Kramers time reversal operator

Kramers time-reversal operator, shown in equation (1), is a product of Pauli spin matrices... 

In other words, Kramers time reversal operator changes the sign of total spin. Also one can verify that Kramers operator K changes the sign of the total momentum,... 

Applying the Kramers time reversal operator we have... 

However A is not a very suitable operator to use directly to spht the ground doublets. It is Hermitian and invariant under even permutations of Si, S2 and S3. But its behaviour under the Kramers, or time-reversal operator ( ) is rather unsatisfactory in that A = — A. 

Very early in the history of quantum theory, H.A. Kramers introduced a time reversal operator [1-3] for V-electron systems. 

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... 

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... 

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... 

Thus K commutes with the wave operator and the wave operator is therefore symmetric under time reversal. Because of the structure of the wave operator, each term f) must be symmetric under time reversal. We now introduce a Kramers-pair basis, and apply the time-reversal operator to the f term, to obtain... 

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. 

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]. 

For Kramers (e.g. one-electron) states where the eigenvalue of T2, A = — 1, the metric gAA is antisymmetric, and so relating to symplectic algebras (in relativistic terms to pure torsion rather than to curvature), rather than symmetric as for non-Kramers systems. The joint action of Hermitian conjugation and time reversal (which is not commutative) is summarized with the above results for these individual operations in Table 1. 

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. 

Using Kramers concept of time-reversal, it is possible to construct symmetry preserving pair creation operators of the form [5,7,8]... 

To signal the transition from a summation over individual orbitals to a summation over Kramers pairs I will employ capital letters, but only under the summation sign Y,pq Y,pq I retain lowercase orbital indices for both cases, as it poses no confusion. We may further insist that the perturbation operators hx have a specific symmetry with respect to time reversal... 

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... 

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. 

Hence, the relativistic analog of the spin-restriction in nonrelativistic closed-shell Hartree-Fock theory is Kramers-restricted Dirac-Hartree-Fock theory. We should emphasize that our derivation of the Roothaan equation above is the pedestrian way chosen in order to produce this matrix-SCF equation step by step. The most sophisticated formulations are the Kramers-restricted quaternion Dirac-Hartree-Fock implementations [286,318,319]. A basis of Kramers pairs, i.e., one adapted to time-reversal s)mimetry, transforms into another basis under quatemionic unitary transformation [589]. This can be exploited not only for the optimization of Dirac-Hartree-Fock spinors, but also for MCSCF spinors. In a Kramers one-electron basis, an operator O invariant under time reversal possesses a specific block structure. 

Time Reversal and Kramers-Restricted Representation of Operators... 

In the previous discussion of the symmetry of the Dirac equation (chapter 6), it was shown that the Dirac equation was symmetric under time reversal, and that the fermion functions occur in Kramers pairs where the two members are related by time reversal. We will have to deal with a variety of operators, and in most cases the methodologies will be developed in the absence of an external magnetic field, or with the magnetic field considered as a perturbation. Consequently, we can make the developments in terms of a basis of Kramers pairs, which are the natural representation of the wave function in a system that is symmetric under time reversal. The development here is primarily the development of a second-quantized formalism. We will use the term Kramers-restricted to cover techniques and methods based on spinors that in some well-defined way appear as Kramers pairs. The analogous nonrelativistic situation is the spin-restricted formalism, which requires that orbitals appear as pairs with the same spatial part but with a and spins respectively. Spin restriction thus appears as a special case of Kramers restriction, because of the time-reversal connection between a and spin functions. 

Our basis of Kramers pairs is 1] , We adopt the convention that general spinors are labeled p, q, r, s, occupied spinors are labeled i, j, k, I, unoccupied spinors are labeled a, b, c, d, and active spinors (where necessary) are labeled t, u, v, w. The time-reversed conjugate of a function or operator is denoted by a bar. We place the bar over the index of a function rather than the function, that is, we use 1] rather than rlf ,. However, the two are equivalent, and we place the bar over the function when there is no index. 

Let us for a moment consider the nonrelativistic case of the second-quantized representation of operators. Here, time reversal flips the spin function between a and p, and in a spin-restricted formalism the basis of Kramers pairs is [(ppa, 4>p(> > where cj) is the spatial part of the orbital. Normally the operators used in nonrelativistic calculations are real and spin free, and we can then use the representation... 

We now return to the question of how time-reversal symmetry relates to doublegroup symmetry and the block structure of operator matrices. For case 2 above, with the two components of the Kramers pair belonging to different irreps, there are no matrix elements of totally symmetric operators between a fermion function and... 

We now turn our attention to the entire space of ny Kramers pairs that transform under the irrep y. We see that operators are represented as 2ny x 2ny matrices. By a suitable reordering, the representation matrix for an operator symmetric under time reversal may be brought to the form... 

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. 

These time-reversal reductions affect the expressions for the second-quantized Kramers-restricted Hamiltonian. Both forms given in chapter 9 contain integrals with an odd number of bars, and in both, these terms vanish if the group has no quaternion irreps. If we want to use the Hamiltonian as expressed in terms of the excitation operators Ep and such as in a coupled-cluster calcula-... 

The symmetry reductions in nonrelativistic methodology come from spin symmetry and from point-group symmetry. In relativistic methodology, time-reversal symmetry is the equivalent of spin symmetry, but it does not provide the same magnitude of reduction as does spin symmetry. This is due to the presence of spin-dependent terms in the Fock operator. Point-group symmetry is intimately connected with time-reversal symmetry in Kramers-restricted relativistic theory, as we saw in chapter 10. 

The only way to get a wave operator that is symmetric under time-reversal symmetry is to impose the restriction from the beginning. While this fixes the relations between the amplitudes, it also forces the occupied and the unoccupied Kramers components of the open shell to be treated equivalently. This equivalence is what introduces the ambiguity in the treatment of the open shell the open-shell Kramers pair must behave as both a particle and a hole, and the result is that the truncated commutator expansions in the coupled-cluster equations are much longer than in closed-shell theory. The alternative is to use an unrestricted wave operator with the Kramers-restricted spinors. The use of the latter provides some reduction in the work due to the relations between the integrals, but not a full reduction (Visscher et al. 1996). 

Because

double excitation in the open-shell space, and because we left excitations within this space out of the excitation operators, the second part of the normalization term is zero, and the energy is given by the left side of the equation. This technique can be used for open-shell Kramers pairs belonging to complex or real irreps, but not to quaternion irreps. In the last case, there are four determinants that are composed of the open-shell spinors, and even though they occur in pairs related by time-reversal symmetry, the Hamiltonian operator connects all four. In the case of complex irreps, the absolute value of the off-diagonal matrix element must be taken, because it will in general be complex. 

In line with what we have said previously, we must in general expect the orbital mixing coefficients Krs to be complex and the k matrices to be anti-Hermitian. In a Kramers-restricted formalism the operator must be invariant under time reversal, and we can incorporate this requirement by expressing the elements of k in terms of the operators ... 

From our experience to date with the transformations, we can immediately foresee a problem. If the transformation is some complicated function of the momentum, we might not be able to separate out the perturbation from the zeroth-order Hamiltonian. This would be unfortunate, because magnetic operators break Kramers symmetry and we would be forced to perform calculations without spin (or time-reversal) symmetry. We might also be forced to perform finite-field calculations. We will address this problem as it arises. 

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