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62b682efe84dd1d29902903d | 8 | To identify the crystal structures, high-power powder X-ray diffraction (HP-PXRD) data were obtained on Rigaku X-ray diffractometer (3 phase, 380 V, 18 kW) equipped with Cu Kα radiation (λ = 1.54 Å) in the 2θ range of 10-80°. The XRD samples were prepared by depositing a thick film of cathode powder on a glass substrate. To reveal the morphology and chemical composition of samples a JEM-2200FS (Cs corrected STEM) HRTEM coupled with an energydispersive X-ray spectrometer (EDX, Oxford-INCA) at an acceleration voltage of 200 kV was used. Samples for the HRTEM analysis were prepared following a conventional standard procedure by dispersing in ethanol and then sonicating for a while. The well-dispersed suspension was dropped on TEM Cu-grid and oven-dried to use for HRTEM analysis. |
62b682efe84dd1d29902903d | 9 | The electrochemical characterizations involved in our study were carried with a three-electrode system on VSP (BioLogic Science Instruments, Inc.) with a graphitic rod and a calibrated Hg/HgO (Figure ) as a counter and reference electrode, respectively. The nickel foam (NF) was engaged as a working electrode. Through a drop-casting method on NF, 1 mg cm -2 loading amount of the catalysts was achieved for all working electrodes. To ensure that experiments are conducted under equilibrium conditions of H2O/O2, the electrolyte (1M KOH) was continuously kept in flowing oxygen environments for at least 20 minutes. The linear sweep voltammograms test was performed at a scan rate of 2 mV s -1 with 95% iR compensation. Chronopotentiometry at fix current density of 10 mA cm -2 is used to evaluate the long term stability of the catalyst. The measured voltage values were converted to the reversible hydrogen electrode (R.H.E.) using the following equation:𝐸𝐸 𝑅𝑅𝑅𝑅𝑅𝑅 = 𝐸𝐸 Hg/HgO + 𝐸𝐸 Hg/HgO 0 + 0.059*pH. Tafel slopes were obtained from η = b log j + C; (where η, b, j, and C represent overpotential, Tafel slope, current density, and intercept, respectively). Finally, the overpotential value was obtained by η = E vs (R.H.E) -1.23. |
62b682efe84dd1d29902903d | 10 | The Vienna Ab initio Simulation Package (VASP) , which implements the projector augmented-wave approach to DFT with PBE GGA functionals is used for all FP calculations. Calculations are spin-polarized and a kinetic energy cutoff of 500 eV is applied. Due to the large size of unit cells, the Γ-centered k-Point grid of (1,2,3) is chosen. (U; J) values of (4:5; 0:6) for Ru and (4:7; 1:0) for Ni are utilized for PBE+U calculations. A huge number of potential energy calculations were needed to find the favorable doping positions and the minimum energy structures. A machine learning potential is built on-the-fly for accelerating the structure search with sparse Gaussian process potential (SGPP) algorithm as implemented in the AutoForce package. |
62b682efe84dd1d29902903d | 11 | After obtaining the unit-cell formula consistent with doping, we searched for the minimum energy structure by switching the relevant atomic positions. A variation of the genetic algorithm (GA) is used. The search starts with a few random parent structures. At each step, tens of children are generated randomly for each parent. Dozens of lowest energy structures among the union of parents and the new generation are kept as parents for the next generation. The search is stopped if the parents set did not change much for a few steps. Several independent searches are carried out. The search is accelerated with a SGPP machine learning model, which is built on-the-fly. If the ML model is estimated to be inadequate at every energy calculation step, exact DFT calculations are carried out and the model is updated. |
627a153a87d01fcf74dfc4ae | 0 | Polyhydroxyurethanes (PHUs) are phosgene and isocyanate-free polyurethanes (PUs) that are foreseen as a promising alternative to conventional polyurethanes (PUs), when European and North American directives will restrict the use of allergenic and carcinogenic isocyanates. They are usually obtained through the polyaddition reaction of diamines onto bis(5-membered cyclic carbonate)s (bis-5CC), resulting in the formation of urethane linkages and hydroxyl groups (OH) (scheme 1). The success of PHUs is largely related to the rapid development of sustainable procedures for the synthesis of 5CC precursors such as the CO2/epoxide coupling chemistry or the carbonatation of biosourced 1,2-diols. However, 5CC aminolysis comes with several drawbacks including (i) slow kinetics and extensive side reactions that hinder high polymerization degree and (ii) a lack of regio-and stereo-control resulting in poorly defined polymers. Many catalysts have already been proposed to accelerate the polymerization kinetics, including N-bases, metal salts and thiourea compounds. While they accelerate the reaction, they currently do not solve the regio-and stereo-regularity issues. Yet, regularity in the arrangement of polymer backbones neighboring units is of paramount importance, as it is critical to their bulk performances. It includes their thermomechanical properties but also, when it applies, their optoelectrical behavior or their biodegradability. Scheme 1 |
627a153a87d01fcf74dfc4ae | 1 | The lack of regiocontrol in the aminolysis of 5CC, i.e. the formation of mixtures of primary, (OH)I, and secondary, (OH)II, alcohols, is a general noted drawback for PHU synthesis . Authors have tentatively addressed the problem by designing suitable monomers. Among them, the group of Endo et al. have demonstrated that the selectivity in favor of (OH)II formation can be increased by increasing the electron-withdrawing ability of the α-or β-substituents of the 5CC. More recently, Kleij et al. have demonstrated that 5CC with bulky substituents can be ring-opened under organocatalytic control (e.g. TBD) with a regioisomer excess up to >99% (e.g. (OH)II). However, to the best of our knowledge, all these methods were tested with model molecules. The preservation of the regiocontrol during the polymerization of bis-5CC derived from these model molecules has not been confirmed yet. More importantly, there are no existing methods to control the degree of regioregularity for the polymerization of a given bis-5CC structure, and thus to explore the impact of the regioregularity onto the properties of the resulting PHUs. As for stereoregularity in PHUs synthesis, it is nearly unaddressed in the literature. This challenge is relegated to the background due to the prevalence of side reactions and low molar masses issues. Yet, the methine carbon of 5CC are chiral centers (Scheme 1) that are inserted in the polymeric chains derived from them. Since the chirality of bis-5CC precursors is usually not controlled, their polyaddition with diamines results in stereo-irregular PHUs. To our knowledge, only one study explored the influence of the stereochemistry of a bis-5CC onto its reactivity and the properties of the resulting PHUs. It was recently reported by Maggliozi et al., who used a procedure of enantioselective crystallization to isolate two crystal structures of diglycerol dicarbonate (DGDC, Scheme 1): the enantiopure meso-DGDC, (R,S), and the racemic mixture of (R,R) and (S,S) DGDC. These two crystals were polymerized in bulk with various diamines (110 °C). Interestingly, the characterizations of the resulting PHUs suggest a slight impact of the stereochemistry of DGDC onto the chain insertion efficiency and on the regioregularity of the chains. PHUs derived from meso-DGDC exhibit slightly higher molar masses, while PHUs obtained from the racemic mixture of (R,R) and (S,S) DGDC come with higher contents of (OH)I. Despite these intriguing results, the procedure to isolate the DGDC stereoisomers is tedious, preventing larger-scale synthesis and systematic studies. Herein, we propose to use enantiopure bis-5CC obtained from the direct carbonatation of sugarbased butadiene tetraols that are enantiopure themselves, namely (i) meso-erythritol and (ii) its (S,S) diastereoisomer, (L)-threitol (Scheme 1). Recently, Dannecker and Meier introduced a simple and sustainable method for the organocatalytic carbonatation of meso-erythritol in the presence of dimethyl carbonate used both as a reagent and a solvent [8]. The corresponding erythritol di(carbonate) (EDC) retains the stereochemistry of the starting tetraol and is obtained in very high yield (90%). We propose to apply the same procedure to (L)-threitol, to obtain a set of two optically pure stereoisomers of butadiene dicarbonates: EDC and its (S,S) diastereoisomer threitol di(carbonate) (TDC). The aminolysis of EDC and its use for the synthesis of PHUs is already well documented in the literature. In particular, several authors reported the straightforward aminolysis of EDC in very mild conditions, including room temperature. This remarkable feature was attributed to the strong mutual inductive effect that the vicinal 5CC exert on each other. However, to the best of our knowledge, there is no report regarding the quantitative measurement of the kinetics rate of the reaction. Schmidt et al. valorized the very high reactivity of EDC for the synthesis of high molecular weight PHUs in bulk, i.e. at high temperature (100 °C), but the reactions are so fast in these conditions, that it is virtually impossible to study their kinetics. Here, we perform a comprehensive study of the aminolysis reaction of EDC and TDC in solvent and at room temperature. The experimental results, supported by DFT calculations, indicate that the kinetics and the regio-orientation of the ring opening of the 5CC are very dependent on the stereochemistry of EDC and TDC respectively. Moreover, for both EDC and TDC, the kinetics rate constant of the aminolysis of the dicarbonate is two orders of magnitude larger than that of the mono-carbonate resulting from the first aminolysis reaction. We use this feature to develop a one-pot, two step polymerization procedure offering sequence-controlled PHUs. In the end, the comparative study of the two vicinal dicarbonates, e.g. EDC and TDC, provides a new family of PHUs with tunable sequence-and regio-regularity. The impact of the chain-regularity of the PHUs onto their thermal properties is investigated. In this work, erythritol di(carbonate) (EDC) was synthesized starting from (R,S) erythritol according to the procedure recently described by Dannecker and Meier (Figure , 84% yield). The same method was applied successfully to L-threitol, the (S,S) diastereoisomer of erythritol (93% yield). This is the first report of the synthesis of L-threitol di(carbonate) using this method. For sake of simplicity, it is called threitol di(carbonate) (TDC) in the rest of the paper. The structure of the two diastereoisomers was confirmed by NMR (Figure ). Moreover, monolithic crystals of EDC and TDC were analyzed by X-ray crystallography, indicating that the (R,S) and the (S,S) isomers are both monoclinic crystals. However, the space group of EDC is P21/n while TDC is associated to the P21 space group. Figure represents their tridimensional structure. The two carbonate cycles of EDC are located on either side of the butane skeleton, meaning that the carbonyl groups point in opposite direction. In the case of TDC, the two carbonate cycles are facing each other. The crystallographic data confirm that the stereochemistry of the dicarbonate is consistent with those of the starting tetraol. The melting temperature, Tm, of the two crystals were measured by Dynamic Scanning Calorimetry, DSC (see Figure ). The melting point of EDC, Tm = 169 °C, is higher than that of TDC's, Tm = 137 °C. These results confirm that the stereoisomery of bis(carbonate) impacts their thermal properties, as already noticed by Magliozzi et al. for the stereoisomers of DGDC. |
627a153a87d01fcf74dfc4ae | 2 | The aminolysis of EDC in solution was succinctly investigated by Goldstein et al. in 1971. They studied the reaction of EDC with butylamine (BA) at 50 °C in DMF (0.2 mol L -1 ), using NMR spectrometry. For [BA] = 2[EDC] (i.e. [NH2] = [CC] where CC = cyclic carbonate), they observed that the reaction follows a second order kinetics up to a conversion of ~ 47% for the global CC functions. Then the reaction stops. It was assumed that after the ring opening of one of the two CC of EDC, the remaining CC was unreactive. This assumption was based on the hypothesis of the mutual influence of the vicinal CC units that exert a strong inductive effect on each other. This effect fades away after the first ring opening, and the remaining CC is much less reactive. Their hypothesis was supported by theoretical calculations of the kinetics rate constant of the CC aminolysis, which was found to be in close agreement with experimental results. However, the structure of the product obtained after the first ring opening reaction was not elucidated and the second ring opening reaction was not studied at all. In the present study, EDC was first reacted with 1 eq. of hexylamine (HA) in DMSO ([EDC] = 1 mol L -1 and [NH2] = 1/2[CC]) at 25 °C. As shown in Figure , many products can be expected depending on the chemo-and the regio-selectivity of the reaction. Mono-(1a, 1a') and/or di-hydroxyurethanes (2a, 2a', 2a'') can be obtained depending on whether only one or two of the CC functions of EDC are involved in the reaction. Moreover, the mono-and dihydroxyurethanes can be made of primary, (OH)I, and/or secondary alcohols, (OH)II, depending on the regioselectivity of the ring opening additions. The reaction progress was monitored by 1 H NMR spectroscopy (see the Supporting Information for experimental details). Figure represents the NMR spectra of the crude at t = 0 and after t = 4h of reaction. The proton signals a and b of EDC decreases rapidly, while new signals g, h, e, f and c increases concomitantly. Further analysis of the crude obtained after 4h of reaction ( 13 C, COSY and HSQC NMR, see the Supporting Information, Figure to S5) indicate that these new signals are attributed to the protons of the mono-hydroxyurethanes enantiomers 1a. The integral indicate that EDC is quantitatively transformed into 1a. Thus, the reaction is 100% chemoselective, with only one of the two CC functions of EDC being ring-opened. This result is in accordance with the prediction of the seminal work of Goldstein et al. To our delight, the reaction is also regiospecific, i.e., the ring opening reaction result in the formation of (OH)II exclusively. Examples of such selectivity in the regiocontrol of the uncatalyzed aminolysis of CC are extremely rare. As far as we know, only two reports show complete regiocontrol in the aminolysis of 5-members CC. They were both reported by the group of Endo et al. for the aminolysis of 𝛼𝛼-trifuoromethyland 𝛽𝛽-chloro-substituted CC. The same group demonstrated that the selectivity in favor of (OH)II formation increased as the electron-withdrawing ability of the 𝛼𝛼or 𝛽𝛽-substituent increases. Based on these results, they stated that the direction of the ring-opening aminolysis of 5-membered CC can be controlled by the electronic effect of substituent introduced on the carbonate ring. Thus, the regiospecificity observed for the aminolysis of EDC is in accordance with the strong mutual inductive effect that the vicinal CC units exert on each other. |
627a153a87d01fcf74dfc4ae | 3 | Figure 3A represent the evolution of the conversion of EDC into 1a (red plot) as a function of time. The slope of the corresponding second-order plot (Figure , red plot) indicates that the second-order rate constant k1,EDC of the ring-opening reaction of EDC is equal to 0.75 L mol -1 s -1 . To our knowledge, very few kinetic rate constants have been measured at 25 °C for the aminolysis of 5-membered cyclic carbonate, because these ring-opening additions are usually very slow close to room temperature. Only 5-membered CC activated by very strong electron withdrawing substituent can display a significant aminolysis rate at room temperature. Our group measured a kinetics rate constant, kester, of about 0.42×10 -4 L mol -1 s -1 at 25 °C for the ring opening addition of HA onto a strongly activated CC with an ester as 𝛽𝛽-substituent (DMSO, 1 mol L -1 ). The kinetics rate constant of the first ring opening of EDC is much larger, k1,EDC ~ 10 4 ×kester, illustrating once again the strong mutual activation of the vicinal CC units. In order to study the subsequent ring-opening aminolysis of the in-situ generated monocarbonate 1a, an additional equivalent of HA was added to the reaction mixture. Again, the reaction progress was monitored by H NMR spectroscopy at 25 °C by monitoring the progress of the integral of signal y. The green plot of Figure indicates that the conversion of CC is much slower this time. The second-order rate constant k2 was calculated to be 0.064 L mol -1 s - 1 , i.e., k2,EDC ~ k1,EDC/10. The reaction was brought to completion by increasing the temperature to 60 °C for a total of 4 hours. The NMR analysis of the crude indicate that both the 2a and 2a' compounds are formed (Figure and Figure to S9). Indeed, 16% of the overall hydroxyl groups in the crude are (OH)I, indicating that 2a' was formed. The ratio of the two regioisomers can be calculated to be 2a:2a' = 68:32. Thus, the second ring opening addition is not regiospecific, however there is a significant orientation of the regiocontrol in favor of (OH)II formation. In their work dedicated to the polymerization of EDC with various diamines, Schmidt et al. measured (OH)I : (OH)II ~ 35 : 65 for PHUs synthetized in DMSO at 100°C, i.e. a regioselectivity oriented towards the formation of (OH)II as well. The ratio measured in our study, for model molecules and at 25 °C -60 °C, indicates a higher orientation of the regioselectivity towards the formation of (OH)II. This might suggest that the regioselectivity is dependent on the nature of the substrates and the reaction conditions. However, it is worth noting that the assignments of the NMR signals of erythritol PHUs are not consistent with ours and were not confirmed by COSY, HSQC and HMBC NMR analysis. |
627a153a87d01fcf74dfc4ae | 4 | The experimental results were compared to mechanistic DFT calculations. Preliminary calculations, presented in the Supporting Information (Figure ), indicate that the overall reaction proceeds according to a non-concerted mechanism, self-catalysed by the amine, in accordance with previous reports of the literature. Figure represents the mechanistic pathways for the first aminolysis of EDC, using ethylamine as a model of hexylamine for sake of simplicity. The initial step corresponds to the nucleophilic attack of the amine onto the carbonate group of CC accompanied by a proton transfer towards the catalytic amine to provide a cyclic amino alkoxide anion interacting with NH3 + by H-bonding, IS1A. It subsequently undergoes ring opening by C-O bond cleavage and a simultaneous proton transfer between NH3 + and the O atom of the carbonate to produce the targeted monohydroxyurethanes interacting by H-bonding with the catalytic amino function, 1a-EA and 1a'-EA, where EA stands for the catalytic ethylamine interacting with the two regioisomers at the end of the reaction. Clearly, 1a-EA, the (OH)II containing regioisomer, is much more stable than 1a'-EA, ΔE(1a'-EA-1a-EA) = 26 kJ mol , indicating that the formation of this regioisomer is thermodynamically favorable. Moreover, the energy difference between the intermediate species, IS1A, and the transition state, TS2, is much smaller in the case of the mechanistic pathway resulting in the formation of 1a-EA (ΔE = 18 kJ mol -1 ). Thus, the formation of 1a-EA is also kinetically favorable. Overall, the calculations are in accordance with the regiospecificity of the reaction observed experimentally, both under thermodynamic and kinetic control. The dihydroxyurethanes 2a and 2a' are obtained according to the same mechanistic pathway (Figure ). However, the energy difference between the two regioisomers is much smaller this time, ΔE(2a'-EA-2a-EA) = 5 kJ mol -1 , suggesting a moderated regioselectivity in favor of 2a formation under thermodynamic control. When considering the energy difference between the intermediate species, IS2A, and the transition state, TS4, the formation of 2a' is much more favorable (ΔE = 20 kJ mol -1 ). Experimentally, 2a:2a' = 68:32, indicating that the regioorientation of the reaction is in accordance with a thermodynamic control. The aminolysis of TDC was studied according to the same protocol than EDC. It was first reacted with 1 eq. of HA in DMSO at 25 °C ([TDC] = 1 mol L -1 and [NH2] = 1/2[CC]). The NMR analysis of the crude indicate that the reaction is chemoselective with only monohydroxyurethanes being formed (Figure and Figure to S14). Moreover, there is a strong regio-orientation in favor of 1b formation, the (OH)II containing regioisomer. However, contrarily to EDC, the reaction is not regiospecific, the regiosiomer 1b' is also formed to a small extent, with 1b : 1b' = 95 : 5. This result indicates a very mild impact of the stereoisomery of the vicinal dicarbonate onto the regioselectivity of the first aminolysis. The second-order rate constant k1,TDC of the ring-opening reaction of TDC is equal to 0.31 L mol -1 s -1 (Figure ). This value is high as compared to the values reported in the literature for the aminolysis of activated CC (cf. kester = 0.42×10 -4 L mol -1 s -1 at 25 °C). However, it is significantly smaller than k1,EDC (~ k1,TDC ×2) indicating an influence of the stereoisomery of the vicinal dicarbonate onto the first-aminolysis kinetics. Goldstein et al. already noticed the influence of stereoisomery onto the kinetics of the aminolysis of carbonated hexitol, i.e. vicinal tricarbonates (50 °C, DMF). The second-order rate constant of the first aminolysis varied according to the following order: isotactic > heterotactic > syndiotactic. They suggested that the evolution might be due to variation of steric factors resulting from different conformational isomerism of the vicinal tricarbonates. Based on this hypothesis, we tentatively used DFT calculation to predict the conformational analysis of EDC and TDC, i.e. the study of the relative energy between their different conformations. Figure represents the relative conformation energy diagram of both EDC and TDC in DMSO at 25°C, as a function of the torsion angle, 𝜃𝜃, between the half-planes 𝜋𝜋A and 𝜋𝜋B, where 𝜋𝜋A is defined by O1, C3 and C2 and 𝜋𝜋B is defined by O4, C2 and C3 (Figure ). The energy diagram of EDC exhibits a conventional profile with two local minima for 𝜃𝜃 = 65°and 295°, corresponding to gauche-conformers, and one absolute minimum, for 𝜃𝜃 = 180°, corresponding to the trans-conformer. The calculation of the population distribution according to a Boltzmann distribution, indicates that 92% of the molecules are in the trans conformation. In this conformation, the two carbonyl groups point in opposite direction and they are easily accessible for aminolysis. TDC exhibits three local energy minima as well, for 𝜃𝜃′ = 60°, 170° and 295°. However, the relative free energy of the gauche conformer associated to 𝜃𝜃′ = 295° is very close to those of the trans conformer (𝜃𝜃′ = 170°). The calculation of the population distribution indicates that 73% of the molecules are in the trans conformation and 27% in the stable gauche conformation for TDC (DMSO at 25°C). For both conformations, the positioning of the substituents results in higher steric interactions as compared to the trans conformer of EDC. The resulting steric bulk inhibits the nucleophilic attack of the carbonyl groups, which explains the difference between the kinetics rate constants of the aminolyses, k1,TDC and k1,EDC, as measured experimentally. Thus, the conformational analysis of vicinal di(carbonate)s illustrates well the impact of stereochemistry onto the reactivity of the CC functions. The subsequent ring-opening aminolysis of the in-situ generated mixture of 1b and 1b' (95:5), was studied by adding an additional equivalent of HA. Again, the 1 H NMR monitoring of the reaction provides the second-order rate constant, k2,TDC, of the second aminolysis reaction: k2,TDC = 0.064 L mol -1 s -1 (Figure ). Interestingly, k2,TDC = k2,EDC, indicating that contrarily to the first aminolysis reaction, the kinetics rate of the second aminolysis is not impacted by the stereochemistry of the vicinal di(carbonate). After completion of the second aminolysis reaction (4h, 60°C), the NMR analysis of the crude indicates that 64% of the overall hydroxyl groups are (OH)II, and 36% are (OH)I (see Figure to S19). In that case, the ratio of the regiosiomers can be estimated to be 2b:2b':2b'' = 30:65:5. Surprisingly, the regioselectivity of the second ring opening addition of TDC is inverted as compared to EDC. Indeed, calculations indicate that 80% of the hydroxyl groups formed during the second ring opening reaction of TDC are (OH)I, against 32% in the case of EDC. These results suggest that the regioselectivity is significantly impacted by the stereochemistry. Magliozzi et al. already noticed a similar effect for the aminolysis of the different enantiomeric forms of DGDC, but to a less extent. Indeed, PHUs derived from the racemic mixture of (R,R) and (S,S)-DGDC contain approximately 25% of (OH)I against 35% for PHUs derived from meso-DGDC. These results suggest that the conformational changes of the carbonates, from one stereoisomer to the other, impact not only the kinetics of their aminolysis, but also the orientation of the nucleophilic attack of the amine, and thus the regioisomery of the resulting hydroxyurethane. The mechanistic DFT calculation for the double ring opening aminolysis of TDC is presented in the Supplementary Information (Figure ). Similarly to EDC, the mechanistic pathway for the formation of the mono(hydroxyurethane)s in interaction with the catalytic ethylamine (EA), 1b-EA and 1b'-EA, indicates that the formation of 1b-EA, the (OH)II containing regioisomer, is much more favorable under thermodynamic control (ΔE(1b'-EA-1b-EA) = 32 kJ mol -1 ). The energy difference between the intermediary species, IS1b, and the transition state, TS2b, is essentially the same for the two regioisomers. Thus, the ratio 1b:1b' = 95:5 is in accordance with the thermodynamic control of the reaction suggested by DFT calculation. When considering the energetic pathway for the formation of the di(hydroxyurethane)s 2b and 2b' (the formation of 2b'' is not considered here), the energy difference between the two regioisomers is small, ΔE(2b-EA-2b'-EA) = 2 kJ mol -1 , suggesting only a slight regioselectivity in favor of 2b' formation, the regioisomer containing one (OH)II group and one (OH)I group. On the other hand, the energy difference between the intermediary species, IS2b, and the transition state, TS4b, indicate a slight regioselectivity in favor of 2b formation. Thus, the experimental ratio, 2b:2b':2b'' = 30:65:5, is in accordance with a thermodynamic control of the second aminolysis of TDC, despite the very small difference between the energy of the two regioisomers. By taking advantage of the remarkable difference of reactivity between the vicinal dicarbonates and their corresponding mono(hydroxyurethane)s in DMSO, PHUs synthesis was envisioned according to a one pot, two steps protocol. The principle, inspired from the results of the model reactions, is illustrated in Figure . It consists in, step 1, reacting 1 eq. of vicinal dicarbonate with 0.5 eq. of a diamine, Ai, at 25 °C, to obtain a bis(cyclic carbonate), BisCC-XAi, where X = E or T, with E standing for EDC and T for TDC, depending on the nature of the vicinal dicarbonate. Subsequently, in step 2, BisCC-XAi is reacted with an additional 0.5 eq. of diamine, at 60 °C, to initiate a polyaddition reaction resulting in the formation of the corresponding PHUs. In step 2, the diamine can be the same as step 1, Ai, or a new diamine, Aj. In the latter case, the corresponding PHU is expected to have an alternated structure, ⋯Ai-diHU-Aj-diHU⋯, where diHU stands for di(hydroxyurethane)s. It is worth noting that a similar strategy was used by Ousaka et al. for the synthesis of sequence-controlled PHUs derived from a petroleum-based spiro bis(six-membered cyclic carbonate). In their study, the diHU linkers were made of primary hydroxyl groups, (OH)I, solely. In our case, the vicinal dicarbonates (EDC vs TDC) offer the additional possibility to tune the structure of the alcohols ((OH)I vs (OH)II). In this study, four BisCC precursors were synthesized by reacting EDC and TDC with two diamines: an aliphatic diamine A1 = 1,6-Hexanediamine, and a short diether diamine A2 = 1,8diamino-3,6-dioxaoctane (Figure ). A typical synthesis proceeds by reacting 0.5 eq. of EDC solubilized in DMSO (1 mol L -1 ) with 0.5 eq. of A1 at 25 °C for 4h, to afford BisCC-EA1. BisCC-EA2, BisCC-TA1 and BisCC-TA2 were successfully synthesized according to the same protocol. In all cases, the 1 H NMR characterization of the crudes indicate that the reaction is brought to completion. The conversions of CC are reported in Table . Moreover, NMR analysis was used to measure the (OH)I : (OH)II ratios of the BisCC. For both BisCC-EA1 and BisCC-EA2, (OH)I : (OH)II = 0 : 100, in accordance with the model reaction of EDC with 0.5 eq. of HA. For BisCC-TA1, and BisCC-TA2, (OH)I : (OH)II ~ 10 : 90, again, in accordance with the model reaction of TDC with 0.5 eq. of HA. In order to validate the molecular structure of the BisCCs and to make sure that there is no oligomerization at this stage, the crudes were all analyzed by size exclusion chromatography (SEC). Figure represents the chromatograms of all the BisCCs. In all cases, a sharp and intense peak is observed for an elution time, te ~ 32 min. The analysis of the peaks (Polystyrene calibration) provides number average molar masses, Mn, that are in close agreement with the theoretical molar masses of the BisCC. They are all reported in Table . The corresponding dispersity, Ð, are very close from 1, confirming the formation of well-defined BisCC. It is worth noting that, on all chromatograms, a second peak is observed for an earlier elution time (te ~ 30.5 min). The corresponding Mn indicates the formation of BisCC dimers resulting from the reaction of two diamines with three vicinal dicarbonates. They represent only 5% of the total mass of the samples. This small fraction is tolerated in the next part of the work. It implies that a known amount of diHU-Ai-diHU-Ai-diHU sequence will be inserted in the polymer synthesized from these crude BisCCs. |
627a153a87d01fcf74dfc4ae | 5 | In the end, the difference of reactivity between the vicinal dicarbonates and their corresponding mono(hydroxyurethane)s in DMSO, enables the synthesis of well-defined BisCC, with good purity, and good control over the nature of the hydroxyl groups they contain. They can be used in step 2, as prepolymers for further polyaddition reaction with another diamine, Aj. |
627a153a87d01fcf74dfc4ae | 6 | Polyhydroxyurethanes (PHU) were synthesized by polyaddition of the BisCC obtained according to the procedure described previously. To do so, 0.5 eq. of amine Ai (where i = 1 or 2) were added to the in-situ generated BisCC-XAi (where X = E or T). Reactions were carried out at 80 °C for a period of 20 h. In all cases, the reaction is brought close to completion (conversions are reported in Table ). The resulting polymers contain a simple repeating unit, diHU-Ai. They are noted poly(XAi), where X stands for the vicinal dicarbonate used in step 1, and Ai stands for the amine used in step 1 and step 2. |
627a153a87d01fcf74dfc4ae | 7 | The crude polymers were fully characterized by 1 H NMR spectroscopy and size exclusion chromatography. NMR spectroscopy was used to calculate the final CC conversion, which was further used to estimate the molecular weight of the polymer according to the Carothers equation. The (OH)I : (OH)II ratio were also calculated based on the NMR spectrum. The number average molar masses, Mn, and the dispersity, Ð, of the polymers were estimated via SEC measurements. The glass transition temperature, Tg, of the polymers and the heat capacity difference at Tg, ∆Cp, were measured by dynamic scanning calorimetry (DSC). Finally, their degradation temperature, Td5% (temperature at 5wt% loss) were measured by thermogravimetric analysis (TGA). The results are listed in Table . The characteristics of poly(EA1) can be compared with the data reported by Schmidt et al. for the one-step synthesis of PHUs obtained by polyaddition of EDC in solvent. They reacted EDC, in DMSO (1 mol L -1 ), with 1 eq. of A1 at 100 °C (20 h). The 1 H NMR analysis indicate that MNMR = 14 800 g mol -1 and (OH)I : (OH)II = 34 : 66. Thus, MNMR is similar to the value measured for our polymer, poly(EA1). The (OH)I : (OH)II ratio is very different from ours and this might be due to the above-mentioned difference in the assignment of the NMR signals. The Tg of the polymer obtained via the one-step protocol of Schmidt et al. is also much smaller, Tg = 9 °C against 52 °C for poly(EA1). This might be due to the presence of residual DMSO in their polymer, or a consequence of significant structural discrepancies between our polymers due to the difference of our respective polymerization protocols. 60 for poly(TA1) and poly(TA2) respectively. Again, the regioselectivity is in close agreement with the results obtained previously for the model reaction of TDC with HA. It is not impacted by the chain insertion process during the polyaddition reaction, suggesting that it is possible to synthesize PHUs with tunable regioregularity simply by tuning the stereochemistry of the vicinal dicarbonate. Indeed, the content of primary hydroxyl group is significantly increased when using TDC, the (S,S) stereoisomer, instead of EDC, its (S,R) diastereoisomer. With 85% of secondary hydroxyl groups, against 60% for the PHUs derived from TDC, The PHU derived from EDC exhibits a higher degree of regioregularity. The DSC thermograms of poly(TA1) and poly(TA2) are plotted in Figure . When compared to the thermograms of poly(EA1) and poly(EA2), the most remarkable difference is that, the heat capacity difference at Tg, i.e. the difference of heat capacity between the glassy and the liquid state, ∆Cp, is much smaller in the case of poly(TA1) and poly(TA2). The regioregularity of the PHUs being directly connected to the stereochemistry of the vicinal dicarbonates, our result suggest that it is possible, to some extent, to tune the thermal properties of PHUs by controlling the stereochemistry of the dicarbonates. |
627a153a87d01fcf74dfc4ae | 8 | In our previous investigations of the impact of DGDC stereoisomery onto the microstructure of PHUs, only moderated variation of the Tg were observed between the PHUs obtained from the enantiopure meso-DGDC, (R,S), vs the racemic mixture of (R,R) and (S,S) DGDC. In this case, the variation of the hydroxyl ratio were of about 10%. For EDC vs TDC, we observe a two-fold increase of the magnitude of the variation (~20%), which might explain the larger variation of the thermal properties of the PHUs. |
627a153a87d01fcf74dfc4ae | 9 | Alternated PHUs were synthesized by reacting 0.5 eq. of amine Aj, where j = 1 or 2, with the in-situ generated BisCC-XAi , where X = E or T, i = 1 or 2 and i ≠ j. Reactions were carried out at 80 °C for a period of 20 h. In all cases, the reaction is brought close to completion (conversions are reported in Table ). The resulting polymers are expected to contain the repeating unit diHU-Ai-diHU-Aj (cf. Figure ). They are noted poly(XAi-alt-XAj), where X stands for the vicinal dicarbonate used in step 1, Ai the amine used in step 1 and Aj the amine used in step 2. |
627a153a87d01fcf74dfc4ae | 10 | For comparison purposes, random copolymers were also synthesized according to a single step protocol: 1 eq. of the vicinal dicarbonate (EDC or TDC) is mixed with 0.5 eq. of amine A1 and 0.5 eq. of amine A2, in DMSO. The reaction is carried out at 25 °C for 4h and 80 °C for 20h. The random copolymers are noted poly(XA1-ran-XA2), where X = E or T. The crude polymers were fully characterized according to the same methods ( 1 H NMR, SEC, DSC and TGA) and the resulting data are listed in Table . |
627a153a87d01fcf74dfc4ae | 11 | As expected, the copolymerization with a second diamine, Aj, different than Ai, offers the same control over the regioregularity of the resulting polymer. The inset of Figure represents a magnification of the signal n', associated to the methine proton of the primary hydroxyl groups, The SEC chromatograms of the alternated and the random polymers are represented in Figure . The corresponding number average molar masses, Mn, SEC, are reported in Table . Again MNMR ~ 5×Mn, SEC, reflecting the underestimation of the molar masses in SEC chromatography. |
627a153a87d01fcf74dfc4ae | 12 | The DSC thermograms are plotted in Figure . Tg = 24 °C and 27 °C for Poly(EA1-alt-EA2) and Poly(EA2-alt-EA1) respectively, indicating that the addition order of the two amines Ai (i = 1 or 2), does not impact the thermal properties of the polymer significantly. Moreover, Tg = 25 °C for Poly(EA1-ran-EA2), suggesting that the thermal properties of these PHUs are essentially dependent on the structure of the diamine residues and almost independent of the sequence regularity. This was also the case for the sequenced-controlled PHUs derived from the symmetric spiro bis(six-membered cyclic carbonate) of Ousaka and Endo. 34 The DSC thermograms of Poly(TA1-alt-TA2), Poly(TA2-alt-TA1), and Poly(TA1-ran-TA2) are plotted in Figure . Again, the heat capacity difference at Tg, ∆Cp, is much smaller than those of their EDC-analogs, suggesting an impact of the regioregularity on the intra-and the inter-molecular interactions of the polymers. Moreover, contrarily to their EDC-analogs, there is a clear influence of the addition order of the amines onto the Tg of the sequence-controlled polymers. Indeed, Tg = 10 °C and 20 °C for Poly(TA1-alt-TA2) and Poly(TA2-alt-TA1) respectively. This result suggests that the positioning of the amine surrogate as compared to the primary hydroxyl groups, (OH)I, essentially generated during step 2, has an impact on the Tg of the polymers. Figure recaps the nature of the neighboring (OH) groups of the Ai and the Aj segments respectively, as a function of the addition order of the diamines, for both EDC and TDC. For TDC, when Aj = A2 (step 2), the oxygen of the ether linkages of A2 can easily interact with the pendant (OH)I, via hydrogen bonding as depicted in Figure . Therefore, there is an increase of the number of intra-molecular interactions, at the expense of the intermolecular arrangements. This can lead to a decrease of the Tg. On the contrary, when Ai =A2 (step 1), the intra-molecular distance between (OH)I and the ether linkages increases. In this case, the closest (OH) neighbors of A2 are the hindered secondary hydroxyl groups generated during step 1, (OH)II, which are less accessible to develop hydrogen bonding with the ether linkages of A2. When EDC is used, instead of TDC, there are much less (OH)I generated during step 2. This might explain why the addition order of the amine has less impact on the Tg of the polymers in this case. In the end, this last result suggest that it is possible to tune the thermal properties of these sequenced-controlled PHUs by playing both with their sequence-and regioregularity. |
627a153a87d01fcf74dfc4ae | 13 | In conclusion, we have deeply explained the high reactivity of vicinal dicarbonates (EDC and TDC) and demonstrated that this reactivity is highly suitable for the synthesis of well-defined novel di(hydroxy-urethane) dicarbonates (BisCC) in soft experimental conditions (25°C, ambient atmosphere, 4h). The selectivity of the hydroxyl group obtained during the ring opening reaction of cyclic carbonate is dictated by the spatial configuration of EDC and TDC. BisCC2 with (OH)II-(OH)II groups are obtained from two diamines under the same experimental condition and with similar selectivity. This result opens the possibility to design a new platform of bio-based 5CCs. Moreover, macromolecular engineering allows the creation of regioregular PHUs, with tuneable and controllable properties, using sustainable products, such as sugar derivative tetraols. A broad range of properties can be obtained from erythritol and threitolbased PHUs, by a very simple selection of diamines and vicinal 5CCs. These simple way of synthesis of poly(hydroxy-urethane)s is promising in order to obtain green and functional polymers at low cost to respond to nowadays challenges. |
64a3a14c6e1c4c986bc566f3 | 0 | Contrast within images is fundamental to their qualitative and quantitative interpretations. Objects with similar illumination source interactions within a sample produce similar detected signal, becoming camouflaged like a chameleon on a leaf. Biological samples have limited intrinsic contrast in the aforementioned imaging modalities. Therefore, an extensive set of extrinsic contrast agents and genetically encoded tools such as Green Fluorescent Protein (GFP) that produce intrinsic contrast are now available. |
64a3a14c6e1c4c986bc566f3 | 1 | For instance, iodine based contrast is used extensively for contrast in X-ray (CT) imaging -either extrinsically intravenously injected or intrinsically (genetically encoded) cell-specific contrast with iodine transporters. Optical microscopy boasts a large palette of contrast tools, ranging from extrinsically applied dyes and stains (e.g., Gram, Eosin, Methylene Blue, Masson's Trichrome) to genetically encoded 'cloneable' contrast agents such a wide variety of fluorescent proteins. In electron microscopy of cells, where contrast depends on electron density (scaling with atomic number Z) intrinsic contrast is especially poor as the major elemental components (C, O, N, P) are very similar in electron density. Furthermore, all contrast tools in electron microscopy, both extrinsic metal stains and intrinsic cloneable approaches only function well in fixed cells. Lack of contrast tools represents a major limitation in imaging of cryogenically preserved frozenhydrated cells in EM. |
64a3a14c6e1c4c986bc566f3 | 2 | Correlative contrast agents -visible in multiple imaging modalities are limited. Semiconductor quantum dots are visible in X-ray CT, fluorescence, and electron microscopy, but tools to localize QDs to specific biomolecules, revealing their location in multiple length scales/resolutions are not stoichiometrically quantitative (e.g., QDantibody conjugates, VIPER. An overarching goal of bio-imaging is to integrate and understand biological processes from length scales ranging from meters to angstroms. Correlative contrast agents that are visible in X-ray, optical, and electron-based images do not yet exist. Such agents could facilitate correlative visual studies across all biologically relevant length scales. This arises both from the ability to correlate images from different modalities and also simply to locate in a meter-scale images the proteins of interest to image in mm and nm scale imaging modalities. |
64a3a14c6e1c4c986bc566f3 | 3 | We developed a cloneable inorganic nanoparticle (cNP) paradigm (Figure , panel C). This paradigm adapts in vitro reductive iNP synthesis to the in vivo context. With in vitro synthesis, soluble metal cations are reduced (e.g., with BH4 -) to insoluble precipitates in the presence of ligands. Ligands cap these precipitates at iNP size, stopping their growth. For in vivo (cNP) synthesis, the metal(loid) precursors are biocoordination complexes present inside cells. Reduction of precursors is by NAD(P)H dependent metal(loid) oxidoreductases. These enzymes transform cNP precursors into bulk phase precipitates, consuming NADPH. Bio-ligands present in the cellular milieu then cap and arrest the growth of metal(loid) precipitates at nano-sized dimensions. If the bio-ligands are peptides or proteins, they can be fused to the enzyme, which in-turn can fuse the cNP to the enzyme. |
64a3a14c6e1c4c986bc566f3 | 4 | We present here a cSeNP (Figure , panel D) as a contrast agent in electron, Xray, and visible light-based imaging modalities. The cSeNP is comprised of a dimeric selenodiglutatione (GS-Se-SG) reducing enzyme (Glutathione Reductase Like Metalloid Reductase, GRLMR) modified to be a single-chain construct with 2 SeNP binding peptides attached (Figure , panel D). Expression of GRLMR in cells grown on SeO3 2- supplemented media results in a striking-red color, attributable to the red allotrope of intracellular SeNPs (Figure , panel B). |
64a3a14c6e1c4c986bc566f3 | 5 | cSeNPs produce distinctive contrast in X-ray and electron imaging modalities, arising from the higher electron density of Se (Z= 34) relative the C, O, N, S, and P (6 < Z < 16) atoms comprising biological samples. SeNPs can react with transition metal cations, forming metal-selenide semiconductor quantum dots (Figure , panel E),[cite dispro paper] which enable distinctive fluorescence contrast. |
64a3a14c6e1c4c986bc566f3 | 6 | This cSeNP nanotechnology fills two gaps in the current bio-imaging contrast toolset. First, because it is visible in X-ray, optical and electron images, it can facilitate correlative imaging studies across all biological length scales. Second, it is a cloneable contrast agent visible in electron microscopy that is produced inside live cells. |
64a3a14c6e1c4c986bc566f3 | 7 | Metallothionein was investigated by multiple groups. We showed that metallothionein can localize high-copy, condensed proteins in cases in fixed cells (not cryo). 14 Ferritin (Ferritag) functions in some cases, but the 0.5 Megadalton size limits use cases. APEX and miniSOG represent non-particulate approaches to cloneable TEM contrast that result in diffuse staining and function only in fixed cells. The VIPER approach delivers exogenous iNPs to proteins of interest. It is demonstrated in fixed cells. Live cell use (required for ECT) may be complicated by cellular transfection, and stoichiometry uncertainties in labeling. (E.g., background with more iNPs than proteins of interest or incomplete representation of proteins of interest when fewer iNPs are present.) The cNP approach represents an improvement because it: (1) functions in vivo; (2) Creates stoichiometrically labeled proteins of interest; (3) Is significantly more compact than ferritin and may be engineered to be significantly smaller than the current construct. |
64a3a14c6e1c4c986bc566f3 | 8 | To assess the utility of the cSeNP for imparting contrast to proteins in cellular electron microscopy, we genetically fused cSeNP DNA to DNA encoding the 'filamentous temperature sensitive protein Z (FtsZ), a bacterial tubulin homologue. FtsZ, along with at least 13 other division proteins, aids in cleaving a parent cell into daughter cells during cell division. FtsZ forms protofilaments observed in two forms. In non-dividing cells, FtsZ is found in membrane associated dispersed helical protofilaments, comprised of 30 -80 FtsZ protomers. During cell division, FtsZ protofilaments localize to the midpoint (cleavage furrow) of the dividing cell, forming a so-called Z ring 95,99 (Figure , panel A). Protofilaments on average lie 16 nm from the cellular inner membrane. Protofilaments can extend to lengths of 100 nm. We hypothesized that expression of cSeNP-FtsZ in E. coli would result in SeNP/filament conjugates, with SeNPs organized by FtsZ at different points in the cell cycle as shown in Figure , panel B. We expressed a c-terminal FtsZ-cSeNP fusion protein in E. coli BL21 cells. The population of FtsZ protomers inside the cell, therefore, is comprised of both genomic FtsZ and recombinant plasmid-expressed FtsZ-cSeNP. FtsZ overexpression interferes with completion of cell division. This results in elongated cells extending up to 20 -30 µm in length. A normal cell is ca. 3 -5 µm). In such extended cells observations of helical protofilaments forming wave-like patterns along the membrane are known. Technical details of cSeNP-FtsZ DNA construction and expression are found in methods (M1) Substantial optimization work (described in SI-cSe_NP_Expression_Optimization) was performed with the highest-throughput method of cellular preservation for EM: drop casting of glutaraldehyde-fixed cells on TEM grids (Methods, M2). In these experiments, we identified conditions where FtsZ-cSeNP resulted in Se and SeNP distributions consistent with expected FtsZ locations (Figure , panels A and B). Figure , panel C shows cells grown in 2 mM SeO32-expressing the cSeNP-FtsZ construct under 100 uM IPTG; Figure , panel D shows cells prepared the same way except without SeO3 2-. In panel C, when SeO3 2-is present, contrast increases at the cleavage furrow, where FtsZ-cSeNP concentration is expected to be high. In addition, putative longitudinal FtsZ-cSeNP filaments are observed, consistent with prior observations of FtsZ overexpression by others. Figure , panel D shows control cells grown without SeO32-. Contrast appears uniform across these cells. Figure , panel E shows elemental mapping of Se in a dividing cell collected by scanning transmission electron microscopy-electron X-ray dispersive spectroscopy (STEM-EDS), confirming that the densities at the cleavage furrow are Se-rich. |
64a3a14c6e1c4c986bc566f3 | 9 | We acquired electron tomograms of plastic sections of E. coli BL21 cells expressing FtsZ-cSeNP, expecting to observe individual cSeNPs organized by FtsZ filaments. See M3 for electron tomography collection methods. A complication we encountered is that the solvents used in typical plastic embedding of cells for EM, such as acetone, dissolve SeNPs (see solvent-screening SI). (Figure SeNP-dissolve, SI). We implemented SeNP crystallization by adding Cu(OAc)2 to particle rich cells as a postfixation step to stabilize the NPs. The Cu 2+ ions react with cSeNPs to form CuSe metal selenides (Figure , panel E). [cite preprint] The resulting CuSe nanoparticles resist solvent dissolution. |
64a3a14c6e1c4c986bc566f3 | 10 | Additionally, we omitted metal stains typically added to plastic sections that react with specific functional groups in biomolecules. For example, osmium tetroxide stains membranes, uranyl acetate stains proteins and nucleic acid, and Reynolds lead is used to enhance contrast from other metal stains. These metal stains were omitted because they may form small granules or beam-induced punctate densities that obscure cSeNP contrast and assignment. The resulting tomograms are of lower than typical contrast. |
64a3a14c6e1c4c986bc566f3 | 11 | Figure shows electron tomography of experimental and control cells. Panel A shows 75 of 340 slices through an E. coli cell expressing FtsZ-cSeNP, grown in 2 mM SeO3 2-. Two features are readily apparent: (1) The outer membrane (OM) and portions of the inner membrane (IM); (2) Punctate densities interpreted as CuSeNPs arising from cSeNPs (blue arrows mark 3 of the 22 punctate densities in this image). The cellular preparation process made the inner-membrane difficult to discern; Membranes also distort during fixation.[cite] Panel B shows a magnified view of the boxed area panel A, with the OM manually segmented in blue. The CuSeNPs (cSeNP) particles have sufficient contrast for automatic segmentation. Panel C shows an isosurface rendering of the boxed section of panel A, except rendering the full X-slice tomogram. The isosurface rendering gives a pink color to all voxels with intensities below a threshold level. Importantly, automatic segmentation of CuSeNPs removes the human bias in their identification. Automatic segmentation CuSeNPs in whole cells is also accomplished with the BeadFinder program of the IMOD software package. Panel D of Figure shows manual membrane and automatic CuSeNP segmentation for the entire 250 nm thick tomogram. Panel E shows a magnified image of the boxed area of Panel D. |
64a3a14c6e1c4c986bc566f3 | 12 | Panels F, G, and H show images of a control cell, grown in the absence of SeO3 2- , but otherwise treated identically to the cell in panel A (including Cu 2+ exposure). There are no obvious punctate densities in this tomogram. Automatic segmentation does not identify voxels of intensity comparable to those in the experimental cell. An additional control, of cells expressing the Se-reducing GRLMR enzyme and grown in SeO3 2- supplemented media was published previously. In that experiment, SeNPs with diameters ranging between 5 and 60 nm were distributed in cells without any apparent organization. FtsZ localizes ~16 nm from the inner membrane,[cite] which is ~30 nm from the reliably segmented OM. Therefore, CuSeNPs that observed about ~45nm of the outer membrane are plausibly associated with FtsZ protofilaments. Figure , panel I shows a histogram of CuSeNP -membrane distances for the cell in panel A. 94% of cSeNPs ( 98 of 104 total) are within 45 nm of the outer membrane. Of those, 37% are closer than 16 nm to the outer membrane. We attribute particles closer than 16nm to the outer membrane to membrane fixation artefacts. Therefore, the vast majority of observed CuSeNPs lie in plausible FtsZ associated locations. |
64a3a14c6e1c4c986bc566f3 | 13 | Figure , panels J through Q, show a variety of renderings of a tomogram of a cell nearing completion of division. Panels K and L show renderings where positive contrast putative FtsZ filaments appear. Panel M shows a rendering of FtsZ filaments (yellow) decorated with cSeNPs that we could pick out. |
64a3a14c6e1c4c986bc566f3 | 14 | As outlined above, cSeNPs can react with transition metals such as Zn, Cd, Ag, and Cu to form highly fluorescent metal selenide nanoparticles (e.g., Quantum Dots). We assessed the cSeNPs converted to fluorescent metal selenides in live cells, followed by their imaging in both fluorescence and electron microscopy. E. coli expressing FtsZ-cSeNP were grown with 10 mM SeO3 2-supplementation for 1 hour. The cells were collected by centrifugation, washed and resuspended with Luria Broth (LB), and allowed to recover for 20 minutes. Zinc Acetate was added to the liquid culture to a concentration of 1mM, and cells were grown for an additional 2 hours. Cells were then collected by centrifugation, fixed with glutaraldehyde, and imaged. The higher magnification electron microscope image in panel F shows E. coli cells with higher contrast densities at the midpoint of 3 dividing cells and at the poles of 3 other cells. These densities correspond to known distributions of FtsZ, and we attribute these densities to SeNPs co-localized to FtsZ. The fluorescence images of panel E show a relatively uniform fluorescence background across the entire cells, with bright emission at the midpoints of 2 of the dividing cells and at the pole of one of the other cells. The overlay image of panel F confirms that all the high florescence regions correspond uniformly to the locations of SeNP density observed in electron microscopy. The control electron image shows cells with a few regions of slightly higher contrast, which we attribute to an excess of protein or inclusion bodies in those locations as an artifact of overexpression. |
64a3a14c6e1c4c986bc566f3 | 15 | X-ray imaging modalities such as Computed Tomography (CT) provide images at length scales of meters with penetration deep into tissues and organisms. Contrast in CT images is measured in terms of X-ray attenuation; the unit for describing X-ray attenuation is the Hounsfield Unit (HU), which provides a quantitative scale for describing radiodensity. The HU scale is defined by the radiodensity of distilled water (0 HU), air (-1000 HU). For biological tissues, typical HU values range from -700 (lungs) to +1900 (bone), with most soft tissues (fat, muscle, organs) having HU values between -120 and 60. Inorganic materials attenuate X-rays much more effectively than biological tissues, with pure copper attenuating at 14,000 HU and gold representing 30,000 HU (an upper measurable limit). For inorganic nanoparticles, X-ray attenuation depends on particle concentration. |
64a3a14c6e1c4c986bc566f3 | 16 | X-ray tomograms of E. coli cells expressing the cSeNP or GRLMR in grown in varying concentrations of SeO3 2-were acquired in a Scanco micro-CT 80 instrument. CT data is presented in Figure . Panel A shows that cells expressing the cSeNP in the indicated concentrations of SeO3 2-attenuate X-rays with an efficiency that depends on the growth condition. Attenuation of 700HU is observed for cells grown in 1mM SeO3 2-. This is significantly greater attenuation than observed for soft tissues, but less than bone. Notably, cells grown in the same concentrations of SeO3 2-but not expressing the cSeNP never attenuate at more than 100 HU. Panel B shows a voxel-parsed trace of X-ray attenuation. Here we observe that cells expressing the cSeNP frown in 2mM SeO3 2-show voxels with minor attenuation peak at 190HU and a major attenuation peak at 1100HU. Corresponding controls (not with in Se) show only an attenuation peak near 100HU, comparable to soft tissue. Panel C shows the attenuation of cSeNP expressing cells grown in concentrations of SeO3 2-ranging from 1μΜ to 10mM. Similar to panel A, attenuation increases with the concentration of SeO3 2-in which the cells were grown. Panel D shows a 9-position microcentrifuge-tube holder. The contents of each position are described in the figure legend. More white in these images corresponds to Se absorption. Panels E and F show tomograms of the samples in tubes 5 and 6. They are colored according to attenuation, with red representing attenuation near 1100HU and blue representing attenuation near 100HU. More details on this rendering are available in the SI. Overall, these tomograms show that cells expressing the cSeNP show clearly distinguishable X-ray contrast relative to cells grown in the same concentrations of SeO3 2- that do not express the cSeNP. |
64a3a14c6e1c4c986bc566f3 | 17 | Expression of the cSeNP in an E. coli model system produces molecular contrast in electron and fluoresence microscopies, as well as cellular contrast in X-ray imaging (where resolutions are not molecular.) This paper demonstrates a 'proof of concept' using the E. coli, a workhorse laboratory model-organism. What is true in E. coli, often translates to more complex model and experimental systems. We are presently working to extend these findings to more complex model systems such as Drosophila. |
60c75882bb8c1abb9c3dca45 | 0 | The human homologue of drosophila tailless gene tll, TLX (NR2E1), is a member of the nuclear receptor (NR) family which act as ligand-dependent transcriptional regulators. In adults, expression of TLX is strongly limited to adult neural stem cells (NSCs) residing in the subventricular zone and dentate gyrus of the hippocampus as well as in retinal progenitor cells . |
60c75882bb8c1abb9c3dca45 | 1 | According to rodent models, TLX is required to maintain NSCs in an undifferentiated proliferating state and its mutations cause disruption of neurogenesis in NSCs . In mice and drosophila, TLX knockout results in severe aggressiveness, abnormal brain development and retinal dystrophies . Moreover, TLX appears to play a crucial role in spatial learning and cognitive functions during adolescence and adulthood . Therefore, dysregulation of TLX has been associated with mental illness including bipolar disorders and schizophrenia . Beyond these roles in neurological homeostasis and brain function, recent reports have suggested a potential tumorigenic activity of the orphan NR due to marked overexpression in glioblastoma and neuroblastoma cell lines . These lines of evidence suggest TLX as an attractive target for neurodegenerative diseases and malignant brain tumors. Endogenous TLX ligands, forming an essential part of the NR function, remain elusive and only few synthetic TLX modulators with limited potency have been described to date . In light of its obvious therapeutic potential, further evaluation of the receptor's function and the discovery of potent TLX ligands are imperative. |
60c75882bb8c1abb9c3dca45 | 2 | Additionally, the structurally related and recently approved anti-Parkinson drug istradefylline emerged as a potent TLX modulator with low nanomolar potency. Mutagenesis studies defined a molecular epitope of TLX modulation by this class of TLX ligands to a region inside the ligand binding domain (LBD) involving interaction with helix 5. We observed also heterodimerization of TLX with retinoid X receptor (RXR) as well as recruitment of the nuclear receptor co-repressor 2 (SMRT). We demonstrated that istradefylline robustly displaced this repressor from TLX and modulated dimerization aligning with its effects on cellular TLX activity. This observation also suggests unprecedentedly a role of TLX as a direct repressor for other nuclear receptors. Overall, our results provide potent TLX modulators to study the receptor's role in health and disease and as a chemical starting point to facilitate future TLX ligand discovery. |
60c75882bb8c1abb9c3dca45 | 3 | As an essential basis to identify and characterize TLX ligands we established a hybrid reporter gene assay to capture TLX activity in a bidirectional fashion by combining Gal4-TLX with the potent transcriptional inducer Gal4-VP16 (see Supporting Information and Figure ). This artificial system reflected the repressor activity of TLX and simultaneously enabled potent control experiments to confirm TLX modulation. We also demonstrated that this VP16/TLX assay was representative of cellular TLX function. When we co-transfected Gal4-TLX with constitutively active NRs (Nurr1 or RORα resembling VP16 as ligand-independent transcriptional inducer) or NRs with low intrinsic activity in Gal4 format, a dose-dependent repression of reporter activity by TLX could be observed (Figure ) corroborating the VP16/TLX assay setting as suitable to discover and profile small molecule modulators of TLX activity. These cellular experiments indicated TLX-mediated effects on the activity of a variety of NRs. To confirm this observation in a cell-free setting, and to determine whether direct interactions are involved, we probed dimerization of TLX with RXR by titrating Tb 3+ -cryptate-labeled TLX LBD with GFP-labeled RXRα LBD in a homogenous time-resolved fluorescence resonance energy transfer (HTRF) assay. We detected strong RXR-TLX heterodimerization with an EC50 value of 120 nM (Figure ) suggesting an unprecedented direct repressor activity of TLX on nuclear receptors and further validating the VP16/TLX assay setting as highly suitable to study TLX modulation by small molecules. |
60c75882bb8c1abb9c3dca45 | 4 | Using the VP16/TLX reporter gene assay, we then screened a drug fragment library comprising 480 small organic molecules (MW range 80-300 g/mol) for TLX modulatory activity and discovered 1-methylxanthine (1) as TLX modulator that diminished TLX-mediated transcriptional repression with an IC50 value of 9±3 µM (Table ) suggesting inverse agonism. Intrigued by this activity, we studied the structurally related xanthines 2-8 for TLX modulation. 2-4 and 7 were inactive on TLX whereas theophylline (5), paraxanthine (6) and caffeine (8, Figure ) counteracted TLX-dependent transcriptional repression as well. Control experiments in absence of the Gal4-TLX hybrid receptor revealed no unspecific effects of the xanthines confirming their TLX-mediated activity (Figure ). |
60c75882bb8c1abb9c3dca45 | 5 | With caffeine (8) as a first TLX modulator tool compound in hand, we studied its effect on TLX repressor activity on NRs to probe its response to ligands. When Gal4-TLX and a human NR in Gal4 format were co-transfected, caffeine (8, 30 µM) reversed the repressor activity of TLX (Figure , Figure ). Such activity was observed for constitutively active NRs (e.g. Nurr1) and for NRs with low intrinsic activity (e.g. RARα, FXR and RXRα). Of note, caffeine (8) selectively modulated TLX amongst NRs (Figure ) except a slight additive activity with the reference agonist T0901317 on LXRs further demonstrating that its effects were TLX mediated. In an attempt to define key residues involved in TLX modulation by caffeine (8) as basis for future TLX ligand discovery, we performed a preliminary mutagenesis study. From the only available TLX X-ray apo-structure 4XAJ we hypothesized a potential binding region for 8 in the cavity inside the TLX LBD protein clamped between residues A189, F226, I230 and L268 (Figure ). |
60c75882bb8c1abb9c3dca45 | 6 | TLXA189E retained the repressor activity of wt-TLX while TLXL268R turned out as almost inactive concerning repression of VP16-induced reporter expression (Figure ). prompting us to study the potential of expanding the 8-phenyl residue with substituents (Table ). Eventually, we combined the 1,3-diethyl substitution pattern of the most active TLX ligand 30 with the favored 8-(furan-3-yl)theophylline motif (29) in compound 33 which exhibited equal potency and slightly higher efficacy compared to 30. Further extension of the alkyl substituents to 1,3dipropyl in 34 enhanced potency but was accompanied by a loss in efficacy. In summary, our preliminary optimization of the xanthine scaffold for TLX modulation succeeded by modification in 8-position and by altering the alkylation pattern. In 8-position, small heterocyclic residues were favored with the 3-furyl substituent as preferred motif and potency tended to increase with larger alkyl substituents in 1-and 3-positions. To elucidate the mechanism by which ligands modulate TLX activity, we probed the response of the TLX LBD on ligand binding in various homogenous time-resolved fluorescence resonance energy transfer (HTRF)-based settings using Tb 3+ -cryptate or sGFP-labeled nuclear receptor ligand binding domains and Tb 3+ -cryptate or fluorescein-labeled co-regulator peptides. An association of the TLX LBD with atrophin has been described previously prompting us to determine ligand effects on the affinity between TLX and the Atro box peptide . The TLX LBD robustly and specifically recruited the Atro box peptide , an interaction incompetent mutant was not bound (Figure ). |
60c75882bb8c1abb9c3dca45 | 7 | Thus, we screened twenty-nine canonical co-regulator interaction motifs for recruitment to TLX in apo-state or in presence of varying concentrations of caffeine (8, Figure ). High HTRF indicated potential binding of nuclear receptor co-repressor 1 (NCoR1) and silencing mediator for retinoid or thyroid-hormone receptors (SMRT, also termed NCoR2) to the TLX LBD. Titration of NCoR1 |
60c75882bb8c1abb9c3dca45 | 8 | and SMRT with the TLX LBD confirmed this assumption indicating similar or even higher affinity compared to the Atro box peptide (Figure and). The TLX-NCoR1 interaction showed no response to TLX modulators 8, 29, 30 or ccrp2 (Figure ), however, the interaction of TLX with the repressor SMRT was markedly decreased by istradefylline (30, Figure ) aligning with the inverse TLX agonism we observed in the VP16/TLX assay. These results suggest that TLX modulation by xanthines is at least in part mediated through the TLX-SMRT interaction. Still, additional mechanisms might involve in TLX modulation by ligands, and we hypothesized heterodimerization of TLX as potential further mediator of ligand effects in line with the repressor activity of TLX. When we titrated Tb 3+ -cryptate-labeled TLX LBD with sGFP-labeled RXRα LBD in presence of TLX ligands, we observed indeed enhanced dimerization for istradefylline (30) and ccrp2 while caffeine (8) and 29 had only weak effects (Figure ). |
60c75882bb8c1abb9c3dca45 | 9 | Following up on our observation that the recombinant LBDs of TLX and RXRα dimerized (Figure ) and that this interaction was responsive to 30 (Figure ), we further probed whether TLX and RXR would also interact in cellular setting. For this, we co-transfected HEK293T cells with Gal4-responsive firefly luciferase, constitutive renilla luciferase, and a VP16-RXRα-LBD fusion construct with or without Gal4-TLX (Figure ). In absence of Gal4-TLX, VP16-RXRα failed to induce reporter transcription but upon addition Gal4-TLX, reporter activity increased with increasing Gal4-TLX doses demonstrating that the interaction between the TLX and RXRα LBDs is relevant in cells as well. When we studied the effect of istradefylline (30) on this interaction (Figure ), we detected an increase in the TLX-RXRα interaction as observed in elevated reporter activity. These findings support our observation of a TLX repressor activity on various nuclear receptors and further suggest that direct LBD interactions are involved in these effects. |
60c75882bb8c1abb9c3dca45 | 10 | To further characterize the effects of xanthines on TLX activity and evaluate TLX modulation in a native cellular setting, we treated human glioblastoma cells (T98G) with caffeine (8), 29, istradefylline (30), or 33, and determined changes in TLX regulated gene expression by quantitative real-time PCR (Figure ). All four studied xanthines 8, 29, 30 and 33 enhanced expression of the NAD-dependent deacetylase sirtuin-1 (SIRT1, Figure ), the cyclin-dependent kinase inhibitor 1 (p21, Figure ), and the solute carrier family 1 member 1 (SLC1a1, Figure ), all of which are known as TLX regulated . Importantly, the expression of TLX (Figure ) was not affected further indicating that the xanthines affect TLX regulated gene expression through direct TLX modulation. The orphan NR TLXexclusively expressed in certain areas of the CNSincreasingly attracts attention for its potential as therapeutic target in neurodegenerative and neurological disorders or for brain tumors. However, studies on TLX biology beyond knockout experiments are hindered by the lack of potent and well-characterized TLX modulators to be employed as tool compounds for functional experiments. Moreover, the molecular mode of TLX activity and its modulation by small molecule ligands remain widely elusive complicating the search for TLX ligands and early drug discovery. To overcome these obstacles in TLX target validation, we have designed a screening system for TLX ligands, screened for TLX modulators and employed these as tools for early functional studies. |
60c75882bb8c1abb9c3dca45 | 11 | As a key in vitro tool, we constructed a robust cellular assay system to mimic the repressor role of TLX by combining the transcriptional repressor Gal4-TLX with the potent ligand-independent transcriptional activator Gal4-VP16. An ability of Gal4-TLX to counter Gal4-VP16 induced reporter gene activity in a dose-dependent fashion allowed tuning of the test setup to observe bidirectional TLX modulation. Despite its artificial character, this cellular test system turned out very valuable for the discovery and preliminary characterization of TLX ligands, and might also be transferable to other repressive nuclear receptors such as the testicular receptors (TR). Importantly, our observations on Gal4-VP16 repression by Gal4-TLX also translated to combinations of TLX with human NRs. This validated the TLX/VP16 setting, but also unprecedentedly revealed TLX acting as a repressor towards various nuclear receptors and dimerization of TLX with RXR in cell-free setting additionally demonstrated direct interaction. |
60c75882bb8c1abb9c3dca45 | 12 | In the VP16/TLX assay, caffeine directly modulated TLX and counteracted its repressor activity with an IC50 value of 9 µM. We also observed an ability of caffeine to reverse TLX-mediated repression of various NRs. Considering that typical caffeine concentrations after coffee consumption or pharmacological caffeine intake peak at 10 mg/L (~ 50 µM) plasma levels and that brain penetration of caffeine is high , this unprecedented molecular activity of caffeine has potential biological relevance. Importantly, habitual caffeine intake has been correlated with reduced risk for Parkinson's Disease and Alzheimer's Disease . As TLX is an essential factor of neural maintenance and neurogenesis , a potential connection of these effects is obvious. |
60c75882bb8c1abb9c3dca45 | 13 | Systematic structural modification of the caffeine chemotype favourably led to a dramatic increase in potency on TLX, exemplified by 29 (IC50 160 nM, 2.9-fold TLX repression), 33 (40 nM, 4.0-fold) and 34 (9 nM, 3.0-fold). In addition, we discovered the recently approved drug istradefylline (30) as a potent TLX modulator (IC50 40 nM, 2.6-fold TLX repression). This activity might prove as important feature of the drug's pharmacological profile since TLX, characterized as essential regulator of NSC homeostasis and neurogenesis , could involve in the pathology and treatment of PD. Furthermore, TLX is a crucial factor for spatial learning and a recent rodent AD model has demonstrated improvements in memory and spatial learning in istradefylline treated animals further suggesting a potential involvement of TLX in the drug's pharmacology. |
60c75882bb8c1abb9c3dca45 | 14 | Our mutagenesis study suggested residues in the core region of the TLX LBD involved in TLX modulation by xanthines. In essence, we found Phe226 and Ile230 located in helix 5 playing a role in mediating the effects of the xanthines. Moreover, consistent with previous studies knockdown has been reported to upregulate expression of both genes, too . Upregulation of SIRT1, in contrast, according to current understanding indicates TLX activation by xanthines. |
60c75882bb8c1abb9c3dca45 | 15 | While our data, hence, demonstrate direct modulation of the TLX LBD by xanthines, the molecular mechanism of this TLX activity will require future attention to confirm the ligand binding site and capture stabilizing or destabilizing effects of ligand binding that contribute to the modulation of TLX activity. We hypothesize that several molecular factors, including but likely not limited to heterodimerization with RXR and SMRT interactions, involve in the regulation of cellular TLX activity by xanthines. The higher potency we have observed for istradefylline in modulating TLX in cells compared to cell-free assays may hence be explained by the sum of several weaker contributions that cooperate in cellular environment. Further elucidation of this coregulatory network of TLX and its response to ligand binding is needed for which the xanthines present as useful tool. |
60c75882bb8c1abb9c3dca45 | 16 | Gly-Ser linker. The resulting plasmid was termed pFTI-CMV (fusion trans-inducing factor plasmid). Gal4-TLX/Gal4-VP16 assay procedure. HEK293T cells were grown in Dulbecco´s modified Eagle´s medium (DMEM, ThermoFisher Scientific), high glucose with 10% fetal calf serum (FCS), sodium pyruvate (1 mM), penicillin (100 U/mL) and streptomycin (100 µg/mL) at 37 °C and 5% CO2. 24 hours before transfection, cells were seeded in 96-well plates (30,000 cells/well) in DMEM with above mentioned supplements. Prior to transfection, medium was changed to Opti-MEM (ThermoFisher Scientific) without supplements. Cells were then transiently transfected with plasmid mixtures containing 100 ng/well pFR-Luc, 1 ng/well pRL-SV40, 6 ng/well of pECE-SV40-Gal4-VP16 and 3 ng/well of pFA-CMV-Gal4-TLX (during assay establishment, these plasmid amounts per well were systematically varied to optimize conditions allowing robust observation of bidirectional TLX modulation). Transient transfection was achieved using Lipofectamine LTX reagent (ThermoFisher Scientific) according to the manufacturer´s protocol. Five hours after transfection cells were treated with Opti-MEM supplemented with penicillin (100 U/mL) and streptomycin (100 µg/mL) additionally containing 0.1% dimethylsulfoxide (DMSO) and the respective test compounds or 0.1% DMSO alone as negative control. Each sample was tested in duplicates and every experiment was conducted at least three times. After 14 h incubation, cells were lysed for luciferase luminescence detection using the Dual-Glo Luciferase Assay System and XhoI. The sequence coding for the RXRα LBD followed by a stop codon was then introduced in frame between the afore inserted restriction site for BamHI and XhoI. For expression of TLX LBD with N-terminal GFP the sequence encoding TLX residues 150-385 was cloned between the same sites. For generation of biotinylated TLX LBD, the pMal vector system (New England Biolabs, NEB, Ipswich, MA, USA) was used. In pMal-c2E, the section between the sequence encoding 10x Asparagine (Asn10) and the SalI restriction site was replaced with a sequence encoding Leu-Gly-Ile-Glu-Leu-Val-[His8-Tag]-Asp-Tyr-Asp-Ile-Pro-Gly-Thr-Leu-[TEV site] |
60c75882bb8c1abb9c3dca45 | 17 | followed by an Avi-Tag and restriction sites for BamHI and XhoI. The sequence encoding TLX followed by two stop codons was cloned in frame between these restriction sites. From this construct, a fusion protein is expressed with N-terminal maltose-binding protein (MBP) followed by an Asn10 linker, a His8-Tag, a cleavage site for TEV protease, an Avi-Tag, and the TLX LBD with unmodified C-terminus. For expression, E. coli T7 express cells (NEB) were co-transformed with pGro7 (TAKARA Bio Inc., Kusatsu, Japan) and the aforementioned expression construct, and selected overnight at 37°C on LB (Luria Broth) agar containing 34 µg/ml chloramphenicol and either 100 µg/ml ampicillin (for pMal) or 35 µg/ml kanamycin (for pET). Culture in liquid LB was inoculated and grown at 37°C with constant shaking at 180 rpm until optical density at 600 nm (OD600) reached 0. Purification was achieved by immobilized metal affinity chromatography (IMAC) using columns packed with Ni Sepharose 6 Fast Flow resin on an ÄKTApurifier FPLC system (GE Healthcare, Chicago, IL, USA). After washing with buffer supplemented with 50 mM imidazole, the protein was eluted with 300 mM imidazole. Afterwards, GFP fusion proteins were processed with His tagged TEV protease overnight while imidazole content was reduced to 10 mM by dialysis against buffer A in order to allow for reverse IMAC. The flow through was concentrated and applied to size exclusion chromatography using a 16/60 Superdex200™ column equilibrated and run in HTRF assay buffer [25 mM HEPES pH 7.5, 150 mM KF, 10% (w/v) glycerol, 5 mM DTT]. Following the initial IMAC purification step, the MBP fusion protein for generation of biotin labeled TLX LBD was processed with MBP-tagged TEV protease during overnight dialysis against buffer A. Afterwards, uncleaved fusion protein, free MBP-Tag, and TEV protease were removed by passaging through a gravity flow column packed with Amylose High Flow resin (NEB). The flow through was then supplemented with 0.5 mM biotin, 0.5 mM ATP, 5 mM MgCl2, and E. coli biotin ligase birA at a molar ratio of approx. 1:10 for enzymatic conjugation of biotin to the lysine residue in the avitag. |
60c75882bb8c1abb9c3dca45 | 18 | After overnight incubation at 4°C, the solution was subjected to a column packed with 5 ml monomeric avidin UltraLink™ resin (Pierce Biotechnology Inc., Rockford, IL, USA). Unlabeled protein and birA were removed by washing for 10 column volumes with buffer A before biotin labeled TLX LBD was eluted using buffer A supplemented with 2 mM biotin. The product was then concentrated and subjected to size exclusion chromatography using a 10/30 Superdex75™ column equilibrated and run in HTRF assay buffer. |
60c75882bb8c1abb9c3dca45 | 19 | Spectra acquisition was carried out on a Bruker 600MHz AVIIIHD spectrometer equipped with a 5 mm a nitrogen-cooled triple resonance probe 1 H/ 19 F [ 13 C, N]-TCI (Prodigy) and high throughput sample changer (SampleJet) for 579 samples with temperature option for sample storage. All spectra were acquired and processed using Bruker software Topspin 3.6.2 and Topspin 4.0.9, respectively. For the TLX-ligand interaction studies, two samples (with and without protein) were prepared. H-1D, water-suppressed proton 1D (zgesgppe , water suppression using excitation sculpting with gradients using perfect echo) was acquired for each of the sample. Interaction studies were performed at a ratio of 1:1 with respect to TLX and the ligand. |
65c0b8cbe9ebbb4db9b50bfa | 0 | The predictive power of molecular dynamics simulations crucially relies on the accuracy of trajectory-based representations of molecular systems to determine their structural and dynamical properties at a reasonable computational cost. A most common use of trajectorybased methods is to mimic the evolution of the nuclei by driving them with classical, quasiclassical, semiclassical, or quantum forces that represent the effect of the electrons, either in the ground state (i.e., for adiabatic dynamics), or including the effect of the excited states (i.e., for nonadiabatic dynamics). On the least computationally demanding end, purely classical trajectory simulations offer access to complex systems consisting of hundreds of atoms. |
65c0b8cbe9ebbb4db9b50bfa | 1 | Even for large systems, there are plenty of situations where quantum dynamical effects are known to be important-and therein lies the challenge. For example, the study of hydrogen transfer in proteins is important for understanding the multitude of biological functions supported by this fundamental reaction-which in some cases even requires consideration of deep proton tunneling. Similarly, hydrogen tunneling dynamics was experimentally revealed in phenol-ammonia clusters activated by UV-photon absorption. Other aromatic biomolecules (e.g., indole and pyrole 3 in the presence of water and ammonia solvents) manifest a behavior that attests to quantum nuclear effects governing the dynamics in excited electronic states. These examples serve to highlight the importance of considering nuclear quantum effects, combined with multiple electronic states when necessary, that cannot be taken into account using only classical trajectories. |
65c0b8cbe9ebbb4db9b50bfa | 2 | There is, however, no longer a need to restrict oneself to purely classical simulations. In the last decades, progress has been made towards pushing the limits imposed by the computational cost of exact or nearly exact quantum mechanical approaches (like time-dependent wavepacket methods, e.g., the multiconfiguration time-dependent Hartree, and even exact spectroscopic methods ). Less expensive but more approximate computational methods are also available, such as semiclassical initial value representation, 8 approximated path-integral methods, nonadiabatic ring-polymer molecular dynamics and nonadiabatic quantum instanton theory -all of which have been applied to nonadiabatic processes. |
65c0b8cbe9ebbb4db9b50bfa | 3 | An alternative route towards completely recovering nuclear quantum effects, while still maintaining a trajectory-based representation of nuclear dynamics, is provided by the socalled quantum trajectory methods (QTMs). In past years, such QTMs have been used to describe the quantum nuclear behaviour of different model and molecular systems. In this work, our aim is to focus on the theoretical and numerical aspects of one particular quantum trajectory formalism, and its application to adiabatic and nonadiabatic problems. |
65c0b8cbe9ebbb4db9b50bfa | 4 | Numerical QTMs that are referred to as synthetic are those that propagate the wavefunction's hydrodynamic fields (density and phase) along the quantum trajectories, and deduce trajectory dynamics at each time step from those field values. We refer the reader to the book of Wyatt for an excellent review. A major numerical benefit of such a trajectorybased representation in a Lagrangian frame is the fact that one essentially ends up with a moving grid (i.e., the trajectories themselves) that follows the probability flow. Using the moving grid to "sample" the nuclear configuration space in this manner thus allows one to substantially reduce the number of grid points needed to accurately describe, for instance, scattering processes. In many practical instances, use of QTMs rather than wavefunctionbased methods may be reasonably expected to provide better-than-exponential scaling with system dimensionality-whereas fixed-grid methods, in contrast, are always characterized by exponential scaling. |
65c0b8cbe9ebbb4db9b50bfa | 5 | Moreover, since the "quantum potential" (whose gradient is the negative quantum force) has an inverse dependence on density, the numerical derivatives become unstable in the vicinity of wavefunction nodes, leading to a breakdown of the simulation-the so-called "node problem." Various treatments for these numerical issues have been proposed in the literature-e.g., the moving least squares (MLS) approach, finite differences for constrained trajectories in an arbitrary Lagrangian-Eulerian (ALE) moving frame, 20 artificial viscosity forces, "bipolar" decompositions of the wavefunction, and semiclassical treatments based on linearized quantum forces. More recently, a fundamentally different QTM-type approach has been developed, based on a complete reformulation of quantum theory in terms of trajectories. In this approach one still works with the same ensemble of quantum trajectories as in the original Bohmian QTM formulation described above. However, instead of using those trajectories to propagate the wavefunction-based hydrodynamic fields, the quantum force is computed directly from the trajectories themselves-without making any reference to the wavefunction itself. That this is even possible is no trivial development. In any event, from a numerical standpoint, the spatial variable x is replaced with a "trajectory-labelling coordinate" (usually denoted C), in terms of which the numerical grid becomes structured and stationary. This, in turn, leads to enormous numerical advantages in terms of evaluating the derivatives needed to compute the quantum force. Indeed, the trajectory-based form of the quantum force is generally found to overcome the node problem, although in the time-dependent case under-sampling of trajectories in areas of low density impacts the accuracy of finite differences schemes. This is discussed in more details below. |
65c0b8cbe9ebbb4db9b50bfa | 6 | The aforementioned trajectory-based reformulation can be derived either from the stationary or from the time-dependent Schrödinger equation, resulting in differing properties for the trajectories and numerical considerations. In a previous work, some of the authors applied the stationary formulation to the study of quantum scattering processes on a chemical abstraction reaction model as well as in an adiabatic quantum capture model of cold and ultra-cold chemistry, in both cases within the Born-Oppenheimer (adiabatic) approximation. In such contexts, the numerical propagation is extremely stable, essentially because the trajectories are solutions of an ordinary differential equation (ODE). |
65c0b8cbe9ebbb4db9b50bfa | 7 | In contrast, time-dependent applications necessarily involve a partial differential equation (PDE). Since, as mentioned, x(C, t) (e.g., for the one-dimensional case) is the sought-for PDE solution, it is in principle necessary to impose both initial conditions and boundary conditions [i.e., x(t) at both C-grid edges] in order to solve the requisite PDE. The problem is that the boundary conditions are not known a priori. Here, we find a noteworthy disadvantage, in comparison with wave-based PDE solutions for which zero Dirichlet boundary conditions may almost always be presumed. In practice, we find that trajectory dynamics in the interior are not sensitively dependent on the choice of boundary conditions, provided the grid interval is sufficiently large, as might be expected. On the other hand, the quantum trajectory dynamics can in many cases become numerically unstable, if there is a scattering potential present. This tends to manifest as errors propagating in from the grid edges, however, rather than originating from nodes-i.e., this is not the node problem. |
65c0b8cbe9ebbb4db9b50bfa | 8 | Various techniques have been developed that provide some improvement to the numerical stability in the time-dependent case, without the need to invoke additional, sometimes costly, computational smoothing procedures such as those mentioned above, nor supplementary approximations for the quantum potential and for the quantum force. Rather, the "tricks" used here are in the vein of a judicious choice of boundary conditions and finite difference discretizations, and are therefore numerically exact. Curiously, the particular choices made here can be interpreted as determining the precise form of the "interworld potential"-according to the "discrete" many-interacting-worlds (MIW) interpretation of quantum mechanics that has sprung up from the trajectory-based reformulation. In the original, "continuous" MIW interpretation, however, these are merely choices for the numerical discretization. |
65c0b8cbe9ebbb4db9b50bfa | 9 | First, in addition to exploiting the numerical techniques described above, we also introduce new methods that appear to greatly improve the numerical stability of the time-dependent QTM calculations. This is an extremely important development from the perspective of practical, widespread adoption of the approach as a generic robust computational tool. Second, we aim to extend the time-dependent formalism to nonadiabatic processes in the same manner as for adiabatic processes. However, in a "standard" formulation of nonadiabatic dynamics, the effect of several electronic states is accounted for via the inclusion of multiple potential energy surfaces (PESs), and couplings among them; 41 this picture is fundamentally different from adiabatic dynamics, where a single electronic state, and thus a single PES, contributes to the "classical force" guiding nuclear dynamics (in addition to the quantum force already discussed). Note that wavefunction-based nonadiabatic quantum trajectory dynamics have certainly been considered in the past. However, these were done in a standard Born-Huang (or related) representation of nonadiabatic dynamics involving multiple PES components-which, for technical reasons, poses a source of fundamental difficulties in the interacting trajectory-based context. Accordingly, in this work, we instead invoke the exact factorization formalism in combination with the interacting quantum trajectories approach presented above [ 36,38]. The exact factorization yields nuclear dynamics under the effect of a single time-dependent classical force accounting for the electronic excited states, thereby avoiding the aforementioned technical difficulty. |
65c0b8cbe9ebbb4db9b50bfa | 10 | Note that some of the present authors have already attempted to combine the standard time-dependent Bohmian QTM formulation with exact factorization to include quantum nuclear effects within a trajectory description of coupled electron-nuclear dynamics. Nonetheless, numerical issues associated with that study prevented the authors from en-visioning extensions to actual practical applications. Accordingly, we now present in this paper-for the first time-a proof-of-principle illustrative study that demonstrates the potential of the new scheme, and in particular, the value of combining the time-dependent interacting-trajectory-based QTM approach with exact factorization. Although only onedimensional examples are considered here, we note further that the "continuous" MIW theory adopted here (unlike the "discrete" version) readily generalizes for many-dimensional applications as well. The paper is organized as follows. Section 2 provides a brief description of the employed methodology for our quantum dynamics calculations, starting with the exact factorization formalism in Section 2.1 and continuing with the interacting quantum trajectories formalism in Section 2.2. In Section 3 we present our numerical results for various types of scattering potentials, focusing on the adiabatic case in Section 3.1 and on the nonadiabatic case in Section 3.2. We present our conclusions and perspectives in Section 4. |
65c0b8cbe9ebbb4db9b50bfa | 11 | describes a system of interacting electrons and nuclei, whose positions are collectively indicated as q and x, respectively. The nuclear kinetic energy operator is expressed in Cartesian coordinates and contains a sum over the 3N n nuclear degrees of freedom, each labeled with the index ν, with spatial derivatives ∂ ν with respect to nuclear positions; M ν are the nuclear masses. The electronic Hamiltonian Ĥel (q, x) is the sum of the electronic kinetic energy and of all interactions. The time evolution of the electron-nuclear system is dictated by the molecular time-dependent Schrödinger equation (TDSE) |
65c0b8cbe9ebbb4db9b50bfa | 12 | Equation ( ) is thus unique up to a gauge encoded in a phase factor e (i/h)θ(x,t) (with θ(x, t) a real function): multiplying the nuclear wavefunction by this factor and the electronic term by its complex conjugate, Eq. (3) remains unaffected and the time-dependent potentials transform as standard gauge potentials. Therefore, this ambiguity has to be eliminated by imposing a choice of gauge. |
65c0b8cbe9ebbb4db9b50bfa | 13 | In Section 2.2 we will work in a gauge where the TDVP is identically zero, which is a possible choice of gauge only for one-dimensional problems (in nuclear space), as those presented in Section 3. Another possible choice of gauge in high-dimensional situations, is to put to zero the TDVP along a "selected" direction in nuclear configuration space. Then, one could combine a classical and a quantum trajectory-based representation of nuclear dynamics, adopting the refined quantum description only along the "selected" direction. |
65c0b8cbe9ebbb4db9b50bfa | 14 | An alternative strategy is to reformulate Section 2.2 including the effect of the TDVP, along with the TDPES. Note that quantum trajectories' velocity field is a gauge-invariant quantity, thus they are the same for any choice of gauge. In any case, it is clear that many interesting routes, currently under investigation, can be undertaken to combine the interacting quantum trajectories formalism with exact factorization. |
65c0b8cbe9ebbb4db9b50bfa | 15 | In this preliminary study, we apply the interacting quantum trajectories approach for a single nuclear degree of freedom in the exact factorization framework; thus the vectorial notation x reduces to the single nuclear coordinate x. As stated in Section 2.1, we choose to work in the gauge where only the TDPES affects nuclear dynamics. With this gauge choice, the nuclear current density reduces to |
65c0b8cbe9ebbb4db9b50bfa | 16 | From the above, we identify M ẋ = ∂S(x, t)/∂x as the velocity field for the quantum trajectories (since we deal with one nuclear degree of freedom we remove all dependencies on the index ν). Inserting the polar form of the nuclear wavefunction in the nuclear TDSE (4), and separating real and imaginary parts, one obtains quantum hydrodynamic equations for the fields ρ(x, t) and S(x, t) in an Eulerian frame of reference, |
65c0b8cbe9ebbb4db9b50bfa | 17 | Note that in Eq. ( ) the spatial derivative with respect to x acts on the nuclear velocity field in square brackets. Finally, taking the x-derivative of the phase equation above, one finds that the local shape of the quantum hydrodynamic fields drives the quantum trajectories via the sum of both classical and quantum potentials as follows: |
65c0b8cbe9ebbb4db9b50bfa | 18 | The same basic equations apply in the trajectory-based formulation, except that as stated, the quantum potential is not computed from Eq. ( ). To understand how Q is defined, it is first necessary to discuss the trajectory labelling coordinate C, in terms of which the PDE solution for the quantum trajectory ensemble is expressed as x(C, t). In principle, there is complete freedom in terms of how C is defined. One of the simplest choices is to take C to be the initial value of a given trajectory at time t = 0-i.e., C = x 0 = x(t = 0). Through probability conservation [i.e., Eq. ( )], one then obtains the following relation for the density at any time t: |
65c0b8cbe9ebbb4db9b50bfa | 19 | where ρ 0 (x 0 ) = ρ(x 0 , t = 0), x = (x 0 , t), and x (x 0 , t) = ∂x(x 0 , t) ∂x 0 | t . Note that the "spatial" derivative of x is taken with respect to the labelling coordinate C, which in this case is just the initial value x 0 . The dimensionless quantity x thus becomes a measure of the relative spacing of nearby trajectories over time, as compared to the initial spacing at t = 0. Inserting the expressions for ρ(x, t) [from Eq. ( )] and for x (x 0 , t) into the above wavefunction-based quantum hydrodynamic equations, all reference to the time-evolved wavefunction are now entirely removed. The resultant trajectory-based dynamical PDE |
65c0b8cbe9ebbb4db9b50bfa | 20 | The trajectory ensemble dynamical PDE of Eq. ( ) is fourth-order in "space" (i.e., x 0 ) and second order in time. Note that all trajectory interactions (which arise from the x 0 derivatives) are due to quantum forces; otherwise, the trajectories would not interact, and Newton's classical ODE would result. |
65c0b8cbe9ebbb4db9b50bfa | 21 | We note that Eq. ( ) is written out explicitly in a form that depends only on x(x 0 , t), constants, and the initial density ρ 0 (x 0 ). The presence of the latter quantity is simply an artifact associated with the coordinate choice C = x 0 . Indeed, the ρ 0 dependence can be easily removed by transforming to a "uniformizing" choice for C, in terms of which The explicit transformation from x 0 to the uniformizing C can be defined as follows: |
65c0b8cbe9ebbb4db9b50bfa | 22 | Note that any two suitable labelling coordinates must be related to via a bijective function (i.e., monotonic, in the one-dimensional case). In addition, we stress that the labelling coordinate (which could also be termed the "Lagrangian coordinate") is necessarily timeindependent along a given trajectory: C(x(x 0 , t)) = C(x(x 0 , 0)) ∀t. Finally, for each trajectory in the ensemble, we have |
65c0b8cbe9ebbb4db9b50bfa | 23 | As discussed, the uniformizing C of Eq. ( ) implies trajectories x(C, t) that all bear the same probability density, in contrast to the x(x 0 , t) ensemble. From Eq. ( ), however, the shape of the density profile ρ(x, t) can be retrieved at any time t as follows: |
65c0b8cbe9ebbb4db9b50bfa | 24 | The classical force is indicated as the spatial (x) derivative of the TDPES of Eq. ( ), which reduces to the adiabatic potential if the Born-Oppenheimer approximation is valid, as was shown in Ref. [ 49]. An explicit uniformizing-C expression for Q is straightforwardly derived by inserting Eq. ( ) into Eq. ( ), and making use of the chain rule: |
65c0b8cbe9ebbb4db9b50bfa | 25 | Finally, we note that a more rigorous derivation of the above quantum trajectory expressions is also possible, based on a Lagrangian/action extremization procedure. In this approach, the Lagrangian is equal to the usual classical one, with the addition of a quantum contribution-i.e., L = T -V -L Q , where T is the kinetic energy. It is important to note that a gauge freedom exists in the definition of L Q . Thus in general, L Q need not be Qalthough this particular choice is allowed, and has the advantage that the resultant action, obtained by integrating the Lagrangian over time, or equivalently |
65c0b8cbe9ebbb4db9b50bfa | 26 | In any numerical implementation, the continuous trajectory labelling coordinate C must be discretized-with specific, discrete values C i corresponding to the individual discrete quantum trajectories used in the calculation. Note that these values do not change over time-the C i grid is thus fixed, rather than moving. It can therefore also be structuredusually such that the grid-point spacing, (C i+1 -C i ) = ∆C, is uniform across the grid, |
65c0b8cbe9ebbb4db9b50bfa | 27 | Having defined the discrete quantum trajectories, x i = x(C i , t), some kind of finitedifference scheme is needed to evaluate C derivatives numerically, in order to compute the quantum forces F Q i , acting on each x i trajectory. In previous numerical applications of this method, the following expression for the quantum force was used, which can yield a reasonably stable numerical propagation (especially for free particles), and also accounts correctly for quantum effects: |
65c0b8cbe9ebbb4db9b50bfa | 28 | A numerical issue arises with the (two) first and (two) last trajectories at the edges of the discrete C i -grid ensemble, since these lack a sufficient number of neighbors to evaluate spatial derivatives using the finite-difference scheme. In reality, this is nothing but the aforementioned boundary condition difficulty. This problem is mitigated by introducing additional fixed virtual trajectories x 0 and x -1 set at -∞, and x n+1 and x n+2 at +∞. Note that (x 0x -1 ) = (x n+2x n+1 ) = +∞ is also presumed. With these choices, reasonable (i.e., non-singular) values for F Q are obtained using Eq. ( )-although as discussed, errors can still propagate in from the edges, particularly if there is a scattering PES present. |
65c0b8cbe9ebbb4db9b50bfa | 29 | To improve numerical stability still further, some additional measures are also implemented here. Consider, as an example, a Gaussian wavepacket, for which the quantum force is linear in x. Clearly, Eq. ( ) yields more accurate F Q values in the interior of the wavepacket than in the periphery, since the trajectories are less densely distributed in regions of low density. Numerical errors near the edges do not necessarily lead to a significant deterioration of the overall propagation, however-again, because probabilities are small in the periphery. If the edge errors are stable at least (even if somewhat large) then as discussed, comparatively large errors at the periphery have little effect in the interior. In any event, errors tend to manifest as trajectories that oscillate around their true course. Numerical instability is signalled by oscillations that grow in magnitude over time, and/or propagate into the interior region. |
65c0b8cbe9ebbb4db9b50bfa | 30 | Though surprisingly effective, this approach is not a panacea, and in any event rather expensive to implement. We therefore also introduce an initial grid-point relaxation procedure, prior to the numerical propagation, which operates as follows. To begin with, for every calculation performed here, the initial wavepacket is Gaussian, and therefore the initial quantum force is known analytically everywhere. As discussed, grid points are distributed uniformly in C, and remain uniform over time. However, even for a Gaussian wavepacket, such a distribution does not lead to numerically exact F Q i values via Eq. ( ). The purpose of the relaxation procedure, therefore, is to shift the trajectories starting positions slightly (especially in the periphery) such that the numerically computed F Q i errors are minimized. |
65c0b8cbe9ebbb4db9b50bfa | 31 | x 1≤j≤n , and so the inverse of the Jacobian matrix J ij = ∂f i ∂x j used in the Newton iterative root search is well defined. Note that the exact quantum force values, F exact Q (x i ) can be obtained analytically from the known Gaussian initial density through Eq. ( ). In particular, the Gaussian form |
65c0b8cbe9ebbb4db9b50bfa | 32 | Yet, initial quantum force errors are substantially diminished, to the extent that subsequent numerical propagation in general becomes much more stable. The initial grid-point relaxation procedure thus strikes a nice balance between dynamical fidelity with (nearly) uniform wavefunction representation. In any event, how we recover the density from the trajectory distribution is detailed below. |
65c0b8cbe9ebbb4db9b50bfa | 33 | In the trajectory-based approach, the relation between x and C is known at a given time only through the discrete values, x i (C i , t). Hence, interpolation is needed to estimate the density at arbitrary x. Our interpolation procedure is as follows: we first generate a monotonic interpolation of C(x, t) by the means of a monotone quintic splines interpolation algorithm. Then, taking the derivative of the resulting interpolating function yields the density at arbitrary points, thanks to the definition of C as ρ(x, t) = ∂C/∂x. The resulting density synthesis strictly obeys norm conservation, which is implicitly guaranteed by the definition of quantum trajectories and use of the trajectory labelling coordinate C. This method is favored over using finite differences to estimate the density from the spacing of trajectories [i.e., taking ρ(x i , t) ≈ 2∆C/(x i+1 -x i-1 ), then interpolating ln [ρ(x, t)] using cubic splines] as that procedure might violate the conservation law because of spline oscillations or "ringing". |
65c0b8cbe9ebbb4db9b50bfa | 34 | Assuming an initial Gaussian density of the form of Eq. ( ), the initial phase is given by S(x, t = 0) = p 0 (xx c ), where x c < 0, and p 0 > 0 is the initial momentum, so that the wavepacket is incident from the left side of the PES barrier. Unless explicitly stated, we use atomic units henceforth. |
65c0b8cbe9ebbb4db9b50bfa | 35 | The numerical procedure based on the propagation of quantum trajectories will be henceforth referred to as the "time dependent quantum trajectories" (TDQT) approach. Given that the trajectories are discretized and coupled through the quantum force as evaluated in Eq. ( ), the dynamical PDE ( ) can be thought of as having been replaced by the following set of coupled ODEs in terms of positions {x i } and momenta {p i = M ẋi } for the discrete ensemble of trajectories: |
65c0b8cbe9ebbb4db9b50bfa | 36 | This set of coupled equations may thus be propagated using an ODE integrator. The time integration is performed using the well-known adaptive time-step Bulirsch-Stoer scheme with error tolerance set to 10 -9 . Moreover, an upper bound is set on the integration step so that no neighboring trajectories spacing should decrease by a factor above 40% from one time step to the next. For comparison, we also performed a wavefunction-based calculation of the time evolution of the corresponding TDSE, using the Crank-Nicholson algorithm. |
65c0b8cbe9ebbb4db9b50bfa | 37 | We are interested in determining the transmission probability over time, which is obtained by integrating the density in the product region defined as [x P ; +∞], for some sufficiently large x P . For the TDQT calculation, this is straightforward, as every trajectory carries the same probability of 1/n. Accordingly, at any given time t, one need only count the number of trajectories, n trans (t), for which x(C i , t) > x P , and compare with n. Note that for this reason, the dynamical simulation cannot be expected to provide a transmission probability resolution finer than 1/n. Indeed, the exact result is only bound to be between n trans (t)/n and (n trans (t) + 1)/n. Hence, a sensible estimate of the transmission probability is obtained as |
65c0b8cbe9ebbb4db9b50bfa | 38 | Table : List of parameters defining: the initial nuclear wavepacket, via γ 0 and x c ; the system and the potentials, via the mass m and a, α, V 0 for the Eckart barriers or b, d for the ramp; the numerical procedure for the TDQT propagation, via the ODE error tolerance and the number of trajectories; the numerical procedure for the Crank-Nicholson integration via the grid boundaries, the spacing dx and the time step dt. For the symmetric and asymmetric Eckart potentials, several calculations were performed by increasing the initial kinetic energy as indicated in Fig. ; for the potential ramp, four |
65c0b8cbe9ebbb4db9b50bfa | 39 | Near perfect agreement is achieved in every case. In particular, absolute differences between the two calculations-which yield an estimate of the TDQT error-is always found to be less than the maximum expected value of ∆P max T = 1/n. For the Eckart PES problems, this value is ∆P max T = 2.5 × 10 -3 ; for the potential ramp problem, it is 3.3 × 10 -3 . |
65c0b8cbe9ebbb4db9b50bfa | 40 | As an illustrative study of the combination of the TDQT approach with exact factorization for nonadiabatic dynamics using the TDPES (7), we simulate the well-known Tully models. These are one-dimensional models in nuclear space (as are those discussed in Section 3.1), and include two coupled electronic states: Tully model 1 presents a singled avoided crossing between the potential energy curves; Tully model 2 is a dual avoided crossing model; Tully model 3 shows an extended coupling region and manifests possible reflections of the incoming wavepacket. In this work we use the original models, whose electronic Hamiltonian Ĥel is given in the diabatic basis representation. |
65c0b8cbe9ebbb4db9b50bfa | 41 | The numerical calculation of the TDPES has been carried out as follows: (1) exact vibronic wavepacket dynamics calculations have been performed with the ElVibRot program 54 on the Tully models; (2) the output of ElVibRot has been analyzed by the Exact Factorization Analysis Code (EFAC) to produce the TDPES. The TDPES can be easily expressed in terms of "standard" (a)diabatic nuclear amplitudes (and their spatial and time derivatives) arising from the Born-Huang representation of the molecular wavefunction. We refer the interested reader to Refs. [ 56-58] for a detailed discussion on those expressions. In any case, ElVibRot outputs the necessary information in the diabatic representation at various time steps throughout the propagation, and EFAC reads this information as input in order to reconstruct the TDPES. Additionally, EFAC imposes the gauge condition, which in the present case is simply taken to be A(x, t) = 0 ∀ t. Following from its definition given in Eq. ( ), and expressing the electronic wavefunction as the ratio of the molecular and nuclear wavefunctions, from Eq. ( ), the TDVP reads |
65c0b8cbe9ebbb4db9b50bfa | 42 | with S(x, t) the phase of the nuclear wavefunction. It is easy to see that imposing the gauge condition A(x, t) = 0 yields an integral equation that defines S(x, t) in terms of the nuclear momentum field 59 -i.e., the first term on the right-hand side of Eq. ( ). |
65c0b8cbe9ebbb4db9b50bfa | 43 | ElVibRot calculations have been performed in the diabatic basis by initializing a Gaussian wavepacket in the lowest-energy electronic state in the negative x region. The Gaussian is centered at x c = -8.0 a 0 , and we considered three different values of the initial momentum p 0 = hk 0 , defined by the values k 0 = 10, 15, 20 a -1 0 , with an initial width γ 0 = 20/k 0 . |
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