Positronic spinors

Aucar et al demonstrated, by means of a four-component relativistic calculation, that the origin of the diamagnetic contribution to any magnetic molecular property is due to contributions from positronic spinors in calculating the response of the system. Several approximations for the calculation of the DSO term were also investigated. As example, the DSO term for the chalcogen hydrides, XH2 (X = O, S, Se and Te), were calculated. 

The Dirac equation (7) can be considered as an equation for the components of the classical electron-positron field >I a(a ), 4 c(a ) (a is the spinor index). The Lagrangian for this classical field can be constructed as ... 

The transformation of the Dirac Hamiltonian to two-component form is accompanied by a corresponding reduction of the wavefunction. As discussed in detail in section 2, the four-component Dirac spinor will have only two nonvanishing components, as soon as the complete decoupling of the electronic and positronic degrees of freedom is achieved, and can thus be used as a two-component spinor. This feature can be exploited to calculate expectation values of operators in an efficient manner. However, this procedure requires that some precautions need to be taken care of with respect to the representation of the operators, i.e., their transition from the original (4 x 4)-matrix representation (often referred to as the Dirac picture) to a suitable two-component Pauli repre-... 

The first step consists in a decoupling of electronic from positronic states by means of the not norm-conserving transformation Wa of Eq. (148). At the same step we project to electronic states, since we are only interested in these, i.e. we perform a projecting transformation that transforms from a 4-component spinor space to a 2-component spinor space. 

The Dirac-Kohn-Sham equation in Eq. (6.68) is solved using the above elements. However, the most stable electronic states are not obtained by directly solving this equation, because the variational principle for electronic states is not established due to the contribution of positronic states. To solve this problem, the large-component and small-component basis spinor functions are balanced using... 

The optimization with respect to the spinors can be accomplished by obeying the minimax principle, and positronic energy states are allowed to relax in this correlation method (as in Dirac-Hartree-Fock-Roothaan calculations)... 

---