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# Dirac‐exact relativistic methods: the normalized elimination of the small component method

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Dirac‐exact relativistic methods, i.e., 2‐ or 1‐component methods which exactly reproduce the one‐electron energies of the original 4‐component Dirac method, have established a standard for reliable relativistic quantum chemical calculations targeting medium‐ and large‐sized molecules. Their development was initiated and facilitated in the late 1990s by Dyall's development of the normalized elimination of the small component (NESC). Dyall's work has fostered the conversion of NESC and related (later developed) methods into routinely used, multipurpose Dirac‐exact methods by which energies, first‐order, and second‐order properties can be calculated at computational costs, which are only slightly higher than those of nonrelativistic methods. This review summarizes the development of a generally applicable 1‐component NESC algorithm leading to the calculation of reliable energies, geometries, electron density distributions, electric moments, electric field gradients, hyperfine structure constants, contact densities and Mössbauer isomer shifts, nuclear quadrupole coupling constants, vibrational frequencies, infrared intensities, and static electric dipole polarizabilities. In addition, the derivation and computational possibilities of 2‐component NESC methods are discussed and their use for the calculation of spin‐orbit coupling (SOC) effects in connection with spin‐orbit splittings and SOC‐corrected energies are demonstrated. The impact of scalar relativistic and spin‐orbit effects on molecular properties is presented. WIREs Comput Mol Sci 2014, 4:436–467. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods
Schematic illustration of positive and negative eigenvalues obtained by DHF calculations. Besides the discrete electronic energy levels for bound states, there is a negative (positronic) and positive (electronic) continuum at −mc2 and +mc2, respectively (left side). To simplify a comparison with nonrelativistic calculations, the rest mass of the electron is subtracted in the D‐Hamiltonian, which corresponds to a shift by mc2 so that the discrete electron energies become negative (right side).
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Deviations (in percentage) of the 2‐component (2c) NESC/SOC/GHF and NESC/SOC(SNSO)/GHF spinor energy splittings from exact 4‐component DHF splittings given in the case Z = 120.
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Contourline diagram of the NESC difference density distribution of the HAt molecule. Solid red (dashed blue) contour lines indicate an increase (decrease) of the density because of scalar relativistic effects. The difference density adopts a shell structure at the At atom, which is caused by sp contraction. The outer red sphere corresponds to the 6s6p shell and and the most inner red sphere to the overlapping 3s3p and 2s2p shells. The 1 s region cannot be seen because it is just 0.1 Å outside the At nucleus. NESC/PBE0 calculations.
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