Kramer pairs

Fig. 8. The torsional potential and energy levels of a methyl-like rotor. The feasible group is isomorphic with C3. The three minima of the potential correspond to the three equilibrium orientations of the rotor in its molecular/crystal surrounding. The torsional levels come in triplets whose individual components transform according to the irreps A, Ea, and E, of C3. The E sublevels come in perfectly degenerate Kramers pairs those of A symmetry are shifted in energy from the Kramers sublevels by the tunnelling quanta the magnitudes of which rapidly grow with...

Presently, it will be a concern to review the basics of crystal field theory as a vehicle to understand the electronic features of transition metal atoms and ions in an octahedral environment. Thus is considered the limited basis of ten spinorbitals of the partially occupied atomic d-shell for the relevant transition metal. A particular choice of basis is made in order to obtain a convenient form for the spin-orbital interaction and to simplify the application of the point group symmetry. The e-type orbitals spin factors, a basis for the four-dimensional irreducible representation U. The Kramers pairs will be used ... 

The examination of the role of the two-electron term in the Hamiltonian shows that elements of the many-electron basis with occupied Kramers pairs of spin orbitals generally will have a larger energy than others and that the ground state configuration conforms with Hund s rule. Kramers pairs are related to the Racah seniority approximate quantum number. The rotationally invariant geminal creator... 

The theory of symmetry-preserving Kramers pair creation operators is reviewed and formulas for applying these operators to configuration interaction calculations are derived. A new and more general type of symmetry-preserving pair creation operator is proposed and shown to commute with the total spin operator and with all of the symmetry operations which leave the core Hamiltonian of a many-electron system invariant. The theory is extended to cases where orthonormality of orbitals of different configurations cannot be assumed. 

Configuration interaction using Kramers pair creation operators 193... 

In the present paper, we shall discuss a method for generating many-electron states of a given symmetry using Kramers pair creation operators and other symmetry-preserving pair creation and annihilation operators. We will first develop the formalism for the case where orthonormality between the orbitals of different configurations can be assumed. Afterwards we will extend the method to cases where this orthonormality is lost, so that the method also can be used in generalized Sturmian calculations [11-13] and in valence bond calculations. 

Thus, the Kramers pair creation and annihilation operators defined by equations (36) and (37) preserve the symmetry of the states on which they act. 

Using the anticommutation relations (32), we can obtain the following commutation relations for the Kramers pair creation and annihilation operators [8] ... 

As an example of the symmetry-preserving Kramers pair creation operators, we can think of the case of D3 symmetry, where they have the form [8] ... 

As a second example, we can think of the case where the one-electron Hamiltonian has spherical symmetry. Then the Kramers pair creation operator corresponding to the shell n and subshell l is given by... 

The commutation relations (42) can be used to normalize the A-electron states obtained by acting on (A — 2)-electron states with Kramers pair creation operators [8], Suppose that P) is a properly normalized (N — 2)-electron state which is annihilated by By, i.e., suppose that... 

This last relationship can be used to normalize the states obtained by creating successively larger numbers of Kramers pairs in a subshell until no more states with the symmetry and seniority of the parent state can be created by further filling of the sub shell. 

CONFIGURATION INTERACTION USING KRAMERS PAIR CREATION OPERATORS... 

In other words, when a Kramers pair creation operator acts on an (N — 2)-electron state 1,4). which is an eigenfunction of the core Hamiltonian, it produces an /V-electron state which is also an eigenfunction of //, with an eigenvalue increased... 

From the argument given above, it can also be seen that the Kramers pair creation operators B t commute with S, since w nl = -JlB v... 

The Dirac spinors are said to constitute a Kramers pair of 4-spinors. If... 

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. 

The Kramers-restricted form of the Hamiltonian that was used in Cl theory is not suitable for Coupled Cluster theory because it mixes excitation and deexcitation operators. One possibility is to define another set of excitation operators that keep the Kramers pairing and use these in the exponential parametrization of the wavefunction. This would automatically give Kramers-restricted CC equations upon rederivation of the energy and amplitude equations. A more pedestrian but simpler alternative is to start from the spin-orbital formulation and inspect the relations that follow from the Kramers relation of the two-electron integrals. This method does also readily give the relations between the Kramers symmetry-related amplitudes. We will briefly discuss the basic steps in this approach, a detailed description of a possible algorithm is given in reference [47],... 

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... 

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... 

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