Breit Operator Research Articles

Scalar Breit interaction for molecular calculations.

Variational treatment of the Dirac-Coulomb-Gaunt or Dirac-Coulomb-Breit two-electron interaction at the Dirac-Hartree-Fock level is the starting point of high-accuracy four-component calculations of atomic and molecular systems. In this work, we introduce, for the first time, the scalar Hamiltonians derived from the Dirac-Coulomb-Gaunt and Dirac-Coulomb-Breit operators based on spin separation in the Pauli quaternion basis. While the widely used spin-free Dirac-Coulomb Hamiltonian includes only the direct Coulomb and exchange terms that resemble nonrelativistic two-electron interactions, the scalar Gaunt operator adds a scalar spin-spin term. The spin separation of the gauge operator gives rise to an additional scalar orbit-orbit interaction in the scalar Breit Hamiltonian. Benchmark calculations of Aun (n = 2-8) show that the scalar Dirac-Coulomb-Breit Hamiltonian can capture 99.99% of the total energy with only 10% of the computational cost when real-valued arithmetic is used, compared to the full Dirac-Coulomb-Breit Hamiltonian. The scalar relativistic formulation developed in this work lays the theoretical foundation for the development of high-accuracy, low-cost correlated variational relativistic many-body theory.

Correlated Dirac-Coulomb-Breit multiconfigurational self-consistent-field methods.

The fully correlated frequency-independent Dirac-Coulomb-Breit Hamiltonian provides the most accurate description of electron-electron interaction before going to a genuine relativistic quantum electrodynamics theory of many-electron systems. In this work, we introduce a correlated Dirac-Coulomb-Breit multiconfigurational self-consistent-field method within the frameworks of complete active space and density matrix renormalization group. In this approach, the Dirac-Coulomb-Breit Hamiltonian is included variationally in both the mean-field and correlated electron treatment. We also analyze the importance of the Breit operator in electron correlation and the rotation between the positive- and negative-orbital space in the no-virtual-pair approximation. Atomic fine-structure splittings and lanthanide contraction in diatomic fluorides are used as benchmark studies to understand the contribution from the Breit correlation.

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.

Atomic data for doubly-excited states [formula omitted] of He-like ions and [formula omitted] of Li-like ions with [formula omitted] and [formula omitted

The wavelengths and radiative transition probabilities for transitions 2lnl′→1snl″, 2lnl′→1s2l″, 1s2lnl′→1s2nl″, 1s2lnl′→1s22l″, and the autoionization decay probabilities for doubly-excited states 2lnl′, 1s2lnl′ were calculated in He-like and Li-like ions with atomic numbers Z=6–36 for the principal quantum number n=2,3 and the orbital quantum one l=0–2. The calculations were carried out by means of the MZ code based on the Z-expansion method. Relativistic corrections were taken into account within the framework of the Breit operator. The main difference with previous calculations by MZ code consists in accounting for the first order corrections in powers of 1/Z, corresponding to the screening effects, in calculations of autoionization rates. New data for comparatively large rates are of about 20%–50% less as compared to previous ones and are in an agreement within 10% with the results of calculations made by the methods based on multi-configuration wave functions with non-relativistic and relativistic orbitals. Some refinements and corrections concerning the energies and radiative transition probabilities were also introduced in the MZ code. In this paper the main formulas used in a modified MZ code and the data needed for the description of dielectronic satellites with n=2,3 are given.

Precision spectroscopy of the hydrogen molecular ionD2+

Expectation values of the Breit operators are calculated for the D-2(+) molecular ion with the vibrational number (nu = 0-4) and angular momentum (L = 0-4). Hyperfine splittings of energy levels of the states (nu = 0-4, L = 3,4) are also determined in the framework of the spin-dependent Breit-Pauli Hamiltonian, which can be considered as an extension of our previous work [ Phys. Rev. A 88, 032519 (2013)]. These theoretical calculations may be used to improve spectroscopic precision of D-2(+) and determine the dissociation or the ionization energy of D-2.

Convergence of the Breit interaction in self-consistent and configuration-interaction approaches

Much discussion in relativistic atomic physics and quantum optics has related to the interaction of gauge and perturbation of the Hamiltonian or Dirac operator. It has been commented that Lorentz and gauge independence requires different forms of the perturbation operator in shifting from one gauge to another. Equally, it has been commented that gauge convergence is not possible without different operator forms in different bases and without the operator being embedded within the self-consistent kernel. We explore the logic and self-consistency of these arguments, applied to the well-known Breit operator in an area of continuing discussion. We find that convergence is now possible to a remarkable degree including a Breit interaction operator in a form consistent with the gauge for length and velocity relativistic forms of the multipole operator, implemented at the configuration-interaction level. Excellent convergence is obtained for Breit interaction energies, interaction mixing coefficients, interaction transition probabilities and eigenenergies and transition probabilities in complex open shells (transition metal K $\ensuremath{\alpha}$ transitions and shake satellites), and forbidden transitions.

Precision spectroscopy of the hydrogen molecular ion HD+

Expectation values of the Breit operators and the $Q$ terms are calculated for HD${}^{+}$ with the vibrational number $v=0$--4 and the total angular momentum $L=0$--4. Relativistic and radiative corrections to some rovibrational transition frequencies are determined. Numerical uncertainty in the ${R}_{\ensuremath{\infty}}{\ensuremath{\alpha}}^{2}$ order correction is reduced to sub-kHz or smaller. Our work provides an independent verification of Korobov's calculations [V. I. Korobov, Phys. Rev. A 74, 052506 (2006); V. I. Korobov, Phys. Rev. A 77, 022509 (2008)].

Relativistic corrections to He I transition ratesThis paper was presented at the International Conference on Precision Physics of Simple Atomic Systems, held at École de Physique, les Houches, France, 30 May – 4 June, 2010.

The relativistic corrections to the theoretical oscillator strengths of light elements such as helium are typically less than 0.1% and usually are ignored. However, they can be important for comparisons with the most accurate experiments, and they rapidly increase in magnitude with increasing nuclear charge. We have begun with the spin-forbidden electric-dipole transitions of neutral helium, using calculations consisting of (1) extremely accurate wave functions without relativistic corrections for both infinite and finite nuclear mass, (2) spin-changing matrix elements through the perturbations of the wave functions by the spin-orbit and spin-other-orbit Breit operators, (3) the use of pseudostates in the sums over all the intermediate states including the continuum, and (4) the inclusion as perturbers of the 1S0 and 3S1 states the pseudostates corresponding to the doubly excited npn′p 3P 0e and npn′p 1P 1e terms, respectively. As examples of these calculations, we present oscillator strengths for the transitions 1 1S0–2 3P1, 2 1S0–2 3P1, 2 3S1–2 1P1, 2 1P1–3 3D1,2, and 2 3P1,2–3 1D2.

THE GENERALIZED BREIT OPERATOR OF A LONG-RANGE TYPE IN THE FRAMEWORK OF THE SECOND-ORDER EFFECTS OF QUANTUM ELECTRODYNAMICS

The problem of interaction of two quasimolecular electrons located at an arbitrary distance from each other and near different atoms (nuclei) is solved. The interaction is considered as a second-order effect of quantum electrodynamics in the coordinate representation. It is shown that a consistent account for the natural condition of the interaction symmetry with respect to both electrons leads to an additional contribution to the relativistic interaction of the two quasimolecular electrons compared with both the standard Breit operator and the generalized Breit operator known previously. The generalized Breit–Pauli operator and the operator of electric dipole–dipole interaction of two quasimolecular electrons located at an arbitrary distance from each other are obtained. Modern methods of accounting for the relativistic and correlative effects in the problem of ion–atom interactions are discussed.

Interaction of two quasimolecular electrons via the field of virtual photons as a second-order effect of quantum electrodynamics

The problem of interaction of two quasimolecular electrons located at an arbitrary distance from each other and near different atoms (nuclei) is solved. The interaction is considered as a second-order effect of quantum electrodynamics in the coordinate representation. It is shown that consistent account of the natural condition of the interaction symmetry with respect to both electrons leads to the additional contribution to the relativistic interaction of the two quasimolecular electrons compared with both the standard Breit operator and the generalized Breit operator obtained previously [O. N. Gadomskii, Usp. Fiz. Nauk 170, 1145 (2000) [Phys. Usp. 43, 1071 (2000)]]. The generalized Breit-Pauli operator for interaction of two quasimolecular electrons located at an arbitrary distance from each other is obtained.

The generalized Breit operator of the interaction between two quasimolecular electrons in the framework of the third-order effects of quantum electrodynamics

The problem of the interaction of two quasimolecular electrons located at an arbitrary distance from each other and near different atoms (nuclei) is solved. Effects of the third order of quantum electrodynamics, which include the exchange of a virtual photon between the electrons and emission (absorption) of a real photon, are considered. The general expression for the matrix elements of the operator of the effective interaction energy of the two quasimolecular electrons with the external radiation field, which allows us to calculate the probabilities of inelastic processes with rearrangement in slow collisions of multiply-charged ions with relativistic atoms, is obtained. Carrying out consistently the procedure of symmetrization of the retardation factor with respect to both the electrons results in the appearance of additional terms in the relativistic operator of the interaction of two quasimolecular electrons in comparison with both the standard and generalized Breit operators known previously.

The quantum electrodynamic problem of two electrons

We solve the problem of the interaction of two quasimolecular electrons at an arbitrary distance from each other, i.e., near different atoms (nuclei). We regard the interaction as a second-order effect of the quantum electrodynamic perturbation theory in the coordinate representation. Taking the natural condition of the symmetry of the retardation factor, the electron spins, and the effects of retardation of the relativistic interaction of the two quasimolecular electrons located near different nuclei into account consistently, we obtain additional terms in the interelectron interaction operator compared with the standard Breit operator.

Atoms and molecules in relativistic quantum mechanics

Relativistic N-electron Hamiltonian can be written as a sum of the one electron Dirac Hamiltonians for an electron moving in the external field of nucleus, the Coulomb repulsion potential energy between electrons and the Breit operator representing the magnetic and retardation corrections to this interaction. Approximating many-electron wave function by a sum of anti- symmetrised products of orthonormal single particle spinors and using it as a trial function in a variational method leads to the Dirac-Fock equations for N-electron system. However the presence of a continuum of negative energy states below the bound states in the spectrum of Dirac Hamiltonian is a source of a several difficulties in atomic and molecular structure calculations: the Brown-Ravenhall disease, variational collapse and variational prolapse. Taking the boundary conditions properly into account, imposing the kinetic balance condition for small and large component basis sets, using correct strategy in developing basis sets and employing the finite nuclear size model results in the workable finite-difference and the basis-set methods of calculation for relativistic atomic and molecular structure.

Relativistic Dirac-Fock exchange and Breit interaction energy functionals based on the local-density approximation and the self-consistent multiplicative constant method

Local approximations, formulated in the context of a relativistic extension of the self-consistent $\ensuremath{\alpha}$ method, are advanced for the case of the relativistic exact Coulomb exchange functional---defined as the Dirac-Fock exchange---and for the Breit expression for the transverse photon exchange, given in terms of the expectation value of the Breit operator. The resulting potentials are local (multiplicative) operators, in both cases. Using these new functionals, fully relativistic calculations have been performed for selected atoms with $Z$ between 10 and 94, within the framework of relativistic density functional theory (RDFT). The total and exchange energies and ionization potentials obtained from these functionals are quite close to those of the relativistic optimized potential method (ROPM). The computational effort involved in the application of this new method is, however, significantly lower.

Four-component relativistic Kohn-Sham theory.

A four-component relativistic implementation of Kohn-Sham theory for molecular systems is presented. The implementation is based on a nonredundant exponential parametrization of the Kohn-Sham energy, well suited to studies of molecular static and dynamic properties as well as of total electronic energies. Calculations are presented of the bond lengths and the harmonic and anharmonic vibrational frequencies of Au(2), Hg(2+)(2), HgAu(+), HgPt, and AuH. All calculations are based on the full four-component Dirac-Coulomb Hamiltonian, employing nonrelativistic local, gradient-corrected, and hybrid density functionals. The relevance of the Coulomb and Breit operators for the construction of relativistic functionals is discussed; it is argued that, at the relativistic level of density-functional theory and in the absence of a vector potential, the neglect of current functionals follows from the neglect of the Breit operator.

Investigation of the relativistic equivalent Hamiltonian in the LS coupling scheme

Armstrong’s method of relativistic description of atoms in LS coupling using the equivalent operator is developed. Its form in terms of standard unit operators acting within a space of one shell is obtained. The interpretation of separate terms of equivalent Breit operator is investigated in a general case of nonequivalent electrons. The relativistic Dirac–Fock equations for the level and term in LS coupling are derived. The equivalent operator is applied to obtain the averages of relativistic operators too.

Inner-shell Excitation Energies and Autoionization Rates for C-, N-, O-, and F-like Ions with Z = 6–54

Absolute energies of the autoionizing states 1s2s22pn [LSJ] with n = 3–6, 1s2s2pn [LSJ] with n = 4–6 and 1s2pn [LSJ] with n = 5–6, and non-autoionizing states 1s22s2pn [LSJ] with n = 2–6, 1s22s2pn [LSJ] with n = 3–6, and 1s22pn [LSJ] with n = 4–6 were calculated for ions with a nuclear charge Z = 6–54 by using the 1/Z perturbation theory method. These states were considered as the states with one or few holes in the Ne-like core state 1s22s22p6. Relativistic corrections were taken into account in the frame of the Breit operators. The autoionization rates for the above-mentioned autoionizing states and ions were computed as well and are shown in grafic forms as Z-dependences. These new atomic data represent an important material, which can be used in different areas of atomic and plasma physics involving the study of multiply-charged ions.

The problem of two electrons in an external field and the method of integral equations in optics

This paper solves the problem of the interaction, via the field of virtual photon field with the emission or absorption of a real photon, of two atomic electrons located at arbitrary distances from one another. The interaction is interpreted as a third-order QED effect in the coordinate representation. The role of intermediate states with positive and negative frequencies is studied. A general expression is derived for the matrix elements of the operator of the effective electron-electron interaction energy for different types of quantum transitions. The expression makes it possible to calculate the probabilities of the corresponding transitions and to examine various patterns of induction of polarizing fields by one atom at the point occupied by the other atom. The exchange of virtual photons between the atoms located at arbitrary distances from one another is shown to lead to additional terms in the operators of spin-orbit and spin-spin coupling of the atomic electrons, over and above those in the corresponding Breit operators. It is shown that there is an important difference between the induction of polarizing fields and the transfer of optical photons. In particular, it is found that when polarizing fields are induced, a situation may arise in which the disappearance (production) of a photon takes place at the point occupied by one atom, while absorption (emission) of the same photon occurs at the place occupied by the other atom. A block diagram of an experimental device that could be used to study this property of polarizing fields is presented. Finally, a method of deriving integral field equations is proposed. The method is based on allowing for polarizing fields, and its effectiveness is demonstrated by the example of electric dipole and spin transitions in the spectrum of interacting atomic electrons.

RELATIVISTIC MANY-BODY CALCULATIONS OF ENERGIES OFn= 3 STATES FOR THE BORON ISOELECTRONIC SEQUENCE,Z= 6–30

Energies of the 2s2(1S)3l, 2s2p(1P)3l, 2s2p(3P)3l, 2p2(3P)3l, 2p2(1D)3l, and 2p2(1S)3l states, withl = s,p, andd, for B-like ions with nuclear chargesZin the range 6–30 are determined to second order in relativistic many-body perturbation theory starting from a frozen-core Dirac–Fock basis. These perturbation theory calculations include the first- and second-order diagram contributions from the Coulomb and Breit operators for diagonal and nondiagonal matrix elements. Corrections for the frequency dependence of the Breit interaction are taken into account in lowest order. The Lamb-shift correction to the energies is also included in lowest order. Contributions of the three-electron interaction are compared with contributions of the one- and two-electron interactions. Comparisons are made with other theoretical calculations and with experimental data. We obtain excellent agreement for term splitting, even for low-Zions.

Four-component Dirac-Hartree-Fock equations for solids; generalization of the relativistic Hartree-Fock equations

Relativistic Hartree-Fock equations will be presented for 1D-, 2D- and 3D-infinite systems. 4-Component spinors with Gaussian basis sets will be used and the so-called kinetic balance beween the small and large components of the spinors is introduced. The resulting somewhat complicated generalized matrix eigenvalue equation for solids is described. It is also shown for 1D- and 2D-systems how the MP2 and MP2- r 12 methods could be applied in their relativistic form. With the help of them, on the one hand, the total energy per unit cell (including correlation effects) can be computed. On the other hand, applying the inverse Dyson equation, the relativistic band structure can also be corrected for correlation. Furthermore, it is argued that to obtain more reliable one-electron functions and energies for the correlation calculations, one has to include into the relativistic Hartree-Fock equations the frequency-dependent Breit operator, the first-order Lamb shift terms (the difference of the self energy and mass correction terms) and the vacuum polarization term in the Uehling potential approximation (all the other quantum electrodynamical effects are order(s) of magnitude smaller). Including all these terms into the expression of the relativistic total energy, the corresponding generalized Dirac-Hartree-Fock equations obtained after performing the variational calculations are presented. Finally a scheme is proposed in which most of the electrons are treated in the standard way (Dirac-Hartree-Fock equations with only Coulomb interactions and calculation of all the other terms with the aid of first-order perturbation theory), while for the core electrons of large Z atoms or ions, the generalized relativistic HF equations are used. Therefore, one applies the solutions of the generalized relativistic HF equations for the construction of the relativistic Slater determinant in the case of core electrons, while for the rest of the electrons the one-electron functions are obtained from the standard Dirac-Hartree-Fock equations.