Dirac standard representation

Using the standard representation of the Dirac matrices (3 and , the residual P,T-odd interaction operator can be written... 

In the standard (Dirac-Pauli) representation, the Dirac equation for an electron in the field of a stationary potential V reads... 

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. 

If p 7 0 Eqs. (21) and (22) are coupled, but the relations between components of the wavefunction are much simpler than in the standard Dirac-Pauli representation. By the elimination of and respectively from Eq. (21) and from Eq. (22), we get two decoupled second-order equations for and ... 

The Pauli form factor also generates a small contribution to the Lamb shift. This form factor does not produce any contribution if one neglects the lower components of the unperturbed wave functions, since the respective matrix element is identically zero between the upper components in the standard representation for the Dirac matrices which we use everywhere. Taking into account lower components in the nonrelativistic approximation we easily obtain an explicit expression for the respective perturbation... 

M. Stanke, J. Karwowski, Recent advances in the theory of chemical and physical systems, In J.-R Julien, J. Maruani, S. Wilson, G. Delgado-Barrio (Eds.), Non-Standard Representations of the Dirac Equation and the Variation Method, Springer, Dordrecht, The Netherlands, 2006, pp. 217-228. 

The standard representation of the four-component relativistic Dirac equa-tion for a single particle in an electrostatic potential V(r) and a vector potential (r) reads... 

It is not difficult to obtain the eigenvectors of the matrix h(p) with the standard methods of linear algebra. We start with the eigenvectors of the Dirac matrix /3, which are particularly easy to find in the standard representation. For example, take the four-dimensional unit vectors... 

The eigenfunctions, k defined above are obviously also eigenfunctions of the Dirac matrix (3 (in t ie standard representation),... 

The vectors and denote the Dirac 4x4 matrices for electron i (in standard representation) and the nuclear spin operator for nucleus a. The constants and Kg are nuclear parameters, while is the nuclear spin quantum number. 

However, the salient features of this approach can hardly be transfered to two-component methods, and we therefore focus in our presentation in the following only on the common standard representation of the Dirac algebra given above. 

Four-component Dirac spinor quantities are given in the standard representation in which the Dirac matrices are (in terms of 2 x 2 blocks)... 

Pi its components form the 4x4 Dirac matrices acting on the ith electron, a may be formulated in the standard representation in terms of the three 2x2 Pauli spin matrices = ([Pg.631]

This particular choice of the (4 x 4)-matrices is neither unique nor is the dimension 4 the only dimension, which allows us to fulfill the required properties. Higher even dimensions are also possible. How one may find other representations of these four parameters will be discussed in section S.2.4.3. The particular choice of Eqs. (5.37) is called the standard representation of the Dirac matrices a. and j6. We give as a further example the Weyl representation... 

Since the dimension of Y must necessarily be the same dimension as the one of the Dirac matrices we understand that it is, in general, an n-component vector of functions if the dimension of the Dirac matrices is n. In the standard representation, the quantum mechanical state Y is a vector of four functions, called a 4-spinor. 

According to the standard representation of the Dirac theory, the one-electron operators considered in chapters 5 and 6 are (4 x 4)-matrices acting on the one-particle Hilbert space... 

Basically, DKH theory starts from the standard representation of the Dirac Hamiltonian introduced in section 5.2. The fact that the Dirac matrices a and j6 are not uniquely defined suggests that a different representation of the Dirac operator might be more advantageous. In order to shed light on this question, we consider the Weyl representation of Eq. (5.38), which produces a Dirac Hamiltonian of a very different structure, namely the following... 

As a side remark, we may note that elimination procedures such as the one just discussed depend on the choice of the Dirac matrices see section 5.2. All that has been said so far holds for the standard representation, for which we have in split notation. 

We demonstrated that by the selection of a representation of the Dirac Hamiltonian in the spinor space one may strongly influence the performance of the variational principle. In a vast majority of implementations the standard Pauli representation has been used. Consequently, computational algorithms developed in relativistic theory of many-electron systems have been constructed so that they are applicable in this representation only. The conditions, under which the results of these implementations are reliable, are very well understood and efficient numerical codes are available for both atomic and molecular calculations (see e.g. [16]). However, the representation of Weyl, if the external potential is non-spherical, or the representation of Biedenharn, in spherically-symmetric cases, seem to be attractive and, so far, hardly explored options. 

We use the standard Dirac representation with the 4x4 Dirac matrices... 

The natural solution of the operator Eq. 4.38 would be in the momentum space due to the presence of momentum p which replaces the standard configuration space formulation by a Eourier transform. The success of the Douglas-Kroll-Hess and related approximations is mostly due to excellent idea of Bernd Hess [53,54] to replace the explicit Eourier transformation by some basis set (discrete momentum representation) where momentum p is diagonal. This is a crucial step since the unitary transformation U of the Dirac Hamiltonian can easily be accomplished within every quantum chemical basis set program, where the matrix representation of the nonrelativistic kinetic energy T = j2m is already available. Consequently, all DKH operator equations could be converted into their matrix formulation and they can be solved by standard algebraic techniques [13]. 

But the most important feature for practical purposes comes with a certain approximation, which is the scalar-relativistic variant of DKH. This one-component DKH approximation, in which all spin-dependent operators are separated by Dirac s relation and then simply omitted, is particularly easy to implement in widely available standard nonrelativistic quantum chemistry program packages, as Figure 12.4 demonstrates. Only the one-electron operators in matrix representation are modified to account for the kinematic or (synonymously) scalar-relativistic effects. The inclusion of the spin-orbit terms requires a two-component infrastructure of the computer program. The consequences of the neglect of spin-orbit effects have been investigate in pilot studies such as those reported in Refs. [626,655] and, naturally, the accuracy depends on the systems under consideration (see also section 14.1.3.2 and chapter 16 for further discussion). 

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