The presentation will be focused on development and applications of relativistic wave function-based approaches aiming to extend the accuracy and applicability of quantum chemistry to heavy elements. An atomic mean-field spin-orbit approach within exact two-component theory, the X2CAMF scheme, is shown to enhance the computational efficiency while retaining the accuracy of the parent four-component Dirac-Coulomb-Breit approach. An efficient implementation of the X2CAMF scheme together with analytic energy gradients for spin-orbit coupled-cluster methods enables accurate calculations of geometries and properties for molecules containing heavy atoms. The applicability of these relativistic quantum-chemical methods is demonstrated with applications in heavy-element chemistry and spectroscopy.

l0pt Figure The laser-induced fluorescence (LIF) excitation spectra of jet-cooled ruthenium monoxide (RuO) molecule in the gas phase have been investigated in the range of 13,800 to 19,250 cm⁻¹. As shown in the figure, a total of sixteen vibronic bands were experimentally observed and grouped into the transition systems from the ground X⁵∆₄ and X⁵∆₃ states to six excited electronic states, labeled as [15.07]3−X⁵∆₄, [16.05]5−X⁵∆₄, [16.43]3−X⁵∆₄, [16.19]4−X⁵∆_4,3, [18.09]4−X⁵∆_4,3, [18.46]3−X⁵∆_4,3. The spin-orbit splitting and the rotational constants in the lower and upper states were obtained accurately by the rotationally and isotopically resolved LIF spectra. In addition, the single-vibronic-level (SVL) emission spectra from the excited states were recorded, and the vibrational constants in the ground X⁵∆₄ and X⁵∆₃ states were obtained. Our results are sufficiently reliable and accurate to guide spectroscopists on further studies of RuO molecule.

Accurate prediction of molecular geometries is a central subject in electronic structure theory. For accurate calculations of vibronic branching ratios in laser coolable molecules, it requires accurate calculations of molecular geometries for both electronic ground states and excited states. Using exact two-component theory with atomic-mean-field (X2CAMF) framework and analytical gradient techniques for spin-orbit coupled-cluster (SO-CC) method, we can obtain molecular equilibrium structures with accurate treatment of electron correlation and relativistic effects. By comparing with the experimental measurements of period-four-element containing diatomic molecules, the calculated bond lengths are accurate to 0.001 Å and the calculated harmonic frequencies are accurate to a few cm⁻¹.

LaO bands appear in the optical spectra of S-type stars. The formation of the elements can be studied by measuring the stellar abundances of heavy metals such as La. For cooler stars, the visible and near-infrared electronic transitions of LaO are more useful than La atomic lines.
We have analyzed the LaO B²Σ⁺-X²Σ⁺ band system up to v=5 in both ground and excited states. The rotational analysis of the B²Σ⁺-X²Σ⁺ transition was carried out using the PGOPHER program. Most of the ground state spectroscopic parameters and hyperfine parameters of the excited B²Σ⁺ state were taken from literature and kept fixed. The equilibrium constants for X²Σ⁺ and B²Σ⁺ states were determined. The line strengths were calculated based on the ab initio transition dipole moment and RKR potential curves. We also provide radiative lifetimes of the B²Σ⁺ state for v=0 to v=4. With this work, we provide a modern line list for the LaO B²Σ⁺-X²Σ⁺ transition that can be used to simulate LaO spectra of cool S-type stars to determine La abundances.
A similar analysis is in progress for the LaO A²Π-X²Σ⁺ transition.

l0pt Figure Many transition metal complexes are popular catalysts for homogeneous organic synthesis. Their instantaneous geometric and electronic configurations and roles in the reaction mechanisms can be directly probed by in-situ K-edge X-ray absorption near-edge structure (XANES) spectroscopy. First-principles modeling is indispensable to translate the frequencies and lineshapes of K-edge absorptions into orbital and structural configurations. In the present study, we performed real-time time-dependent density functional theory (RT-TDDFT) calculations for (2,6-dimethylphenyl)imino)vanadium(V) trichloride and its methyl-substituted derivatives, and obtained time-dependent electronic densities and transition dipole moments by solving the time-dependent Kohn-Sham equations under an applied electromagnetic field. Compared to traditional linear-response TDDFT (LR-TDDFT), RT-TDDFT allows a significant rearrangement of electronic densities after photoexcitations and provides a broadband spectrum in the frequency domain after the Fourier transform. Based on our RT-TDDFT calculations, we managed to reproduce the pre-edge peaks for these species and assigned them to the dipole-allowed transitions of electrons from 1s orbital to the 3d4p hybridized orbitals of vanadium. Both characters align with the results from LR-TDDFT. In addition, RT-TDDFT leads to important features from the shoulder peaks, which correspond to the dipole-allowed, density-rearranging transitions of electrons from 1s orbital of vanadium to its 4p orbitals or the 3p orbitals of chlorine. These shoulder peak features have never been provided by LR-TDDFT. From the present study, we provided a proof-of-concept that the next-generation RT-TDDFT approach is a versatile and powerful computational tool for the prediction and interpretation of X-ray spectroscopy of transition metal complexes.

Transition metals represent a space of continual interest due to their complex electronic structure and the diverse range of possible ligands and oxidation states. Studies of transition metal complexes often rely on X-ray spectroscopies (usually at the L or M edges) and computational methods to explain spectral features and help design new experiments. Most computational approaches either account for relativistic effects at the scalar level, by omitting spin-orbit coupling terms, or completely neglect them sacrificing accuracy in favor of a lower computational cost. On the other hand, explicit ab-initio treatments of spin-orbit couplings are costly and labor intensive, restricting their applications to smaller atomic systems. In the present work, we propose a simplified approach based on linear-response time-dependent density functional theory (LR-TDDFT) and the relativistic two-component zeroth order regular approximation (ZORA) to generate spin-orbit couplings of closed-shell molecular systems. The proposed approach was validated by computing the L-edge absorption spectra of several first and second row transition metal complexes. The method reproduces experimental data with satisfactory accuracy at a fraction of the cost of exact two-component or fully relativistic methods.

Progress in laser cooling and trapping molecules has lead to a renewed interest in alkaline earth monoalkoxide (MOR) free radicals as promising candidates for direct laser-cooling of polyatomic molecules. Theoretical understanding of these molecules is challenging due to the presence of quasi-degenerate electronic states in these molecules. In addition to that, pseudo-Jahn Teller interactions and spin-orbit coupling also play a very important role. Understanding these couplings and their effects on the molecular spectra will provide critical information for future direct laser cooling of MORs and similar radicals. In this talk we discuss the theoretical intricacies involved in calculating the spin-vibronic spectra of such molecules from first principles. A Hamiltonian has been developed in a spin-vibronic representation for molecules with quasi-degenerate electronic states. We will describe calculating the parameters in this Hamiltonian using ab-initio methods and the software developed for solving such Hamiltonians (SOCJT3). Our discourse includes both frequency calculations and relative transition intensities from first principles for both excitation and emission spectra which can be compared to experimentally observed using LIF and DF spectra of CaOCH₃, CaOC₂H₅, and iso-CaOC₃H₇ to be reported in succeeding talk. Typical Franck-Condon factor calculations done under the Born-Oppenheimer approximation reproduce the dominant features of these spectra, but the inclusion of Jahn-Teller and pseudo-Jahn-Teller couplings and spin-orbit interactions in the calculations not only improves the accuracy of simulation but also leads to additional vibronic transitions that help to explain the finer structure observed in the spectra. A major limitation of these methods is the amount of computational resources required as the molecules become larger. Efforts to minimize the amount of resources required as well as approximations involved in simulating the spin-vibronic spectra for larger molecules like iso-CaOC₃H₇ have also been discussed.

We report a combined experimental and computational study of spin-vibronic structure and transition intensities of the lowest electronic states of nonlinear alkaline earth monoalkoxide (MOR) radicals, including calcium methoxide (CaOCH₃), calcium ethoxide (CaOC₂H₅), and calcium isopropoxide [CaOCH(CH₃)₂]. Experimentally, laser-induced fluorescence/dispersed fluorescence (LIF/DF) and cavity ring-down (CRD) spectra of the à ² E – X̃ ² A₁ electronic transition of CaOCH₃ (C_3v), the Ã₁ ² A^′′ / Ã₂ ² A^′ – X̃ ² A^′ transition of CaOC₂H₅ (C_s), and the Ã₁ ² A^′ / Ã₂ ² A^′′ – X̃ ² A^′ transition of [CaOCH(CH₃)₂] were recorded under jet-cooled conditions. An essentially constant Ã₂ − Ã₁ energy separation for different vibronic levels is observed in the LIF spectrum of each radical, attributed to the spin-orbit (SO) interaction and, in the case of the two C_s molecules, the zero-point-energy-corrected "difference potential’’. The complete active space self-consistent field (CASSCF) and the coupled-cluster (CC) methods are used to calculate electronic transition energies and vibrational frequencies and to predict parameters governing the spin-vibronic energy level structure and simulate the recorded LIF/DF spectra. The Jahn-Teller (JT), pseudo-Jahn-Teller (pJT), and SO interactions, especially those between the Ã₁ / Ã₂ and the neighboring B̃ states, induce a number of off-diagonal Franck-Condon (FC) matrix elements leading to additional vibronic transitions. The spin-vibronic Hamiltonian presented in the preceding talk has been employed for the spectral simulation. Computational and experimental results on all three free radicals will be compared, and the implications for future laser cooling experiments will be discussed.

Ultracold molecules have been widely utilized in various fields of study from the search for new physics in the ultracold regime to ultracold chemistry. Systems at such ( < 1μK) temperatures are usually made with laser cooling techniques, i.e. photoassociation and stimulated Raman adiabatic passage, which utilize closed optical schemes with rovibronic levels of electron-excited states serving as intermediate steps to create ultracold molecules from pre-cooled atoms. This process requires full and detailed data for the ground and excited states in a wide range of internuclear distances, R. Sufficiently accurate data requires going beyond the adiabatic approximation, calculating non-adiabatic interaction matrix elements (IME), i.e. the spin-orbit (SO) and L-coupling (LC) IMEs, with special attention paid to correct long-range behavior. Here, the asymptotic behavior near the dissociation limit (DL) of IMEs is studied focusing on the transition-dipole moment (TDM), SO and LC IMEs Phys. Chem. Chem. Phys. 20, 1889–1896 (2018); Phys. Rev. A 99, 012507 (2019).; Phys. Chem. Chem. Phys. 23(9), 5187–5198 (2021). These were calculated for LiNa, LiK, LiRb, LiCs using spin-averaged wavefunctions corresponding to Hund’s case (a) and effective core pseudopotentials. The electronic correlation is accounted for using a 2 valence electron multi-reference configuration interaction calculation. Core-polarization potentials take the core–valence effect into account. Where possible, theoretical curves were compared to ones derived from experiment. The leading asymptotic trends for the TDMs, and SO and LC IMEs, have been determined for three groups of state pairs: (a) dipole allowed transitions between an excited state and one converging to the first DL; (b) forbidden transitions between two states converging to the same DL > 1; (b) forbidden transitions between two states converging to different DL > 1. Thus, for the TDMs type (a) pairs converge as R⁻³ to the atomic dipole moment, while type (b,c) pairs converge to zero as R⁻⁴. The SO IMEs converge: as R⁻⁷ to zero for (a); as R⁻⁶ to the atomic SO splitting for (b); as R⁻³ to zero for (c). Finally, the LC functions: approach infinity linearly for (a); converge as R⁻⁶ to a constant for (b); converge as R⁻³ to zero for (c).

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Footnotes:

Phys. Chem. Chem. Phys. 20, 1889–1896 (2018); Phys. Rev. A 99, 012507 (2019).; Phys. Chem. Chem. Phys. 23(9), 5187–5198 (2021)..

Core-valance separated delta-coupled-cluster (CVS-∆CC) with spin-free exact two-component theory in its one-electron variant (SFX2C-1e) has been shown to provide quantitative description of core-ionization energies for second-row elements [1]. Here we extend the applicability of CVS-∆CC calculations to K-edge core-ionization energies for third-row elements. We develop a revised CVS scheme to make it applicable in larger basis sets. Basis-sets effects have been demonstrated to be important. The use of uncontracted cc-pCVTZ basis sets for target atom and cc-pVTZ sets for the other atoms appears to be an efficient and accurate approach (cc-pCVTZ-unc*). High-level relativistic (HLR) corrections beyond the SFX2C-1e, including two-electron picture change, spin-orbit coupling, Breit interaction and QED effects have been taken into account and shown to play an important role. SFX2C-1e CVS-∆CCSD(T)/cc-pCVTZ-unc* calculations augmented with high-level relativistic corrections can provide highly accurate K-edge core-ionization energies of third-row elements with deviation of less than 0.5 eV from experimental values.