Group theoretical properties

The transformation properties with respect to spatial and time inversion are also important. The spatial inversion operator for Dirac 4-spinors is given by where. Q l (r) = (—r) is the ordinary parity operator. Then 

Time inversion of a Dirac 4-spinor requires the same operator to act on both 

The Dirac spinors are said to constitute a Kramers pair of 4-spinors. If 

By incorporating these symmetries in the 4-spinor basis functions, as we have done in our BERTHA code [50-54], we can make substantial computational economies in computing interaction integrals. The angular stracture of Dirac 4-spinors described here is also exploited by the major computer package TSYM, which utilizes projection operators to construct relativistic molecular symmetry orbitals for double valued representations of point groups [77-79]. 

The DIRAC package [55,56], devised by Saue and collaborators, rather than exploiting the group theoretical properties of Dirac spherical 4-spinors as in BERTHA, treats each component in terms of a conventional quantum chemical basis of real-valued Cartesian functions. The approach used in DIRAC, building on earlier work by Rosch [80] for semi-empirical models, uses a quaternion matrix representation of one electron operators in a basis of Kramers pairs. The transformation properties of these matrices, analysed in [55], are used to build point group transformation properties into the Fock matrix. 

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Oppenheimer approximation, 517-542 Coulomb interaction, 527-542 first-order derivatives, 529-535 second-order derivatives, 535-542 normalization factor, 517 nuclei interaction terms, 519-527 overlap integrals, 518-519 permutational symmetry, group theoretical properties, 670—674... 

Dirac bra-ket notation, permutational symmetry, group theoretical properties, 672-674... 

In this section we apply the technique described above in order to perform in-depth analysis of the problems of symmetry reduction and construction of exact invariant solutions of the SU(2) Yang-Mills equations in the (l+3)-dimensional Minkowski space of independent variables. Since the general method to be used relies heavily on symmetry properties of the equations under study, we will briefly review the group-theoretic properties of the SU(2) Yang-Mills equations. 

R. Dagis and I. B. Levinson, Group Theoretical Properties of Adiabatic Potentials in Molecules, Optika i Spektroskopiia Sbomik Statei, Nauka, Leningrad, 1967, Vol. 3, pp. 3-8. 

Though all the arguments sofar are built on the hydrogenic behaviour of (41), Schaffer (32) has recently demonstrated that the more general, group-theoretical properties of orbitals in octahedral symmetry are sufficient for several of the results obtained. 

These commutation relations are an unequivocal sign of symmetry for the Hamiltonian operator, H. Such symmetry is made clear through a careful study of group theoretical properties of the bilinear forms a]uj. As in similar cases, one can carry out certain transformations of these bilinear forms to convert them into irreducible tensors with respect to rotations [7,19] (see also a much more detailed discussion in Section II.C.2). The final result is that the bilinear forms a dj can be viewed as generators of the n-dimensional unitary group U(n). The underlying rotational structure of the harmonic oscillator is then recovered by means of a... 

It is also important to separate the dynamical properties from the group theoretical properties. An example from SU S) will illustrate this very well. Let Vj j = , ,8) be any set of currents or operators which transform as an octet imder 517(3), i.e. 

One of the most important aspects of spin functions is their group theoretical properties. In particular, the set of spin functions, for a given N and S, forms a basis for an irreducible representation of the group of permutations of N objects. This group is of order Nl and is often referred to as the symmetric group . It is denoted by the symbol Sn- Thus if P" denotes a permutation of the N spin coordinates [Pg.2675]

The value of the branching-diagram method is that it leads to spin eigenfunctions that provide standard irreducible representations (irreps) of the group when the permutations are applied to spin variables, as in (4.2.1) and for many pmposes it is possible to exploit the group-theoretical properties of the basis functions without ever explicitly constructing them. This approach is developed in Sections 4.3-4.6 and in later chapters. 

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