Quaternion Symmetry

Salomon, Y. and Avnir, D. (1999) Continuous symmetry measures Finding the closest C2-symmetric object or closest reflection-symmetric object using unit quaternions. J. Comput. Chem. 20, 772-780. 

Table 16.22. Quaternion and Cayley-Klein parameters for the symmetry operators of the point group S4.

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... 

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]. 

The covariance groups underlying the tensor forms of the respective Einstein and the Maxwell held equations are reducible. This is because they entail reflection symmetry, not required by relativity theory, as well as the required continuous symmetry of the Einstein group E. When the Einstein held equations are factorized, they yield the irreducible form, which are then in terms of the quaternion and spinor variables, rather than the tensor variables. Such a generalization must then extend the physical predictions of the usual tensor forms of general relativity of gravitation and the standard vector representation of the Maxwell theory (both in terms of second-rank tensor helds, one symmetric and the other antisymmetric) because the new factorized variables have more degrees of freedom than did the earlier version variables. 

Since there is no reflection symmetry in the quaternion formulation, the reflected quaternion q 1 must be distinguishable from. The conjugate differential fine element to ds is ds = q dxVL. The product of the quaternion and conjugate quaternion line elements is then the real-number-valued element that corresponds to the squared differential element of the Riemannian geometry ... 

An alternative way of handling time reversal symmetry is through the use of quaternion algebra [82,84,85]. A (real) quaternion number is written as... 

T. Saue, H. J. A. Jensen. Quaternion symmetry in relativistic molecular calculations The Dirac-Hartree-Fock method. /. Chem. Phys., 111(14) (1999) 6211-6222. 

An important realization of quaternions is to be found in the Pauli matrices. The set of matrices I2, ia, ioy, ia c) is isomorphic to the set of quaternion units 1, i, j, k. This isomorphism has been exploited in computational schemes for the construction of symmetry spinors (Saue and Jensen 1999). 

The reduction of the number of integrals due to time-reversal symmetry follows a similar pattern to that of the one-electron integrals. We consider the case of D jt and subgroups, where all irreps belong to one of the three types quaternion, complex, or real. The analysis for groups that have more than one irrep type follow. 

Classification by time-reversal relations overlaps the symmetry classification. For groups without quaternion irreps, integrals in the classes (// //) and must... 

If both components of both open-shell Kramers pairs are in the same nondegenerate irrep (the quaternion case), as is the case when there is no symmetry, all four open-shell determinants must be included in the wave function, and we must resort to a genuine multiconfiguration DHF method. 

Based on this ordering, it is possible to rewrite the equations as a quaternion problem, which turns out to provide considerable computational advantages, in particular in the handling of symmetry (Hafner 1980, Saue and Jensen 1999). For the present purposes we will stay with the usual ordering by component size, which is more convenient for displaying the structure of the equations, and also has the advantage of familiarity. 

As for the SCF method, the main reductions due to double-group symmetry follow from the nature of the irreps. For quaternion irreps, all amplitudes are in principle nonzero for complex and real irreps the odd-bar amplitudes vanish, giving a reduction of a factor of 2 and for real irreps the imaginary part of the amplitudes is zero, giving another reduction of a factor of 2. 

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. 

The demonstration [1] that both Lorentz transformation and quantum spin are the direct result of quaternion rotation implies that aU relativistic and quantum structures must have the same symmetry. This is the basis of cosmic self-similarity. The observation that the golden mean features in many known self-similarities confirms that r represents a fundamental characteristic of space-time curvature. The existence of antimatter and the implied CPT symmetry of space-time favors... 

The concept of quaternion groups i.s useful for the understanding of icosahedral quasicrystals and shows how the existence of icosahedral quasicrystals is a natural consequence of the use of quaternions to represent symmetry groups. In this connection a real quaternion is defined as an ordered quadruple of four real numbers w,x, y, z). subject to the following rules of addition and multiplication where q = (w, x, y, z) and q = (w, r. /, c ) ... 

The following shows how symmetry point groups (Section 3) can be described by quaternion groups ... 

Quasicrystals may thus be regarded as a special type of incommensurate system, which may be described by space groups of dimension larger than three in a similar way to modulated crystal phases and incommensurate composite structures. In the case of icosahedral quasicrystals the above model based on the icosahedral quaternion group //, fits well into the idea of 6D space groups. Each of the three standard coordinates, namely x, y, and z, corresponds to two coordinates in 6D space, namely a rational and an irrational coordinate corresponding to the rational and irrational portions of variables, of the form a -H o V5, where a and d are integers. Projection of the lattice points of this 6D space of icosahedral symmetry into conventional 3D space leads to the icosahedral quasicrystal lattice. 

We used the DIRAC program suite. Time-reversal symmetry [30] and Abelian point groups, including C2v, are fully exploited in the DIRAC program with the help of quaternion algebra. [31] We briefly summarize quaternion algebra for the case of C2v... 

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