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67b7c567fa469535b9379533 | 122 | For excitons confined within semiconductor nanostructures there are different non-radiative escape mechanisms. Escape can occur via the loss of the exciton, uncorrelated electronhole pairs or single charge carriers. Each alters the effective activation energy which is accounted for by the fitting factor. Sullivan & Cole used Equation 69 to examine quantum rings, and found that the qualitative behaviour of quantum rings is well describes by the thermal emission of excitons whilst quantum dots emit uncorrelated electron-hole pairs. The interplay of the radiative and non-radiative rates of exciton loss was vital in explaining the observed behaviour. At low temperatures, the charge carriers primarily reside in their grounds states that can radiatively recombine. Thus, in the low temperature regime, radiative recombination dominates. However, this rate decline as temperature increases due to the increased thermal energy available to the charge carriers allowing them to reach dark states that do not permit radiative recombination. As temperature continues to rise and charge carriers have access to even more thermal energy, it becomes possible for emission into the wetting layer. Thus a steady increase in the non-radiative rate as a function of temperature is observed. The total exciton loss rate is the sum of these two rates and, for a particular quantum ring geometry (Figure ), it can be seen that not until temperature surpasses 200 K does the non-radiative rate become the dominant source of loss in the system. |
67b7c567fa469535b9379533 | 123 | If one considers a dimer interaction in the Kasha model, we recall that as two quantum bodies come closer together, the localised energy levels begin to diverge aggressively. In addition to Frenkel states, strong charge transfer states enter into the energetic continuum; as well charge transfer interactions interfere with the vibronic coupling between two electronic states. In this regime, the coupling may not always be weak; therefore perturbation methods may not be valid. As such, dimer interactions are nontrivial, and alternative quantum methods need to be considered. |
67b7c567fa469535b9379533 | 124 | The density matrix reorganisation group method[183-185] (Equation ) can be used to calculate nearly-exact full quantum dynamics, and the corresponding internal conversion rate constants, on molecular aggregates. If it is assumed that were the aggregate system in thermal equilibrium, the aggregate could be expressed in terms the Frenkel-Holstein model , cast as : |
67b7c567fa469535b9379533 | 125 | where â is the electronic creation/annihilation operator, b is the vibrational creation/annihilation operator, V EP is the electronphonon coupling strength, for all aggregate fragments α. Using an azulene dimer as a case study, Wang and co-workers used the energy gap law (Equation ) and found internal conversion rate constants between 10 11 -10 12 s -1 as a function of energy. |
67b7c567fa469535b9379533 | 126 | Ross and co-workers , and earlier in Metiu's doctoral dissertation , were among the first to assume non-adiabatic loss is not simply due to the non-adiabatic coupling. In fact, Metiu postulated that the total coupling was a linear combination of the electronic coupling V mn , V mr,ns , and the spin-orbit coupling V SO : |
67b7c567fa469535b9379533 | 127 | In the case of a dimer, excitons could not only decay to the ground state while conserving spin, but also spontaneously transfer to the coupled fragment. In this picture, transfer could be mediated through contributions from a spin-orbit coupling term in the transfer matrix. Martynov and co-workers found that Plotnikov's model (Equation ) was able to capture the internal conversion behaviour in a series of Tetrathiafulvalene chromophores at varying temperatures (Figure ), capturing physical effects that would be expected within Kasha's exciton model . Specifically, splitting of the energy levels, with each split corresponding to a Frenkel exciton residing on a single molecular fragment, would be expected to result in favourable pathways from one fragment compared to the other. Interestingly, temperature effects were minimal if not negligible, with the largest shift between 100 and 500 Kelvin of ∼6%. |
67b7c567fa469535b9379533 | 128 | Aggregated-induced emission as a photophysical mechanism of interest is also intimately connected to internal conversion; compounds which display pronounced photophysical properties when aggregated are aggregation-induced emitters. With the first models proposing control over loss mechanisms based on intramolecular rotational restriction as a method to promote the mechanism , it becomes obvious that were otherwise free normal modes, like twisting or rotational modes, to be restricted or locked in place, internal conversion could be damped with respect to the monomer phase. A good example of this process are compounds with biaryl groups ; in the monomer phase they are free to rotate and can be labelled as the promoting mode in internal conversion, while in the solid state, this rotation is forbidden. See Ref. 398 for a review on good review on aggregation-induced phenomena. |
67b7c567fa469535b9379533 | 129 | Wu and co-workers[57] used Fermi's Golden Rule in the time domain (Equation ), and reported rates of ∼10 11 s -1 and ∼10 8 s -1 for 3-(2-cyano-2-phenyl-ethenyl-Z)-NH-indole chromophores in the gas phase and in n=75 QM/MM clusters, respectively. Temperature was found to be largely independent to internal conversion, with rate constant remaining in the same order of magnitude, but slowing by 57% in the case of the gas phase chromophore. Further, aggregation was found to strongly affect the rate of internal conversion by 3 orders of magnitude, while displaying a minimal effect on radiative processes. Jin and co-workers note very large differences in the rate of internal conversion from the monomer phase to the aggregation phase for some systems, in most cases at least 2 orders of magnitude. However others they note the reverse behaviour, where some systems become more photostable due to aggregation by the same factor. |
67b7c567fa469535b9379533 | 130 | By now it should be very obvious that despite there not yet being a singular that is both concise and robust; what we can do now compared to a few short decades ago is staggering. The eb and flow of research in the field is ongoing and vital in the pursuit of complete quantum control. Specifically in the case of internal conversion, so much is already possible in terms of discrete control. The most cutting edge applications include its use as a channelling mechanism to guide excitons to important excited states , applying it towards battery design , and even fight certain cancers . The nuances are effectively endless, with newer combined studies on how it can be controlled, such as through exciton-polariton coupling, heavy-atom effects, and the role of complex potential energy surface interactions. These considerations offer critical insights into refining our understanding of IC, both as a fundamental process and as a tool for future technological applications. |
67b7c567fa469535b9379533 | 131 | Even with the new systems now available to study, internal conversion has yet to catch up to more nuanced applications. Take for example an electromagnetic field: while fluctuations and their effects are a key feature of quantum electrodynamics , very little is understood as per a fields interaction with the internal conversion mechanism. Couto & Kowalewski note that the tuning of cavity strength and resonance frequencies can alter the exciton dynamics of a chromophore. Huang and co-workers were able to determine a relation between the coupling strength of a plexciton (hybrid emitter-nanocavity pseudo-particle) and internal conversion, however only in terms of a specific decay channel. Bennett and co-workers highlighted how internal conversion in a given system could be altered by using a cavity. Ulusoy and co-workers examined how light-matter coupling could be used to modify known photochemical properties via polaritons (photon strongly coupled to electric dipole), and found that direct excitation of the upper polaritonic states could allow for a pathway to drastically slow internal conversion rates. However, beyond that, there is minimal work on the subject matter, as the state of the art continues to evolve. |
67b7c567fa469535b9379533 | 132 | Schäfer and co-workers noted that quantum electrodynamics affected molecular systems via non-adiabatic coupling. In other words, quantum electrodynamics was reported to directly affect internal conversion. In swift response, Tsai and co-workers developed a generalised method which could examine internal conversion under weak light-matter coupling conditions, providing a full derivation of their work. Here we will summarise it as concisely as possible. |
67b7c567fa469535b9379533 | 133 | where p is the photon creation/annihilation operator. Here, the Hamiltonian terms for the atomic system of interest within the cavity, of the photon, and of the system-photon interaction are labelled, where E g is the photonic frequency, g are the number of photon modes within the cavity, V is the effective mode volume, λ is the two transverse polarisation directions, μ is the dipole moment operator, and e pp is the overall photonic polarisation. The effective mode volume can also be used to quantify the degree of light-matter coupling directly . |
67b7c567fa469535b9379533 | 134 | Where the typical electrostatic Hamiltonian neglects electromagnetic fluctuations , Tsai and co-workers instead choose to consider an atomic system of interest within an optical cavity as a collection of charges, neutral, and coupled to a unique photonic mode. Acknowledging that a full description can be defined within a weak-coupling regime , a number of approximations can be made; a long-wavelength approximation , the length gauge , and a unitary phase transformation ; assuming the photonic mode is larger in wavelength than the atomic system of interest, and the electric displacement field is coupled to the dipole moment, yielding Equation . |
67b7c567fa469535b9379533 | 135 | Then, the Condon approximation can the be applied to treat for photon-photon and photon-electron interactions. Duschinsky rotations and complex normal mode couplings are omitted as a way to account for quantum electrodynamical effects on the internal conversion mechanism. This allows for the vibronic and cavity based contributions to the rate constant to be effectively decoupled, at least to a degree. Therefore, the internal conversion rate constant within a cavity can be recast as : |
67b7c567fa469535b9379533 | 136 | where W system is the rate constant for the system of interest, be it chromophore, quantum dot, thin film, etc., and W cavity is the internal conversion rate constant within an electromagnetic vacuum. Tsai and co-workers used the formalism used by Lin (Equation ), however as per the separability between the system rate and cavity interaction in Equation , any method should be applicable. In particular, the two rate components can be calculated, and readily compared, with the only free parameter being the effective mode volume. Previous studies have shown that the use of Fermi's Golden rule (Equation ), even in the strong-coupling regime with in a cavity, is applicable. Finally, W cavity can be expressed in full as : |
67b7c567fa469535b9379533 | 137 | where e is the electronic charge, ⃗ R is the unit vector of the optical transition dipole moment, e pp,q is the q th photonic mode polarisation, and Z q is the nuclear charge. The vibronic component is the usual nuclear-electronic coupling, while the vibrational-photon component describes the alignment between each normal mode and the light polarisation. The vibrational energy E vib corresponds to the energy difference between the initial and final vibrational states for all normal modes . It is worth highlighting that when both initial and final states are resonating in the absence of light (W cavity ), the atomic system component is off-resonance (W system ). In other words, both components are out of phase with respect to each other. Therefore, vibrational energy difference part of the density of states is different between both components of Equation . |
67b7c567fa469535b9379533 | 138 | Using this methodology, a number of considerations need to be highlighted. Firstly, while in non-cavity internal conversion there are no photonic degrees of freedom (the zero-photon state), this is not the case here. The photonic degrees of freedom are important here and cannot be neglected; in this formalism it is the vacuum fluctuation of photons rather than the promoting modes themselves that participate in the transition . Via this process, the energy which would otherwise be lost is instead partially converted into photonic energy, yielding a final electronic-photonic state. |
67b7c567fa469535b9379533 | 139 | Using Equation , Tsai and co-workers studied internal conversion occurring in some anthracene derivatives and perylene within a cavity, and were able to demonstrate that the electrodynamic internal conversion rate constant can be significantly perturbed by adjusting V , resulting in drastic changes to the light-matter coupling. For 9-cyanoanthracene, the combined internal conversion rate varied between 0.2-7×10 8 s -1 for a V range of a thousand to a million cubic nanometers, while for perylene it varied between 1.5-100×10 6 s -1 for V between five thousand to three and a half million cubic nanometers. Lower values of V also result in dominant W cavity terms, with this property more apparent in perylene than 9-cyanoanthracene. At certain effective mode volumes, internal conversion of the system was found to equal internal conversion due to the cavity W system = W cavity , for which an analytical expression for this crossing point was found . For 9-cyanoanthracene this is around fourty thousand cubic nanometers, while for perylene it is around two hundred thousand cubic nanometers. Experimental V values seen in the literature vary greatly , between a hundred to a hundred million cubic nanometers; therefore even simple cavity manipulation is a viable method to control internal conversion. It is also worth noting that while based on Equation 14 one would assume that even in a cavity, vibronic coupling would be the driving force of internal conversion, instead the orientation of each normal mode with respect to the polarisation axis can have a powerful effect. |
67b7c567fa469535b9379533 | 140 | One can get a feel for the degree of molecular rigidity, and therefore a qualitative probing of cavity-type internal conversion within the system, by examining the combined Huang-Rhys factor (Equation ). A series of anthracene derivatives were examined by Tsai and co-workers , and it was found that each yielded different Y values and displayed similar photophysical properties, but with rate constants differing by more than an order of magnitude. Simply put, the size of Y correlates directly with the degree of wavefunction overlap. However, it should be emphasised that this is only the case for electrodynamic internal conversion, where the reverse is true for non-cavity based internal conversion, since greater overlap results directly stronger wavefunction overlap and therefore less vibronic coupling, as shown by Manian and co-workers . This does however, offer an interesting opportunity for cavity tuning, whereby electromagnetic fluctuations could be shifted to minimise internal conversion. Rather, even the most minute of cavity alterations can have drastic effects on the internal conversion mechanism. |
67b7c567fa469535b9379533 | 141 | It should be noted that there are a number of situations where this formalism is not valid. Firstly, in the case of strong light-matter coupling , Fermi's Golden rule (Equation ) becomes invalid. In this case, an exact non-adiabatic quantum electrodynamics model will be required . Secondly, Equation 74 is analogous to an electrodynamical treatment of the energy gap law, and therefore in the case where the vibronic coupling is too large, perturbation theory may not be valid . Thirdly, a number of external factors have not yet been included in the electrodynamical internal conversion cast. These include, but are not limited to, cavity dielectric effects or optical cavity loss phenomena, which have all been shown to strongly influence light-matter interactions . |
67b7c567fa469535b9379533 | 142 | Couto & Kowalewski instead examined the nature of polariton-exciton internal conversion using a quantum nuclear wavepacket propagation method. Here, the Schrödinger equation, is solved numerically using the Chebychev method with the ground state being forced to the S 1 potential energy surface . Using meso-tertbutyl-boron-dipyrromethene as a case study, they found that changes in the energetic landscape and population transfer rates could easily be achieved through tuning of the cavity properties through analysis of the polartonic potentials and the resulting wavepackets, highlighting that the linchpin is the interplay between cavity resonance and field strength. Where ultra-strong coupling was observed, internal conversion was significantly damped, with the wavepacket being projected onto the ground state energetic surface, therefore drastically increasing the photoluminescence quantum yield of a given system. |
67b7c567fa469535b9379533 | 143 | Polariton-exciton effects can also be probed using the energy gap law (Equation ), whereby cavity coupling to the system of interest is the dominant mechanism, and all other internal conversion mechanisms are higher-order processes . This casts the rate constant in terms of the final dark state and numerous saddle-point approximations : where subscripts LP and UP refer to the lower and upper polaritonic states respectively, and X is the number of molecules in the cavity. Assuming a continuum of dark states with discrete polaritonic states, or that the majority of the system energy is distributed through dark modes, internal conversion should be dominant. While this was tested on an arbitrary system by Poh and co-workers , tuning was shown to be instrumental in its control. |
67b7c567fa469535b9379533 | 144 | Avramenko & Rury also noted that the energy gap law can be used to form a microscopic model of an exciton-polariton coupled system. Looking closely at polaritonic effects and how they could control ultrafast internal conversion in zinc(II) tetraphenyl porphyrin, their results (Figure ) suggest external physical parameters of polariton formation affects the internal conversion rate. Specifically, it may be that polariton coupling strongly effects the electronic energies, or the strong lightmatter coupling may influence the vibronic couplings directly. This of course agrees with much of the previously highlighted work, however their delve into their results provides further evidence for this observation. |
67b7c567fa469535b9379533 | 145 | Briefly on the subject of deuteration: C-H stretching normal modes are often the most common accepting modes for aromatic hydrocarbons . Since it is most often mid-high energy normal modes with the largest Huang-Rhys factors, they are therefore more likely to participate in a given transition. Atomic deformations due to C-C modes, for example, while larger than those of C-H-type modes, are unable to combat the effect of a significantly smaller frequency. This ends up with large configurational indexes , thus smaller contributions are observed to the rate constant. In addition, nuclear deformation due to electronic excitation, for highly symmetric compounds like most aromatic hydrocarbons, in order to maintain symmetry are observed through stretching, breathing, and twisting normal modes. It is therefore interesting to highlight the effect of deuteration; the much smaller energy of C-D modes requires more quanta per normal mode . Another way to examine this result is to consider the mass of hydrogen Vs. deuterium. Huang-Rhys factors are sensitive to nuclear mass; since deuterium is twice as heavy as hydrogen, the energy of the mode can be expected to weaken by a factor of ∼ √ 2. |
67b7c567fa469535b9379533 | 146 | Valiev and co-workers , when attempting to probe the effects of deuteration found their methods were unable to yield results correlating positively with experiment. This was most pronounced in the case of anthracene derivatives (Table ), which should have experienced an increase in internal conversion rather than a decrease . Fischer and coworkers calculate rates of 1.6 × 10 3 s -1 and 4.4 × 10 5 s -1 for naphthalene and deuterated naphthalene in the vapour phase. It was also noted that these rates agreed well with experimental observations with respect to excess energy and deuteration. |
67b7c567fa469535b9379533 | 147 | While most are content with internal conversion sitting in the domain of the harmonic approximation, Hoche and coworkers found instead that under the assumption of the energy gap law, internal conversion should be expected to drop exponentially with respect to the emission energy. However, this is not observed in the merocyanine 4-(dicyanomethylene)-2-tert-butyl-6-[3-(3-butyl-benzothiazol-2-ylidene)1-propenyl]-4H-pyran chromophore. Instead, a drop is observed as the emission energy decreases due to solvatochromic effects. While it is possible that this could be due to some degree of molecular complexation, resulting in a solute-solvent superstructure locking in or perturbing important transition participating modes, they postulate the possibility of internal conversion to a conical intersection, which could lead back to the reactant well, or to the formation of a photoproduct. |
67b7c567fa469535b9379533 | 148 | Under the assumption that hitting the conical intersection will always result in internal conversion, it can be assumed that the exciton dynamics are dependent on only the barrier energy and vibrational energy of the system. This can be calculated using Kramers' barrier model , which described the crossing of an exciton between two potential energy wells across a barrier, and in simple terms relates the well frequency ω w with the barrier of height E w and frequency ω u within a medium with a friction constant γ F , cast as : |
67b7c567fa469535b9379533 | 149 | Here, ω u should correspond to the transition participating mode with a negative force constant at the lowest transition state between the conical intersection, and G F is a semi-empirical global fitting parameter which Hoche and coworkers found by optimising the sample Pearson correla-tion coefficient between experimental and simulated data, yielding a correction of 0.192 eV. The friction coefficient can be calculated using Stokes Law, cast as : |
67b7c567fa469535b9379533 | 150 | where η is the dynamic viscosity of the solvent, R is the radius of a single solvent molecule, and M is the molecular mass. It should be noted that Kramers' model is one dimensional; a mode at both the Franck-Condon point and the transition state of highest similarity should be used. Using this formalism, Hoche and co-workers found rates were strongly dependant on the solvent used. The fastest rate was observed for methylcyclohexane, with a rate of 3 × 10 12 s -1 , and slows down to 2.2 × 10 7 s -1 when using dimethyl sulfoxide. In this paradigm, it is observed that in the lower polarity solvents, the conical intersection transition dominate loss, harmonic transitions are dominant in higher polarity solvents (Figure ). The compared harmonic rates are calculated using a Fermi's Golden Rule methodology. Blacker and co-workers similarly examined internal conversion at a conical intersection in nicotinamide adenine dinucleotide and nicotinamide adenine dinucleotide phosphate, and noted an exponential relationship between viscosity and the rate constant for both the short and long components of the well known bi-exponential fluorescent spectra (Figure ). |
67b7c567fa469535b9379533 | 151 | This comprehensive review has looked at highlighting this work's namesake, that is to say each of the successes and failures in the state of the art of internal conversion from first principles. By exploring a variety of different methods which can each calculate rates of internal conversion, beginning from the formalisms' inception, to more modern day fine-tuning and benchmarking works, we have highlighted that while no robust model exists, many are quite successful and can be employed on most systems. Each has their own strengths and weaknesses, as well as degree of truncation, from complete to approximate. |
67b7c567fa469535b9379533 | 152 | The utility of internal conversion is largely untapped as of yet, due to the difficulties associated with modelling it. However, it has already been proven an effective suite of control in numerous applications. What researchers need to do is continue their work! There is so much more than can be done, and while arduous, the results will be well worth it. To aid in this desire, it is our hope this review will aid those in chemical physics to better choose an appropriate model, resulting in a deeper insight into the science, and that this work inspires others to continue to push the boundaries of what is possible... |
67d92d8181d2151a02cf1d15 | 0 | The use of chemical oxidants including ozone, chlorine, and chlorine dioxide in water treatment processes is essential to mitigate waterborne diseases and abate inorganic and organic micropollutants. Chemical oxidants have been used successfully for decades for the provision of safe drinking water and, more recently, for wastewater treatment. Their reactions with matrix components and organic pollutants create transformation and disinfection (by)products which can cause adverse effects on human and environmental health. Providing efficient disinfection/oxidation while minimizing toxic product formation is particularly challenging for wastewater treatment and potable water reuse purposes where concentrations of dissolved organic matter and organic pollutants are typically much faster. Evaluating the efficacy of pollutant abatement in chemical oxidation processes and identifying potentially hazardous transformation products are thus paramount for the implementation of adequate water treatment strategies. To date, there is a large data base on reactions and mechanisms of oxidation reactions, however, comprehensive experimental descriptions of all reaction pathways pertinent to oxidative water treatment would be extremely laborious and experimental evidence remains inherently incomplete. Computational support is therefore required to complement the experimental characterizations of oxidation processes. In fact, chemoinformatic (i.e., datadriven) pathway prediction tools and first-principles quantum chemical calculations have already assisted the experimental elucidation of kinetics and reaction pathways. Previously, empirical reaction rules for reactive functional groups of compounds with unknown reactivity towards oxidants were applied to predict the formation of transformation products. Furthermore, quantum chemical descriptors for structure activity relationships were established, allowing access to chemical structures, their Gibbs energies, and barrier heights for the characterization of reaction pathways. However, quantum chemical construction of potential energy surfaces (PES) requires significant manual input and guidance, limiting computational analyses to specific reaction coordinates. These circumstances have so far prevented a more expansive, systematic, and possibly predictive evaluation of reactions involving chemical oxidants and organic micropollutants. |
67d92d8181d2151a02cf1d15 | 1 | These restrictions can now be overcome with high-throughput computational chemistry which offers new avenues to elucidate reaction pathways and mechanisms through chemical reaction network (CRN) explorations. Supported by the steady increase of computing power and availability of high performance computing clusters, quantum chemistry based reaction network explorations are developing rapidly and they make extensive use of algorithms for their automated and autonomous execution. Several mechanism-exploration procedures have been implemented in recent years (see compilation in ref ). In the Software for Chemical Interaction Networks (SCINE), which we will use in this study, such CRN explorations are carried out in three principle steps. First, the reaction space is explored by searching elementary steps which interconnect the molecular structures of reactants, intermediates, transition states and products. To do so, first-principle heuristics, that is features of the electronic wave function, facilitate the generation of reaction coordinates and thus prediction of reactive sites in molecular structures for uni-and bimolecular reactions. Not only are these algorithms agnostic to the functional groups in the studied molecules, but they also allow for exploring all possible reactive combinations to obtain a comprehensive picture of the reaction network. Second, the minimum and transition-state structures and corresponding energies generated during the initial CRN explorations may require a refinement given that these explorations are typically carried out with fast electronic structure methods such as semipirical ones (e.g., xTB ) and density functional theory (DFT). These quantum chemical characterizations of the CRN can involve computationally more demanding coupled cluster (CC) methods for electronic energy calculations. Third, the obtained reaction Gibbs energies need to be evaluated modeling experimental conditions in terms of shortest reaction path to determine reaction mechanisms, and microkinetic simulations to predict the concentrations of all the compounds in the CRN. Most recent developments address the need for efficient navigation of CRN explorations and restrict the combinatorial explosion of reaction paths by allowing user-defined interventions (e.g., by expert knowledge, on-the-fly evaluation of concentration fluxes, interactive structure manipulation). The goal of this study was to evaluate how automated CRN explorations can support the elucidation of mechanisms and pathways of reactions of chemical oxidants with organic compounds and provide a conceptual basis for future applications. We focused our study on reactions of ozone with olefins in aqueous solution for purposes of both practical relevance and systematic development of the computational methodology for CRN explorations. |
67d92d8181d2151a02cf1d15 | 2 | Reactions of ozone with ethenes through the 1,3-dipolar cycloaddition (Criegee) mechanism are furthermore prototypical for reactions of electron-rich olefins and aromatic compounds in treated waters and their kinetics and mechanisms are well-documented with experimental data. Moreover, ozone has biradical and thus multireference character and its computational identification and optimization of molecular structures and predicting the reaction energies may not be accurate with fast DFT methods, as illustrated in studies of the Criegee mechanism. Instead, single-reference coupled cluster methods are recommended, though they can be challenging to use given the large number of molecular structures in CRNs. The specific objectives of this study were threefold. (i) We identified the optimal computational methodology for automated CRN explorations with the Chemoton software 29 through systematic evaluation of the Criegee mechanism for ozone and ethene against benchmark studies of Wheeler et al. (ii) We tested the predictive capabilities of automated CRN explorations by investigating two model reactions of ozone with ethene and tetramethylethene, respectively. These processes were selected as benchmarks because they both proceed through the Criegee mechanism but generate different stable products under identical experimental conditions. Here, we performed the CRN evaluation with methods identified in specific objective (i) and evaluated their utility for future automated CRN explorations involving ozone and organic compounds. (iii) Finally, we analysed the outcome for the lowest energy path reaction mechanism between the reactants and the experimentally observed products with Pathfinder 32 as well as through microkinetic modeling. |
67d92d8181d2151a02cf1d15 | 3 | Density functional theory and coupled cluster calculations were performed with the ORCA software package version 5.0.3. We also tested three semiempirical methods, namely DFTB3, 43 PM6 44 and GNF2-xTB. We have not considered the application of multireference configuration interaction methods because of the two following limitations. First, molecular orbitals that form the complete active space must be consistent for all the molecular structures along a reaction path, hence new methodologies are currently under development to provide direct orbital selection mapping. Second, multireference methods are notorious for being among the most time-demanding calculations in quantum chemistry. Given that pioneering studies demonstrated that CC succeeded in both optimizing the molecular structures and predicting the reaction energies of the Criegee mechanism for ozone and ethene, we have followed their recommendation. DFT structure optimizations were done with Perdew-Becke-Ernzerhof PBE, the hybrid PBE0, and the long-range LC-PBE 50 density functionals, with the Ahlrichs def2-TZVP basis set. Grimme's D3 dispersion correction was employed for the PBE and PBE0 calculations. Single point calculations were performed with canonical coupled cluster, CCSD(T), and the Domain-based Local Pair Natural Orbital approximation of coupled cluster, DLPNO-CCSD(T) with the correlation consistent basis sets. Because the reaction network contains both open-and closed-shell compounds, we employed the DLPNO approach, instead of the DPNO, to ensure that we can combine their absolute electronic energies when calculating reaction energies. We employed the default protocol in ORCA, where UHF alpha and beta orbitals are transformed to quasi-restricted orbitals, thus removing much of the spin contamination. Solvation in water was introduced using the Conductor-like Polarizable Continuum Model, CPCM, both for structure optimization and single point calculations. Stationary points were characterized with analytic frequency calculations. Gibbs energy corrections were computed using LC-PBE, at 298.15 K and 1 atm, and using the ideal gas-rigid rotor-harmonic oscillator (IGRRHO) model. Final Gibbs energies were calculated combining the electronic energy from DLPNO-CCSD(T), and the Gibbs energy corrections from the DFT method. In order to employ molar concentrations in the rate constant, the reference state was modified to 298.15 K and 1.0 M (standard state in solution). Given that the temperature remains constant between both states, this was done by simply increasing the barrier by 7.90 kJ•mol -1 . |
67d92d8181d2151a02cf1d15 | 4 | We carried out autonomous reaction network explorations with ethene and ozone as well as tetramethylethene and ozone using the SCINE Chemoton module. An introduction to the terminology, the principles of CRN construction, and the characterization of CRNs is provided in Section S1 of the Supporting Information. Briefly, CRN explorations generate four datasets. (i) A structure is a single identity defined by a set of atoms distributed in the three-dimensional space with a determined spin and charge. (ii) A compound is a group of structures with same number of atoms, charge and spin but different electronic energies. (iii) |
67d92d8181d2151a02cf1d15 | 5 | To generate transition state guesses for uni-and bimolecular reactions, we employed the algorithm implemented in Chemoton named Newton Trajectory scan (NT2), which is inspired by the Artificial Force Induced Reaction (AFIR), and the use of Newtonian forces for computing an effective PES. We restricted the size of the compounds in the two networks as follows. NT2 trials that led to activation barriers greater than 250 kJ•mol -1 (according to the method which was used for the exploration) were dismissed, as the subsequent reaction would be unlikely to happen. The chemical reaction network of ethene and ozone (CRN-E) contained a maximum of 2 carbon atoms, 4 hydrogen atoms, and 3 oxygen atoms (i.e., the size of the ozonide) to reduce the computational cost of the exploration (see further discussion below and Section S2.2). CRN-T for reactions of tetramethylethene and ozone was made up of a maximum of 6 carbon atoms, 14 hydrogen atoms, and 3 oxygen atoms. This threshold has two hydrogen atoms more than the tetramethylethene-ozonide, in order to allow the search for the relevant side product, 2,3-dimethylbutane-2,3-diol. We stopped the explorations once the expected, experimentally confirmed products were found, and the subsequent evaluations of the networks led to the correct mechanistic and kinetic results. |
67d92d8181d2151a02cf1d15 | 6 | ReaDuct, for communicating with ORCA's generation of input and output files, Molassembler, for handling molecular structures, and Utilities 66 which contains a broad variety of functionalities essential for all SCINE modules. We monitored the progress of the reaction exploration with the open-source graphical user interface Heron, and the reaction networks can be interactively visualized as standalone HTML files generated with VizChemoton. A detailed description of the parameters employed in the exploration is provided in Section S2.2. |
67d92d8181d2151a02cf1d15 | 7 | CRN-E and CRN-T were characterized with reaction pathway identification and kinetic analyses. An analysis of equilibrium species distributions based on thermodynamic stability was not conducted, as ozonation of olefins is driven by fast kinetics. We employed Pathfinder and its built-in cost function to determine the most favorable reaction mechanisms. For the kinetic analyses we used microkinetic simulations with KiNetX's original Matlab code (see parameters in Section S2.7). Initial concentrations for both the pathway identification and kinetic analysis were set to 4.97•10 -5 M of ozone, 4.97•10 -3 M of the model olefin, and excess of water. |
67d92d8181d2151a02cf1d15 | 8 | To determine the adequate quantum chemistry method for the reaction exploration, we evaluated the performance of a broad range of methods, focusing on the first step of the ozonolysis reaction. This 1,3-dipolar cycloaddition of ozone to the double bond of the olefin leads to an ozonide intermediate. Figure shows the mechanism of the ozonolysis, known as the Criegee mechanism, for the two olefins, ethene and tetramethylethene, studied in this work. We used as reference the data reported by Wheeler et al. on the 1,3-dipolar cycloaddition reactions, which consisted of the molecular structures of ozone, ethene, and the TS, as well as the corresponding activation energy (14.2 kJ•mol -1 ). Moreover, Figure depicts the expected products for the aqueous ozonolysis of ethene (formaldehyde and αhydroxymethylhydroperoxide) and tetramethylethene (acetone and hydrogen peroxide). The final products of the ozonolysis and their yields reported in Dowideit and Von Sonntag 37 are used in the final section of the work to validate the outcome of the CRN exploration in kinetic simulations. |
67d92d8181d2151a02cf1d15 | 9 | We tested three semiempirical methods, DFTB3, PM6 and GFN2-xTB, and three versions of the Perdew-Becke-Ernzerhof functional, PBE, PBE0, and LC-PBE, to optimize the transition state structure leading to the ozonide in the first part of the method benchmarking step. We did not employ CC methods to optimize molecular structures because calculations would become too slow. Figure shows the the results of structure and energy benchmarks that we conducted to identify the optimal quantum chemistry methodology for ensuing CRN explorations. The data in Figure demonstrate that no single quantum chemistry method simultaneously offers low deviations in molecular structures and electronic energies. Therefore, Figure : Ozonolysis reaction of ethene and tetramethylethene through the ozonide intermediate (1). This Criegee mechanism 70 proceeds either directly with an heterolytic cleavage forming a carbonyl compound and an alkyliumperoxolate (3) or via a zwitterionic intermediate (2), the latter leading to aldehyde and α-hydroxyl-alkylperoxide (5) in presence of water as solvent, and to the Criegee ozonide (4) in apolar solvents. it is necessary to define a composite approach, where molecular structures are optimized first with one method, and electronic energies are subsequently calculated with a second method. |
67d92d8181d2151a02cf1d15 | 10 | Figure shows the root mean squared deviation (RMSD) of the aforementioned methods respect to the reference TS structure of Wheeler et al. DFTB3 provides a molecular structure with a RMSD above 1.04 Å, followed by PM6 (0.59 Å). GFN2-xTB outperforms the two previous semiempirical methods with a RMSD of 0.20 Å. Neither PBE nor PBE0 succeeded in converging the TS structure, whereas LC-PBE not only locates the TS structure, but also provides the lowest RMSD (0.03 Å). Considering that commonly used RMSD thresholds range from 0.5 to 2.0 Å, 71 only GFN2-xTB and LC-PBE are meaningful candidates for running the reaction exploration. The former is on average 100 times faster than the latter, however, LC-PBE is approximately 10 times more accurate than GFN2-xTB. |
67d92d8181d2151a02cf1d15 | 11 | The excellent performance of LC-PBE is explained by the accurate electronic structures for zwitterionic compounds predicted with range-separated functionals. This point is particularly relevant since not only does ozone exhibit partly zwitterionic character, but also many products of the reaction network are zwitterions. Considering that standard quantum chemistry methods (PBE and PBE0) failed to locate the TS and that semiempirical methods are persistently inaccurate for complex bond-breaking processes, we employed LC-PBE for finding and optimizing the molecular structures in our CRN explorations. |
67d92d8181d2151a02cf1d15 | 12 | We have evaluated the performance of 14 quantum chemistry methods (3 semiempirical methods, 3 DFT functionals, 4 DLPNO CC, and 4 canonical CC) to predict the activation energy of the ozonide. Figure shows how the electronic energies calculated with the 14 methods, using single point calculations with the reference structures from Wheeler et al. , deviate from the reference value of 14.2 kJ•mol -1 . Deviations vary substantially for the 3 semiempirical methods from -83.6 kJ•mol -1 with GFN2-xTB to 0.08 kJ•mol -1 with PM6. |
67d92d8181d2151a02cf1d15 | 13 | We attribute PM6's high accuracy to its parametrization with CC energy data of its non-covalent interactions. The energy deviations of DFT calculations, that is PBE, PBE0 and LC-PBE are -29.9, -19.1, and -10.1 kJ•mol -1 , respectively. This energy trend follows the Jacob's ladder principle by which a more precise treatment of the exchange-correlation term enhances the prediction of the electronic energy. However, none of the activation energies calculated with these methods are within the chemical accuracy range of ±4.18 kJ•mol -1 . The poor accuracy of TS energy calculations with DFT functionals in this type of reactivity agrees with a previous benchmark study. Only data obtained from DLPNO and canonical CC resulted in reliable TS activation energies, as depicted with a green rectangle in Figure . Despite the fact that DLPNO is an approximation of the canonical CC, the former demonstrates a similar accuracy to the latter, even when pvtz CCSD(T) is approximately three times slower than pvtz DLPNO-CCSD(T) (Figure ). In fact, the wall clock time of the canonical CC rapidly scales exponentially for molecules with more than 20 atoms, whereas DLPNO CC scales linearly even with molecules up to 1000 atoms. Moreover, complete basis set extrapolations (CBS) shows a higher deviation than the triple and quadruple zeta basis sets. Data in Figure also illustrates that pvqz DLPNO-CCSD(T) is 4.1 times more accurate than pvtz DLPNO-CCSD(T), however, it is also 4.7 times more expensive. |
67d92d8181d2151a02cf1d15 | 14 | The CRN exploration initiated through the reaction of ethene with ozone, with LC-PBE for structures and DLPNO-CCSD(T) for energies, generated 595 compounds and 1350 reactions (Table ). The serial computing time for the CRN-E exploration, that is the time invested in all the calculations of the CRN exploration on a hypothetical single processor core, 34 was 28500 days (Table ). Given that computations are run in parallel, the effective total computing time using an average of 250 cores was 114 days. |
67d92d8181d2151a02cf1d15 | 15 | We assessed the plausibility of structures generated for CRN-E in analogy to the computational benchmarking process with experimental evidence for the detected reaction products. Table shows that the exploration for CRN-E found the the key compounds pertinent to the ozonolysis reactions, namely the ozonide, formaldehyde, methyliumperoxolate, α-hydroxymethylhydroperoxide, and hydrogen peroxide. Chemoton successfully deduced the intermediates of the 1,3-dipolar cycloaddition mechanism summarized in Figure without any human intervention. We note that water, which is essential for the reaction of the alkyliumperoxolate (3 in Figure ), was not included at the start of the exploration but generated by Chemoton. Lastly, water was found to be involved in 59 reactions, thus demonstrating the consistency of the autonomous exploration procedure. |
67d92d8181d2151a02cf1d15 | 16 | Representing the complete network of CRN-E in a single yet meaningful Figure is not doable given the large number of reactions and compounds (Table ). Here, we illustrate the initial steps with the involved compounds, flasks, and transition states (Figures and) leading to the ozonide and the reactions of the ozonolysis reaction path (Figures and). A complete representation of the CRNs with all compounds and reactions is available in the HTML files generated with the browser-based visualization module VizChemoton. Note that the identification of a compound a priori does not imply relevance. The latter is requires the evaluation of the complete reaction network for reaction mechanisms and kinetics as is done below. starts by pushing together ozone and ethene to form flask A1, which was already found by Wheeler et al. as vdW complex. This pre-reactant complex forms three different compounds through three distinct transition states. The three-dimensional structures of the transition states are depicted in the orange boxes of Figure . The exploration procedure identified the most critical compound, the ozonide, C3, through reaction R1, in good agreement with experiments and previous theoretical investigations. Figure 3 also depicts a second reaction, R2, which leads to the formation of the known secondary zwitterion in the Criegee mechanism (2 in Figure ). Reaction, R3 shows the formation of the 1ethenyltrioxidane, C5, which has not been not been proposed for ozonolysis of ethene. |
67d92d8181d2151a02cf1d15 | 17 | The exploration for tetramethylethene and ozone, CRN-T, generated 388 compounds and 441 reactions (Table ). The serial computing time for the CRN-T exploration was 11600 days, and the total computing time, using an average of 250 cores, was reduced to approximately 47 days. We compared some structures generated for CRN-T with the initial steps leading to the tetramethylethene ozonide intermediate in analogy to the CRN-E evaluation. CRN-T contains the key compounds depicted in Figure , thus confirming that the product hypothesis is also met for this exploration. The CRN-T exploration also found water as compound. Figure shows an analogous representation of Figure for CRN-T, where the key ozonide compounds and its transition state were successfully identified. However, one experimentally reported side-product was not found, which corresponds to the product of the partial oxidation of tetramethylethene: 2,3-dimethylbutane-2,3-diol. In fact, the reaction exploration did not find any diol compound. |
67d92d8181d2151a02cf1d15 | 18 | We tested the hypothesis that the CRN-T did not find any diol and, presumably, other intermediates and products because of the inherently vast chemical space. This chemical space would not be accessible within the time frame of our computations with LC-PBE based structure searches. To that end, we ran a separate, exploratory reaction exploration starting from tetramethylethene and ozone using GFN2-xTB instead of LC-PBE (abbreviated CRN-T-xTB in Table ). The computing time running on 250 cores was approximately 6 days (1436 days of serial computing), and during that short period of calculation time, it found 133426 reactions and 68721 compounds. Chemoton indeed generated eight different diols (Figure ), but only after CRN-T-xTB included over 40000 reactions vs. 441 reaction with CRN-T, that is s 100-fold increased depth of the exploration. Further qualitative evidence for the operational limitations of LC-PBE-based CRN explorations was obtained from the consideration of typical intermediates of the aqueous ozone chemistry. CRN-T-xTB found additional important products associated with the general decay of ozone not included in CRN-T such as the hydroxyl radical, the superoxide ion and the ozone radical (Figure ). Furthermore, Figure illustrates that singlet and triplet dioxygen, hydrogen peroxide, and water were also present in CRN-T, but involved in a 2-to almost 400-fold lesser frequency than in reactions of CRN-T-xTB. These intermediates did not matter for the reaction of ethene and tetramethylethene with ozone studied here, but would do so in ozonation processes with more complex substrates than olefins. The exploration of the reactions of these ozone decay intermediates, however, was beyond the scope of this study. |
67d92d8181d2151a02cf1d15 | 19 | The DLPNO-CCSD(T) energy calculations for the 1350 reaction in CRN-E enabled us to obtain 2700 forward and backward reaction rates constants for the reactions of 595 compounds in the network. Figure shows the unimodal exponential distribution of reaction rate constants according to their activation energies as confined by the initial boundary conditions of the CRN exploration. The distribution covers a range of approximately 250 kJ•mol -1 , which is equivalent to 44 orders of magnitude in reaction rate constants. The peak of the distribution is roughly at 25 kJ•mol -1 and the majority of reactions have an energy of <80 kJ•mol Consequently, most reactions have a half-life of 12 seconds at room temperature according to the Eyring equation. We have divided the reaction types in three main groups according to whether the stoichiometry of the reaction (i.e., ν reactants -ν products ) is positive (indicative of an association reaction), negative (dissociation) or zero (bond rearrangement). Table 1 shows that CRN-E has 1710 associations, 520 dissociations and 470 bond rearrangements. |
67d92d8181d2151a02cf1d15 | 20 | A similar ration between the three reaction types, that is the predominance of association 336 reactions was also found for CRN-T despite the smaller number of compounds and reactions. While Chemoton allows for the autonomous construction of CRNs, the large uncertainty of quantum chemistry methods for predicting absolute values of rate constants still remains. To exemplify issues with absolute rate constants for the CRNs studied here, we have depicted two sets of the experimental and quantum chemical rate constants, k exp O 3 and |
67d92d8181d2151a02cf1d15 | 21 | , for the reaction of ozone with ethene and tetramethylethene, perspectively, in Figure . The experimentally determined value of k exp O 3 for ethene is shown as a red vertical line, . The value of the theoretical k QC O 3 for reactions R1 and R2 of Figure , however, are three and two orders of magnitude different (brown and purple lines in Figure ). The mismatch in the absolute value of the rate constants is also observed in the exponential decay of ozone, as it reacts slower than in experiments (Figure ). The mismatch between experimental and theoretical data is even more pronounced for tetramethylethene. Because this reaction is too fast to be determined experimentally, the lower limit of k exp O 3 is shown as a dashed red vertical line in Figure . |
67d92d8181d2151a02cf1d15 | 22 | to reactions R1 and R2 of ozone and tetramethylethene (Figure ) to yield the vdW flask and the ozonide, depicted as purple and a red vertical lines, respectively. Another important consideration is that association reactions suffer from an entropic penalty. On the one hand, the formation of the vdW flask, k QC-R1 O 3 |
67d92d8181d2151a02cf1d15 | 23 | , that is the reaction of vdW flask to form the ozonide compound (brown vertical line for k R2 O 3 in Figure ), the outcome is reversed. These pieces of evidence highlight the fundamental challenge of predicting accurate absolute reaction rate constants with quantum chemistry methods. It thus cannot be the goal of CRN explorations to provide and evaluate individual activation barriers for selected reactions in terms of bimolecular rate constants. |
67d92d8181d2151a02cf1d15 | 24 | Reaction Pathway Identification for the compound's availability for a reaction, 32 is highlighted in dark blue color, while all other reaction pathways are depicted in grey. In this reaction pathway identification, ozone with ethene and tetramethylethene, respectively, were defined as sources whereas the hydroxyalkylhydroperoxides were defined arbitrarily as targets which will be further decompose in water. The selection of the same type of target for both CRNs allows for a direct comparison of the reaction mechanisms with experimental evidence (i.e., for reactions leading to compound 5 in Figure ). |
67d92d8181d2151a02cf1d15 | 25 | Figure shows the most favorable reaction mechanism from ozone and ethene (source) to α-hydroxymethylhydroperoxide (target). The other 24 alternative mechanisms are depicted in grey because Pathfinder classifies them as unlikely due to their greater compound costs (see elementary steps of reaction paths in Section S2.6.1). Mechanism 1 depicted in Figure is spontaneous, with a total reaction Gibbs energy of -365 kJ•mol -1 . This reaction pathway starts with the barrierless formation of the vdW complex (Int1 in Figure ). Despite the fact that the Pathfinder mechanism is not the lowest in energy (there are other pathways in Figure with lower energy), it is indeed the shortest and the most favorable when considering both the activation barriers (i.e., kinetics) and initial concentrations of the reactants. Moreover, the alternative mechanisms show that there is a consistent trend in terms of total reaction Gibbs energy. For example, mechanism 25 (see Section S2.6.1) has a cost of 159.9 compared to 124.8 for mechanism 1. The difference stems from the fact that mechanism 25 includes an additional reaction step. For CRN-E, the formation of the α-hydroxymethylhydroperoxide is exergonic for all the reaction pathways shown in Figure . |
67d92d8181d2151a02cf1d15 | 26 | The most favorable reaction mechanism from ozone and tetramethylethene to 2-hydroxy-2-propylhydroperoxide is shown in Figure . The 24 alternative mechanisms are depicted in grey and exhibit greater compound costs. For instance, mechanism 25 illustrated in Section S2.6.2 has a cost of 288.5 as compared to 104.4 for mechanism 1. This large difference is not only caused by the extra reaction step of mechanism 25, but also by the formation of hydrogen which is not observed in experiments. Mechanism 1 starts with the barrierless formation of the vdW complex (Int1) and then undergoes a 1,3-dipolar cycloaddition through TS1 with an activation energy of 4 kJ•mol -1 to generate the ozonide (Int2). Note that the activation barrier for TS1 from tetramethylethene is 30 kJ•mol -1 lower than TS1 from ethene, in good agreement with the higher reactivity of tetramethylethene reported in experiments (see k exp CRN-E, it is not for CRN-T. This interpretation perfectly agrees with the experimental data, as α-hydroxymethylhydroperoxide is one of the main products whereas 2-hydroxy-2propylhydroperoxide is not. |
67d92d8181d2151a02cf1d15 | 27 | While the reaction pathway identification relies on an explicit definition of source and target compounds, microkinetic simulations are literally exploratory within the network of generated compounds, flasks, and transition states. Here, we employed KiNetX 33 to carry out microkinetic analyses for CRN-E and CRN-T and the outcome is depicted in Figure . |
67d92d8181d2151a02cf1d15 | 28 | Figure shows the reaction kinetics of compounds exceeding femto-molar concentrations in CRN-E that included all compounds and reactions included in Table . Typical intermediates of ethene ozonolysis (i.e., the ozonide, α-hydroxymethyl hydroperoxide, methylium peroxolate) are formed continuously until they decay as ozone is consumed completely (between 1 to 100 seconds). Formaldehyde and α-hydroxymethyl hydroperoxide are the final products of this reaction and they form in the correct equimolar yields. This accurate reproduction of experimental findings 37 was obtained even though reaction rate constants, such as those for the decay of ozone (k QC O 3 in Figure ), deviate orders of magnitude from measured rate constants. Our data thus confirms the error cancellation in transition state energy calculations invoked above. |
67d92d8181d2151a02cf1d15 | 29 | Note that we combined DLPNO-CCSD(T) electronic energies with LC-PBE entropy corrections in our microkinetic simulations to arrive at Gibbs energies. When reaction rates are exclusively calculated with LC-PBE Gibbs energies, the resulting kinetic simulations lead to the formation of the ozonide as predominant final product (Figures S11A and S12A for CRN-E and CRN-T, respectively). None of these calculations match experimental evidence. These observations reinforces results from our QC method benchmarking (Fig- |
67d92d8181d2151a02cf1d15 | 30 | ure ) which show that LC-PBE electronic energies do not ensure chemical accuracy in simulations of ozone chemistry. When reaction rate constants are calculated using DLPNO-CCSD(T) electronic energies only, the microkinetic simulations form the correct compounds but in flasks instead of compounds (Figures and). Electronic energies do not favor dissociation reactions of flasks due to the lack of the entropic contribution. |
67d92d8181d2151a02cf1d15 | 31 | Figure shows the kinetics of compounds in CRN-T from a microkinetic simulation carried out in analogous fashion than CRN-E (see Table for number of compounds and reactions and Table for simulation parameters). This microkinetic analysis is based on an exploration data with a confined set of identified compounds (see above). Even though the limited depth of CRN-T is a limitation to the network's evaluation, these results allow for critical insights into the interpretation of automatic CRN explorations. In CRN-T, tetramethylethene and ozone react to acetone as primary product through decomposition of the tetramethylethene-ozonide via heterolytic O-O bond cleavage in agreement with experiments. The fact that the hydroxy-alkyl hydroperoxide (2-hydroxypropyl-2-hydroperoxide) |
67d92d8181d2151a02cf1d15 | 32 | does not become the predominant product is also consistent with experiments. However, formation of two unexpected compounds, 3,3,6,6-tetramethyl-1,2,4,5-tetroxane, simply re-ferred to as tetroxane (Gibbs energy of -227 kJ•mol -1 , Figure ) and propenylperoxypropaneperoxol (-277 kJ•mol -1 , Figure ) vs. -218 kJ•mol -1 for the ozonide (Figure ) hint at limited extent of reaction progress or lacking depth of CRN-T exploration. We hypothesize that this outcome is also due to an artifact with regard to the role that water plays in the hydrolysis of the alkyliumperoxolate (i.e., the 2-peroxo propylate zwitterion in CRN-T) to hydrogen peroxide and acetone (3 in Figure ). |
67d92d8181d2151a02cf1d15 | 33 | As discussed in Section S2.11, the transition state leading to the formation of hydrogen peroxide from 2-peroxopropylate strongly depends on the position of the attacking water (Figure ). The need for aligning water to the hydrolysis reaction coordinate artificially increases the activation energy thus ultimately hindering the reaction. We note that this issue does not systematically affect reaction of water in general. For example, in the hydrolysis reaction with 2-peroxopropylate to form 2-hydroxy-2-propylhydroperoxide, water attacks from an optimal angle, leading to a smooth trajectory on the potential energy surface (Figure ). Our observations point to intrinsic limitations of the implicit solvation model (CPCM) to describe the local solvation effects. Ongoing developments for the implementation of updated solvation methodology into Chemoton are currently underway. |
67d92d8181d2151a02cf1d15 | 34 | Our work demonstrates how quantum chemistry-based CRN exploration offers novel avenues to elucidate reactions of ozone with organic compounds in aqueous solution. Experimental observations for ozonation of ethene were predicted correctly through reaction pathway identification and microkinetic modeling of major product yields once the suited computational methodology had been identified. Furthermore, insights from reactions of tetramethylethene with ozone revealed bottlenecks that need to be addressed in improved computational and data processing workflows for more expansive CRN exploration of chemical oxidation processes. Examples include the efficient combination of fast semiempirical electronic structure calculations with DFT and CC methods to arrive at reliable structures and minimum/TS energies of structurally more complex compounds than substituted ethenes, the management of the ensuing large datasets for evaluation and visualization, as well as the handling of water as both solvent and reactant. |
67d92d8181d2151a02cf1d15 | 35 | Once established, CRN evaluations of chemical oxidation processes can support the reevaluation of reactions of many organic compounds of concern in water and wastewater treatment processes, and the predictive assessment of chemical reactivity of chemicals in the product design phase. Most reactions of micropollutants with ozone are, in fact, understood incompletely. Even though sophisticated chemical analyses, such as non-target analysis by high resolution mass spectrometry generates numerous structural features of organic compounds, 87 such approaches are unable to provide a comprehensive view of the identity of the numerous intermediates and products (e.g., from drinking water ozonation processes ). |
67d92d8181d2151a02cf1d15 | 36 | Moreover, (eco)toxicity evaluations recommend circumventing the assignment of the observed toxicity to chemicals in polluted and treated waters because only a minor fraction of the observed toxicity can be allocated to individual compounds in mixtures of pollutants and their transformation products. Chemical reaction networks generated through high-throughput computational chemistry have great potential to address this knowledge gaps by providing atomistic evidence on transient chemical species and overlooked products. For example, intermediates reacting at diffusion controlled rates in ozonation processes typically require concentrations exceeding approximately 10 -12 M to be relevant during water treatment operations. 2 Such cut-off criteria would support the systematic search of compounds unaccounted so far in microkinetic modeling exercises. Moreover, the numerous compounds acquired in CRN databases might prove highly beneficial for prediction of high resolution mass spectra with deep learning-based approaches. Finally, predictions of potentially harmful or toxic compounds or compound (sub)structures from CRN explorations of oxidative water treatment could become pivotal for the evaluation of environmental impacts of chemicals in early design phases. |
6675ff7301103d79c5ccfdb9 | 0 | spectrometry platforms are utilized. Many of these workflows, however, have focused on individual synthetic RNAs or RNAs isolated and purified from biological sources with relatively little heterogeneity compared to RNAs in biological samples. In addition, the sample requirements for MS/MS, typically requiring concentrations in the range of hundreds of nanomolar, can be out-of-step with the amount of a single RNA that can be effectively isolated from a biological source. To begin addressing these challenges in mass spectrometry workflows, we have recently reported a nanoscale on-line desalting platform for nucleic acids, offering fast clean-up in a manner suitable for high-throughput analysis of RNAs, overcoming some of the obstacles to MS analysis. This foundational study focused on analyzing a single, isolated tRNA species with isoforms that could be readily resolved in the m/z domain. While tRNA can be readily extracted from cells, there are a significant number of possible species and isoforms. tRNAs are further diversified by an array of post-transcriptional modifications and non-canonical nitrogenous bases frequently incorporated into tRNAs, presenting a substantial degree of heterogeneity for analysis by MS. Furthermore, lacking an effective orthogonal chromatographic separation method prior to on-line desalting for tRNAs -owing to their high hydrophilicity and relative homogeneity in sequence length (70-80 nt)resolution of individual tRNA isoforms from a tRNA extract in the m/z domain is challenging. To extend the utility of MS/MS characterization of tRNA isoforms, we have developed an MS/MS strategy employing two fragmentation methodologies to provide information on potential precursor species present in an MS/MS scan and characterization of the fragments produced. |
6675ff7301103d79c5ccfdb9 | 1 | The most frequently employed activation method employed for MS/MS analysis of RNA is collisionally activated dissociation (CAD), which has been shown to routinely yield abundant terminal w-, c-, and y-type fragments and high sequence coverage for individual homogeneous RNAs. For heterogeneous populations of RNA where precursor ions overlapping in the m/z domain may be co-isolated and co-activated, CAD may result in chimeric spectra, confounding identification. Although modern fragment identification workflows can identify well-resolved fragments in heterogeneous spectra, confident characterization of each RNA, including sequences and modifications, can be difficult without exact knowledge of the identities of the precursor species that were initially isolated prior to MS/MS. Ultraviolet photodissociation (UVPD) can overcome this obstacle by providing additional information about the composition of the precursor ion population along with production of additional diagnostic fragment ions, complementing characterization by CAD. Upon UV photoactivation, negatively charged nucleic acids generally yield a series of high-abundance electron photodetachment (EPD) products. These EPD products are intact precursor ions that have lost one or more electrons and exhibit lower charge states than the original precursor, resulting in an array of charge-reduced precursors distributed across a higher m/z region. In an MS/MS experiment undertaken on a population of overlapping precursors of different charge states, the dispersed charge-reduced EPD product ions can be more easily identified by their nowresolved isotopic distributions which were previously overlapping. This process is analogous to charge-reduction workflows in proteomics, which enable analysis of heterogeneous precursor and fragment populations, often via ion-ion reactions in the gas phase. Extending this chargereduction methodology to tRNA analysis facilitates identification of intact, charge-reduced precursors in UVPD spectra and subsequent assignment of corresponding fragment ions in the CAD spectra of the same m/z isolation window. |
6675ff7301103d79c5ccfdb9 | 2 | In the present study, we applied UVPD and CAD to narrow 2 m/z windows across an MS1 spectrum of a desalted yeast tRNA extract in a gas-phase fractionation approach. The specific isolation windows were selected to cover the region of the MS1 spectrum with the most abundant precursor ions. Charge-reduced EPD product ions observed in the UVPD spectra were identified by their observed isotopic distributions compared to theoretical isotope patterns modelled based on a database of characterized yeast tRNA sequences. Once precursor ions were identified in a UVPD spectrum, their theoretical fragment ions were identified in a parallel CAD spectrum acquired for the same 2 m/z region in the MS1 spectrum. Additionally, to assess the confidence with which the identified tRNAs were characterized by MS/MS, we implemented a scoring calculation -analogous to the P-score commonly employed in top-down proteomics experiments -to provide a consistent metric by which top-down MS characterization can be assessed across each spectrum and tRNA isoform. |
6675ff7301103d79c5ccfdb9 | 3 | Yeast tRNA extract was acquired from Sigma Aldrich (Roche 10109495001) and desalted using piperidine-based mobile phases on a Dionex RSLCnano Ultimate 3000 as described previously. A short desalting column (5 cm by 100 µm ID fused silica with a homemade Kasil frit) was packed in-house using Waters XBridge Phenyl (3.5 µm, 130 Å) pH-resistant stationary phase. For each run, 0.5 µL of a 10 µM solution of yeast tRNA extract was injected and desalted with an isocratic gradient of 80% B (A: water, 10 mM piperidine; B: acetonitrile, 10 mM piperidine) at a flow rate of 1 µL/min. This nanoflow desalting approach resulted in a 10-minute run, enabling high-throughput analysis. |
6675ff7301103d79c5ccfdb9 | 4 | MS1 and MS/MS spectra were recorded using either an Orbitrap Lumos mass spectrometer (Thermo Scientific Instruments) outfitted with a 193 nm excimer laser, enabling UVPD, or an Orbitrap Eclipse mass spectrometer (Thermo Scientific Instruments) for CAD acquisition. All MS1 and MS/MS spectra were acquired at 240,000 resolution during the elution period from 3 min to 8 min. A spray voltage of 1600 V was applied to a fused silica nanoelectrospray emitter (New Objective, 50 µm ID, 30 µm tip) via a stainless-steel tee and platinum electrode. MS/MS scans were acquired by performing quadrupole isolation of 2 m/z wide segments of the MS1 spectrum, with 0.5 m/z overlap of adjacent windows, prior to UVPD or CAD. The specific 2 m/z isolation width was selected to restrict the number of precursors per window, thereby limiting the heterogeneity in the MS/MS spectra. The 0.5 m/z overlap with adjacent windows was selected to ensure effective isolation of precursors at the edge of each isolation window. Windows spanned the range from m/z 859 to m/z 886, where the most abundant tRNA isoforms were observed, for a total of 25 windows. For UVPD, isolated precursor populations were irradiated with a single 0.75 mJ laser pulse applied during a 2 ms period. All UVPD was performed in the high-pressure linear ion trap. CAD spectra were acquired on an Orbitrap Eclipse mass spectrometer utilizing a normalized collision energy of 25 or 30 and a q value of 0.14, resulting in 50 total CAD spectra (corresponding to each of the twenty-five 2 m/z isolation/activation windows). |
6675ff7301103d79c5ccfdb9 | 5 | To inform subsequent fragment ion searches of CAD spectra, charge-reduced precursors generated via EPD during UV photoactivation were identified in the MS/MS spectrum for each isolation window. tRNA sequences for S. cerevisisae were downloaded from the T-Psi-C database and converted via the MODOMICS nomenclature to a sequence format implemented previously for manipulation in R. A spreadsheet of these species can be found in the Supporting Information. |
6675ff7301103d79c5ccfdb9 | 6 | From this sequence database, theoretical precursor isotopic envelopes were generated using the IsoSpecR package. Based on the average molecular weight of each tRNA sequence, theoretical average m/z values were generated for various potential charge states of each tRNA sequence. Those predicted charge states that fell within one of the 2 m/z windows defined above were then searched in the corresponding UVPD spectra. From the expected charge determined from the theoretical average m/z, theoretical isotopic distributions of EPD charge-reduced precursor ions were generated for up to three charge losses -corresponding to detachment of up to 3 electronsand assigned in the UVPD spectrum. For example, a 23.9 kDa tRNA in the 27-charge state, [M-27H] Once precursor candidates were identified in this way, theoretical fragment ions were generated for the corresponding precursor charge state and searched the CAD spectra acquired for the same specific 2 m/z window in which the precursor was identified. All fragment ions were identified using Nucleo-SAFARI as described previously, utilizing a 10-ppm m/z tolerance and a 20% intensity tolerance. Fragment ions were validated based on their predicted isotopic abundances and observed centroided intensity values of each isotope. Scores were assigned for the characterization of each tRNA based on the identified fragments using a method analogous to the P-score implemented in many top-down proteomics workflows, with some minor differences. First, only sequential fragments from either terminus -e.g. a continuous series of a, b, c, or d ions or a continuous series of w, x, y, or z ions -were counted as tag sequences. Second, given the identification of fragments in the m/z domain, the likelihood of identifying a random residue at unit resolution was calculated based on the mean charge state of the tag sequence rather than only corresponding to the mean mass of each residue. Details on the scoring process in Nucleo-SAFARI are described elsewhere. P-scores were utilized to assess which of the tRNAs were identified most confidently and with the best characterization. A schematic of the full workflow is shown in Figure . A total of 376 searches were executed across the 25 isolation windows, a number derived from performing searches on 122 tRNA precursors, each for two CAD conditions (25 NCE and 30 NCE), and the fact that 35 precursors were identified in more than one isolation window and thus subjected to additional searches. |
6675ff7301103d79c5ccfdb9 | 7 | To date, there are 56 tRNA sequences derived from S. cerevisiae in the T-psi-C database, a collection of tRNA sequences and their modifications obtained from high throughput tRNA sequencing (see a histogram summarizing the mass distribution of the isoforms in Figure ). This set of sequences is used for generation of theoretical precursor and fragment ion isotopic distributions. The heterogeneity of the desalted tRNA extract is significant, as illustrated in the MS1 spectrum in Figure . There are multiple overlapping tRNA isoforms throughout the MS1 spectrum, as shown in Figure and with expanded detail in Figure for the m/z range spanning 849 to 887, making it challenging to assign compositions of precursors with confidence. A set of isolation windows, each 2 m/z wide, is overlaid on the MS1 spectrum in Figure , indicating the windows that are selected for isolation and subsequent UV photoactivation to promote charge reduction of the most abundant precursor ions between m/z 849-887. In this manner, precursor ion isotopic distributions corresponding to each of the 56 yeast tRNA sequences recorded in the T-Psi-C database were identified with or without possible 3' terminal adenosine loss, totaling 112 sequences. In total, 122 sets of theoretical fragment ions based on identified tRNA precursor ions and their corresponding charge states, some of which were identified in more than one charge state, were generated from this set of 112 sequences. Here, CAD (30 NCE) was applied to the precursor ions isolated between m/z 883.5-885.5, corresponding to the second isolation window from the right shown in Figure . This window contains the Asp GUC species identified in the UVPD spectrum of the same isolation window (Figure ). Fragment ions generated from the database sequence (without a 3' terminal adenosine) were identified directly in the m/z domain, resulting in the sequence coverage map shown in Figure . A sequence coverage of 65.8% was achieved for this tRNA, resulting in a significant P-score of 1.30e-122, indicative of high confidence in identification of the tRNA. Moreover, fragment ions -mainly c and y fragments -bracket many residues near the 5' and 3' termini, respectively, localizing several covalent modifications present in the predicted sequence. These modifications include number of co-isolated species in same charge state (26-) with substantial overlap in the m/z domain. Theoretical isotope distributions of potential precursor identifications for six tRNAs are overlaid on the array of charge-reduced precursors in Figure , resulting in six potential tRNA identities (accession tdbR00000362, 370, 443, 444, 454, and 555). Although not all match the observed isotopic abundances, all predicted isotopes match within a 10 ppm m/z tolerance of the centroided m/z values. Additional precursors at m/z values (e.g., m/z 906-907) in Figure were unmatched, which may be indicative of additional tRNA isoforms that are yet uncharacterized. |
6675ff7301103d79c5ccfdb9 | 8 | Several of the fragment ions are expanded in the insets to illustrate their isotope distributions, and these identified fragment ions are assigned and color-coded to match the tRNA identities. In the case of the c14 6-ion, the corresponding tRNAs, both being Thr IGU (tdbR00000443 and tdbR00000444), only differ by two nucleotides, and exhibit many fragments of identical chemical composition. |
6675ff7301103d79c5ccfdb9 | 9 | Likewise, specific fragment ions like the identified z66 5-and y63 5-ions display nearly identical isotopic distributions; however, the observed ion of m/z 762 is likely attributable to the Val IAC tRNA (tdbR00000464) rather than Tyr GPA tRNA (tdbR00000555) based on the more confident precursor identification in 4A, additional confident fragment assignments in 4B, and higher overall abundances of identified fragment ions in 4B compared to the alternative Tyr GPA tRNA (tdbR00000555). Figure displays a summary of fragment identification results from fragment ion searches undertaken for the six precursors. Sequence coverage maps and the corresponding P-scores for the identified fragments of each tRNA are displayed in Figures . A summary of sequence coverage versus calculated P-score is displayed in Figure for all 376 searches. Sequence coverages and P-scores for these 376 searches can be found in the Supporting Information. To assess the quality of top-down annotation for this highly heterogenous tRNA sample, decoy searches were carried out by performing a fragment ion search of each CAD spectrum for a tRNA that was not identified as a charge-reduced precursor in the corresponding UVPD spectrum of that same isolation window. The outcomes of these searches, displayed in red in Figure , allowed generation of a cutoff P-score, where only tRNAs exhibiting higher P-scores were considered accurate identifications based on their fragment ions. This strategy was more stringent than a traditional scrambled sequence search, owing to the highly heterogeneous MS/MS spectra, the high sequence homology across tRNAs -particularly at their 3' termini -and the potential for differential modification of tRNAs to influence the species present in a given spectrum, which may summarizing characterization of tRNAs across the 376 fragment ion searches undertaken. Decoy searches (red) were carried out on each CAD spectrum (50 total), and P-scores of experimental tRNA fragment identifications (blue) were compared to the P-score of the decoy (red). Validated spectra with more confident P-scores compared to the decoys are shown in bold blue; those with P-scores less confident than the decoys are gray. Of the 376 fragment ion searches, 174 exhibited higher Pscores than the corresponding decoy search of the same CAD spectrum, and the average sequence coverage among these is 46.1%. |
6675ff7301103d79c5ccfdb9 | 10 | Previous results for characterization of modified tRNAs by CAD revealed high-abundance w ions that appeared to be specific to a methylated guanine nucleobase present in the sequence of a yeast phenylalanine tRNA. Orthogonal sequencing approaches determined that this nucleobase was a 7-methylguanine, which contains a fixed positive charge, potentially enhancing this type of fragmentation. Previous MS-based studies also found facile base-loss of 7-methylguanine nucleobases upon collisional activation. Positional methylation isomers of adenine and cytidine, including 1-methyladenine and 3-methylcytidine, can likewise exhibit fixed positive charges, and thus might also be anticipated to enhance fragmentation. Of the tRNAs present in the T-Psi-C database, 26 are reported to contain at least one of these modifications (e.g., 7-methylguanine, 1-methyladenine, 3-methylcytidine). One of the most well-characterized of these tRNAs is Val IAC tRNA (tdbR00000464) noted as one of the six tRNAs in |
60c7418fbdbb892f24a3837f | 0 | The latter effect, which arises due to the coupling of the angular momentum of an electron and its spin, is of special importance. Although small in magnitude for molecules composed of light atoms, spin-orbit coupling (SOC) leads to the mixing of otherwise non-interacting states, e.g., singlets and triplets, and splits electronic degeneracies, e.g., between degenerate Π or ∆ states or between different M s components of spin multiplets. Consequently, SOCs open new reaction channels (via inter-system crossing, ISC) and change the spectroscopic behavior by lighting up dark states via intensity borrowing. SOCs lead to noticeable changes in the wave functions and electronic properties, and are particularly important in open-shell systems. |
60c7418fbdbb892f24a3837f | 1 | Spin-related properties, such as SOCs and spin-spin interactions, determine magnetic behavior of molecules, which is central for understanding spectroscopic signatures of unpaired electrons (e.g., EPR spectroscopy), their macroscopic magnetic properties (magnetozabilities), and magnetic relaxation times. The ability to model these properties computationally is a key prerequisite for the design of novel materials. For example, single-molecule magnets (SMMs), molecules with several unpaired electrons that have a high-spin ground state in the absence of an applied magnetic field , can be used as building blocks to create novel, light-weight, and tunable magnetic materials or as building blocks for quantum information storage and quantum computations . The key challenge in realizing the full potential of SMMs is the ability to tune and control their magnetic behavior. |
60c7418fbdbb892f24a3837f | 2 | In the context of SMMs uses in information storage, the key quantities needed for the parameterization of these Hamiltonians are energy levels corresponding to different spin states and properties such as zero-field splittings (ZFS), hyperfine couplings (HFS), g-tensors, and asymmetric Dzyaloshinksii-Moriya (DM) interactions. All these quantities depend on SOCs directly on indirectly. |
60c7418fbdbb892f24a3837f | 3 | For example, SOC significantly contributes to ZFS, which can be estimated perturbatively from the SOC between a selected state with other closely lying states . ZFS can be (approximately) decomposed into single-ion anisotropy and exchange anisotropy tensors, giving rise to effective Hamiltonians . These quantities ultimately determine the barrier for the reversal of magnetization and magnetic relaxation times. |
60c7418fbdbb892f24a3837f | 4 | Here we consider only such perturbative scheme in which SOCs are computed as ma-trix elements of the BP Hamiltonian using non-relativistic wave functions . Different implementations of full and approximate SOC treatments have been reported for different types of wave functions including those obtained by complete activespace self-consistent field (CASSCF) , restricted active space self-consistent field (RASSCF) , multi-reference configuration interaction (MRCI) , coupled-cluster (CC) response , equation-of-motion CC (EOM-CC) , and multi-reference CC (MRCC) within the Mk-MRCCSD formulation . DFT implementations have also been reported . |
60c7418fbdbb892f24a3837f | 5 | While the working expressions for calculating the matrix elements of the BP Hamiltonian between the pairs of non-relativistic states have been throughly described in previous works, one important aspect of the theory has not yet been fully developed. To construct experimentally relevant quantities, such as rates of ISC crossing, oscillator strengths, or magnetic anisotropies, one needs SOCs computed for all multiplet components. For example, the expression for the SOC constant (SOCC), the quantity that enters the Fermi golden rule expression of the ISC rate, is given by the following expression : |
60c7418fbdbb892f24a3837f | 6 | Our previously reported implementation of SOC calculations within the EOM-CCSD framework enables calculations between the selected target EOM states. In a typical non-relativistic EOM-CC calculation (and in many other excited-state calculations), one computes only one component of the target multiplet because other components are exactly degenerate. For example, traditionally only M s =0 components of triplet states are computed. This is obviously insufficient for computing the full SOCC matrix. One possible strategy is to explicitly compute two other components (M s =±1), which requires additional coding efforts and increases the cost of the calculations. In addition, a special care needs to be taken to synchronize the phases of all computed states . Alternatively, one can apply coordinate rotations (which is equivalent to changing the quantization axis) and extract the SOC matrix elements in the original coordinate frame from the matrix elements in the rotated frames. However, practical application of this approach is also complicated by the phase problem mentioned above: the phases of the wave functions computed in different coordinate frames are arbitrary. In addition, frame rotations may require disabling point-group symmetry, thus further increasing the cost of calculations. |
60c7418fbdbb892f24a3837f | 7 | Here we describe how to generate all required SOC matrix elements from a minimal amount of calculations. The theory is formulated in terms of reduced one-particle density matrices, such that it is ansatz-agnostic and can be applied to any electronic structure method that can furnish required transition densities. We illustrate this approach by using the EOM-CC method with single and double excitations (EOM-CCSD) as an example. |
60c7418fbdbb892f24a3837f | 8 | In essence, the approach is grounded in Wigner-Eckart's theorem that outlines the relationships between the matrix elements involving multiplet components. In the context of SOC, Wigner-Eckart's theorem has been exploited within configuration interaction formalism: for example, an algorithm for generating the entire SOC matrix from only three reduced matrix elements has been described by Fedorov . However, as explained below, Fedorov's approach requires access to several multiplet components (M s =0,±1), a feature which is not commonly available in excited-state codes, and is also affected by the phase issue. |
60c7418fbdbb892f24a3837f | 9 | A more general strategy based on the application of Wigner-Eckart's theorem to transition density has been outlined by McWeeny . Here we follow McWeeny's proposal and develop second-quantization formulation of this approach. The essential components of our formalism are very similar to the implementation of SOC calculations within the restricted active space state interaction (RAS-SI) framework in the Molcas program . In this approach, one needs to compute only one multiplet component (the M s =0 state) from which the reduced spinless density matrix is generated. This density matrix is used to compute a reduced matrix element and the entire SOC matrix is obtained by the application of Wigner-Eckart's theorem. The phase problem is avoided by construction, because the density matrix corresponding to a reduced matrix element is computed from only one transition between the pair of target states (or between the reference and the target EOM state). The theory is illustrated by application to the EOM-CCSD wave functions. As numerical exam-ples, we report calculations for a set of small open-shell molecules, considered as candidates for laser cooling experiments, a set of diradicals used as benchmark systems in previous studies , several transition-metal ions, and a Fe(II) SMM with a large spin-reversal barrier (magnetic anisotropy) for which a synthetic analogue has been synthesized . |
60c7418fbdbb892f24a3837f | 10 | The structure of the paper is as follows. In the next section, we outline essential features of the EOM-CCSD method and the calculation of the state and transition properties within EOM-CC. We then describe spin-tensor formalism and apply it to derive the key equations for calculation of the SOC matrix elements via reduced density matrices and Wigner-Eckart's theorem. We also discuss the implementation of the algorithm within the EOM-CC suite of methods in the Q-Chem electronic structure package . We discuss additional aspects of the theory in the case of open-shell references. Section III presents illustrative calculations. |
60c7418fbdbb892f24a3837f | 11 | where Φ 0 is a reference determinant, which defines the many-body vacuum state and, consequently, the occupied and virtual orbital spaces , e T |Φ 0 is a CC wave function, and R is a general excitation operator . Explicit form of R depends on the level of correlation included in the model (e.g., it generates up to doubly excited configurations at the EOM-CCSD level) and on the type of the target states. For example, in EOM-CCSD for excitation energies , R is spin-and particle-conserving, i.e., of 1h1p and 2h2p types at the CCSD level, with an additional constraint that in each operator the number of α holes equals the number of α particles and the number of β holes equals the number of β particles. In EOM-CCSD for ionized (EOM-IP-CCSD) or electron-attached (EOM-EA-CCSD) states, R s are not particle conserving. In EOM-IP-CCSD, R is of 1h and 2h1p types and in EOM-EA-CCSD, R is of 1p and 2p1h types. In the spin-flip variant (EOM-SF-CCSD), operators R change the spin-projection by flipping the spin of an electron. |
60c7418fbdbb892f24a3837f | 12 | The normalization and phases within the right and left sets are arbitrary, but the biorthogonality condition, Eq. ( ), fixes the relative norms and phases: i.e., if one changes the sign and norm of R K , the sign and norm of L K changes accordingly. In many implementations, the diagonalization is carried out independently for the right and left vectors and the biorthogonality condition, which synchronizes the phases, is applied a posteriori (in the case of degenerate eigenstates, one needs to rotate the degenerate eigenstates such that the overlap matrix between the left and right degenerate manifolds becomes a unit matrix). |
60c7418fbdbb892f24a3837f | 13 | Because of a non-Hermitian nature of EOM-CC, γ pq (I, I ) = γ qp (I , I), which leads to different numerical results for Ψ I |A|Ψ I and Ψ I |A|Ψ I A common resolution of this problem is to take a geometric average . However, this treatment should be modified for complex-valued tensor properties, in order to preserve phase consistency and to generalize it for a matrix. Below (section II C) we introduce a spinless density matrix, which enables generation of all phase-consistent SOCs between the selected pairs of states. We also propose a different resolution of the averaging problem. |
60c7418fbdbb892f24a3837f | 14 | However, one can consider properties of a spin operator Ŝ alone. It has three Cartesian components and transforms under rotations as a vector. Operators involving spin may obey other transformation rules. For example, the Ŝ2 operator, formed as a dot product of the two spin operators, is a scalar operator: it has only one component and it does not change upon the rotation of the coordinate frame. The spin-orbit operator of the BP Hamiltonian is |
60c7418fbdbb892f24a3837f | 15 | Representation theory tells us that the objects (wave functions or operators) transform according to some representations of the transformation group. These representations are irreducible if they cannot be decomposed into other representations. This leads to irreducible tensor operators, which are especially useful for second-quantized formulations. It is convenient to define irreducible spin-tensor operators ÔS,M through commutation relations : |
60c7418fbdbb892f24a3837f | 16 | Here S denotes the spin value and M denotes spin projection, which goes from -S, -S + 1, . . . , S -1, S. Any set of operators satisfying these relations are called irreducible spintensor operators. For example, the S = 0 case corresponds to singlet operators, which commute with S z , S + , and S -. Any spin-independent operator, such as a dipole moment operator, is a singlet operator. Spin-orbit operator depends on spin, and as shown below, it is composed from triplet spin-tensor operators. In particular, McWeeny writes down both angular momentum and spin in irreducible tensor form for spin-orbit operator in Eqs. (2.1) and (2.2) in Ref. (see also exercise 2.10 in ). |
60c7418fbdbb892f24a3837f | 17 | The one-electron part requires only one-electron transition density matrix for the final SOC matrix elements. The two-electron part requires the SO two-electron integrals and twoelectron transition densities, which makes the full calculation expensive. However, the twoelectron contribution can be effectively approximated by considering only the separable part of the two-electron transition density, such that the two-electron SO integrals are first contracted with the density of the reference determinant (usually Hartree-Fock determinant), and the resulting mean-field-like one-electron operator is then folded into the one-electron part. This spin-orbit mean-field approximation (SOMF) is commonly used in calculations of SOCs ; it is described in Section II D. |
60c7418fbdbb892f24a3837f | 18 | Here the matrix element between the bra state with spin S and spin projection M and the ket state with spin S and spin projection M is expressed through a Clebsh-Gordan coefficient S M ; SM |S M and a reduced matrix element S ||O S,• ||S . Here, the superscript S, • signifies the reduced matrix element, which does not depend on spin projection. |
60c7418fbdbb892f24a3837f | 19 | where I and I enumerate multiplets. This strategy has been used, for example, within the CI framework , where the full SO matrix between I and I spin manifolds was constructed from three (or two, if symmetry is used) reduced matrix elements. An application of this approach to EOM-CCSD would require calculation of target states of different spin projections (with consistent phases). |
60c7418fbdbb892f24a3837f | 20 | Being a reduced matrix element, u pq does not depend on the spin projections of the states or the excitation operator. To emphasize this property, we drop the corresponding index in excitation operator and replace it with a dot: T 1,• pq . If the transition between the target states (or between the reference and target states) is a spin-flip transition, the spin-tensor form of the one-particle transition density matrix is the transition density matrix γ up to a sign, see Eq. ( ) and . If the transition is spin-conserving, the transition density matrix always can be decomposed into the singlet and triplet components in any spin-adapted basis (i.e., atomic orbitals): |
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