Two-electron Contributions Research Articles

Convergence-acceleration approach to partial-wave expansion of two-electron self-energy contributions to the Lamb shift

Convergence-acceleration approach to partial-wave expansion of two-electron self-energy contributions to the Lamb shift

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.

Efficient evaluation of the Breit operator in the Pauli spinor basis.

The frequency-independent Coulomb-Breit operator gives rise to the most accurate treatment of two-electron interaction in the non-quantum-electrodynamics regime. The Breit interaction in the Coulomb gauge consists of magnetic and gauge contributions. The high computational cost of the gauge term limits the application of the Breit interaction in relativistic molecular calculations. In this work, we apply the Pauli component integral-density matrix contraction scheme for gauge interaction with a maximum spin- and component separation scheme. We also present two different computational algorithms for evaluating gauge integrals. One is the generalized Obara-Saika algorithm, where the Laplace transformation is used to transform the gauge operator into Gaussian functions and the Obara-Saika recursion is used for reducing the angular momentum. The other algorithm is the second derivative of Coulomb interaction evaluated with Rys-quadrature. This work improves the efficiency of performing Dirac-Hartree-Fock with the variational treatment of Breit interaction for molecular systems. We use this formalism to examine relativistic trends in the Periodic Table and analyze the relativistic two-electron interaction contributions in heavy-element complexes.

Linear-Scaling Implementation of Multilevel Hartree-Fock Theory.

We introduce a new algorithm for the construction of the two-electron contributions to the Fock matrix in multilevel Hartree–Fock (MLHF) theory. In MLHF, the density of an active molecular region is optimized, while the density of an inactive region is fixed. The MLHF equations are solved in a reduced molecular orbital (MO) basis localized to the active region. The locality of the MOs can be exploited to reduce the computational cost of the Fock matrix: the cost related to the inactive density becomes linear scaling, while the iterative cost related to the active density is independent of the system size. We demonstrate the performance of this new algorithm on a variety of systems, including amino acid chains, water clusters, and solvated systems.

Performance of an atomic mean-field spin–orbit approach within exact two-component theory for perturbative treatment of spin–orbit coupling

ABSTRACT Development of an atomic mean-field (AMF) spin–orbit approach within the spin-free exact two-component theory in its one-electron variant (SFX2C-1e) together with pilot applications are reported. The effective one-electron spin–orbit integral matrix in the four-component representation is assembled as a direct sum of one-centre spin–orbit integral matrices with the mean-field two-electron contributions evaluated using atomic SFX2C-1e Hartree–Fock density matrices. It is then transformed into two-component representation using analytic SFX2C-1e energy derivative formulation. The resulting two-component spin–orbit integral matrix is by design suitable for use in perturbative calculations of spin–orbit coupling, treating SFX2C-1e wavefunctions as unperturbed states. The accuracy of the present AMF approach has been demonstrated using benchmark calculations of spin–orbit splittings for representative diatomic radicals at the equation-of-motion coupled-cluster singles and doubles level. To demonstrate the applicability and accuracy of the present perturbative spin–orbit scheme in calculations of challenging heavy-element containing systems, a thorough computational investigation of six low-lying electronic states of ThO + is reported.

Spin-Orbit Matrix Elements for a Combined Spin-Flip and IP/EA approach.

We present a practical approach for computing the Breit-Pauli spin-orbit matrix elements of multiconfigurational systems with both spin and spatial degeneracies based on our recently developed RAS-nSF-IP/EA method (Houck, S. E.; et al. J. Chem. Theory Comput. 2019, 15, 2278). The spin-orbit matrix elements over all the multiplet components are computed using a single one-particle reduced density matrix as a result of the Wigner-Eckart theorem. A mean field spin-orbit approximation was used to account for the two-electron contributions. Basis set dependence as well as the effect of including additional excitations is presented. The effect of correlating the core and semicore orbitals is also examined. Surprisingly accurate results are obtained for spin-orbit coupling constants, despite the fact that the efficient wave function approximations we explore neglect the bulk of dynamical correlation.

Relativistic Calculation of the Nuclear Recoil Effect on the g Factor of the 2P3/2 State in Highly Charged B-like Ions

The nuclear recoil effect on the $^2 P_{3/2}$-state $g$ factor of B-like ions is calculated to first order in the electron-to-nucleus mass ratio $m/M$ in the range $Z=18$--$92$. The calculations are performed by means of the $1/Z$ perturbation theory. Within the independent-electron approximation, the one- and two-electron recoil contributions are evaluated to all orders in the parameter $\alpha Z$ by employing a fully relativistic approach. The interelectronic-interaction correction of first order in $1/Z$ is treated within the Breit approximation. Higher orders in $1/Z$ are partially taken into account by incorporating the screening potential into the zeroth-order Hamiltonian. The most accurate to date theoretical predictions for the nuclear recoil contribution to the bound-electron $g$ factor are obtained.

Релятивистский расчёт эффекта отдачи ядра для g-фактора состояния ^2P-=SUB=-3/2-=/SUB=- многозарядных бороподобных ионов

The nuclear recoil effect on the $^2 P_{3/2}$-state $g$ factor of B-like ions is calculated to first order in the electron-to-nucleus mass ratio $m/M$ in the range $Z=18$--$92$. The calculations are performed by means of the $1/Z$ perturbation theory. Within the independent-electron approximation, the one- and two-electron recoil contributions are evaluated to all orders in the parameter $\alpha Z$ by employing a fully relativistic approach. The interelectronic-interaction correction of first order in $1/Z$ is treated within the Breit approximation. Higher orders in $1/Z$ are partially taken into account by incorporating the screening potential into the zeroth-order Hamiltonian. The most accurate to date theoretical predictions for the nuclear recoil contribution to the bound-electron $g$ factor are obtained.

General framework for calculating spin-orbit couplings using spinless one-particle density matrices: Theory and application to the equation-of-motion coupled-cluster wave functions.

Standard implementations of nonrelativistic excited-state calculations compute only one component of spin multiplets (i.e., Ms = 0 triplets); however, matrix elements for all components are necessary for deriving spin-dependent experimental observables. Wigner-Eckart's theorem allows one to circumvent explicit calculations of all multiplet components. We generate all other spin-orbit matrix elements by applying Wigner-Eckart's theorem to a reduced one-particle transition density matrix computed for a single multiplet component. In addition to computational efficiency, this approach also resolves the phase issue arising within Born-Oppenheimer's separation of nuclear and electronic degrees of freedom. A general formalism and its application to the calculation of spin-orbit couplings using equation-of-motion coupled-cluster wave functions are presented. The two-electron contributions are included via the mean-field spin-orbit treatment. Intrinsic issues of constructing spin-orbit mean-field operators for open-shell references are discussed, and a resolution is proposed. The method is benchmarked by using several radicals and diradicals. The merits of the approach are illustrated by a calculation of the barrier for spin inversion in a high-spin tris(pyrrolylmethyl)amine Fe(II) complex.

Density Functional Calculations of Electron Paramagnetic Resonance g- and Hyperfine-Coupling Tensors Using the Exact Two-Component (X2C) Transformation and Efficient Approximations to the Two-Electron Spin-Orbit Terms.

A two-component quasirelativistic density functional theory implementation of the computation of hyperfine and g-tensors at exact two-component (X2C) and Douglas-Kroll-Hess method (DKH) levels in the Turbomole code is reported and tested for a series of smaller 3d1, 4d1, and 5d1 complexes, as well as for some larger 5d7 Ir and Pt systems in comparison with earlier four-component matrix-Dirac-Kohn-Sham results. A main emphasis is placed on efficient approximations to the two-electron spin-orbit contributions, comparing an existing implementation of two variants of Boettger's "scaled nuclear spin-orbit" (SNSO) approximation in the code with a newly implemented atomic mean-field spin-orbit (AMFSO) approximation. The different variants perform overall comparably well with the four-component data. The AMFSO approximation has the added advantage of being able to include the spin-other-orbit contributions arising from the Gaunt term of relativistic electron-electron interactions. These are of comparably larger importance for the 3d complexes than for their heavier homologues. The excellent agreement between X2C and four-component electron paramagnetic resonance parameter results provides the opportunity to treat large systems efficiently and accurately with the computationally more expedient two-component quasirelativistic methodology.

A one-electron variant of direct perturbation theory for the treatment of scalar-relativistic effects

ABSTRACTThe different importance of scalar-relativistic two-electron contributions in second-order direct perturbation theory (DPT2) and the spin-free one-electron variant of exact two-component theory (SFX2C-1e) is analysed. The apparent discrepancy is traced back to the fact that SFX2C-1e is not ignoring the total DPT2 two-electron contribution rather just a so-called commutator term which originates from a rewrite of the small-component density (matrix) in terms of the related, though different kinetic-energy density (matrix). This commutator term is shown to be significantly smaller (10% and less) and to have, unlike the total DPT2 two-electron contribution, a negative sign. Based on our findings, we propose a one-electron variant of direct perturbation theory, referred to as DPT-1e, and report on its implementation for the computation of energies and first-order properties at the second-order level, i.e. DPT2-1e. Numerical results are presented for the hydrogen halide series HX, X=F, Cl, Br, I, and At, as well CuF and CuCl in order to investigate its performance in comparison to DPT2 and SFX2C-1e.

Nuclear recoil effect on the g factor of highly charged Li-like ions

The nuclear recoil effect on the g factor of highly charged Li-like ions is evaluated in the range Z=10-92. The calculations are performed using the 1/Z perturbation theory. The one-electron recoil contribution is evaluated within the fully relativistic approach with the wave functions which account for the screening effects approximately. The two-electron recoil contributions of the first and higher orders in 1/Z are calculated within the Breit approximation using a four-component approach.

Hybrid CPU/GPU Integral Engine for Strong-Scaling Ab Initio Methods.

We present a parallel integral algorithm for two-electron contributions occurring in Hartree-Fock and hybrid density functional theory that allows for a strong scaling parallelization on inhomogeneous compute clusters. With a particular focus on graphic processing units, we show that our approach allows an efficient use of CPUs and graphics processing units (GPUs) simultaneously, although the different architectures demand conflictive strategies in order to ensure efficient program execution. Furthermore, we present a general strategy to use large basis sets like quadruple-ζ split valence on GPUs and investigate the balance between CPUs and GPUs depending on l-quantum numbers of the corresponding basis functions. Finally, we present first illustrative calculations using a hybrid CPU/GPU environment and demonstrate the strong-scaling performance of our parallelization strategy also for pure CPU-based calculations.

A Hartree–Fock study of the confined helium atom: Local and global basis set approaches

Two different basis set methods are used to calculate atomic energy within Hartree–Fock theory. The first is a local basis set approach using high-order real-space finite elements and the second is a global basis set approach using modified Slater-type orbitals. These two approaches are applied to the confined helium atom and are compared by calculating one- and two-electron contributions to the total energy. As a measure of the quality of the electron density, the cusp condition is analyzed.

Spin-orbit couplings within the equation-of-motion coupled-cluster framework: Theory, implementation, and benchmark calculations.

We present a formalism and an implementation for calculating spin-orbit couplings (SOCs) within the EOM-CCSD (equation-of-motion coupled-cluster with single and double substitutions) approach. The following variants of EOM-CCSD are considered: EOM-CCSD for excitation energies (EOM-EE-CCSD), EOM-CCSD with spin-flip (EOM-SF-CCSD), EOM-CCSD for ionization potentials (EOM-IP-CCSD) and electron attachment (EOM-EA-CCSD). We employ a perturbative approach in which the SOCs are computed as matrix elements of the respective part of the Breit-Pauli Hamiltonian using zeroth-order non-relativistic wave functions. We follow the expectation-value approach rather than the response-theory formulation for property calculations. Both the full two-electron treatment and the mean-field approximation (a partial account of the two-electron contributions) have been implemented and benchmarked using several small molecules containing elements up to the fourth row of the periodic table. The benchmark results show the excellent performance of the perturbative treatment and the mean-field approximation. When used with an appropriate basis set, the errors with respect to experiment are below 5% for the considered examples. The findings regarding basis-set requirements are in agreement with previous studies. The impact of different correlation treatment in zeroth-order wave functions is analyzed. Overall, the EOM-IP-CCSD, EOM-EA-CCSD, EOM-EE-CCSD, and EOM-SF-CCSD wave functions yield SOCs that agree well with each other (and with the experimental values when available). Using an EOM-CCSD approach that provides a more balanced description of the target states yields more accurate results.

Shape-Dependent Electronic Excitations in Metallic Chains

The electronic excitations of silver chains with different geometries (linear, circle, arc and zigzag chains) have been investigated at the time-dependent density functional theory level, by solving the equation of motion of the reduced single-electron density matrix in the real-time domain. A scaling parameter 0 ≤ λ ≤ 1 has been introduced to adjust the two-electron contributions during propagation in the time domain in a way that allows us to distinguish different electronic excitations — plasmon and single-particle excitations. The dipole responses, in particular the plasmon resonances of these metallic chains to an external δ-pulse, show a strong dependence on their geometric structures. In most cases, the dipole responses of these chains possess great tunability when altering their geometric parameters — the radius of the circle and arc chains, and the bond angle of the zigzag chains. Some excitations in these chains also show a wide tunable excitation energy range, more than 1 eV, making it possible to fine-tune the excitations of the metallic chains at an atomic scale.

Testing Three-Body Quantum Electrodynamics with TrappedTi20+Ions: Evidence for aZ-dependent Divergence Between Experiment and Calculation

We report a new test of quantum electrodynamics (QED) for the w (1s2p(1)P(1)→1s(2)(1)S(0)) x-ray resonance line transition energy in heliumlike titanium. This measurement is one of few sensitive to two-electron QED contributions. Systematic errors such as Doppler shifts are minimized in our experiment by trapping and stripping Ti atoms in an electron beam ion trap and by applying absolute wavelength standards to calibrate the dispersion function of a curved-crystal spectrometer. We also report a more general systematic discrepancy between QED theory and experiment for the w transition energy in heliumlike ions for Z>20. When all of the data available in the literature for Z=16-92 are taken into account, the divergence is seen to grow as approximately Z(3) with a statistical significance on the coefficient that rises to the level of 5 standard deviations. Our result for titanium alone, 4749.85(7) eV for the w line, deviates from the most recent ab initio prediction by 3 times our experimental uncertainty and by more than 10 times the currently estimated uncertainty in the theoretical prediction.

Correlation and relativistic effects within the trivalent Lu-like sequence

Energies of the three-electron-above the closed-shell lutetium-like ions ([Xe]4f145d3, [Xe]4f145d26s, [Xe]4f145d26p and [Xe]4f145d6s6p states) with Z = 74–100 are determined using second-order relativistic many-body perturbation theory. Our calculations start from a Er-like V(N − 3) Dirac–Fock potential ([Xe]4f14, where [Xe] = 1s22s22p63s23p63d104s24p64d105s25p6). Second-order Coulomb and Breit–Coulomb interactions are included. Correction for the frequency dependence of the Breit interaction is taken into account in the lowest order. The Lamb shift correction to energies is also included in the lowest order. The three-electron contributions to the energy are compared with the one- and two-electron contributions. They are found to contribute about 10–20% of the total second-order energy. The ratios of the third- and second-order corrections to the one-electron contributions are found to be about 5–10%. A detailed discussion of the various contributions to the energy levels is given for Lu-like tungsten (Z = 74). Comparisons are made with available experimental data. Trends of excitation energies including splitting of the doublet and quartet terms as functions of nuclear charge Z = 73–100 are illustrated graphically for some states. These are the first ab initio calculations of excitation energies in Lu-like ions. The results for the W3 + ion are particularly important for fusion application.

Calculation of the electronic structure and spectra for bacteriochlorophyll analogs

Geometric structures and excited electronic states for free bases of bacteriochlorin (H2BC) and tetraazabacteriochlorin (H2TABC) as well as for their magnesium complexes (MgBC and MgTABC), analogs of bacteriopheophytin a (H2BPhea) and bacteriochlorophyll a (MgBPhea), have been calculated by a DFT method and by an INDO/Sm method (the INDO/S method with parameterization modified by the authors), respectively. The factors responsible for the observed bathochromic shift of the long-wavelength Q x (0–0) band of MgBPhea relative to H2BPhea, $ {{\updelta }}{E_{{Q_x}}} \cong - 300\;{\text{c}}{{\text{m}}^{ - 1}} $ , have been clarified. Contributions of one- and two-electron interactions to the resulting shift of the Q x (0–0) band have been analyzed in detail for the H2BC/MgBC, H2TABC/MgTABC, and porphine (H2P)/Mg porphine (MgP) pairs. It is shown that the bathochromic shift under consideration for the tetrahydro derivatives is caused by a decrease of the orbital energy gap e1–e−1 between the lowest unoccupied and highest occupied molecular orbitals. The variation of δ(e1–e−1) is large and amounts to –1660 and –920 cm–1 for the H2BC/MgBC and H2TABC/MgTABC pairs, respectively. The two-electron contributions, both into the energy of electronic configurations and due to the superposition of the configurations, produce a compensating hypsochromic effect such that the shifts $ {{\updelta }}{E_{{Q_x}}} $ are –260 and –150 cm–1 for the H2BC/MgBC and H2TABC/MgTABC pairs, respectively. It is also shown that the calculated electronic spectra for the considered molecules agree quantitatively with the experimental absorption spectra.

Calculation of the 4,5-dihydro-1,3,2-dithiazolyl radical g tensor components by the coupled-perturbed Kohn-Sham hybrid density functional and configuration interaction methods: a comparative study

The g tensor components of the 4,5-dihydro-1,3,2-dithiazolyl (H2DTA•) radical, which is a basic building block for molecular magnets and spintronic devices, is calculated by the coupled-perturbed Kohn-Sham (CPKS) hybrid density functional (HDF) and multireference configuration interaction-sum over states (MRCI-SOS) techniques. In both methods, the diagonalized g tensor principal axes are found to be aligned with the radical's inertial axes. The tensor components are in very good agreement with those determined experimentally by electron paramagnetic resonance (EPR) spectroscopy. The MRCI technique produced g tensor components that are more accurate than those obtained by the CPKS-HDF method. Nonetheless, to get reasonable MRCI results, one must include the in-plane and out-of-plane interactions in an unbiased way. The minimum reference space that satisfies these conditions is generated from a complete active space of nine electrons in six orbitals [CAS(9,6)] and contains a(1), a(2), b(1) and b(2) type orbitals. In addition, the number of roots in the MRCI-SOS g tensor expansion should include all excited states that range from 0 to 56,000 cm(-1). The most accurate results are obtained using an MRCI-SOS/CAS(13,9) calculation. These g tensor components are within the experimental accuracy range of 1000 ppm. The one- and two-electron contributions to the g tensor components are separated and individually analyzed. The very good agreement with experiment opens the door for further accurate calculations of spin Hamiltonian tensors of larger DTA• radicals.