Using ‘designer’ coherences to control electron transfer in a model bis(hydrazine) radical cation: can we still distinguish between direct and superexchange mechanisms?

We have simulated two mechanisms, direct and superexchange, for the electron transfer in a model Bis(hydrazine) Radical Cation, which consists of two hydrazine moieties coupled by a benzene ring. The computations, that are inspired by the attochemistry approach, focus on the electron dynamics arising from a coherent superposition of four cationic states. The electron dynamics, originating from a solution of the time dependent Schrödinger equation within the Ehrenfest method, is coupled to the relaxation of the nuclei. Both direct (ca. 15 fs dynamics) and superexchange (ca. 2 fs dynamics) mechanisms are observed and turn out to lie on a continuum depending on the strength of the coupling of the benzene bridge electron dynamics with the hydrazine chromophore dynamics. This contrasts with the chemical pathway approach where the direct mechanism is completely non-adiabatic via a conical intersection, while the superexchange mechanism involves an intermediate radical with the unpaired electron localized on the benzene ring. Thus, with the attochemistry-inspired electron dynamics approach, one can distinguish direct from superexchange mechanisms depending on the strength of the coupling of two types of electron dynamics: the slow hydrazine dynamics (ca. 15 fs) and the fast benzene linker dynamics (ca. 2 fs). In this model bis(hydrazine) radical cation, only when the intermediate coupler is in an anti-quinoid state, does one see the coupling of the bridge and hydrazine chromophore dynamics.

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Introduction
A chemical pathway is a mechanism consisting of a sequence of chemical structures, viz.minima, transition states, and conical intersections (in the case of non-adiabatic reactions).The chemical pathway idea, which is a central concept for modelling reactivity and elucidation of reaction mechanisms, assumes a slow nuclear motion in which the electronic structure changes in concert with nuclear motion.This idea was extended to include non-adiabatic effects in Marcus theory [1][2][3], where the pre-exponential factor in transition state theory is associated with an electronic coupling related to the energy difference between the two lowest states at the 'transition state' [4] (meaning highest occupied and lowest unoccupied states).Such a generalization was the basis of the computational study of electron transfer processes using the chemical pathway approach [5][6][7].In the case of electron transfer in the Bis(hydrazine) radical cation [5] (see figure 1), the pathway approach leads to two different electron transfer mechanisms, each of them with a distinct sequence of molecular structures and a different evolution of the (de)localization of the unpaired electron as the transfer proceeds.
The recent development of the field of attochemistry [8][9][10] provides a complementary approach to the chemical pathway perspective.The experimental aspects of attophysics have been recently recognized with the award of the 2023 Nobel prize in physics [11].The attochemistry approach involves a coherent superposition of adiabatic electronic eigenstates.The superposition behaves like an electronic state in its own right (rather than an average or mean) with a characteristic distribution of the unpaired electron.In contrast with the chemical pathway approach, which focuses on the potential surface and its critical points, in the attochemistry approach, one starts with the electronic motion via the solution of the time dependent Schrödinger equation (TDSE) and then the nuclei respond to the gradient of the electron dynamics.Accordingly, the nuclear motion is 'steered' by the electron dynamics.Given the complementarity of the chemical pathway and the attochemistry approaches, it is of interest to compare the results provided by each one of them for a given model electron transfer process.
The purpose of this paper is to study the electron transfer process in the bis(hydrazine) radical cation model system using the attochemistry approach and compare the results with those previously obtained using the chemical pathway approach [5].According to the chemical pathway results, the electron transfer between the two hydrazine moieties can take place by means of two different mechanisms, namely the direct and superexchange channels.As shown in figure 2, these mechanisms have different molecular structures along the pathway from A to B, which are used to denote the reactant and product of the electron transfer, respectively.The direct pathway involves a conical intersection (CI) hyperline (see E-F in figure 2(a)), separating reactants (A) and products (B) (i.e. the crossing is unavoided); thus, the electron transfer is completely non-adiabatic.The superexchange pathway instead involves intermediate anti-quinoid or quinoid structures (along the 'ridge' C-D in figure 2(b)), where the unpaired electron lies on the benzene radical cation ring.Here we have two transition states (namely, quinoid TS 1 and anti-quinoid TS 2 , which can be pictured as D, C structures in figure 2(b)) and an isolated CI point (whose electronic structure is similar to that along the CI 'seam' E-F in figure 2(a)).Note that in figure 2 we use Valence Bond (VB-like) structures in two ways: (1) as an indication of the molecular structure (bond lengths) and ( 2) as an indication of the electronic structure (i.e.where the bonds and charges are).This latter picture is at the forefront of the discussion on electron dynamics that follows.
With the results of the present work, we will demonstrate that all the mechanisms observed in the static chemical pathway picture can be simulated within the electron dynamics in the attochemistry approach.Specifically, it will be shown, that the particular coherent superposition of adiabatic electronic states that are chosen as the initial state to solve the TDSE, defines the type of mechanism that is operative.It thus follows that a spectrum of mechanisms for electron transfer can be generated depending on the electronic structure (i.e. the distribution of the unpaired electron across the molecule) of the initial coherent superposition of states.

Theoretical background
We begin with a brief resume of the approach we have used [12,13] for the solution of electron dynamics within the TDSE.Assuming a wavefunction that is a coherent superposition of stationary states {ϕ i (R)} with complex coefficients z i (t), the initial wavefunction is then propagated in time using equation ( 1) Note that the expansion runs over N ϕ stationary adiabatic states {ϕ i (R)}.The ϕ i (R) adiabatic states used in equation ( 1) are obtained from a complete active space configuration interaction (CASCI) formalism as a superposition of N χ configuration spin-adapted functions, CSFs, {χ k (R)}.Notice that here, we use CASCI to denote only the CI part of a complete active space self-consistent field (CASSCF) computation.Each χ k (R) function is, in turn, expressed in terms of the active orbitals involved in the CASCI calculation.The timedependent probability density for a superposition of two states i and j is then given by: ) . ( The oscillatory terms of the form exp [−i (E i (t)−E j (t)) t/h] are the quantum mechanical origin of electron dynamics [14].Invoking the Planck-Einstein relation (E = hν), we would  Schematic qualitative ground state potential energy surfaces [5] for electron transfer in a model bis(hydrazine) radical cation, where the two hydrazine units are coupled by a benzene ring.(a) Direct mechanism and (b) superexchange mechanism.In the sketches of the molecules, carbon and hydrogen atoms bonded to the nitrogen atoms and benzene ring are omitted for clarity.Note also that dashed line means aromaticity.The color-coding green to red is intended to give a quantitative representation of the relative energy (e.g. the CI seam in (a) is higher in energy than the isolated CI point in (b)).Note that the X and Y variables are different in (a) and (b), while Z corresponds always to energy.The electronic structure at the CI 'point' in (b) is similar to that along the CI 'seam' E-F in (a).then expect oscillations to occur at frequencies ν as a function of the energy gap between states (the period τ of the oscillation is the inverse of ν) The initial conditions for solving the TDSE correspond to the initial position of the unpaired electron involved in the electron transfer at some reference geometry.One way to specify this position is by providing a given initial [15,16] superposition of adiabatic electronic states (as used in this work).Alternatively, one could use a localized orbital representation  3 for MOs (in which a hole has been created) of the active space employed to generate the set of {χ k }.Note that, in this particular case, the dimension of {ϕ i } and {χ k } is the same (N ϕ = Nχ = 4).[17] and start with a specific electronic configuration (i.e. a configuration state function), which is obviously not a stationary eigenstate and, thus, is implicitly a superposition of adiabatic states.Accordingly, one either chooses a superposition of eigenstates or a specific combination of configuration state functions; in either case, one has an initial state that is not an eigenfunction.Yet, another approach can be formulated using real-time dependent density functional theory (DFT) [18].However, real-time DFT depends on the choice of the initial MOs, while the (attochemistry) approach here employed does not.It would be more difficult to create/control the different and distinct dynamics using real-time DFT or using a single configuration state function as initial conditions rather than using a specific coherent superposition.
In this work we shall explore three different initial conditions for the attochemistry-inspired electron dynamics aimed at illustrating both the direct and superexchange mechanisms.These were constructed from three different coherent superpositions of the four lowest adiabatic states (ϕ i (t) with i = 1−4 in table 1) formed as a linear combination of configuration spin adapted functions (CSF, χ k ) resulting from creating a hole in each of the four active molecular orbitals, MOs, shown in figure 3. The three initial states that were created are: 4).There are other ways one might choose the initial conditions (as discussed above [17,18]); however, choosing superpositions enables one to focus on the nature of the electron dynamics (ca.15 fs and ca. 2 fs) driven by the energy differences via equation ( 3).
In the first initial state (Ψ 1 , see figure 4(a)), the unpaired electron is essentially localized on one of the two hydrazine moieties (note there is some minor contribution delocalized over the entire benzene ring).According to the wavefunction defining Ψ 1 , ∆E (ϕ 4 − ϕ 3 ) = 0.28 eV which results in the ca.15 fs hydrazine•••hydrazine dynamics (see tables 1, 2 and equation ( 3)).The 15 fs dynamics thus 'defines' the direct mechanism.In the second simulation (Ψ 2 , see figure 4(b)), the initial state contains ϕ 2 as well as ϕ 3 + ϕ 4 and mixes the quinoid state.The spin density is localized on one hydrazine and on the four C atoms nonbonded to N. Here, ∆E (ϕ 4 − ϕ 3 ) = 0.28 eV and the average Table 2. Energies of the adiabatic states extracted from table 1, energy differences between adiabatic states corresponding to a given initial condition Ψ 1−3 , and period of the corresponding oscillation (calculated using equation ( 3)).Energies in eV and periods in fs.Note an average of energy differences for Ψ 2 and Ψ 3 is used.Also, the period (τ ) is approximated to 15 fs and 2 fs since the order of magnitude is what matters.
, leading to two distinct electron dynamics: ca. 15 fs dynamics of the hydrazine chromophores and ca. 2 fs dynamics of the bridge.The involvement of the 2 fs electron dynamics of the bridge defines the superexchange mechanism.Finally, Ψ 3 contains ϕ 1 rather than ϕ 2 and involves the anti-quinoid state.The spin density is localized mainly on one hydrazine and on the C atom directly bonded to that N. For Ψ 3 , we have ∆E (ϕ 4 − ϕ 3 ) = 0.28 eV and the average ∆E ( ϕ 3/4 − ϕ 1 ) = 2.06 eV (see table 2), resulting again in ca.15 fs electron dynamics of the hydrazines and 2 fs dynamics of the bridge.It thus follows that Ψ 2 and Ψ 3 both involve the ca. 2 fs dynamics of the benzene bridge and 15 fs dynamics of the N-N chromophores.The question is whether the bridge dynamics is coupled to the hydrazine chromophores and which state of the bridge (quinoid or antiquinoid) is involved.Now let us briefly mention the nuclear motion associated with electron dynamics.For each value of t in equation ( 1), one may compute the gradient and second derivatives with respect to nuclear motion [13].The nuclei can then be propagated either classically or quantum mechanically using the gradients and second derivatives.Our implementation in a CASSCF formalism has been described elsewhere [12,13].The exact factorization method is very similar [20,21].Notice that the potential energy surface is now time dependent, although one can always project onto stationary states.The frequency of oscillation (equation ( 3)) in the electron dynamics changes as the nuclei move because the energy gap in equation ( 3) changes with time.We wish to emphasize that the computed gradient is not the average gradient of the states in the superposition; rather, we use the full expressions for the gradient and hessian from [13].These were derived from the general formulation of Almlöf and Taylor [22].In contrast, the meanfield approach would involve the average gradient of the two adiabatic states and would not include the cross interstate mixing terms.We have discussed the effect of these cross-mixing terms in depth elsewhere [15,16,23].
Finally, we should point out that the present computation will not recover the decoherence that might result from a 'spread' of initial geometries corresponding to many coupled trajectories [24].We have developed methods to deal with many coupled trajectories [12].However, we are here interested mainly in very short dynamics (a few periods only) before such decoherence occurs.

Computational details
Calculations were carried out at 4-SA-CASSCF(7,4)/6-31g(d) level (4 active orbitals and 7 active electrons) using a development version of Gaussian [25].The wavefunction (CASCI part of the CASSCF computation) was propagated (TDSE) using the Ehrenfest method.In other words, in a CASSCF computation the CI part of the problem is not diagonalized; here, rather than diagonalization, the wavefunction is propagated according to the TDSE [13] (the orbitals themselves satisfy the second order [26] CASSCF equations).For each simulation, the initial Ψ (R, t = 0) wavefunction was formed as a superposition of {ϕ i } eigenstates built from the hole configurations {χ k } (both shown in table 1), which in turn were formed from MOs in figure 3.In all cases, our initial conditions involve a choice of initial geometry and the initial electronic structure (i.e.superposition).For the initial geometry, one could choose a geometry near the chemical pathway transition state (figure 2(a)) or near the conical intersection seam (figure 2(b)), i.e. a geometry which lies on the ridge of figure 2. In these computations we choose the initial geometry to be the ground state neutral geometry of bis(hydrazine), whose benzene central ring is aromatic, because it resembles the geometry of the conical intersection.For electronic structure initial conditions, we choose several initial superpositions.Since the ϕ 4 + ϕ 3 superposition is present in all the initial conditions, this effectively localizes the electron approximately on one or the other hydrazine, leaving different contributions from the benzene bridge ring in Ψ 2−3 , as shown in figure 4.This initial setup mirrors what might be used in an attosecond experiment starting by ionization from the neutral.Obviously, other choices are possible.For example, one could change the relative phases of the components of the superposition, e.g.ϕ 4 + iϕ 3 , but this would only change the starting point on the periodic electron dynamics.Also, one might use an initial geometry that resembled the 'reactants'.We choose our geometry to lie approximately on the ridge because we want to focus on the role of the bridge.In the following, the results of our simulations referring to Ψ 1 (R, t), Ψ 2 (R, t), and Ψ 3 (R, t) will be discussed.

Results and discussion
We begin by analyzing the electron dynamics of bis(hydrazine) radical cation when using Ψ 1 (R, t = 0) as the initial state.The time-resolved evolution of the spin population of the N atom lone pairs 3 and 21 (see figure 1 for atom numbering) is shown in figure 5(a), while the corresponding data for the coupler benzene ring is shown in figure 5(b).Notice that the oscillations of the Mulliken spin populations in the ring have the same period (τ 1 ∼ 15 fs) as the N atom lone pairs because they involve the same two adiabatic states (ϕ 3 and ϕ 4 in table 1).As mentioned above, this ca.15 fs electron dynamics 'defines' the direct mechanism.The effect of relaxing the geometry gives rise to a small variation of the energy difference, ∆E (ϕ 4 − ϕ 3 ), of about 2.5 kcal mol −1 = 0.1 eV (see figure 5(d)).This decreases the period by around 3.0 fs, from 15 to 12 fs.Note that the N-N bond distances both initially decrease overall (see figure 5(c)) because one has removed an electron from a delocalized but slightly anti-bonding lone pair (either MOa or MOb in figure 3).However, because of the electron dynamics (figure 5(a)), an intermediate asymmetry in N-N distances is observed, which is associated with the electron movement from one chromophore to the other (and back) and lasts until a full period of 12.0 fs.The amplitude of the electron dynamics on the benzene ring is small (ca.0.03 in figure 5(b)) compared to that corresponding to N 3 and N 21 (ca.0.60 in figure 5(a)) and the electron dynamics of the bridge and of the hydrazines are not coupled.
At t = 3.6 fs, which corresponds to 1/4 of the period of the oscillations (t = 3.6 fs = 1/4 τ 1 ), the instantaneous spin densities on the N lone pairs are equal.We will define this particular state, which has equal spin population on each of the N-N chromophores, as 'electronic transition state' (e-TS).The spin densities of such e-TS (see figure 6(a)) faithfully represent the electron density associated with the ridges E-F in figure 2(a).Therefore, the spin distribution of the e-TS state and the overall form of the spin density oscillations (see figures 6(a), 5(a) and (b), respectively) clearly indicate that the electron dynamics initiated by Ψ 1 can be linked to the direct mechanism, in which the electron transfer from one chromophore to the other occurs without involving the benzene ring (mechanism shown in figure 2(a)).
We shall now focus on the electron dynamics prompted upon initializing the cationic system in the state Ψ 2 , which involves the ϕ 2 , ϕ 3 and ϕ 4 adiabatic electronic states (see figure 4(b)).Looking at the coefficients of each ϕ i =2−4 state in table 1, it can be inferred that the period of the oscillation of the N spin densities is driven by the energy difference between E (ϕ 4 ) − E (ϕ 3 ) = 0.28 eV (τ 2 ∼ 15 fs), while the period of C atom electron density oscillation is driven by the (average)   9, however, there is a partial cancellation every 2 fs starting from 1/8 τ 3 , which blurs the e-TS).Here, the bridge electron dynamics is active and perturbs the hydrazine electron dynamics.
energy difference E ( ϕ 3/4 ) − E (ϕ 2 ) = 1.81 eV (ca. 2 fs) (see table 2).In figure 7(a), one can see that the period for the N-N electron transfer (15 fs reduced to 12 fs by nuclear relaxation) is similar to that for the direct mechanism, as expected, since the energies (table 1 and equation ( 3)) are the same.Of course, the amplitude of the oscillations on the N atoms is smaller (ca.0.45 in figure 7(a)) compared to the direct mechanism because there is delocalization on the benzene acting as a coupler (ca.0.10 in figure 7(b)).One also sees the asymmetric N-N bond length change (see figure 7(c); same as in figure 5(c)).The period of ca. 2 fs for the C atoms is much shorter (only the atoms on one side of the ring are shown for clarity in figure 7(b)) and electron dynamics involves mainly C atoms 15 and 17 (also C10 and C12 by symmetry), where the π-bond weakens upon removal of one electron.Although one observes the expected 2 fs benzene electron dynamics involving the benzene ring (see figure 7(b)), the bridge electron dynamics is passive and does not perturb the hydrazine electron dynamics.Therefore, using Ψ 2 as initial condition creates the bridge dynamics, but this does not couple to the hydrazine chromophore dynamics.Therefore, given this situation, the superexchange becomes silent and so effectively becomes equivalent to the direct mechanism.
In this case, the geometry and energetic changes are driven by the relaxation of the bonds in the ring (see figure 7(d)).We can see that the C15-C17 bond elongates (i.e. an electron has been removed from the quinoid Kekulé double bonds).The in-phase spin density oscillation on C15 and C17 is driving the corresponding C15-C17 bond length extension.Thus, the only effect of the benzene coupler is in the dynamics of the benzene ring itself.
A close inspection of the e-TS state (again, at t = 3.6 fs = 1/4 τ 2 , see figure 6(b)) of the electron dynamics initiated using the Ψ 2 state reveals that the spin density is delocalized over the whole molecule.The contribution of the spin populations is larger in the benzene ring (summation of the spin density on the four C atoms not bonded to the hydrazine units) than in each individual N 2 chromophore (see figure 6(b)).Clearly, this state corresponds to quinoidal structure D in figure 2(b).However, comparison of the spin density at t = 0 and e-TS shows that the benzene contribution barely changes upon electron dynamics.As mentioned above, the bridge electron dynamics is passive and does not perturb the hydrazine electron dynamics, as extracted from the analysis of the oscillations of the spin densities (figures 7(a) and (b)).Therefore, the electron dynamics initiated by Ψ 2 can be associated with the direct mechanism of electron transfer.
The energy E (ϕ 4 ) − E (ϕ 3 ) change over the first full period (ca.12 fs) of the electron dynamics is approximately 2.0 kcal mol −1 (see figure 5(d)), so the period of the electron dynamics is hardly affected.The change in total energy is found to be ca.7 kcal mol −1 .Thus, the relaxation energies for the direct mechanism explored using two different initial conditions (Ψ 1 and Ψ 2 ) are small and similar (as expected since they are a component of the reorganization energy of Marcus theory in bis(hydrazine)).Using Ψ 3 (R, t = 0) as initial condition, the picture obtained by the electron dynamics evolution changes drastically.In this case, we see both the hydrazine dynamics (ca.15 fs) and the benzene ring dynamics (ca. 2 fs).Further, they perturb each other, and we see 'true' superexchange.
Firstly, notice that the spin density oscillates for N3 and N21 with an overall period similar to the previous two cases (ca.15 fs, see figure 8(a)).This is shown by the envelope sketched in figure 8(a).The fast oscillation that is superimposed arises from the interaction with the 2 fs from the ring dynamics (see figure 8(b)).This is analyzed in figure 9, where we show both ring and hydrazine dynamics with spin density presented at ca. 1 fs intervals over a full 1 /2 period of the hydrazine dynamics (i.e.transfer from one hydrazine to the other).One can see that C9 and C14 (bonded to N atoms) spin densities oscillate in concert slowly driving the spin density from one hydrazine to the other.Thus, we have the superexchange-like involvement of the benzene ring in the hydrazine•••hydrazine dynamics.This involves the antiquinoid state rather than the quinoid state.The fact that the electron density of the anti-quinoid structure is more localized on the C9 and C14 atoms, which are directly bonded to the N atoms of the hydrazine units, is the obvious reason why the anti-quinoid state (arising from Ψ 3 ) couples and the quinoid state (arising from Ψ 2 ) does not.
As for bond distances, there is an asymmetric N-N bond length change (see figure 8(c); same as in figures 5(c) and 7(c)).Also, there is the relaxation of the bonds in the ring (see figure 8(d)).Here, the spin density oscillation between C15 and C17 is out-of-phase and removal on an electron from the anti-Kekulé structure (anti-bonding between C15 and C17) causes the C17-C15 bond to shorten (note that, at t = 0, all C-C bonds have similar length, corresponding to the neutral ground state of the aromatic benzene ring).Now let us comment briefly on the energetics.The process in figure 2(a) represents an attempt to view an inherently non-adiabatic process using adiabatic potential surfaces, so we can only examine these energies in a very qualitative fashion.The energetics computed in our simulations are not easily comparable to the pathway approach.This dilemma has been discussed recently elsewhere [27].The relaxation energies we compute cannot be easily related to the activation energies seen in the pathway approach.Within the pathway framework, the energy difference between the seam (E, F in CI hyperline) and the charge localized minima (A, B) is ca.40 kcal mol −1 (see figure 2(a) and [5]).The energy of the ϕ 3 ± ϕ 4 superposition lies some 40 kcal mol −1 above S 0 .However, the relaxation energies we compute are 'local' as we now discuss.
In figure 5(d), the evolution of the energy difference during the first full period between the two highest energy states, namely, E (ϕ 4 ) − E (ϕ 3 ), is 2.5 kcal mol −1 .This drives the change in the period of the electron dynamics, which is very small.Then, there is the change in the total energy of the coherent superposition between t = 0 and t = 20 fs, which is approximately 8 kcal mol −1 (similar in all our simulations).This is the same order of magnitude as the energies separating the local minima and transition states in the pathway approach (this would be only a small component of the Marcus re-organization energy).In order to consider the energetics further, one would need to allow the simulations to run much longer and to allow for further decoherence by mixing with other vibrations during this relaxation.
Finally, let us give a 'flavor' of the actual charge transfer process that is taking place.We have only run the calculations starting from initial conditions Ψ 1−3 for a few complete periods.The differences of the values of the spin density at the first two maxima in the spin density oscillations (τ /2, and τ ) of, for instance, N3, serve to illustrate the magnitude of the electron transfer from one hydrazine unit to the other (see table 3).It is clear that there is charge transfer in all three scenarios, the most efficient associated with the superexchange mechanism Ψ 3 (∆ρ N3 = 0.03).Of course, if the simulations were allowed to run longer and if the decoherence that might result from a 'spread' of initial geometries corresponding to many coupled trajectories was included, one would see the eventual collapse of the dynamics associated with charge transfer.

Conclusions
We have documented electron transfer in a model Bis(hydrazine) radical cation focusing on the electron dynamics arising from a coherent superposition of four cation states, which might result from a laser experiment.From a practical point of view, the computations are quite cheap and involve wavefunction propagation in a CASCI formalism (rather than diagonalizing the CI Hamiltonian) together with gradient and hessian computation (which takes most of the computer time).
No geometry optimizations are required.
The computations of the attochemistry electron dynamics for the superexchange mechanism yield a picture that is similar to the static pathway picture in figure 2(b).Yet, the electron dynamics 'picture' is much simpler: the ca.15 fs dynamics defines the direct mechanism and the involvement of the ca. 2 fs electron dynamics of the bridge defines the superexchange mechanism.In general, both mechanisms are activated.In the limit, where the bridge is not involved in the dynamics, namely Ψ 1 , one has the limiting case for direct dynamics.In general, e.g.Ψ 2 and Ψ 3 , one has bridge dynamics.Here, the nature of the electronic state of the bridge (quinoid vs. anti-quinoid) determines whether the hydrazine•••hydrazine dynamics couples to the bridge dynamics or not.The main, longer time, effect of this coupling is the change in the geometry of the benzene bridge itself.These geometry relaxations are like those in the pathway approach.
The electron dynamics in the ring is what might be expected of the benzene radical cation and involves the quinoid or anti-quinoid structures.However, the most important conclusion is that the distinction between adiabatic and non-adiabatic 'pathways' has here disappeared along with the involvement of any conical intersection (figures 5, 7 and 8).The only difference between the electron dynamics associated with the two mechanisms is the magnitude and nature of the oscillations of the spin densities due to the electron dynamics on the benzene ring and its nuclear relaxation (compare figures 5(b) and 7(b)/ 8(b)).
We should also comment on the relationship to chemical ideas associated with electron transfer in bridged compounds (see for example the reviews [28,29]).In this case, the role of the bridge is associated with the electronic coupling.In contrast, the effect that we have focused on in this paper is the fact that there are two distinct mechanisms for electron transfer, direct and superexchange, that can be documented within the same molecule.These two mechanisms are part of a continuum that can be spanned by changing the initial conditions.These conditions could be explored experimentally in an attosecond laser experiment.
Thus, the picture that emerges from this treatment is remarkably simple: the difference between the direct and superexchange mechanisms lies only in the electron dynamics in the benzene ring (i.e. the coupler) and the concomitant nuclear geometry relaxation (or lack of it) within the ring.No conical intersections, intermediates, or transition states need to be invoked.Therefore, some of the concepts used in a static pathway picture (Born-Oppenheimer) become blurred when incorporating the attochemistry electron dynamics.

Figure 1 .
Figure 1.Geometry of the bis(hydrazine) radical cation used in this work: (a) 3D structure and (b) active heavy atom framework with atom labels used in subsequent discussion.Note in (b) carbon and hydrogen atoms bonded to the nitrogen atoms and to the benzene ring are omitted for clarity.The 'active' lone pair chromophores are the N atoms 3 and 21.

Figure 2 .
Figure2.Schematic qualitative ground state potential energy surfaces[5] for electron transfer in a model bis(hydrazine) radical cation, where the two hydrazine units are coupled by a benzene ring.(a) Direct mechanism and (b) superexchange mechanism.In the sketches of the molecules, carbon and hydrogen atoms bonded to the nitrogen atoms and benzene ring are omitted for clarity.Note also that dashed line means aromaticity.The color-coding green to red is intended to give a quantitative representation of the relative energy (e.g. the CI seam in (a) is higher in energy than the isolated CI point in (b)).Note that the X and Y variables are different in (a) and (b), while Z corresponds always to energy.The electronic structure at the CI 'point' in (b) is similar to that along the CI 'seam' E-F in (a).

Figure 3 .
Figure 3. Schematic representation of the MO in which a hole has been created and thus is singly occupied among the four MOs active space corresponding to the here-employed configuration state functions, CSF.(a) CSF 1 (χ 1 ) corresponds to the VB structure A (MOa), while (b) CSF 2 (χ 2 ) is VB structure B (MOb) in.Figure 2. Similarly, (c) CSF 3 (χ 3 ) corresponds to the quinoid D (MOd) structure and (d) CSF 4 (χ 4 ) to the anti-quinoid C (MOc) form in figure 2.

Figure 2 .
Figure 3. Schematic representation of the MO in which a hole has been created and thus is singly occupied among the four MOs active space corresponding to the here-employed configuration state functions, CSF.(a) CSF 1 (χ 1 ) corresponds to the VB structure A (MOa), while (b) CSF 2 (χ 2 ) is VB structure B (MOb) in.Figure 2. Similarly, (c) CSF 3 (χ 3 ) corresponds to the quinoid D (MOd) structure and (d) CSF 4 (χ 4 ) to the anti-quinoid C (MOc) form in figure 2.

Figure 4 .
Figure 4. Wavefunctions Ψ 1−3 resulting from the coherent superposition of adiabatic states ϕ i =1−4 used as initial conditions to propagate the TDSE within the attochemistry approach.The numerical values displayed on the bis(hydrazine) chemical structure indicate the t = 0 Mulliken spin density either projected [19] onto N atomic sites (in black) or partial summations onto the coupler benzene ring (in blue).Specifically, (a) Ψ 1 locates the spin density mainly on one hydrazine and delocalizes some minor contribution over the entire benzene ring.(b) Ψ 2 contains ϕ 2 and mixes the quinoid state.The initial spin density is on one hydrazine and on the four C atoms non-bonded to N. (c) Ψ 3 contains rather than ϕ 2 and mixes the anti-quinoid state.The spin density is again mainly on one hydrazine and on the three C atoms directly bonded to that N.

Figure 5 .
Figure 5. Electron and nuclear dynamics for the direct mechanism.(a) Spin density for the lone pair N atoms 3 (in purple) and 21 (in green).(b) Spin density for all coupler carbon ring atoms.Note the amplitude is negligible compared to (a).The 3.6 fs highlighted in (a-b) identifies a 1/4 period of hydrazine electron dynamics (τ 2 ∼ 15 fs).(c) Terminal N-N distances between N2-N3 in purple and N21-N22 in green.(d) Evolution of the energy difference during the first full period between the two highest energy states, namely, E (ϕ 4 ) − E (ϕ 3 ) (see table 1 for t = 0 values resulting in ∆E = 0.28 eV = 6.45 kcal mol −1 ).See figure 1 for atom numbering.

Figure 6 .
Figure 6.Mulliken spin density distribution at the electronic transition state (1/4 of the period of the direct ca.15 fs dynamics) using as initial conditions (a) Ψ 1 , (b) Ψ 2 , and (c) Ψ 3 .The numerical values displayed on the bis(hydrazine) chemical structure indicate the Mulliken spin density either projected [19] onto N atomic sites or partial summations onto the coupler benzene ring.Specifically, (a) e-TS from Ψ 1 resembles the electronic structure of the ridge E-F in the direct mechanism (figure 2(a)), (b) e-TS from Ψ 2 resembles the quinoidal intermediate D of the superexchange mechanism but it is also related to the direct mechanism since the bridge electron dynamics is passive and non-perturbing, and (c) e-TS from Ψ 3 resembles the anti-quinoidal intermediate C of the superexchange mechanism (the ring dynamics has a significant spin density amplitude that oscillates between 0.20 and 0.03 following the 2 fs dynamics, as shown in figure9, however, there is a partial cancellation every 2 fs starting from 1/8 τ 3 , which blurs the e-TS).Here, the bridge electron dynamics is active and perturbs the hydrazine electron dynamics.

Figure 7 .
Figure 7. Electron and nuclear dynamics using Ψ 2 (R, t = 0) as initial condition.(a) Spin density for the lone pair N atoms 3 (in purple) and 21 (in green).The 3.6 fs highlight shows 1/4 period of hydrazine electron dynamics (τ 2 ∼ 15 fs).(b) Spin density for half of the carbon ring atoms for simplicity.Note the amplitude of C15 and C17 (and C10, C12 by symmetry) is not negligible compared to (a).The 2 fs highlight indicates the period of benzene electron dynamics.(c) Terminal N-N distances between N2-N3 in purple and N21-N22 in green.(d) C-C bond lengths inside the coupler benzene ring, namely, C9-C17 in purple, C17-C15 in green, and C14-C15 in blue.Note dashed lines are equivalent to solid lines by symmetry.See figure 1 for atom numbering.

Figure 8 .
Figure 8. Electron and nuclear dynamics using Ψ 3 (R, t = 0) as initial condition.(a) Spin density for the lone pair N atoms 3 (in purple) and 21 (in green).See 3.75 fs inset to show 1/4 period of hydrazine electron dynamics (τ 3 ∼ 15 fs).Lines are a guide for the eye to visualize the full period of the hydrazine electron dynamics.(b) Spin density for the carbon ring atoms.Note the amplitude of C9 and C14 is not negligible compared to (a).See 2 fs to show a full period of benzene electron dynamics.(c) Terminal N-N distances between N2-N3 in purple and N21-N22 in green.(d) C-C bond lengths inside the coupler benzene ring, namely, C9-C17/C9-C10 in purple, C17-C15/C10-C12 in green, and C14-C15/C14-C12 in blue.See figure 1 for atom numbering.

Figure 9 .
Figure 9. Spin density snapshots taken every ca. 1 fs from t = 0 to t = τ 3 /2 showing the interplay between the benzene electron dynamics and the hydrazine electron dynamics.Only N atoms and C atoms bonded to them are shown since their spin density dominates.

Table 1 .
Adiabatic states {ϕ i } as linear combinations of CSFs {χ k } resulting from CASCI calculations.See figure

Table 3 .
Spin density gain of N3 after one full period extracted at τ /2, and τ fs, which correspond to the maximum value of the spin density.