4.9.6 Scalar Relativistic Effects

Relativistic effects play a major role in several physical and chemical phenomena, such as the properties of heavy elements and the proper characterization of the most inner energy levels probed in X-Ray espectroscopy experiments. Solving the four component Dirac equation, which describes both electrons and its anti-particles (positrons), is computationally expensive. Since most chemical proceses can be explained by solely taking the electronic wavefunction into account, several ways of effectively decoupling the electronic and positronic degrees of freedom have been proposed.

The exact two-component (X2C) hamiltonian ⁵³⁰ Ilias M., Saue T.
J. Chem. Phys.
(2007), 126, pp. 064102. Link ^{, 747} Liu W., Peng D.
J. Chem. Phys.
(2009), 131, pp. 031104. Link ^{, 1052} Saue T.
ChemPhysChem
(2011), 12, pp. 3077. Link ^{, 722} Li Z., Xiao Y., Liu W.
J. Chem. Phys.
(2012), 137, pp. 154114. Link provides one route for achieving such decoupling. The method relies on solving the more tractable one electron four-component Dirac Hamiltonian in a restricted kinetic balance (RKB) ⁶⁵² Kutzelnigg W.
Int. J. Quantum Chem.
(1984), 25, pp. 107. Link form to obtain the decoupling unitary transformations that will be used to modify the one-electron matrix elements, such as the kinetic energy and nuclear-attraction, to account for relativistic effects. A key ingredient to the X2C transformation matrices is to compute

which is accomplished by noting that the the momentum operator is the generator of translations and its effects on a basis function can be captured by taking appropriate derivatives of such functions. It should be noted that, in order to properly capture the effects of the small components to the electronic wavefunction through X2C, decontracted basis sets are required. Full details of the finite difference X2C algorithm are provided in Ref.  252 Cunha L. A. et al.
J. Phys. Chem. Lett.
(2022), 13, pp. 3438. Link . An example on how to include scalar relativistic effects to model K-edge X-Ray spectroscopy can be found in Section 7.13.4.