Electrons Kramers’ theorem

II electronic states, 638-640 vibronic coupling, 628-631 triatomic molecules, 594-598 Hamiltonian equations, 612-615 pragmatic models, 620-621 Kramers doublets, geometric phase theory linear Jahn-Teller effect, 20-22 spin-orbit coupling, 20-22 Kramers-Kronig reciprocity, wave function analycity, 201 -205 Kramers theorem ... 

This is due to the so-called Kramers theorem, which establishes that all electronic energy levels containing an odd number of electrons are at least doubly degenerate. 

For this reason, ESR is much more easily done in Kramers ions which have an odd number of electrons and at least a two fold degeneracy in the absence of an applied magnetic field (Kramers theorem). This restricts one to those paramagnetic iron materials which have half integer spin (1/2, 3/2, 5/2) and eliminates those iron materials with spins of 0, 1, 2 unless special efforts are made which we won t discuss here. 

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. 

State, which must be totally symmetric. The second emerges from the extended Kramers theorem, which imposes half-integer J rotational states to be degenerate. Thus, the lowest rotational state for the electronic ground state of Li3 corresponds to J = j, and must be degenerate. 

The (3d) system (or the equivalent d hole) is subject to Kramers theorem which states that when a system is composed of an odd number of electrons (or holes) it is not possible for electric fields to remove degeneracies completely — at least two-fold degeneracies must remain. 

The primary effect of the crystal field on any manifold of levels of a 4f ion is to raise the 2J + I degeneracy, giving a set of levels with an overall splitting of a few hundred wave-numbers. If the ion has an odd number of electrons, then by Kramers theorem a two-fold degeneracy must remain in each level, but for a non-Kramers ion there is no such restriction. All the degeneracy may be lifted by a field of low... 

It is seen from the values of the degeneracy number e that Kramers theorem that an even number of independent wave functions always participates in the energy level of a system with an odd number of electrons is valid. Kramers degeneracy can only... 

In general the crystalline field is not of perfectly cubic symmetry. Under the action of lower-S3unmetry field and the spin-orbit coupling between d electrons, each ground level listed in Table I is split into several levels, although at least twofold degeneracy remains if the number of electrons is odd (Kramers theorem). ... 

We see that, for many-electron functions, a double time reversal produces a phase that depends on the number of electrons. If N is odd, there is a change of phase if N is even, there is no change of phase. This is a manifestation of Kramers theorem a system with an odd number of fermions behaves like a fermion, but a system with an even number of fermions behaves like a boson. 

One important point to keep in mind is that Kramers theorem always applies, which for systems with equal numbers of spin-up and spin-down electrons implies inversion symmetry in reciprocal space. Thus, even if the crystal does not have inversion symmetry as one of the point group elements, we can always use inversion symmetry, in addition to all the other point group symmetries imposed by the lattice, to reduce the size of the irreducible portion of the Brillouin Zone. In the example we have been discussing, C2 is actually equivalent to inversion symmetry, so we do not have to add it explicitly in the list of symmetry operations when deriving the size of the IBZ. 

Kramers degeneracy theorem states that the energy levels of systems with an odd number of electrons remain at least doubly degenerate in the presence of purely electric fields (i.e. no magnetic fields). This is a consequence of the time-reversal invariance of electric fields, and follows from an application of the antiunitary T-operator to the wavefunction of an odd number of electrons [51]. 

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