Relativistic Coupled-cluster Method Research Articles

A reduced cost four-component relativistic unitary coupled cluster method for atoms and molecules.

We present a four-component relativistic unitary coupled cluster method for atoms and molecules. We have used commutator-based non-perturbative approximation using the "Bernoulli expansion" to derive an approximation to the relativistic unitary coupled cluster method. The performance of the full quadratic unitary coupled-cluster singles and doubles method (qUCCSD), as well as a perturbative approximation variant (UCC3), has been reported for both energies and properties. It can be seen that both methods give results comparable to those of the standard relativistic coupled cluster method. The qUCCSD method shows better agreement with experimental results due to the better inclusion of relaxation effects. The relativistic UCC3 and qUCCSD methods can simulate the spin-forbidden transition with easy access to transition properties. A natural spinor-based scheme to reduce the computational cost of relativistic UCC3 and qUCCSD methods has been discussed.

Optical cycling in charged complexes with Ra–N bonds

The extension of laser cooling and trapping techniques to polyatomic molecular ions would have advanced scientific applications such as search of physics outside of the Standard Model, ultracold chemistry etc. We apply the Fock space relativistic coupled cluster method to study low-lying electronic states of molecular ions with Ra–N bonds, namely RaNCH+, RaNH3+ and RaNCCH3+. Prospects of laser cooling of these species are estimated, and the peculiarities of unpaired-electron distributions are analyzed from the point of view of the molecular electronic structure. RaNH3+ and RaNCCH3+ are the first symmetric top molecular ions expected to be suitable for direct laser cooling.

Cholesky Decomposition-Based Implementation of Relativistic Two-Component Coupled-Cluster Methods for Medium-Sized Molecules.

A Cholesky decomposition (CD)-based implementation of relativistic two-component coupled-cluster (CC) and equation-of-motion CC (EOM-CC) methods using an exact two-component Hamiltonian augmented with atomic-mean-field spin-orbit integrals (the X2CAMF scheme) is reported. The present CD-based implementation of X2CAMF-CC and EOM-CC methods employs atomic-orbital-based algorithms to avoid the construction of two-electron integrals and intermediates involving three and four virtual indices. Our CD-based implementation extends the applicability of X2CAMF-CC and EOM-CC methods to medium-sized molecules with the possibility to correlate around 1000 spinors. Benchmark calculations for uranium-containing small molecules were performed to assess the dependence of the CC results on the Cholesky threshold. A Cholesky threshold of 10-4 is shown to be sufficient to maintain chemical accuracy. Example calculations to illustrate the capability of the CD-based relativistic CC methods are reported for the bond-dissociation energy of the uranium hexafluoride molecule, UF6, with up to quadruple-ζ basis sets, and the lowest excitation energy in the solvated uranyl ion [UO22+(H2O)12].

Ab initio potential energy surfaces and spectroscopic and radiative properties of the low-lying states of the radium monohydroxide RaOH radical

Herein, we present the state-of-the-art ab initio studies of the lower states of radium monohydroxide RaOH radical and its deuterated RaOD isotopologue. The potential energy surfaces of the ground and five low-lying excited states are calculated using the Fock-space relativistic coupled cluster method. The vibrational energy levels and all the fundamental frequencies are calculated for the first time using the potential energy surfaces and taking into account the interaction of the modes. Spectroscopic parameters such as electronic term energies, equilibrium internuclear distances, transition and permanent dipole moments, Franck–Condon factors, and radiative lifetimes are predicted. The probable vibrational laser cooling schemes are also proposed.

Relativistic coupled-cluster analysis of the second-order effects on the hyperfine structure in Cs133

Using the single and double approximated relativistic coupled-cluster method, we calculate the first-order hyperfine structure constants for the $6{p}_{3/2}, 8{p}_{3/2}, 6{d}_{3/2}, 7{d}_{3/2}$, and $7{d}_{5/2}$ states in $^{133}\mathrm{Cs}$, and also evaluate the second-order magnetic dipole--magnetic dipole, magnetic dipole--electric quadrupole effects caused by the off-diagonal hyperfine interaction. With these calculations, we reanalyze some recent high-precision experimental measurements of the hyperfine splitting intervals of $^{133}\mathrm{Cs}$. We find that the second-order magnetic dipole--magnetic dipole effects are especially sizable at the order of kHz in the $B$ constant for the ${d}_{3/2,5/2}$ states. Our calculations can provide a reference for future higher precision experimental measurements.

Precision theoretical determination of electric-dipole matrix elements in atomic cesium

We compute the reduced electric-dipole matrix elements $\ensuremath{\langle}n{S}_{1/2}||D||{n}^{\ensuremath{'}}{P}_{J}\ensuremath{\rangle}$ with $n=6,7$ and ${n}^{\ensuremath{'}}=6,7,...,12$ in cesium using the most complete to date ab initio relativistic coupled-cluster method which includes singles, doubles, perturbative core triples, and valence triples. Our results agree with previous calculations at the linearized single double level but also show large contributions from nonlinear singles and doubles as well as valence triples. We also calculate the normalized ratio ${\ensuremath{\xi}}_{n,{n}^{\ensuremath{'}}}\ensuremath{\equiv}(1/\sqrt{2})\ensuremath{\langle}n{S}_{1/2}||D||{n}^{\ensuremath{'}}{P}_{1/2}\ensuremath{\rangle}/\ensuremath{\langle}n{S}_{1/2}||D||{n}^{\ensuremath{'}}{P}_{3/2}\ensuremath{\rangle}$ which is important for experimental determination of matrix elements. The ratios ${\ensuremath{\xi}}_{6,n}$ display large deviations from the nonrelativistic limit which we associate with Cooper-like minima. Several Appendices are provided where we document the procedure for constructing finite basis sets and our implementation of the random-phase approximation and Brueckner-orbitals method.

Using parity-nonconserving spin-spin coupling to measure the Tl nuclear anapole moment in a TlF molecular beam

An experiment utilizing a TlF molecular beam is being developed by the CeNTREX collaboration to search for hadronic interactions that violate both time-reversal ($\mathcal{T}$) and parity ($\mathcal{P}$) invariance. Here, we propose to use the same beam to look for a $\mathcal{T}$-invariance conserving but $\mathcal{P}$-nonconserving (PNC) effect induced by the anapole moment of the Tl nucleus, via a vector coupling of the two nuclear spins in TlF. To measure the nuclear anapole moment, the dc electric and magnetic fields in CeNTREX are replaced by rf fields resonant with a nuclear spin-flip transition. We adapt the relativistic coupled-cluster method in combination with relativistic density functional theory for the calculation of the molecular PNC spin-spin vector coupling constant that links the experimental signal with the anapole moment. The value of the $\mathcal{P}$-conserving spin-spin coupling constant calculated within the same approach is found to be in good agreement with available experimental data.

LIBGRPP: A Library for the Evaluation of Molecular Integrals of the Generalized Relativistic Pseudopotential Operator over Gaussian Functions

Generalized relativistic pseudopotentials (GRPP) of atomic cores implying the use of different potentials for atomic electronic shells with different principal quantum numbers give rise to accurate and reliable relativistic electronic structure models of atoms, molecules, clusters, and solids. These models readily incorporate the effects of Breit electron–electron interactions and one-loop quantum electrodynamics effects. Here, we report the computational procedure for evaluating one-electron integrals of GRPP over contracted Gaussian functions. This procedure was implemented in a library of routines named LIBGRPP, which can be integrated into existing quantum chemistry software, thus enabling the application of various methods to solve the many-electron problem with GRPPs. Pilot applications to electronic transitions in the ThO and UO2 molecules using the new library and intermediate-Hamiltonian Fock space relativistic coupled cluster method are presented. Deviations of excitation energies obtained within the GRPP approach from their all-electron Dirac–Coulomb–Gaunt counterparts do not exceed 50 cm−1 for the 31 lowest-energy states of ThO and 110 cm−1 for the 79 states of UO2. The results clearly demonstrate that rather economical tiny-core GRPP models can exceed in accuracy relativistic all-electron models defined by Dirac–Coulomb and Dirac–Coulomb–Gaunt Hamiltonians.

Molecular enhancement factors for the P,T -violating electric dipole moment of the electron in BaCH3 and YbCH3 symmetric top molecules

High-precision tests of fundamental symmetries are looking for the parity- ($\mathcal{P}$), time-reversal- ($\mathcal{T}$) violating electric dipole moment of the electron (eEDM) as proof of physics beyond the Standard Model. Particularly, in polyatomic molecules, the complex vibrational and rotational structure gives the possibility to reach high enhancement of the $\mathcal{P},\mathcal{T}$-odd effects in moderate electric fields, and with the possibility of increasing the statistical sensitivity by using laser cooling. In this work, we calculate the $\mathcal{P},\mathcal{T}$-odd molecular enhancement factor of the eEDM (${W}_{\mathrm{d}}$) and of the scalar-pseudoscalar interaction (${W}_{\mathrm{s}}$) necessary for the interpretation of future experiments on the promising candidates ${\mathrm{BaCH}}_{3}$ and ${\mathrm{YbCH}}_{3}$. We employ high-accuracy relativistic coupled cluster methods and systematically evaluate the uncertainties of our computational approach. Compared to other Ba- and Yb-containing molecules, ${\mathrm{BaCH}}_{3}$ and ${\mathrm{YbCH}}_{3}$ exhibit larger ${W}_{\mathrm{d}}$ and ${W}_{\mathrm{s}}$ associated to the increased covalent character of the $M$--C bond. The calculated values are $3.22\ifmmode\pm\else\textpm\fi{}0.12\ifmmode\times\else\texttimes\fi{}{10}^{24}\frac{h\text{Hz}}{e\text{cm}}$ and $13.80\ifmmode\pm\else\textpm\fi{}0.35\ifmmode\times\else\texttimes\fi{}{10}^{24}\frac{h\text{Hz}}{e\text{cm}}$ for ${W}_{\mathrm{d}}$, and $8.42\ifmmode\pm\else\textpm\fi{}0.29\phantom{\rule{4pt}{0ex}}h\mathrm{kHz}$ and $50.16\ifmmode\pm\else\textpm\fi{}1.27\phantom{\rule{4pt}{0ex}}h\mathrm{kHz}$ for ${W}_{\mathrm{s}}$, in ${\mathrm{BaCH}}_{3}$ and ${\mathrm{YbCH}}_{3}$, respectively. The robust, accurate, and cost-effective computational scheme reported in this work makes our results suitable for extracting the relevant fundamental properties from future measurements and also can be used to explore other polyatomic molecules sensitive to various violations of fundamental symmetries.

Determination of quadrupolar dispersion coefficients of the alkali-metal atoms interacting with different material media

In the present work, we determine the ${C}_{5}$ coefficients along with their uncertainties due to quadrupole polarization effects of all the alkali atoms interacting with a metal (Au), a semiconductor (Si), and four dielectric materials (${\mathrm{SiO}}_{2}, {\mathrm{SiN}}_{x}$, yttrium aluminum garnet, and sapphire). The required dynamic electric quadrupole ($E2$) polarizabilities are evaluated by calculating $E2$ matrix elements of a large number of transitions in the alkali atoms by employing a relativistic coupled-cluster method. A significant contribution towards the long-range van der Waals potential is made by the quadrupole polarization effects. Our finding shows that contributions from the ${C}_{5}$ coefficients to the atom-wall interaction potentials are pronounced at short distances (1--10 nm). The ${C}_{3}$ coefficients of a Fr atom interacting with the above material media are also reported. These results could be useful in understanding the interactions of alkali atoms trapped in different material bodies during high-precision measurements.

Two-Photon Polarizability of Ba+ Ion: Control of Spin-Mixing Processes in an Ultracold 137Ba+ − 87Rb Mixture

In this work, we present a scheme of a two-photon interaction to calculate magic wavelengths for the 62S12 − 52D32,52 clock transitions of Ba+ ion employing the relativistic coupled-cluster method. These magic wavelengths can be essential inputs to achieve better accuracy in the future ionic clock experiments. In this paper, we further show an application of a two-photon interaction to the spin-mixing processes, |0,0⟩↔|+1,−1⟩ and |0,0⟩↔|−1,+1⟩, of an ultra-cold spin-1 mixture of 137Ba+ ions and 87Rb atoms. We determine the protocols for selecting these spin-mixing oscillations by changing the strength and frequencies of the externally applied magnetic field and laser beams, respectively.

Relativistic coupled cluster calculations of the electron affinity and ionization potential of Nh(113)

Theoretical calculations based on the Dirac–Coulomb–Breit relativistic coupled cluster method have been carried out for the electron affinities and ionization potentials (IPs) of the superheavy element nihonium (Nh) and its lighter homologues In and Tl. The In and Tl calculations are in agreement with measurement within uncertainties. For Nh, where experiment is yet unknown, we predict the IP of 7.569(48) eV and electron affinity of 0.776(30) eV.

Ionization potentials and electron affinities of Rg, Cn, Nh, and Fl superheavy elements

The successive ionization potentials (IPs) and electron affinities (EAs) for superheavy elements with $111\ensuremath{\le}Z\ensuremath{\le}114$, namely, Rg, Cn, Nh, and Fl, are reexamined using the relativistic Fock-space coupled-cluster method with nonperturbative single (S), double (D), and triple (T) cluster amplitudes (FS-CCSDT). For most of the considered quantities, the triple-amplitude contributions turn out to be important. The Breit and frequency-dependent Breit corrections are evaluated by means of the configuration-interaction method. The quantum-electrodynamics corrections to the IPs and EAs are taken into account within the model-QED-operator approach. The obtained results are within 0.10 eV uncertainty.

A reduced cost four-component relativistic coupled cluster method based on natural spinors.

We present the theory, implementation, and benchmark results for a frozen natural spinors based reduced cost four-component relativistic coupled cluster method. The natural spinors are obtained by diagonalizing the one-body reduced density matrix from a relativistic second-order Møller-Plesset calculation based on a four-component Dirac-Coulomb Hamiltonian. The correlation energy in the coupled cluster method converges more rapidly with respect to the size of the virtual space in the frozen natural spinor basis than that observed in the standard canonical spinors obtained from the Dirac-Hartree-Fock calculation. The convergence of properties is not smooth in the frozen natural spinor basis. However, the inclusion of the perturbative correction smoothens the convergence of the properties with respect to the size of the virtual space in the frozen natural spinor basis and greatly reduces the truncation errors in both energy and property calculations. The accuracy of the frozen natural spinor based coupled cluster methods can be controlled by a single threshold and is a black box to use.

Relativistic coupled-cluster calculations of the energies, Landé g factors, radiative transition properties including line strengths, oscillator strengths, and transition rates for Th3+

Energies, Landé g factors, line strengths, transition probabilities, and oscillator strengths for electric dipole (E1), electric quadrupole (E2), and magnetic dipole (M1) transitions arising from the nS1/2 (n=7-10), nP1/2,3/2 (n=7-9), nD3/2,5/2 (n=6-9), nF5/2,7/2 (n=5-8), and nG7/2,9/2 (n=5,6) energy levels in Th3+, are calculated using the singles and doubles approximation of the relativistic coupled-cluster (CCSD) method. The data are compared with available experimental and theoretical results, showing a good agreement. To the best of our knowledge, many results are reported for the first time. This work reports a high-precision atomic property calculation of Th3+ and is of great significance for future spectral analysis and precision experiments targeting various applications.

Precision Calculation of Hyperfine Constants for Extracting Nuclear Moments of ^{229}Th.

Determination of nuclear moments for many nuclei relies on the computation of hyperfine constants, with theoretical uncertainties directly affecting the resulting uncertainties of the nuclear moments. In this work, we improve the precision of such a method by including for the first time an iterative solution of equations for the core triple cluster amplitudes into the relativistic coupled-cluster method, with large-scale complete basis sets. We carried out calculations of the energies and magnetic dipole and electric quadrupole hyperfine structure constants for the low-lying states of ^{229}Th^{3+} in the framework of such a relativistic coupled-cluster single double triple method. We present a detailed study of various corrections to all calculated properties. Using the theory results and experimental data, we found the nuclear magnetic dipole and electric quadrupole moments to be μ=0.366(6)μ_{N} and Q=3.11(2) eb, respectively, and reduce the uncertainty of the quadrupole moment by a factor of 3. The Bohr-Weisskopf effect of the finite nuclear magnetization is investigated, with bounds placed on the deviation of the magnetization distribution from the uniform one.

Relativistic coupled-cluster calculation of hyperfine-structure constants of Th3+229 and evaluation of the electromagnetic nuclear moments of Th229

$^{229}$Th is a promising candidate for developing a nuclear optical clock and searching the new physics beyond the standard model. Accurate knowledge of the nuclear properties of $^{229}$Th is very important. In this work, we calculate hyperfine-structure constants for the first four states of $^{229}$Th$^{3+}$ using the relativistic coupled-cluster method based on the Gauss basis set. The no-pair Dirac-Coulomb-Breit Hamiltonian with the lowest-order quantum electrodynamics (QED) correction is the starting point, together with all linear and non-linear terms of single and double excitations are included in coupled-cluster calculation. With the measured value of the hyperfine-structure constants [Phys. Rev. Lett. 106. 223001(2011)], we get the magnetic dipole moment, $\mu=0.359(9)$, and the electric quadrupole moment, $Q=2.95(7)$, of the $^{229}$Th nucleus. Our magnetic dipole moment is perfectly consistent with the recommended values, $\mu=0.360(7)$, from the all-order calculation by Safronova \textit{et. al.}[Phys.Rev.A 88, 060501 (2013)], but our electric quadrupole moment is smaller than their recommended value, $Q=3.11(6)$, about 5\%. Our results show that the non-linear terms of single and double excitations, which were not included in the all-order calculation by Safronova \textit{et. al.}, are very crucial to produce a precise $Q$ value of $^{229}$Th. Additionally, we also present magnetic octupole hyperfine-structure constants and some important non-diagonal hyperfine transition matrix elements, which are required for further extracting the magnetic octupole moment $\Omega$ of $^{229}$Th nucleus.

Radiative transition properties including line strengths, oscillator strengths, and transition rates for Ra II

In this paper, we calculate the energy and radiative transition properties of singly charged Ra using the singles and doubles approximated relativistic coupled-cluster method. The excitation energies of the (7 - 12)S1/2, (7 - 12)P1/2,3/2, (6 - 10)D3/2,5/2 states in Ra+ are reported. The results are compared with available experimental results, showing a good agreement. Accurate values along with their uncertainties of the electric-dipole (E1), electric-quadrupole (E2), and several important magnetic-dipole (M1) matrix elements for transitions in singly charged Ra are determined by the relativistic coupled-cluster method at different approximations. Using these computed E1, E2, and M1 matrix elements, we also provide the related radiative transition properties of Ra+, including line strengths, oscillator strengths, and transition rates. Furthermore, we also investigate the role of electron correlation in these properties. To the best of our knowledge, many results are reported for the first time. Present work reports a high-precision atomic property calculation of Ra+ and is beneficial for present and future experiments using Ra+ targeting various applications such as atomic clocks and studies of parity non-conservation.

Ab initio calculations of the hyperfine structure of Ra+ and evaluations of the electric quadrupole moment Q of the 209,211,221,223Ra nuclei

The relativistic coupled-cluster method has been employed to calculate the energies, magnetic dipole, electric quadrupole and magnetic octupole hyperfine-structure constants of the low-lying states of Ra+. The validity of our calculations is substantiated by comparing the calculated binding energies and magnetic dipole hyperfine-structure constants with the corresponding available experimental results. The combination of our theoretical results with the available experimental values of the electric quadrupole hyperfine-structure constant, makes it possible to extract the electric quadrupole moments Q of the 209,211,221,223Ra nuclei. Our Q(221Ra) = 1.968(34) and Q(223Ra) = 1.248(22) are consistent with the referenced values 1.978(106) and 1.254(66) from a semi-empirical analysis (1988 Z. Phys. D 11 105), but Q(211Ra) = 0.33(2) is smaller than the referenced value 0.48(4) by about 30%. Furthermore, we also performed a procedure for assessing the contributions of magnetic octupole hyperfine interaction to the hyperfine splitting considering the preliminary value of magnetic octupole moment from the single-particle nuclear structure model. The sensitivity of hyperfine-structure interval measurements in 223Ra+ that can reveal the effect caused by the nuclear octupole moment are found to be on the order of kHz.

Relativistic coupled-cluster calculation of hyperfine-structure constants of La 2+

Using the singles and doubles approximated relativistic coupled-cluster method, we calculate hyperfine-structure constant A and B of 24 low-lying states of La2+. The results of the lowest four states 5d 3/2,5/2 and 4f 5/2,7/2 are compared with recently measured values (Olmschenk et al 2017 Phys. Rev. A 96 032502). Reasonable agreement is found for hyperfine-structure constant A of 5d 3/2 and 4f 5/2,7/2 states, but the hyperfine-structure constant A of 5d 5/2 and the hyperfine-structure constant B have a large deviation. To understand the role of the electron correlation effect, the intermediate results from lower-order many-body perturbation-theory and the linear version of coupled-cluster with single and double approximation are also presented. In the framework of coupled-cluster theory, the contribution trends from individual correlation effects are also investigated.