Breit-Pauli Hamiltonian Research Articles

Atomic structure, electron-impact excitation and collisional-radiative modelling for Ar II

The spectra from singly ionized argon Ar II has significant diagnostic capability in the characterization and modelling of both magnetically-confined fusion and astrophysical plasmas. The literature has several pre-existing data sets for Ar+ but this paper presents the results from 3 new atomic structure and electron-impact scattering models in order to better constrain the differences in atomic data and how they impact well-known plasma diagnostics. Several independent atomic structure methodologies are employed to calculate the energy levels and transition probabilities for each model. The first approach employs a relativistic Dirac–Coulomb Hamiltonian model, the second approach uses a semi-relativistic Breit–Pauli Hamiltonian with the mass-velocity, Darwin and spin–orbit corrections, and in a third case an ICFT approach. Three atomic structure models provide a foundation for Dirac R-matrix, a semi-relativistic ICFT (Intermediate Coupling Frame Transformation) and a Breit–Pauli R-Matrix with Pseudostates (BPRMPS) calculation. Synthetic spectra utilizing these three data sets are compared against measurements taken at the Compact Toroidal Hybrid (CTH) stellarator, and the total radiative power loss is also benchmarked against previous calculations.

Multiexcitonic and optically bright states in subunits of pentacene crystals: A hybrid DFT/MRCI and molecular mechanics study.

A hybrid quantum mechanics/molecular mechanics setup was used to model electronically excited pentacene in the crystal phase. Particularly interesting in the context of singlet fission (SF) is the energetic location of the antiferromagnetically coupled multiexcitonic singlet state, 1(TT), and the ferromagnetically coupled analog in relation to the optically bright singlet state. To provide photophysical properties of the accessible spin manifold, combined density functional theory and multi-reference configuration interaction calculations were performed on pentacene dimers and a trimer, electrostatically embedded in the crystal. The likelihood of a quintet intermediate in the SF process was estimated by computing singlet-quintet electron spin-spin couplings employing the Breit-Pauli Hamiltonian. The performance of the applied methods was assessed on the pentacene monomer. The character of the optically bright state and the energetic location of the 1(TT) state depend strongly on the relative orientation of the pentacene units. In the V-shaped dimers and in the trimer, the optically bright state is dominated by local and charge transfer (CT) excitations, with admixtures of doubly excited configurations. The CT excitations gain weight upon geometry relaxation, thus supporting a CT-mediated SF mechanism as the primary step of the SF process. For the slip-stacked dimer, the energetic order of the bright and the 1(TT) states swaps upon geometry relaxation, indicating strong nonadiabatic coupling close to the Franck-Condon region-a prerequisite for a coherent SF process. The multiexcitonic singlet, triplet, and quintet states are energetically too far apart and their spin-spin couplings are too small to bring about a noteworthy multiplicity mixing.

MRCI Study of the Electronic Structure and Transition Properties of a Tin Dimer.

The ground and excited states of Sn2 are calculated using the multireference configuration interaction method combined with Davidson correction (MRCI+Q). The influence of the spin-orbit coupling (SOC) effect on the electronic structure is also considered by the state interaction method of Breit-Pauli Hamiltonian. In the calculations, the potential energy curves and spectroscopic constants of 23 Λ-S states and 31 Ω states of Sn2 are obtained. The prominent spectral features in the visible region, new constants, and potential energy curves are discussed. The intensity of weak magnetic and quadrupole transitions in the near IR spectra is also calculated. From a computational point of view, we predict that the weak v'(0-2)-v″(0-5) bands of the magnetic b1Σg,0++-X3Σg,1(Ms=±1)- transition may be detected experimentally; the sub-bands (0, 0), (1, 0), and (2, 0) of the a1Δg,2-X3Σg,1(Ms=±1)- transition also may be observed in experiments since they are not overlapped by the strong electric dipole transition in the same IR region. According to the SOC matrix elements and contributions of the 15Πu0+, 15Πu1 (|Σ| = 0), and 15Πu1 (|Σ| = 2) states to the predissociation line width of the 13Σu- -X3Σg1- transition, the broading and other predissociation features of the 13Σu- state are analyzed. From our calculations, it follows that the strong coupling between the bound 13Σu- state and the repulsive 15Πu state causes the predissociation of the 13Σu- state at the vibrational levels v' ≥ 8. In addition, our results suggest that the previously observed bands of Sn2 in the visible range of 19000-20000 cm-1 should be reassigned into the mixing transitions among the X3Σg,1--23Σu,0-+ and X3Σg,0+--23Σu,1+ manifold. The results are expected to provide new comprehensive information for better understanding the spectra and dynamics of the electronic excited states of the Sn2 molecule.

Review of the hyperfine structure theory of hydrogen molecular ions

The study of high-precision spectroscopy for hydrogen molecular ions enables the determination of fundamental constants, such as the proton-to-electron mass ratio, the deuteron-to-electron mass ratio, the Rydberg constant, and the charge radii of proton and deuteron. This can be accomplished through a combination of high precision experimental measurements and theoretical calculations. The spectroscopy of hydrogen molecular ions reveals abundant hyperfine splittings, necessitating not only an understanding of rovibrational transition frequencies but also a thorough grasp of hyperfine structure theory to extract meaningful physical information from the spectra. This article reviews the history of experiments and theories related to the spectroscopy of hydrogen molecular ions, with a particular focus on the theory of hyperfine structure. As far back as the second half of the last century, the hyperfine structure of hydrogen molecular ions was described by a comprehensive theory based on its leading-order term, known as the Breit-Pauli Hamiltonian. Thanks to the advancements in non-relativistic quantum electrodynamics (NRQED) at the beginning of this century, a systematic development of next-to-leading-order theory for hyperfine structure has been achieved and applied to <inline-formula><tex-math id="M8">\begin{document}$\text{H}_2^+$\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$\text{HD}^+$\end{document}</tex-math></inline-formula> in recent years, including the establishment of the <inline-formula><tex-math id="M10">\begin{document}$m\alpha^7\ln(\alpha)$\end{document}</tex-math></inline-formula> order correction. For the hyperfine structure of <inline-formula><tex-math id="M11">\begin{document}$\text{H}_2^+$\end{document}</tex-math></inline-formula>, theoretical calculations show good agreement with experimental measurements after decades of work. However, for HD<sup>+</sup>, discrepancies have been observed between measurements and theoretical predictions that cannot be accounted for by the theoretical uncertainty in the non-logarithmic term of the <inline-formula><tex-math id="M13">\begin{document}$m\alpha^7$\end{document}</tex-math></inline-formula> order correction. To address this issue, additional experimental measurements are needed for mutual validation, as well as independent tests of the theory, particularly regarding the non-logarithmic term of the <inline-formula><tex-math id="M14">\begin{document}$m\alpha^7$\end{document}</tex-math></inline-formula> order correction.

A systematic study on spectroscopic and transition properties of the phosphorus mononitride cation

The potential energy curves (PECs) of 28 Λ–S states of the phosphors mononitride cation have been constructed by using the internally contracted multireference configuration interaction (icMRCI) method plus Davidson correction (+Q). The effect of core correlation and scalar relativistic treatment of electron, and the basis-set extrapolation to the complete basis set limit are considered. The spin–orbit coupling is computed via the full Breit-Pauli Hamiltonian. The transition dipole moments (TDMs) for the dipole-allowed and spin-forbidden transitions are calculated. Utilizing the PECs and TDMs, the spectroscopic and transition properties of bound states are then evaluated. The current spectroscopic parameters, vibrational band origins, Einstein coefficients, and Franck-Condon factors well match the available measured values. And the radiative lifetimes for most excited states are estimated for the first time. These results can be useful in the analysis and assignment of molecular spectra or as guidelines for detecting these states spectroscopically.

Spectroscopic Constants and Predissociation of the low-lying states in selenium dimer

High-level ab initio calculations on the electronic structure and spectroscopic properties for the low-lying electronic states of selenium dimer (Se2) are performed by using internally contracted multireference-configuration interaction method (icMRCI) including Davidson correction (+Q) and scalar relativistic effect (SR). The spin-orbit coupling (SOC) effect is introduced in the computations via state-interacting technique with the full Breit-Pauli Hamiltonian. The potential energy curves (PECs) of 22 Λ-S states of Se2 are computed. After taking SOC effect into account, PECs of 28 Ω states generated from the Λ-S states with the energy under 36,000cm−1 are yielded. Moreover, from the calculated PECs, the spectroscopic constants of the bound Λ-S and Ω states are determined, and good agreement with available experimental and theoretical works is found. By virtue of the calculated SOC matrix elements and the PECs of the Λ-S states, the spin-orbit induced predissociation mechanism of the B3Σu− state is discussed. Our computations would provide helpful information on spectroscopic properties of the low-lying electronic states for the Se2, and deep understanding of the predissociation mechanism for the B3Σu− state.

Spin-orbit couplings within spin-conserving and spin-flipping time-dependent density functional theory: Implementation and benchmark calculations.

We present a new implementation for computing spin-orbit couplings (SOCs) within a time-dependent density-functional theory (TD-DFT) framework in the standard spin-conserving formulation as well in the spin-flip variant (SF-TD-DFT). This approach employs the Breit-Pauli Hamiltonian and Wigner-Eckart's theorem applied to the reduced one-particle transition density matrices, together with the spin-orbit mean-field treatment of the two-electron contributions. We use a state-interaction procedure and compute the SOC matrix elements using zero-order non-relativistic states. Benchmark calculations using several closed-shell organic molecules, diradicals, and a single-molecule magnet illustrate the efficiency of the SOC protocol. The results for organic molecules (described by standard TD-DFT) show that SOCs are insensitive to the choice of the functional or basis sets, as long as the states of the same characters are compared. In contrast, the SF-TD-DFT results for small diradicals (CH2, NH2 +, SiH2, and PH2 +) show strong functional dependence. The spin-reversal energy barrier in a Fe(III) single-molecule magnet computed using non-collinear SF-TD-DFT (PBE0, ωPBEh/cc-pVDZ) agrees well with the experimental estimate.

A theoretical study on the electronically excited-state spectroscopic properties of phosphorus nitride

ABSTRACT The potential energy curves of 103 Ω states generated from the 39 Λ–S states of PN have been calculated using the internally contracted multireference configuration interaction method with the Davidson modification. Core-valence correlation and scalar relativistic corrections, as well as the basis-set extrapolation to the complete basis set limit are considered. The spin–orbit coupling is computed using the state interaction approach with the Breit-Pauli Hamiltonian. The spectroscopic parameters and molecular constants of bound and quasibound Λ–S and Ω states are evaluated. Our calculated spectroscopic results agree quite well with the available experimental data. The interactions among different electronic states in curve crossing regions have been discussed with the help of computed spin–orbit coupling matrix elements. The perturbations and predissociation phenomena of the A1 , b3 , D1Δ, E1Σ+, and 21 states and so on have been revealed.

Variational vs perturbative relativistic energies for small and light atomic and molecular systems.

Variational and perturbative relativistic energies are computed and compared for two-electron atoms and molecules with low nuclear charge numbers. In general, good agreement of the two approaches is observed. Remaining deviations can be attributed to higher-order relativistic, also called non-radiative quantum electrodynamics (QED), corrections of the perturbative approach that are automatically included in the variational solution of the no-pair Dirac-Coulomb-Breit (DCB) equation to all orders of the α fine-structure constant. The analysis of the polynomial α dependence of the DCB energy makes it possible to determine the leading-order relativistic correction to the non-relativistic energy to high precision without regularization. Contributions from the Breit-Pauli Hamiltonian, for which expectation values converge slowly due the singular terms, are implicitly included in the variational procedure. The α dependence of the no-pair DCB energy shows that the higher-order (α4Eh) non-radiative QED correction is 5% of the leading-order (α3Eh) non-radiative QED correction for Z = 2 (He), but it is 40% already for Z = 4 (Be2+), which indicates that resummation provided by the variational procedure is important already for intermediate nuclear charge numbers.

Ab initio composite strategies and multireference approaches for lanthanide sulfides and selenides.

The f-block ab initio correlation consistent composite approach was used to predict the dissociation energies of lanthanide sulfides and selenides. Geometry optimizations were carried out using density functional theory and coupled cluster singles, doubles, and perturbative triples with one- and two-component Hamiltonians. For the two-component calculations, relativistic effects were accounted for by utilizing a third-order Douglas-Kroll-Hess Hamiltonian. Spin-orbit coupling was addressed with the Breit-Pauli Hamiltonian within a multireference configuration interaction approach. The state averaged complete active space self-consistent field wavefunctions obtained for the spin-orbit coupling energies were used to assign the ground states of diatomics, and several diagnostics were used to ascertain the multireference character of the molecules.

Fine and hyperfine structures of pionic helium atoms

The fine and hyperfine structures of pionic helium metastable states is calculated within the formalism of the Breit–Pauli Hamiltonian by using the variationally generated wave functions in Hylleraas coordinates. Our results not only verify the existing values of Hori et al. [Phys. Rev. A 89, 042515 (2014)] for the fine structure of π 4He+, but also determine the hyperfine structure of π 3He+.

Radiative lifetimes of the doublet states of 107Ag16O: A configuration interaction study

Potential energy curves and a new set of spectroscopic parameters have been calculated for the X 2 Π , A 2 Σ + , and C 2 Π states. The RKR potential for the D 2 Π state is also included in our analysis. The dissociation energy of 15,970 cm −1 was derived for the ground electronic state, in good agreement with the most recent experimental measurement. The obtained rotationless radiative lifetimes for the lower states A 2 Σ + , C 2 Π , and D 2 Π of AgO are in the orders of microseconds. The spin–orbit coupling effect is accounted for using the Breit-Pauli Hamiltonian. Theoretical A v values have been calculated for the X 2 Π ( v = 0–15) and C 2 Π ( v = 0–5), showing the inverted character of these states.

On the Breit interaction in an explicitly correlated variational Dirac-Coulomb framework.

The Breit interaction is implemented in the no-pair variational Dirac-Coulomb (DC) framework using an explicitly correlated Gaussian basis reported in the previous paper [P. Jeszenszki, D. Ferenc, and E. Mátyus, J. Chem. Phys. 156, 084111 (2022)]. Both a perturbative and a fully variational inclusion of the Breit term are considered. The no-pair DC plus perturbative Breit and the no-pair DC-Breit energies are compared with perturbation theory results including the Breit-Pauli Hamiltonian and leading-order non-radiative quantum electrodynamics corrections for low Z values. Possible reasons for the observed deviations are discussed.

Study of the most relevant spin-orbit coupling descriptions of magnetic excitations in a series of lanthanide complexes.

We present a number of computationally cost-effective approaches to calculate magnetic excitations (i.e. crystal field energies and magnetic anisotropies in the lowest spin-orbit multiplet) in lanthanide complexes. In particular, we focus on the representation of the spin-orbit coupling term of the molecular Hamiltonian, which has been implemented within the quantum chemistry package CERES using various approximations to the Breit-Pauli Hamiltonian. The approximations include the (i) bare one-electron approximation, (ii) atomic mean field and molecular mean field approximations of the two-electron term, (iii) full representation of the Breit-Pauli Hamiltonian. Within the framework of the CERES implementation, the spin-orbit Hamiltonian is always fully diagonalized together with the electron repulsion Hamiltonian (CASCI-SO) on the full basis of Slater determinants arising within the 4f ligand field space. For the first time, we make full use of the Cholesky decomposition of two-electron spin-orbit integrals to speed up the calculation of the two-electron spin-orbit operator. We perform an extensive comparison of the different approximations on a set of lanthanide complexes varying both the lanthanide ion and the ligands. Surprisingly, while our results confirm the need of at least a mean field approach to accurately describe the spin-orbit coupling interaction within the ground Russell-Saunders term, we find that the simple bare one-electron spin-orbit Hamiltonian performs reasonably well to describe the crystal field split energies and g tensors within the ground spin-orbit multiplet, which characterize all the magnetic excitations responsible for lanthanide-based single-molecule magnetism.

K-Shell photoabsorption in Si11+: Relativistic contributions via Breit-Pauli R-matrix calculations

As a part of the investigation of the entire Si isonuclear sequence for x-ray spectral diagnostics, we focus on the Li-like Si11+ ion, a stable component that features prominently in x-ray astrophysical spectra. We perform R-matrix calculations for the photoabsorption cross section of Si11+ including relativistic contributions via the use of a Breit-Pauli Hamiltonian. Relativistic effects—predominantly spin-orbit splitting of resonances and mass-velocity and one-body Darwin global shifts of energies —are shown to be tractable and of importance for accuracy in x-ray spectral modeling data banks.

Super ccCA (s-ccCA): an approach for accurate transition metal thermochemistry

Computational chemistry provides a powerful route to determine thermochemical properties. For transition metal thermochemistry, typically, high-level quantum methodologies are required. Ab initio composite methods – which aim to replicate the predictions possible from a high-level method/advanced basis set with predictions from a series of lower-level methods/basis sets to reduce computational cost – have proven to be effective for transition metal species, with better than chemical accuracy for transition metal energetics (<12 kJ mol−1), on average. While useful, to provide a more robust computational approach, Super ccCA (s-ccCA) is introduced herein and varies from its predecessor, ccCA, by utilising higher-level coupled-cluster corrections along with a spin–orbit contribution from a Breit-Pauli Hamiltonian. In this work, s-ccCA has been utilised for the prediction of dissociation energies of 3d and 4d molecules. A set of borides, sulphides and carbides in conjunction with three early first-row transition metals (Sc, Ti, V) and three second-row transition metals (Y, Zr, Nb) were studied with this new composite method. The energies calculated herein were compared with the experiment and shown to be in excellent agreement. The energetic predictions show that for cases where a balance of static and dynamic correlation is of paramount importance, s-ccCA offers an effective approach.

Calculation of spin-orbit couplings using RASCI spinless one-particle density matrices: Theory and applications.

This work presents the formalism and implementation for calculations of spin-orbit couplings (SOCs) using the Breit-Pauli Hamiltonian and non-relativistic wave functions described by the restricted active space configuration interaction (RASCI) method with general excitation operators of spin-conserving spin-flipping, ionizing, and electron-attaching types. The implementation is based on the application of the Wigner-Eckart theorem within the spin space, which enables the calculation of the entire SOC matrix based on the explicit calculation of just one transition between the two spin multiplets. Numeric results for a diverse set of atoms and molecules highlight the importance of a balanced treatment of correlation and adequate basis sets and illustrate the overall robust performance of RASCI SOCs. The new implementation is a useful addition to the methodological toolkit for studying spin-forbidden processes and molecular magnetism.

Accurate MRCI study of the low-lying electronic states of the AlS molecule

In this paper, ab initio potential energy curves of the first two lowest dissociation channels of AlS have been calculated by using the complete active space self-consistent field (CASSCF) and the internally contracted multireference configuration interaction (icMRCI) method. 13 Λ–S states (X2Σ+, A'2Π, A2Σ+, a4Σ+, b4Δ, c4Σ−, D2Δ, E2Σ−, B2Π, d4Π, e4Π, 22Σ− and 24Σ−) of AlS molecule and the transition properties between these states are reported. The transition properties of A'2Π3/2_ X2Σ+1/2, and A'2Π1/2_ X2Σ+1/2 are studied with the effect of spin–orbit coupling (SOC) using the Breit-Pauli Hamiltonian. The accuracy of these potential energy curves is improved the extrapolation of the limit of the complete basis set. The spectroscopic parameters and transition properties reported in this paper are expected to provide useful guidance for future experiments, astronomical observations and theoretical research.

Equation-of-Motion Coupled-Cluster Theory to Model L-Edge X-ray Absorption and Photoelectron Spectra.

We present an extension of the equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) theory for computing X-ray L-edge spectra, both in the absorption (XAS) and in the photoelectron (XPS) regimes. The approach is based on the perturbative evaluation of spin-orbit couplings using the Breit-Pauli Hamiltonian and nonrelativistic wave functions described by the fc-CVS-EOM-CCSD ansatz (EOM-CCSD within the frozen-core core-valence separated (fc-CVS) scheme). The formalism is based on spinless one-particle density matrices. The approach is illustrated by modeling XAS and XPS of several model systems ranging from Ar to small molecules containing sulfur and silicon.

Theoretical study on predissociation of B3Σu- of sulfur dimer.

A high level ab initio study on electronic states of sulfur dimer S2 is carried out by utilizing multi-reference configuration interaction plus Davidson correction (MRCI+Q) method. The modification to the electronic structures caused by spin-orbit coupling (SOC) effect is taken into consideration by the state-interacting method with the full Breit-Pauli Hamiltonian. The potential energy curves (PECs) of 19 Λ-S states and 52Ω states generated from the Λ-S states are calculated. With the aid of calculated SOC matrix elements and the PECs of the Ω states, we discuss the predissociaiton of the B3Σu- state. The variation of line width of B3Σu-- X3Σg- transitions with vibrational quantum number v' of the B3Σu- are determined from Fermi golden rule calculations. Our study indicate that the predissociation of the B3Σu- state is induced by the strong spin-orbit coupling with different electronic excited states, resulting in the abnormal dependence of the dissociation rate on vibrational states.