Quaternions components

A disadvantage of the Euler angle approach is that the rotation matrix contains a total of six trigonometric functions (sine and cosine for each of the three Euler angles). These trigonometric functions are computationally expensive to calculate. An alternative is to use quaternions. A quaternion is a four-dimensional vector such that its components sum to 1 0 + 1 + <72 + = 1- quaternion components are related to the Euler angles as follows ... 

From the above discussion it becomes clear that in order to eliminate the spin-orbit interaction in four-component relativistic calculations of magnetic properties one must delete the quaternion imaginary parts from the regular Fock matrix and not from other quantities appearing in the response function (35). It is also possible to delete all spin interactions from magnetic properties, but this requires the use of the Sternheim approximation [57,73], that is calculating the diamagnetic contribution as an expectation value. 

We then turn to the question of how to eliminate the spin-orbit interaction in four-component relativistic calculations. This allows the assessment of spin-orbit effects on molecular properties within the framework of a single theory. In a previous publication [13], we have shown how the spin-orbit interaction can be eliminated in four-component relativistic calculations of spectroscopic properties by deleting the quaternion imaginary parts of matrix representations of the quaternion modified Dirac equation. We show in this chapter how the application of the same procedure to second-order electric properties takes out spin-forbidden transitions in the spectrum of the mercury atom. Second-order magnetic properties require more care since the straightforward application of the above procedure will extinguish all spin interactions. After careful analysis on how to proceed we... 

In 1843, during a flash of inspiration while walking with his wife, Hamilton realized that it took four (not three) numbers to accomplish a 3-D transformation of one vector into another. In that instant, Hamilton saw that one number was needed to adjust the length, another to specify the amount of rotation, and two more to specify the plane in which rotation takes place. This physical insight led Hamilton to study hypercomplex numbers (or quaternions) with four components, sometimes written with the form Q = + ad + a + aji where the as are... 

The non-Abelian component of the field tensor is defined through a metric that is a set of four quaternion-valued components of a 4-vector, a 4-vector each of whose components can be represented by a 2 x 2 matrix. In condensed notation ... 

There exist generally covariant four-valued 4-vectors that are components of q, and these can be used to construct the basic structure of 0(3) electrodynamics in terms of single-valued components of the quaternion-valued metric q1. Therefore, the Sachs theory can be reduced to 0(3) electrodynamics, which is a Yang-Mills theory [3,4]. The empirical evidence available for both the Sachs and 0(3) theories is summarized in this review, and discussed more extensively in the individual reviews by Sachs [1] and Evans [2]. In other words, empirical evidence is given of the instances where the Maxwell-Heaviside theory fails and where the Sachs and 0(3) electrodynamics succeed in describing empirical data from various sources. The fusion of the 0(3) and Sachs theories provides proof that the B(3) held [2] is a physical held of curved spacetime, which vanishes in hat spacetime (Maxwell-Heaviside theory [2]). 

In Eq. (5), the product q q is quaternion-valued and non-commutative, but not antisymmetric in the indices p and v. The B<3> held and structure of 0(3) electrodynamics must be found from a special case of Eq. (5) showing that 0(3) electrodynamics is a Yang-Mills theory and also a theory of general relativity [1]. The important conclusion reached is that Yang-Mills theories can be derived from the irreducible representations of the Einstein group. This result is consistent with the fact that all theories of physics must be theories of general relativity in principle. From Eq. (1), it is possible to write four-valued, generally covariant, components such as... 

These are cyclic relations between single-valued metric field components in the non-Abelian part [Eq. (6)] of the quaternion-valued P v. Equation (16) can be put in vector form... 

In general, all the off-diagonal elements of the quaternion-valued commutator term [the fifth term in Sachs Eq. (4.19)] exist, and in this appendix, it is shown, by a choice of metric, that one of these components is the Ba> field discussed in the text. The B<3) field is the fundamental signature of 0(3) electrodynamics discussed in Vol. 114, part 2. In this appendix, we also give the most general form of the vector potential in curved spacetime, a form that also has longitudinal and transverse components under all conditions, including the vacuum. In the Maxwell-Heaviside theory, on the other hand, the vector... 

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. 

As quaternions have disappeared from the common chemists mathematical background, we recall here the basic properties of this beautiful mathematical instrument. Quaternions are objects having four components. The first one can be treated as a real scalar, whereas the other three can be considered components of a three-dimensional vector. These objects are customarily represented in a form similar to that of complex numbers ... 

Any 3-dimensional vector v can be treated as a special case of a quaternion with the zero scalar component... 

By singling out the contributions eq. (3.53) up to the second order with respect to the (small) components of the vector parts of the quaternions q and p, we obtain the first order correction ... 

In the previous section we used quaternions to construct a convenient parameterization of the hybridization manifold, using the fact that it can be supplied by the 50(4) group structure. However, the strictly local HOs allow for the quaternion representation for themselves. Indeed, the quaternion was previously characterized as an entity comprising a scalar and a 3-vector part h = (h0, h) = (s, v). This notation reflects the symmetry properties of the quaternion under spatial rotation its first component ho = s does not change under spatial rotation i.e. is a scalar, whereas the vector part h — v — (hx,hy,hz) expectedly transforms as a 3-vector. These are precisely the features which can be easily found by the strictly local HOs the coefficient of the s-orbital in the HO s expansion over AOs does not change under the spatial rotation of the molecule, whereas the coefficients at the p-functions transform as if they were the components of a 3-dimensional vector. Thus each of the HOs located at a heavy atom and assigned to the m-th bond can be presented as a quaternion ... 

The quaternion representation of the HOs is useful also for analysis of the symmetry properties of the energy components arising within the SLG-based semiempirical theories. Using the quaternion notation eq. (3.58) we get for the one-center molecular... 

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. 

The symmetry group of relativity theory tells the story. For the irreducible representations of the Poincare group (of special relativity) or the Einstein group (of general relativity) obey the algebra of quaternions. The basis functions of the quaternions, in turn, are two-component spinor variables [17]. 

This quaternion differential is a generalization of the Riemannian metric. The 4-vector quaternion fields qn (x) then replace the second-rank, symmetric tensor fields gM v(x) as the fundamental metric of the spacetime. The metric field q (x) is a 4-vector, whose four components are each quaternion-valued. This is then a 16-component field, rather than the 10-component metric tensor field g v of the standard Riemannian form. 

The two-component spinor form of electromagnetic held theory (17 ) is generalized in the curved spacetime by (1) globally extending the Pauli matrices to the quaternion elements, o41 > q (x), and (b) generalizing the ordinary... 

The starting point then to achieve the factorization of the Einstein equations is the factorized differential line element in the quaternion form, ds = q,1(x)dxll, where qyi are a set of four quaternion-valued components of a 4-vector. Thus ds is, geometrically, a scalar invariant, but it is algebraically a quaternion. As such, it behaves like a second-rank spinor of the type v / v /, where / is a two-component spinor variable [17]. 

The squared bracket in Eq. (36) denotes the behavior of the quaternion field with respect to its vector degrees of freedom alone. The covariant derivatives of the two-component spinor variables are as follows v(/ p = (0p + Op) ]/ and the spin-affine connection has two alternative (equivalent) forms [17] ... 

The solutions of the latter equations are the 10 components of the symmetric second-rank metric tensor g iv. The solutions of the factorized equations (46a) [or (46b) are the 16 components of the quaternion metrical field qp (or q ). We will now see that this 16-component metrical quaternion field, indeed, incorporates the gravitational and the electromagnetic fields in terms of their earlier tensor representations. Gravitation entails 10 of the components in the symmetric second-rank tensor g iv. Electromagnetism entails 6 of the components (the 3 components of the electric field and the three components of the magnetic field), as incorporated in the second-rank antisymmetric tensor Fpv. 

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