Comparison of restricted, unrestricted, inverse, and dual kinetic balances for four-component relativistic calculations

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Various kinetic balances for constructing appropriate basis sets in four-component relativistic calculations are examined in great detail. These include the well-known restricted (RKB) and unrestricted (UKB) kinetic balances, the less-known dual kinetic balance (DKB) as well as the unknown inverse kinetic balance (IKB). The RKB and IKB are complementary to each other: The former is good for positive-energy states, whereas the latter good for negative-energy states. The DKB combines the good of both RKB and IKB and even provides full variational safety. However, such an advantage is largely offset by its complicated nature. The UKB does not offer any particular advantages as well. Overall, the RKB is the simplest ansatz. Although the negative-energy states by a finite RKB basis are in error of O(c0), there is no objection to using them as intermediates for a sum-over-states formulation of perturbation theory, provided that the magnetic balance is also incorporated in the case of magnetic properties. In particular, the RKB is also an essential ingredient for formulating two-component relativistic theories, while all the others are simply incompatible. As such, the RKB should be regarded as the cornerstone of relativistic electronic structure calculations.