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67b7c567fa469535b9379533 | 22 | Formally, there are an infinite number of ordered terms. However, if we consider the first order term to be small with respect to the zeroth order term, we can say H 1 is a perturbation upon H 0 . Hence, Fermi's Golden rule operates within the weak coupling limit, which can be expressed as : |
67b7c567fa469535b9379533 | 23 | where ρ (E mr -E ns ) is the density of states. For more complex systems, a Boltzmann weighting factor P r can be employed, however for many cases, the lack of is fine . Due to the lack of computational chemistry during early years of the methods' development, vibronic coupling could not be calculated directly, however, at low temperatures, the rate is maximised when the energy gap and rigidity parameters are nearly equal . Instead, it was approximated from the promoting mode (strong participating mode) frequency . Lin took the derivation further; assuming a promotingaccepting mode paradigm, Equation 12 can be recast in terms of the overlap between these two modes K j and G j ′ : |
67b7c567fa469535b9379533 | 24 | Since K j is dependant on the electronic/nuclear terms, and G j ′ on the vibrational terms, this can be more simply explained as the rate of internal conversion between a promoting electronic state, and the accepting vibrational states of another electronic state; where the rate of coupling/overlap of these two components determines the speed of the mechanism. The evaluation of K j and G j ′ can be performed using Mehler's formula . Formally, the coupling matrix elements can be expanded with respect to the normal modes {Q j }, cast as : |
67b7c567fa469535b9379533 | 25 | The second order derivative term vanishes when adopting a linear approximation; it varies slowly with the normal mode coordinates . When the first-order derivative of the initial electronic state displays minimal variance with respect to Q j , the Condon approximation, or the course adiabatic approximation, can be applied . Further, cross terms between different vibrational normal modes can be neglected by assuming a single promoting mode . This is the formal casting of the Born-Oppenheimer coupling . The first order derivative term Equation , is known to have 3 problems . |
67b7c567fa469535b9379533 | 26 | To further understand these limitations, many authors reported on a one-dimensional case study of benzene on the vibrational potential along specific C-H stretching modes . The findings of these works conclusively show that for specific systems, specifically case studies with minimal transition participating modes, a non-Condon treatment yielded significant increases in the overall rate constant, by up to a factor of 3. |
67b7c567fa469535b9379533 | 27 | where J j mn is the pure electronic term, and F k (mr, ns) are the Franck-Condon factors. This formalism can be simplified via selection rules, which are similar to those for vibronic coupling . The pure electronic matrix elements becomes non-vanishing for modes corresponding to similar representations within the molecular point group, via the direct product representation ϕ m × ϕ n . Specifically , the internal conversion selection rule is determined by ψ ns ∂ ∂Qj ψ mr and can be calculated in a similar fashion to that of the optical transition dipole moment . In the case of non-degenerate electronic states, only one vibrational normal mode could be expected to interfere with this interaction, in which case the matrix element can be reduced to a single electronic-Franck-Condon product term. These coupling matrix elements can also be displayed as a sum of products between electronic and Franck-Condon overlap terms . |
67b7c567fa469535b9379533 | 28 | As per Equations 13 and 14, Li and co-workers report rates between 4.8×10 3 s -1 for the 1 A 1g ← 1 B 2u transition of benzene, depending on the promoting mode. They do, however, remark that their results may be underestimated from the true rate constant, due to the combination of a large energy gap, and small Huang-Rhys factors and vibronic coupling matrix elements, and that their results should instead be interpreted as a lower bound of possible rates. Li and co-workers study the same transition in benzene, and find rates of 2.85 × 10 3 s -1 and 1.91 × 10 3 s -1 , with respect to the promoting mode. Valiev and co-workers built on this and calculated internal conversion in terms of quantum Green's functions for some cyanines, and found rates on the order of ∼10 10 s -1 . Marconi and co-workers studied a series of aza-stilbenes and calculated rate constants on the order of 10 10 s -1 , while Lei and co-workers studied a series of different thermally activated delayed emitters and calculated rates on the order of 10 5 s -1 for five different 9,10-dihydro- 9,10-diboraanthracene derivatives. Yin and co-workers studied 1,1-dimethyl-2,3,4,5-tetraphenylsilole in detail and calculated a rate of 1.8 × 10 11 s -1 . Lin and co-workers examine formaldehyde and calculated rates of 1.21 × 10 7 s -1 and 3.79 × 10 6 s -1 , depending on how many states are averaged over . Interestingly, the former compares best to experimental observations. Hornburger and co-workers noted that when using a local mode method, internal conversion in benzene was more sensitive excess vibrational energy within a harmonic model compared to a morse one; 10 7 s -1 compared to up to 10 5 s -1 . They however highlighted that a key problem remained: for a successful treatment, the electronic and vibrational components must be separated. Further, Hornburger and co-workers found that a one-dimensional non-Condon treatment results in a much slower rate, since energy is distributed along all modes rather than a single transition participating mode; the importance of the single mode is governed by qualities of the final electronic state normal modes, analogous to the inclusion of large occupations numbers , and is therefore influenced by anharmonic or inter-vibronic coupling effects. They therefore concluded that a Condon treatment was viable with respect to specific uncertainties. |
67b7c567fa469535b9379533 | 29 | When taking Duschinsky-type phenomena into consideration, internal conversion can he altered by more than 50% in some cases. Some work has been done to incorporate these effects , the rate constant is expressed as a linear combination of matrix elements, where it is assumed that there is only one promoting mode and it is pure/unmixed. Mebel and co-workers first looked at internal conversion for the 1 B 1u → 1 A g transition in ethylene, and found two promoting modes, each with respective rate constant of between 1×10 6 -8×10 7 s -1 and 1×10 9 -1.26×10 10 s -1 , for a temperature range between 0-1500 K. They also examined the 1 B 1 → 1 A 1 internal conversion in allene, with a focus on the effect of Duschinsky vs. non-Duschinsky rates, and found that between the same temperature range, an increase by more than 50% can be observed as a function of temperature. |
67b7c567fa469535b9379533 | 30 | Here, the vibrational wavefunctions ψ s ∂ ∂Qj ψ r are Boltzmann distributed, and therefore is valid for internal conversion in a tightly bound system. By taking the density of states as a Dirac delta function and applying a Fourier transform to Fermi's Golden Rule (Equation ), shifting from the frequency domain to the time domain, cast as : |
67b7c567fa469535b9379533 | 31 | where Z p is the partition function, ω is the energy gap in the frequency domain, and Tr [ρ ic (t, T )] is the trace of the thermal vibrational correlation function. Full analytical derivations of the correlation function can be found in Refs. . Reimers reported on solutions for calculating Duschinsky rotations and vector displacements, required for the correlation function. It is worth highlighting that sometimes Equation 19 is cast with the partition function and density absorbed into the coupling term. One of the most well known implementations of this methodology is the Momap software package , which be believe to be one of the only licensed software packages which can calculate internal conversion. Calculations on azulene for example yield a rate 2.3 × 10 10 s -1 for the S 1 → S 0 pathway, comparing very well to experiment. Further calculations by Veys & Escudero on multiple azulene derivatives found various rates on the order of 10 11 s -1 for the S 1 → S 0 transition, and 10 9 s -1 for the S 2 → S 1 transition. Importantly, the correlate well with experimental observations. We do note that to our knowledge, Momap does not yet account for Herzberg-Teller contributions to the vibronic coupling. |
67b7c567fa469535b9379533 | 32 | External to this implementation, Wang and co-workers[161] used the time domain treatment to examine internal conversion in azulene, calculating an analytically exact rate of 2.17×10 10 s -1 and 2.44×10 10 s -1 for temperatures of 0 K and 300 K, respectively. Further, to break down the method, they examined two special cases. The first within the harmonic approximation was to understand how many bond dimensions were required to yield qualitatively accurate results without considering the whole system; where a bond dimension here is the number of degrees of freedom allowed for the ansatz. While they calculated for dimensions up to 60, convergence was fast, while a bond dimension of 20 yielding quantitatively accurate rates of 2.11×10 10 s -1 and 2.32×10 10 s -1 for temperatures of 0 K and 300 K, respectively. The bond dimensions of 60 yielded rates of 2.16×10 10 s -1 and 2.41×10 10 s -1 for temperatures of 0 K and 300 K, respectively. The second special case was to examine the degree of anharmonicity in azulene; this was done by calculating both potential energy surfaces separately and the excited state within the harmonic approximation, while the ground state potential was examined as a morse potential (specifically, a one-mode representation). The resulting rates suggested that anharmonic tendancies in azulene accounted for between 30-40% of the photophysical behaviour, at least with respect to the S 1 → S 0 transition. Peng and co-workers highlight the importance of mode mixing, with a qualitative increase in the internal conversion rate in ethylene from ∼1.12 × 10 7 s -1 to ∼1.58 × 10 7 s -1 . |
67b7c567fa469535b9379533 | 33 | Miyazaki & Ananth used a singularity-free casting to calculate the rate constants for the smaller polyacenes and azulene, and find rates of 2.7 × 10 9 s -1 , 1.1 × 10 7 s -1 , 4.2 × calculate the rates of a number of pyrazine derivatives, and find rates of 4.45 × 10 9 s -1 and 3.29 × 10 5 s -1 at room temperature for 2,3dicyano-5,6-diphenylpyrazin and 2,3-dicyanopyrazino phenanthrene, respectively. Importantly, they probe Duschinsky effects in both systems, and find them to be vital in the description of the former, and much less to in the latter. Interestingly, they also probe temperature effects, with (again) the former being more sensitive to temperature effects (an order of magnitude shift between 20 K to 300 K). Humeniuk and co-workers found that the broadening function used can drastically effect the rate of internal conversion, with this dependence weakening at the rates become large. A Voigt profile was found to yield significantly large rate constants when compared to those calculated using a Gaussian profile (Table ), whereby a strong relationship can be observed between the rate constant and the width of the Lorentzian, as opposed to the structure of the vibrational continuum itself, with this effect becoming more pronounced for those systems in which Duschinsky rotations were more important. Adjacently, Wenzel & Mitric calculate internal conversion for a series of molecular compounds within the extended thawed Gaussian approximation as well as the vertical and adiabatic harmonic models, and found rate constants would vary drastically depending on the choice of density functional and the Gaussian lineshape function (Table ). Unlike those results reported by Humeniuk and co-workers , variability in rate constants was within an order of magnitude. Rybczński and co-workers[163] examined a series of benzothiazole-difluoroborate substituted derivatives, and found rate constants between 0.26 -7.87 × 10 8 s -1 , depending on the specific substitution. They also noted a strong dependence on this method to the broadening function employed, with order of magnitude shifts for 4 of the 9 systems examined. |
67b7c567fa469535b9379533 | 34 | Zhou and co-workers used the same implementation for a series of triphenyl amine derivatives, with rate constants ranging between 10 5 -10 10 S -1 depending on the substitution, providing a suite of control for internal conversion in the monoradical species. However, no clear pattern was observed between the eight studied sub-species. Inai and co-workers calculated the rate constants for bisphosphoryl-bridged stilbenes and their thiophene-fused counterparts; and calculated a rate of 6.1×10 3 s -1 when using density functional theory. However, the rate constant dropped to 9.0×10 2 s -1 when the energies were calculated at the coupled cluster (singlets-doublets) level of theory. For the other stilbene derivatives they studied, rate constants between 10 5 -10 8 s -1 were reported. Hayashi and co-workers examine the case of ethylene, and found rates ultrafast rates on the order of 10 10 s -1 for the higher excited states, and slower rates on the order of 10 6 s -1 for the lower excited states. Veys and co-workers highlighted a strong dependence on the full-width half-maximum of the broadening function, in some cases resulting in 2 orders of magnitude difference in the final rate across 7 different chromophores. |
67b7c567fa469535b9379533 | 35 | There are some advantages to employing such a methodology . Firstly, it is entirely analytical, and therefore technically exact, at least with respect to the quantum chemical information used to solve it. Second, vibrational mixing effects are included, which is very important in the case of low energy normal modes; these modes have been shown to mix upon electronic excitation. Thirdly, the entire vibrational continuum is taken into account as opposed to only the promoting mode. Fourthly, computing the rate in the time domain rather than the frequency domain is often computationally less expensive. Conversely, this method cannot be used accurately for internal conversion rate constants faster than 10 12 s -1 . However, there are also disadvantages to relying entirely on the lineshape. For example, Xu and co-workers noted that temperature effects were overtly broadening vibronic contributions in the spectral lineshape and width of some helicene derivatives, yielding a rate constant of < 34 s -1 , much smaller than should be expected. |
67b7c567fa469535b9379533 | 36 | Soon after, Siebrand redefined the Franck-Condon factors, and was able to show that they are dominated by distortions in the case of large energy gaps, or displacements in the case of small energy gaps. This lead to an exponential-like relation between the Franck-Condon factors and the energy gap, cast as : |
67b7c567fa469535b9379533 | 37 | As discussed previously, Hornburger and co-workers noted that a non-Condon type method was acceptable, disagreeing with many previous studies , and concluded that a Condon treatment was acceptable for most systems. However, for the case where a first-order treatment was not acceptable, the theory was still lacking. Following this, Nitzan & Jortner pushed for a more generalised formalism, outside of the Condon approximation. While the theory proposed by Herzberg & Teller was available at the time, no one yet understood exactly how this could be applied to internal conversion. By assuming each molecular frequency was roughly identical (an average frequency), a generating function can be integrated with the steepest descent method, yielding a decay rate formalism in terms of the vibration-less : |
67b7c567fa469535b9379533 | 38 | where ω is the average energy of the vibration-less level, ∆ 2 j is the dimensionless displacement of the j th normal mode, and η is a complex "correction" factor . This more generalised treatment, however, was limited to the weak coupling regime. The strong coupling limit was speculated to be treatable outside of the Condon approximation provided that the system is limited to a single oscillator model. For the non-Condon formalism, it was observed that for systems which could mimic the qualities of larger systems, decay would be underestimated by up to 3 orders of magnitude . Using Siebrand's reevaluation (Equation ), Kowaka and co-workers found very fast lifetimes, corresponding to ∼5×10 6 s -1 , for all transitions from the 1 B 3u state of pyrene, due to strong vibronic coupling with the higher 1 B 2u state. They also reported a 1 B 3u ← 1 B 2u rate of ∼2×10 9 s -1 . |
67b7c567fa469535b9379533 | 39 | It should be pointed out that the only difference between the Condon the non-Condon formalism is the energy dependence on the nuclear coordinate ; modelling of the energetic landscape seems to require some consideration towards the adiabatic wavefunctions rather than just the nuclear coordinates. In works where the Condon approximation was used , the relationship between the single oscillator model in the weak coupling limit can be summmarised simply as : |
67b7c567fa469535b9379533 | 40 | A further simplification can be taken where for large energy gaps, the first term on the right-hand side of Equation 22 can be ignored; this is formally a generalised relation of the energy gap law in non-radiative decay . In addition, analysis of optical selection processes within the Condon approximation , which assumed any electronic coupling was independent of the initial vibronic state, assuming that the electronic coupling matrix element is independent of the initial vibronic state. While a shift of less than 2% was seen in the vibronic level of S 1 benzene due to the derived non-Condon correction term , it is apparent that an internal conversion decay probability is strongly dependent on the amount of excess vibrational energy, since higher normal modes may be allowed to participate in the transition. |
67b7c567fa469535b9379533 | 41 | Shizu & Kaji examined internal conversion in benzophenone across all vibrational normal modes, for both the singlet and triplet manifolds using the correlation-type formalism (Equation ). For the S 1 → S 0 transition, a rate constant of 3.3×10 6 s -1 was recorded, while on the triplet manifold, "uphill" rate constants from the T 1 state were no larger than 100 s -1 . For the near-degenerate T 2 → T 3 transition, the "uphill" transition was much faster, at ∼10 10 s -1 [170] Thermally allowed transitions across the triplet manifold otherwise were quite fast, of the order of 10 10 s -1 or larger. They then compared this with a cost-effective rate constant expression, in which a series of kinetic equations are derived for a given system, cast as : |
67b7c567fa469535b9379533 | 42 | where x j is the vibrational quantum number for the j th oscillators, ϵ 0 is the vacuum permittivity, c is the speed of light, a B is the Bohr radius, e is the elementary charge, and HWHM is the half width half maximum of the spectral lineshape. This method has been reported to drastically simplify the modelled of exciton dynamics . The resulting rate constants from this methodology appear qualitatively similar to their counterparts. Shizu & Kaji noted that for the S 1 → S 0 transition in benzophenone, the rate increased by several orders of magnitude to 1.1×10 8 s -1 . On the triplet manifold, while triplet "uphill" transitions from the T 1 state were no larger than 10 3 s -1 , thermally allowed transitions were also noted to have increased by one to two orders of magnitude. It should be noted that the authors reported rate constants at a number of different excited state geometries, for each transition, not just at the Franck-Condon point. Here, we have reported only the rate constant from the Franck-Condon point, as we believe them to be the most relevant. |
67b7c567fa469535b9379533 | 43 | Shizu and co-workers[171] used the same method (Equation 24) and found picosecond rate constants. For a series of ortho, meta, and para bridged tetracene dimers, rate constants of 2.02×10 11 s -1 , 1.36×10 8 s -1 , and 3.31×10 9 s -1 were reported. Two different para-configurations were examined (planar vs. free-rotated), and a difference of 3 orders of magnitude in the rate constant was observed, favouring the more flexible rotated configuration. Shizu and co-workers then juxtaposed a tetracene dimer allowed to equilibrate with both full quantum mechanical and split quantum mechanical molecular mechanical techniques; both techniques yielded geometries with very similar internal conversion speeds, with a quantum mechanical geometry rate of 8.74×10 11 s -1 , and a quantum mechanical molecular mechanical geometry rate of 5.05×10 11 s -1 . In all tetracene dimer cases, numerous small energy transition participating modes (∼50 cm -1 ) were observed, leading to the very fast rates of internal conversion. |
67b7c567fa469535b9379533 | 44 | The partial derivative of a generating function provides information as to the dynamics of a system (in this case formally the same as Equation ). The generating function can be solved through the employment of some attenuation factor, typically with some complex time component. With a generating function of the form : |
67b7c567fa469535b9379533 | 45 | Ren and co-workers utilised the n-mode representation method to construct the initial and final state potential energy surfaces and the density matrix reorganisation group method in conjunction with an autocorrelation function (Equation ) to model S 1 → S 0 internal conversion in azulene. They found a rate constant of 0.79×10 10 s -1 at 0 K and 1.00×10 10 s -1 at 300 K. Wenzel & Mitrić used the extended thawed Gaussian approximation to calculate rate constants as a function of anharmonicity. On an arbitrary system, it was shown that the rate decreased from 0.394 E h ℏ to 0.003 E h ℏ as the anharmonicity was increased by a factor of 4. This was found to compare well to the more standard adiabatic and vertical harmonic methods used typically . Islampour and co-workers examined internal conversion in trans,trans-1,3,5,7-octatetraene also using a variable n-mode model for a number of different transition pathways, and varied the number of modes from 1 up to the full number of normal modes. In general, a drastic improvement can be observed as the number of included normal modes increased (Figure ). For the 2 1 A g →1 1 A g transition, they began at a low of 4.95×10 4 s -1 for a single mode model, and increased up to 2.94×10 6 s -1 for all modes, while for the 1 1 B u →2 1 A g transition, the rates increase from 1.45×10 8 s -1 for a single mode and increased up to 1.59×10 10 s -1 when including all b u -type normal modes. Further, modes which were displaced-distorted normal modes resulted in the rate constant stabilising and reducing by on average a factor of 2. Wang and co-workers study a series of 1,2,3,5-tetrakis(carbazol-9-yl)-4,6-dicyanobenzene derivatives, and calculate rate constants on the order of 10 7 s -1 for five derivatives. Peng & Shuai noted that for 1,1,3,4tetraphenyl-2,5-bis(9,9-dimethylfluoren-2-yl)silole, the rate of internal conversion decreased by two orders of magnitude between gas to solid phase, due to the change in electronic character of the potential energy surface of low energy normal modes between the two phases. Peng and co-workers calculate a rate of 7.5 × 10 5 s -1 in anthracene. |
67b7c567fa469535b9379533 | 46 | Assuming the initial and final state are well separated, the Franck-Condon factors decay exponentially with the energy gap between the two levels . The temperature dependence and activation energy in an Arrhenius-like examination of internal conversion is highly semi-empirical, and is defined by the formation or cleavage of chemical bonds. The prediction properties are taken from fitting parameters of experimental data in the form of a pre-exponential factor A IC and the internal conversion activation energy E A . This can be cast as : |
67b7c567fa469535b9379533 | 47 | Kohn and co-workers used the Arrhenius form to study 15 naphthalene derivatives, with resulting rate constants between 6.52-11.29×10 10 s -1 with respect to the derivative, with a pre-factor term of 2.02×10 12 s -1 extracted from experimental measurements. Here, only 2 systems were found to be of the same order of magnitude as experiment, while all but 3 other derivatives were observed to be overestimated with respect to experimental measurements. Lin and co-workers examined 48 Boron-dipyrromethene derivatives using Equation , with resulting rate constants between 0.01-3438×10 8 s -1 with respect to the derivative. The statistics for the latter study are much more transparent; of the training set, 7 predicted rate constants were of the same order of magnitude, while of the remaining 41 rate constants, 30 were overestimated while 18 were underestimated. For the test set, 5 rates were of the same magnitude to experiment, with 27 underestimated and 17 overestimated rate constants. Frank and co-workers examined loss in carotenoids, and found a fitted rate constant of 1.15×10 11 s -1 , however their work was unable to distinguish between different types of non-radiative decay. |
67b7c567fa469535b9379533 | 48 | As shown by the small success rate reported above, there are a number of profits and consequences to using this method. Even through its simplicity, it predicts the strong temperature dependence typical to internal conversion , and also predicts the relationship between E mn and W (m → n). However, the Arrhenius method is entirely semi-empirical, and therefore one cannot reliably predict unknown systems without some experimental input. In particular, the use of this method requires an analysis of the transition state to yield an appropriate value for E A . Secondly, assuming you can obtain an appropriate value for A IC from fitting functions, Equation 29 then becomes entirely dependent on E A , and omits all other photophysical qualities. This way, important interactions like strong vibronic coupling due to overpowering normal modes, level inversion effects, or solvent stabilisation effects, may not be captured correctly. However, this method could be used as a powerful predictive tool to analyse molecular devivatives of a species of interest, particularly in the space of machine learning , assuming the descriptors were sufficient enough to overcome the discussed limitations. |
67b7c567fa469535b9379533 | 49 | Here, E mn is assumed to be much larger than the average vibrational energy, which allows the simplification that only high-frequency normal modes need be analysed. In terms of a physical system, this can be thought of as a series of weakly coupled oscillators, where each oscillator exhibits roughly the same frequency, the energy of which corresponds to the energy of the highest frequency transition participating mode ω MAX . In this picture, the energy gap law can be cast as : |
67b7c567fa469535b9379533 | 50 | where ∇ is the average origin shift across all oscillators. To investigate the validity of the expected linearily of Equation , Cynwat & Frank fitted experimental internal conversion measurements for various carotenoids species, and found a high fidelity exponential trend (linear in the case of ln (W )), with a negative gradient of 0.755 cm -1 s -1 . Further, this trend was able to predict the rates of numerous other carotenoids. This was further experimentally validated by Maciejewski and co-workers . Further investigation by Andersson and coworkers , who examined the rate constants for various carotenes, found that multiple terms in Equation 30 could be considered as constants. In addition to the pre-exponential terms, E λ was found to depend weakly on the length of carotene chain. Indeed, the rate of internal conversion was found to increase by a factor of ∼300 when the chain length was increased from five to eleven, implying some degree of invariance with respect to the former exponential term. Further it was highlighted that coupling for the ployacene series that the coupling was similar, and therefore insentive to the size of the chromophore . In the limit of the natural logarithm, it is the former exponential term that is dominant. Following, a simplified form of Equation 30 can be cast : |
67b7c567fa469535b9379533 | 51 | where C p is the pre-exponential factor. Using Equation 31, Andersson and co-workers[199] modelled the S 1 → S 0 internal conversion pathway for a series of carotenes. As a function of E mn , they reported negative exponential gradients of 1.315 cm -1 s -1 and 0.940 cm -1 s -1 solvated in nhexane toluene and 3-methylpentane, respectively. For larger carotenoids, Ehlers and co-workers found a gradient of 0.893×10 -3 cm -1 s -1 . Bachilo and co-workers instead studied internal conversion using the simplified energy gap law method using diphenylpolyenes, and reported rates between 0.003-2.040×10 8 s -1 with respect to the length of conjugation along the chromophore. Interestingly, the shortest species they studied was found to not fit into the exponential pattern typically expected, being several orders of magnitude faster compared to the next few in the series. Frank and co-workers highlighted this exponential relationship for a series of carotenoids decreasing from ∼9 × 10 11 s -1 with respect to the energy gap. |
67b7c567fa469535b9379533 | 52 | Using this model, there are some significant limitations worth noting. For one, only undistorted oscillators are considered; in other words, normal modes without near-identical frequencies are omitted, thereby ignoring any distortion effects. This was probed by Islampour and co-workers , who found that the difference between distorted and undistorted oscillators could be more than several orders of magnitude when it comes to E mn . Orlandi and co-workers noted behaviour similar for the polyacenes, while Andersson and co-workers highlighted that distortion could influence the interpretation of their results. Secondly, the influence of temperature is different when omitting undistorted oscillators. Typically it would be expected that internal conversion approach zero as temperature drops ; that is not the case here. In fact, Andersson and coworkers noted that the observed temperature dependence was minimal with respect to E mn . Minimal temperature effects due to the weak coupling limit agrees with the original tenets of the energy gap law , however it is worth recalling that it is applicable for low temperatures (when the oscillator energy is much larger than the Boltzmann temperature). Thirdly, anharmonic effects are omitted, which are expected to be significant for certain systems. However, in the low-energy scheme, a harmonic model remains valid. In the case of Andersson and co-workers , carotenes can easily be considered within the high-energy regime. Therefore, the internal conversion rate constants for the S 1 → S 0 transition can be expected to shift within an anharmonic model. However, in the case of a derivative series, if the effect is roughly linear, Equation 31 holds. Fourthly, in the case of carotenoids , shifts in the conjugation length of the system, the degree of methyl-substitutions, or the number of -ionone substituents can result in an asymmetry of global constants. In other words, while chemically similar, enough of a difference can be observed in the pre-exponential term leading to misleading results . |
67b7c567fa469535b9379533 | 53 | Internal conversion calculated for nucleobases yield erroneous results using the energy gap law ; it cannot explain the experimentally observed subpicosecond internal conversion due to the minimal correlation between the energy gap of the bases and fluorescence lifetimes. Cadet & Vigny report that the adiabatic S 1 energies for each nucleoside are as Adenosine > Uridine > Thymidine > Cytidine > Guanosine. Interestingly, nucleosides Uridine and Adenosine, with the shortest S 1 fluorescence lifetime, also yield the largest S 1 → S 0 energy gaps. |
67b7c567fa469535b9379533 | 54 | Semi-empirical in nature, Marcus theory has also been used to calculate the rate internal conversion, which suggests the vibronic transition scale to be significantly smaller than other vibronic energy scales. Building on the Arrhenius expression (Equation ), the activation energy is replaced with the Gibbs free energies of reaction, and the pre-exponential factor is formally defined similar to Fermi's Golden Rule. In addition, if it is assumed that the nuclear modes are harmonic and can be treated classically, this density of states can be examined analytically . In its full form, it is cast as : |
67b7c567fa469535b9379533 | 55 | where E 0 λ is the Gibb's free energy of the system. A number of investigations have reported couplings calculated using the generalised Mulliken-Hush model , while other works employ the fragment methods (excitation or spin) . These methods are typically used to treat charge or energy transfer problems rather than internal conversion; however it has been shown that they can be used to calculate internal conversion under certain conditions ; specifically, when the electron-hole overlap is small and the exciton radius is large. |
67b7c567fa469535b9379533 | 56 | Marcus theory is typically used for donor-acceptor systems, when the molecular interface is important to the photophysics, or when charge transfer states are of import. Bai and co-workers depending on the molecular configuration. Of the three studied conformations (face-on, edge-on, and end-on), the fastest rate constant was for the face-on geometry, with a rate of 5.71×10 6 s -1 . Slowest was for the edge-on geometry with a rate of 1.94×10 -1 s -1 , with end-on being in the middle (1.10×10 3 s -1 ). This sensitivity is likely a symptom of the strong dispersion between the two molecular fragments, with energy levels shifting as a function of the orientation. However, based on the orbital configuration for each of the geometries (Figure ), it is apparent that the electron density has a stronger degree of overlap for a face-on configuration compared to an edge-on geometry. There may also be some interchromophore coupling between the two moieties increasing the vibronic coupling of the dimer system, based on the curving geometry of the planar fragment. |
67b7c567fa469535b9379533 | 57 | Chen and co-workers examine internal conversion from the lowest lying charge transfer state to the ground state at a Tris(2,2'-bipyridine)ruthenium(II) 314.5 cm -1 2.769 5.75 × 10 6 s -1 1.17 × 10 7 s -1 pentacene to C 60 interface, and found multiple dependencies on the number of pentacenes surrounding the C 60 fragment, as well as their configuration (Figure ). In a herringbone configuration, rates would vary between 10 9 -10 11 s -1 for edge-on configurations, while the spread was much less for face-on configurations, between 10 10 -10 11 s -1 . Conversely, for co-facial configurations, face-on geometries peak at speeds of ∼10 10 s -1 for three pentacene chromophores, while edge-on geometries appear to continually increase in speed as the number of pentacenes increase, from 10 9 -10 11 s -1 . This behaviour was attributed to the interplay between the electronic energy gap and vibronic coupling term caused by hole delocalisation and consequential migration from the interface. Bhattacharya and co-workers examined donor-acceptor-donor Pechmann dye analogues, and found interfacial internal conversion rate constants ranging between 10 -4 -10 11 s -1 , depending on the employed substitution, highlighting the extreme sensitivity and control that can be elicited through the right kind of engineering. Zhao and co-workers performed a similar examination on a novel dithienyl-diketopyrrolopyrrole-selenophene donor and -phenyl-C 61 -butyric acid methyl ester acceptor system, and found an rate constant of 3.767×10 |
67b7c567fa469535b9379533 | 58 | heterjunctions, and reported a series of internal conversion rate constants <<1 s -1 . They go on to say that the resulting rate constants are small entirely due to the large shift in the Gibbs free energy, and that while extraordinarily small these rates are confirmed by weak bi-molecular and trap-assisted recombination experiments. However, as the same method yielded qualitatively appreciable results for a similar system it is more likely that the generalised Mulliken-Hush model was not applicable to the this particular system. In addition, non-radiative rate constants that are small in the Franck-Condon limit are more likely to display either increased anharmonic behaviour or activity in the Herzberg-Teller regime ; analysis of contributions to these regimes may be required. |
67b7c567fa469535b9379533 | 59 | It is worth noting that Marcus' model has been known to fail for systems that fall within the so called "inverted region" ; as E 0 λ becomes more negative the rate slows down, opposing the exergonicity. It is also worth highlighting that the ionisation potential and electron affinity can be approximated as the highest occupied and lowest unoccupied molecular orbitals, respectively ; whereby the generalised Mulliken-Hush model as the coupling is not valid for many systems. A good example of this was shown by Biswas and co-workers where the rate of internal conversion was less than <1 s -1 for a series of fused heteroacenes. |
67b7c567fa469535b9379533 | 60 | Marcus theory (Equation ) is valid when any nuclear changes occurring only involve modes with smaller energies than the molecular thermal energy within the classical limit . As such, there are plenty of cases where the use of Marcus theory yield results that are sceptical at best. Zhang and co-workers examined PM6 in a number of donor-acceptor configurations and differing geometries, but universally found internal conversion rate constants of < 1 s -1 . They argues that this behaviour corresponds to a bi-molecular decay at the heterojunction interface, however even considering this as a possibility, the result is still too impossibly small. It is much more likely that this system is not viable for modelling through Marcus theory; internal conversion may be dominated by Herzberg-Teller contributions, as there is no consideration for vibrational effects. |
67b7c567fa469535b9379533 | 61 | The Marcus-Jortner-Levich model builds on classical Marcus theory, while considering the semiclassical form for the rate constant. It is able to incorporate quantum tunneling effects, and overcomes complications concerning the "inverted region" in Marcus theory by factoring in vibronic coupling between states within the harmonic approximation, and separates E λ into an inner sphere E λ,V and outer sphere E λ,S . In terms of a full active space treatment, whereby all modes are considered, can be cast as : |
67b7c567fa469535b9379533 | 62 | However, the full active space method is computationally expensive due to the exponentially scaling nature of the problem, and is therefore infeasible for most systems. A more general method would be to use a single effective quantum mode ω eff ; an amalgamation of all participating normal modes, their energies, and their quantum effects. This one-effective mode rendition has been shown to be a successful approximation with respect to experiment , and can be cast as ] |
67b7c567fa469535b9379533 | 63 | Lui & Troisi used the effective-mode Marcus-Jortner-Levich model (Equation ) to study internal conversion in a poly(3-hexylthiophen) to -phenyl-C 61 -butyric acid methyl ester interface and found a rate of 1.93×10 9 s -1 by setting E λ to 0.11 eV. Moreover, they studied the relationship between the rate constant and E λ , and reported a gradient of 6.76 × 10 5 eV -1 . Cerdá and co-workers examine a similar problem for a truxene-tetrathiafulvalene to hemifullerene (C 30 H 12 ) interface, and found internal conversion rate constants from the three lowest charge transfer states to the ground state of 2.6×10 9 s -1 , 1.7×10 5 s -1 , and 2.3×10 4 s -1 , respectively. Bozzi & Rocha used the one-effective mode form of the Marcus-Levich-Jortner model to calculate rate constants for a small number of chromophores, and compared it to classical Marcus theory. Their results (Table ) agree well with experiment for both the one-effective mode and classical Marcus model, however the classical Marcus model is observed to deviate further from experiment than the one-mode Marcus-Levich-Jortner model. In general, classical Marcus theory was found to yield faster rate constants than compared to the effective-mode Marcus-Levich-Jortner model. |
67b7c567fa469535b9379533 | 64 | Here, I (E mn ) and I 0 (E mn ) are the densities of states for the transition state and reactant molecule, respectively. Application of this method to formaldehyde by Miller noted rates on the order of 10 6 s -1 . As well, they noted the importance of tunnelling effects only below the nanosecond scale, otherwise their contributions are minimal. Zhang and co-workers used this method to calculate ring deformation in 2-aminopyridine, with a rate constant of 8.69 × 10 11 s -1 . Interestingly, neither hydrogen transport or ring opening could compete with ring deformation, however the authors noted applicability issues. |
67b7c567fa469535b9379533 | 65 | In building on the work of Robinson-Bixon-Jortner, Plotnikov suggested three approximations : the vibrational structure is decoupled from medium, the system is assumed to be at zerotemperature condition, such that vibrational energies are smaller than thermal energies , and the initial electronic state of the transition is in the zeroth-vibrational state. Understanding that fast vibrational loss is typically on the order of ∼10 11 s -1 offers important simplifications to the model , since radiative and non-radiative processes normally occur between electronically excited states in vibrational equilibrium. For ultra-fast internal conversion, the latter approximation may not be valid, since internal conversion will be competing directly with vibrational relaxation , and may therefore be multiple orders of magnitude faster than fluorescence, like in low pressure vapours . From these rationalisations, the rate of internal conversion can be expressed as : |
67b7c567fa469535b9379533 | 66 | where Λ s is the energy resonance defect between the initial and final states, and Γ ns is the relaxation width of the vibrational state n. The resonance energy defect is smaller than or equal to ∼100 cm -1 when both initial and final electronic states are in their zeroth-vibrational state . Green's functions were initially used to cast the harmonic and anharmonic vibrational terms since at the time alternative methods to model the aromatic C-H stretching modes were not available. Green's functions are applicable when coupling is comparably smaller than either Γ ns or Λ {n} . Lin noted that most of the electronic energy in internal conversion is transferred to the vibrational continuum rather than straight to phonon-based relaxation. Physically, this represents the case where non-radiative transitions originate from a Boltzmann distribution of vibrational levels. Another way to interpret this is that the electronic energy from the initial state is transferred via vibronic quanta to the final state . |
67b7c567fa469535b9379533 | 67 | This way, normal modes can be categorised into one of two paradigms : either a mode with some non-zero coupling between some initial zeroth-vibrational ground state and some final n th -vibrational excited state, or a mode displays zero coupling, and acts as an energy reservoir and inhibits quantumnumber reversibility (opposite-direction transition via identical vibrational quantum numbers). Importantly, the former participates in the transition and the latter does not. The coupling between initial and final electronic states, when the initial state is fixed at the zeroth-vibrational level ⟨ψ ns |H| ψ m0 ⟩ can be found using the one-electron nonadiabaticity operator , cast using internal coordinates Qj within the Franck-Condon approximation as : |
67b7c567fa469535b9379533 | 68 | Franck-Condon Factors Each component of the overall vibronic coupling can be terms in terms of either an electronic-nuclear or electronic-vibrational coupling terms. The product overlap term is known as the Franck-Condon factors , and effectively dictates the degree of coupling a given mode has with respect to its degree of overlap with all other modes. As far as determining which of these modes are participating and which are preventing reversibility, it needs to be performed for each physical case, and is therefore phenomenological. However, this can be obtained from experimental data . The electronic-nuclear component to Equation 41 is cast in the same from as Equation , while the electronic-vibrational component can be cast using harmonic oscillator functions, shown as : |
67b7c567fa469535b9379533 | 69 | A significant truncation can be assumed here for the sake of computational brevity. If the symmetry point group goes unchanged, changes to the atomic configuration due to normal mode transitions are non-zero only for symmetric modes . Lin further confirms that only transition participating modes are relevant . In other words, since the Franck-Condon factors decrease as the occupation increase, selection rules can then be applied. This means that the (0,1) transition can be assumed as the only non-zero term in the transition integral. This simplification of course fails for many systems, and should be used carefully. These equations can also be cast in terms of a morse potential rather than a harmonic one, which allows for the treatment of anharmonic effects. In particular, it is derived to tackle -H-type anharmonicity, which has been shown to be very important for high/mid-high energy oscillators . That formalism can be found in Refs. 41, 42, 75, and 43. |
67b7c567fa469535b9379533 | 70 | Plotnikov noted that for transitions with energy gaps larger than 0.25 eV, the vibrational continuum is dense, such that at least one energy resonance defect term is zero, dominating the overall contribution to the final internal conversion rate constant. Further, contributions due to anharmonic high frequency modes can be observed as small despite the vibrational quantum number ; as per Equation , a small occupation will result in a small contributing factor, rapidly decreasing as the quantum number increases. Therefore, the rate constant can be recast as : |
67b7c567fa469535b9379533 | 71 | q |c q,m c q,n | is on the same order as the ratio between the number of C-H bonds in the system to the square of the number of electrons in the system (≈ 1 7 for benzene); therefore decreasing as system size increases. Equation 44 is a simplified representation of Equation , however here only C-H transition participating modes are considered , drastically simplifying the problem. Konyshev and co-workers applied this simplified model to tetraoxa circulene at three different levels of theory, and found an internal conversion rate on the order of 10 6 s -1 . Valiev and co-workers examine tetraphenylporphyrin coupled to both ethylene-diaminetetraacetic acid and diethylenetriamine-pentaacetic acid with internal conversion rate constants of 6.7 × 10 7 s -1 and 8.7 × 10 7 s -1 , respectively, while later calculating rates on the order of 10 6 s -1 for numerous chromophores. |
67b7c567fa469535b9379533 | 72 | From Equation , a relation can be drawn between the energy gap and the rate constant (Figure ). It can be seen that two regimes can be defined. Firstly, Kasha's regime, where Kasha's rule is obeyed, and a range in which the Ermolaev-Sveshnikova rule (large energy gaps yield weak internal conversion) is obeyed. It should be noted that there is also an intermediary regime, where internal conversion directly competes with fluorescence. |
67b7c567fa469535b9379533 | 73 | Artyukov and co-workers further simplified Equation 44 by reducing it to a collection of three terms: a vibrational factor Q rs , the energy gap, and the number of C-H TABLE III. Tabulated rate constants calculated for a series of three chromophores, correlating to a rate in units of s -1 . Rates were calculated using Equation bonds, given simply as : |
67b7c567fa469535b9379533 | 74 | Here, Q mn can be expressed as a function of the relevant molecular orbital expansion coefficients tied to each C-H-type normal mode, yielding accurate rates to within 5% of experiment . This solidified the dependence of the Plotnivok-Robinson-Bixon-Jortner model on the accuracy of the structure of the π-electronic system of interest . One can also cast Q mn as : |
67b7c567fa469535b9379533 | 75 | The resulting work of Artyukov and co-workers , using a semi-empirical partial neglect of differential overlapping method , found that S 1 internal conversion was comparable with intersystem crossing across three acridine chromophores. The calculated interexcited state rate constants (Table ) are several orders of magnitude faster compared to the excitedground transition, however as per Kasha's exciton model , this is common behaviour. Similarly fast rates for the laurdan fluorescent probe chromophore were shown by Zharkova and coworkers . Their earlier work on bi-polyacene systems suggested energy transfer between a donor naphthalene fragment to an acceptor anthracene fragment as equivalent to internal conversion; with very fast internal conversion pathways (∼10 8 -10 13 s -1 ). Valiev and co-workers examined the case of internal conversion in free-base porphyrin, and found a rate of 1.12×10 8 s -1 . They also examined tetrabenzoporphyrin and tetraphenylporphyrin, and found corresponding rates of 8.3 × 10 7 s -1 and 1.91 × 10 8 s -1 , respectively. Artyukhov and co-workers performed a comprehensive study of internal conversion in the polyacenes for the three lowest singlet excited states, and yielded rate constants in reasonable agreement with experimental observations. Baryshnikov and co-workers calculated internal conversion for a myriad of different compounds also correlating well with experiment. Samsonova and co-workers performed an extended analysis on some acridine compounds, and found that the S 1 → S 0 was generally slower than fluorescence, with ultra-fast interexcited state transitions at least 10 10 s -1 . |
67b7c567fa469535b9379533 | 76 | Valiev and co-workers noted that the density of states can be fitted and approximated as per Figure . Here, the criteria for the strongest transition participating mode is probed, and estimated within one of three regimes as a function of E mn . Either the strongest mode has an energy larger than 1000 cm -1 and a Huang-Rhys factor between 0.1 and 0.5, has an energy smaller than 1000 cm -1 and a Huang-Rhys factor greater than 0.1 and 0.5, or has a Huang-Rhys factor larger than 0.5. The former regime was used by Plotnikov & Dolgikh . |
67b7c567fa469535b9379533 | 77 | Applied to a 3 (MeO) 2 cluster by Valiev and co-workers found rate constants between 10 12 -10 14 s -1 for various S m → S 0 transitions. Merzlikin and co-workers calculate internal conversion for azatrioxa circulene and heptamethine cyanine, and found rates of 1 × 10 7 s -1 and 1 × 10 respectively. Interestingly, these authors also looked at mixing the electric dipole operator with the nonadiabatic operator, and found that while the contribution was weak (<1 s -1 ), a strong electric field could strong perturb the rate of internal conversion, though this phenomenon has been known for some time . Baryshnikov and co-workers studied five porphyrin derivatives, and for all but one calculated rate constants comparing well with experiment. |
67b7c567fa469535b9379533 | 78 | Shi and co-workers took the Plotnikov-Robinson-Jortner model, and looked to develop a model to allow for high thoroughput virtual screening for material candidates. For such a purpose, the focus would be on speed rather than accuracy, with a focus only on capturing trends . To allow for this, they used a simplified form of Equation by replacing the vibronic coupling with an effective coupling of a linear combination of all non-adiabatic couplings for all non-zero normal modes, cast as : |
67b7c567fa469535b9379533 | 79 | Shi and co-workers concluded that the use of this method was effective, in that the experimental trends were well reproduced (Figure ). Here, each calculated rate constant is underestimated by at least one order of magnitude, and in many cases the disagreement is by several orders of magnitude. More specifically, the closest agreement observed is for deuterated azulene, where the calculated rate is 1.66% of the experimental rate. They go on to discuss that the quantitative discrepancies between theory and experiment are ultimately due to the limitation of the harmonic approximation, however in the context of virtual screening, the approximation provides a robust means. Martynov and co-workers applied this method in a much more rigorous manner on the polyacenes, calculating Γ ns using 3 separate methods (Pekarian, Hybrid, Gaussian). While the Gaussian function yielded a null rate FIG. . Tabulated calculated and experimental rate constants for the S1 → S0 transition calculated for a series of twelve chromophores. Rates were calculated using Equation . Repurposed from Ref. for numerous transitions, the Hybrid linewidth underestimated the rate. The Pekarian model yielded the most positive results: 1.7×10 -4 s -1 , 1.5×10 3 s -1 , 7.0×10 3 s -1 , and 1.3×10 6 s -1 for naphthalene, anthracene, tetracene, and pentacene S 1 → S 0 transitions, respectively. Some of the higher-order internal conversion pathways were also mapped, but these are difficult to correlate with experiment. |
67b7c567fa469535b9379533 | 80 | The approach of Artyukhov & Galeeva based on the work of Plotnikov, while important to the state of the art, is severely limited. Applicable only to biological systems with light atoms (H, C, N , O, F , S, Cl), the Plotnikov-Robinson-Jortner model cannot be used for organometallic systems for example. Further, use of the model has found issues with treatment of complex chromophores like porphyrins and circulenes , since the partial neglect of differential overlapping method is lacking in its ability to model excitation energies. In looking for ways to scaffold this issue, Valiev and co-workers first examined the photophysics of a series of complex chromophores, and highlighted the difference between rates calculated using density functional theory and extended multi-configuration quasi-degenerate second-order perturbation theory , compared to second-order coupledcluster calculations when the vertical excitation energies were calculated using first principles or semi-empirical methods. Similar work was performed by Baryshnikov and co-workers in the same vein. Following this work, propulsion towards hybridisation with modern quantum chemical techniques was undertaken to incorperate first principle non-adiabatic coupling matrix elements, then incorporating more technical effects like Herzberg-Teller contributions , and anharmonic effects , entirely removing the need for semi-empiricism. |
67b7c567fa469535b9379533 | 81 | The Valiev-Plotnikov-Robinson-Bixon-Jortner model builds directly on the previous formalism, where the rate constant is expressed as Equation 39 and the coupling as Equation . As previously, the transition is between some initial electronic state in a zeroth-vibrational state to a final electronic state in an s th -vibrational state. However, now the theory diverges; if we assume that the relaxation width Γ ns depends only weakly on the vibrational quantum number, and that it is also much larger than the resonance energy defect, similarly noted by Plonikov , Equation 39 can be recast and simplified to : |
67b7c567fa469535b9379533 | 82 | Occupational Coupling Equation 49 has two major components: the energy condition, and the occupational coupling. The occupational coupling is similar to the previous vibronic matrix coupling element, expressed using the non-adiabaticity operator (Equation ), in which one term can be omitted. However, now instead of remaining within an internal coordinate framework, we can instead incorporate density functional theory, and with it the Cartesian coordinate system. This is similar to Equation , except we now include an internal conversion vibrational matrix B qj , which consists of the mass weighted normal modes of all transition participating modes. We can recast this as : |
67b7c567fa469535b9379533 | 83 | Here, the electronic-vibrational component is calculated the same as previously (Equations 42 and 43), however the electronic-nuclear component is calculated using modern quantum chemical techniques . Specific examples can be found within density functional theory, such as the method developed by Chernyak & Mukamel building on the work of Send & Furche . Outside of density functional theory, the couplings could be computed directly using a state-averaged gradient within a multi-configurational based method . |
67b7c567fa469535b9379533 | 84 | It is also worth noting that when using modern quantum chemistry packages, the origin shift for each normal mode ∇ j FIG. . Visual representation of vibrational occupational quanta resulting in internal conversion between two electronic states. Here, for an S1 → S0 transition, a configuration of 6 normal modes with some arbitrary occupational quanta define the transition. can be calculated directly . Therefore, it is possible to recast Equation as : |
67b7c567fa469535b9379533 | 85 | The energy condition defines the vibrational occupational quanta as coinciding with the transition of interest. In considering the physical ramifications of this approximation, the mechanism of internal conversion is built of weighted occupations through the various transition participating normal modes between the two electronic states (Figure ) ie. a bosonic configuration problem. This can easily be estimated using Lagrange's method . Using a Lagrange function of the form : |
67b7c567fa469535b9379533 | 86 | Valiev and co-workers found that for most molecules a value of 1.6 × 10 14 s -1 for S 1 → S 0 transitions is appropriate. The results calculated using Equation (Table ) infer rate constants are within a reasonable degrees of error with respect to experimental values, for almost all case study chromophores. Free-base porphyrin was shown to show strong disagreement when compared to experiment. Since the first singlet excited state of this chromophore is shown to be optically dark , calculations within the Franck-Condon regime are not adequate. A similar disagreement is observed for some of the polyacenes, since the S 1 states in many of the smaller variants are not pure, but mixed states . Earlier work by Baryshnikov and co-workers agreed with the results on the circulene poor internal conversion rate constants. Valiev & Kurten applied this model to photocatalysis models to probe hydrogen shifts of peroxyl radicals, and found internal conversion to dominate the reaction cross-section. |
67b7c567fa469535b9379533 | 87 | where H and L are the configuration interaction coefficients with indicies corresponding to virtual and occupied molecular orbitals, and K mn is a vibrational factor associated with the product of the Franck-Condon factors and the matrix elements of the electronic-nuclear components and the vibrational wavefunctions of initial and final states. The electronic-nuclear matrix element can be expanded as a linear combination between basis functions of the molecular orbital expansion, allowing for truncation of important matrix elements as opposed to for the entire system. Their previous work showed that for the lower excited states transitions are typically mediated by -H-type normal modes, and is therefore reminiscent of the approximation previously made by Plotnikov (Equation ). Using Equation , Valiev and co-workers yielded rate constants in good agreement compared to using a more complete configuration space and incorporating additional effects (Tables and). The S 2 → S 1 internal conversion rate constant was found to be much smaller compared to a more complete calculations, which they rationalised to be evidence of anti-kasha fluorescence. It was concludes that by using this simplified method for estimation of the overall non-adiabatic coupling that difficulties should be expected when treating FIG. . Visualisation of the linearly expanded configuration space methodology developed in Ref 44. For some arbitrary system, computation of a high dimensional phase space, shown here as a Calabi-Yau surface, can be shifted from an exponential to a polynomial problem by searching for configuration vectors where the sum of all elements of that vector equals n. The deep green segment signifies the section of the CS for which n solves a Lagrangian multiplier, while the light green segments illustrate the rates for all other values of n ± x. In principle, integration of the curve equals the rate constant for the system. Calabi-Yau surface is adapted from Ref. deuterated systems, or of course systems with minimal hydrogen bonds, since the full configuration space is simplified to assume all transition participating modes are found only in hydrogen bond-type normal modes. |
67b7c567fa469535b9379533 | 88 | Using a series of tris(2,4,6-trichlorophenyl)methyl derivatives, they found rates between 10 7 -10 8 s -1 with respect to the derivative flavour. Here, the electronic-nuclear matrix elements were calculated using density functional theory , while the defect energy was reported to be not greater than 100 cm -1 . When accounting for Herzberg-Teller contributions to internal conversion, Valiev and co-workers expanded the vibronic matrix element along the transition participating normal modes, using the first and second order terms. This method is very similar to that proposed by Lin , who expanded the Hamiltonian as a Taylor series in terms of the normal coordinates up to the second order term : |
67b7c567fa469535b9379533 | 89 | where H o and H so are the kinetic/potential potential energies for the system electrons and the spin-orbit coupling operator, respectively; and the superscripts refer to qualities that are evaluated at the equilibrium nuclear configuration. For cases where the spin-orbit coupling matrix elements are much smaller than the non-adiabatic coupling matrix elements, such as for internal conversion, spin-orbit coupling may be safely ignored. Interestingly, this infers that there are cases where spin-orbit coupling may be required to correct mis-estimated vibronic couplings. Valiev and co-workers do not consider the spinorbit coupling contribution, and cast the Herzberg-Teller-type internal conversion matrix element V HT m0,n{s1,s2,s3,...,s 3N -6 } |
67b7c567fa469535b9379533 | 90 | Incorporation of these methods into internal conversion rate calculations by Valiev and co-workers shows an increase by several orders of magnitude for many of the probed systems (Table ). In particular, Herzberg-Teller contributions yield significant improvement to unstable systems like the freebase porphyrin derivatives. However, it was reported that the inclusion of anharmonic effects yield the greatest improvement. |
67b7c567fa469535b9379533 | 91 | The ShaMan model model builds on the Valiev-Plotnikov-Robinson-Bixon-Jortner model, but reevaluates the energy condition in terms of a fully sampled bosonic configuration problem. While Equation is one such method to solve this energy condition, it presents only one single solution of a possible continuum of solutions, and is therefore unrealistic if taken alone. Why is this the case? Quantum mechanics is a statistical representation of the world, in the simplest of terms, meaning that from such a single configuration, there should be expected some degree of oscillation through some wide mathematical space of configurations, or configuration space. Vibrational energies fluctuate as the temperature of a system increases, and atoms themselves vibrate, resulting in a small range of energies to solve for, rather than a fixed value. It is plausible to infer that this average range of energies δE mn is directly related to the ensemble thermal energy of a system, cast as : |
67b7c567fa469535b9379533 | 92 | This results in a small continuum of configurations that satisfy the energy condition. Scanning the configuration space for all possible solutions to the energy condition is analogous to the 0-1 knapsack problem in mathematics , and is therefore an NP-complete problem, meaning that no solution exists that is both exact and computationally reasonable. However, Shaw and co-workers propose three possible treatments: a brute force scan, stochastic sampling, and expand the configurations linearly. Importantly, linear expansion allows for compartmentalisation of each configuration based on its respective magnitude (index), and shifts the scaling from exponential to polynomial. This redefines the full configuration space from some N-dimensional surface to a two dimensional skewed-Gaussian (Figure ). A Lagrangian provides the maximum index, corresponding to the Gaussian peak, and provides the dominant configurations to the rate constant. Without the expansion, much of the configuration space is left out, and in many cases the rate of internal conversion can often be underestimated significantly . A good analogy for this is approximating the integral of a Gaussian-like structure as a delta function; the approximation holds if the set of configurations that satisfy the energy condition have indices that are all similar in value, but fails in its description if the indices vary. |
67b7c567fa469535b9379533 | 93 | Manian and co-workers used the ShaMan model, and the continuum energy condition (Equations 49 and 61), focusing on S 1 → S 0 transitions, and found that at the Franck-Condon limit, the rate constants for anthracene, tetracene, and pentacene, were small to negligible. However, contributions due to Herzberg-Teller effects increased the rate constants significantly (Table ). In the case of the polyacenes, second order contributions are observed to dominate internal conversion processes, while for di-keto pyrrolopyrrole and perylene diimide, their contributions are much smaller. Comparing to the Valiev-Plotnikov-Robinson-Bixon-Jortner method, corresponding rate constants were artificially raised due differences between the continuous maximum rather than the discrete maximum. |
67b7c567fa469535b9379533 | 94 | In order to then further investigate the nature of inter-excited state internal conversion, Manian and co-workers examined the thermal accessibility between each transition approximated as a Boltzmann factor. Transitions in which additional energy is required by the system (E m < E n ) as those high energy states are thermally inaccessible by the exciton residing on a lower electronic state. This switching function B m,n is only used in the case of an "uphill" transition, and set to unity in the case of a "downhill" transition, formally cast as : |
67b7c567fa469535b9379533 | 95 | Use of Equation by Manian and co-workers to study the exciton dynamics of di-keto pyrrolopyrrole show internal conversion to be the primary loss mechanism for every electronic state save for the S 1 state (Table ). Detailed examination of these rates highlighted that Franck-Condon the nuclear-electronic couplings between same-spin states was calculated from the residues of the imaginary component of the non-adiabatic matrix element as shown by Send & Furche , this work used the spinflip TDDFT method, as implemented in the Q-chem software package . The difference between the two methods is a factor of ∼1.5, explained by Manian and co-workers as being due to the residue method not accounting for the final state correctly. |
67b7c567fa469535b9379533 | 96 | It should be clearly noted that use of a perturbative method within an harmonic paradigm, like that of the ShaMan model, can lead to displaced rate constants as larger amplitudes between oscillations can lead to larger deviations from the unperturbed Hamiltonian, such as is the case of large energy gaps. While these displacements can be minimised through cohesive coupling between vibrational degrees of freedom and the electronic states , in the case where coupling between states is very large Equation will break down, and only be able to estimate the speed of ultra-fast rate constants . If we recall the conditions in which the Plotnikov-Robinson-Bixon-Jortner model (Equation ) is valid, it becomes apparent that any rate constant faster than ∼ 10 14 s -1 should be interpreted tenuously, and in context to photophysical characteristics of the system's initial and final states. |
67b7c567fa469535b9379533 | 97 | In their initial works, Manian and co-workers noted that for perylene diimide the Herzberg-Teller contribution to internal conversion initially calculated was grossly overestimated, due in no small part to 5 high energy normal modes with nonzero Huang-Rhys factors. High energy modes have larger contributions to the Hessian, and therefore over-contribute to the rate constant. In the case of perylene diimide, two modes with significant Huang-Rhys factors of energy larger than 0.4 eV were found, corresponding to normal modes of the outermost hydrogens. Initially suggesting that this issue could be fixed through application of an alternate quantum chemical method, they argued that modes above a certain threshold energy could be truncated, yielding a rate constant of ∼10 4 s -1 . This hypothesis is bolstered when considering the molecular symmetry of ground state perylene diimide, and the fact they noted 53 transition participating normal modes for perylene diimide. Comparison of this work to work by Shi and co-workers shows the ShaMan model to perform better on the polyacenes. As well, we note that while Valiev and co-workers note the failure of the harmonic approximation in cases such as for the polyacenes, we highlight that Manian and co-workers reported fairly accurate photophysics due to the inclusion of Herzberg-Teller contributions over a Morse potential. |
67b7c567fa469535b9379533 | 98 | Probing of these hypotheses was conducted through a study in the exciton dynamics of perylene diimide and tetracene hetero-and homo-geneous substrates , whereby the rate constants for both chromophores were found to stabilise significantly through the use of a dispersion correction. However, in a follow up work, Manian and co-workers found that the issue may also be due to how different density functionals treat the electronic-nuclear coupling between two electronic excited states, where they tested the ShaMan method on base perylene using 15 different density functionals across two different triple-ζ basis sets. Here, they found that while for internal conversion, rate constants across each density functional/basis set combination would vary, in some cases by several orders of magnitude (Table ). Interestingly, this serves as an additional reason as to the disagreement previously observed but not discussed by Valiev and co-workers . found a similar pattern of behaviour in their calculated rate constants, concluding that the calculated vibronic coupling matrix elements nonadiabatic coupling elements were subject to this affect, among other factors. Further probing found the secondorder corrected rate constant to heavily depend on the method used to obtain the electronic Hessian. From these results, 5 density functional/basis set combinations were found to yield an appreciable rate constant for the S 1 → S 0 transition when compared directly to experimental results. Interestingly, Veys and co-workers noted a similar disagreement with respect to the choice of density functional, however their calculations highlighted a relatively consistent disagreement whereas Manian and co-workers highlighted multiple orders of magnitude differences. |
67b7c567fa469535b9379533 | 99 | A followup work on the subject matter[325] examined a more complete exciton dynamics picture by also including the S 2 state, and found that in general the internal conversion between excited states was overestimated by several orders of magnitude in all cases. However, a fitting factor could be utilised to damp the overestimated vibronic coupling, yielding a single density functional/basis set combination that was of a reasonable value. |
67b7c567fa469535b9379533 | 100 | A number of issues were observed with the ShaMan model. Firstly, comparison to the literature is difficult; Manian and co-worker's study on perylene's incorrectly mapped excited state manifolds . Secondly, it is clear from these results that density functional theory is drastically overestimating the vibronic coupling between two excited states. Density functional theory is formally exact with respect to the density functional approximation used ; therefore, any failures are part of the functional rather than the method itself . Consequentially, density functional theory may not be a viable method to calculate vibronic coupling between two excited states until either a new functional is design to cater specifically to this problem, or until there is an exact solution for the density functional approximation; however, the latter is unlikely to occur in the foreseeable future . Conversely, one could use multi-reference methods to calculate the couplings, however not only is this exceedingly resource intensive, rates were still shown to fall short likely due to the lack of solvent effects. |
67b7c567fa469535b9379533 | 101 | do Casal and co-workers note that this benchmarking was performed primarily at the density functional theory level, and for only one system, and infer how a method independent from the electronic structure theory may yield more positive results. However, outside of very recent non-adiabatic dynamics studies , there are few and far between as fas as alternative methods that are accurate and account for solvent effects, as seen in the comparison of electronic-nuclear couplings calculated using density functional theory with a solvent model vs. those calculated using complete active space self consistent field theory . |
67b7c567fa469535b9379533 | 102 | If we were to compare the results of Bozzi & Rocha with those of Valiev and co-workers , one would notice that internal conversion for naphthalene predicted by Valiev and coworkers was significantly smaller than those predicted by Rozzi & Rocha, with the latter's results matching with experiment most closely. Table clearly shows that there are a myriad of photophysical factors contributing to internal conversion in naphthalene specifically, with a Franck-Condon treatment failing spectacularly in its description of naphthalene. Based on the minimum energy conical intersection geometry used by Bozzi & Rocha, it may be that the harsh nuclear shifts fulfil a Huang-Rhys anharmonic occupation that the ShaMan model otherwise could not. However, it is difficult to speculate without an indepth analysis. |
67b7c567fa469535b9379533 | 103 | Treatment of internal conversion in complex systems is nothing new, if not much more difficult from first principles. For many systems, simple approximations can be made to incorporate complex electronic interactions. The most common and prolific is that of the polarisable continuum model, whereby a solvent is simulated using a series of point charges acting as a dielectric cavity. However, there are also many systems which are highly sensitive to their environment, where this approximation breaks down. In this case alternative methods are required, and due to their computational complexity may be semi-empirical or entirely within classical mechanics. While this review is opting to focus on first principle treatments of internal conversion, here we will briefly discuss semi-classical methods to examine excitonic recombination. See Ref. for a comprehensive review. |
67b7c567fa469535b9379533 | 104 | Trajectory surface hopping was first discussed by Bjerre & Nikitin in 1967 as a semiclassical method to treat nonadiabatic dynamics, followed by a formal proposal by Tully & Preston in 1971. While many of the early studies employed simplified transition probabilities due to computational constraints (Landau-Zener approach) , numerous strides were made, before on-the-fly and generalised approaches were made available in many standard commercial software packages. Similar implementations by Xu and co-workers noted femtosecond deactivation rates for azomethane, while Cui & Thiel used a generalised surface hopping model to highlight ultra-fast internal conversion in acrolein on the femtosecond timescale. |
67b7c567fa469535b9379533 | 105 | Mirón & Lebrero incorporated Plotnikov-Robinson-Jortner theory into quantum mechanical/molecular mechanical simulations to study indole in different solvents. They realised that due to the inseparability between the nuclear and electronic degrees of freedom, the effect of the solvent on the electronic component could be considered constant across a bulk with respect to different environments. As bulk simulations are in some way cast in terms of classical mechanics, Equation 39 could be instead cast as : |
67b7c567fa469535b9379533 | 106 | where the nonadiabatic coupling vector V m,s takes into account the electronic degrees of freedom, while nuclear quantum effects are taken into account via the experimental fitting parameters α ex and β ex . V m,s was calculated here based on an ab initio nonadiabatic model by Tapavicza and co-workers which casts the potential energy surface relevant to the transition in terms of the ground state. They also note that V m,s can be calculated using trajectory surface hopping , and directly substituted as the coupling term. Their resulting simulations found equal internal conversion rates of 7.55 × 10 7 s -1 for both L a and L b states. More rigorous methods are of course also available, such as factorised general coupled-trajectory model . However, despite this methods available detail, it remains highly laborious and computationally expensive. |
67b7c567fa469535b9379533 | 107 | Huang & Rhys were among the first to discuss internal conversion in the solid state, studying F-centres. Within the Born-Oppenheimer approximation, a concise description of both the F-centre and the atomic lattice shows that transitions can occur between degenerate states . Importantly, Fermi's Golden rule is valid as long as wavefunction orthogonality is maintained. From here, an expression for the transition probability of internal conversion in solid state materials can be cast as : |
67b7c567fa469535b9379533 | 108 | where ω l is the average frequency, V is the unit cell volume, x l is the average quantum vibrational number, ϵ s is the static dielectric constant, ϵ ∞ is the high frequency dielectric constant, and R p is an overlap integral. This assumes the excited electron to be uniformly distributed across the entire F-centre. Huang & Rhys noted that when plotting the ratio of internal conversion to fluorescence (Figure ), the dominance of internal conversion is exponentially related to temperature except for higher temperatures, while at temperatures of ∼100 K internal conversion becomes the submissive mechanism by at least a factor of a thousand. In term of internal conversion in modern semiconductor physics, much of the early behaviour was described using the Shockley-Read-Hall model , in which internal conversion is reinterpreted as a trapping mechanism. Generally, one can use the conventional Fermi's Golden Rule formalism (Equation ) to describe internal conversion in the solid state. However, the direct formalism to calculate the coupling term can be adapted from the Shockley-Read-Hall model. One such method is the adiabatic approximation , whereby the initial and final states are described entirely by the Born-Oppenheimer approximation. Lin & Wang calculated the rate for gallium nitride semiconductor blocks, but found their results compared poorly to experiment. The coupling can also be considered in terms of a singular collective of phonon modes alongside the transition degrees of freedom. This one-dimensional quantum formula represents the atomic degrees of freedom through a generalised configuration rather than a specific one. |
67b7c567fa469535b9379533 | 109 | Marcus theory (Equation ) is another approach, however it is more tenuous to employ in the solid state as initial and final electronic states are localised, while in practice it is more likely one is delocalised. While this did not deter some , their results were less than stellar. Marcus theory can also be extended to phonon degrees of freedom as well as the electronelectron coupling. This can also be called quantum charge transfer theory , and while it does have a significantly different temperature relation compared to classical Marcus theory, phonon quantum tunnelling effects can be included in the physical description of the system. Marcus theory can be described by a Landau-Zener transition, but the Born-Oppenheimer approximation can break down for some systems, similar to how it fails at conical intersections in molecular systems; the initial and final potential energy surfaces are connected under the diabatic approximation, ie. conical intersection. To circumvent this issue, the wavefunction can be described as two uncrossed states. This allows a new coupling constant to be defined; known as static coupling theory . Alkauskas and coworkers studied gallium nitride semiconductors using the Shockley-Read-Hall model, but found underestimated results compared to experiment, arguing that static coupling would have worked better. |
67b7c567fa469535b9379533 | 110 | Following this are numerous examples in the state of the art of researchers using techniques typically used for smaller molecular systems to examine solid state systems. Li and co-workers use Fermi's Golden Rule (Equation ) to examine diphenyldibenzofulvene, whereby they diabatisise their electronic states to yield 3.816×10 10 s -1 for S 1 recombination, and 3.429 × 10 10 s -1 for S 2 recombination. As well, a number of other theories, such as the energy gap law and the ShaMan model, are applicable to condensed matter systems, but have not been attempted in the current state of the art due to the computational costs. |
67b7c567fa469535b9379533 | 111 | When considering non-adiabatic molecular dynamics simulations in terms of Fermi's Golden Rule (Equation ), Duncan & Prezhdo noted that an approximate relation between the degree of vibronic coupling and the optical band gap could be drawn. Specifically, if the density of states were to be approximated as the the inverse of the energy band between the initially photoexcited electron or hole state and the corresponding bandgap state E optical , respectively, and the perturbative coupling term to be taken as the vibronic coupling V vibronic singularly, then a simplified expression could be used to model the photophysics of a given nanostructure system. This can be cast in a simplified form : |
67b7c567fa469535b9379533 | 112 | in stark contrast to the traditional employment of Fermi's golden rule, where the coupling term would otherwise be the exact electronic coupling use of this method allows for a computationally simple estimation of internal conversion. This model assumes that the low-lying excited state's evolution/projection and nonequilibrium dynamics can be driven by a ground state trajectory, and that the primary contribution to the vibronic coupling is the electronic wavefunction components depend on nuclear coordinates . The latter approximation is made by assuming any off-diagonal matrix elements of the first-order energy gradient with respect to the nuclear coordinates are not able to be calculated analytically. Nam and co-workers employed a non-adiabatic molecular dynamics model for the excited state trajectories, and found rate constants on the order of 10 9 -10 11 (Figure ), with rate constants steadily decreasing as the titanium dioxide structure size increased. Valero and co-workers diverged from the non-adiabatic molecular dynamics, and instead opted for Born-Oppenheimer ab initio molecular dynamics, to allow for larger systems to be examined which could otherwise not be. Using the Turbomole software package[361] to approximate the vibronic coupling terms, internal conversion rate constants were calculated for titanium dioxide, highlighting the differences between the three methods of excited state trajectory propagation. When compared to those calculated by . Internal conversion rate constants for the S1 → S0 transition in titanium dioxide nanostructures of varying size, using the ground state, Born-Oppenheimer ab initio molecular dynamics, and nonadiabatic molecular dynamics trajectory methods, respectively. Rates were obtained from Ref. 356. co-workers , we see that rate constants are faster using the ground state method in all but the n=29 case, while the ab initio method yields slower rates in all cases except for n=78. Duncan & Prezhdo examined how the average density of states impacts internal conversion, and for titanium dioxide, they reported rate constants between 2-5×10 13 s -1 , where the density of states is calculated through integration as a function of the dye-energy distribution. Liu and co-workers examined the case of two silicon nanoclusters at two different temperatures for varying cluster sizes, and found that not only did internal conversion slow down as cluster size increased, but also as temperature increased from 100 K and 300 K, from 10 6 s -1 at its slowest, to 10 10 s -1 at its fastest. We note here that the smaller the size of the nanostructure the greater the separation between its energy levels; therefore if we do not consider or omit surface or trap states from our description of the system, the effect of temperature on internal conversion will be suppressed as a function of the size. This is because the intraband transition will be much larger than the thermal energy. |
67b7c567fa469535b9379533 | 113 | Due to its applicability to weakly coupled donor-acceptor states, Sarkar and co-workers used the simplified Fermi's Golden Rule method (Equation ) to model porphyrin nanoring structures, and found rate constants on the order of 10 10 s -1 for two different structures. Conversely, Shrabanti and coworkers found significantly slower rate constants for hexameric zinc-centered porphyrin nanorings, between 10 6 -10 7 s -1 depending on the molecular species. Nam and coworkers highlighted that the degree of non-radiative loss could be controlled through use of oxygen vacencies in titanium dioxide nanoparticles. Specifically, whether those defects were of singlet or triplet ground state quality. Rate constants for sin-glet ground states were on the order of 10 10 s -1 , compared to between 10 8 -10 9 s -1 depending on the defect for triplet ground states. This is likely in relation to the spin-forbidden nature of triplet-singlet transitions, but proves interesting nonetheless. |
67b7c567fa469535b9379533 | 114 | Within a generating function regime (Equation ), one can calculate the rate of internal conversion in quantum dots. Consider a system within a strong coupling regime: here the sum of the average number of phonons N j is much greater than one ( j N j ≫ 1). If we then apply a Fourier transform to the delta function, we can truncate the expansion with respect to each respective frequency at the second-order term . The rate of internal conversion in quantum dots can then be expressed as : |
67b7c567fa469535b9379533 | 115 | In work by Xie and co-workers , where the coupling was calculated using the generalised Mulliken-Hush model the clusters display sufficiently small electron-hole overlap and large excitonic radii, allowing the generalised Mulliken-Hush model to be applicable. Using a four-point model for their reorganisation energy, rate constants of 2.55×10 10 s -1 , 1.48×10 8 s -1 , 1.61×10 13 s -1 , and 2.00×10 12 s -1 , were found for cadmium selanide clusters 3-D 3h , 4-D 4h , 4-T d , and 3-C 3v , respectively (Figure ). Later work by Xie and co-workers found for larger cadmium selanide quantum dots rates of 4.80×10 11 s -1 , 2.65×10 14 s -1 , 3.63×10 14 s -1 , and 3.66×10 7 s -1 , for compounds A, B, C, and D, respectively. For these compounds, it was determined that the size and conformation of each quantum dot that was the determining factor with respect to the speed of internal conversion. However, in the latter study, the internal conversion is correlated with the local softness ; either exalted by cadmium's electrophilicity, or damped by the selanide's nucleophilicity. Swenson and co-workers note the importance of binding geometries with respect to the ligand. With respect to CdSe nanoclusters, less rigid structures result in larger changes in Huang-Rhys factors, increasing the rate of internal conversion, in some cases by several orders of magnitude (Figure ). Du and co-workers use the formalism to study ZnSe dots; no pattern was clearly noted. Internal conversion was seen to oscillate wildly as cluster size increased. Clusters of size 4, 6, and 13 were reported to have rates on the order of 10 13 s -1 , while those of size 3, 5, and 11 all had rates on the order of 10 8 s -1 . Further in-depth analysis did however find a strong relationship between the cluster size and rate of internal conversion based on the reorganisation energy. |
67b7c567fa469535b9379533 | 116 | In quantum dots, internal conversion can also occur via electronic-to-vibrational energy transfer; as the name suggests, an excited electronic energy may relax by transfer- ring energy to normal modes , with the oscillation modulating the energy as a function of the energy gap . This mechanism falls outside of the Born-Oppenheimer approximation , coupling the adiabatic electronic states through nuclear motion . Similar to molecular internal conversion, excess energy is taken by multiple phonons where necessary , resulting in a shift in the vibrational modes between the initial and final electronic states. Conversely, for small energy gaps, single-phonon mechanisms are preferable, regardless of the Huang-Rhys factors . In the literature, the Huang-Rhys coupling strength is directly related to the size of the quantum dot. Han & Bester for example show that lattice phonon coupling is inversely proportional to the quantum dot size, while Schaller and co-workers reported highly efficient multi-phonon processes due to nonadiabatic electron-phonon overlap in lead selenide quantum dots. The one-effective mode Marcus-Levich-Jortner model (Equation 34) can also be used to predict the properties of graphene quantum dots (GQDs). Cui & Xue used the generalised Mulliken-Hush model to calculate the vibronic couplings between the first optically bright state and the electronic ground state, and found the fastest internal conversion rate constant for the GQD-4 system (Figure ), with a rate of 2.73 × 10 8 s -1 . Following this derivative in speed were GQDs -2, -5, -1, and -3, with rates of 1.00 × 10 6 s -1 , 1.81 × 10 5 s -1 , 2.59 × 10 4 s -1 , and 1.61 × 10 3 s -1 , respectively. Further, pure GQD was studied and rate constants for transitions between the first, second, and third excited states to the electronic ground state were found to be 2.22 × 10 6 s -1 , 1.56 × 10 7 s -1 , and 7.63 × 10 8 s -1 , respectively. Li and co-workers examined the differences between classical Marcus theory, the Marcus-Jortner-Levich model, and a fully quantum mechanical treatment for silicon quantum dot donor-acceptor systems, and found Marcus-type models underperformed, with rate constants of <1 s -1 in both cases when compared to the fully quantum treatment of 1.5×10 2 s -1 , but the Marcus-Jortner-Levich model was almost 30 orders of magnitude larger than classical Marcus. Interestingly, when including a dangling bond in the molecular ) to study internal conversion in tri-nuclear gold(I) carbeniate quantum dots, and found an ultrafast recombination rate in the monomer phase on the order of 10 10 s -1 . Importantly, the found that the level of theory used to calculate the photophysical properties had a strong effect on the final rate, where using coupled cluster methods yielded a faster rate of 2 × 10 10 s -1 , multi-configuration quasidegenerate second-order perturbation theory was slower by an order of magnitude. In the dimer phase however, the rate was damped significantly to <1 s -1 . |
67b7c567fa469535b9379533 | 117 | A further curiosity within the field of semiconductor nanostructures is the quantum ring. These structures are not as easily treated due to their photo-and electrophysical complexities , primarily due to the nature of the many-fermion problem . A particularly interesting property of quantum ring systems is their circumvention of the standard relation to temperature. While most all systems in nature, especially typical quantum dot systems , exhibit retardation of their photoluminescence quantum yields, quantum rings have to the contrary been observed to display increased yields . This makes treatment of these and similar quantum species difficult. |
67b7c567fa469535b9379533 | 118 | If it is assumed that the electron-hole substituents are in thermal equilibrium, then it can be said that the photoluminescence bandstructure also decays with an averaged time constant weighted with respect to its thermal occupation . In the case of internal conversion being the dominant mechanism, a three dimensional energy continuum therefore behaves similarly to trap states. In the case of a thin well with a single confined level, this is analogous to the rate of thermal ionisation within the two-level system . This can be cast as : |
67b7c567fa469535b9379533 | 119 | where E c is the confinement energy of the ground state, and Γ c is the scattering rate. It should be noted that Γ c must be constant in order for this methodology to remain valid. Gurioli and co-workers , who showed that any photoluminescence decay was intimately related to any non-radiative loss within the system of interest and can be provided through measurement of the decay time, confirming that internal conversion was the primary loss mechanism in quantum well structures. Specifically, they show that the standard Arrhenius fitting often used was only applicable in the case where any temperature dependence was weak to negligible, while Colocci and co-workers noted that loss mechanisms could not be negligible in any way, especially when considering pure radiative recombination was found to possess a linear dependence on excitation power, contrary to the expected dependences on temperature. Despite this model omitting several well documented phenomena, such as sub-band warping and non-parabolicity , this method still manages to capture the primary features of internal conversion qualitatively, and is able to avoid unnecessary complications. |
67b7c567fa469535b9379533 | 120 | If it is assumed that any energy expelled due to internal conversion is absorbed by the wetting later, then Equation 68 can be recast to account for the individual electron and hole characteristics, and is determined by the thermal escape rate weighted by a Boltzmann distribution describing its thermal occupation : |
67b7c567fa469535b9379533 | 121 | Here, N and Z denote the number of states and partition functions, respectively, for the electron and hole components, and v is a thermal emission fitting factor. The probability of an electron or hole undergoing internal conversion here is proportional to the energy difference between its energy and that of the wetting layer (activation energy). The thermal fitting factor can be considered in terms of how the activation energy is affected by the dominant form of loss in the quantum system . The damping factor of v can be used to differentiate between the different forms of thermal emission that tells which sort of thermal escape you have ; scaling of v when compared to experiment can allow for the distinction between excitonic, uncorrelated pairs, or single charge carrier losses. |
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