Spinor rotation operator

Starting from the Dirac-Coulomb approximation, a set of Dirac-Kohn-Sham equations may again be derived. In chapter 8, a spinor-rotation procedure was used to derive the relativistic Fock operator. A similar procedure applied to the present case shows that the gradient of the energy has elements the form... 

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

Example 12.6-1 The point group C2v = E C2z perpendicular axes, all operations except E are irregular and there is consequently only one doubly degenerate spinor representation, Ei/2. Contrast C 21, = E C2z / [ in which rrh is az = IC2z and thus an improper binary rotation about... 

This operation represents a quarter turn on a great circle or rotation of ir radians about an axis. An intermediate position is given by the spinor ... 

We are not going to review here the transformation properties of spatial wave functions under the symmetry operations of molecular point groups. To prepare the discussion of the transformation properties of spinors, we shall put some effort, however, in discussing the symmetry operations of 0(3)+, the group of proper rotations in 3D coordinate space (i.e., orthogonal transformations with determinant + 1). Reflections and improper rotations (orthogonal transformations with determinant -1) will be dealt with later. 

We can employ an analog of the spatial transformation operator [138] for analyzing the transformation properties of a spinor under coordinate rotations. The evaluation of the corresponding 2D transformation matrices is simplified if we rewrite... 

It has to be noted that the relation between the elements of 0(3)+ (also called SO(3), the group representing proper rotations in 3D coordinate space) and SU(2) (the special unitary group in two dimensions) is not a one-to-one correspondence. Rather, each R matches two matrices u. Molecular point groups including symmetry operations for spinors therefore exhibit two times as many elements as ordinary point groups and are dubbed double groups. 

According to this matrix formulation the quaternion, also known as a rotation operator or spinor transformation, becomes... 

What happens with the interaction between the rotational and spin symmetries once the system is characterized as being defined by at least different spinors Wigner and von Neumann [10] combined both types of symmetries with the permutation aspect [11]. They intuitively reached the idea using atomic spectroscopy that the H operator has to be constructed by two terms H, resulting from the spatial motion of the single electron only (and the electromagnetic interaction with the field of the atomic core), and (//2), which has to visualize the electron spin. For simplicity, we can consider the eigenvalue problem of the spinless wave function i r without the second term as... 

To one who is familiar with nonrelativistic quantum mechanic it may appear quite clear what is meant by a spherically symmetric potential— any potential that actually only depends on x, so that it is invariant under any rotation applied to the system. It is indeed true that a scalar function

R ) for all rotation matrices R, if and only if it is a function of r = [x]. But a general potential in quantum mechanics is given by a Hermitian matrix, and the unitary operators (86) representing the rotations in the Hilbert space of the Dirac equation can also affect the spinor-components. Hence it is not quite straightforward to tell, which potentials are spherically symmetric. 

Factorization of the Cl space is difficult in the relativistic case. The first problem is the increase in number of possible interactions due to the spin-orbit coupling. The second problem is the rather arbitrary distinction in barred and unbarred spinors that should be used to mimic alpha and beta-spinorbitals. Unlike the non-relativistic case the spinors can not be made eigenfunctions of a generally applicable hermitian operator that commutes with the Hamiltonian. If the system under consideration possesses spatial symmetry the functions may be constrained to transform according to the representations of the appropriate double group but even in this case the precise distinction may depend on arbitrary criteria like the choice of the main rotation axis. 

The spinor basis enables us to obtain a two-dimensional matrix representation of the point-group operations. Let us first limit ourselves to the relationships for the proper rotations. The counterclockwise rotation of the vector over an angle a with the pole at the positive z-axis is given by the matrix ... 

For the actual construction of the double group as a group of operators, we need a convention to connect the spatial operators to the spinor matrices. As we have seen in Sect. 7.2, the four possible parametric descriptions of a given rotation yield two different choices for the Cayley-Klein parameters. Hence, our convention should define how to characterize unequivocally the parameters of a rotation. It will consist of two criteria the rotation angle must be positive, and the pole from which the rotation is seen as counterclockwise must belong to the positive hemisphere in the nx,tiy, tiz parameter space. This is the hemisphere above the equatorial plane, i.e., with > 0. In the (rix, Wy)-plane, we include the half-circle of points with positive -value, i.e., with = 0, > 0, and also the point with ny = l,nx = 0, and... 

As a straightforward example, we take the double group of D2. The standard drawing puts the twofold rotational axes in the positive hemisphere, and the corresponding spinor matrices are easily obtained from Eq. (7.34). The results are given in Table 7.3. Eor each operation of G, there are two operations in the double group, R and iiR. Note that the Bethe operation, commutes with every element of the group. Armed with this set of matrices, one can easily construct the multiplication... 

This character can be zero only for a = n and, hence, for binary rotations with n = 2. To examine whether or not the matrix for a binary rotation can be class-conjugated to minus itself, we may limit ourselves to the study of one orientation of the rotation axis, say C. Indeed, in SU 2) any orientation can always be transformed backward to this standard choice by a unitary transformation. The problem thus reduces to finding a spinor operation X represented by a matrix X with Cayley-Klein parameters ax,bx, which transforms (C ) into minus itself ... 

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