Search NASA⌕ Search

SEARCH · Search NASA

Results for “Quantum mechanics”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 307 records · Page 17
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Entanglement entropy and non-local duality: Quantum channels and quantum algebras

Here, we investigate the transformation of entanglement entropy under dualities, using the Kramers–Wannier duality present in the transverse field Ising model as our example. Entanglement entropy between local spin degrees of freedom is not generically preserved by the duality; instead, entangled states may be mapped to states with no local entanglement. To understand the fate of this entanglement, we consider two quantitative descriptions of degrees of freedom and their transformation under duality. The first involves Kraus operators implementing the partial trace as a quantum channel, while the second utilizes the algebraic approach to quantum mechanics, where degrees of freedom are encoded in subalgebras. Using both approaches, we show that entanglement of local degrees of freedom is not lost; instead it is transferred to non-local degrees of freedom by the duality transformation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Simultaneous Measurements of Noncommuting Observables: Positive Transformations and Instrumental Lie Groups

We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function, defined relative to the invariant measure of the instrumental Lie group. This diffusion can be analyzed using Wiener path integration, stochastic differential equations, or a Fokker-Planck-Kolmogorov equation. This way of considering instrument evolution we call the Instrument Manifold Program. We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover that we call the universal instrumental Lie group, which is independent not just of states, but also of Hilbert space. The universal instrument is generically infinite dimensional, in which case the instrument’s evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, and it can be analyzed within the differential geometry of the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, position and momentum, and the three components of angular momentum. As these measurements are performed continuously, they converge to strong simultaneous measurements. For a single observable, this results in the standard decay of coherence between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Multicomponent Cholesky Decomposition: Application to Nuclear–Electronic Orbital Theory

The Cholesky decomposition technique is commonly used to reduce the memory requirement for storing two-particle repulsion integrals in quantum chemistry calculations that use atomic orbital bases. However, when quantum methods use multicomponent bases, such as nuclear–electronic orbitals, additional challenges are introduced due to asymmetric two-particle integrals. This work proposes several multicomponent Cholesky decomposition methods for calculations using nuclear–electronic orbital density functional theory. To analyze the errors in different Cholesky decomposition components, benchmark calculations using water clusters are carried out. The largest benchmark calculation is a water cluster (H 2 O) 27 where all 54 protons are treated quantum mechanically. Furthermore, this study provides energetic and complexity analyses to demonstrate the accuracy and performance of the proposed multicomponent Cholesky decomposition method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

$T\bar{T}$ in JT Gravity and BF Gauge Theory

JT gravity has a first-order formulation as a two-dimensional BF theory, which can be viewed as the dimensional reduction of the Chern-Simons description of 3d 3 d gravity. We consider {T\overbar{T}} T T ¯ -type deformations of the (0+1) ( 0 + 1 ) -dimensional dual to this 2d 2 d BF theory and interpret the deformation as a modification of the BF theory boundary conditions. The fundamental observables in this deformed BF theory, and in its 3d 3 d Chern-Simons lift, are Wilson lines and loops. In the 3d 3 d Chern-Simons setting, we study modifications to correlators involving boundary-anchored Wilson lines which are induced by a {T\overbar{T}} T T ¯ deformation on the 2d 2 d boundary; results are presented at both the classical level (using modified boundary conditions) and the quantum-mechanical level (using conformal perturbation theory). Finally, we calculate the analogous deformed Wilson line correlators in 2d 2 d BF theory below the Hagedorn temperature where the principal series dominates over the discrete series.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Creating large Fock states and massively squeezed states in optics using systems with nonlinear bound states in the continuum

The quantization of the electromagnetic field leads directly to the existence of quantum mechanical states, called Fock states, with an exact integer number of photons. Despite these fundamental states being long-understood, and despite their many potential applications, generating them is largely an open problem. For example, at optical frequencies, it is challenging to deterministically generate Fock states of order two and beyond. Here, we predict the existence of an effect in nonlinear optics, which enables the deterministic generation of large Fock states at arbitrary frequencies. The effect, which we call an n-photon bound state in the continuum, is one in which a photonic resonance (such as a cavity mode) becomes lossless when a precise number of photons n is inside the resonance. Based on analytical theory and numerical simulations, we show that these bound states enable a remarkable phenomenon in which a coherent state of light, when injected into a system supporting this bound state, can spontaneously evolve into a Fock state of a controllable photon number. This effect is also directly applicable for creating (highly) squeezed states of light, whose photon number fluctuations are (far) below the value expected from classical physics (i.e., shot noise). In conclusion, we suggest several examples of systems to experimentally realize the effects predicted here in nonlinear nanophotonic systems, showing examples of generating both optical Fock states with large n (n > 10), as well as more macroscopic photonic states with very large squeezing, with over 90% less noise (10 dB) than the classical value associated with shot noise.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Revealing quantum effects in highly conductive δ-layer systems

Abstract Thin, high-density layers of dopants in semiconductors, known as δ -layer systems, have recently attracted attention as a platform for exploration of the future quantum and classical computing when patterned in plane with atomic precision. However, there are many aspects of the conductive properties of these systems that are still unknown. Here we present an open-system quantum transport treatment to investigate the local density of electron states and the conductive properties of the δ -layer systems. A successful application of this treatment to phosphorous δ -layer in silicon both explains the origin of recently-observed shallow sub-bands and reproduces the sheet resistance values measured by different experimental groups. Further analysis reveals two main quantum-mechanical effects: 1) the existence of spatially distinct layers of free electrons with different average energies; 2) significant dependence of sheet resistance on the δ -layer thickness for a fixed sheet charge density.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Simultaneous Optimization of Nuclear–Electronic Orbitals

Accurate modeling of important nuclear quantum effects, such as nuclear delocalization, zero-point energy, and tunneling, as well as non-Born-Oppenheimer effects, requires treatment of both nuclei and electrons quantum mechanically. The nuclear–electronic orbital (NEO) method provides an elegant framework to treat specified nuclei, typically protons, on the same level as the electrons. In conventional electronic structure theory, finding a converged ground state can be a computationally demanding task; converging NEO wavefunctions, due to their coupled electronic and nuclear nature, is even more demanding. Herein, we present an efficient simultaneous optimization method that uses the direct inversion in the iterative subspace method to simultaneously converge wavefunctions for both the electrons and quantum nuclei. In conclusion, benchmark studies show that the simultaneous optimization method can significantly reduce the computational cost compared to the conventional stepwise method for optimizing NEO wavefunctions for multicomponent systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Attosecond XUV probing of vibronic quantum superpositions in Br 2 +

Ultrafast laser excitation can create coherent superpositions of electronic states in molecules and trigger ultrafast flow of electron density on a few femtosecond timescale. While recent attosecond experiments have addressed real-time observation of these primary photochemical processes, the underlying roles of simultaneous nuclear motions and how they modify and disturb the valence electronic motion remain uncertain. In this work, we investigate coherent electronic-vibrational dynamics induced among multiple vibronic levels of ionic bromine (Br 2 + ), including both spin-orbit (X 2 Π 3/2,g – X 2 Π 1/2,g ) and valence (A 2 Π 3/2,u – 4 Σ 3/2,u ) electronic superpositions, using attosecond transient absorption spectroscopy. Decay, revival, and apparent frequency shifts of electronic coherences are measured via characteristic quantum beats on the Br-3d core-level absorption signals. Quantum-mechanical simulations attribute the observed electronic decoherence to broadened phase distributions of nuclear wave packets on anharmonic potentials. Molecular vibronic structure is further revealed to be imprinted as discrete progressions in electronic beat frequencies. These results provide a future basis to interpret complex charge-migration dynamics in polyatomic systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Integrated Quantum-Classical Protocol for the Realistic Description of Solvated Multinuclear Mixed-Valence Transition-Metal Complexes and Their Solvatochromic Properties

Linear cyanide-bridged polymetallic complexes, which undergo photoinduced metal-to-metal charge transfer, represent prototypical systems for studying long-range electron-transfer reactions and understanding the role played by specific solute–solvent interactions in modulating the excited-state dynamics. Here, to tackle this problem, while achieving a statistically meaningful description of the solvent and of its relaxation, one needs a computational approach capable of handling large polynuclear transition-metal complexes, both in their ground and excited states, as well as the ability to follow their dynamics in several environments up to nanosecond time scales. Here, we present a mixed quantum classical approach, which combines large-scale molecular dynamics (MD) simulations based on an accurate quantum mechanically derived force field (QMD-FF) and self-consistent QMD polarized point charges, with IR and UV–vis spectral calculations to model the solvation dynamics and optical properties of a cyano-bridged trinuclear mixed-valence compound (trans-[(NC) 5 Fe III (μ-CN)Ru II (pyridine) 4 (μ-NC)Fe III (CN) 5 ] 4– ). We demonstrate the reliability of the QMD-FF/MD approach in sampling the solute conformational space and capturing the local solute–solvent interactions by comparing the results with higher-level quantum mechanics/molecular mechanics (QM/MM) MD reference data. The IR spectra calculated along the classical MD trajectories in different solvents correctly predict the red shift of the CN stretching band in the aprotic medium (acetonitrile) and the subtle differences measured in water and methanol, respectively. By explicitly including the solvent molecules around the cyanide ligands and calculating the thermal averaged absorption spectra using time-dependent density functional theory calculations within the Tamm–Dancoff approximation, the experimental solvatochromic shift is quantitatively reproduced going from water to methanol, while it is overestimated for acetonitrile. This discrepancy can likely be traced back to the lack of important dispersion interactions between the solvent cyano groups and the pyridine substituents in our micro solvation model. The proposed protocol is applied to the ground state in water, methanol, and acetonitrile and can be flexibly generalized to study excited-state nonequilibrium solvation dynamics.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Photoinduced charge transfer in Zn(II) and Au(III)-ligated symmetric and asymmetric bacteriochlorin dyads: A computational study

The excited-state properties and photoinduced charge-transfer (CT) kinetics in a series of symmetrical and asymmetrical Zn- and Au-ligated meso–meso-connected bacteriochlorin (BChl) complexes are studied computationally. BChl derivatives, which are excellent near-IR absorbing chromophores, are found to play a central role in bacterial photosynthetic reaction centers but are rarely used in artificial solar energy harvesting systems. The optical properties of chemically linked BChl complexes can be tuned by varying the linking group and involving different ligated metal ions. We investigate charge transfer in BChl dyads that are either directly linked or through a phenylene ring (1,4- phenylene) and which are ligating Zn or Au ions. The directly linked dyads with a nearly perpendicular arrangement of the BChl units bear markedly different properties than phenylene linked dyads. In addition, we find that the dielectric dependence of the intramolecular CT rate is very strong in neutral Zn-ligated dyads, whereas cationic Au-ligated dyads show negligible dielectric dependence of the CT rate. Rate constants of the photo induced CT process are calculated at the semiclassical Marcus level and are compared to fully quantum mechanical Fermi’s golden rule based values. The rates are calculated using a screened range separated hybrid functional that offers a consistent framework for addressing environment polarization. We study solvated systems in two solvents of a low and a high scalar dielectric constant.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

Discerning Influences from Enthalpy and Entropy at Aqueous Interfaces Involved in Biomass Conversions in Porous Catalysts

Project Summary: The goal in this proposal is to learn how solvent influences the enthalpies and entropies of catalytic species in zeolite pores. Specifically, the physical, chemical, and structural features of solvent, catalytic species, and zeolite pores that determine enthalpies and entropies of solvation in solution phase biomass conversions will be interrogated using multiscale simulations and machine learning. Multiscale simulations are based off of existing strategies and employ quantum mechanics and classical molecular dynamics, providing an excellent balance between chemical accuracy and computational expense. They are capable of calculating enthalpies and entropies of solvation separately and have been validated in prior work to achieve high accuracy compared to their parent methods.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Structural and physical properties of 99 complex bulk chalcogenides crystals using first-principles calculations

Chalcogenide semiconductors and glasses have many applications in the civil and military fields, especially in relation to their electronic, optical and mechanical properties for energy conversion and in enviormental materials. However, they are much less systemically studied and their fundamental physical properties for a large class chalcogenide semiconductors are rather scattered and incomplete. Here, we present a detailed study using well defined first-principles calculations on the electronic structure, interatomic bonding, optical, and mechanical properties for 99 bulk chalcogenides including thirteen of these crytals which have never been calculated. Due to their unique composition and structures, these 99 bulk chalcogenides are divided into two main groups. The first group contains 54 quaternary crystals with the structure composition (A2BCQ4) (A = Ag, Cu; B = Zn, Cd, Hg, Mg, Sr, Ba; C = Si, Ge, Sn; Q = S, Se, Te), while the second group contains scattered ternary and quaternary chalcogenide crystals with a more diverse composition (AxByCzQn) (A = Ag, Cu, Ba, Cs, Li, Tl, K, Lu, Sr; B = Zn, Cd, Hg, Al, Ga, In, P, As, La, Lu, Pb, Cu, Ag; C = Si, Ge, Sn, As, Sb, Bi, Zr, Hf, Ga, In; Q = S, Se, Te; x=1, 2, 3; y=0, 1, 2, 5; z=0, 1, 2 and n=3, 4, 5, 6, 9). Moreover, the total bond order density (TBOD) is used as a single quantum mechanical metric to characterize the internal cohesion of these crystals enabling us to correlate them with the calculated properties, especially their mechanical properties. This work provides a very large database for bulk chalcogenides crucial for the future theoretical and experimental studies, opening opportunities for study the properties and potential application of a wide variety of chalcogenides.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

BETO 2021 Peer Review - Biochemical Process Modeling and Simulation (BPMS)

The Biochemical Process Modeling and Simulation project aims to reduce the cost and time of research by applying theory, modeling, and simulation to the most relevant bottlenecks in the biochemical process. We use molecular modeling, quantum mechanics, metabolic modeling, fluid dynamics, and reaction-diffusion methods in close collaboration with pretreatment, hydrolysis, upgrading, and TEA. The project's outcomes are increased yields and efficiency of the biochemical process, added value to products, and reduced price of fuels by specifically targeting catalytic efficiency, reactor design, enzyme efficiency, and microbial design.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2020

On the Planckian bound for heat diffusion in insulators

In an insulator, thermal transport at high temperature is expected to be dominated by entirely classical phonon dynamics. In apparent tension with this expectation, recent experimental observations have led to the conjecture that the transport lifetime, τ, is subject to a Planckian bound from below, namely, τ ≳ τ Pl ≡ ℏ / (k B T). Here, we argue that this Planckian bound is due to a quantum-mechanical bound on the sound velocity: v s < v M . The ‘melting velocity’ v M is defined in terms of the melting temperature of the crystal, the interatomic spacing and Planck’s constant. We show that for several classes of insulating crystals, both simple and complex, τ / τ Pl ≈ v M / v s at high temperatures. The velocity bound therefore implies the Planckian bound.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Influence of uranyl complexation on the reaction kinetics of the dodecane radical cation with used nuclear fuel extraction ligands (TBP, DEHBA, and DEHiBA)

Specialized extractant ligands – such as tri-butyl phosphate (TBP), N,N-di-(2-ethylhexyl)butyramide (DEHBA), and N,N-di-2-ethylhexylisobutryamide (DEHiBA) – have been developed for the recovery of uranium from used nuclear fuel by reprocessing solvent extraction technologies. These ligands must function in the presence of an intense multi-component radiation field, and thus it is critical that their radiolytic behaviour be thoroughly evaluated. This is especially true for their metal complexes, where there is negligible information on the influence of complexation on radiolytic reactivity, despite the prevalence of metal complexes in used nuclear fuel reprocessing solvent systems. Here we present a kinetic investigation into the effect of uranyl (UO 2 2+ ) complexation on the reaction kinetics of the dodecane radical cation (RH˙ + ) with TBP, DEHBA, and DEHiBA. Complexation had negligible effect on the reaction of RH˙ + with TBP, for which a second-order rate coefficient (k) of (1.3 ± 0.1) × 10 10 M -1 s -1 was measured. For DEHBA and DEHiBA, UO 2 2+ complexation afforded an increase in their respective rate coefficients: k(RH˙ + + [UO 2 (NO 3 ) 2 (DEHBA) 2 ]) = (2.5 ± 0.1) × 10 10 M -1 s -1 and k(RH˙ + + [UO 2 (NO 3 ) 2 (DEHiBA) 2 ]) = (1.6 ± 0.1) × 10 10 M -1 s -1 . This enhancement with complexation is indicative of an alternative RH˙ + reaction pathway, which is more readily accessible for [UO 2 (NO 3 ) 2 (DEHBA) 2 ] as it exhibited a much larger kinetic enhancement than [UO 2 (NO 3 ) 2 (DEHiBA) 2 ], 2.6× vs. 1.4×, respectively. Complementary quantum mechanical calculations suggests that the difference in reaction kinetic enhancement between TBP and DEHBA/DEHiBA is attributed to a combination of reaction pathway (electron/hole transfer vs. proton transfer) energetics and electron density distribution, wherein attendant nitrate counter anions effectively ‘shield’ TBP from RH˙ + electron transfer processes.

38 RADIATION CHEMISTRY, RADIOCHEMISTRY, AND NUCLEA↗
DOE OSTI2025

Coupling to rotational manifolds to improve gas-phase pump–probe spectroscopic models

The physical picture of gas-phase optical transitions is normally presented as an isolated two-level system balanced by upward and downward processes. Isolated models assume a phenomenological treatment of collisional dephasing but do not strictly account for collisional population exchange with the rotational baths. While this assumption is valid under low-intensity conditions, where excitation is rate-limiting, isolated models can deviate from Beer’s Law at sufficient pressures and monochromatic intensities when both collisional broadening and power broadening are comparable to (or greater than) lifetime broadening, which are not uncommon conditions for cavity enhanced spectroscopies in the mid-IR spectral range. Although this problem has been addressed by rate-equation models for linear absorption measurements, a general treatment for multi-level quantum mechanical models suitable for non-linear absorption measurements (two-photon/two-color/pump–probe) is lacking. Isolated models require physical parameter inputs that disagree with expected values by at least an order of magnitude. These non-physical models undermine the ability to predict non-linear signal strengths under untested conditions and thereby limit the potential to optimize the sensitivity of non-linear spectroscopies and to expand their analytical applications (e.g., new analytes and/or buffer gases, changes in cavity free-spectral-range, changes in intracavity powers or wavelengths, and accurate investigation of physical phenomena). In this study, we derive bath-coupled models for gaseous pump–probe spectroscopy by application of the quantum Lindblad equation and detailed balance. Bath-coupled models are shown to fit data consistently across variations in intensity and agree with all physically expected values.

Cavity ring-down spectroscopy↗
DOE OSTI2020

Enabling Large-Scale Condensed-Phase Hybrid Density Functional Theory Based Ab Initio Molecular Dynamics. 1. Theory, Algorithm, and Performance

By including a fraction of exact exchange (EXX), hybrid functionals reduce the self-interaction error in semilocal density functional theory (DFT) and thereby furnish a more accurate and reliable description of the underlying electronic structure in systems throughout biology, chemistry, physics, and materials science. However, the high computational cost associated with the evaluation of all required EXX quantities has limited the applicability of hybrid DFT in the treatment of large molecules and complex condensed-phase materials. To overcome this limitation, we describe a linear-scaling approach that utilizes a local representation of the occupied orbitals (e.g., maximally localized Wannier functions (MLWFs)) to exploit the sparsity in the real-space evaluation of the quantum mechanical exchange interaction in finite-gap systems. In this work, we present a detailed description of the theoretical and algorithmic advances required to perform MLWF-based ab initio molecular dynamics (AIMD) simulations of large-scale condensed-phase systems of interest at the hybrid DFT level. We focus our theoretical discussion on the integration of this approach into the framework of Car–Parrinello AIMD, and highlight the central role played by the MLWF-product potential (i.e., the solution of Poisson’s equation for each corresponding MLWF-product density) in the evaluation of the EXX energy and wave function forces. We then provide a comprehensive description of the exx algorithm implemented in the open-source Quantum ESPRESSO program, which employs a hybrid MPI/OpenMP parallelization scheme to efficiently utilize the high-performance computing (HPC) resources available on current- and next-generation supercomputer architectures. Furthermore, this is followed by a critical assessment of the accuracy and parallel performance (e.g., strong and weak scaling) of this approach when AIMD simulations of liquid water are performed in the canonical (NVT) ensemble. With access to HPC resources, we demonstrate that exx enables hybrid DFT-based AIMD simulations of condensed-phase systems containing 500–1000 atoms (e.g., (H₂O)₂₅₆) with a wall time cost that is comparable to that of semilocal DFT. In doing so, exx takes us one step closer to routinely performing AIMD simulations of complex and large-scale condensed-phase systems for sufficiently long time scales at the hybrid DFT level of theory.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

In situ x-ray absorption investigations of a heterogenized molecular catalyst and its interaction with a carbon nanotube support

A highly active heterogenized molecular CO 2 reduction catalyst on a conductive carbon support is investigated to identify if its improved catalytic activity can be attributed to strong electronic interactions between catalyst and support. The molecular structure and electronic character of a [Re +1 (tBu-bpy)(CO) 3 Cl] (tBu-bpy = 4,4'-tert-butyl-2,2'-bipyridine) catalyst deposited on multiwalled carbon nanotubes are characterized using Re L 3 -edge x-ray absorption spectroscopy under electrochemical conditions and compared to the homogeneous catalyst. The Re oxidation state is characterized from the near-edge absorption region, while structural changes of the catalyst are assessed from the extended x-ray absorption fine structure under reducing conditions. Chloride ligand dissociation and a Re-centered reduction are both observed under applied reducing potential. The results confirm weak coupling of [Re(tBu-bpy)(CO) 3 Cl] with the support, since the supported catalyst exhibits the same oxidation changes as the homogeneous case. However, these results do not preclude strong interactions between a reduced catalyst intermediate and the support, preliminarily investigated here using quantum mechanical calculations. Thus, our results suggest that complicated linkage schemes and strong electronic interactions with the initial catalyst species are not required to improve the activity of heterogenized molecular catalysts.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗