Positive-energy spinors

If one constructs the projection operators from the same independent particle basis as is used in the expansion of then the effect is simply to limit the expansion of to include only two-electron configurations involving positive energy spinors. Defining bj, to be a creation operator for the single-particle positive-energy state, the equation... 

Here, the summation extends over both negative and positive energy spinors. Pnn > ) and Qr if) are the large and small radial components and are expanded in G spinors, x il and xfi, that satisfy the boundary conditions associated with the finite nucleus [Ishikawa et al. (1997b)),... 

Many-electron wave functions correct to oi may be expanded in a set of CSFs that spans the entire N-electron positive-energy space j (7/J 7r), constructed in terms of Dirac one-electron spinors. Individual CSFs are eigenfimctions of the total angular momentum and parity operators and are linear combinations of antisymmetrized products of positive-energy spinors (g D(+ ). The one-electron spinors are mutually orthogonal so the CSFs / (7/J 7r) are mutually orthogonal. The un-... 

Because we have restricted the iV-particle space to states composed of positive-energy spinors, the transformation reduces the number of integrals by a factor of 4. Just as in the nonrelativistic case, the transformation over the spinor indices can be performed as a sequence of four matrix multiplications, and as a result the cost of the integral transformation scales as n , where n is the numbCT of basis functions. 

If p is a positive-energy spinor, this matrix element is O(c ), and the contribution from the sum over positive-energy states is also O(c ). If is a negative-energy spinor, this matrix element is and the contribution from the sum over negative-energy... 

Before doing so we note a certain peculiarity of negative energy spinors. Let (p,a) be a positive energy solution of the Dirac equation corresponding to helicity s so that... 

Here frs and (ri-l tM> are, respectively, elements of one-electron Dirac-Fock and antisymmetrized two-electron Coulomb-Breit interaction matrices over Dirac four-component spinors. The effect of the projection operators is now taken over by the normal ordering, denoted by the curly braces in (15), which requires annihilation operators to be moved to the right of creation operators as if all anticommutation relations vanish. The Fermi level is set at the top of the highest occupied positive-energy state, and the negative-energy states are ignored. 

In this notation the presence of two upper and two lower components of the four-component Dirac spinor fa is emphasized. For solutions with positive energy and weak potentials, the latter is suppressed by a factor 1 /c2 with respect to the former, and therefore commonly dubbed the small component fa, as opposed to the large component fa. While a Hamiltonian for a many-electron system like an atom or a molecule requires an electron interaction term (in the simplest form we add the Coulomb interaction and obtain the Dirac-Coulomb-Breit Hamiltonian see Chapter 2), we focus here on the one-electron operator and discuss how it may be transformed to two components in order to integrate out the degrees of freedom of the charge-conjugated particle, which we do not want to consider explicitly. 

The subspace of positive energies does not contain strictly localized spinors. There is no wave function that vanishes everywhere in an open region of space. All positive-energy wave packets are essentially spread over all of space. Still, there are wave packets which are approximately localized in the same sense as a Gaussian wave packet, i.e., they vanish faster than any inverse power of x, as x goes to infinity. Examples of such Gaussian-type wave packets are in Figures 3 and 4. 

The Dirac equation represents a proper relativistic form of the characteristic equation for energy. It is fulfilled for state vectors in a form of a four-component spinor. Since the upper two-component spinor dominates in the positive energy solutions for an electron, a decomposition of the Dirac equation is appropriate. 

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