Spinor Rotations

Moreover, the second-generation MCSCF parametrizes the wave function in a way that enables the simultaneous optimization of spinors and Cl coefficients, in this context then called orbital or spinor rotation parameters and state transfer parameters, respectively. Then, a Newton-Raphson optimization method is employed which also requires the second derivatives of the MCSCF electronic energy with respect to the molecular spinor coefficients (more precisely, to the orbital rotation parameters) and to the Cl coefficients. As we have seen, in Hartree-Fock theory the second derivatives are usually not calculated to confirm that a solution of the SCF procedure has indeed reached a minimum with respect to the large component and not a saddle point. Now, these general MCSCF methods could, in principle, provide such information, although it is often not needed in practice. 

We would expect the open-open Fock matrix elements to correspond to redundant spinor rotation parameters because the energy should be invariant to rotations of equivalent spinors. However, if we construct the spinor rotation gradient, we see that there is a term that survives in the diagonal of the gradient ... 

The imaginary part of the integral is not necessarily zero, so the diagonal spinor rotation gradient is not zero. To explain this rather peculiar situation, we must turn to the application of symmetry. 

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. 

Turning now to the case of polarization/rotation modulation, or continuous rotation of iT i 2rl2 corresponding to a continuous rotation of Cj through 20, there is a rotation of the resultant through 0. This correspondence is a consequence of the A 1A = 7 relation, namely, that if the unitary transformation of A or A 1 is applied separately the identity matrix will not be obtained. However, if the unitary transformation is applied twice, then the identity matrix is obtained and from this follows the remarkable properties of spinors that corresponding to two unitary transformations of, for example, 27i, namely, 471, one null vector rotation of 271 is obtained. This is a bisphere correspondence and is shown in Fig 3b. This figure also represents the case of polarization/rotation modulation—as opposed to static polarization/rotation. 

In order to address polarization/rotation modulation—not just static polarization/rotation—an algebra is required which can reduce the ambiguity of a static representation. Such an algebra which is associated with 2, r, and that reduces the ambiguity up to a sign ambiguity, is available in the twistor formalism [28]. In this formalism, polarization/rotation modulation can be accomodated, and a spinor, K, can be represented not only by a null direction indicated by 2, q, or C, but also a real tangent vector L indicated in Fig. 4. 

In order to establish the homomorphism between SU(2) and SO(3), we will consider first the dual u v) of the spinor basis u v). Note that no special notation, apart from the bra and ket, is used in C2 to distinguish the spinor basis (u v from its dual u v). A general rotation of the column spinor basis in C2 is effected by (see Altmann (1986), Section 6.7)... 

The IRs of G comprise the vector representations, which are the IRs of G, and new representations called the spinor or double group representations, which correspond to half-integral j. The double group G contains twice as many elements as G but not twice as many classes g, and g,- are in different classes in G except when g,- is a proper or improper BB rotation (that is, a rotation about a binary axis that is normal to another binary axis), in which case g, and gt are in the same class and (gj, (xg,) are necessarily zero in spinor... 

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... 

Example 12.6-2 The classes of Td are E 4C3 3C2 6S4 6binary rotations are BB rotations. The six dihedral planes occur in three pairs of perpendicular improper BB rotations so both 3C2 and 6rrd are irregular classes. There are therefore Nv 5 vector representations and Ns = 3 spinor representations. 

D3 = E 2C3 3C2. There are no BB rotations so that the groups both consist of three regular classes. There are therefore three vector representations (/Vv = Nc) and three spinor representations (NS = NTC = NC). The dimensions of the Nv vector representations are /V = 1 1 2 (because g 6) and of the Ns spinor representations also /s = 1... 

My aim has been to give in these tables only the most commonly required information. For character tables for n > 6, Cartesian tensor bases of rank 3, spinor bases, rotation parameters, tables of projective factors, Clebsch-Gordan coefficients, direct product... 

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 ... 

The spinor that describes the spherical rotation satisfies Schrodinger s equation and specifies two orientations of the spin, colloquially known as up and down (j) and ( [), distinguished by the allowed values of the magnetic spin quantum number, ms = . The two-way splitting of a beam of silver ions in a Stern-Gerlach experiment is explained by the interaction of spin angular momentum with the magnetic field. 

Representation theory of molecular point groups tells us how a rotation or a reflection of a molecule can be represented as an orthogonal transformation in 3D coordinate space. We can therefore easily determine the irreducible representation for the spatial part of the wave function. By contrast, a spin eigenfunction is not a function of the spatial coordinates. If we want to study the transformation properties of the spinors... 

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... 

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