Second-order Many-body Perturbation Research Articles

Thermal quasiparticle theory.

The widely used thermal Hartree-Fock (HF) theory is generalized to include the effect of electron correlation while maintaining its quasi-independent-particle framework. An electron-correlated internal energy (or grand potential) is postulated in consultation with the second-order finite-temperature many-body perturbation theory (MBPT), which then dictates the corresponding thermal orbital (quasiparticle) energies in such a way that all fundamental thermodynamic relations are obeyed. The associated density matrix is of a one-electron type, whose diagonal elements take the form of the Fermi-Dirac distribution functions, when the grand potential is minimized. The formulas for the entropy and chemical potential are unchanged from those of Fermi-Dirac or thermal HF theory. The theory thus stipulates a finite-temperature extension of the second-order Dyson self-energy of one-particle many-body Green's function theory and can be viewed as a second-order, diagonal, frequency-independent, thermal inverse Dyson equation. At low temperatures, the theory approaches finite-temperature MBPT of the same order, but it may outperform the latter at intermediate temperatures by including additional electron-correlation effects through orbital energies. A physical meaning of these thermal orbital energies is proposed (encompassing that of thermal HF orbital energies, which has been elusive) as a finite-temperature version of Janak's theorem.

Impact of correlations on nuclear binding energies

A strong effort will be dedicated in the coming years to extend the reach of ab initio nuclear-structure calculations to heavy doubly open-shell nuclei. In order to do so, the most efficient strategies to incorporate dominant many-body correlations at play in such nuclei must be identified. With this motivation in mind, the present work analyses the step-by-step inclusion of many-body correlations and their impact on binding energies of Calcium and Chromium isotopes. Employing an empirically-optimal Hamiltonian built from chiral effective field theory, binding energies along both isotopic chains are studied via a hierarchy of approximations based on polynomially-scaling expansion many-body methods. More specifically, calculations are performed based on (i) the spherical Hartree–Fock–Bogoliubov mean-field approximation plus correlations from second-order Bogoliubov many-body perturbation theory or Bogoliubov coupled cluster with singles and doubles on top of it, along with (ii) the axially-deformed Hartree–Fock–Bogoliubov mean-field approximation plus correlations from second-order Bogoliubov many-body perturbation theory built on it. The corresponding results are compared to experimental data and to those obtained via valence-space in-medium similarity renormalization group calculations at the normal-ordered two-body level that act as a reference in the present study. The spherical mean-field approximation is shown to display specific shortcomings in Ca isotopes that can be understood analytically and that are efficiently corrected via the consistent addition of low-order dynamical correlations on top of it. While the same setting cannot appropriately reproduce binding energies in doubly open-shell Cr isotopes, allowing the unperturbed mean-field state to break rotational symmetry permits to efficiently capture the static correlations responsible for the phenomenological differences observed between the two isotopic chains. Eventually, the present work demonstrates that polynomially-scaling expansion methods based on unperturbed states that possibly break (and restore) symmetries constitute an optimal route to extend ab initio calculations to heavy closed- and open-shell nuclei.

The performance of approximate equation of motion coupled cluster for valence and core states of heavy element systems

The equation of motion coupled cluster singles and doubles model (EOM-CCSD) is an accurate, black-box correlated electronic structure approach to investigate electronically excited states and electron attachment or detachment processes. It has also served as a basis for developing less computationally expensive approximate models such as partitioned EOM-CCSD (P-EOM-CCSD), the second-order many-body perturbation theory EOM (EOM-MBPT(2)) and their combination (P-EOM-MBPT(2)) [S. Gwaltney et al., Chem. Phys. Lett. 248, 189–198 (1996)]. In this work we outline an implementation of these approximations for four-component based Hamiltonians and investigate their accuracy relative to EOM-CCSD for valence excitations, valence and core ionisations and electron attachment, and this for a number of systems of atmospheric or astrophysical interest containing elements across the periodic table. We have found that across the different systems and electronic states of different nature considered, partition EOM-CCSD yields results with the largest deviations from the reference, whereas second-order based approaches tend show a generally better agreement with EOM-CCSD. We trace this behaviour to the imbalance brought about by the removal of excited state relaxation in the partition approaches, with respect to degree of electron correlation recovered.

Modewise Johnson–Lindenstrauss embeddings for nuclear many-body theory

In this work, we explore modewise Johnson-Lindenstrauss embeddings (JLEs) as a tool to reduce the computational cost and memory requirements of nuclear many-body methods. JLEs are randomized projections of high-dimensional data tensors onto low-dimensional subspaces that preserve key structural features. Such embeddings allow for the oblivious and incremental compression of large tensors, e.g., the nuclear Hamiltonian, into significantly smaller random sketches that still allow for the accurate calculation of ground-state energies and other observables. Their oblivious character makes it possible to compress a tensor without knowing in advance exactly what observables one might want to approximate at a later time. This opens the door for the use of tensors that are much too large to store in memory, e.g., complete two-plus three-nucleon Hamiltonians in large, symmetry-unrestricted bases. Such compressed Hamiltonians can be stored and used later on with relative ease. As a first step, we analyze the JLE's impact on the second-order Many-Body Perturbation Theory (MBPT) corrections for nuclear ground-state observables. Numerical experiments for a wide range of closed-shell nuclei, model spaces and state-of-the-art nuclear interactions demonstrate the validity and potential of the proposed approach: We can compress nuclear Hamiltonians hundred- to thousand-fold while only incurring mean relative errors of 1\% or less in ground-state observables. Importantly, we show that JLEs capture the relevant physical information contained in the highly structured Hamiltonian tensor despite their random characteristics. In addition to the significant storage savings, the achieved compressions imply multiple order-of magnitude reductions in computational effort when the compressed Hamiltonians are used in higher-order MBPT or nonperturbative many-body methods.

Stochastic evaluation of four-component relativistic second-order many-body perturbation energies: A potentially quadratic-scaling correlation method.

A second-order many-body perturbation correction to the relativistic Dirac-Hartree-Fock energy is evaluated stochastically by integrating 13-dimensional products of four-component spinors and Coulomb potentials. The integration in the real space of electron coordinates is carried out by the Monte Carlo (MC) method with the Metropolis sampling, whereas the MC integration in the imaginary-time domain is performed by the inverse-cumulative distribution function method. The computational cost to reach a given relative statistical error for spatially compact but heavy molecules is observed to be no worse than cubic and possibly quadratic with the number of electrons or basis functions. This is a vast improvement over the quintic scaling of the conventional, deterministic second-order many-body perturbation method. The algorithm is also easily and efficiently parallelized with 92% strong scalability going from 64 to 4096 processors.

Benchmarking calculations of wavelengths and transition rates with spectroscopic accuracy for W xlviii through W lvi tungsten ions

Atomic properties of $n=3$ levels for ${\mathrm{W}}^{47+}\ensuremath{-}{\mathrm{W}}^{55+}$ ions ($Z=74$) are systematically calculated using two different and independent methods, namely, the second-order many-body perturbation theory and the multiconfiguration Dirac-Hartree-Fock method combined with the relativistic configuration interaction approach. Wavelengths and transition rates for electric- and magnetic-dipole transitions involving the $n=3$ levels of ${\mathrm{W}}^{47+}\ensuremath{-}{\mathrm{W}}^{55+}$ are calculated. In addition, we discuss in detail the importance of the valence and core-valence electron correlations, the Breit interaction, the higher-order frequency-dependent retardation correction, and the leading quantum electrodynamical corrections for transition wavelengths. Spectroscopic accuracy is achieved for the present calculated wavelengths, and most of them agree with experimental values within $0.05%$. Our calculated wavelengths, combined with collisional radiative model simulations, are used to identify the yet unidentified 25 observed lines in the extremely complex spectrum between $27\phantom{\rule{0.16em}{0ex}}\AA{}$ and $34\phantom{\rule{0.16em}{0ex}}\AA{}$ measured by Lennartsson $et\phantom{\rule{4pt}{0ex}}al.$ [Phys. Rev. A 87, 062505 (2013)]. We provide additional data for 472 strong electric-dipole transitions in the wavelength range of 17--50 \AA{}, and 185 strong magnetic-dipole transitions between $36\phantom{\rule{0.16em}{0ex}}\AA{}$ and $4384\phantom{\rule{0.16em}{0ex}}\AA{}$, with a line intensity greater than $1\phantom{\rule{0.16em}{0ex}}\mathrm{photon}/\mathrm{s}$. These can provide benchmark data for future experiments and theoretical calculations.

Stochastic Vector Techniques in Ground-State Electronic Structure.

We review a suite of stochastic vector computational approaches for studying the electronic structure of extended condensed matter systems. These techniques help reduce algorithmic complexity, facilitate efficient parallelization, simplify computational tasks, accelerate calculations, and diminish memory requirements. While their scope is vast, we limit our study to ground-state and finite temperature density functional theory (DFT) and second-order many-body perturbation theory. More advanced topics, such as quasiparticle (charge) and optical (neutral) excitations and higher-order processes, are covered elsewhere. We start by explaining how to use stochastic vectors in computations, characterizing the associated statistical errors. Next, we show how to estimate the electron density in DFT and discuss effective techniques to reduce statistical errors. Finally, we review the use of stochastic vectors for calculating correlation energies within the second-order Møller-Plesset perturbation theory and its finite temperature variational form. Example calculation results are presented and used to demonstrate the efficacy of the methods.

Monte Carlo MP2-F12 for Noncovalent Interactions: The C60 Dimer.

A scalable stochastic algorithm is presented that can evaluate explicitly correlated (F12) second-order many-body perturbation (MP2) energies of weak, noncovalent, intermolecular interactions. It first transforms the formulas of the MP2 and F12 energy differences into a short sum of high-dimensional integrals of Green's functions in real space and imaginary time. These integrals are then evaluated by the Monte Carlo method augmented by parallel execution, redundant-walker convergence acceleration, direct-sampling autocorrelation elimination, and control-variate error reduction. By sharing electron-pair walkers across the supermolecule and its subsystems spanned by the joint basis set, the statistical uncertainty is reduced by one to 2 orders of magnitude in the MP2 binding energy corrected for the basis-set incompleteness and superposition errors. The method predicts the MP2-F12/aug-cc-pVDZ binding energy of 19.1 ± 4.0 kcal mol-1 for the C60 dimer at the center distance of 9.748 Å.

Relativistic coupled-cluster calculation of the electric dipole polarizability and correlation energy of Cn, Nh+ , and Og: Correlation effects from lighter to superheavy elements

We employ a fully relativistic coupled-cluster theory to calculate the ground-state electric dipole polarizability and electron correlation energy of superheavy elements Cn, Nh$^+$ and Og. To assess the trend of electron correlation as function of $Z$, we also calculate the correlation energies for three lighter homologs--Zn, Cd and Hg; Ga$^+$, In$^+$ and Tl$^+$; Kr, Xe and Rn--for each superheavy elements. The relativistic effects and quantum electrodynamical corrections are included using the Dirac-Coulomb-Breit Hamiltonian with the corrections from the Uehling potential and the self-energy. The effects of triple excitations are considered perturbatively in the theory. Furthermore, large bases are used to test the convergence of results. Our recommended values of polarizability are in good agreement with previous theoretical results for all SHEs. From our calculations we find that the dominant contribution to polarizability is from the valence electrons in all superheavy elements. Except for Cn and Og, we observe a decreasing contribution from lighter to superheavy elements from the Breit interaction. For the corrections from the vacuum polarization and self-energy, we observe a trend of increasing contributions with $Z$. From energy calculations, we find that the second-order many-body perturbation theory overestimates the electron correlation energy for all the elements considered in this work.

M1 transition energy and rate in the ground configuration of Ag-like ions with 62 ⩽ Z ⩽ 94

Systematic calculations and assessments are performed for the magnetic dipole (M1) transition energies and rates between the levels in the ground configuration 4d104f along the Ag-like isoelectronic sequence with 62 ⩽ Z ⩽ 94 based on the second-order many-body perturbation theory implemented in the Flexible Atomic Code. The electron correlations, Breit interaction and quantum electrodynamics effects are taken into account in the present calculations. The accuracy and reliability of our results are evaluated through comprehensive comparisons with available measurements and other theoretical results. For transition energies, our results are in good agreement with the recent experimental data obtained from the electron beam ion traps within 0.18%. The maximum discrepancy between our results and those obtained with the large-scale multiconfiguration Dirac–Hartee–Fock calculations by Grumer et al. [Phys. Rev. A 89 062501 (2014)] is less than 0.13% along the isoelectronic sequence. Furthermore, the corresponding M1 transition rates are also reported. The present results can be used as the benchmark and useful for spectra simulation and diagnostics of astrophysical and fusion plasmas.

Benchmarking calculations with spectroscopic accuracy of level energies and wavelengths in W LVII–W LXII tungsten ions

Atomic properties of n=3 states of the W56+ - W61+ ions are systematically investigated through two state-of-the-art methods, namely, the second-order many-body perturbation theory, and the multi-configuration Dirac–Hartree–Fock method combined with the relativistic configuration interaction approach. The contributions of valence-valence and core-valence electron correlations, the Breit interaction, the higher-order retardation correction beyond the Breit interaction through the transverse photon interaction, and the quantum electrodynamical corrections to the excitation energies are studied in detail. The excitation energies and wavelengths obtained with the two methods agree with each other within ≈0.01%. The present results achieve spectroscopic accuracy and provide a benchmark test for various applications and other theoretical calculations of W56+ - W61+ ions. They will assist spectroscopists in their assignment and direct identification of observed lines in complex spectra.

Determination of the dipole polarizability of the alkali-metal negative ions

We present electric dipole polarizabilities ($\alpha_d$) of the alkali-metal negative ions, from H$^-$ to Fr$^-$, by employing four-component relativistic many-body methods. Differences in the results are shown by considering Dirac-Coulomb (DC) Hamiltonian, DC Hamiltonian with the Breit interaction, and DC Hamiltonian with the lower-order quantum electrodynamics interactions. At first, these interactions are included self-consistently in the Dirac-Hartree-Fock (DHF) method, and then electron correlation effects are incorporated over the DHF wave functions in the second-order many-body perturbation theory, random phase approximation and coupled-cluster (CC) theory. Roles of electron correlation effects and relativistic corrections are analyzed using the above many-body methods with size of the ions. We finally quote precise values of $\alpha_d$ of the above negative ions by estimating uncertainties to the CC results, and compare them with other calculations wherever available.

Study of laser-driven multielectron dynamics of Ne atom using time-dependent optimised second-order many-body perturbation theory

We calculate the high-harmonic generation (HHG) spectra, strong-field ionisation, and time-dependent dipole-moment of Ne using explicitly time-dependent optimised second-order many-body perturbation method (TD-OMP2) where both orbitals and amplitudes are time-dependent. We consider near-infrared (800 nm) and mid-infrared (1200 nm) laser pulses with very high intensities (, , and ), required for strong-field experiments with the high-ionisation potential (21.6 eV) atom. We compare the result of the TD-OMP2 method with the time-dependent complete-active-space self-consistent field method and the time-dependent Hartree-Fock method. Further, we report the implementation of the TD-CC2 method within the chosen active space, which is also a second-order approximation to the TD-CCSD method, and present results of time-dependent dipole-moment and HHG spectra with an intensity of at a wavelength of 800 nm. It is found that the TD-CC2 method is not stable in the case with a higher laser intensity, and it does not provide a gauge-invariant description of the physical properties, which makes TD-OMP2 a superior choice to reach out to larger chemical systems, especially for the study of strong-field dynamics. The obtained results indicate that the TD-OMP2 method shows moderate performance, overestimating the response of Ne, while TDHF underestimates it..

Time-dependent optimized coupled-cluster method for multielectron dynamics. III. A second-order many-body perturbation approximation.

We report successful implementation of the time-dependent second-order many-body perturbation theory using optimized orthonormal orbital functions called time-dependent optimized second-order many-body perturbation theory to reach out to relatively larger chemical systems for the study of intense-laser-driven multielectron dynamics. We apply this method to strong-field ionization and high-order harmonic generation of Ar. The calculation results are benchmarked against ab initio time-dependent complete-active-space self-consistent field, time-dependent optimized coupled-cluster double, and time-dependent Hartree-Fock methods, as well as a single active electron model to explore the role of electron correlation.

Energy levels, transition rates and electron impact excitation rates for B-like Kr XXXII

Energy levels and transition rates for electric-dipole, electric-quadrupole, electric-octupole, magnetic-dipole, and magnetic-quadrupole transitions among the levels arising from the n≤ 5 configurations in B-like Kr XXXII are calculated by using two state-of-the-art methods, namely, the multi-configuration Dirac–Hartree–Fock (MCDHF) approach and the second-order many-body perturbation theory (RMBPT). Our results are compared with several available experimental and other theoretical values. Electron-impact excitation (EIE) collision strengths are calculated via the independent process and isolated resonance approximation using distorted-wave (denoted by IPIRDW). Radiation damping effects on the resonance excitation contributions are included. Effective collision strengths are calculated as a function of electron temperature by assuming a Maxwellian electron velocity distribution. Spectral line intensities are modelled by using collision radiative model, and several line pairs pointed out might be useful for density diagnostics.

Extensive and accurate energy levels, transition rates, lifetimes, hyperfine structures and isotope shifts for the Rb XXXVI spectrum of plasma interest

We present relativistic multiconfiguration Dirac-Hartree-Fock (MCDHF) spectrum calculations for Rb XXXVI with correlations within the n = 8 complex. An extensive and accurate set of energy levels, wavelengths, line strengths, weighted oscillator strengths, lifetimes, isotope shifts, hyperfine structure and Landé gJ factors has been calculated for lower 127 odd- and even-parity states arising from the 1s2 and 1snl(n=1−8,0≤l≤n−1) configurations in He-like-Rb. Transition rates for electric-multipole (dipole (E1), quadrupole (E2), and octupole (E3)) and magnetic-multipole (dipole (M1), quadrupole (M2), and octupole (M3)) have been also calculated. For the accuracy of our results, we have implemented parallel calculations using a Flexible Atomic Code (FAC) by introducing the Second-Order Many-Body Perturbation Theory (MBPT) method. Additionally, Breit interaction and leading quantum electrodynamic effects (QED) have been included as perturbations in extensive relativistic configuration interaction (RCI) calculations. The results arising in the two sets of calculations MCDHF and MBPT are quite close. Our results are compared with other available theory reported in the literature as well as the experimental data listed in the Atomic Spectra Database (ASD) of the National Institute of Standards and Technology (NIST) online database. Good agreement between them has been found. The present complete and consistent results can be used in line identification, plasma modeling and diagnostics of astrophysical plasma.

An adiabatic connection for doubly-occupied configuration interaction wave functions.

An adiabatic connection (AC) is developed as an electron correlation correction for doubly occupied configuration interaction (DOCI) wave functions. Following the work of Pernal [Phys. Rev. Lett. 120, 013001 (2018)], the working equations of the approach, termed AC-DOCI, are rooted in the extended random phase approximation (ERPA) and require knowledge of only the ground-state two-electron reduced density matrix (2RDM) from the DOCI. As such, the AC is naturally suited to modeling electron correlation in variational 2RDM (v2RDM)-based approximations to the DOCI. The v2RDM-driven AC-DOCI is applied to the dissociation of molecular nitrogen and the double dissociation of water; the approach yields energies that are similar in quality to those from second-order multireference perturbation theory near equilibrium, but the quality of the AC-DOCI energy degrades at stretched geometries. The exact adiabatic connection path suggests the assumption that the one-electron reduced-density matrix is constant along the AC path is invalid at stretched geometries, but this deficiency alone cannot explain the observed behavior. Rather, it appears that the ERPA's single-particle-transition ansatz cannot, in general, provide good approximations to the 2RDM along the AC path. The AC-DOCI is also applied to a set of 45 reaction energies; for these systems, the approach has an average accuracy that is comparable to that of single-reference second-order many-body perturbation theory.

Extended calculations of energy levels, radiative properties, and lifetimes for nitrogen-like Zr XXXIV

Highly accurate calculations are performed for the lowest 272 fine-structure levels arising from the 2s22p3, 2s2p4, 2p5, 2s22p23l, 2s2p33l, and 2p43l configurations in N-like Zr XXXIV using two state-of-the-art methods, second-order many-body perturbation theory and multiconfiguration Dirac–Fock. Complete and consistent atomic data, including excitation energies, wavelengths, E1, E2, M1, M2 line strengths, oscillator strengths, transition rates, and lifetimes for these 272 levels, are provided. Comparisons are made between the present two data sets, as well as with other available results. The present data are accurate enough for identification and deblending of emission lines involving the n=3 levels and are also useful for modeling and diagnosing fusion plasmas. These data can be considered as a benchmark for other calculations.

Extended calculations of energy levels, radiative properties, and lifetimes for oxygen-like Mo XXXV

Employing two state-of-the-art methods, second-order many-body perturbation theory and multiconfiguration Dirac–Fock, highly accurate calculations are performed for the lowest 318 fine-structure levels arising from the 2s22p4, 2s2p5, 2p6, 2s22p33l, 2s2p43l, 2p53l, and 2s22p34l configurations in O-like Mo XXXV. Complete and consistent atomic data, including excitation energies, lifetimes, wavelengths, and E1, E2, M1 line strengths, oscillator strengths, and transition rates among these 318 levels are provided. Comparisons are made between the present two data sets, as well as with other available experimental and theoretical values. The present data are accurate enough for identification and deblending of emission lines involving the n=3,4 levels and are also useful for modeling and diagnosing fusion plasmas. These data can be considered as a benchmark for other calculations.

Energy levels, weighted oscillator strengths, transition rates, lifetimes, hyperfine structures and isotope shifts of He-like-Sn deduced from the relativistic multiconfiguration Dirac–Hartree–Fock and second-order many-body perturbation theory calculations

We disclose relativistic multiconfiguration Dirac–Hartree–Fock (MCDHF) spectrum calculations for SnXLIX. Energy levels, weighted oscillator strengths, isotope shifts, hyperfine structure and Landé gJ factors are calculated for 127 odd- and even-parity states as well as lifetimes and transition rates between these states. To scrutinize the accuracy of our results, we have implemented parallel calculations using a Flexible Atomic Code (FAC) by introducing the Second-Order Many-Body Perturbation Theory (MBPT) method. Additionally, the Breit interaction and leading quantum electrodynamic effects (QED) are included as perturbations in extensive relativistic configuration interaction (RCI) calculations. We signal that, our calculations for SnXLIX are made for the first time and they provide to date the most accurate and complete atomic data related to this ion.