Normalized quaternions

It follows that a turn is the geometric realization of a normalized quaternion, that the addition of turns is the geometric realization of the multiplication of quaternions of unit norm, and that the group of turns is isomorphous with the group of normalized quaternions. Furthermore, eqs. (1) and (2) provide the correct parameterization of a normalized quaternion as... 

The coordinate map given by the variables (cD+, cD ) is a significant improvement as compared to eq. (3.25). Nevertheless, an explicit expression for an h matrix in its terms is still a clumsy combination of the trigonometric functions of two triples of reparametrizing angles w . It is known however that in the case of the SO(3) group [8] its quaternion [27] parameterization has the advantage that the matrix elements of SO(3) rotation matrices, when expressed in terms of the components of the normalized quaternion, are quadratic functions of these components. 

It is easy to check that the rows and columns of this matrix are orthogonal and its determinant equals unity. The independent complex matrix elements in eq. (3.45) are known as Cayley-Klein parameters of the rotation group. Also, one can see that for quaternions connected by the relation r = ri o r2 the corresponding 2x2 matrices are connected by the same relation with replacement of the quaternion multiplication by the usual matrix product. This establishes isomorphism between the SU(2) group and the group of normalized quaternions HP which can be continued to the homomorphism on 50(3). 

The product of two rotations is a rotation. Obtain an expression for the Cayley-Klein parameters of the product as a function of the parameters of its factors. Is the product commutative The SU 2) matrices may also be identified as normalized quaternions. 

Mathematically the hybridization of the 5p-basis at a given atom is defined by a SO (4) rotation. Each HO is thus a normalized quaternion with a scalar and a 3-vector parts The set of four vector parts... 

A general rotation R( o n) in ft3 requires the specification of three independent parameters which can be chosen in various ways. The natural and familiar way is to specify the angle of rotation and the direction of the unit vector n. (The normalization condition on n means that there are only three independent parameters.) A second parameterization R(a b) introduced above involves the Cayley-Klein parameters a, b. A third common parameterization is in terms of the three Euler angles a, (3, and 7 (see Section 11.7). Yet another parameterization using the quaternion or Euler-Rodrigues parameters will be introduced in Chapter 12. 

Our purpose is to construct an analogous parameterization for the 50(4) group. In order to reach this goal we mention that there exists a similar homomorphism between the 50(2) group of 2 x 2 unitary matrices with complex elements with the unit determinant and the 50(3) group. The correspondence establishes as follows for a rotation 1Z in the three dimensional space one can choose a quaternion representation r = (rn.r). This quaternion is normalized and it defines a 2 x 2 matrix ... 

The normalization condition for the two quaternions involved allows us to write ... 

A very elegant statement concerning the properties of hybridization tetrahedra belongs to Kennedy and Schaffer [30] in any hybridization tetrahedron two planes formed by any two pairs of HOs are orthogonal. It can be easily proven using the quaternion representation for the scalar product of the vectors normal to two said planes, the following chain of equalities holds (numeration is obviously arbitrary) ... 

In Section V it will be shown that the quaternion structure of the fields that correspond to the electromagnetic field tensor and its current density source, implies a very important consequence for electromagnetism. It is that the local limit of the time component of the four-current density yields a derived normalization. The latter is the condition that was imposed (originally by Max Bom) to interpret quantum mechanics as a probability calculus. Here, it is a derived result that is an asymptotic feature (in the flat spacetime limit) of a field theory that may not generally be interpreted in terms of probabilities. Thus, the derivation of the electromagnetic field equations in general relativity reveals, as a bonus, a natural normalization condition that is conventionally imposed in quantum mechanics. 

Thus the symmetric second-rank metric tensor g v of Einstein s formulation of general relativity corresponds to the symmetric sum from the quaternion theory, (—+ qvq ). [The factor (—5) is chosen in anticipation of the normalization of the quaternion variables.] Thus we see that ds is a factorization of the standard Riemannian squared differential metric ds2 = g dx dxv. 

For the resolution of the orientation parameters the representation of the 3x3 rotation matrix R in terms of normalized Hamilton quaternions a, 6, c is essential, namely... 

Because

double excitation in the open-shell space, and because we left excitations within this space out of the excitation operators, the second part of the normalization term is zero, and the energy is given by the left side of the equation. This technique can be used for open-shell Kramers pairs belonging to complex or real irreps, but not to quaternion irreps. In the last case, there are four determinants that are composed of the open-shell spinors, and even though they occur in pairs related by time-reversal symmetry, the Hamiltonian operator connects all four. In the case of complex irreps, the absolute value of the off-diagonal matrix element must be taken, because it will in general be complex. 

It should be noted that the transformation matrix becomes unbounded for / 7t/2. This is the reason for taking other parameterizations of the rotation matrix if /3 tends towards tt/2. Such a reparameterization introduces discontinuities which can be avoided when using a redundant set of rotation coordinates. One typically uses quaternions often also called Euler parameters. These are four coordinates instead of the three angles and one additional normalizing equation, see Ex. 5.1.10. This normalizing equation describes a property of the motion, a so-called solution invariant. Differential equations with invariants will be discussed in Sec. 5.3. 

The molecular spinors are expanded in terms of the four quaternion units (1, i, j, k). Two-dimensional contour maps of the large components are created in this work for the molecular spinors so to illustrate the nodal structure, and we review the relation between the quaternion representation and the normal four-component complex representation. [32] Each quaternion unit belongs to one of the boson irreducible representations (boson irreps) of C2v provided that the small components are neglected. 

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