Kramers Partner

In the non-relativistic domain one-electron operators can be classified as triplet and singlet operators, depending on whether they contain spin operators or not. In the relativistic domain the spin-orbit interaction leads to an intimate coupling of the spin and spatial degrees of freedom, and spin symmetry is therefore lost. It can to some extent be replaced by time-reversal symmetry. We may choose the orbital basis generating the matrix of Hx to be a Kramers paired basis, that is each orbital j/p comes with the Kramers partner = generated by the action of the time-reversal operator We can then replace the summation over individual orbitals in (178) by a summation over Kramers pairs which leads to the form... 

For two-particle states the wave function must transform according to a boson irrep. The simplest case is that of a two-particle state made up of a spinor and its Kramers partner. In the absence of other degeneracies this corresponds to a closed shell, and we would expect this product to transform as the totally symmetric irrep. 

We partition both the molecular spinors and the basis set into Kramers pairs. This partitioning reduces to a partitioning of the large and small components into Kramers pairs, (rjrf, rlrf) for the molecular spinors and (x, xf) for the basis functions, where X is L or S. We cannot assume a priori that the molecular spinors for one of the Kramers partners can be expanded solely in the corresponding basis spinors, and we must therefore retain the full expansion until we have deduced the proper restrictions ... 

If the Kramers partners are in different rows of the same, doubly degenerate irrep, as in the group then the matrix elements are real and the gradient is zero. Likewise, the Hamiltonian matrix element between the two wave functions is zero, because the two configurations span two boson irreps and the -I- and - linear combinations have specific boson symmetries. This situation is somewhat similar to the nonrelativistic case of and I states, which are linear combinations of two determinants, or to the open-shell singlet case. 

This relation gives us a connection between the t amplitudes and the i amplitudes, but it does not provide any relation between t amplitudes for the Kramers partners such as we get for the closed-shell case. This stands to reason, because the amplitudes represent the configuration mixing due to correlation, and we cannot expect the correlation to be the same for a and spin in an open-shell doublet. The incorporation of spin-orbit interaction makes no change to this picture, in which the Pauli repulsion between spin-orbitals of the same spin is transferred to spinors of the same irrep row. 

The same considerations also apply to the case of two open shells where the product of the fermion irreps for the open shells belongs to a doubly degenerate boson irrep. In this case the reference is a single determinant, related to its partner by the time-reversal operator. Because there is no symmetry between the open shells, we cannot derive relations between the amplitudes for Kramers partners. 

These combinations therefore give pure bonding and antibonding diatomic molecular spinors, which also have pure spin. For homonuclear diatomic molecules, the spinors have a definite inversion symmetry a = ofg, a = a , ti = 7i , n = Ttg, and so on. The Kramers partners of these spinors are found by interchanging the spin parts. The first one is... 

The capability for relativistic hybridization is present for molecules of other than linear symmetry. The symmetry spinors for groups lower than cubic can be classified in terms of the m.j quantum numbers for spinors at the high symmetry point (or axis). Since the j quantum number is not part of the classification, relativistic hybridization can always take place. For example, in Dih the fermion irreps are e /2, e ji, and es/2, for which the Kramers partners have Tw modb = 1/2, 3/2, 5/2, respectively. The i and j = l + j spinors for a given rtij both belong to the same irrep. In... 

Taking TIH as an example, we can expand the molecular spinor combination in terms of spin-orbitals, and collect together the two-electron determinants that form the ground state. Let us write the spinor and its Kramers partner as ... 

In the last chapter you heard Mike describe the tortures of going up and down on different medications, each of which had distinctive side effects. A few years ago Peter Kramer, the author of Listening to Prozac wrote a book about human relationships entitled Should You Leave The book explores perennially significant questions about intimacy and autonomy— How do we choose our partners How well do we know them How do mood states affect our assessment of them and theirs of us When should we work to improve a relationship, and when should we walk away In essence, Kramer could have raised these same questions in his book on Prozac. In the following instances, however, experiments were abject failures and the only real option was to walk away ... 

Because the open-shell spinor is different for the two partners, the Kramers-unrestricted wave operator cannot be the same. At the least it must differ in the terms that involw the open-shell Kramers pair. Hence we must have two excitation operators, T and t, such that... 

---