Two-component Formalism Research Articles

Relativistic effects on the nuclear magnetic shielding tensor

A new approach for calculating relativistic corrections to the nuclear magnetic shieldings is presented. Starting from a full relativistic second order perturbation theory expression a two-component formalism is constructed by transforming matrix elements using the elimination of small component scheme and separating out the contributions from the no-virtual pair and the virtual pair part of the second order corrections to the energy. In this way we avoid a strong simplification used previously in the literature. We arrive at final expressions for the relativistic corrections which are equivalent to those of Fukui et al. [J. Chem Phys. 105, 3175 (1996)] and at some other additional terms correcting both the paramagnetic and the diamagnetic part of the nuclear magnetic shielding. Results for some relativistic corrections to the shieldings of the heavy and light nuclei in HX and CH3X (X=Br,I) at both random phase and second order polarization propagator approach levels are given.

Interactions between relativistic subsystems: consequences of the two-component approximation

The operator for interactions between two relativistic subsystems is derived in the framework of the two-component ‘‘for electrons only’’ approximation to the Dirac–Coulomb (DC) theory. The consequences of the transition from the Dirac picture to the two-component scheme are analysed. It is shown that upon this transition the interaction operator undergoes the so-called change of picture transformation. This analysis shows that the two-component methods require that all terms of the DC hamiltonian are transformed in the same way. The same conclusion applies to all relativistic hamiltonians used in two-component many-electron theories. The long-range limit of the interaction operator in the two-component formalism is analysed and shown to convert into products of multipole operators in the changed picture. The presented formalism can be used in both perturbation and supermolecular (variational) approaches to the theory of intermolecular interactions. In comparison with non-relativistic theories, the only difference amounts to the evaluation of matrix elements with wave functions of the two-component formalism. Whenever the spin–orbit coupling effects are negligible, the present theory can be further reduced to the one-component (scalar, spin-averaged) form.

Transverse dipole spin response of elliptic quantum dots

The transverse dipole spin response of ellipticquantum dots is analysed as a function of an applied staticmagnetic field and dot deformation. Interaction effects arenumerically studied within the time-dependent local-spin-densitytheory using a two-component spinorial formalism.

Infinite-order two-component theory for relativistic quantum chemistry

A method for the iterative algebraic generation of the numerically accurate two-component Hamiltonian for the use in relativistic quantum chemistry is presented. The separation of the electronic and positronic states of the Dirac Hamiltonian is accomplished by the algebraic solution for the Foldy–Wouthuysen transformation. This leads to the two-component formalism whose accuracy is primarily limited by the choice of basis functions. Its performance is tested in calculations of the most sensitive 1s1/2 energy for increasing values of the nuclear charge. These calculations show that the electronic part of the Dirac eigenspectrum can be obtained from the two-component theory to arbitrarily high accuracy. Moreover, if needed, the positronic states can be separately determined in a similar way. Thus the present method can be also used for the evaluation of quantum electrodynamic corrections in the finite basis set approximation.

Two-component methods of relativistic quantum chemistry: from the Douglas–Kroll approximation to the exact two-component formalism

The two-component methods of relativistic quantum chemistry based on the Foldy–Wouthuysen (FW) transformations of the Dirac hamiltonian are reviewed. Following the strategy designed by Douglas and Kroll, the FW transformation is carried out in two steps. The first amounts to performing the exact free-particle FW transformation. At variance with other approaches, the second step is written in the form, which results in a nonlinear operator equation. This equation can be solved iteratively, leading to two-component hamiltonians of arbitrarily high accuracy in even powers of the fine structure constant. All these hamiltonians can be classified according to their completeness with respect to the leading order in the fine structure constant. On passing to the basis set representation one obtains the usual Douglas–Kroll hamiltonian and all possible higher-order approximations. By a simple modification of the operator equation which determines the block-diagonalizing transformation one can obtain numerical infinite-order solutions, i.e. one can obtain the exact numerical solution for the separation of the pure electronic part of the Dirac spectrum. This gives the exact two-component method for the use in relativistic quantum chemistry. The computational aspects of this approach are discussed as well. The transition from the Dirac formalism to any two-component approximation is accompanied by the change of all operators, including those which correspond to external perturbations and lead to properties of different orders. This so-called change of picture problem is given particular attention and its importance for certain operators is identified.

Relativistic scalar Aharonov-Bohm scattering

We discuss the scattering of relativistic spin zero particles by an infinitely long and arbitrarily thin solenoid. The exact solution of the first-quantized problem can be obtained as a mimic of the nonrelativistic case, either in the original Aharonov-Bohm way or by using the Berry magnetization scheme. The perturbative treatment is developed in the Feshbach-Villars two-component formalism for the Klein-Gordon equation and it is shown that it also requires renormalization as in the Schrodinger counterpart. The results are compared with those of the field theoretical approach, which corresponds to the two-body sector of the scalar Chern-Simons theory.

UTILITY OF GALILEAN SYMMETRY IN LIGHT-FRONT PERTURBATION THEORY: A NONTRIVIAL EXAMPLE IN QCD

Investigations have revealed a very complex structure for the coefficient functions accompanying the divergences for individual time(x+)-ordered diagrams in light-front perturbation theory. No guidelines seem to be available to look for possible mistakes in the structure of these coefficient functions emerging at the end of a long and tedious calculation, in contrast to covariant field theory. Since, in light-front field theory, the transverse boost generator is a kinematical operator which acts just like the two-dimensional Galilean boost generator in nonrelativistic dynamics, it may provide some constraint on the resulting structures. In this work we investigate the utility of Galilean symmetry beyond tree level in the context of coupling constant renormalization in light-front QCD using the two-component formalism. We show that for each x+-ordered diagram separately, the underlying transverse boost symmetry fixes relative signs of terms in the coefficient functions accompanying the diverging logarithms. We also summarize the results leading to coupling constant renormalization for the most general kinematics.

Microscopic two-component multistep direct theory for continuum nuclear reactions

We present a multistep direct reaction theory for analyzing nucleon-induced reactions to the continuum for incident energies up to 200 MeV. Two principal advances in multistep direct theory are studied. (1) A microscopical approach is given for calculating distorted-wave Born approximation (DWBA) transitions to the continuum, where transitions to all accessible 1$p$1$h$ shell model states are explicitly determined. These states, obtained from a simple noninteracting Nilsson model, are assumed to be spread according to Gaussian distributions. In this approach, therefore, state densities are not used. We also provide a link with more conventional methods that utilize particle-hole state densities, and present a more accurate technique for sampling the DWBA strength. (2) A two-component formulation of multistep direct reactions is given, where neutron and proton excitations are explicitly accounted for in the evolution of the reaction, for all orders of scattering. We show that the attractive convolution structure for multistep processes persists within a two-component formalism, and conveniently automatically generates the many reaction pathways that can occur in the Feshbach-Kerman-Koonin expansion of the multistep cross section when neutron and proton excitations are followed. This formalism is particularly important for the simultaneous analyses of neutron and proton emission spectra. The multistep direct theory is applied, along with theories for multistep compound, compound, and collective reactions, to analyze experimental emission spectra for a range of targets and energies. Particular attention is paid to a complete and comprehensive analysis of all important decay channels and reaction mechanisms. We show that the theory correctly accounts for measured neutron and proton emission angle-integrated spectra, as well as angular distributions. Additionally, we note that these microscopic and two-component developments facilitate more fundamental studies into effective nucleon-nucleon interactions in multistep calculations.

Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion

The concept of eigenfunction expansions for the wave equation is generalized to open systems, in which waves escape to the outside. These non-conservative systems are non-Hermitian in the usual sense. It is shown that the natural framework is an eigenfunction expansion within a two-component formalism that treats the wavefunction and its conjugate momentum together. Provided the system approaches spatial infinity rapidly `without tails', and possesses spatial discontinuities, the expansion in terms of the eigenfunctions (which are now quasinormal modes) is shown to be valid.

Large-N analysis of the (2+1)-dimensional Thirring model.

We analyze the (2 + 1)-dimensional vector-vector type four-Fermi interaction (Thirring) model in the framework of the $\frac{1}{N}$ expansion. By solving the Dyson-Schwinger equation in the large-$N$ limit, we show that in the two-component formalism the fermions acquire parity-violating mass dynamically in the range of the dimensionless coupling $\ensuremath{\alpha}$, $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}{\ensuremath{\alpha}}_{c}\ensuremath{\equiv}\frac{1}{16}\mathrm{exp}(\frac{\ensuremath{-}N{\ensuremath{\pi}}^{2}}{16})$. The symmetry breaking pattern is, however, in a way to conserve the overall parity of the theory such that the Chern-Simons term is not induced at any orders in $\frac{1}{N}$. ${\ensuremath{\alpha}}_{c}$ turns out to be a nonperturbative UV-fixed point in $\frac{1}{N}$. The $\ensuremath{\beta}$ function is calculated to be $\ensuremath{\beta}(\ensuremath{\alpha})=\ensuremath{-}2(\ensuremath{\alpha}\ensuremath{-}{\ensuremath{\alpha}}_{c})$ near the fixed point, and the UV-fixed point and the $\ensuremath{\beta}$ function are shown to be exact in the $\frac{1}{N}$ expansion.

On quantum electrodynamics of two-particle bound states containing spinless particles

We develop here the general treatment arising from the Bethe-Salpeter equation for a two-particle bound system in which at least one of the particles is spinless. It is shown that a natural two-component formalism can be formulated for describing the propagators of scalar particles. This leads to a formulation of the Bethe-Salpeter equation in a form very reminiscent of the fermion-fermion case. It is also shown, that using this two-component formulation for spinless particles, the perturbation theory can be systematically developed in a manner similar to that of fermions. Quantum electrodynamics for scalar particles is then developed in the two component formalism, and the problem of bound states, in which one of the constituent particles is spinless, is examined by means of the means of the Bethe-Salpeter equation. For this case, the Bethe-Salpeter equation is cast into a form which is convenient to perform a Foldy-Woutyhuysen transformation which we carry out, keeping the lowest-order relativistic corrections to the nonrelativistic equation. The results are compared with the corresponding fermion-fermion case. It is shown, as might have been expected, that the only spin-independent terms that occur for the fermion-fermion system which do not occur for bound scalar particle cases, is the zitterbewegung contribution. The relevance of the above considerations for systems that are essentially bound by electromagnetic interactions, such as kaonic hydrogen, is discussed.

Relativistic and correlation effects for element 105 (hahnium, Ha): a comparative study of M and MO (M = Nb, Ta, Ha) using energy-adjusted ab initio pseudopotentials

Nonrelativistic and quasirelativistic energy-adjusted ab initio pseudopotentials are presented for element 105 (hahnium, Ha) together with corresponding energy-optimized valence basis sets. The method of energy adjustment of pseudopotentials is extended to a two-component formalism and to multiconfiguration wave functions. The accuracy of the pseudopotential scheme is demonstrated by a comparison of atomic valence-only results to corresponding all-electron data. Atomic multiconfiguration self-consistent field and multireference configuration interaction calculations for M and M[sup +] (M = Nb, Ta, Ha) are compared with available experimental data. Corresponding molecular calculations, which included spin-orbit coupling, have been performed for the low-lying states of HaO and are compared to the results from corresponding calculations of the lighter homologs NbO and TaO. 41 refs., 1 fig., 8 tabs.

Comment on: Relativistic linear combination of Gaussian-type orbitals density functional method based on a two-component formalism with external field projectors

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Reply to the Comment on: Relativistic linear combination of Gaussian-type orbitals density functional method based on a two-component formalism with external field projectors

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Relativistic linear combination of Gaussian-type orbitals density functional method based on a two-component formalism with external field projectors

A two-component scalar relativistic method which is based on the no-pair projection operator formalism of quantum electrodynamics has been studied within the framework of local density functional theory. This projection operator formalism, first proposed by Sucher for wave function based methods, provides a stable approach to the relativistic electronic structure problem without the well-known deficiencies of some variational procedures for the Dirac equation. Several approximations of increasing accuracy have been investigated. Besides the method based on free-particle projectors and on external field projectors for the singular nuclear potential, projectors on the full electronic potential are presented here for the first time. Atomic all-electron calculations within this framework give excellent results compared to a fully numerical solution of a scalar relativistic approximation to the corresponding one-particle Dirac problem. Calculations for cerium and for lead are analyzed in detail. About 96% of the relativistic shift of the one-electron energies and about 98% of the relativistic effect on the total energy are covered within this method. The method allows for a rather straightforward generalization to molecular systems with multinuclear potentials.

One-loop three-gluon vertex and power counting in the light-cone gauge

The divergent part of the one-loop three-gluon vertex is evaluated in the framework of the two-component formalism for the light-cone gauge. The result is consistent with multiplicative renormalizability for pure Yang-Mills gauge theories. The calculation entails a discussion on some of the peculiar properties of naive power counting exhibited by this special gauge. I employ the Mandelstam prescription implemented via dimensional regularization to regulate Feynman integrals.

Relativistic electronic-structure calculations employing a two-component no-pair formalism with external-field projection operators.

A no-pair formalism employing external-field projection operators correct to second order in the potential is used to calculate the 1s energies of one-electron atoms and ground-state properties of the bromine and silver atoms in the framework of the multireference double-excitation configuration-interaction (MRD-CI) method. It is found that the relativistic two-component method that has been used reproduces the one-particle energies of the Dirac equation to order (Z\ensuremath{\alpha}${)}^{3}$. The operator is bounded from below and can be used variationally in relativistic electron-structure calculations of many-electron atoms and molecules. The relativistic correction to the total energy recovers 97% of the relativistic correction of the Dirac-Hartree-Fock (DHF) result in the case of the bromine atom and more than 99% in the case of the silver atom. The relativistic correction of the ionization potential of silver has been calculated to be 0.47 eV at the CI level, in good agreement with DHF results, the correlation contribution in the relativistic case being 0.42 eV. The remaining discrepancy of the absolute value of 6.85 eV (DHF 6.34 eV) to experiment (7.57 eV) is attributed to basis-set deficiencies. The corresponding CI value of the electron affinity (relativistic CI value 1.05 eV, nonrelativistic 0.90 eV) is in much better agreement with experiment (1.30 eV). It is found that correlation contribution and relativistic effects are nonadditive.

Two Yang-Mills theories in the light-cone gauge:: Complete one-loop counterterms

The Mandelstam-Leibbrandt prescription is used to study one-loop structures of the two-component (LCM2) and four-component (LCM4) formalisms of the same Yang-Mills theory in the light-cone gauge. The complete one-loop counter lagrangians are constructed by computing the one-loop two-, three- and four-point vertices. LCM2 is renormalizable order-by-order in g with δ L counter = (Z − 1) L, Z = 1 + 11g 2C 2/48π 2ϵ . For LCM4, both the two- and three-vertices generate anomalous counterterms which, however, cancel upon summation so that the total δ L counter is the same as LCM2. Slavnov-Taylor identities are satisfied in LCM4; they do not exist in LCM2. The method of analytic regularization is used in computation; all invariant and tensor integrals are evaluated using a single representation for light-cone invariant two-point integrals. The calculation is exceedingly simple in LCM2, far less so in LCM4.

Relativistic Hamiltonian with square root in the path integral formalism

This paper discussed the quantization of the relativistic Hamiltonian with square root. It is shown that calculation of this integral does not give the Green's function of the Klein-Gordon equation but leads to the propagator of a scalar field in the two-component Feshbach-Villars formalism. The difficulties in calculating path integrals in the presence of nontrivial external fields are discussed.

Two-component Fermi-liquid theory: Equilibrium properties of liquid metallic hydrogen

Recent studies point to the possibility that condensed hydrogen will undergo a transition from an insulating molecular crystal phase to a metallic liquid phase at zero temperature and high pressure. Liquid metallic hydrogen (LMH) comprising interpenetrating proton and electron fluids, would constitute a two-component Fermi liquid having both long-range, species-dependent bare interactions and a very high component-mass ratio. We examine the low-temperature equilibrium properties of LMH (assuming that it is "normal") by means of a generalization to the case of two components of the phenomenological Landau Fermi-liquid theory. The general two-component formalism is discussed in some detail. Estimates for the relevant phenomenological Landau parameters for LMH are made, and results for low-temperature specific heat, compressibility, thermal expansion coefficient, and spin susceptibility are given. The specific heat and thermal expansion coefficient are found to be vastly greater in the liquid than in the corresponding solid due to the presence of proton quasiparticle excitations in the liquid. The possibility of a negative expansion coefficient of the liquid cannot be dismissed.