Symmetry operators double-group

Although the techniques that incorporate double-group symmetry presented here are primarily aimed at four-component calculations, they are equally applicable to two-component calculations in which the spin-dependent operators are included at the SCF stage of a calculation. 

We have seen how time-reversal symmetry and double-group symmetry are intimately connected in the matrices of one- and two-electron operators. These two symmetries are just as intimately connected in the many-electron Hamiltonian matrix. 

Although we may defer the introduction of double-group symmetry by performing the correlated calculations first, we still need to take the symmetry of the spin-orbit operator into account in the selection of the -particle expansion space. 

The scheme outlined above (Sjpvoll et al. 1997) has been implemented in the program LUCIA. The program also exploits both double-group symmetry and time-reversal symmetry. The main computational costs over a nonrelativistic Cl arise from the presence of vector operators, from the need to use complex arithmetic, and from the extended interaction space due to the fact that the spin-orbit operators connect determinants of different spin multiplicity. 

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. 

Now the trace of the new representation should correspond to a character of the irreducible representation of the double group. By inspecting all classes of symmetry operations, the given z-th energy level is unambiguously classified according to the character table of the double group. 

It has to be noted that the relation between the elements of 0(3)+ (also called SO(3), the group representing proper rotations in 3D coordinate space) and SU(2) (the special unitary group in two dimensions) is not a one-to-one correspondence. Rather, each R matches two matrices u. Molecular point groups including symmetry operations for spinors therefore exhibit two times as many elements as ordinary point groups and are dubbed double groups. 

The corresponding double group Civ comprises eight symmetry operations ... 

As is outlined in Appendix A, the U can straightforwardly be determined, making use of the determinantal form (10) of Hso- In particular, an associated unitary 2x2 matrix can be found for each of the 12 elements of D3. The resulting group of order 24, the so-called spin double group is the symmetry group of the SO operator (10). 

Using the methods discussed in the tutorial (Sect. 2), the group of symmetry operations of Hso can be constructed explicitly. As is shown in Appendix B, the symmetry group of Hso of (53) is T, the spin double group of the tetrahedral rotation group, is of order 48. T also is the symmetry group of the total Hamiltonian H = Hes + Hso-... 

In addition to the cylindrical symmetry, the Hamiltonian is time-reversal invariant, that is, it commutes with the time-reversal operator t of (15). The symmetry group of //so is the spin double group of Cooi . 

Table 1 The symmetry operations of the spin double group Abbreviation e =...

In principle, all four-component molecular electronic structure codes work like their nonrelativistic relatives. This is, of course, due to the formal similarity of the theories where one-electron Schrbdinger operators are replaced by four-component Dirac operators enforcing a four-component spinor basis. Obviously, the spin symmetry must be treated in a different way, i.e. it is replaced by the time-reversal symmetry being the basis of Kramers theorem. Point group symmetry is replaced by the theory of double groups, since spatial and spin coordinates cannot be treated separately. 

Table B.4. Double group characters table for the Ta point group. The numbers before the symmetry operations correspond to the number of geometrically different axes or symmetry planes. Some of the operations of the double group belong to the same class as those of the original group. When more than one IR is indicated, the first one corresponds to the notation of Mulliken [11], the second one to Koster et al. [9] and the one in parentheses to [3]...

Factorization of the Cl space is difficult in the relativistic case. The first problem is the increase in number of possible interactions due to the spin-orbit coupling. The second problem is the rather arbitrary distinction in barred and unbarred spinors that should be used to mimic alpha and beta-spinorbitals. Unlike the non-relativistic case the spinors can not be made eigenfunctions of a generally applicable hermitian operator that commutes with the Hamiltonian. If the system under consideration possesses spatial symmetry the functions may be constrained to transform according to the representations of the appropriate double group but even in this case the precise distinction may depend on arbitrary criteria like the choice of the main rotation axis. 

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