Kramers Replacement Operators

One way of explicitly building time reversal symmetry into the formalism is to introduce Kramers replacement operators [83,81] in analogy with the singlet and triplet excitation operators of the non-relativistic domain [80]. Using (181), we may rewrite the property (179) operator as... 

For the two-electron integrals, we want to divide the integrals into symmetry classes, as for the nonrelativistic integrals. We also want to divide the integrals into classes by time reversal symmetry, as we did for the one-electron integrals. Because of the structure of the Kramers-restricted Hamiltonian in terms of the one- and two-particle Kramers replacement operators, we hope to obtain a reduction in the expression for the Hamiltonian from time-reversal symmetry. The classification by time-reversal properties is also important for the construction of the many-electron Hamiltonian matrix, whose symmetry properties we consider in the next section. 

The formalism can be straightforwardly extended to two-electron operators with the introduction of Kramers double replacement operators [81]. However, the multitude of terms arising in subsequent derivations finally leads to a rather cumbersome formalism. In the author s opinion it is better to derive general formulas and then consider the structure due to time reversal symmetry after the derivation. 

This expression represents a simplification of the original expansion in the primitive Kramers pairs basis. Although there is not a reduction in the size of the one-particle basis, we need only consider half the matrix elements, and there is therefore a 50% reduction in the amount of work. The lack of reduction might be expected because the matrix elements are potentially spin-dependent. The X q operators are called Kramers single-replacement operators, and they define what we will call a Kramers basis. 

There is an even more compact way of defining the Kramers single-replacement operators, Xfq, which also has the advantage of displaying the permutational symmetry. To do this, we introduce two auxiliary operators. One is the bar-reversal operator, Kp, which is the time-reversal operator for a spinor with index p. The effect of this operator is... 

The Kramers single-replacement operators, can now be expressed in terms of these auxiliary operators. 

The reduction of products of second-quantized operators makes use of commutators of the replacement operators. The various commutators of Kramers single-replacement operators are given by application of bar reversal to the basic commutator relation... 

Using these auxiliary operators, it is easy to define Kramers double-replacement operators in an analogous manner ... 

In the FKKE proposed by Barkai and Silbey [30] in contrast, the memory or fractional derivative term is supposed to act only on the dissipative part of the normal Klein-Kramers operator, so that Barkai and Silbey s equation [where, however, rotational quantities (angle [Pg.373]

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. 

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... 

---