Foldy Wouthuysen Research Articles

Applicability of the lowest-order two-electron Breit–Pauli relativistic correction in many-electron heavy and super-heavy elements

The applicability of the lowest-order two-electron Breit–Pauli (2e-BP) relativistic correction incorporated with the infinite-order Foldy–Wouthuysen (IOFW) transformation was examined in heavy and super-heavy elements. Comparing with the non-relativistic Coulomb repulsion term, in cases of moderately heavy noble-gas atoms (He to Ar), the 2e-BP correction gave the SCF energies practically equivalent to those obtained with the Dirac–Fock/Coulomb method. The spin-orbit splittings were also corrected reasonably. On the other hand, in heavy noble-gas atoms (Kr to Rn), the 2e-BP correction tended to overshoot energy lowering, and in a super-heavy atom (E118), evidences of collapse of the 2e-BP correction were observed.

Quantum study of Foldy–Wouthuysen–Tani theory on lattice

We study here a quantum version of Foldy–Wouthuysen–Tani (FWT) transformation and compare the similarities and differences between the quantum and the classic FWT theories. Then the improvement of action on lattice is discussed. The result shows that it is not necessary to improve the covariant difference along the time direction on lattice. Finally we discuss briefly the structure of physical vacuum and give a model independent of field condensate.

Quasirelativistic theory I. Theory in terms of a quasi-relativistic operator¶

In this first part of a general analysis of a quasi-relativistic theory, i.e. a relativistic theory for electrons only, the transformation of the Dirac operator to an operator with two-component spinor solutions is studied, while a forthcoming second part will be devoted to a transformation at matrix level. We start with a simple derivation of the key relation between the upper (ϕ) and lower (χ) components of the Dirac bispinor (ψ) for both electrons and positrons. The three possible choices of a non-hermitian quasi-relativistic Hamiltonian L, a hermitian one with non-unit metric, and a hermitian one L + with unit metric are compared. The eigenfunctions of the first two are the upper components ϕ of ψ, while those of L + are the Foldy–Wouthuysen-type spinorsφ. Some general properties of quasi-relativistic Hamiltonians and their eigenfunctions are discussed, especially the behaviour near the position of a nucleus. Exact solutions, and even variational ones are only obtained if the orbital basis describes the weak singularities at the positions of the nuclei correctly. A new derivation of the quasi-relativistic version of direct perturbation theory (DPT) is given, followed by the theory of quasi-relativistic effective Hamiltonians, both non-hermitian and hermitian ones. The classical Foldy–Wouthuysen transformation is then presented as the singular limit of quasi-relativistic effective Hamiltonians with the model space extended to the entire space of positive-energy states. Finally, the problems that arise for a Douglas–Kroll transformation or the regular approximation at operator level are studied in detail. In part 2, it will be shown that everything becomes much simpler if one performs the transformation from relativistic to quasi-relativistic theory at matrix level. ¶This paper is dedicated to Andrzej Sadlej on the occasion of his 65th birthday.

The Foldy–Wouthuysen transformation technique in optics

We describe the standard Foldy–Wouthuysen transformation technique along with its applications in optics and the new results it leads to. The use of the Foldy–Wouthuysen transformation technique is central in the non-traditional prescriptions of Helmholtz optics and Maxwell optics, respectively, resulting in the wavelength-dependent modifications of light beam optics, even at the paraxial level. It is further shown that the wavelength-dependent modifications in the newly developed formalisms of light beam optics are in close anology with the quantum theory of charged-particle beam optics.

Exact decoupling of the Dirac Hamiltonian. I. General theory.

Exact decoupling of positive- and negative-energy states in relativistic quantum chemistry is discussed in the framework of unitary transformation techniques. The obscure situation that each scheme of decoupling transformations relies on different, but very special parametrizations of the employed unitary matrices is critically analyzed. By applying the most general power series ansatz for the parametrization of the unitary matrices it is shown that all transformation protocols for decoupling the Dirac Hamiltonian have necessarily to start with an initial free-particle Foldy-Wouthuysen step. The purely numerical iteration scheme applying X-operator techniques to the Barysz-Sadlej-Snijders (BSS) Hamiltonian is compared to the analytical schemes of the Foldy-Wouthuysen (FW) and Douglas-Kroll-Hess (DKH) approaches. Relying on an illegal 1/c expansion of the Dirac Hamiltonian around the nonrelativistic limit, any higher-order FW transformation is in principle ill defined and doomed to fail, irrespective of the specific features of the external potential. It is shown that the DKH method is the only valid analytic unitary transformation scheme for the Dirac Hamiltonian. Its exact infinite-order version can be realized purely numerically by the BSS scheme, which is only able to yield matrix representations of the decoupled Hamiltonian but no analytic expressions for this operator. It is explained why a straightforward numerical iterative extension of the DKH procedure to arbitrary order employing matrix representations is not feasible within standard one-component electronic structure programs. A more sophisticated ansatz based on a symbolical evaluation of the DKH operators via a suitable parser routine is needed instead and introduced in Part II of this work.

Calculation of nuclear magnetic shieldings: infinite-order Foldy–Wouthuysen transformation

A scheme for calculating nuclear magnetic shieldings using the infinite-order Foldy–Wouthuysen (FW) transformation proposed by Barysz and Sadlej (BS) is presented. The nuclear magnetic shieldings of hydrogen halides are calculated by three variant BS schemes; a double finite perturbation method for the external magnetic flux density (B 0) and the nuclear magnetic dipole moment ( μ M ) (BS/FPT-2), a single finite perturbation method for B 0 with analytical differentiation of energy with respect to ( μ M ) (BS/FPT-1), and an approximate analytical differentiation method with respect to both B 0 and μ M (BS/CHF). Although the BS/FPT-2 method is exact theoretically, the actual computation for heavy nuclei includes large error due to reduction of the number of significant figures. The BS/FPT-1 and BS/CHF approaches, on the other hand, yield reasonable values for all of the shieldings. Although several results could not be obtained by the BS/FPT-2 method, no serious contradictions were recognized among these three results. From a comparison of our results with values in the literature, our shieldings for the halogen nuclei are lower than those determined by the four-component relativistic random phase approximation (4-RPA), but the reason for this is not obvious.

A parabolic equation for penetrable rough surfaces: using the Foldy–Wouthuysen transformation to buffer density jumps

The goal of this effort is to design a parabolic equation (PE) that can be fully integrated into modern rough surface scattering theory. The Foldy–Wouthuysen transformation is used to design a PE that addresses this challenge. The paradigm employed is based on a non-relativistic theory of the quantum Lamb shift, in which a parabolic equation (the Schrödinger equation) was used to model a field near a rough surface (the world line of the hydrogen nucleus advected by vacuum fluctuations). With the acoustic field serving as the prototypical classical field, the PE derived using the Foldy–Wouthuysen transformation exploits higher-order boundary conditions to buffer density discontinuities in a manner precisely dictated by the formalism. This is significant because the techniques used in rough surface scattering theory ultimately rely on perturbation theory, conformal mappings, a local method of images, or some similar distortion of the range-independent problem. This type of distortion is not possible with techniques currently employed at a density jump, and so ad hoc rules have been used instead. The new PE allows interfaces where the density jumps (such as the ocean bottom) to be distorted into rough ones, and so it is fully compatible with rough surface scattering theory.

Foldy–Wouthuysen transformation for relativistic particles in external fields

A method of Foldy–Wouthuysen transformation for relativistic spin-1/2 particles in external fields is proposed. It permits the determination of the Hamilton operator in the Foldy–Wouthuysen representation with any accuracy. Interactions between a particle having an anomalous magnetic moment and nonstationary electromagnetic and electroweak fields are investigated.

CONFORMAL COUPLING AND FOLDY–WOUTHUYSEN TRANSFORMATION

The propagation of a free scalar field ϕ with mass m in a curved background is generally described by the equation (gμν∇μ∇ν + m2 + ξR)ϕ = 0. There exist some arguments in the literature that seem to favor the conformal coupling to the detriment of the minimal one. However, the majority of these claims are inconclusive. Here we show that the exact Foldy–Wouthuysen transformation for spin-0 particle coupled to a wide class of static spacetime metrics exists independently of the value of ξ. Nevertheless, if the coupling is of the conformal type, the gravitational Darwin-like term has an uncomplicated structure and it is proportional to the corresponding term in the fermionic case. In addition, an independent computation of this term, which has its origin in the zitterbewegung fluctuation of the boson's position with the mean square <(δr)2> ≈ 1/m2, gives a result that coincides with that obtained using the aforementioned exact transformation with ξ = 1/6.

DIRAC EQUATION IN SPACETIMES WITH NON-METRICITY AND TORSION

Dirac equation is written in a non-Riemannian spacetime with torsion and non-metricity by lifting the connection from the tangent bundle to the spinor bundle over spacetime. Foldy–Wouthuysen transformation of the Dirac equation in a Schwarzschild background spacetime is considered and it is shown that both the torsion and non-metricity couples to the momentum and spin of a massive, spinning particle. However, the effects are small to be observationally significant.

The generalized Douglas–Kroll transformation

We derive the most general parametrization of the unitary matrices in the Douglas–Kroll (DK) transformation sequence for relativistic electronic structure calculations. It is applied for a detailed analysis of the generalized DK transformation up to fifth order in the external potential. While DKH2–DKH4 are independent of the parametrization of the unitary matrices, DKH5 turns out to be dependent on the third expansion coefficient of the innermost unitary transformation which is carried out after the initial free-particle Foldy–Wouthuysen transformation. The freedom in the choice of this expansion coefficient vanishes consistently if the optimum unitary transformation is sought for. Since the standard protocol of the DK method is the application of unitary transformations to the one-electron Dirac operator, we analyze the DKH procedure up to fifth order for hydrogenlike atoms. We find remarkable accuracy of the higher-order DK corrections as compared to the exact Dirac ground state energy. In the case of many-electron atomic systems, we investigate the order of magnitude of the higher-order corrections in the light of the neglect of the DK transformation of the two-electron terms of the many-particle Hamiltonian. A careful analysis of the silver and gold atoms demonstrates that both the fourth- and fifth-order one-electron DK transformation yield a smaller contribution to the total electronic energy than the DK transformation of the two-electron terms. In order to improve significantly on the third-order correction DKH3, it is thus mandatory to include the DK transformation of the two-electron terms as well as the spin-dependent terms before proceeding to higher orders in the transformation of the one-electron terms. However, an analysis of the ionization energies of these atoms indicates that already DKH3 yields a highly accurate treatment of the scalar-relativistic effects on properties.

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.

THE SEMICLASSICAL RELATIVISTIC DARWIN POTENTIAL FOR SPINNING PARTICLES IN THE REST-FRAME INSTANT FORM: TWO-BODY BOUND STATES WITH SPIN 1/2 CONSTITUENTS

We extend previous results on the extraction of the Darwin potential to all orders in c-2 from the radiation gauge Lienard–Wiechert solution for the system of N positive-energy scalar particles plus the electromagnetic field in the Wigner-covariant rest-frame instant form of dynamics to the case of N positive-energy spinning particles. This is done in the semiclassical approximation of using Grassmann-valued electric charges for regularizing the Coulomb self-energies and extracting the unique semiclassical action-at-a-distance interaction hidden in any Green function used for the Lienard–Wiechert solution. By describing semiclassically also the spin of the particles with Grassmann variables, by means of a semiclassical Foldy–Wouthuysen transformation applied the the Dirac-like constraints of the manifestly Lorentz covariant spinning particles, we determine the coupling of positive-energy spinning particles to the electric field in the semiclassical approximation. Then we follow the same procedure developed for scalar particles and, in the sector where there is no in-radiation, we determine the effective semiclassical interparticle potential. Besides the relativistic Darwin term there are spin-orbit and spin-spin terms in the potential. Quantization of the lowest order (in c-2) part of the closed form of the effective Hamiltonian in the case N = 2 reproduces exactly the standard result of the reduction of the Bethe–Salpeter equation for the bound states of two spin 1/2 constituents of arbitrary mass (hydrogen atom, positronium, muonium).

Systematic treatment of relativistic effects accurate through arbitrarily high order in α2

A systematic method for the generation of two-component relativistic Hamiltonians for use in relativistic quantum chemistry is presented and discussed. The free particle Foldy–Wouthuysen transformation of the Dirac Hamiltonian is performed prior to the determination of the block-diagonalizing unitary transformation. The latter can be determined iteratively through arbitrarily high leading order with respect to α (fine structure constant). Certain freedom in the initialization of the iterative solution leads to the whole class of two-component Hamiltonians h2k which are exact through the order of α2k and differ in contributions of all higher orders in α2. The efficiency of different iterative schemes is analyzed. Also the relation between the present method and the Douglas–Kroll approximation is investigated. The performance of two-component Hamiltonians for k=2, 3, and 4 is studied numerically in calculations of energies of the 1s1/2 level in heavy hydrogen-like ions. Their performance in calculations of the valence-determined atomic and molecular properties is investigated by computing the ionization potential of Au and spectroscopic constants of the AuH molecule. The total energy of these systems strongly depends on the level of exactness with respect to α2. However, for moderately relativistic systems the α4-class Hamiltonian derived in this paper is found to be sufficient for accurate calculations of valence-determined properties.

A new strategy for applying the parabolic equation to a penetrable rough surface

The range-dependent parabolic equation (PE) propagates an auxiliary field that roughly corresponds to the square root of the downrange flux. At the ocean bottom, the interface is broken into stair steps, and this auxiliary field is conserved at the vertical interfaces. Because of its success in matching benchmark solutions, this method is widely used to model standard propagation problems in underwater acoustics. However, when the interface is rough at the wavelength scale, the physics of the problem changes, and the standard approach ceases to be practicable. This talk presents a new approach specifically designed to handle this important problem. The Foldy–Wouthuysen transformation generates the PE for this auxiliary field, complete with a full suite of boundary conditions. In the discrete problem, the boundary conditions are used to evaluate the Hamiltonian on the interface, and the result is used to perform the next downrange step. At lowest order, the new theory streamlines the stair step method, and frees it from ad hoc impositions such as those spawned by the density jump at the ocean bottom. The approach also systematically provides higher-order corrections. [Work supported by ONR.]

Transformation of operator equations describing the interaction of relativistric particles with an electric field

A method is developed for solving relativistic wave equations for the Hamiltonian in the Foldy—Wouthuysen representation. These equations are recast into a quadratic form, whereby we get rid of square roots of operators. Applications of the method are exemplified by considering the interaction of a particle with a field characterized by a harmonic potential.

Relativistic dynamics of two spin-half particles in a homogeneous magnetic field

Relativistic dynamics of two spin-1/2 particles in an external, homogeneous magnetic field is investigated here. The problem is important for a preliminary understanding of the effect of magnetic field on atoms and molecules at the relativistic level. The relativistic Hamiltonian is formulated in three distinct forms which involve the Bethe–Salpeter interaction, generalized Breit interaction and projected Breit interaction. The total pseudomomentum of the two-particle system is conserved in each case, and its components are distinct in the zero-charge sector. This permits the separation of the center of mass motion from the Hamiltonian of the neutral two-particle system. The resulting Hamiltonian operator describes the movement of the two particles in relative coordinates. It is further simplified by using suitable unitary transformations so as to reduce the one-particle operator for the first particle into a diagonal form. The effective equation of motion for the movement of the second particle in relative coordinates is then identified. A second set of transformations convert the two-particle relative Hamiltonian into a form where the one-particle operator for each spin-1/2 particle is completely diagonalized and separable into positive and negative energy states. The correspondingly transformed interaction operators can be written in an order by order expansion from which the odd terms are removable by using suitable Foldy–Wouthuysen type transformations in a systematic way. The resulting Hamiltonian operator reduces to previously known expressions when the magnetic field is switched off. Thus the two sets of transformations which convert the one particle parts completely into separable as well as diagonal forms also transform the interaction operator to generate terms consistently through order v2/c2. The field dependence lies entirely in the diagonalized one-particle parts, which is a consequence of the initial choice of interaction operators. Our results also include expressions corresponding to the interaction operator being projected. The Bethe–Salpeter and projected Breit cases lead to the same interaction operators for a hydrogen atom in the nonrelativistic limit. In the same limit the methodology directly yields the anomalous Zeeman interaction term, some correction to it, and terms which can account for nuclear magnetic resonance. All these terms are embedded in the final two-particle Hamiltonian operator. These, along with the previously known, field-independent, terms which describe the hyperfine interactions, can account for electronic and magnetic resonance spectroscopies on the basis of the same Hamiltonian.

The relativistic scheme for eliminating small components Hamiltonian: Analysis of approximations

The derivation of the recently proposed one-component relativistic Hamiltonian, and the resulting relativistic scheme by eliminating small components (RESC) method of Nakajima and Hirao, are analyzed in terms of the Foldy–Wouthuysen transformation of the Dirac Hamiltonian. This approach reveals the meaning of different approximations used in the derivation of the RESC Hamiltonian and its close relation to approximate relativistic Hamiltonians resulting from the free-particle Foldy–Wouthuysen transformation. Moreover, the present derivation combined with what is called the classical approximation in Nakajima and Hirao’s approach shows that there is a whole family of the RESC-type Hamiltonians. Some of them, including the original RESC Hamiltonian, are analyzed numerically. It is documented that neither of the RESC-type Hamiltonians offers variational stability. As a consequence the RESC methods may suffer from the variational collapse for heavier systems. On the other hand the energy differences (e.g., ionization potentials) computed within the RESC approach turn out to be close to the values obtained in the Douglas–Kroll scheme.

Relativistic electric and magnetic property operators for two-component transformed Hamiltonians

A general procedure is presented for the derivation of property operators for electric and magnetic perturbations for Hamiltonians derived from the Dirac Hamiltonian by a partially block-diagonalizing unitary transformation. The procedure involves a regularized expansion in powers of p2/m2c2. Property operators are expressed in terms of the solid spherical harmonics. Expressions for the free-particle Foldy–Wouthuysen, Douglas–Kroll, and Barysz–Sadlej–Snijders transformations are compared with the well-known Pauli results. Explicit examples of a constant electric field and a constant magnetic field are given. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 412–421, 2000