Time reversal nonrelativistic 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. 

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. 

Now let us turn to time reversal. In nonrelativistic physics, time reversal leaves positions invariant, but changes the sign of all velocities. If K is the time-reversal operator, we must have... 

In the nonrelativistic Hamiltonian we need not worry about spin. With r being real and p purely imaginary in our standard representation, the complex conjugation operator, /Co, can be used to effect time reversal. For b and d real numbers, the effect of Ko is... 

Extending our treatment to the two-component relativistic case, we have a situation analogous to that previously encountered for inversion there may be parts of a total time-reversal operator that do not give explicit effects when applied in the nonrelativistic realm, but which are essential for the relativistic treatment. Taking the same approach as for inversion, we write the total time-reversal operator as... 

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. 

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... 

The operator with si = 2 = 1 is identical in form to the nonrelativistic two-particle excitation operator. To arrive at expressions for the time-reversed operators, any index in this expression can be replaced with the barred index. The final expression for the Dirac-Coulomb Hamiltonian is... 

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 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 scheme outlined above (Sjpvoll et al. 1997) has been implemented in the program LUCIA. The program also exploits both double-group symmetry and time-reversal symmetry. The main computational costs over a nonrelativistic Cl arise from the presence of vector operators, from the need to use complex arithmetic, and from the extended interaction space due to the fact that the spin-orbit operators connect determinants of different spin multiplicity. 

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