Transformed Dirac Equation

Transformed Dirac equations are convenient starting points for the derivation of quasi-relativistic Hamiltonians. The transformed Dirac equations can be obtained by using approximate solutions for the small components as ansatze for the wave function. The ansatz can be deduced from the lower half of the Dirac equation by an approximate elimination of the small component. 

The zeroth-order regular approximation (ZORA) Hamiltonian can be derived from the upper part of the transformed Dirac equation (20). By using the ZORA ansatz for the small component (5) and assuming that the upper and the lower components are equal, the final ZORA equation for the upper component becomes... 

It has already been noted the resemblance of the classical Schrddinger equation wavefianctions, when expressed in EH spaces, with relativistic spinors. Dirac equation can be found in many sources, see for example references [5,58,69,70,77,79,81-83]. It is well know how Dirac equation can be transformed as well as the problems appearing when trying to solve it for large atomic and molecular systems. Among all the possible transformations Dirac equation can suffer, the one described in Messiah s treatise [58] conforms the best to the EH space formalism here discussed. Indeed, whenever the relativistic spinors, y/, ... 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

We now consider how to eliminate the spin-orbit interaction, but not scalar relativistic effects, from the Dirac equation (25). The straightforward elimination of spin-dependent terms, taken to be terms involving the Pauli spin matrices, certainly does not work as it eliminates all kinetic energy as well. A minimum requirement for a correct procedure for the elimination of spin-orbit interaction is that the remaining operator should go to the correct non-relativistic limit. However, this check does not guarantee that some scalar relativistic effects are eliminated as well, as pointed out by Visscher and van Lenthe [44]. Dyall [12] suggested the elimination of the spin-orbit interaction by the non-unitary transformation... 

The Dirac equation is invariant to Lorentz transformations [8], a necessary requirement of a relativistic equation. In the limit of large quantum numbers the Dirac equation reduces to the Klein-Gordon equation [9,10]. The time-independent form of Dirac s Hamiltonian is given by... 

Here, X is to be determined by imposing that the resulting transformed Dirac Hamiltonian is block diagonal. It is fairly easy to see that this leads to an equation... 

In this case, X is to be determined by requiring that the off-diagonal blocks of the resulting transformed Dirac Hamiltonian vanishes. It can be shown that the equation for X is identical to the one we obtained in the case of a unitary transformation as given in equation (38). In this case, the effective Hamiltonian hn and wave function xp, can be written as... 

However, it is more appropriate to provide theoretical justifications for such use. In this respect, first, we introduce the third category of decoupling of positive and negative states commonly known as the direct perturbation theory . This approach does not suffer from the singularity problems described previously. However, the four-component form of the Dirac equation remains intact. The new Hamiltonian requires identical computational effort as for the Dirac equation itself, hence it is not an attractive alternative to the Dirac equation. However, it is useful to assess the accuracy of approximate two-component forms derived from the Dirac equation such as Pauli Hamiltonian. Consider the transformation... 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

Abstract. An application of the Rayleigh-Ritz variational method to solving the Dirac-Coulomb equation, although resulted in many successful implementations, is far from being trivial and there are still many unresolved questions. Usually, the variational principle is applied to this equation in the standard, Dirac-Pauli, representation. All observables derived from the Dirac equation are invariant with respect to the choice of the representation (i.e. to a similarity transformation in the four-dimensional spinor space). However, in order to control the behavior of the variational energy, the trial functions are subjected to several conditions, as for example the kinetic balance condition. These conditions are usually representation-dependent. The aim of this work is an analysis of some consequences of this dependence. 

Relativistic variational principles are usually formulated as prescriptions for reaching a saddle point on the energy hypersurface in the space of variational parameters. The results of the variational calculations depend upon the orientation of the saddle in the space of the nonlinear parameters. The structure of the energy hypersurface may be very complicated and reaching the correct saddle point may be difficult [14,15]. If each component of the wavefunction is associated with an independent set of nonlinear parameters, then changing the representation of the Dirac equation results in a transformation of the energy hypersurface. As a consequence, the numerical stability of the variational procedure depends on the chosen representation. 

Relativity becomes important for elements heavier than the first row transition elements. Most methods applicable on molecules are derived from the Dirac equation. The Dirac equation itself is difficult to use, since it involves a description of the wave function as a four component spinor. The Dirac equation can be approximately brought to a two-component form using e.g. the Foldy-Wouthuysen (FW) transformational,12]. Unfortunately the FW transformation, as originally proposed, is both quite complicated and also divergent in the expansion in the momentum (for large momenta), and it can thus only be carried out approximately (usually to low orders). The resulting equations are not variationally stable, and they are used only in first order perturbation theory. 

In standard quantum field theory, particles are identified as (positive frequency) solutions ijj of the Dirac equation (p — m) fj = 0, with p = y p, m is the rest mass and p the four-momentum operator, and antiparticles (the CP conjugates, where P is parity or spatial inversion) as positive energy (and frequency) solutions of the adjoint equation (p + m) fi = 0. This requires Cq to be linear e u must be transformed into itself. Indeed, the Dirac equation and its adjoint are unitarily equivalent, being linked by a unitary transformation (a sign reversal) of the y matrices. Hence Cq is unitary. 

Equation (3.125) is the required transformed Hamiltonian, and we see that in the representation in which (3 is diagonal, the Dirac equation decomposes into uncoupled equations for the upper and lower components of the wave function, i.e. for electron and positron wave functions. Setting (3 equal to +1 gives the positive energy (electron) states, whilst (3 equals -1 gives the negative energy (positron) states. 

In other words, it is the distance, ct, travelled by light in a given time interval which fulfills the role of the fourth coordinate, rather than the time interval itself. The special theory of relativity requires that after a Lorentz transformation the new form of all laws of physics is the same as the old form. The Dirac equation, for example, is invariant under a Lorentz transformation. 

SO coupling is a relativistic effect. The theory of the interaction of the magnetic moments of the electron spin and the orbital motion in one- and two-electron atoms has been formulated independently by Heisenberg and Pauli [12,13], shortly before the advent of the four-component Dirac theory of the electron [14]. Breit later has added the retardation correction [15]. The resulting Breit-Pauli SO operator, which can more elegantly be derived from the Dirac equation via a Foldy-Wouthuysen transformation [16], was thus well known for atoms since the early 1930s [17]. 

A representation of the one-electron Dirac equation which is decoupled in the electronic and the charge-conjugated degrees of freedom is achieved by a unitary transformation of (3.1)... 

A truncation of the expansion (3.5) defines the zero- and first-order regular approximation (ZORA, FORA) (van Lenthe et al. 1993). A particular noteworthy feature of ZORA is that even in the zeroth order there is an efficient relativistic correction for the region close to the nucleus, where the main relativistic effects come from. Excellent agreement of orbital energies and other valence-shell properties with the results from the Dirac equation is obtained in this zero-order approximation, in particular in the scaled ZORA variant (van Lenthe et al. 1994), which takes the renormalization to the transformed large component approximately into account, using... 

The fine structure was calculated by Pirenne [110] and independently by Berestetski [4J. Minor errors are corrected, and numerical results are given by Ferrell [45]. The approach used by these authors is to write down the Dirac equations for the two particles, and the interaction terms as they are expressed in quantum field theory. The equations can be transformed so that the particle spins appear explicitly. The interaction terms are found to comprise the Coulomb energy, the Breit interaction, and a term analogous to the Fermi expression for... 

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