Group double

Spin and spatial coordinates do not factorize in Dirac-based theories (in contrast to nonrelativistic quantum mechanics). Hence, ordinary point group symmetries are not a proper approach to understanding half-spin systems. Bethe [590] solved this problem by distinguishing between the null rotation and the rotation by 2n in spinor physics, the concepts, however, were introduced much earlier to mathematics (see the book by Altmann [591] for a historical perspective and a detailed account on double groups). 

In Section 9.3 we showed that for an orbital or state wave function having angular momentum quantum number / (or L) the character of the representation for which this forms a basis, under a symmetry operation that consists of rotation by an angle a, is given by 

We then applied this formula to various types of single-electron wave functions, for example s,p,d, /, g, and to wave functions for various Russell-Saunders terms characterized by integral values of the quantum number L. 

There are, however, many cases of interest in which we may want to determine the splitting of a state that is well characterized by its total angular momentum, J. This will in fact be the only thing of importance in the very heavy elements, for example, the rare earth ions, where states of particular L cannot be used since the various free-ion states of different J are already separated by much greater energies than the crystal field splitting energies. 

Now J - L + 5, and for ions with an odd number of electrons S and hence J must be half-integral numbers. For states in which J is an integer, the characters can be obtained by using the above formula with / replaced by J. However, when J is half-integral a difficulty arises. We know that a rotation by 2n is an identity operation and therefore it should be true that 

It can easily be seen that this is true when L or J is an integer. However, when J is half-integral we have 

In order to find a normal constant of motion for a one-electron atom we have combined the spin and orbital angular momentum of an electron into a total angular momentum j. This has profound consequences for the description of the symmetry of atomic systems. 

The great orthogonality theorem must still hold for the double group, as must the various orthogonalities between rows and columns of the character table. Also, the sum of squares of the dimensions of the irreducible representations (irreps) must equal the order of the group. If the nonrelativistic group has ki irreps with dimensions 

The double group retains these irreps to describe basis functions that do not change sign under E, and these will appear as the first ki irreps of the double group. Thus, expansion of the group just adds new irreps such that 

We will consider three examples of double groups and their character tables. The simplest of these is for the molecule with no spatial symmetry, which belongs to the point group Ci. Including spin symmetry and the E operation we get the C double 

We consistently try to use hats , for example tw, for operators Aroughout the text. However, hats are conventionally not used with symmetry operations in the context of character tables. 

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When Cj symmetry is present, irreducible representations of the double group (see Table II) so that = 0 by symmetry. In this case, there are only... 

Species of Spin Functions for Some Important Double Groups... 

Case (b). This case includes no single-group representations, and all real one-dimensional, double-group representations. This includes only the following. 

Case (c). This case includes all single- and double-group representations with complex character. 

Dominant-diagonal theorem, 58 Doob, J. L., 171,174,269 Dorodnitzin, A., 388 Double-group representations, 737 Double groups, 727... 

When spin-orbit coupling is introduced the symmetry states in the double group CJ are found from the direct products of the orbital and spin components. Linear combinations of the C"V eigenfunctions are then taken which transform correctly in C when spin is explicitly included, and the space-spin combinations are formed according to Ballhausen (39) so as to be diagonal under the rotation operation Cf. For an odd-electron system the Kramers doublets transform as e ( /2)a, n =1, 3, 5,... whilst for even electron systems the degenerate levels transform as e na, n = 1, 2, 3,. For d1 systems the first term in H naturally vanishes and the orbital functions are at once invested with spin to construct the C functions. 

For the calculation of the g values only the latter second-order term and the first order spin term, which contain j3H, are significant, and in the former either OMg and nMg, or n Mg and 0 Ms must belong to the same double group irreducible representation for the contribution to be non-vanishing. 

However, as mentioned above, T c)3) will be orthogonal to all the k states, and T ) is nonzero. This implies that the number of total states of the same eigenvalue E is (k + 1), which contradicts our initial hypothesis. Thus, we conclude that k must be even, and hence proved the generalized Kramers theorem for total angular momentum. The implication is that we can use double groups as a powerful means to study the molecular systems including the rotational spectra of molecules. In analyses of the symmetry of the rotational wave function for molecules, the three-dimensional (3D) rotation group SO(3) will be used. 

Once these double-group character tables are known, the procedure is exactly the same as that followed in the preceding sections when single character tables were used. Let us now tackle the problem of understanding the spectrum given in Figure 7.9 for Sm + ions in YAB. 

The first step is to construct the representations and corresponding to the excited and terminal 2/9/2 states, in the D3 double group. This... 

Representation of orbital spinors of symmetric molecules in terms of relativistic double groups [6]. 

After we have worked out the characters for all of the new operations, CyR, of the double group, we will then collect them into classes, using the same rule as for simple groups, namely, that all operations having the same characters are in the same class. Thus in general we shall find the following classes in double rotation groups ... 

It is easy to convince oneself that the only combination of positive integers satisfying this equation is 1, 1, 1, 1, 2, 2, 2. Thus there are four one-dimensional and three two-dimensional irreducible representations of the double group Di... 

The direct products of representations of double groups can be taken in the usual way and reduced to sums of irreducible representations. 

In order to illustrate the utility of double groups let us consider several examples. Suppose that we have an ion with one d electron in a planar complex. Real examples of this case are represented by complexes of Ou(II) and Ag(II) (where we virtually have one positron, but this behaves as one electron except in the signs of the energies). In each case there will be two states with / values l i = 2 i, namely, J — i and / = . Using 9.3-2, we find that these form bases for the following representations ... 

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