Two-component Formalism Research Articles

Calculation of electric field gradients with the exact two-component (X2C) quasi-relativistic method and its local approximations.

When calculating electric field gradients (EFGs), relativistic and electron correlation effects are crucial for obtaining accurate results, and the commonly used density functional methods produce unsatisfactory results, especially for heavy elements and/or strongly correlated systems. In this work, a stand-alone program is presented, which enables calculation of EFGs from the molecular orbitals supplied by an external high accuracy quantum chemical calculation and includes relativistic effects through the exact two-component (X2C) formalism and efficient local approximations to it. Application to BiN and BiP molecules shows that a high precision can be achieved in the calculation of nuclear quadrupole coupling constants of 209Bi by combining advanced ab initio methods with the X2C approach. For seventeen iron compounds, the Mössbauer nuclear quadrupole splittings (NQS) of 57Fe calculated using a double-hybrid functional method are in very good agreement with the experimental values. It is shown that, for strongly correlated molecules, the double-hybrid functionals are much more accurate than the commonly used hybrid functionals. The computer program developed in this study furnishes a useful utility for obtaining EFGs and related nuclear properties with high accuracy.

Reducing Exact Two-Component Theory for NMR Couplings to a One-Component Approach: Efficiency and Accuracy.

The self-consistent and complex spin-orbit exact two-component (X2C) formalism for NMR spin-spin coupling constants [ J. Chem. Theory Comput. 17, 2021, 3874-3994] is reduced to a scalar one-component ansatz. This way, the first-order response term can be partitioned into the Fermi-contact (FC) and spin-dipole (SD) interactions as well as the paramagnetic spin-orbit (PSO) contribution. The FC+SD terms are real and symmetric, while the PSO term is purely imaginary and antisymmetric. The relativistic one-component approach is combined with a modern density functional treatment up to local hybrid functionals including the response of the current density. Computational demands are reduced by factors of 8-24 as shown for a large tin compound consisting of 137 atoms. Limitations of the current ansatz are critically assessed for Sn, Pb, Pd, and Pt compounds, i.e. the one-component treatment is not sufficient for tin compounds featuring a few heavy halogen atoms.

Relativistic self-consistent GW : Exact two-component formalism with one-electron approximation for solids

We present a formulation of relativistic self-consistent $GW$ for solids based on the exact two-component formalism with one-electron approximation (X2C1e) and nonrelativistic Coulomb interactions. Our theory allows us to study scalar relativistic effects, spin-orbit coupling, and the interplay of relativistic effects with electron correlation without adjustable parameters. Our all-electron implementation is fully ab initio and does not require a pseudopotential constructed from atomic calculations. We examine the effect of the X2C1e approximation by comparison to the established four-component formalism and reach an excellent agreement. The simplicity of X2C1e enables the construction of higher order theories, such as embedding theories, on top of perturbative calculations.

Gravitational form factors and mechanical properties of a quark at one loop in light-front Hamiltonian QCD

We calculate the gravitational form factors (GFFs) and pressure, shear, and energy distributions for a quark state dressed with a gluon at one loop in QCD. We use the light-front Hamiltonian approach. In the light-front gauge, we use a two-component formalism to eliminate the constrained fields. The state may be thought of as a perturbative model for a relativistic spin-$1/2$ composite system having a gluonic degree of freedom. We compare the results with model calculations for a nucleon.

NMR Coupling Constants Based on the Bethe–Salpeter Equation in the GW Approximation

We present the first steps to extend the Green's function GW method and the Bethe-Salpeter equation (BSE) to molecular response properties such as nuclear magnetic resonance (NMR) indirect spin-spin coupling constants. We discuss both a nonrelativistic one-component and a quasi-relativistic two-component formalism. The latter describes scalar-relativistic and spin-orbit effects and allows us to study heavy-element systems with reasonable accuracy. Efficiency is maintained by the application of the resolution of the identity approximation throughout. The performance is demonstrated using conventional central processing units (CPUs) and modern graphics processing units (GPUs) for molecules involving several thousand basis functions. Our results show that a large amount of Hartree-Fock exchange is vital to provide a sufficient Kohn-Sham starting point to compute the GW quasi-particle energies. As the GW-BSE approach is generally less accurate for triplet excitations or related properties such as the Fermi-contact interaction, the admixture of the Kohn-Sham correlation kernel through the contracted BSE (cBSE) method improves the results for NMR coupling constants. This leads to remarkable results when combined with the eigenvalue-only self-consistent variant (evGW) and Becke's half and half functional (BH&HLYP) or the CAM-QTP family. The developed methodology is used to calculate the Karplus curve of tin molecules, illustrating its applicability to extended chemically relevant molecules. Here, the GW-cBSE method improves upon the chosen BH&HLYP Kohn-Sham starting points.

Two-Component Spinorial Formalism Using Quaternions for Six-Dimensional Spacetimes

In this article we construct and discuss several aspects of the two-component spinorial formalism for six-dimensional spacetimes, in which chiral spinors are represented by objects with two quaternionic components and the spin group is identified with $SL(2;\mathbb{H})$, which is a double covering for the Lorentz group in six dimensions. We present the fundamental representations of this group and show how vectors, bivectors, and 3-vectors are represented in such spinorial formalism. We also complexify the spacetime, so that other signatures can be tackled. We argue that, in general, objects built from the tensor products of the fundamental representations of $SL(2;\mathbb{H})$ do not carry a representation of the group, due to the non-commutativity of the quaternions. The Lie algebra of the spin group is obtained and its connection with the Lie algebra of $SO(5,1)$ is presented, providing a physical interpretation for the elements of $SL(2;\mathbb{H})$. Finally, we present a bridge between this quaternionic spinorial formalism for six-dimensional spacetimes and the four-component spinorial formalism over the complex field that comes from the fact that the spin group in six-dimensional Euclidean spaces is given by $SU(4)$.

Relativistic two-component projection-based quantum embedding for open-shell systems.

In this work, we present a relativistic quantum embedding formalism capable of variationally treating relativistic effects, including scalar-relativity and spin-orbit coupling. We extend density functional theory (DFT)-in-DFT projection-based quantum embedding to a relativistic two-component formalism, where the full spin magnetization vector form is retained throughout the embedding treatment. To benchmark various relativistic embedding schemes, spin-orbit splitting of the nominally t2g valence manifold of W(CO)6, exchange coupling of [(H3N)4Cr(OH)2Cr(NH3)4]4+, and the dissociation potential curve of WF6 are investigated. The relativistic embedding formalism introduced in this work is well suited for efficient modeling of open-shell systems containing late transition metal, lanthanide, and actinide molecular complexes.

Relativistic Effects in Magnetic Circular Dichroism: Restricted Magnetic Balance and Temperature Dependence

Magnetic circular dichroism of transition metal complexes and open-shell systems are challenging to simulate and analyze, mainly due to the interplay of spin-orbit couplings and finite-magnetic-field included Zeeman effects with the complex selection rules dictated by the circularly polarized light. In this work, we introduce an ab initio relativistic two-component formalism based on the restricted magnetic-balanced Hamiltonian for simulating MCD spectra. Both homogeneous finite magnetic field and relativistic effects are included variationally in the ground state reference. Finite-field London orbitals are used to enforce the constrained gauge-origin independence in the calculation using localized atomic orbitals. Through benchmark studies of AuCl4-, Pt(CN)42-, and Mo(CN)83-, we discuss how relativistic effects are manifested in MCD for both closed-shell and open-shell molecular complexes and how the interplay between spin-orbit coupling and magnetic field modulates the MCD selection rules. Finally, an investigation on temperature-dependent MCD is carried out and compared to experiment.

High energy scattering of Dirac particles on smooth potentials

The derivation of the Glauber type representation for the high energy scattering amplitude of particles of spin 1/2 is given within the framework of the Dirac equation in the Foldy–Wouthuysen (FW) representation and two-component formalism. The differential cross-sections on the Yukawa and Gaussian potentials are also considered and discussed.

Composite vibrational spectroscopy of the group 12 difluorides: ZnF2, CdF2, and HgF2.

The vibrational spectra of group 12 difluorides, MF2 (M = Zn, Cd, Hg), were investigated via coupled cluster singles, doubles, and perturbative triples, CCSD(T), including core correlation, with a series of correlation consistent basis sets ranging in size from triple-zeta through quintuple-zeta quality, which were then extrapolated to the complete basis set (CBS) limit using a variety of extrapolation procedures. The explicitly correlated coupled cluster method, CCSD(T)-F12b, was employed as well. Although exhibiting quite different convergence behavior, the F12b method yielded the CBS limit estimates closely matching more computationally expensive conventional CBS extrapolations. The convergence with respect to basis set size was examined for the contributions entering into composite vibrational spectroscopy, including those from higher-order correlation accounted for through the CCSDT(Q) level of theory, second-order spin-orbit coupling effects assessed within four-component and two-component relativistic formalisms, and vibrational anharmonicity evaluated via a perturbative treatment. Overall, the composite results are in excellent agreement with available experimental values, except for the CdF2 bond-stretching frequencies compared to spectral assignments proposed in a matrix isolation infrared and Raman study of cadmium difluoride vapor species [Loewenschuss et al., J. Chem. Phys. 50, 2502 (1969); Givan and Loewenschuss, J. Chem. Phys. 72, 3809 (1980)]. These assignments are called into question in the light of the composite results.

Second-order electron-correlation and self-consistent spin-orbit treatment of heavy molecules at the basis-set limit

Second-order perturbation theory using explicitly correlated wave functions has been introduced into a quasirelativistic two-component formalism. The convergence of the correlation energy is as much improved as for the nonrelativistic Hamiltonian, achieving basis-set-limit results in a moderate-size basis set. Equilibrium distances and vibrational frequencies of small molecules of the 6th period of the periodic system of the elements have been calculated, demonstrating the improved behavior of the explicitly correlated wave functions. Taking advantage of density-fitting techniques, the explicitly correlated approach is an economical and appealing alternative to conventional two-component second-order perturbation theory in a large one-particle basis.

Self-consistent treatment of spin–orbit interactions with efficient Hartree–Fock and density functional methods

Efficient self-consistent field (SCF) schemes including both scalar relativistic effects and spin-orbit (SO) interactions at Hartree-Fock (HF) and density functional (DFT) levels are presented. SO interactions require the extension of standard procedures to two-component formalisms. Efficiency is achieved by using effective core potentials (ECPs) and by employing the resolution-of-the-identity approximation for the Coulomb part (RI-J) in pure DFT calculations as well as also for the HF-exchange part (RI-JK) in the case of HF or hybrid-DFT treatments. The procedures were implemented in the program system TURBOMOLE; efficiency is demonstrated for comparably large systems, such as Pb54. Relevance of SO effects for electronic structure and stability is illustrated by treatments of small Pb and Po clusters with and without accounting for SO effects.

Explaining the derivation and functioning of the exact infinite-order two-component Dirac theory in terms of the Hückel model

The transition from the four-component Dirac theory to the exact two-component formalism is considered by using a simple algebraic model of the Dirac hamiltonian. This model is found to correspond to the four-center Huckel matrix and permits to replace the complex operator algebra by very easy matrix operations. This mapping of operators onto number matrices shows how the exact two-component relativistic hamiltonians are derived. It also explains certain conceptual aspects of the relation between four-component and two-component hamiltonians and their eigenfunctions. The generalized Huckel model of a four-center heteroatomic π-electron system can be used to analytically analyze the essential features of a variety of different exact and approximate two-component methods of relativistic quantum chemistry.

COMMENT ON THE CHOICE OF TIME IN A TWO-COMPONENT FORMULATION OF THE WHEELER–DEWITT EQUATION

The two-component formalism in quantum cosmology is revisited with a particular emphasis on the identification of time. Its relation with the appearance of imaginary eigenvalues is established. It is explicitly shown how a good choice of the global time prevents this peculiarity.

Two-component wave formalism in spherical open systems

We study wave evolution in open dielectric spheres by expanding the wave field and its conjugate momentum?the two components?in terms of relevant quasi-normal modes (QNMs), which are complete under appropriate conditions. We first establish a novel outgoing boundary condition at the surface of a sphere for waves emanating from its interior. A proper definition of inner product for two-component outgoing wavefunctions, involving only the waves inside the sphere and a surface term, can then be defined in general. The orthogonality relation of QNMs and hence a unique expansion in terms of the QNM basis are found, which can be applied to solve for the evolution of waves inside open dielectric cavities. Furthermore, a time-independent perturbation for QNMs can also be developed.

Theoretical study of the electronic spectra of square-planar platinum (II) complexes based on the two-component relativistic time-dependent density-functional theory

In the present work the electronic spectra of [PtCl(4)](2-), [PtBr(4)](2-), and [Pt(CN)(4)](2-) are studied with a recently proposed relativistic time-dependent density-functional theory (TDDFT) based on the two-component zeroth-order regular approximation and a noncollinear exchange-correlation (XC) functional. The contribution to the double group excited states in terms of singlet and triplet single group excited states is estimated through the inner product of the transition density matrix obtained from two-component and scalar relativistic TDDFT calculations to better understand the double group excited states. Spin-orbital coupling effects are found to be very important in order to simulate the electronic spectra of these complexes. The results show that the two-component TDDFT formalism can afford excitation energies with high accuracy for the transition-metal systems studied here when use is made of a proper XC potential.

A simplified relativistic time-dependent density-functional theory formalism for the calculations of excitation energies including spin-orbit coupling effect

In the present work we have proposed an approximate time-dependent density-functional theory (TDDFT) formalism to deal with the influence of spin-orbit coupling effect on the excitation energies for closed-shell systems. In this formalism scalar relativistic TDDFT calculations are first performed to determine the lowest single-group excited states and the spin-orbit coupling operator is applied to these single-group excited states to obtain the excitation energies with spin-orbit coupling effects included. The computational effort of the present method is much smaller than that of the two-component TDDFT formalism and this method can be applied to medium-size systems containing heavy elements. The compositions of the double-group excited states in terms of single-group singlet and triplet excited states are obtained automatically from the calculations. The calculated excitation energies based on the present formalism show that this formalism affords reasonable excitation energies for transitions not involving 5p and 6p orbitals. For transitions involving 5p orbitals, one can still obtain acceptable results for excitations with a small truncation error, while the formalism will fail for transitions involving 6p orbitals, especially 6p1/2 spinors.

The calculation of excitation energies based on the relativistic two-component zeroth-order regular approximation and time-dependent density-functional with full use of symmetry

In the present work, we propose a relativistic time-dependent density-functional theory (TDDFT) based on the two-component zeroth-order regular approximation and a noncollinear exchange-correlation (XC) functional. This two-component TDDFT formalism has the correct nonrelativistic limit and affords the correct threefold degeneracy of triplet excitations. The relativistic TDDFT formalism is implemented into the AMSTERDAM DENSITY FUNCTIONAL program package for closed-shell systems with full use of double-group symmetry to reduce the computational effort and facilitate the assignments. The performance of the formalism is tested on some closed-shell atoms, ions, and a few diatomic molecules containing heavy elements. The results show that the fine structure of the excited states for most atoms and ions studied here can be accurately accounted for with a proper XC potential. For the excitation energies of the molecules studied here, the present formalism shows promise and the error encountered is comparable to that of nonrelativistic TDDFT calculations on light elements.

Two-component relativistic methods for the heaviest elements.

Different generalized Douglas-Kroll transformed Hamiltonians (DKn, n=1, 2,...,5) proposed recently by Hess et al. are investigated with respect to their performance in calculations of the spin-orbit splittings. The results are compared with those obtained in the exact infinite-order two-component (IOTC) formalism which is fully equivalent to the four-component Dirac approach. This is a comprehensive investigation of the ability of approximate DKn methods to correctly predict the spin-orbit splittings. On comparing the DKn results with the IOTC (Dirac) data one finds that the calculated spin-orbit splittings are systematically improved with the increasing order of the DK approximation. However, even the highest-order approximate two-component DK5 scheme shows certain deficiencies with respect to the treatment of the spin-orbit coupling terms in very heavy systems. The meaning of the removal of the spin-dependent terms in the so-called spin-free (scalar) relativistic methods for many-electron systems is discussed and a computational investigation of the performance of the spin-free DKn and IOTC methods for many-electron Hamiltonians is carried out. It is argued that the spin-free IOTC rather than the Dirac-Coulomb results give the appropriate reference for other spin-free schemes which are based on approximate two-component Hamiltonians. This is illustrated by calculations of spin-free DKn and IOTC total energies, r(-1) expectation values, ionization potentials, and electron affinities of heavy atomic systems.

Spin–orbit coupling and the far infrared response of quantum dots in magnetic fields

The influence of the spin–orbit (SO) coupling on the far infrared response of quantum dots is investigated as a function of the intensity of the coupling, the applied magnetic field and the dot deformation. Analytical results are obtained in the limit of weak magnetic field and SO coupling intensity. Interaction effects are numerically studied within the time-dependent local-spin-density theory using a two-component spinorial formalism.