Kramers pairs 2-spinor basis

The DIRAC package [55,56], devised by Saue and collaborators, rather than exploiting the group theoretical properties of Dirac spherical 4-spinors as in BERTHA, treats each component in terms of a conventional quantum chemical basis of real-valued Cartesian functions. The approach used in DIRAC, building on earlier work by Rosch [80] for semi-empirical models, uses a quaternion matrix representation of one electron operators in a basis of Kramers pairs. The transformation properties of these matrices, analysed in [55], are used to build point group transformation properties into the Fock matrix. 

The basic theory of second quantization is found in most advanced textbooks on quantum mechanics but inclusion of relativity is not often considered. A good introduction to this topic is given by Strange [10] in his recent textbook on relativistic quantum mechanics. We will basically follow his arguments but make the additional assumption that a finite basis of Im Kramers paired 4-spinors is used to expand the Dirac equation. This brings the formalism closer to quantum chemistry where use of an (infinite) basis of plane waves, as is done in traditional introductions to the subject, is impractical. 

Restriction to Kramers-paired basis spinors gives... 

Simple open shell cases may also be treated via this kind of perturbation theory. The high spin case with one electron outside a closed shell is of course easy when an unrestricted formalism is used. Dyall also worked out equations for the restricted HE formalism and the more complicated case of two electrons in two Kramers pairs outside a closed shell [32]. Also in this method the crucial step remains the efficient formation of two-electron integrals in the molecular spinor basis. 

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. 

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. 

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... 

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 ... 

We can estimate the increased relative cost of relativistic integral transformations over nonrelativistic integral transformations as follows. For this purpose we assume that the integral transformation is performed with one triangular pair index and one square pair index to exploit the efficiency of matrix multiplications. We also assume a nonrelativistic basis of size n that matches the relativistic 2-spinor Kramers-pair basis. 

Open-shell coupled-cluster theory can be formulated in terms of the one-particle spinor or spin-orbital basis. However, spin-restricted or Kramers-restricted open-shell theories are complicated by the ambiguous role of the open-shell orbital or Kramers pair. We develop here the basic outline of open-shell Kramers-restricted coupled-cluster theory. 

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