Relating transition-state spectroscopy to standard chemical spectroscopic processes

Transition-state spectra are mapped out using generalized adiabatic electron-transfer theory. This simple model depicts diverse chemical properties, from aromaticity, through bound reactions such as isomerizations and atom-transfer processes with classic transition states, to processes often described as being “non-adiabatic”, to those in the “inverted” region that become slower as they are made more exothermic. Predictably, the Born-Oppenheimer approximation is found inadequate for modelling transition-state spectra in the weak-coupling limit. In this limit, the adiabatic Born-Huang approximation is found to perform much better than non-adiabatic surfacehopping approaches. Transition-state spectroscopy is shown to involve significant quantum entanglement between electronic and nuclear motion.

Not all chemical phenomena involve transition states, however. Non-radiative transitions and avoided crossings also give rise to ultrafast processes and can equally be studied using transition-state spectroscopic techniques [51][52][53][54][55][56]. Such transitions can arise either because the coupling between diabatic states goes to zero at a conical intersection, or else because the transitionstate becomes converted into a point of inflection on the potential-energy surface of a highly exothermic (or endothermic) reaction. The later scenario is called the "inverted" region of chemical kinetics [57]. However, transition states can also disappear when tautomers are replaced by "delocalized" resonance hybrids [58,59], providing another asymptotic limit of transition-state spectroscopy.
The transition-state spectroscopy of non-dissociative reactions, as well as the spectroscopy of non-adiabatic, inverted, and delocalized processes, are all describable using the adiabatic electron transfer developed by Hush initially to describe electron-transfer and atom-transfer reactions [58][59][60][61][62][63][64][65][66]. This approach has recently been generalized to include all types of isomerization processes and reactions between bound species, including treatment of aromatic chemistry [67]. Achievements include the first explanation as to why NH 3 is tetrahedral whilst PH 3 -BiH 3 are octahedral [68], understanding properties of Born-Oppenheimer breakdown as well as the calculation methods used to examine it [69], and modelling the entanglement developed during chemical reactions [70,71]. Here, we consider key results predicted for transition-state spectroscopy and related non-adiabatic processes, focusing on how it relates to other chemical spectroscopies.

The basic model used in adiabatic electron-transfer theory.
In its simplest analytical form as applied here, adiabatic electron-transfer theory assumes parabolic surfaces and includes only one type of nuclear motion, motion along the reaction coordinate. These assumptions lead to three limitations in its applicability to transition-state spectroscopy: i) often motions other than the reaction coordinate are critical to observed phenomena [4,[72][73][74], including continuous motions like solvent friction [75]; ii) conical intersections intrinsically involve at least two nuclear coordinates; and iii) many reactions of interest involve bimolecular processes that cannot easily be discussed using harmonic potentials. However, the basic model does provide an underlying basis for critical aspects of these more complicated phenomena. In particular, the connection between chemical bonding and transition-state spectroscopy [76] can be discussed, and its inherent quantum density-matrix based description links smoothly to modern more thorough approaches [77].

2a. Diabatic descriptions of chemical and spectroscopic processes.
In its original formulation, adiabatic electron-transfer theory expresses chemical structure, thermodynamics, kinetics, and spectroscopy in terms of two coupled harmonic diabatic potentialenergy surfaces. Diabatic surfaces were first used to discuss transition states, avoided crossings, and non-adiabatic reactions by London [78,79] and quickly applied by Eyring and Polanyi to produce the analytical LEP potential-energy surface [80] for general triatomic molecules [81], as well as by Horiuti and Polanyi [82] for proton transfer reactions and Wall and Glockler [83] for isomerization reactions. These early approaches were often based on Morse diabatic functions that facilitated bond dissociation, with Hush [84] later introducing the simpler harmonic surfaces for bound reactions. The basic equations can be written in three formally equivalent forms, two of which are that are diabatic with the other being adiabatic (see later). Each representation forms simple descriptions of some observed phenomena but complex descriptions of other phenomena, and while accurate numerical solutions yield the same answers, the degree of difficulty in obtaining accurate solutions varies accordingly [67,69,85,86]. For studying chemical kinetics and transition states, the most common diabatic representation used was introduced by Hush [60][61][62]84] and is known as the localized diabatic description, formally represented by the Hamiltonian where m Q α ω =  is a linear vibronic-coupling constant. Expressed in this form, adiabatic electrontransfer theory is often called the pseudo Jahn-Teller effect [88], providing a traditional way for considering molecular spectroscopy. The labels G and T stand for the ground state and its "twin" state and describe diabatic spectroscopic states rather than reactants or products, the type of labels usually applied in considering aromatic and other spectroscopies. The twin state of the ground state is an excited state that has properties directly related to those of the ground state-knowing one state is tantamount to knowing the other. This concept was first introduced by Shaik [89][90][91][92][93] but he thought the twin state to always by a state formed by single excitation from the ground state. Indeed, such singly excited states are the twin states for electron transfer problems, forming the basis of intervalence charge-transfer (IVCT) spectroscopy [64], but in general the twin state is actually the state of highest-possible excitation amongst the frontier orbitals [67]. Recognizing the actual nature of the twin state, and how other states like single-excitation states relate to it and to the ground state, allows adiabatic electron-transfer theory to be immediately generalized to isomerization reactions and aromatic process modelling, completing the vision of quantum chemistry pioneers like London, Eyring, and Polanyi.

2b. Adiabatic representation.
Both diabatic descriptions lead without approximation to the same adiabatic description of the chemical and spectroscopic properties of the system. This is obtained by applying the Born-Oppenheimer transformation to either L H or D H , diagonalizing these at each individual nuclear coordinate Q to create [69,94,95] the diagonal correction and the kinetic energy (or second derivative) correction where [59] The coefficients a and b are used to specify the density matrix of the system, 2 ( ) b Q specifies the product component of the adiabatic state at each nuclear coordinate, with 2 x ( ) b Q approximating the product character of the transition state [61,62,79,97]. The cusp diameter is so named as transformation of the diabatic Hamiltonians into the adiabatic basis introduces a pitchforkbifurcation cusp catastrophe [98][99][100][101] that manifests derivative discontinuities in ( ) Q ε ± whenever 0 J = . For finite J , the nature of the lower surface changes from largely reactant-like to largely product-like when the coordinates change by c Q near the crossover point x Q [61,62,79,97].
During this changeover, the transition-state region is crossed in a process well recognized in the Reaction Force model of chemical reactivity [102][103][104][105]. The emphasis on the crossover region as the critical element of general electron-transfer processes sets adiabatic electron-transfer theory apart from its alternative instantaneous-transition alternatives [106], focusing on the properties of the transition region and the charged transferred at all points along the reaction coordinate, allowing for Zewail's observation of transition-state spectroscopy.

2c. Occurrence of a transition state on the lower Born-Oppenheimer surface.
The lower Born-Oppenheimer adiabatic surface ( ) Q ε − shows a transition state whenever is the electronic energy gap. In the weak-coupling limit of 2 / 1 J λ  , and is referred to the normal region of chemical kinetics; when no transition state exists it is referred to as being the inverted region, an important scenario championed by Marcus [57,106]. When there exists a transition state, the barrier height in the weak-coupling limit be approximated by the energy of the localized diabatic states at their crossing point a quantity also taken to be indicative of system properties even in the inverted region and treated if indeed a transition state did exist. For general symmetric reactions with 0 E D =, transition states exist whenever [58,59,108] This is known as the localized regime, with the alternative being the delocalized regime.
Resonance stabilized aromatic molecules like benzene are examples of systems in the delocalized regime, with its hypothetical Kekulé "reactant" and "product" structures not appearing as local minima on ( ) Q ε − [67]. In the localized regime the barrier height is given by [69] 2 ‡ ( 2 ) 4 The parameter space over the whole model in which ( ) Q ε − shows a transition state (Eqn. (14)) is shown in Fig. 1. The barrier height is in general close to [69] 2d. The Born-Huang adiabatic approximation.
The Born-Oppenheimer approximation Eqn. (5) is not the only way to simplify Eqn. (3) into an adiabatic form, an alternative being the Born-Huang approximation [79, 94, 95 , 96, 97] ( ) Here, the diagonal correction ( ) to the Born-Oppenheimer approximation from Eqn. (7) is retained whilst the non-adiabatic coupling terms ( ) that couple the ground and excited adiabatic states remain neglected. From Eqns. (6)- (7), the effect of the diagonal correction is to add within the intersection region at in parallel to both Born-Oppenheimer surfaces. In the delocalized region its contribution is negligible as c Q becomes wider than the length scale of chemical interest and † E DD becomes much less than both the electronic and vibrational energy scales of the system. However, in the weak-coupling limit the DC spike becomes much higher than the energy ‡ E D at the crossover point. As a result, the lower Born-Huang adiabatic surface almost always manifests a transition state, making Zewail's concept of transition-state spectroscopy directly relevant to most electron-transfer reactions performed in the inverted region.

2e. Failure of the Born-Oppenheimer approximation.
Using the Born-Oppenheimer description, processes in the inverted region are usually described as being non-adiabatic as they involve radiationless transitions between the two different potential-energy surfaces. The reactant and product surfaces do not intersect and so there is no transition state available to facilitate the reaction. Typically, however, use of the alternate Born-Huang adiabatic description typically re-creates a transition state, with the reaction then describable as occurring via nuclear tunneling through the high but narrow DC spike [69], facilitating electron transfer.
In the weak-coupling limit, failure of transition-state theory based on the Born-Oppenheimer approximation to describe electron-transfer reactions is usually expressed in terms of the Levich-Dogonadze rate equation [109][110][111][112] where k is called the transmission coefficient, The maximum physically meaningful value for k is unity, with perceived values greater than this indicating that the weak-coupling approximation used by Levich and Dogonzdze no longer holds. When k = 1, Born-Oppenheimer-based transition state theory is adequate to describe the reaction. However, instead of viewing k as specifying a radiationless transition rate between the two localized diabatic surfaces or else 1k as specifying a radiationless transition rate between the Born-Oppenheimer adiabatic surfaces, similar results can be obtained using the Born-Huang approximation [69] for the ground state without the need to include the second surface.
A common method used for understanding radiationless transitions in general chemical reactions involves simulating surface hopping between Born-Oppenheimer surfaces [113,114]

Applications based on accurate numerical solutions of spectroscopic properties
We investigate the spectra expected for electron-transfer processes happening near the transition state.  which, in these trajectories, is constrained to be 0.1 throughout. As / E ω D  approaches 1 it is likely that Jahn-Teller effects associated with conical intersections will contribute significantly to observed processes, making any one-dimensional model inappropriate. Smaller values than 0.1 represent systems with deep wells supporting many bound vibrations, but no qualitative changes in behaviour are expected and so this dimension is not explored herein. Basic properties of the 16 systems considered are summarized in Table 1, including the activation energy (using the crossover energy approximation Eqn. (16) in the inverted regime), the Born-Huang spike height (Eqn. (21)) and the cusp diameter (Eqn. (10)).

3a. Representation of the dipole operator specifying electronic and vibrational transitions.
To determine spectra, the electric dipole operator of the system needs to be specified for each electronic basis set. In the localized diabatic basis, the dipole operator is assumed to be where R is the distance through which the active electrons shift during the reaction, a form that is highly appropriate to electron-transfer problems as this is then just the distance through which the electron is transferred [86,[115][116][117]. In the delocalized diabatic basis, this operator becomes a form that is highly appropriate to understanding the intense electronic transitions observed in aromatic molecular spectroscopy. In the Born-Oppenheimer basis, the dipole operator becomes coordinate dependent: When applied using the Born-Oppenheimer or Born-Huang approximations, the diagonal elements of this matrix generate traditional infrared vibrational spectra and associated (perhaps microwave) vibrational tunneling transitions, whilst the off-diagonal terms facilitate Franck-Condon and Herzberg-Teller (etc.) electronic transitions.
All results shown in Fig. 2 and later are fully converged numerically and hence are independent of the actual Hamiltonian and corresponding dipole-moment operator used. However, in addition we also show spectra obtained using the approximation to Eqn. (26): The full Hamiltonians all embody mixing between vibrational and electronic motions and so result in spectra that can differ somewhat from basic expectations. Transition-state spectroscopy can also investigate low-energy electronic processes of the same order as vibrational energy spacings, further confusing the distinction between processes considered to be either primarily electronic or primarily nuclear in origin. As a result, spectral line intensities can simultaneously gain strength from both the diagonal elements and the off-diagonal elements of the dipole matrix, with the two effects interfering with each other as they sum to give line transition moments. The approximation in Eqn. (26) includes only the electronic transition component to the total intensity, facilitating the identification of electronic spectra, vibrational spectra, and any interferences between them. In Fig.  2, the total cross-section is shown in blue whilst its electronic component is shown in red, giving the allusion of magenta colouring whenever the electronic and vibrational contributions do not interfere and the total spectrum is dominated by just its electronic component.

3b. Time-dependent representation of molecular spectra and properties.
As the time spent in the vicinity of a transition state is usually limited, time-dependent methods provide the natural way for considering transition-state spectroscopy. In these, initial wavepackets (0) ψ are constructed and their time dependence followed, with the spectra calculated from this. In this way, the spectra depend explicitly on the initial wavepacket chosen. In standard equilibrium spectroscopy, the initial wavepacket is often just the stationery state of the system identified as the ground vibrational level of the ground electronic state, though this may be thermally weighted if necessary [17]. Here we consider simple wavepackets that would describe the ground-state of the diabatic surfaces should they be centred on the diabatic potential minima, but their initial positions are chosen depicting wavepacket energies in the vicinity of the transition state. It is possible that in experiments such initial wavepackets could be created from either some external spectroscopic process or else following reaction from some initial chemical species. Under these initial conditions they are not eigenstate of any potential and hence evolve in time according  (1)). The dynamics expressed in terms of the localized diabatic reactant and product surfaces is shown using red and blue colouring to indicate how much of the wavepacket is on each diabatic surface at any point in time. In parallel, the composition of the wavepacket projected onto the ground-state and excited-state Born-Oppenheimer surfaces is shown in green and magenta, respectively. Here, the initial wavepackets are located purely on the Born-Oppenheimer ground-state surface and are mixtures of the reactant and product surfaces, but these natures evolve during the trajectories.

3c. Numerical solution method.
A wide variety of methods are available for evaluating Eqn. (29). In principle, the dynamics can be solved using explicit time-dependent methods that integrate the differential equation specified by the time-dependent Schrödinger equation, something that can often be approximated accurately by classical or semiclassical means. In this way solutions can be obtained for large systems using full Hamiltonians rather than simple model ones like our Hamiltonians. However, dynamics around conical intersections is intrinsically chaotic and hence converging such approaches to the desired accuracy is always difficult, especially in the weak-coupling limit in which k gets small. We use a method that maximizes convergence but is of high cost, as can be done typically only for simple model Hamiltonians. The full details are described in detail elsewhere [69]. Basically, a harmonic-oscillator vibrational basis is used for each electronic basis state centred at 0 Q = . This is truncated at 256 levels per state, sufficient for all examples considered herein to achieve full convergence of even the poorly convergent calculations (required computer time proportional to 6 c Q − [69]) performed using the Born-Oppenheimer electronic basis. All eigenstates of the appropriate Hamiltonian are then obtained by full matrix diagonalization, allowing an analytical representation of the time dependence of the initial wavepacket that gives stable solutions throughout chaotic domains. A mixture of analytically determined and numerically evaluated matrix elements is used to construct the Hamiltonians, with no convergence issues arising. p ω .
Collected in Table 1 are a variety of properties summarizing key features of the calculated dynamics. The transmission coefficient 2π k (Eqn. (22)) indicates the fraction of the wavepacket transferred from reactants to products after 1 vibrational period. When large, this indicates that classical transition-state theory is adequate in describing reaction kinetics. Note, however, that it is always short of unity as the quantum wavepackets embodies classical states both slightly above and below any classical transition-state barrier, so some degree of non-reaction is always expected.
Also, π f + gives the fraction of the wavepacket transferred to the excited adiabatic state after half of a period, crudely indicating the significance of non-adiabatic dynamical processes. Finally, Table 1 also includes the quantum entanglements developed between the electronic and nuclear motions after one period of vibration, projected onto the localized diabatic states, 2π L S , as well as onto the Born-Oppenheimer electronic states, 2π BO S . The projection onto the localized diabatic states is in-principle experimentally measurable using spatially resolved detectors and could be utilized in some quantum information-processing device [71,[118][119][120], whilst that revealed in the Born-Oppenheimer basis provides a unique and sensitive way of understanding the failure of the Born-Oppenheimer approximation [70].
The calculated spectral densities (a)-(d) in the weak-coupling limit with 2 / J λ = 0.03 depict vibrational tunnelling transitions from the reactant well to the product well. While these transitions are nominally considered to be vibrational, because of the electron-vibration coupling, these pull the electron with them to facilitate electron transfer if unconsidered factors induce decoherence [121]. All states contained in the wavepacket contribute to this tunnelling. The transmission factors are all very low for (a)-(c) (0.03 -0.04) but the initial wavepacket for (d) already has a significant overlap with the product state and so the tabulated measure 2π k is poorly between the reactant and product states that could be used in a quantum information processor with spatial detectors remains very low, however [122], indicating the unsuitability of processes in this regime for quantum computing applications.
All four cases lead to intense electronic spectra (Fig. 2). A component coming from vibrational transitions is also found that grows in relative strength as the reaction becomes more endothermic, perhaps reaching a maximum at around 0 / E λ = 0.25 (case (m)) owing to increasing initial wavepacket displacement covering an energy range in which the ground-state surface is highly anharmonic. The symmetric system (n) shows a single line in the electronic spectrum representing the origin of the electronic transition, with a weak additional infrared transition coming from the mismatch of the wavepacket with the significantly anharmonic ground-state potential. This situation mimics the standard spectroscopy of aromatic molecules. Of course Franck-Condon effects associated with motions other than the reaction coordinate broaden this line into a band when multiple nuclear motions are included in calculations.
Trajectories The calculated spectra therefore show a wide range of features that could be observed in transition-state spectroscopy, linking smoothly to commonly observed tunneling transitions, infrared transitions, and electronic spectroscopic transitions in appropriate limits. Figure 5 shows the time-domain functions ( ) c t (Eqns. (28)-(29) associated with the total spectra from Fig. 2. These also show a very wide range of responses, both regular periodic-looking motion and rather irregular chaotic-looking motion. In these simple calculations in which only one nuclear motion is included, quantum coherence of the dynamics is assured. The consequences of this coherence become apparent through looking at the irregular features of the transition-state spectra.
In calculations on complex molecules, the Born-Oppenheimer description is often used to calculate spectral and chemical properties. Fig. 3 shows results obtained using this (BO) (Eqn. (5)), the Born-Huang approximation (BH) (Eqn. (20)), and the first-derivative approximation (FD) typically used as part of surface hopping approaches where they are compared to results from the full calculation (FC) obtained using all three Born-Oppenheimer correction terms. Significant deviations are found only for systems in the weakcoupling limit with 2 / J λ = 0.03, with the Born-Oppenheimer approximation performing very well for 2 / J λ = 0.3 and above. This is because only in the weak-coupling limit does the Born-Huang spike height † E DD become appreciable (see Table 1, and Fig 3). Results performed using the adiabatic Born-Huang approximation always outperform those obtained using standard nonadiabatic surface-hopping approaches. This result is typical of those obtained for many properties comparing the BH and FD approaches [69], an exception being that only the FD method can manifest entanglement between the Born-Oppenheimer states and so would become advantageous for some observed property that is directly dependent on entanglement [70]. Mostly the results shown in Fig. 4 indicate that the BH approximation does qualitatively describe transition state spectra, and that many processes considered to be "non-adiabatic" using the Born-Oppenheimer description are indeed "adiabatic" when using the Born-Huang one. Nevertheless, accurate calculations performed to investigate transition-state spectroscopy in the weak-coupling limit should always be performed in a diabatic basis or else must include all three Born-Oppenheimer corrections.

3e. Results for trajectories starting at the transition state.
We also consider the alternate set of trajectories commencing near the transition state, as would be produced following fluorescence from the excited-state potential minimum down to the ground state. Spectra analogous to Fig. 2 are shown in Fig. 6, with the trajectories obtained using the full calculation in the Born-Oppenheimer basis shown in Fig. 7. Again a wide mix of vibrational and electronic transitions can be produced, with the trends in Fig. (6) having many parallels to those in Fig. (2). For all cases except those with 2 / J λ = 0.03 in the weak-coupling limit, the electronic spectrum contains only a single peak corresponding to the opposite process to the emission that created the initial wavepacket. In the weak coupling case, the electronic absorption becomes spread out and too weak to discern amongst the vibrational transitions on the ground state. These transitions grow rapidly in intensity, becoming more strongly allowed than the reverse electronic absorption. The vibrational transitions associated with the transition-state spectroscopy are ordered in the inverted region but chaotic otherwise. It is clear that all spectra contain much detailed information relating to the system properties.

Conclusions
Within the scope of the basic one-dimensional generalized adiabatic electron-transfer theory model, a wide variety of transition-state spectra and dynamics can be produced. The model reveals how transition-state spectroscopy reduces to standard molecular spectroscopy in the strong-coupling delocalized regime, and how it can lead to very complex results for both typical transition states as well as for processes in the non-adiabatic and inverted regimes. Vibrational absorption and emission spectra may be very weak, but, throughout the parameter space of the model, they contain important information usable to interpret system properties. Approximate treatments based on surface hopping are found always to give poor descriptions of dynamics and spectra evaluated using full quantum dynamics, with inclusion in approximate methods of only the typically neglected diagonal term without any non-adiabatic coupling typically performing much better than surface hoping approaches that ignore it.   Table 1.