Time reversal four-component operator

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. 

For four-component wave functions it is the spin operator E that changes sign under time reversal, and by simple extension of the algebra above we can write the four-component operator as... 

The development of techniques that incorporate time-reversal symmetry presented here are primarily aimed at four-component calculations, but they are equally applicable to two-component calculations in which the spin-dependent operators are included at the self-consistent field (SCF) stage of a calculation. 

The relativistic basis is no longer the set of products of orbital functions with a and spin functions, but general four-component spinors grouped as Kramers pairs. Likewise, the operators are no longer necessarily spin free. If we apply the time-reversal operator to matrix elements of we can derive some relations between matrix elements... 

Two results follow immediately from this derivation. First, the four parts of the 2-spinor must fulfil the same relation, because we could equally well have done an elimination of the large component, analogously to what we did for the large component at the outset. The same argument could then have been carried through for Second, the symmetry of the time-reversed partner of follows easily because application of the time-reversal operator just corresponds to a spin flip, effectively swapping the two complex spatial components of the 2-spinor ... 

For the case of two reversible equilibria connecting both electrochemical reaction steps (Eq. II. 1.30), a scheme with four components. A, A+, B, B", can be drawn with four reversible processes connecting them. This kind of reaction scheme, the so-called square scheme, is commonly observed for redox state dependent isomerisation [110], ligand exchange [111], and protonation processes [112] and can be seen to be based on an extended version of the ECE scheme. Under limiting conditions, when chemical steps are in equilibrium on the voltammetric time scale (with a sufficiently small potential gap between A+/A and b+/b> and appropriately slow scan rate), a simple reversible voltammetric response can be observed even when the square scheme is operative. In other circumstances, two scan rate dependent processes may be observed. 

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