The Dirac operator in a matrix representation in a kinetically balanced basis is transformed to a quasirelativistic Hamiltonian matrix, that has the same electronic eigenstates as the original Dirac matrix. This transformation involves a matrix X, for which an exact identity is derived, and which can be constructed either in a noniterative way or by various iteration schemes, without requiring an expansion parameter. The convergence behavior of five different iteration schemes is studied numerically, with very promising results.

Two routes for deriving the exact two-component Hamiltonians are compared. In the first case, as already known, we start directly from the matrix representation of the Dirac operator in a restricted kinetically balanced basis and make a single block diagonalization. In the second case, not considered before, we start instead from the Foldy–Wouthuysen operator and make proper use of resolutions of the identity. The expressions are surprisingly different. It turns out that a mistake was made in the former formulation when going from the Dirac to the Schrödinger picture. The two formulations become equivalent after the mistake is corrected.

The basic ideas of relativistic quantum chemistry are highlighted, with the most important ingredients summarised as follows. (1) The restricted kinetic balance (RKB) condition, being both necessary and sufficient, serves as the cornerstone for the matrix representation of the Dirac-based Hamiltonian. (2) The concept of matrix transformation plays the key role in formulating two-component relativistic theories. Some popular ones, albeit presented as operators, are de facto matrix formulations in terms implicitly of the RKB condition. They merely make simple things complicated as a one-step block-diagonalization of the matrix Dirac equation can do the whole job. (3) The computational efficiency for both four- and two-component relativistic theories can be gained by means of the simple chemical idea of ‘from atoms to molecule’ without recourse to mathematical tricks. The two branches of relativistic theories have thus been made fully equivalent in all the aspects of simplicity, accuracy, and efficiency. It is concluded that the best relativistic electrons-only Hamiltonian has been found, which can be combined with any know-how correlation methods for electronic structure calculations of all the atoms in the Periodic Table. Most amazingly, the new quantum mechanical equation serves as a seamless bridge between the Schrödinger and Dirac equations. In short, the ‘relativity problem’ in chemistry has been solved.

The exact one-electron matrix quasirelativistic theory [Kutzelnigg and Liu, J. Chem. Phys. 123, 241102 (2005)] is extended to the effective one-particle Kohn-Sham scheme of density functional theory. Several variants of the resultant theory are discussed. Although they are in principle equivalent, consideration of computational efficiency strongly favors the one (F¯+) in which the effective potential remains untransformed. Further combined with the atomic approximation for the matrix X relating the small and large components of the Dirac spinors as well as a simple ansatz for correcting the two-electron picture change errors, a very elegant, accurate, and efficient infinite-order quasirelativistic approach is obtained, which is far simpler than all existing quasirelativistic theories and must hence be regarded as a breakthrough in relativistic quantum chemistry. In passing, it is also shown that the Dirac-Kohn-Sham scheme can be made as efficient as two-component approaches without compromising the accuracy. To demonstrate the performance of the new methods, atomic calculations on Hg and E117 are first carried out. The spectroscopic constants (bond length, vibrational frequency, and dissociation energy) of E1172 are then reported. All the results are in excellent agreement with those of the Dirac-Kohn-Sham calculations.