Spin-Orbit Coupling and Other Relativistic Effects in Atoms and Molecules

Theoretical spin-orbit laser cooling for AlZn molecule

A spin-orbit coupling electronic structure study of the AlZn molecule is conducted to investigate the molecular properties of the low-lying electronic states and their feasibility toward direct laser cooling. This study uses the complete active-space self-consistent field level of theory, followed by the multireference configuration interaction method with Davidson correction (+Q). The potential energy and dipole moment curves and the spectroscopic constants are computed for the low-lying doublet and quartet electronic states in the 2S+1Λ± and Ω(±) representations. The transition dipole moments, the Franck-Condon factors, the Einstein coefficient, the radiative lifetimes, the vibrational branching ratio, and the slowing distance are determined between the lowest spin-orbit bound electronic states. These results show that the molecule AlZn has a high potential for laser cooling through the X2Π1/2 → (2)2Π1/2 transition by utilizing four lasers at a wavelength in the ultraviolet region, reaching a sub-microkelvin temperature limit.

Machine Learning Isotropic g Values of Radical Polymers

Methods for electronic structure computations, such as density functional theory (DFT), are routinely used for the calculation of spectroscopic parameters to establish and validate structure-parameter correlations. DFT calculations, however, are computationally expensive for large systems such as polymers. This work explores the machine learning (ML) of isotropic g values, giso, obtained from electron paramagnetic resonance (EPR) experiments of an organic radical polymer. An ML model based on regression trees is trained on DFT-calculated g values of poly(2,2,6,6-tetramethylpiperidinyloxy-4-yl methacrylate) (PTMA) polymer structures extracted from different time frames of a molecular dynamics trajectory. The DFT-derived g values, gisocalc, for different radical densities of PTMA, are compared against experimentally derived g values obtained from in operando EPR measurements of a PTMA-based organic radical battery. The ML-predicted giso values, gisopred, were compared with gisocalc to evaluate the performance of the model. Mean deviations of gisopred from gisocalc were found to be on the order of 0.0001. Furthermore, a performance evaluation on test structures from a separate MD trajectory indicated that the model is sensitive to the radical density and efficiently learns to predict giso values even for radical densities that were not part of the training data set. Since our trained model can reproduce the changes in giso along the MD trajectory and is sensitive to the extent of equilibration of the polymer structure, it is a promising alternative to computationally more expensive DFT methods, particularly for large systems that cannot be easily represented by a smaller model system.

Coherent Mixing of Singlet and Triplet States in Acrolein and Ketene: A Computational Strategy for Simulating the Electron-Nuclear Dynamics of Intersystem Crossing

We present a theoretical study of intersystem crossing (ISC) in acrolein and ketene with the Ehrenfest method that can describe a superposition of singlet and triplet states. Our simulations illustrate a new mechanistic effect of ISC, namely, that a superposition of singlets and triplets yields nonadiabatic dynamics characteristic of that superposition rather than the constituent state potential energy surfaces. This effect is particularly significant in ketene, where mixing of singlet and triplet states along the approach to a singlet/singlet conical intersection occurs, with the spin-orbit coupling (SOC) remaining small throughout. In both cases, the effects require many recrossings of the singlet/triplet state crossing seam, consistent with the textbook treatment of ISC.

Strong Relativistic Effects in Lanthanide-Based Single-Molecule Magnets

Lanthanide-based single-molecule magnets (SMMs) are promising building blocks for quantum memory and spintronic devices. Designing lanthanide-based SMMs with long spin relaxation time requires a detailed understanding of their electronic structure, including the crucial role of the spin-orbit coupling (SOC). While traditional calculations of SOC using the perturbation theory applied to a solution of the nonrelativistic Schrödinger equation are valid for light atoms, this approach is questionable for systems containing heavy elements such as lanthanides. We investigate the accuracy of the perturbation estimates of SOC by variationally solving the Dirac equation for the [DyO]+ molecule, a prototype of a lanthanide-based SMM. We show that the energy splittings between the M J states involved in spin relaxation depend on the interplay between strong SOC and dynamic electron correlation. We demonstrate that this interplay affects the resonances between the spin and vibrational transitions and, therefore, the spin relaxation time.

Spin-orbit configuration interaction study of spectral properties of PbO

Relativistic calculations of the structural and spectral properties of the PbO molecule can provide fundamental information about the importance of a proper treatment of angular momentum coupling among electrons in order to achieve accurate computational results for spectral properties. Specifically, the nature of these couplings in PbO is expected to be intermediate between theLS- andjj-coupling limits because of its light/heavy element composition. This article reports potential energy curves, transition energies, electric dipole transition moments, permanent dipole moments and spectroscopic constants of PbO calculated using a multireference single plus double excitations spin-orbit configuration interaction approach in the context of relativistic effective core potentials and their concomitant spin-orbit coupling operators. The calculated results are in general agreement with both available experimental results as well as earlier calculations. New values for properties of excited states are also reported. It is noteworthy that certain properties show larger deviations from previous calculations. These deviations are attributed to direct and indirect relativistic effects resulting from diatomic electron-electron angular momentum coupling effects, which are included consistently in the calculations reported herein.

Theoretical study of laser cooling of the TlF+ molecular ion

The feasibility of the thallium monofluoride TlF+ molecular ion towards laser cooling is brought into focus through an electronic structure study. Ab initio calculations are carried out to investigate the four lowest-lying electronic states, X2Σ+, (1)2Π, (2)2Σ+ and (2)2Π, including the spin-orbit coupling effect by employing the Complete Active Space Self Consistent Field (CASSCF) method at the Multireference Configuration Interaction (MRCI) level of theory while invoking Davidson correction (+Q). Potential energy and permanent dipole moment curves are used to determine the corresponding spectroscopic constants and some other equilibrium parameters. Vibrational parameters of vibrational states and transition dipole moments between possible transitions are computed. The calculated parameters are then used to conduct a theoretical study focusing on the potential possibility of TlF+ ionic molecule to be laser cooled on the (2)2Π1/2(ν')-X2Σ+1/2(ν'') transition based on Di Rosa's criteria. With the results obtained being promising, a laser cooling optical cycling scheme is proposed to illustrate the number of pump lasers needed with the corresponding wavelengths that were found to lie within the ranges covered by a specific scientific laser.

Electronic Structure Calculations with the Spin Orbit Effect of the Low-Lying Electronic States of the YbBr Molecule

This work presents an electronic structure study employing multireference configuration interaction MRCI calculations with Davidson correction (+Q) of the ytterbium monobromide YbBr molecule. Adiabatic potential energy curves (PECs), dipole moment curves, and spectroscopic constants (such as R _e, ω_e, B _e, D _e, T _e, and μ_e) of the low-lying bound electronic states are determined. The ionic character of the YbBr molecule at the equilibrium position is also discussed. With spin-orbit effects, 30 low-lying states in Ω = 1/2, 3/2, 5/2, 7/2 representation are probed. The electronic transition dipole moment is calculated between the investigated states and then used to determine transition coefficients, for example, the Einstein coefficient of spontaneous emission A _ij and emission oscillator strength f _ij . Vibrational parameters such as E _ν, B _ν, D _ν, R _min, and R _max of the low vibrational levels of different bound states in both Λ and Ω representations are also calculated. Upon calculating the Franck-Condon factors, they are found to be perfectly diagonal between three couples of low-lying excited states. Vibrational Einstein coefficients and radiative lifetimes are computed as well for the lowest vibrational transitions. Most of the data reported in this work are presented here for the first time in the literature. Very good accordance is obtained in comparison with the previously reported constants by means of experimental methods.

All-electron relativistic computations on the low-lying electronic states, bond length, and vibrational frequency of CeF diatomic molecule with spin-orbit coupling effects

Ab initio all-electron computations have been carried out for Ce⁺ and CeF, including the electron correlation, scalar relativistic, and spin-orbit coupling effects in a quantitative manner. First, the n-electron valence state second-order multireference perturbation theory (NEVPT2) and spin-orbit configuration interaction (SOCI) based on the state-averaged restricted active space multiconfigurational self-consistent field (SA-RASSCF) and state-averaged complete active space multiconfigurational self-consistent field (SA-CASSCF) wavefunctions have been applied to evaluations of the low-lying energy levels of Ce⁺ with [Xe]4f¹ 5d¹ 6s¹ and [Xe]4f¹ 5d² configurations, to test the accuracy of several all-electron relativistic basis sets. It is shown that the mixing of quartet and doublet states is essential to reproduce the excitation energies. Then, SA-RASSCF(CASSCF)/NEVPT2 + SOCI computations with the Sapporo(-DKH3)-2012-QZP basis set were carried out to determine the energy levels of the low-lying electronic states of CeF. The calculated excitation energies, bond length, and vibrational frequency are shown to be in good agreement with the available experimental data. © 2018 Wiley Periodicals, Inc.

Theoretical Study on the Reaction Mechanism of Ti with CH3CN in the Gas Phase

To gain a deeper understanding of the reaction mechanisms of Ti with acetonitrile molecules, the triplet and singlet spin-state potential energy surfaces (PESs) has been investigated at B3LYP level of density functional theory (DFT). Crossing points between the different PESs and possible spin inversion processes are discussed by spin-orbit coupling (SOC) calculation. In addition, the bonding properties of the species along the reaction were analyzed by electron localization function (ELF), atoms in molecules (AIM) and natural bond orbital (NBO). The results showed that acetonitrile activation by Ti is a typical spin-forbidden process; larger SOC (by 220.12 cm(-1)) and the possibility of crossing between triplet and singlet imply that intersystem crossing (ISC) would occur near the minimum energy crossing point (MECP) during the transfer of the hydrogen atom.

Perturbational treatment of spin-orbit coupling for generally applicable high-level multi-reference methods

An efficient perturbational treatment of spin-orbit coupling within the framework of high-level multi-reference techniques has been implemented in the most recent version of the Columbus quantum chemistry package, extending the existing fully variational two-component (2c) multi-reference configuration interaction singles and doubles (MRCISD) method. The proposed scheme follows related implementations of quasi-degenerate perturbation theory (QDPT) model space techniques. Our model space is built either from uncontracted, large-scale scalar relativistic MRCISD wavefunctions or based on the scalar-relativistic solutions of the linear-response-theory-based multi-configurational averaged quadratic coupled cluster method (LRT-MRAQCC). The latter approach allows for a consistent, approximatively size-consistent and size-extensive treatment of spin-orbit coupling. The approach is described in detail and compared to a number of related techniques. The inherent accuracy of the QDPT approach is validated by comparing cuts of the potential energy surfaces of acrolein and its S, Se, and Te analoga with the corresponding data obtained from matching fully variational spin-orbit MRCISD calculations. The conceptual availability of approximate analytic gradients with respect to geometrical displacements is an attractive feature of the 2c-QDPT-MRCISD and 2c-QDPT-LRT-MRAQCC methods for structure optimization and ab inito molecular dynamics simulations.

Potential energy surfaces for the HBr(+) + CO2 → Br + HOCO(+) reaction in the HBr(+) (2)Π3/2 and (2)Π1/2 spin-orbit states

Quantum mechanical (QM) + molecular mechanics (MM) models are developed to represent potential energy surfaces (PESs) for the HBr(+) + CO2 → Br + HOCO(+) reaction with HBr(+) in the (2)Π3/2 and (2)Π1/2 spin-orbit states. The QM component is the spin-free PES and spin-orbit coupling for each state is represented by a MM-like analytic potential fit to spin-orbit electronic structure calculations. Coupled-cluster single double and perturbative triple excitation (CCSD(T)) calculations are performed to obtain "benchmark" reaction energies without spin-orbit coupling. With zero-point energies removed, the "experimental" reaction energy is 44 ± 5 meV for HBr(+)((2)Π3/2) + CO2 → Br((2)P3/2) + HOCO(+), while the CCSD(T) value with spin-orbit effects included is 87 meV. Electronic structure calculations were performed to determine properties of the BrHOCO(+) reaction intermediate and [HBr⋯OCO](+) van der Waals intermediate. The results of different electronic structure methods were compared with those obtained with CCSD(T), and UMP2/cc-pVTZ/PP was found to be a practical and accurate QM method to use in QM/MM direct dynamics simulations. The spin-orbit coupling calculations show that the spin-free QM PES gives a quite good representation of the shape of the PES originated by (2)Π3/2HBr(+). This is also the case for the reactant region of the PES for (2)Π1/2 HBr(+), but spin-orbit coupling effects are important for the exit-channel region of this PES. A MM model was developed to represent these effects, which were combined with the spin-free QM PES.

Mechanistic exploration of the catalytic cycles for the CO oxidation by O2 over FeO(1-3) application of the energetic span model

Carbon monoxide (CO) and oxygen (O2) catalyzed by small neutral iron oxide clusters (FeO(1-3)) was investigated at the density functional level of theory using the Becke-Perdew-Wang functional (BPW91). Three reaction pathways along with singlet, triplet and quintet states were calculated for ascertaining the presence of some spin inversion during the catalytic cycle. The catalytic cycle was found to be "two state reactivity" resulting from the crossing among the multistate energetic profiles. The Landau-Zener equation was used to calculate the thermally-averaged spin transition probabilities for the non-adiabatic surface crossing reaction. In order to predict the efficiency of catalyst the energetic span model developed by Kozuch was implemented, whereas this model is not suitable for handling the diabatic reaction, this feature we must take into consideration. To this end, a kinetic assessment is carried out with an expansion of the energetic span model, including the spin-crossing effects. This approximation enables one to measure the efficiency of catalytic cycle including spin-crossing effects by quantum mechanical computation.

Toward spin-orbit coupled diabatic potential energy surfaces for methyl iodide using effective relativistic coupling by asymptotic representation

The theoretical treatment of state-state interactions and the development of coupled multidimensional potential energy surfaces (PESs) is of fundamental importance for the theoretical investigation of nonadiabatic processes. Usually, only derivative or vibronic coupling is considered, but the presence of heavy atoms in a system can render spin-orbit (SO) coupling important as well. In the present study, we apply a new method recently developed by us (J. Chem. Phys. 2012, 136, 034103, and J. Chem. Phys. 2012, 137, 064101) to generate SO coupled diabatic PESs along the C-I dissociation coordinate for methyl iodide (CH3I). This is the first and mandatory step toward the development of fully coupled full-dimensional PESs to describe the multistate photodynamics of this benchmark system. The method we use here is based on the diabatic asymptotic representation of the molecular fine structure states and an effective relativistic coupling operator. It therefore is called effective relativistic coupling by asymptotic representation (ERCAR). This approach allows the efficient and accurate generation of fully coupled PESs including derivative and SO coupling based on high-level ab initio calculations. In this study we develop a specific ERCAR model for CH3I that so far accounts only for the C-I bond cleavage. Details of the diabatization and the accuracy of the results are investigated in comparison to reference ab initio calculations and experiments. The energies of the adiabatic fine structure states are reproduced in excellent agreement with ab initio SO-CI data. The model is also compared to available literature data, and its performance is evaluated critically. This shows that the new method is very promising for the construction of fully coupled full-dimensional PESs for CH3I to be used in future quantum dynamics studies.

Accurate relativistic small-core pseudopotentials for actinides. energy adjustment for uranium and first applications to uranium hydride

The options to adjust accurate relativistic energy-consistent pseudopotentials for actinides are explored using uranium as an example. The choice of the reference data and the core-valence separation is discussed in view of a targeted accuracy of 0.04 eV or better in atomic energy differences such as excitation energies and ionization potentials. A new small-core pseudopotential attributing 60 electrons to the core has been generated by an energy adjustment to state-averaged multiconfiguration Dirac-Hartree-Fock/Dirac-Coulomb-Breit Fermi nucleus reference data of 100 nonrelativistic configurations of U to U(7+) corresponding to 30190 reference J levels. At the finite-difference multiconfiguration Hartree-Fock level the mean absolute errors are 0.002 and 0.024 eV for the configurations and J levels, respectively. A first molecular application to uranium monohydride UH yields very satisfactory agreement with results from all-electron calculations based on the Douglas-Kroll-Hess Hamiltonian.

CH4 activation by W atom in the gas phase: a case of two-state reactivity process

Gas-phase methane activation by tungsten (W) atoms was studied at the density functional level of theory using the hybrid exchange correlation functional B3LYP. Four reaction profiles corresponding to the septet, quintet, triplet, and singlet multiplicities were investigated in order to ascertain the presence of some spin inversion during the methane activation. Methane activation mediated by W atoms was found to be a spin-forbidden process resulting from the crossing among the multistate energetic profiles. On the basis of the Hammond postulate, this is a typical two-state reactivity (TSR) reaction. The minimum energy crossing points lead to decrease in the barrier heights of TS01, TS12, TS23, and TS24 that correspond to the first, second, and third hydrogen transfer and the reductive elimination step of H(2), respectively. The spin-orbit coupling is calculated between electronic states of different multiplicities at the crossing points (MECPs) to estimate the intersystem crossing probabilities, and the probability of hopping from one surface to the other in the vicinity of the crossing region is calculated by the Landau-Zener type model.