The transformed Dirac equation

When equation (6) is used as an ansatz for the small components, the wave functions of the transformed and untransformed Dirac equations are related through 

By inserting equation (10) into the energy expression for the Dirac equation (1) one obtains the effective four-component Hamiltonian HDirac) 

To simplify the expression for the effective Hamiltonian (11) and (12), one can separate out a constant of one from /(r) as (/(r) — 1) + 1, where /(r) — 1 is describing the ansatz difference between the kinetic-energy balance condition and the general ansatz. By using the identity 

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Transformed Dirac equations are convenient starting points for the derivation of quasi-relativistic Hamiltonians. The transformed Dirac equations can be obtained by using approximate solutions for the small components as ansatze for the wave function. The ansatz can be deduced from the lower half of the Dirac equation by an approximate elimination of the small component. 

The zeroth-order regular approximation (ZORA) Hamiltonian can be derived from the upper part of the transformed Dirac equation (20). By using the ZORA ansatz for the small component (5) and assuming that the upper and the lower components are equal, the final ZORA equation for the upper component becomes... 

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