Time reversal adapted basis

We are now in a position to explain the results of Table I. As a consequence of the degeneracy of , at a conical intersection there are four degenerate functions tl/f, tb and Ttbf = Ttb = tb j. By using Eq. (Ic), an otherwise arbitrary Flermitian matrix in this four function time-reversal adapted basis has the form... 

Appendix C. Eigenvalues and Eigenvectors of a 4x4 Hermitian Matrix in a Time Reversal Adapted Basis 109... 

For odd electron molecules, the construction of a time reversal adapted basis is facilitated by the symmetry properties of the CSFs. The time reversal adapted CSF basis functions are given in terms of the... 

T is expanded in a basis of time-reversal adapted configuration state functions [8] (TRA-CSFs, x F)... 

It is useful to see how time reversal symmetry is reflected in [/(/) Recall that for the time reversal adapted crude adiabatic states basis functions the Kramers degenerate pairs are (1,3) and (2,4, that is, T j —... 

S and Mg and carry an irreducible representation A of the spatial point group. When the spin-orbit interaction is included in the Hamiltonian, it is convenient to also require that they be time-reversal adapted.A basis is time reversal adapted, provided TOa is in the basis whenever 0a is. Here T, the time reversal operator, is given by °° ° ... 

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. 

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