Crossover between the Adiabatic and Nonadiabatic Limits of Thermal Electron Transfer: Verifying the Landau-Zener Formula

The semiclassical models of nonadiabatic transition were proposed first by Landau and Zener in 1932, which has been widely used in study of electron transfer (ET); however, experimental substantiation of the Landau-Zener formula remains challenging. Herein, 10 employing the Marcus-Hush theory, thermal ET in mixed-valence complexes {[Mo 2 ]-(ph) n -[Mo 2 ]} + ( n = 1-3) is investigated and the Landau-Zener analysis performed in the adiabatic and nonadiabatic limits. Evidently, the Landau-Zener formula is valid in the adiabatic regime in a broader range of conditions than the theoretical limitation known as the narrow avoided-crossing. The intermediate system is identified with an overall transition probability 15 ( k el ) of ~ 0.5, which is resolved by the contributions from the single and the first multiple passage. The results unify the contemporary ET theories under the semiclassical framework. The obtained insights help to understand and control the ET processes in biological and chemical systems.

molecular electronics 7 and solar energy conversion. 8 According to the Marcus theory, 1,3,5 ET rate is governed by three physical parameters: the Gibbs free energy change (ΔG°), the reorganization energy (λ) and the electronic coupling (EC) matrix element (Hab). Quantities l and Hab are the nuclear and electronic factors, respectively, which affect ET process due to the differences in time scales of nuclear vibrations and electron transmission. Both intramolecular and intermolecular ET 5 reactions may occur adiabatically and nonadiabatically, depending on the interplay of the atomic and electronic dynamics of the system and medium. Comparison between the electron hopping frequency (nel) and nuclear vibrational frequency (nn) determines ET in the two regimes, that is, 2,9 adiabatic: nel >>nn; nonadiabatic: nel << nn Nonadiabatic transition of reactions from reactant to product was described first by Landau and 10 Zener in 1930s. 10,11,12 The semiclassical Landau-Zener (LZ) model discriminates quantitatively the nonadiabatic and adiabatic limits by three parameters: adiabatic parameter g (eq 1a) nonadiabatic transition probability P0 (eq 1b ) and electronic transmission coefficient kel (eq 1c), 1 When g >> 1, the adiabatic limit is realized and for thermal ET kel ≈ 1, while the nonadiabatic limit prevails with g << 1. 1,2,13 By definition of g (eq 1a), it is clear that nonadiabatic transition depends upon the electronic and nuclear factors, represented by Hab and nn, respectively. According the LZ can be possibly accomplished in elemental ET reactions, if an array of systems with the electronic dynamics spanning a broad range of time scales with respect to the nuclei is developed.
Photoinduced ET is generally in the nonadiabatic regime, while thermal ET occurs usually adiabatically with nel >> nn. Testing the LZ model (eq 1) also encounters the technique problems. 15 For example, time-resolved spectroscopy 3 and spectral line-broadening analysis 17 This Hab parameter can be used to calculate the electronic transmission frequency (nel) (eq 3), the 15 adiabatic parameter (g) (eq 1a) and the optical or the thermal ET kinetics based on semiclassical theory at the high temperature limit. 5,21 ( This approach was first proposed by Taube in 1986; 22 unfortunately, it has not succeeded for many decades. The reason for this is that few MV molecular systems exhibit characteristic IVCT bands 20 that allow optical derivations of the ET dynamics and kinetics, 9,23,24 although many efforts have been devoted since the Cruetz-Taube ion [(NH3)5Ru(pz)Ru(NH3)5] 5+ was synthesized in 1970s. 18,22,25 Given the characteristic IVCT bands, mixed-valence D-B-A molecular systems with a quadruply-bonded Mo2 unit 26 as the donor and a Mo2 unit having a bond order 3.5 as the acceptor 25 are desirable experimental models for study of thermal and optical ET, in which single electron migration is ensured and the transferring electron is specified to be one of the d electrons. 27,28,29 Hereby, nine MV complexes of three series with a general formula [Mo2(DAniF)3]2(µ-4,4'-  (Fig. 1), were investigated as a testbed of the LZ theory. In Marcus theory, the total reorganization energy l is divided into lin and lout, corresponding to the intramolecular (lin) and solvent (lout) nuclear motions. 5 [EE¢-(ph)n-EE¢] + complexes have small lin, as evidenced by the very low IVCT energy for [SS-ph-SS] + (2640 cm -1 ) 27 in comparison with the Cruetz-Taube 5 complex (6369 cm -1 ). 18 Thus, the adiabaticity of the systems is effectively solvent-controlled because the lin is generally assumed to be independent of the bridge length. 30  [SS-(ph)3-SS] + , are identified with an overall transition probability of ~ 0.5 that is achieved through the single and the first multiple passage. This work has validated, for the first time, the LZ model and revealed the energetic and dynamic details of a system crossing over the two limits, 15 which are not well described by this model. 12,15 The results and conclusions unify the contemporary ET theories under the semiclassical framework. 31 Using the published procedure for preparation of the ph and ph2 bridged analogues, 27 Table 1). 27,28,32 For [EE¢-(ph)n-EE¢] + with the same ancillary DAniF ligands, the vertical δ → δ* transition occurs at similar wavenumber, ca. ~450 nm; 26 however, this band is masked sometimes by the other electronic transitions. 27  The LMCT band for the MV complexes arises from charge transfer from the p orbital of bridging ligand to the d orbital of the cationic Mo2 center, thus, corresponding to hole transfer in the opposite direction. Simultaneous presence of the MLCT and LMCT bands facilitates the through- 5 bond superexchange, 33 leading to strong EC between the two Mo2 centers. 32 The MV dimers with ph and ph2 bridges present a characteristic IVCT band, from which the spectral parameters, transition energy (EIT), molar extinction coefficient (εIT) and half-high bandwidth (Δν1/2), are extracted (Table 1) Table 1). The 15 more delocalized [SS-(ph)3-SS] + has a EIT comparable to that for the organic MV D-B-A system with a ph3 bridge (6700 cm -1 ). 34 It is interesting that the IVCT bands for [EE¢-(ph)3-EE¢] + are narrower than those of the ph2 analogues (Table 1). This is phenomenal because IVCT band broadening is expected for weaker coupling systems according to Δn 0 1/2 = 2[4ln(2)lRT] 1/2 . 9,35 The EC constants (Hab) are calculated from the Mulliken-Hush expression (eq. 2). 9,19 In 20 application of eq. 2 for [EE¢-(ph)n-EE¢] + , the length of the bridge has been used as the effective ET distance, considering that the δ electrons are fully delocalized over the [Mo2] coordination shell. Therefore, for the ph, ph2 and ph3 series, the geometrical lengths of the bridge "-(C6H4)n-", 5.8, 10.0 and 14.3 Å, respectively, are adopted to be rab for the given systems. 27,28 The Hab data are listed in Table 1. Large decrease of Hab is found for the [EE¢-(ph)3-EE¢] + complexes (Table 1)  independent of the nuclear geometries. This phenomenon conforms to the Condon approximation, manifesting a system transition from adiabatic to nonadiabatic, but contradicts the theoretical outcomes with calculated matrix elements. 37 Optical analysis indicates that these two systems belong to the weak coupling Class II, while [OO-(ph)3-OO] + should be assigned to Class I, in terms of the Robin-Day's scheme. 9,38  The MV [Mo2]-bridge-[Mo2] complex constitutes uniquely an effective "one-particle" donoracceptor system. 13 In such as a system, adopting a semiclassical two-state LZ model, 10,11 the ET initial (fI)and final (fF) diabatic states can be approximated by the d orbtials of the donor and acceptor, namely, dD and dA, respectively. Assuming that the diabatic and adiabatic states essentially coincide in the vicinity of the electronic equilibrium configurations, linear 5 combinations of dD and dA generate two first-order or adiabatic states (eq 4), 29 Then, we have the nonadiabatic mixing matrix element where h is an effective one-electron Hamiltonian. 13,29 The energies of the adiabatic states, obtained by solving the two-state secular determinant, are given by eq 5, 24,35 (5a) where DG° = 0 for the current symmetrical systems. These two adiabatic states are represented by 15 the upper (V1) and lower (V2) PESs separated by 2Hab at X = 0.5.
system, features small curvatures of the diabatic parabolic potential curves. The adiabatic PESs coincide with the diabatic PESs in the conical region with the upper (V1) and lower (V2) surfaces meeting almost at the diabatic crossing point ( Fig. 2A). In series [SS-(ph)n-SS] + (Fig. 2B), the PESs for [SS-ph-SS] + are dramatically different from those of the systems with longer bridges. It shows nearly a flat lower V2 surface with two very shallow wells at the reactant and product 5 equilibriums. The separation between V1 and V2 at X = 0 corresponds to the low Frank-Condon transition energy (EIT = l), close to the adiabatic spacing (2Hab) at X = 0.5 ( Table 1). The transition state energy (DG*) is only 79 cm -1 , much less than the thermal energy level kBT (207 cm -1 at 298 K). This causes the thermal energy level unevenly populated around the reactant equillibrium; consequently, Frank-Condon transition generates a "half cutting-off" IVCT band (Table 1) 2C), with the same ph2 bridge, shows that the S chelating atoms enhance effectively the EC by 15 lowering l and increasing 2Hab. According to Marcus,3,5 in the nonadiabatic limit, the thermal activation energy DG* = (l + DG°) 2 /4l; in the adiabatic limit, DG* is reduced by Hab. 9,13 Since (l + DG°) 2 /4l in the nonadiabatic limit is a value of the lowest (i.e., zeroth) order in Hab, we can reasonably approximate DG* by eq 7 for the adiabatic-nonadiabatic borderline regime when Hab is sufficiently small. 2 DG* = (l + DG°) 2 /4l -Hab  Table 1 lists the values of (l/4 -Hab) for each of the systems, in comparison with DG*. Obviously, such a correlation does not exist for strongly coupled systems, for example, [EE¢ph-EE¢] + (Table 1) (Table 1). These results represent the energetic features of systems in transition from the adiabatic to nonadiabatic limit. The adiabatic ET rate constants, ket(ad), for the MV systems are calculated from the classical transition state formalism 1,5,9 (eq. 9) with a preexponential factor kelnn and activation energy (DG*) from the Marcus-Hush theory (eq. 10). 9 10 The accuracy of the optically determined rate constants is confirmed by IR-band broadening analysis recently. 42 In this work, a transmission coefficient (kel = 1-0.11) calculated from the LZ formula (eq. 1) ( Table 1) is used to derive ket(ad) from eq 9. Given the low-frequency solvent modes νout in 10 12 -10 13 s -1 in classical theory, an averaged nuclear frequency, νn = 5 10 12 s -1 is generally adopted. 9,23 This is further justified in the present systems in which the nonadiabatic 15 transition is governed by solvent thermal fluctuations. In the nonadiabatic limit, comparison of eq 11 to eq 9, in conjunction with eq 1a, gives κ = 2(2pg) and DG* = l/4, the Marcus activation erengy. 5,13 This indicates implicitly that the adiabatic and nonadiabatic limits are bridged through the intermediate of the LZ model, which can be exploited to test the connection of the existing ET rate expressions in the two limits. 20 For the ph and (ph)2 bridged series (Table 1), the electron frequencies (νel) are in the order of 10 13 -10 14 s −1 , higher than the nuclear vibrational frequency (νn) (10 12 -10 13 s −1 ) by one order of magnitude.
[SS-ph-SS] + has the highest ET rate with ket(ad) = 3.4 ´ 10 12 s -1 , close to the adiabatic limit (5 ´ 10 12 s -1 ), in accordance with its optical behavior as a Class II-III MV system. However, the rate constant derived from eq 11, ket(nonad) = 1.4 ´ 10 14 s -1 , is significantly larger than nn, 25 indicating the irrationality of the nonadiabatic treatment for this system ( Table 1). The deviation of ket(nonad) from ket(ad) decreases with increase of the nonadiabaticity. It is remarkable that for

transient systems, [OO-(ph)2-OO] + and [EE¢-(ph)3-EE¢] + , ket(ad) = ket(nonad) with small
analytical errors, and the data fall in the range of 10 8 -10 9 s -1 (Table 1). Similarly, ket ~ 10 9 s -1 is reported for the ph3 bridged organic radical system. 34 It is noted that the rate constant for [SS-(ph)3-SS] + is about 5 times larger than that of [OS-(ph)3-OS] + (Table 1), despite the similar Hab values. The high sensitivity of ket on Hab is consistent with the increased nonadiabaticity for these 5 systems. 1,2,5,15,30 In contrast, the strongly coupled series [EE¢-ph-EE¢] + shows Hab independence of the ket, (Table 1), as expected. 2,10,11,15,30 Importantly, the kinetic data demonstrate that the adiabatic and nonadiabatic regimes are smoothly bridged by the crossover regime, which can be well described by the Marcus theory. 5,24 The nonadiabatic treatments using solely the average lowfrequency nuclear mode (nn) on the thermal ET occurring at the intersection of the adiabatic PESs 10 generate precisely consistent outcomes in both the nondiabetic limit and the transient regime.
Therefore, this work shows that the adiabatic and nonadiabatic ET rate expressions are applicable in the respective ET dynamic limits, and work equally well with accordant results for the LZ intermediates, although a single theory that rigorously treats the two limits is not available. 16 [OO-(ph)3-OO] + should be placed in the nonadiabatic regime. This is confirmed by the 0.14 kel value, which is close to the nonadiabatic pre-exponential factor (0.16) calculated from k = 4pg. 1,2 While g >> 1 and g << 1 characterize the adiabatic and nonadiabatic limits, respectively, the intermediate is not explicitly classified in the LZ model. For [OS-(ph)3-OS] + (g = 0.065) and [SS-5 (ph)3-SS] + (g = 0.084), g is much less than 1, from which the systems might be assigned to the nonadiabatic limit. However, for both, kel » 0.5, meaning that about 50% of the transition attempts that reach the transition state through thermal fluctuation can successfully complete the ET process. We have seen that these systems present dynamic and energetic properties in the avoided area that are distinct from those in the adiabatic and nonadiabatic limits.  (Fig. 4C), for which g < 0.15, in consistent with the theoretic value 0.2 given by Sumi. 43 When g > 0.5, kel deviates from the linear dependence on g and approaches unity for g >1, showing the g-dependence of k as theoretically predicated. 2,43 Therefore, the experimental results are generally in accordance with theoretic results, but give a narrower and more precise window of g For systems with P0 < kel, involvement of multiple passages in thermal ET reactions is anticipated. For the intermediate systems, it is assumed that two channels, the single passage and 25 the first multiple passage, operate for nonadiabatic transition, as described by Fig. 3B. In the first channel, the electron makes a transition from the reactant to the product state through the crossing point, giving the probability P0. In the second channel, the electron passes the crossing point three time to complete the reaction. The first (step 1) and third (step 3) crossing take place on the reactant and product diabatic PESs, respectively, which have the same probability, (1-P0). Electron hopes from the reactant to the product PES through the second transition (step 2) with the same probability as the first channel (P0). This multiple passage gives the transition probability of (1-P0)P0(1-P0). 2 For [OS-(ph)3-OS] + and [SS-(ph)3-SS] + , a transition probability of 0.15 and 0.14 is obtained from the multiple passage, respectively. The total probabilities in this two-channel scheme, ca. 0.47 and 0.55, are close to the overall transmission probabilities (kel) 0.48 and 5 0.58 (Table 1), respectively. This means that for these two systems, 98% and 95% of the successful nonadiabatic hopping events proceed through the first and second channels, with the first channel playing the dominant role. For [OS-(ph)3-OS] + , the overall transition probability is slightly small, in comparison with [SS-(ph)3-SS] + , but the multiple passage contribution is relatively large due to the increased nonadiabaticity. It is believed that this two-channel operation can be the typical 10 behavior for thermal ET systems on the adiabatic-nonadiabatic borderline. For [OO-(ph)3-OO] + , the single and multiple passages are nearly equally important, each of which contributes a transition probability of ~ 0.07. Evidently, this system has entered the nonadiabatic regime through the intermediate. The small kel (0.14) visulizes the failure of thermal ET through nonadiabatic transtion due to the high activation erengy, DG*» l/4 (Table 1). However, this does not mean no 15 electron self-exchagne occurring between the donor and acceptor. For this long-bridge, weakly coupled system, the nonadiabatic ET may occur through optical transiton, 9 the highly energetic pathway at the same ET rate as for the thermal ET pathway. 21 The multiple trajectory model, developed based on the Fermi Golden rule, 31 is the core of the quantum mechanism for nonadiabatic reactions. 1,2,43 However, trajectory analysis of ET in experimental systems is rarely 20 seen.
The LZ model was developed to deal with nonadiabatic coupling in the vicinity of the avoied crossing where DG* >> 2Hab. The theory cannot tell what happens if the kinetic energy is comparable to the interaction energy. 2,12 This is the case represented by [EE¢-ph-EE¢] + , for which DG* < 2Hab. Supprisingly, even for the mostly strongly coupled [SS-ph-SS] + , with the activation 25 energy (DG* = 79 cm -1 ) much smaller than the coupling energy (Hab = 864 cm -1 ), the thermal ET can be well described by the LZ parameters, that is, g = 5.32 (> 1), P0 = 1 and kel =1. Moreover, theoretically, application of the LZ model is limited by the requirment of narrow avoded crossing, that is, that the minimal spacing between the adiabatic PESs at the avoided region should be much smaller than the spacing far from the coupling region. 12 Again, taking [SS-ph-SS] + as an example, the separations between the surfaces V1 and V2 at the reactant equilibrium, i.e., EIT = 2640 cm -1 and at the transtion state, i.e., 2Hab = 1728 cm -1 , are in the same order of magnitude, which breakdowns the narrow avoided-crossing approxiamtion. Collectively, this study provides the first precise scaling picture showing system transition from the adiabatic to nonadaibatic ET limits through the intermediate, which demostrates the validity of the Landau-Zener model. The 5 experimental results extend the theoretic limitation in application of the LZ formula, and offers the details of the energetics, dyanamics and kinttics for system crossover. 10