Scalar Relativistic Corrections Research Articles

The structure of ScC2 (X̃2A1): A combined Fourier transform microwave/millimeter-wave spectroscopic and computational study.

Pure rotational spectra of Sc13C2 (X̃2A1) and Sc12C13C (X̃2A') have been measured using Fourier transform microwave/millimeter-wave methods. These molecules were synthesized in a DC discharge from the reaction of scandium vapor, produced via laser ablation, with 13CH4 or 13CH4/12CH4, diluted in argon. The NKa,Kc = 10,1 → 00,0, 20,2 → 10,1, 30,3 → 20,2, and 40,4 → 30,3 transitions in the frequency range of 14 GHz-61 GHz were observed for both species, each exhibiting hyperfine splittings due to the nuclear spins of 13C (I = 1/2) and/or Sc (I = 7/2). These data have been analyzed with an asymmetric top Hamiltonian, and rotational, spin-rotation, and hyperfine parameters have been determined for Sc13C2 and Sc12C13C. In addition, a quartic force field was calculated for ScC2 and its isotopologues using a highly accurate coupled cluster-based composite method, incorporating complete basis set extrapolation, scalar relativistic corrections, outer core and inner core electron correlation, and higher-order valence correlation effects. The agreement between experimental and computed rotational constants, including the effective constant (B + C), is ∼0.5% for all three isotopologues. This remarkable agreement suggests promise in predicting rotational spectra of new transition metal-carbon bearing molecules. In combination with previous work on Sc12C2, an accurate structure for ScC2 has been established using combined experimental (B, C) and theoretical (A) rotational constants. The radical is cyclic (or T-shaped) with r(Sc-C) = 2.048(2) Å, r(C-C) = 1.272(2) Å, and ∠(C-Sc-C) = 36.2(1)°. The experimental and theoretical results also suggest that ScC2 contains a C2 - moiety and is largely ionic.

Rovibrational transition properties of the X1Σ+ and A1Π states of carbon monosulfide

Carbon monosulfide was detected in outer space by rovibrational spectroscopy of the X 1Σ+ state and A 1Π – X 1Σ+ system. This work calculated the potential energy curves and dipole moment functions of the X 1Σ+ 0+ and A 1Π1 states, and computed the transition dipole moments between the two states employing the CASSCF method, followed by the valence icMRCI approach. Core-valence correlation and scalar relativistic corrections were included. The extrapolation of potential energies to the complete basis set limit was performed. The spin-orbit coupling effect was included. The Einstein A coefficients, band origins, and oscillator strengths were calculated for the rovibrational transitions when J ≤ 150. The rovibrational transitions of the X 1Σ+ 0+ and A 1Π1 states became very weak when Δυ ≥ 6. The Einstein A coefficients of vibronic emissions of the A 1Π1 – X 1Σ+ 0+ system were large, indicating that the emissions were able to be measured easily through spectroscopy. Several rovibrational transitions of the A 1Π1 – X 1Σ+ 0+ system were analysed in detail. The distribution of radiative lifetime varying as rotational quantum number was calculated. The results obtained in this work agree well with the available experimental values.

Transition properties between low-lying electronic states of SiO+

In this work, we investigate the potential energy curves (PECs), transition dipole moments (TDMs) and spin-orbit couplings for low-lying electronic states of SiO+ based on ab initio calculations. The PECs of seventeen low-lying electronic states are calculated by the internally contracted multireference configuration interaction (icMRCI) method with the Davidson correction, as well as the basis set extrapolation, core-valence (CV) correction and scalar relativistic correction. The Schrödinger equation of nuclear movement is solved over the PEC to obtain the rotational and vibrational energy levels, which are used to fit the spectroscopic parameters. The Einstein coefficients, Franck-Condon factors and oscillator strengths for dipole-allowed transitions are calculated with the PECs and TDMs. The radiative lifetimes of some excited states are also obtained. Our calculated radiative lifetime of 67.8 ns for B2Σ+(υ′ = 0) is in excellent agreement with the recent experimental measurement of 66 ± 2 ns. The spin-orbit coupling integrals related to the X2Σ+, A2Π, B2Σ+, 14Σ+, 14Π and 14Σ− states are calculated using the Breit-Pauli Hamiltonian. In addition, the effects of spin-orbit couplings on the potential energies of the Λ-S electronic states are discussed.

Dirac's equation and its implications for density functional theory based calculations of materials containing heavy elements

Electronic structure calculations based on density functional theory (DFT) give quantitatively accurate predictions of properties of most materials containing light elements. For heavy materials, and in particular for $f$-electron systems, DFT based methods can fail both qualitatively and quantitatively for two distinct reasons: their failure to describe confinement effects arising from localized $f$-electron behavior and their incomplete or approximate treatment of relativity. In addition, different methods for incorporating relativistic effects, which give identical results in most light materials, can give different predictions in heavy elements. In order to develop a quantitative capability for calculating the properties of these materials, it is essential to separate the predictions of the underlying equations from the uncertainty introduced in approximations used in computation. Working toward that goal, we have developed a code, called dirac-fp, which is based directly on solving the Dirac-Kohn-Sham equations and uses the full-potential linear muffin-tin orbital (FP-LMTO) approach to electronic structure. In order to assess the performance of dirac-fp, we perform calculations on three different face-centered cubic materials using different approximate treatments of relativity: the scalar relativistic (SR) approach commonly used in most solid-state DFT codes, the scalar relativistic plus spin-orbit coupling corrections (SR+SO) approach which includes spin-orbit coupling self-consistently using the SR states inside the muffin tins, and the Dirac-Kohn-Sham (Dirac) approach implemented in dirac-fp. Performing calculations on thorium, in which relativistic effects should be strong, aluminum, in which relativistic effects should be negligible, and gold, in which relativistic effects play an intermediate role, we find that the Dirac approach is able to provide theoretically consistent results in the electronic structure and ground-state properties across all three materials.

PtX as the limit of high oxidation states in oxide-nitride species.

A series of pentaatomic species has been investigated theoretically with relativistic DFT using the M06-L functional with both ZORA scalar relativistic correction, and including spin-orbit coupling effects. The distorted quasi-octahedral local minima for PtNO3+, PtN2O2 and PtN3O- corresponding to decavalent Pt were found to be unstable with respect to the elimination of O2, NO or N2. However, barriers surrounding these minima suggest that these species could be achieved under low-temperature conditions, similar to what was predicted for PtO42+ dications. Decavalent platinum sets the upper limit for high oxidation states of the chemical elements.

Molecular opacities of low-lying states of oxygen molecule**Project supported by the National Key Research and Development Program of China (Grant No. 2017YFA0402300), the National Natural Science Foundation of China (Grant Nos. 11934004, 11404180, and 11604052), and the China Postdoctoral Science Foundation (Grant No. 2018M631404).

The , A′3Δu, , 13Πg, and electronic states of oxygen molecule (O2) are calculated by the multi-configuration self-consisted filed (MRCI) + Q method with the scalar relativistic correction and core–valence correlation correction. The obtained spectroscopic constants of the low-lying bound states are in excellent agreement with measurements. Based on the accurately calculated structure parameters, the opacities of the oxygen molecule at the temperatures of 1000 K, 2000 K, 2500 K, and 5000 K under a pressure of 100 atm (1 atm = 1.01325 × 105 Pa) and the partition functions between 10 K and 104 K are obtained. It is found that with the increase of temperature, the opacities for transitions in a long wavelength range are enlarged because of the larger population on excited electronic states at the higher temperatures.

Basis set convergence of high-order coupled cluster methods up to CCSDTQ567 for a highly multireference molecule

We examine the basis-set convergence of post-CCSD(T) contributions to the dissociation energy of the highly multireference C2(1Σg+) molecule. At the infinite basis set limit we obtain: CCSDT–CCSD(T)/cc-pV{6,7}Z = −2.268, CCSDT(Q)–CCSDT/cc-pV{6,7}Z = 3.420, CCSDTQ–CCSDT(Q)/cc-pV{5,6}Z = −1.151, CCSDTQ(5)–CCSDTQ/cc-pV{Q,5}Z = 0.412, CCSDTQ5–CCSDTQ(5)/cc-pV{T,Q}Z = −0.053, CCSDTQ5(6)–CCSDTQ5/cc-pV{D,T}Z = 0.060, CCSDTQ56–CCSDTQ5(6)/cc-pVTZ(1d) = 0.003, CCSDTQ56(7)–CCSDTQ56/cc-pVTZ(1d) = 0.002, and CCSDTQ567–CCSDTQ56(7)/cc-pVTZ(1d) = 0.001 kcal/mol. These post-CCSD(T) contributions add to 0.427 kcal/mol. Including the CCSD(T)/CBS energy, inner-shell, scalar relativistic, spin-orbit, DBOC, and ZPVE corrections from W4.3 theory results in a CCSDTQ567/CBS D0 value of 144.08 kcal/mol, which agrees to within overlapping uncertainties with the experimental ATcT value of 144.006 ± 0.06 kcal/mol.

Vibronic emissions and radiative lifetimes for a3Σ+, b3Π, c3Σ+, d3Δ, and e 3Σ– states of silicon monoxide

In this study, the potential energy curves of the a3Σ+, b3Π, c3Σ+, d3Δ, and e3Σ– states of the silicon monoxide are calculated and the transition dipole moments between them are computed using the complete active space self–consistent field, which is followed by the valence internally contracted multireference configuration interaction approach. The Davidson correction, core–valence correlation and scalar relativistic corrections are considered. The extrapolation of the potential energies to the complete basis set limit is included. The band origins, Einstein A coefficients, and Franck–Condon factors of all vibronic emissions between these five states are calculated. The rotationless radiative lifetimes are approximately 10–1 to 10–4 s for the b3Π state, 10–2 to 10–4 s for the d3Δ state, and 10–3 to 10–4 s for the e3Σ– state. The radiative lifetime quickly becomes shorter as the vibrational level increases for the three states. The lifetime is of the order of 100 ns for all vibrational levels of the c3Σ+ state, indicating that the spontaneous vibronic emissions originating from this state occurs easily. Overall, the vibronic emissions of the b3Π – a3Σ+, d3Δ – b3Π, and e3Σ– – b3Π systems are weak, suggesting that they are difficult to measure through spectroscopy. The vibronic emissions of the c3Σ+ – a3Σ+ and c3Σ+ – b3Π systems are strong, indicating that they are easy to measure in a spectroscopy experiment. The radiative lifetime with the variation of rotational angular momentum quantum number is discussed briefly for υ ≤ 15 and J ≤ 30 of the b3Π, c3Σ+, d3Δ, and e3Σ– states. The results indicate that the radiative lifetime changes only over a small range for each triplet state. The transition properties calculated in this study can be used to explain why only a few experimental transition properties are currently available for these triplet states.

Structure and quantum chemical study of crystalline platinum(II) acetate

Calculations by the hybrid functional PBE0 with scalar relativistic corrections and the QTAIM method revealed the metal-metal bonding in the molecule of platinum(II) acetate Pt4(μ-OAc)8, which stabilizes the molecule by 50 kcal mol−1.

High-Level Ab Initio Predictions for the Ionization Energy, Bond Dissociation Energies, and Heats of Formation of Vanadium Methylidyne Radical and Its Cation (VCH/VCH+).

The ionization energy (IE) of VCH, the 0 K V-CH/VC-H bond dissociation energies (D0s), and the heats of formation at 0 K (ΔHf0°) and 298 K (ΔHf298°) for VCH/VCH+ are predicted by the wave function-based CCSDTQ/CBS approach. This composite-coupled cluster method includes full quadruple excitations in conjunction with the approximation to the complete basis set (CBS) limit. The contributions of zero-point vibrational energy, core-valence (CV) correlation, spin-orbit coupling, and scalar relativistic corrections are taken into account. The present calculations show that adiabatic IE(VCH) = 6.785 eV and demonstrate excellent agreement with an IE value of 6.774 7 ± 0.000 1 eV measured with two-color laser-pulsed field ionization-photoelectron spectroscopy. The CCSDT and MRCI+Q methods which include CV correlations give the best predictions of harmonic frequencies: ω2 (ω2+) (bending) = 534 (650) and 564 (641) cm-1 and the V-CH stretching ω3 (ω3+) = 835 (827) and 856 (857) cm-1 compared with the experimental values. In this work, we offer a streamlined CCSDTQ/CBS approach which shows an error limit (≤20 meV) matching with previous benchmarking efforts for reliable IE and D0 predictions for VCH/VCH+. The CCSDTQ/CBS D0(V+-CH) - D0(V-CH) = -0.012 eV and D0(VC+-H) - D0(VC-H) = 0.345 eV are in good accord with the experimentally derived values of -0.028 4 ± 0.000 1 and 0.355 9 ± 0.000 1 eV, respectively. The present study has demonstrated that the CCSDTQ/CBS protocol can be readily extended to investigate triatomic molecules containing 3d-metals.

Vibronic emissions between the X 1Σ+, A 1Π, C 1Σ−, D 1Δ, and E 1Σ+ states of silicon monoxide

The potential energy curves of the X 1Σ+, A 1Π, C 1Σ−, D 1Δ, and E 1Σ+ singlet states of silicon monoxide are calculated using the complete active space self-consistent field method, followed by the valence internally contracted multireference configuration interaction approach. The transition dipole moments between these states are computed. To calculate the potential energy curves accurately, core-valence correlation and scalar relativistic corrections are accounted for and the extrapolation of the potential energies to the complete basis set limit is included. The rotationless radiative lifetimes are of the order of 10 ns for all vibrational levels of the A 1Π and E 1Σ+ states, suggesting that spontaneous vibronic emissions originating from the two states occur easily. The band origins, Einstein A coefficients, and Franck–Condon factors of all the vibronic emissions are calculated. Overall, the Einstein A coefficients of numerous spontaneous emissions from the A 1Π–X 1Σ+ and E 1Σ+–X 1Σ+ systems are large, suggesting that these vibronic emissions are strong and therefore easy to measure through spectroscopy. The Einstein A coefficients of all the vibronic emissions from the E 1Σ+–A 1Π system are small, predicting that the E 1Σ+–A 1Π transition is weak and therefore difficult to detect through experiment, regardless of the total radiative lifetimes of the E 1Σ+ state. The distribution of radiative lifetime with the variation of the rotational quantum number J is discussed briefly for the υ ≤ 15 and J ≤ 30 levels of the A 1Π and E 1Σ+ states. At a certain υ, the radiative lifetime slowly increases as the J increases for the A 1Π state, whereas it is almost constant for any J of the E 1Σ+ state.

Laser cooling of the SiO+ molecular ion: A theoretical contribution

The feasibility of laser-cooling SiO+ molecular ion is studied by using ab initio methods. The electronic structures of 14 Λ-S states and 30 Ω states of SiO+ are computed by means of high-level configuration interaction method (including Davidson correction) with scalar relativistic correction. The calculated spectroscopic constants of bound states of SiO+ are in excellent agreement with available experimental results. The perturbation on vibrational levels of B2Σ+ is studied with the help of a spin-orbit coupling matrix elements which includes B2Σ+-14Σ−, B2Σ+-14Π, and B2Σ+-12Σ− interacting states pairs. The calculated Franck-Condon factors and branching ratio of the B2Σ+-X2Σ+ transition are highly diagonal. The radiative lifetime of B2Σ+ state is evaluated to be 72 ns, which is sufficiently short for laser cooling. On the basis of the B2Σ+-X2Σ+ transition, we propose a possible laser cooling scheme by using two lasers with wavelengths around 400 nm.

Erratum: Explicitly correlated wave functions of the ground state and the lowest quintuplet state of the carbon atom [Phys. Rev. A 99 , 052512 (2019)

Variational, nonrelativisitic energies have been calculated for the ground state ($^3P_g$) and the lowest quintuplet state ($^5S_u$) of the carbon atom, with wavefunctions expressed in the basis of symmetry-projected, explicitly correlated Gaussian (ECG) lobe functions. New exact limits of these energies have been estimated, amounting to $-37.844906(4)$ and $-37.691751(2)$ hartree. With finite nuclear masses and leading, scalar relativistic corrections included, respective experimental excitation energy of $^{12}C$ has been reproduced with accuracy of about 7 cm$^{-1}$.

Explicitly correlated wave functions of the ground state and the lowest quintuplet state of the carbon atom

Variational, nonrelativistic energies have been calculated for the ground state (${}^{3}{P}_{g}$) and the lowest quintuplet state (${}^{5}{S}_{u}$) of the carbon atom, with wave functions expressed in the basis of symmetry-projected, explicitly correlated Gaussian lobe functions. New exact limits of these energies have been estimated, amounting to $\ensuremath{-}37.844906(4)$ and $\ensuremath{-}37.691751(2)$ hartree. With finite nuclear masses and leading, scalar relativistic corrections included, respective experimental excitation energy of $^{12}\mathrm{C}$ has been reproduced with accuracy of about 7 ${\mathrm{cm}}^{\ensuremath{-}1}$.

Thermochemistry of Uracil, Thymine, Cytosine, and Adenine.

The goal of this study is to give reliable and accurate thermochemical data for uracil, thymine, cytosine, and adenine. The gas-phase heats of formation of these compounds were determined with the diet-HEAT-F12 protocol, which uses explicitly correlated coupled-cluster calculations along with anharmonic vibrational, scalar relativistic, and diagonal Born-Oppenheimer corrections. The thermochemically relevant tautomers of cytosine were also investigated. To derive heats of formation in the gas and solid phases as well as sublimation enthalpies, the thermochemical network approach was utilized, i.e., the available literature data were collected, reviewed, and combined with our theoretical calculations. Solvation free energies were also determined with various methods and compared to experimental data.

Spontaneous emissions between the a3Π, b3Σ-, c3Σ+, d3Π, and e3Σ- states of the AlO+ cation.

The potential energy curves are calculated for the a3Π, b3Σ-, c3Σ+, d3Π, and e3Σ- states of the AlO+ cation using the valence internally contracted multireference configuration interaction approach. The transition dipole moments are computed between these states. To improve the reliability and accuracy of the potential energy curves, several corrections are included, namely, Davidson correction, core-valence correlation and scalar relativistic corrections, as well as extrapolation of the potential energies to the complete basis set limit. The rotationless radiative lifetimes of all vibrational levels are of the orders of 10-1 to 105 s for the b3Σ- state, which are so long that the transitions generating from the b3Σ- state are almost impossible to occur. Those are several ten to several hundred ns for the e3Σ- state, and several μs for the c3Σ+ and d3Π states, which indicate that the transitions originating from the e3Σ-, c3Σ+, and d3Π states occur easily. The largest Einstein coefficient of the transitions from the b3Σ--a3Π system is approximately 10-1, suggesting that these transitions are too weak to be recorded through spectroscopy. The Einstein coefficients of many transitions from the c3Σ+-a3Π, d3Π-a3Π, d3Π-c3Σ+, and e3Σ--a3Π systems are large, indicating that these transitions are able to be measured via spectroscopy. The transition properties reported in this study can be employed to detect the spectroscopy from the AlO+ cation both in outer space and in experiment.

Theoretical study on low-lying electronic states of CP radical: Energy levels, Einstein coefficients, Franck-Condon factors and radiative lifetimes

The state-of-the-art ab initio-based valence internally contracted multireference configuration-interaction (icMRCI) method, including the Davidson correction, core-valence correction and scalar relativistic correction and the basis-set extrapolation, is used to calculate the potential energy curves (PECs) of the X2Σ+, B2Σ+, A2Π, 12Σ−, 12Δ, 22Δ, 22Σ−, 14Σ+, 24Σ+, 14Π, 14Δ, 14Σ−, 24Δ, 24Σ− 16Σ+ and 26Σ+ electronic states for CP radical. We also calculate the transition dipole moments (TDMs), Einstein coefficients and Franck-Condon factors for nineteen dipole allowed transitions between these sixteen states. The PECs are used to fit spectroscopic parameters, which are in excellent agreement with previous experimental and theoretical ones. The computed Franck-Condon factors also agree quite well with accurate semi-empirical results for the A2Π-X2Σ+ and B2Σ+-X2Σ+ systems. Such good agreement implies that the results are accurate enough to assist identification of the spectra from astrophysical sources. Large amounts of energy levels and transition data of high accuracy are provided in this work for CP radical of astrophysical interest, where experimental data are still scarce.

Energy levels, transition dipole moment, transition probabilities and radiative lifetimes for low-lying electronic states of PN

The valence internally contracted multireference configuration-interaction (icMRCI) method is used to compute potential energy curves (PECs) of the X1Σ+, A1Π, C1Σ−, D1Δ, 21Π, a3Σ+, b3Π, d3Δ, e3Σ−, 23Δ, 23Σ−, 15Σ+ and 15Π states for PN, together with the Davidson, core-valence (CV) and scalar relativistic corrections, as well as the basis-set extrapolation. Transition dipole moments (TDMs) of fifteen dipole-allowed transitions between the thirteen states are calculated by the icMRCI approach with the aug-cc-pV6Z basis set. The vibrational band origins, Einstein coefficients and Franck-Condon factors of all spontaneous emissions for the fifteen band systems are determined, seeking to theoretically predict the strong emissions at least of the order of 103 s−1 for Einstein coefficients. Comparing with experimental measurements, our calculations can well reproduce the band origins and Franck-Condon factors of the A1Π-X1Σ+ system. Similar accuracy is assumed for the other band systems. Many emissions for the A1Π-X1Σ+, 21Π-A1Π, 21Π-X1Σ+, 21Π-C1Σ−, 21Π-D1Δ, b3Π-a3Σ+, e3Σ−-b3Π, 23Δ-13Δ, 23Σ−-13Σ−, 23Σ−-b3Π and 15Π-15Σ+ systems are found to be strong according to our calculated Einstein coefficients, whereas the emissions are weak for the 23Δ-b3Π system. Radiative lifetimes for the first 15 vibrational levels are evaluated to be about tens of nanoseconds for the 21Π state, about several hundred nanoseconds for the A1Π state, about several to tens of microseconds for the b3Π, 23Δ, 23Σ− and 15Π states and about several to several hundred microseconds for the e3Σ− state. The results can be used as guidelines for line identification and diagnostics of astrophysical plasma.

Radiative transition probabilities between low-lying electronic states of N2

ABSTRACTThis work mainly investigates the transition dipole moments (TDMs) and radiative transition probabilities of dipole-allowed transitions between the , , , , , , , , and states of N2. Many of these transition properties are previously unknown. For completeness, another 14 electronic states that correlate to four lowest dissociation limits are also calculated. The potential energy curves (PECs) are calculated at the valence internally contracted multireference configuration-interaction (icMRCI) level of theory, along with the Davidson correction, the core-valence (CV) correction and the scalar relativistic correction, as well as the basis-set extrapolation. These corrections, especially the CV correction, greatly improve the accuracy of the PECs, as shown by the excellent agreement of the fitted spectroscopic parameters with the available experimental data. In order to verify the accuracy of transition properties, we calculate the Einstein coefficients of the extensively studied band transition systems and compute the radiative lifetimes of states, which are in good agreement with the experimental data. Similar accuracy can be assumed for the previously undetermined band transition systems. The large Einstein coefficients of these band systems can provide guidelines for observing such newly predicted band transitions in the appropriate spectroscopy experiments.

Interatomic potentials for ground and excited states of Ar+He.

The potential energy curves (PECs) of the ground and excited states that correlate in the atomic limit with Ar([Ne]3s 23p 6,1S), Ar([Ne]3s 23p 54s 1, 3P, 1P), and Ar([Ne]3s 23p 54p 1, 3D, 3P, 3S, 1D, 1P, 1S) are calculated at the multireference configuration interaction (MRCI+Q) theoretical level with extrapolations to the complete basis set limit using all-electron correlation consistent triple-, quadruple-, and quintuple-zeta basis sets. Scalar relativistic corrections are calculated using second-order Douglas-Kroll-Hess Hamiltonian with the corresponding basis sets contracted for scalar relativistic Hamiltonians. For these calculations, the 3s orbitals of the Ar atom are not included in the active space but are correlated through single and double excitations. Spin-orbit eigenstates are computed by diagonalizing the Breit-Pauli matrix between internal configurations with no electrons in external orbitals and added to the scalar relativistic results. A total of 32 molecular PECs are computed with spin-orbit contributions, which correlate with 1s1, 1s5-2, and 2p10-1 atomic Ar energies in Paschen notation. Important features of the PECs and system crossings are discussed.