Modeling of Spin‐Forbidden Reactions

A spin-forbidden reaction is a reaction in which the total electronic spin-state changes. The standard transition-state theory that assumes a reaction occurs on a single potential energy surface with spin-conservation cannot be applied to a spin-forbidden reaction directly. In this work, we derive the crossing coefficient based on the Wentzel-Kramers-Brillouin (WKB) theory to quantify the effect of intersystem crossing on the kinetics of spin-forbidden reactions. Acrylates and methacrylates, by themselves, can generate free radicals that initiate polymerization at temperatures above 120 °C. Previous studies suggest that a triplet diradical is a key intermediate in the self-initiation. The formation of a triplet diradical from two closed-shell monomer molecules is a spin-forbidden reaction. This study provides a quantitative analysis of singlet-triplet spin crossover of diradical species in self-initiation of acrylates and methacrylates, taking into account the effect of intersystem crossing. The concept of crossing control is introduced and demonstrated computationally to be a new likely route to generate monoradicals via monomer self-initiation in high temperature polymerization.

Theoretical study of spin–orbit coupling and zero-field splitting in the spin-forbidden two-state reaction between cobaltacyclopentadiene and isocyanate

In this paper we develop a “diatomic in molecules semiempirical ligand field” (DIMSELF) method to calculate ground and excited many-body potential energy surfaces for an arbitrary transition metal ion in an arbitrary complex system. This method is not restricted to a high-symmetry environment and is meant to be inexpensive and suitable for nonadiabatic excited states dynamics on-the-fly. Within the approximations employed, the method includes full CI (configuration interaction) and SO (spin-orbit) interactions, essential to the description of nonradiative transitions such as those of myoglobin in the presence of carbon monoxide. We test our method against high level ab initio calculations for a simple model system of myoglobin’s heme pocket. Finally, we discuss our results and compare with previous calculations in the literature. Transition metal complexes play a fundamental role in many technological and biological areas. They act as catalysts in the oil industry during the processes of cracking and reforming, they are also fundamental for oxygen transport in blood, and they are key to plant respiration. Some of these processes can be well understood without the need to include the small contribution of spin-orbit (SO) interaction which, for the most part, is of the order of the error one performs in ab initio calculations of these kind of systems. In some cases, however, nonradiative transitions occur between states with different spins. Of particular interest are heme proteins, a biologically important group of molecules that have a unique common active site: an iron-protoporphyrin-IX complex, where changes in spin state can be an integral part of the protein function. One of the most striking biological manifestations of this modulation is the binding of molecular oxygen and carbon monoxide (CO) 1-9 to the five-coordinated ferrous heme site of globins. Another wellstudied example is the situation of several spin changes in the enzymatic activation cycles of cytochromes P450. These transitions are solely due to SO coupling and are otherwise forbidden. Ab initio calculations are performed within the framework of Hund’s Case A and hence assume that S, the total spin of the system, is a good quantum number. Spin-orbit coupling matrix elements could then be added, and rediagonalization of the Hamiltonian matrix would provide us with the spin coupled eigenstates of the system. Obtaining several electronic excited states for a given spin multiplicity is complex, and for mediumsized systems only configuration interaction for single excitations (CIS) is affordable. However, a more complete description of excited states might be required for the study of dynamics. For large systems, where the environment can be introduced by mixing quantum chemistry with molecular mechanics, electronically excited states are rarely computationally accessible and only the fundamental ground state for each spin multiplicity might be computed. For nonsymmetric systems, even at the density functional theory (DFT) level of theory, several attempts with different initial guesses should be considered in order to achieve the true ground state. The computational effort is drastically increased when several nuclear configurations are required for a complete study of the dynamics, and calculations on-the-fly, including excited states and SO coupling, are simply out of the question. Therefore, a semiempirical approach that includes excited states and SO coupling is very desirable. The method we develop in this paper is a generalization of Gerloch’s 10 cellular AOM (angular overlap model) version of ligand field theory, combined with a semiempirical version of Ellison’s 11-13 DIM (diatomics in molecules). Neither ligand field theory nor the semiempirical version of DIM are exact ab initio methods; not all the parameters can be obtained by analytical methods and they ought to be fitted either to experimental data or other calculations. On the other hand, once these parameters are known, fast computation of ground- and excited-state potentials can be performed without having to recalculate wave functions and their matrix elements. We test this method against a simplified model of the heme

This paper offers educational insight into the Dirac equation, examining its historical context and contrasting it with the earlier Schrödinger and Klein-Gordon equations. The comparison highlights their Lorentz transformation symmetry and potential probabilistic interpretations. We explicitly solve the free particle dynamics in Dirac's model, revealing the emergence of solutions with negative energy. In this discussion, we examine the Dirac Sea Hypothesis and explore the inherent helicity of the solutions. Additionally, we demonstrate how the Dirac equation accounts for spin and derive the Pauli equation in the non-relativistic limit. Through the Foldy-Wouthuysen transformation, we reveal how the equation incorporates spin-orbit interaction and other relativistic effects, ultimately leading to the hydrogen fine structure. A section on relativistic covariant notation is included to emphasize the invariance of the Dirac equation, along with more refined formulations of both the KG and Dirac equations. Designed for undergraduate students with an interest in the Dirac equation, this resource provides a historical perspective without being purely theoretical. Our approach underscores the significance of a pedagogical method that combines historical and comparative elements to profoundly understand the role of the Dirac equation in modern physics.

We study the two-body and three-body bound states in ultracold atomic mixtures with one of the atoms subjected to an isotropic spin-orbit (SO) coupling. We consider a system of two identical fermions interacting with one SO coupled atom. It is found that there can exist two types of three-body bound states, Efimov trimers and universal trimers. The Efimov trimers are energetically less favored by the SO coupling, which will finally merge into the atom-dimer threshold as increasing the SO coupling strength. Nevertheless, these trimers exhibit a new kind of discrete scaling law incorporating the SO coupling effect. On the other hand, the universal trimers are more favored by the SO coupling. They can be induced at negative s-wave scattering lengths and with smaller mass ratios than those without SO coupling. These results are obtained by both the Born-Oppenheimer approximation and exact solutions from three-body equations.

Extensive ab initio study of the electronic states of CP radical including spin–orbit coupling

We investigate the coherent transport properties of electrons in a side terminal of a four-terminal nano device made of quantum wires with Rashba spin-orbit (SO) coupling in the presence of magnetic filed. We numerically calculate the charge and spin conductances dependent on SO coupling strength and reduced flux. The results imply that the coherent transport in this device is determined by the interplay of Rashba SO coupling and magnetic filed. For specific values of magnetic flux and SO coupling strength, a complete blocking can take place. It also shows that there is a de-blocking phenomenon induced by the interplay of magnetic flux and SO coupling. Such a 4-terminal and multi-channel structure may provide more options of controlling methods for the coherent charge and spin transport.

The two-state reaction mechanism of Pt4+ with CH4 on the quartet and doublet potential energy surfaces has been investigated at the B3LYP level. Crossing points between the potential energy surfaces are located using different methods, and possible spin inversion processes are discussed by means of spin−orbit coupling (SOC) calculations. As a result, after the C−H insertion intermediates, two distinct H2 elimination reaction paths have been found, and the second C−H bond broken is regarded as the rate-determining step for two reaction paths. For the first pathway, there is a minimal energy crossing point (MECP) between the two surfaces, where the reacting system will change its spin multiplicities from the quartet state to the doublet state near this crossing region because the magnitude of the spin multiplicity mixing increases in a small energy gap between high- and low-spin states and will greatly enhance the probability of the intersystem crossing. The single P1ISC and double P2ISC passes estimated at MECP (SOC = 201.25 cm−1) are approximately 0.51 and 0.76, respectively. Not only will the reaction overcome a spin-change-induced barrier (ca. 6 kcal/mol) but it also will overcome an adiabatic barrier (ca. 24.7−20.8 kcal/mol), which will greatly reduced the efficiency of the reaction system. As for the second reaction path, similarly, there is a crossing seam between CP-2 and CP-3, the spin-change-induced barrier is very high, about 15.86−18.86 kcal/mol, which also indicates a less favorable process kinetically. Therefore, the lack of a thermodynamic driving force is an important factor contributing to the low efficiency of the reaction system. These conclusions are consistent with the experimental observations.

A simple procedure is described for calculating spin-orbit and nonadiabatic interactions by an extension of the diatomics-in-molecules approach. The method can be easily applied to a large number of polyatomic systems and will hopefully be of value in obtaining qualitative and quantitative descriptions of many important molecular collision processes which involve more than one potential energy surface. Results are presented for two such processes, the quenching of Li(2 2P) by H2 and the reaction of F atoms with H2.

The quasi-bound states which appear as a consequence of the Rashba spin-orbit (SO) coupling, introduce a strongly irregular behavior of the spin-FET conductance at large Rashba parameter. Moreover, the presence of the bulk inversion asymmetry, i.e. the Dresselhaus SO coupling, may compromise the spin-valve effect even at small values of the Rashba parameter. However, by introducing stray electric fields in addition to the SO couplings, we show that the effect of the SO induced quasi-bound states can be tuned. The oscillations of the spin-resolved conductance become smoother and the control of the spin-FET characteristics becomes possible. For the calculations we employ a multi-channel scattering formalism, based on the R-matrix method extended to spin transport, in the presence of Rashba and Dresselhaus SO couplings.

We address the impact of the spin-orbit (SO) coupling on the localization-delocalization-transition (LDT) in a spin-orbit coupled Bose-Einstein condensate in a bichromatic potential. We find that SO coupling significantly alters the threshold depth of the one of sublattices above which the lowest eigenstates transform from delocalizated into localized. For some moderate coupling strengths the threshold is strongly reduced, which is explained by the SO coupling-induced band flattening in one of the sub-lattices. We explain why simultaneous Rabi and SO coupling are necessary ingredients for LDT threshold cancellation and show that strong SO coupling drives the system into the state where its evolution becomes similar to the evolution of a one-component system. We also find that defocusing nonlinearity can lead to localization of the states which are delocalized in the linear limit.

The potential energy curves (PECs) of 23 Ω states generated from the 12 electronic states (X1 Σ +, 21 Σ +, 11 Σ −, 11 Π, 21 Π, 11 Δ, 13 Σ +, 23 Σ +, 13 Σ −, a3 Π, 23 Π and 13 Δ) are studied for the first time. All the states correlate to the first dissociation channel of the SiBr+ cation. Of these electronic states, the 23 Σ + is the repulsive one without the spin-orbit coupling, whereas it becomes the bound one with the spin-orbit coupling added. On the one hand, without the spin-orbit coupling, the 11 Π, 21 Π and 23 Π are the rather weakly bound states, and only the 11 Π state possesses the double well; on the other hand, with the spin-orbit coupling included, the a3 Π and 11 Π states possess the double well, and the 13 Σ + and 13 Σ − are the inverted states. The PECs are calculated by the CASSCF method, which is followed by the internally contracted MRCI approach with the Davidson modification. Scalar relativistic correction is calculated by the third-order Douglas-Kroll Hamiltonian approximation with a cc-pVTZ-DK basis set. Core-valence correlation correction is included with a cc-pCVTZ basis set. The spin-orbit coupling is accounted for by the state interaction method with the Breit-Pauli Hamiltonian using the all-electron aug-cc-pCVTZ basis set. All the PECs are extrapolated to the complete basis set limit. The variation with internuclear separation of the spin-orbit coupling constant is discussed in brief. The spectroscopic parameters are evaluated for the 11 bound electronic states and the 23 bound Ω states, and are compared with available measurements. Excellent agreement has been found between the present results and the experimental data. It demonstrates that the spectroscopic parameters reported here can be expected to be reliably predicted ones. The Franck-Condon factors and radiative lifetimes of the transitions from the a3 Π 0 + and a3 Π 1 states to the X1 Σ + 0+ state are calculated for several low vibrational levels, and some brief discussion has been made.

Supposing the spin-triplet superconducting state of Sr$_2$RuO$_4$, the spin-orbit (SO) coupling associated with relative motion in Cooper pairs is calculated by extending the method for the dipole-dipole coupling given by Leggett in the superfluid $^{3}$He. It is shown that the SO coupling works only in the equal-spin pairing (ESP) state to make the pair angular momentum $\hbar{\vec L}$ and the pair spin angular momentum ${\rm i}{\vec d}\times{\vec d}^{*}$ parallel with each other. The SO coupling gives rise to the internal Josephson effect in a chiral ESP state as in superfluid A-phase of $^3$He with a help of an additional anisotropy arising from SO coupling of atomic origin which works to direct the {\bf d}-vector into $ab$-plane. This resolves the problem of the anomalous relaxation of $^{17}$O-NQR and the structure of {\bf d}-vector in Sr$_2$RuO$_4$.

Coupled-Channel Studies of Rotational and Vibrational Energy Transfer by Collision

Classical transition state theory (TST) provides the rigorous basis for the application of molecular dynamics (MD) to infrequent events, i.e., reactions that are slow due to a high energy barrier. The TST rate is simply the equilibrium flux through a surface that divides reactants from products. In order to apply MD to infrequent events, corrections to the TST rate that account for recrossings of the dividing surface are computed by starting trajectories at the dividing surface and integrating them backward and forward in time. Both classical TST and conventional MD invoke the adiabatic approximation, i.e., the assumption that nuclear motion evolves on a single potential energy surface. Many chemical rate processes involve multiple potential energy surfaces, however, and a number of ‘‘surface-hopping’’ MD methods have been developed in order to incorporate nonadiabatic transitions among the potential energy surfaces. In this paper we generalize TST to processes involving multiple potential energy surfaces. This provides the framework for a new method for MD simulation of infrequent events for reactions that evolve on multiple potential energy surfaces. We show how this method can be applied rigorously even in conjunction with phase-coherent surface-hopping methods, where the probability of switching potential energy surfaces depends on the history of the trajectory, so integrating trajectories backward to calculate the recrossing correction is problematic. We illustrate this new method by applying it in conjunction with the ‘‘molecular dynamics with quantum transitions’’ (MDQT) surface-hopping method to a one-dimensional two-state barrier crossing problem.