Operators Kramers-pair creation

Let R be an element of the group of symmetry operations which leave the one-electron Hamiltonian operator h invariant. Then 

If (f i, is a symmetry-adapted eigenfunction of h belonging to a due-degenerate set of states y, then 

These relationships can also be expressed in second-quantized form [4-8] by introducing Fermion creation and annihilation operators, which obey the anticommutation relations 

In this notation, the one-electron spin-orbital I pk) becomes 

In second-quantized notation, the relationship shown in equation (30) becomes 

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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. 

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... 

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... 

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

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] ... 

The two products of creation operators are in nonrelativistic theory termed a and p strings. To avoid confusion with these operators, but also to retain the analogy between Kramers pairs and spin-orbital pairs, we will term these A and B strings ... 

The subscripts I and J on the strings are indices of the particular set of Kramers pairs from which the creation operators that make up the string are taken. The spinors that make up the Kramers pairs can, in this notation, be labeled A and B spinors. The many-particle state can now be written... 

For a closed-shell system, it is often convenient to reorder the creation operators so that the operators for each Kramers pair are together. Thus, we define a Kramers-pair... 

It may be thought that quaternions could be used in the creation and annihilation operators to define a new basis in which all matrices were block-diagonal in the Kramers pairs. However, because of the noncommutative algebra, the step in which the creation operator is permuted over the matrix element to separate the two does not produce the desired result. Therefore, quaternions are useful only at the matrix algebra stage, and not in the formalism. 

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