Correlation Function Formalism Research Articles

Theoretical Study of the Photocyclization Reaction-Induced Dual Aggregation-Induced Emission Phenomenon.

Photochromic molecules with aggregation-induced emission (AIE) effects are of great value and prospective in various practical applications. To explore its inherent mechanism, the open isomer ap-BBTE and the closed isomer c-BBTE were chosen to perform the theoretical calculation using the quantum mechanics/molecular mechanics model combined with thermal vibration correlation function formalism. The calculations show that the photocyclization (PC) reaction from ap-BBTE to c-BBTE facilitates an improvement in the AIE effect. It is found that the fluorescence quantum yield (ΦF) enhancement of ap-BBTE is attributed to the restriction of the low-frequency rotational motion of the benzothiophene moiety and the high-frequency stretching vibrations of the C-C bond between the benzothiophene and benzylbis(thiadiazole) vinyl groups after aggregation. For c-BBTE, the increase in ΦF upon aggregation is mainly due to the suppression of the high-frequency stretching vibration of the C-C bond between the benzothiophene and the benzobis(thiadiazole) vinyl groups. In addition, the AIE effect was also enhanced from ap-BBTE to c-BBTE, which is consistent with the experimental phenomenon. The corresponding emission spectrum red-shifted from ap-BBTE to c-BBTE in both dilute solution and the crystalline state due to the improved intramolecular conjugation of c-BBTE. Moreover, the PC reaction from ap-BBTE to c-BBTE easily occurs in an excited state with a low energy barrier transition state by forming a C-C bond between benzothiophene groups effectively in dilute solution. Our calculations provide theoretical guidance for the further rational design of efficient AIE luminogens.

Effect of choosing coordinate systems on computationally predicting nonradiative transition rates of flexible thermally activated delayed fluorescence molecules

AbstractUnderstanding nonradiative transition mechanisms is important in various situations. However, compared with radiative processes where temporal profiles of photon emission can be monitored in a straightforward manner, experimentally accessing the rate information may not be an easy task with nonradiative transitions. Hence, applying theoretical tools toward predicting the rates can be a useful tactic. Such predictions become very useful in designing optoelectronic materials as in the molecules adopted for constructing organic light‐emitting diodes (OLEDs). The correlation function formalism is a method that can fulfill the purpose of designing OLED materials. The formalism requires information regarding the vibrational normal modes of the two electronic states before and after the transition. Because the method is also based on harmonic oscillator approximation, it can actually fail to provide high reliability when there is a large geometric distortion between the initial and the final states. In fact, the harmonic normal mode picture is more prone to lose reliability in the Cartesian coordinates than in the internal ones even at a small distortion, and hence adopting internal coordinates may be more preferable for practical calculations. This is because normal mode mixing becomes less severe when molecular coordinates are described with internal degrees of freedom such as bond stretching, bending, and torsion. In this regard, how much more reliable the nonradiative rate predictions in OLED materials become with the use of internal coordinate system deserves a close inspection. In this account, we review on the derivation of the correlation function formalism and provide how it can be adapted toward the use of the internal coordinates. As a demonstration, we evaluate the intersystem crossing and the internal conversion rates of a series of thermally activated delayed fluorescence (TADF) molecules with both Cartesian and internal coordinate systems. Overall, handling transitions involving substantial structural changes is improved indeed with the internal coordinates. However, limitations are still apparent for the TADF systems with a flexible donor–acceptor type construct especially when the inevitable inter‐domain twisting takes place with the electronic transition. Future prospect for handling the issue is commented as a concluding remark.

Conformational Flexibility-Controlled Piezochromic Behavior of Organic Fluorophores

The pressure-responsive behaviors of organic fluorophores are often irregular and highly system-dependent. Unraveling the pressure effects on them is the key to the precise design of piezochromic materials. Here, we demonstrate that the different conformational flexibilities of organic fluorophores can lead to their distinct piezochromic behaviors. By combining quantum mechanics/molecular mechanics models and thermal vibration correlation function formalism, we show that two organic fluorophores, dibenzo[b,d]thiophene 5,5-dioxide (DBTS) and carbazole (Cz), with similar chemical structures, display different conformations and conformational flexibilities, which determine the responsiveness of their piezochromic behaviors. It is found that pressurization continuously improves the planarity between the sulfonyl group and the aromatic skeleton in cross-shaped DBTS, leading to red-shifted emission. In addition, the fluorescence quantum efficiency (FQE) of DBTS increases monotonically in a wide range of 0–7 GPa and then shows a dramatic reduction upon further compression to 12 GPa. The non-monotonical FQE of DBTS is attributed to the continuously decreased radiative decay rate constant (kr) and non-monotonically changed non-radiative decay rate constant (kic). The initial decrease in kic is due to the restriction of aromatic skeleton vibrations in the high-frequency region, and the subsequent increase in kic is ascribed to the acceleration of the bending and rotational vibrations of the sulfonyl group in the low-frequency region. In contrast, pressurization hardly influences the conformation of planar Cz, resulting in a slight change in emission and FQE, which is well consistent with experimental results. This study opens an efficient shortcut to the rational design of advanced piezochromic luminescent materials.

Primordial stochastic gravitational wave background anisotropies: in-in formalization and applications

Primordial non-Gaussianities of the scalar(tensor)-tensor-tensor type supporting a non-trivial squeezed component are known to induce anisotropies in the stochastic gravitational wave background. We derive the explicit form of such anisotropies by making use, for the first time in this context, of the in-in formalism for cosmological correlation functions. After illustrating the general method and using it for the minimal single-field slow-roll case, we apply it to multi-field models, providing both a tree-level and a one-loop example. First, we make contact with previous results on anisotropies due to the presence of an extra spin-2 field during inflation. Secondly, we calculate the 1-loop scalar-tensor-tensor three-point function in the context of so-called supersolid inflation. The corresponding gravitational wave anisotropy is induced atop a gravitational signal that may be sufficiently large for detection.

Nonequilibrium reaction rate theory: Formulation and implementation within the hierarchical equations of motion approach.

The study of chemical reactions in environments under nonequilibrium conditions has been of interest recently in a variety of contexts, including current-induced reactions in molecular junctions and scanning tunneling microscopy experiments. In this work, we outline a fully quantum mechanical, numerically exact approach to describe chemical reaction rates in such nonequilibrium situations. The approach is based on an extension of the flux correlation function formalism to nonequilibrium conditions and uses a mixed real and imaginary time hierarchical equations of motion approach for the calculation of rate constants. As a specific example, we investigate current-induced intramolecular proton transfer reactions in a molecular junction for different applied bias voltages and molecule-lead coupling strengths.

A computational scheme for evaluating the phosphorescence quantum efficiency: applied to blue-emitting tetradentate Pt(II) complexes.

Phosphorescent organic light-emitting diodes (PhOLEDs) are leading candidates for displays or lighting technologies. Recently, blue phosphorescent tetradentate Pt(II) complexes have been attracting extensive attention due to their high phosphorescence quantum efficiency and numerous chemical structures on account of flexible ligand frames and modifications. Using quantum chemistry coupled with our thermal vibration correlation function (TVCF) formalism, we investigated the triplet excited state energy surface and the decay processes involving both direct vibrational relaxation and minimum energy crossing point (MECP) via the transition state (3TS) to the ground state (S0) for 16 recently experimentally reported blue-emitting tetradentate Pt(II) emitters containing fused 5/6/6 metallocycles. We found that (i) in most cases, the direct vibrational relaxation deactivations dominated the triplet non-radiative decay because either the 3TS is too high or the MECP is not reachable. Hence, results from the TVCF formalism agreed well with the experiments for the phosphorescence quantum efficiency; (ii) only when both 3TS and MECP are low, for instance, for PtON1-oMe, deactivations via MECP dominated the triplet non-radiative decay.

Rate of Insulin Dimer Dissociation: Interplay between Memory Effects and Higher Dimensionality

We calculate the rate of dissociation of an insulin dimer into two monomers in water. The rate of this complex reaction is determined by multiple factors that are elucidated. By employing advanced sampling techniques, we first obtain the reaction free energy surface for the dimer dissociation as a function of two order parameters, namely, the distance between the center-of-mass of two monomers (R) and the number of cross-contacts (Q) among the backbone Cα atoms of two monomers. We then construct an orthogonal 2D reaction energy surface by introducing the reaction coordinate X to denote the minimum energy pathway and a conjugate coordinate Y that spans the orthogonal direction. The free energy landscape is rugged with multiple maxima and minima. We calculate the rate by employing not only the non-Markovian multidimensional rate theory but also several other theoretical approaches. The necessary reaction frequencies and the frictions are calculated from the time correlation function formalism. Our best estimate of the rate is 0.4 μs-1. Our study reveals interesting opposite influences of dimensionality and memory in determining the rate constant of the reaction. We gain interesting insights into the dimer dissociation process by looking directly at the trajectories obtained from molecular dynamics simulation.

Aggregation-Enhanced Thermally Activated Delayed Fluorescence Efficiency for Two-Coordinate Carbene-Metal-Amide Complexes: A QM/MM Study.

The two-coordinate carbene-metal-amide complexes have attracted a great deal of attention due to their remarkable thermally activated delayed fluorescence (TADF) properties, giving them promise in organic light-emitting diode application. To reveal the inherent mechanism, we take CAAC-Cu(I)-Cz and CAAC-Au(I)-Cz as examples to investigate the photophysical properties in solution and solid phases by combining quantum mechanics/molecular mechanics approaches for the electronic structure and the thermal vibration correlation function formalism for the excited-state decay rates. We found that both intersystem crossing (ISC) and its reverse (rISC) are enhanced by 2-4 orders of magnitude upon aggregation, leading to highly efficient TADF, because (i) the metal proportion in the frontier molecular orbitals increases, leading to an enhanced spin-orbit coupling strength between S1 and T1, and (ii) the reaction barriers for ISC and rISC are much lower in solution than in aggregate phases through a decrease in energy gap ΔEST and an increase in the relative reorganization energy through bending the angle ∠C2-Cu-N1 for T1. We propose a pump-probe time-resolved infrared spectroscopy study to verify the mechanism. These findings can clarify the ongoing dispute over the understanding of the high TADF quantum efficiency for two-coordinate metal complexes.

Theoretical Characterizations of TADF Materials: Roles of ΔG and the Singlet-Triplet Excited States Interconversion.

The thermally activated delayed fluorescence (TADF) phenomenon has attracted increasing attention because it can harvest 100% of the electro-pumped carriers to form singlet bound excited state for fluorescence. It is generally believed that the small energy gap between S1 and T1 (ΔEST) is essential for TADF to facilitate the reverse intersystem crossing (rISC). However, for a few donor-acceptor (D-A) organic compounds with small ΔEST, the TADF phenomenon is absent, indicating that ΔEST might not be a good molecular descriptor. Here, using our self-developed thermal vibration correlation function (TVCF) formalism in combination with quantum chemistry calculations, we revisit the key factors that dominate the TADF property for 11 D-A systems with small ΔEST. Based on our theoretical results in comparison to experiments, we conclude that the activation energy ΔG is a good molecular descriptor to characterize the TADF performance because a significantly better linear relationship is observed between ΔG and the rISC rate constant (krISC) compared to that between ΔEST and krISC. These findings provide deeper understanding of the TADF mechanism, shedding light on the molecular design of high-performance TADF materials.

Theory of Long-Lived Room-Temperature Phosphorescence in Organic Aggregates.

ConspectusRoom-temperature phosphorescence (RTP) with a long afterglow from purely organic molecular aggregates has recently attracted many investigations because traditionally only inorganic and transition-metal complexes can emit phosphorescence at room temperature. Purely organic molecules can exhibit phosphorescence only at cryogenic temperatures and under inert conditions in solution. However, recently, a number of organic compounds have been found to demonstrate bright RTP upon aggregation, sometimes with a remarkable morphology dependence. We intended to rationalize such aggregation-induced organic RTP through theoretical investigation and quantum chemistry calculations by invoking intermolecular interaction effects. And we have identified the molecular descriptors for the molecular design of RTP materials.In this Account, we started with the proposition of the mechanism of intermolecular electrostatic-interaction-induced RTP at the molecular level by using molecular dynamics simulations, hybrid quantum mechanics, and molecular mechanics (QM/MM) coupled with the thermal vibration correlation function (TVCF) formalism we developed earlier. The effective intermolecular electrostatic interactions could stem from a variety of interactions in different organic RTP crystals, such as hydrogen bonding, π-halogen bonding, anion-π+ interaction, and d-pπ bonds and so forth. We find that these interactions can change the molecular orbital compositions involved in the lowest-lying singlet and triplet excited states that are responsible for phosphorescence, either through facilitating intersystem crossing from the excited-state singlet to the triplet and/or suppressing the nonradiative decay process from the lowest triplet to the ground state. This underlying RTP mechanism is believed to be very helpful in systematically and comprehensively understanding the aggregation/crystal-induced persistent organic RTP, which has been applied to explain a number of experiments.We then propose the molecular descriptors to characterize the phosphorescence efficiency and lifetime, respectively, derived from fundamental photophysical processes and requirements to obey the El-Sayed rule and generate phosphorescence. For a prototypical RTP system consisting of a carbonyl group and π-conjugated segments, the excited states can be regarded as an admixture of n → π* (with portion α) and π → π* (with portion β). The intersystem crossing (ISC) rate of S1 → Tn is mostly governed by the modification of the product of α and β, and the nonradiative rate of T1 → S0 is determined by the β value of T1. Thus, we employ γ = α × β and β to describe the phosphorescence efficiency and lifetime, respectively, which have been successfully applied in the molecular design of efficient and long-lived RTP systems in experiments. The molecular descriptors outlined in this Account, which are easily obtained from simple quantum chemistry calculations, are expected to play important roles in the machine-learning-based molecular screening in the future.

Thermal Vibration Correlation Function Formalism for Molecular Excited State Decay Rates

Thermal Vibration Correlation Function Formalism for Molecular Excited State Decay Rates

Computational Protocol To Predict Anti-Kasha Emissions: The Case of Azulene Derivatives.

In this contribution, we present a computational protocol to predict anti-Kasha photoluminescence. The herein developed protocol is based on state-of-the-art quantum chemical calculations and excited-state decay rate theories (i.e., thermal vibration correlation function formalism), along with appropriate kinetic models which include all relevant electronic states. This protocol is validated for a series of azulene derivatives. For this series, we have computed absorption and emission spectra for both their first and second excited states, their radiative and nonradiative rates, as well as fluorescence yields from the two different excited states. All the studied azulene derivatives are predicted to exclusively display anomalous anti-Kasha S2 emission. A quantitative agreement for the herein computed excited-state spectra, lifetimes, and fluorescence quantum yields is obtained with respect to the experimental values. Given the increasing interest in anti-Kasha emitters, we foresee that the herein developed computational protocol can be used to prescreen dyes with the desired aforementioned anomalous photoluminescence properties.

Spatial correlations of extended cosmological structures

ABSTRACT Studies of large-scale structures in the Universe, such as superstructures or cosmic voids, have been widely used to characterize the properties of the cosmic web through statistical analyses. On the other hand, the two-point correlation function of large-scale tracers such as galaxies or haloes provides a reliable statistical measure. However, this function applies to the spatial distribution of point-like objects, and therefore it is not appropriate for extended large structures that strongly depart from spherical symmetry. Here we present an analysis based on the standard correlation function formalism that can be applied to extended objects exhibiting arbitrary shapes. Following this approach, we compute the probability excess Ξ of having spheres sharing parts of cosmic structures with respect to a realization corresponding to a distribution of the same structures in random positions. For this aim, we identify superstructures defined as future virialized structures (FVSs) in semi-analytic galaxies in the MPDL2 MultiDark simulation. We have also identified cosmic voids to provide a joint study of their relative distribution with respect to the superstructures. Our analysis suggests that Ξ provides useful characterizations of the large-scale distribution, as suggested from an analysis of subsets of the simulation. Even when superstructure properties may exhibit negligible variations across the subsets, Ξ has the sensitivity to statistically distinguish sub-boxes that depart from the mean at larger scales. Thus, our methods can be applied in analysis of future surveys to provide characterizations of large-scale structure suitable to distinguish different theoretical scenarios.

Cyano substitution effect on the emission quantum efficiency in stilbene derivatives: A computational study

Abstract A series of cyano-substituted stilbene derivatives exhibit quenched fluorescence in solution, in sharp contrast to their highly emissive parent counterparts. By employing the polarizable continuum model (PCM) method coupled with thermal vibration correlation function (TVCF) formalism, we comparatively investigated the electronic structures and excited-state dynamics of cyano-substituted para-distyrylbenzene (CN-DSB), 1,2-bis(pyridylphenyl)ethane (CN-BPPE) and the corresponding non-substituted DSB and BPPE lack of the cyano groups at their vinylene moieties in solution. The results show that the non-radiative decay rate constants of the CN-substituents are greatly enhanced by more than 4 orders of magnitude, while the radiative decay rate constants experience slight reduction, resulting in the very tiny fluorescence quantum efficiency. By analyzing the reorganization energies projected onto both normal modes and internal coordinates, the non-radiative consumption ways via molecular torsional motions of the single even double bonds adjacent to the CN group are active upon CN-substitution. Our study presents a rational explanation for the experimental CN-substitution induced quenching phenomena in stilbene derivatives, which is beneficial for the design of new fluorescent emitters and probes.

Theoretical investigations on the electron-vibration couplings and description of opto-electronic properties of materials

One of the challenging topics for physical chemistry is the theoretical modeling of atomistic dynamics at quantum mechanical level, including the processes for chemical bond breaking and formation. If the atomic motions are simply slight deviations away from the equilibrium, their couplings with electrons will determine the physical properties related to mechanical, including opto-, electric/magnetic, and thermal functions. Taking organic light-emitting materials as examples, we developed a vibration correlation function formalism to elucidate the essential role of electron-vibration couplings in evaluating the optical spectrum and light-emitting quantum efficiency starting from electronic structures. Such formalism has been successfully applied to explain the exotic aggregation induced emission phenomena and help molecular design of highly efficient organo-metallic ligand compounds. The predicted aggregation causing isotope effect has been validated by experiment.

Excitonic coupling effect on the nonradiative decay rate in molecular aggregates: Formalism and application

We present here an analytical thermal vibration correlation function formalism to calculate the nonradiative decay rate constant (knr) considering excitonic coupling effect (ECE) for molecular aggregates based on split-operator approximation. Combining with first-principles calculations, we found that knr is enhanced by ECE for both H- and J-aggregates. In addition, ECE is found to be minor for the AIEgens (aggregation-induced emission luminogens).

Revising Intramolecular Photoinduced Electron Transfer (PET) from First-Principles

Photoinduced electron transfer (PET) plays relevant roles in many areas of chemistry, including charge separation processes in photovoltaics, natural and artificial photosynthesis, and photoluminescence sensors and switches. As in many other photochemical scenarios, the structural and energetic factors play relevant roles in determining the rates and efficiencies of PET and its competitive photodeactivation processes. Particularly, in the field of fluorescent sensors and switches, intramolecular PET is believed (in many cases without compelling experimental proof) to be responsible of the quench of fluorescence. There is an increasing experimental interest in fluorophore's molecular design and on achieving optimal excitation/emission spectra, excitation coefficients, and fluorescence quantum yields (importantly for bioimaging purposes), but less efforts are devoted to fundamental mechanistic studies. In this Account, I revise the origins of the fluorescence quenching in some of these systems with state-of-the-art quantum chemical tools. These studies go beyond the common strategy of analyzing frontier orbital energy diagrams and performing PET thermodynamics calculations. Instead, the potential energy surfaces (PESs) of the lowest-lying excited states are explored with time-dependent density functional theory (TD-DFT) and complete active space self-consistent field (CASSCF) calculations and the radiative and nonradiative decay rates from the involved excited states are computed from first-principles using a thermal vibration correlation function formalism. With such a strategy, this work reveals the real origins of the fluorescence quenching, herein entitled as dark-state quenching. Dark states (those that do not absorb or emit light) are often elusive to experiments and thus, computational investigations can provide novel insights into the actual photodeactivation mechanisms. The success of the dark-state quenching mechanism is demonstrated for a wide variety of fluorescent probes, including proton, cation and anion targets. Furthermore, this mechanism provides a general picture of the fluorescence quenching which englobes intramolecular charge-transfer (ICT), ratiometric quenching, and those radiationless mechanisms believed to be originated by PET. Finally, this Account provides for the first time a computational protocol to quantitatively estimate this phenomenon and provides the ingredients for the optimal design of fluorescent probes from first principles.

NMR signal for particles diffusing under potentials: From path integrals and numerical methods to a model of diffusion anisotropy.

We study the influence of diffusion on NMR experiments when the molecules undergo random motion under the influence of a force field and place special emphasis on parabolic (Hookean) potentials. To this end, the problem is studied using path integral methods. Explicit relationships are derived for commonly employed gradient waveforms involving pulsed and oscillating gradients. The Bloch-Torrey equation, describing the temporal evolution of magnetization, is modified by incorporating potentials. A general solution to this equation is obtained for the case of parabolic potential by adopting the multiple correlation function (MCF) formalism, which has been used in the past to quantify the effects of restricted diffusion. Both analytical and MCF results were found to be in agreement with random walk simulations. A multidimensional formulation of the problem is introduced that leads to a new characterization of diffusion anisotropy. Unlike the case of traditional methods that employ a diffusion tensor, anisotropy originates from the tensorial force constant, and bulk diffusivity is retained in the formulation. Our findings suggest that some features of the NMR signal that have traditionally been attributed to restricted diffusion are accommodated by the Hookean model. Under certain conditions, the formalism can be envisioned to provide a viable approximation to the mathematically more challenging restricted diffusion problems.

Organic light-emitting diodes: theoretical understanding of highly efficient materials and development of computational methodology

Abstract Theoretical understanding of organic light-emitting diodes started from the quest to the nature of the primary excitation in organic molecular and polymeric materials. We found the electron correlation strength, bond-length alternation as well as the conjugation extent have strong influences on the orderings of the lowest lying excited states through the first application of density matrix renormalization group theory to quantum chemistry. The electro-injected free carriers (with spin 1/2) can form both singlet and triplet bound states. We found that the singlet exciton formation ratio can exceed the conventional 25% spin statistics limit. We proposed a vibration correlation function formalism to evaluate the excited-state decay rates, which is shown to not only give reasonable estimations for the quantum efficiency but also a quantitative account for the aggregation-induced emission (AIE). It is suggested to unravel the AIE mechanism through resonance Raman spectroscopy.

Anharmonic Infrared Spectroscopy through the Fourier Transform of Time Correlation Function Formalism in ONETEP.

Density functional theory molecular dynamics (DFT-MD) provides an efficient framework for accurately computing several types of spectra. The major benefit of DFT-MD approaches lies in the ability to naturally take into account the effects of temperature and anharmonicity, without having to introduce any ad hoc or a posteriori corrections. Consequently, computational spectroscopy based on DFT-MD approaches plays a pivotal role in the understanding and assignment of experimental peaks and bands at finite temperature, particularly in the case of floppy molecules. Linear-scaling DFT methods can be used to study large and complex systems, such as peptides, DNA strands, amorphous solids, and molecules in solution. Here, we present the implementation of DFT-MD IR spectroscopy in the ONETEP linear-scaling code. In addition, two methods for partitioning the dipole moment within the ONETEP framework are presented. Dipole moment partitioning allows us to compute spectra of molecules in solution, which fully include the effects of the solvent, while at the same time removing the solvent contribution from the spectra.