concept of regularized states is typically applied to systems with 3 or 4 atoms (H₃ [30], NO₂ [31], SO₂ [32], C₂H₂ [33]) while our treatment of larger systems (such as aromatic and heteroaromatic molecules with 10–12 nuclear degrees of freedom and 2–5 strongly coupled electronic states) relies on the QVC approach.

6.1.2.2 Classification of Two-State Intersections

For apparent reasons the most common type of surface intersections will comprise two states. Here we give a brief survey on their different topologies and symmetries. Three-state intersections (and higher ones) will be discussed below.

Quite generally, the nonadiabatic (or derivative) couplings diverge whenever two different molecular PES become degenerate (intersect). The very notion “conical intersection” implies not only a point or seam of degeneracy but also a topology in the immediate vicinity, namely, a lifting in first order of the nuclear displacements from the degenerate subspace [3, 6, 17]. The phrase “glancing intersection” is used for cases where this degeneracy is lifted in second order only [34, 35], the most prominent example being the Renner–Teller effect in linear molecules (with the bending angle as active coordinate) [36]. Apart from this special situation, a surface degeneracy will mostly be associated with a first-order lifting, i.e. be of the conical type [37]. This is also the generic situation for symmetry-induced degeneracies in nonlinear molecules, where the JT theorem “guarantees” the conical shape as already indicated above.

The appearance of surface intersections can also be understood as a generalization of the famous von Neumann–Wigner noncrossing rule, which states that in diatomic molecules the potential curves of states of the same symmetry do not cross [38]. The simplified proof given by Teller [39] casts the conditions for degeneracy into two algebraic equations, which cannot be normally satisfied at the same time when only a single parameter (the internuclear distance) is to be varied. For a polyatomic system with N atoms and 3N-6 (or 3N-5) internal coordinates this rule is immediately relaxed, and 3N-8 (or 3N-7) coordinates can still be varied after two coordinates have been fixed to fulfill the two aforementioned conditions for degeneracy. The subspace of degeneracy (“intersection space” or “seam”) is thus of rather high dimension, while that in which the degeneracy is lifted (“branching space” or “branching plane”) has dimension two [3, 5–8, 17]. It emerges naturally that surface intersections are an ubiquitous phenomenon and play an important role for excited-state dynamics as induced by photoexcitation in the visible or UV spectral range.

It is instructive to address the different types of intersections from the point of view of symmetry (see, for example, [35]). The key vibronic coupling constant λ, which governs the first-order interaction between different electronic states $\varPsi _{1}(Q_{0})$ and $\varPsi _{2}(Q_{0})$ at the nuclear geometry Q ₀, can be written as

$$\displaystyle{ \lambda \,=\,\langle \varPsi _{1}(Q_{0})\vert \frac{\partial H} {\partial Q}\vert \varPsi _{2}(Q_{0})\rangle }$$

(6.2)

Since the molecular Hamiltonian H transforms totally symmetric, λ does not vanish by symmetry only if the following selection rule is fulfilled [3]:

$$\displaystyle{ \varGamma _{1} \otimes \varGamma _{2} \otimes \varGamma _{Q} \supset \varGamma _{A}. }$$

(6.3)

Here the irreducible representations Γ on the l.h.s. denote those of the electronic states and nuclear mode, in an obvious notation, and Γ _A stands for the totally symmetric irreducible representation of the respective point group. Several cases can be distinguished.

Without any symmetry, or if Γ ₁ = Γ ₂ is nondegenerate, the coupling mode Q is totally symmetric (possibly trivially so). No restriction on the point of degeneracy is given by symmetry alone. This is also called accidental intersection and plays a central role, for example, in organic photochemistry (see the contribution by Robb and coworker [40]).

If Γ ₁ and Γ ₂ differ, the coupling mode Q is non-totally symmetric (coined Q _u ). It can thus not coincide with (first-order) Franck–Condon active modes which are characterized by the decomposition of Γ ₁ ⊗Γ ₁ or Γ ₂ ⊗Γ ₂ and are totally symmetric (in Abelian point groups). The latter are also called tuning modes in the LVC scheme which forms the body of our early work in the field. The conical intersection is usually termed symmetry-allowed in such a case since it normally occurs (for a single coupling mode) in the subspace Q _u  = 0 where Γ ₁ ≠ Γ ₂ and a free crossing is possible.

The situation becomes somewhat more complicated (but also richer) in non-Abelian point groups. Limiting ourselves to two-state coupling scenarios, we have Γ ₁ = Γ ₂ in case of (double) degeneracy. This amounts to a symmetry-induced (also called symmetry-enforced) degeneracy or intersection. By virtue of the JT theorem [13, 41], there exist always non-totally symmetric modes to fulfill the selection rule (6.3). This directly leads to the JT or symmetry-induced conical intersections mentioned earlier. An exception is provided by linear molecules, where such modes do not exist and a coupling between the components of a doubly degenerate (for example, Π) state is provided only by terms of second order in the (bending mode) displacements [41]. These second-order terms necessarily cause the intersection to be of glancing rather than conical type.

As a final distinction, we mention a difference between “trigonal” (such as C _3v or D _3h ) and “tetragonal” (such as C _4v , D ₄, D _2d ) point groups. In the first case doubly degenerate modes are JT active, in the second case nondegenerate, non-totally symmetric modes are JT active [3]. In the former case the JT-split PESs have cylindrical or (more generally) threefold symmetry, in the latter case the symmetry is twofold, thus reflecting the different symmetry of the underlying nuclear framework.

6.1.2.3 Dynamics at Conical Intersections

It has already been reiterated above that nonadiabatic couplings generally diverge at degeneracies of different molecular PESs. A vast amount of numerical results has shown that these degeneracies are almost invariably of the conical type. This is also supported by general theoretical reasoning [37]. In what follows I will therefore identify degeneracies with conical intersections, also bearing in mind that in all specific cases reported below the intersections are of the conical type.

From semiclassical reasoning it is expected that nonadiabatic coupling effects are strong for energies at and above that of a CoIn. At a CoIn, nearly isoenergetic vibrational eigenfunctions of the intersecting PES have similar classical turning points, rendering their overlap integral substantial. For lower energies either the vibrational states are not isoenergetic or their overlap becomes exponentially small. This general situation was already exemplified by the study of the ethene radical cation mentioned above [12]. Strong nonadiabatic couplings manifest themselves in distinct ways, in either the energy or time domain. Concerning the vibronic structure of electronic spectra they lead to a thorough mixing of upper and lower-surface vibrational wavefunctions [3]. This implies a highly irregular line structure, and any regular FC patterns will be completely lost. If the CoIn occurs well outside the FC zone, the interacting electronic states give rise to well-separated spectral bands; for the higher-energy band, the spectral intensities of the vibronic eigenstates will derive from the admixture of the upper-surface vibrational levels and their density from the lower-surface levels (being of much larger density than the upper-surface levels). Under low resolution the spectral envelope will look diffuse, and in extreme cases this will hold even under high resolution (provided the intrinsic lifetime broadening exceeds the average line spacings). Concerning the time domain, the system undergoes an ultrafast internal conversion (IC) process from the upper to the lower PES, proceeding on the same time scale as the nuclear vibrations. If the CoIn, on the other hand, is not classically accessible to the nuclear motion, and a “surface hop” becomes possible only through tunneling, the nonradiative transition is slowed down and may become much slower than vibrational periods, ultimately reaching the golden rule limit of traditional radiationless decay theory.

The heteroaromatic radical cations furan, pyrrole, thiophene represent a beautiful series of molecules to illustrate this dependence [42, 43]. Their two lowest electronic states exhibit a characteristic change in their vertical ionization potentials such that their difference substantially decreases in the series. As a consequence also the minimum energy of the resulting CoIn decreases, not only absolutely but also in relation to the upper adiabatic PES, see Fig. 6.5 for a schematic drawing. For furan the CoIn occurs well above ( ∼ 0.5 eV) the upper state minimum, for pyrrole and thiophene it is very close (within 0.01 and 0.04 eV). Thus the IC process in the latter cases is expected to compete with the vibrational motion, while in the former it should be slower. This is nicely confirmed by the electronic populations obtained from ab initio-based WP propagations and displayed in Fig. 6.6. A typical stretching vibrational period amounts to ∼ 20 fs, which is the time scale of the ${}^{2}B_{1} {-}^{2}A_{2}$ population transfer (IC) for pyrrole and thiophene. In the case of furan, on the other hand, the IC time scale is considerably slower ( ∼ 150 fs) which is completely in line with the higher energy of the surface crossing in Fig. 6.5. Consequently, also the PE spectral bands of the three systems differ in a characteristic way [42, 43]. In furan the nonadiabatic couplings lead to small line splittings and moderate broadening effects only (forming non-overlapping quasi-resonances), while in pyrrole and thiophene they cause a complete redistribution of spectral lines with no correlation between unperturbed vibrational levels and vibronic eigenstates. The high density and irregularity of vibronic eigenstates in these two examples is similar to the cases of $\mathrm{C}_{2}\mathrm{H}_{4}^{+}$ (see Sect. 6.1.1) and excited states of Bz⁺ to be discussed below.

Fig. 6.5

Cut through the lowest PES of the furan, pyrrole, and thiophene radical cations along the coordinate of an effective mode. The latter is given by a straight line connecting the minimum of the ${}^{2}A_{2}$ ionic ground state (taken to be the zero of energy in all cases) to the minimum of the CoIn between the ground and first excited (${}^{2}B_{1}$ ) ionic states. The dotted lines refer to thiophene, the dashed ones to pyrrole and the full lines to the furan radical cation

Fig. 6.6

Electronic populations of the ground and first excited electronic states of the three radical cations given in the panels, following a vertical excitation to the upper (² B ₁) electronic state. Consistent with the energetic positions of the surface crossings in the preceding figure, the transition occurs within a single vibrational period for pyrrole and thiophene, but is considerably slower in the case of the furan radical cation

6.1.2.4 Two-State vs. Three-State Coupling Scenarios

Degeneracies or near-degeneracies of more than two electronic states may play a role in two distinctly different ways. They can (1) affect the dynamics sequentially, i.e. as series of two-state (quasi-)degeneracies, which the system encounters separately during its time evolution and which both play a comparable role. An example of this type has been reported to occur in highly excited states of formaldehyde, severely affecting its VUV absorption spectrum [44]. (2) More than two electronic states may come close or intersect also at a single point in nuclear coordinate space. These three-state intersections are of particular interest and should appear frequently as accidental intersections (see above) in non-Abelian point groups, when a doubly degenerate (by symmetry) electronic state may cross freely with a nondegenerate state. (In principle, though less likely in practice, also two doubly degenerate states of different symmetry may cross freely thus producing a fourfold degeneracy.) A typical scenario is given by the first and second excited states of the benzene radical cation (Bz⁺), which are doubly degenerate and nondegenerate, respectively (of ${}^{2}E_{2g}$ and ${}^{2}A_{2u}$ symmetry in the D _6h molecular point group) [45, 46]. This system features the coexistence of two different pairs of electronic (component) states with two different nonadiabatic coupling mechanisms and an ${}^{2}A_{2u} {-}^{2}E_{2g}$ energy gap which changes sign as a function of the C = C stretching coordinate Q ₂. For low energies there is nonadiabatic coupling within the ${}^{2}E_{2g}$ manifold, which is characterized by a sparse and rather regular level structure such as occurs for the ground electronic state of Bz⁺ (Fig. 6.4). For high energies the crossing point of the ${}^{2}A_{2u} {-}^{2}E_{2g}$ PESs plays a decisive role for the dynamics. This amounts to a triple degeneracy as is displayed by the perspective drawing of Fig. 6.7. The vibronic structure of the PE spectrum [45, 47] becomes similarly complex and erratic as discussed above for the ethene, pyrrole and thiophene radical cations.

Fig. 6.7

Perspective drawing of a rotationally symmetric three-state (“triple”) conical intersection such as occurs, for example, at an accidental crossing of a doubly degenerate state with a nondegenerate state. The coordinates are the components of a doubly degenerate mode, causing a linear coupling between the electronic states

The triple intersection of Fig. 6.7 deserves special mentioning with respect to low-energy vibronic motion. It has been pointed out above that the surface topology near a degeneracy (conical or not) is not of central importance for the strength of the nonadiabatic coupling effects. For low-energy motion this shape does matter, and conical intersections are highlighted by the appearance of the geometric phase phenomenon [48, 49]. Very briefly, the conical nature of the PES topography necessarily leads to a sign change of the real electronic wavefunction whenever the point of degeneracy is encircled in a closed loop. Since the whole wavefunction must be single valued, the nuclear wavefunction also has to undergo a sign change, which amounts to a different boundary condition for this (“pseudorotational”) motion [50]. In other words, the vibronic eigenvalues are affected by the presence of the CoIn, although it may be high in energy and not be in reach for the nuclear motion. This is counterintuitive, but not contradicting the above statements about the adiabatic approximation, because the product nature of the molecular wavefunction is not affected. Rather, it may be termed adiabatic separation with modified boundary conditions.

As is well known, glancing intersections as occur in the Renner–Teller effect do not exhibit the geometric phase phenomenon [51], and neither do triple intersections like the one in Fig. 6.7 [52]. This can be traced to the different surface topography near the degeneracy (glancing intersections) and to the cancellation of phase contributions from the different degenerate pairs (triple intersections). The situation is illustrated by the drawings of Fig. 6.8 which shows a time-dependent WP evolving on a double (right panels) or triple (left panels) intersection [53]. Otherwise the PESs in question are identical in the two sets of panels. By virtue of the geometric phase there is destructive interference in the case of a two-state conical intersection but constructive interference for a three-state intersection. This shows up nicely in the lowerhalf-plane, opposite to that where the initial WP is located, and leads to a completely different shape of the WP for longer propagation times (see lowest panels).

Fig. 6.8

Comparison of time-evolution on doubly (right panels) and triply (left panels) intersecting molecular PES. The lowest adiabatic PES on which the evolution starts is the same in all panels. The effect of constructive (no geometric phase) and destructive (geometric phase) interference shows up in the third panels from above and leads to a completely different shape of the WP for longer times (lowest panels)

I mention that this type of surface topography and WP evolution plays a role for the two-photon ionization spectrum of the sodium trimer, and considerations of the type presented here have led to a thorough re-interpretation of the experimental spectrum [52, 54]. For a more complete overview over different coupling scenarios regarding interactions within the degenerate state and between the degenerate and a nondegenerate state, I refer to [55].

6.1.2.5 Vibronic Coupling and Localization Phenomena

We briefly mention that near-degenerate electronic states may also emerge for systems with weakly coupled subsystems, such that the symmetry-adapted linear combinations of the locally excited states show only a rather small energetic separation. Despite this weak interaction between locally excited subsystems the vibronic interactions in the whole aggregate or molecule may be substantial. Two prominent cases are (1) molecules with several equivalent core hole sites; here, the vibrational motion may destroy the nominal equivalence of the sites and lead to excitation of asymmetric modes in odd quanta in the spectrum (proceeding on diabatic PES with localized core holes) [56–58]. Another case are (2) dimeric systems with weakly coupled subsystems (e.g. through hydrogen bonding or long covalent linkers). Again, asymmetric modes can be excited, which may lead here to a substantial reduction of the excitonic energy splittings and amount to nonadiabatic tunneling on the lowest (double-well) adiabatic PES of the dimer [59–61].

6.2 Applications in Spectroscopy

In the previous section we have already discussed a few examples of two-state vibronic coupling effects in spectroscopy. They all relied on the LVC or QVC approach and were based on extended ab initio calculations for the underlying system parameters (frequencies and coupling constants). The latter is a generic element of our approach and applies to all examples presented here; for details we refer to the original publications.

In the present section I discuss examples where the treatment has been extended in two different ways, either going beyond the QVC approach or including more than two strongly coupled states in the analysis.

Fig. 6.9

Comparison of experimental (upper panel) and theoretical (lower panel) first photoelectron band system of ozone, corresponding to the lowest electronic states of the radical cation. Peak A in the experimental recording is assumed to be a hot band

Over the years, a series of related triatomic molecules and cations has been investigated by the Heidelberg group, notably NO₂ [31], $\mathrm{O}_{3}^{+}$ [62], and SO₂ [32, 63], and also H₃ [30, 64]. NO₂ and SO₂ are famous (and notorious) showcases in molecular spectroscopy with exceedingly complex spectra which could only incompletely be assigned despite intense efforts by several groups. An even partly systematic list of references is well beyond the scope of this short article. The heavy mixing of vibrational wavefunctions of different molecular PES is an immediate cause for such complications. Consider the photoelectron spectrum of ozone in Fig. 6.9, which shows a comparison of experimental (upper panel) with theoretical (lower panel) results [62]. In the theoretical treatment the line spectrum actually computed has been phenomenologically broadened with Lorentzians (FWHM = 0. 04 eV) to account for finite experimental resolution. Overall, the experiment is well reproduced, in particular, the regular peak progression at lower energies and the irregular structures at higher energies. Two electronic states contribute to the spectral intensity, the ${}^{2}A_{1}$ ground and ${}^{2}B_{2}$ first excited states of $\mathrm{O}_{3}^{+}$. The regular progression can be understood as excitation of the bending mode because of different equilibrium geometries in the neutral and ionic ground states. The situation can be visualized in Fig. 6.10 which shows contour line drawings of the relevant PES for C_2v geometries. The ground and first excited states of $\mathrm{O}_{3}^{+}$ have their minima near bond angles of 130^∘ and 105^∘, respectively, while the centre of the FC zone occurs near 117^∘. Following the FC principle, extended bending progressions are expected not only for the ${}^{2}A_{1}$ state as observed, but also for the higher-energy ${}^{2}B_{2}$ state. For the latter state the experimental spectral profile shows distinct irregularities, and the theoretical results reveal their origin, namely, a highly irregular underlying line structure beyond an energy of ∼ 12.8 eV. This is just the minimum energy of a seam of conical intersections also displayed in Fig. 6.10. It is symmetry-allowed according to the classification of Sect. 6.1.2.2, and further exemplifies corresponding statements of Sect. 6.1.2.3 about the energy range of strongly nonadiabatic motion. The seam is substantially bent and affected by the anharmonicity of the C_2v PES of the system. This has been fully included in the numerical computations, while the coupling has been taken to be a linear function of the asymmetric stretch coordinate (LVC scheme).

Fig. 6.10

Contour line drawing of the C _2v PES of the ozone radical cation. The single full (bent) line represents the seam of ${}^{2}A_{1} {-}^{2}B_{2}$ conical intersections

In later work, the concept of regularized diabatic states (Sect. 6.1.2.1) has been more fully adopted to treat the cases of NO₂ and SO₂. In Fig. 6.11 we show the dependence of the vibronic coupling constant λ on the symmetric stretch coordinate or bond length r (which is basically the seam coordinate) near the seam minimum for NO₂ [31]. It is seen to increase markedly for increasing bond length r which corresponds to an increase of the energy along the seam and to a larger energy after photoexcitation. It rationalizes the (originally surprising) fact that for lower energies (namely, in the photodetachment spectrum) the spectral irregularities are considerably smaller than in the visible absorption spectrum of NO₂ which probes the dynamics for higher energies and which is well known for its notorious complexity [65]. The UV absorption spectrum of SO₂ is similarly intricate and could recently be well reproduced by us theoretically, see Fig. 6.12 [32]. Under low resolution, it features a seemingly regular progression of the bending mode, extending until  ∼ 34,000 cm⁻¹ [66]. However, there is a very complex line structure under the spectral envelope which becomes itself most irregular in the higher energy part of the spectrum. The work of [32] is the first in the literature achieving the remarkable agreement with experiment shown, and I refer to this reference for all further details of the analysis, including a comparison with uncoupled-surface spectra.

Fig. 6.11

Variation of the vibronic coupling constant λ along the ${}^{2}A_{1} {-}^{2}B_{2}$ conical intersection seam of NO₂

Fig. 6.12

Comparison of experimental and theoretical UV absorption spectrum of SO₂

Fig. 6.13

Cut through the lowest 8 PESs of the benzene radical cation along the coordinate Q ₁₈ of a doubly degenerate mode. The filled symbols denote the individual ab initio data points, the interconnecting lines the fitted model PES. The crossings at the origin Q ₁₈ = 0 represent cuts through JT intersections, those at Q ₁₈ ≠ 0 are cuts through pseudo-JT intersections

Let us now address a multi-state coupling problem where up to five interacting states have been included in the treatment, namely, the benzene radical cation Bz⁺. Figure 6.13 shows results of ab initio calculations for the lowest five electronic states of Bz⁺ along the two components of the JT active coordinate Q ₁₈ [46]. The latter is doubly degenerate and so are three of the electronic states, amounting to eight component states or PESs in total. The individual symbols denote the ab initio data points, while the contiguous (full and dashed) lines stand for the fitted model curves resulting from the LVC approach. Overall, an impressive set of mutually crossing PES is revealed with a whole network of CoIns which lets one expect a rich variety of vibronic phenomena. The crossings at the origin Q ₁₈ = 0 represent cuts through JT intersections while those for Q ₁₈ ≠ 0 involve a degenerate (at the D_6h point) and a nondegenerate state, which we term pseudo-JT effect or intersection [45].

Fig. 6.14

Second and third PE spectral bands of benzene as computed with the present vibronic coupling approach. The upper panel corresponds to the (${}^{2}E_{2g}$ ) first excited state, the lower panel to the (${}^{2}A_{2u}$ ) second excited state. The composite theoretical band compares favourably with experiment

The (${}^{2}E_{1g}$ ) ground state intersection represents the JT effect in this state which has already been documented in the spectrum of Fig. 6.4. The next electronic state (${}^{2}E_{2g}$ first excited state) is likewise subject to the JT effect with similar implications for the spectrum, see Fig. 6.14 (upper panel) [47]. There is moderate excitation of the JT active modes with a rather sparse level structure which has been assigned in the original work. Moreover, Fig. 6.13 gives evidence of a slightly higher, nondegenerate (${}^{2}A_{2u}$ ) state which approaches and also crosses the ${}^{2}E_{2g}$ state along the C = C stretching coordinate Q₂. This, among others, leads to the triple intersection mentioned in Sect. 6.1.2.4. We thus have a pseudo-JT intersection and coupling between the ${}^{2}A_{2u}$ and ${}^{2}E_{2g}$ states which, in conjunction with the JT coupling in the ${}^{2}E_{2g}$ state, leads to the coexistence and interplay of two different interaction mechanisms for the different pairs of component states [45]. Ultimately, this also renders the motion in the nondegenerate state highly nonadiabatic and leads to a complex vibronic structure in the corresponding PE band, see lower panel of Fig. 6.14 [47]. There, numerous individual lines are seen (despite being somewhat small in the drawing) to contribute to the broad and structureless spectral envelope. Their density exceeds that of the unperturbed upper-surface vibrational levels by about two orders of magnitude. The computed spectral profile for the composite band is in fair agreement with the observations. The very high density and complexity of the vibronic lines resembles the situation, for example, in the ethene, pyrrole, and thiophene cations mentioned above; however, the coexistence of different coupling mechanisms renders the situation even more complex in the present example.

The situation becomes further intriguing after realizing that, in addition, some branches of the JT-split ${}^{2}E_{1g}$ and ${}^{2}E_{2g}$ PES intersect each other. This leads to an ultrafast IC process in Bz⁺ comprising all five component states of the ${}^{2}E_{1g}$, ${}^{2}E_{2g}$, and ${}^{2}A_{2u}$ electronic manifold. More details and, in particular, the effect which fluorination has on these multi-state nonradiative transitions will be discussed in the next section.

6.3 Applications in Photophysics and Photochemistry

In the present section we address time-dependent phenomena governed by CoIns and strong nonadiabatic couplings, notably the electronic population dynamics of strongly coupled PES and the consequences it has on observable quantities. It has already been demonstrated above in Sect. 6.1.2.3 that, if the CoIn is in reach energetically, the IC process may proceed on the time scale of nuclear vibrations (otherwise it is slower). However, it should be emphasized that the actual nonradiative transition can be even faster. A striking example is provided by the Rydberg de-excitation of triatomic hydrogen: here the population transfer from the upper to the lower adiabatic sheet of the JT-split ground state PES of H₃ proceeds within only 4 fs [30, 67]! The reason is that for the Rydberg de-excitation, starting from a near-D_3h geometry, the initial state in the decay process is generated directly at the JT ground-state intersection seam. It does not need to time-evolve towards the CoIn, and the population transfer gives a direct measure of the nonradiative transition itself. In most other cases the time evolution of the electronic population includes the time needed for the system to arrive at the CoIn, and it is thus more an upper limit to the time needed for the actual “surface hop”.

Fig. 6.15

Time-dependent electronic populations of the $\tilde{C}{(}^{2}A_{2u}) -\tilde{ B}{(}^{2}E_{2g}) -\tilde{ X}{(}^{2}E_{1g})$ manifold of Bz⁺ following a vertical transition to the $\tilde{C}{(}^{2}A_{2u})$ state of the system. The full calculation, with all degeneracies retained (upper panel) is compared with an approximate one where the electronic and vibrational degeneracies are suppressed (lower panel)

6.3.1 Ultrafast Internal Conversion and Its Competition with Fluorescence

Let us now return to Bz⁺, discussed in Sect. 6.2. For this prototypical system the ${}^{2}E_{2g}$ and ${}^{2}A_{2u}$ set of states has also been treated by time-dependent methods and the aforementioned population transfer from the upper to the lower state been found to proceed within 20–30 fs (one period of the C = C stretching mode is 33 fs) [45]. This parallels the pyrrole and thiophene cases and other systems treated by us over the years. Moreover, also the complete set of all above CoIns comprising the lowest five component states of Bz⁺ (within the LVC approach) could be treated quantum dynamically [47] using the powerful MCTDH wavepacket propagation method [68, 69]. The resulting time-dependent electronic populations are presented in Fig. 6.15. They reveal a step-wise nonradiative transition—within 100–200 fs—from the (dipole-allowed) ${}^{2}A_{2u}$ over the (dipole-forbidden) ${}^{2}E_{2g}$ state to the ${}^{2}E_{1g}$ ground state of the radical cation. This is remarkable in view of the large energy gaps of ∼ 3 eV involved, the indirect couplings operative and the sub-ps dynamics resulting from the multi-state interactions [47]. It provides a natural explanation for the absence of detectable fluorescence in Bz⁺ [70, 71] which cannot compete with the efficient IC process identified. We mention in passing that the whole set of CoIns identified for Bz⁺ comprises also higher energy electronic states included in Fig. 6.13, and a (more approximate) modeling has been undertaken to account for them also [72].

Fig. 6.16

Correlation diagram between ionization potentials of benzene and its mono-, di- and a trifluoro-derivative. Some of the molecular orbitals are also shown for comparison

A further intriguing perspective is offered by the fluoro derivatives of Bz⁺. It has been known for a long time experimentally that the monofluoro derivative is non-fluorescent like the Bz⁺ parent cation [73]. The difluoro derivatives constitute a limiting case, while the trifluoro and all further, more completely fluorinated cations do exhibit (strong) fluorescence [71, 73, 74], which has actually been used to gather substantial information on the structural and spectroscopic properties of the systems (see, for example, [75–79]).

Fig. 6.17

Time-dependent electronic populations of the lowest five electronic states of the mono- and 1,2,3-trifluorobenzene radical cations (lower and upper panel, respectively). The initial electronic state is visible from the curves

It is gratifying that the present coupling mechanism can naturally account for these interrelations. Consider the correlation diagram of Fig. 6.16 between the orbitals and state energies of Bz⁺ and some of its fluoro derivatives [80]. There are two noteworthy trends visible in the figure, first the expected lifting of the degeneracies due to the loss of symmetry upon asymmetric fluorination and, second, a gradual but substantial shift to higher energy of the $\tilde{{B}}^{2}E_{2g}$ —derived states. The latter relate to ionization out of a σ orbital which is C–F bonding and stabilized energetically by fluorination. On the other hand, the π-type ionization potentials, correlating with the $\tilde{{C}}^{2}A_{2u}$ and $\tilde{{X}}^{2}E_{1g}$ states of Bz⁺, stay nearly the same as for the parent system. One should be aware that in Bz⁺ the $\tilde{{C}}^{2}A_{2u}$ and $\tilde{{X}}^{2}E_{1g}$ states are not coupled directly, but rather indirectly, through the $\tilde{{B}}^{2}E_{2g}$ state as intermediate [46]. (The same holds for the fluoro derivatives because of high energies of the corresponding CoIn seams in all species [80–82].) The increase in relative energy of the $\tilde{{B}}^{2}E_{2g}$ —derived states in the fluoro derivatives causes a similar energetic increase of the respective seams of CoIns and thus an effective weakening of the coupling between their two lowest ($\tilde{{X}}^{2}E_{1g}$ -derived) states and their next higher excited states. Consequently also the population transfer to the ground state is slowed down and becoming increasingly less efficient for an increasing number of F atoms in the compound [80, 81, 83]. I refer to the original literature for the actual numbers and all further details, and confine myself to presenting the time-dependent populations of Fig. 6.17 as representative examples. The figure shows the populations of the lowest five electronic states of the mono- and 1,2,3-trifluoro derivative [80, 81], which correspond to those shown in Fig. 6.15 for the parent cation (due to the degeneracies for Bz⁺ only three curves are shown there). The initial state is the B₁ state for F-Bz⁺ (same as for the parent system in Fig. 6.15) and the next higher state (B₂ state) for the trifluoro derivative in question. The reason for the latter choice is to be able to show any nontrivial dynamics at all, because for the B₁ state as initial state the electronic populations stay virtually constant over time.

Comparing Fig. 6.17 with Fig. 6.15 we easily notice the characteristic change of the IC process with increasing fluorination. For the monofluoro derivative the populations evolve in a similar way as in the parent system Bz⁺, and the upper state for a potential dipole-allowed transition, the B ₁ state, again decays on a femtosecond time scale. For the trifluoro derivative 3F-Bz⁺ the situation is entirely different, and the B ₁ population stays constantly large. The competition between the IC process and fluorescence thus disappears according to the theoretical treatment and, in line with this finding, 3F-Bz⁺ shows clear emission, as do the higher fluoro-derivatives of Bz⁺ [71, 73, 74]. The reason, as already stated above, is the energetic increase of the B ₂ state and hence, of the CoIn between its PES and that of the A ₂ and B ₁ states deriving from the ground state of Bz⁺ (see Fig. 6.16). The difluoro cations represent an intermediate case where only the 1,3 (meta) isomer shows weak emission [73]. Even this more subtle difference seems to be captured qualitatively by our treatment [83]. I conclude this discussion by mentioning that there are further characteristic changes upon (partial) fluorination which have to do more with the symmetry breaking in general than with fluorination in particular [80, 81, 83]. The interested reader is referred to the original literature for more details.

6.3.2 Elementary Photochemical Transformations

So far the discussion has focussed on rather small-amplitude displacements affecting photophysical properties like fluorescence behaviour. Now I address radiationless processes associated with larger-amplitude displacements or photochemical transformations. This is indeed the subject of ongoing interest in the literature and of intense work by several groups. In our work, we have, for example, considered the quantum dynamics of nonadiabatic cis–trans isomerization of acetylene following excitation to the S ₂ state [33]. This state features a CoIn seam with the bending mode potentials of the lower-lying S ₁ state. Even through this symmetry-lowering the two states do not interact unless the torsional mode is activated and any symmetry is lost in the system. The equilibrium geometry of the S ₁ state is trans-bent while the S ₂ state takes a cisoid shape (C _s symmetry). From there the IC process takes the system over the out-of-plane distortion to the trans-bent structure of the S ₁ state where it preferentially remains, although not entirely. The complexity of the UV absorption spectrum of acetylene is severely affected by the nonadiabatic interactions, although the pronounced anharmonicity associated with the deep wells of the bending mode potentials also plays an important role [84].

Fig. 6.18

Potential energy curves of the five lowest singlet states of pyrrole along the N–H stretching normal coordinate Q₂₄

Fig. 6.19

Time-dependent electronic populations (upper panel) and reactive flux (lower panel) for the relevant states of pyrrole after excitation to the B ₂ state of Fig. 6.18

The radical cations of the five-membered heterocycles furan, pyrrole, and thiophene have already been discussed above. The photophysics and photochemistry of the neutral species have received considerably more attention in the literature. Concerning furan, I only mention the theoretical (dynamical) studies on the ring-opening process following UV photoexcitation [85–87]. In pyrrole, in addition N-H photodissociation comes into play as a further radiationless deactivation channel (see, for example, [88–90]). Following earlier work by Domcke and co-workers [91, 92] we have extended the quantum-dynamical treatment by considering also the higher-excited states populated by a dipole-allowed transition from the ground state. Figure 6.18 shows the potential curves of the lowest five singlet states of pyrrole along the N–H stretching normal coordinate Q₂₄. Near the ground state equilibrium geometry there are two π −σ ^∗ excited states, followed by two π −π ^∗ excited states [92, 93]. The latter carry most of the oscillator strength, the former decrease in energy for increasing N–H bond length and cross with the nominal $A_{1}(S_{0})$ state in the asymptotic region. All four excited states are interconnected through a set of CoIns similar to the case of Bz⁺ discussed above. Within the QVC approach the excited states would be decoupled from the $A_{1}(S_{0})$ state, but by virtue of the anharmonic potentials along Q ₂₄ there can be a sequence of IC processes comprising all five states in question, extending over an energy range of ∼ 6 eV and leading to N–H photodissociation on three different channels, see Fig. 6.18. This is indeed confirmed by extensive WP calculations using the MCTDH method [68, 69], for which a selected result is shown in Fig. 6.19 [94]. The electronic populations in the upper panel show indeed that the population is transferred within less than 50 fs from the optically bright π −π ^∗ state to one of the π −σ ^∗ states where it can dissociate. Moreover, a small fraction undergoes a further transition to the $A_{1}(S_{0})$ state where it can also dissociate, but with a considerably smaller excess energy. This is of interest because photofragmentation is known to lead to slow and fast H atoms in the system [89]. Using well-known techniques [95, 96], the reactive flux can be extracted from the present WP calculations, and the results are shown in the lower panel of Fig. 6.19 [94]. The figure indeed gives evidence of N–H photodissociation on the three channels visible in Fig. 6.18. Consistent with the smaller population of the $A_{1}(S_{0})$ state, also the reactive flux on this state is smaller than for the other states. The flux on the $A_{1}(S_{0})$ state can be considered to represent slow H atoms while that on the π −σ ^∗ states represents fast H atoms. From the results of [94] their ratio, and in particular its energy dependency can be determined. In line with observations, the ratio of slow to fast H atoms is found to increase with increasing energy of the system. This agreement is intriguing, although the absolute values (of the ratio) are considerably too small in comparison with experiment [89]. Presumably other vibrational redistribution mechanisms, such as occurring through the ring-puckering CoIns emphasized by Lischka et al. [93], should be included to arrive at a more quantitative agreement regarding the absolute numbers. To conclude this section I mention the extensive work by Domcke and Sobolewski to demonstrate the vital importance of nonadiabatic photochemistry governed by CoIns for the photostability of biologically relevant molecules and related processes [97, 98].

6.4 Outlook and Future Perspectives

In the present contribution I have attempted to give a condensed overview over our work on the quantum dynamics on vibronically coupled PES, considering pedagogic, historic, and systematic aspects in a balanced way. Time-independent and time-dependent phenomena have been addressed. While the examples discussed were necessarily not exhaustive, a set of representative cases has been selected which should cover the most important aspects and their interrelations. In particular, the quantum nature of the phenomena is apparent whenever the discrete level structure of the vibronically coupled states plays a role. Also regarding electronic populations, and other time-dependent quantities not explicitly addressed here, a general-purpose, reliable, and accurate treatment requires to consider the quantum nature of the nuclear motion.

Special realizations of vibronic coupling systems, such as occur for well-localized, weakly coupled molecular subsystems have been touched upon only briefly, in Sect. 6.1.2.5. Another special case, that of light-induced CoIns [99, 100] has not been discussed above, but is expected to emerge as highly relevant in strong laser fields in the future. This may, of course, also apply to other realizations of specific systems, and likewise regarding their experimental detection.

General lines of future developments may concern, for example, applications in photobiology and the related design of methods dealing with increasingly larger systems. The use of the MCTDH method has already defined new limits in applications of the QVC scheme, where systems with more than 20 nuclear degrees of freedom can be treated accurately (including the introductory example of the butatriene cation discussed in Sect. 6.1) [101–103]. To carry the developments systematically further, more extended strategies may prove helpful or even crucial. As done in electronic structure theory, the whole system of interest may be divided into a central part and external parts treated with successively decreasing accuracy. For example, the inner part should be treated with the highest-level methods, allowing for finer details to be covered accurately. The next outer part could be described by the LVC or QVC approach, the subsequent one be considered as an “environment”. For the latter, efficient effective-mode schemes have been introduced recently which capture the influence of many modes by a hierarchy of 3, 6 or 9 (etc.) modes which describes the evolution of the system for increasingly longer times (see, for example, [104–106]). For the most remote parts of the whole “aggregate” even more approximate schemes may prove useful. In line with the treatment of the nuclear dynamics, also the electronic-structure methodology is to be adapted. In this way it is hoped that the range of applicability of the quantum treatment, and the phenomena to be covered, can be continuously extended and the field may prove useful for many years to come.