The concept of energy density has been formulated in terms of stress tensor in general relativity. The spin vorticity of electron has been hidden in the energy-momentum tensor and plays a significant role in the dynamics of electron. The dynamics of electron spin is driven by the antisymmetric component of the stress tensor of electron through the vorticity. The symmetric component of the stress tensor of electron drives the tensorial energy density of chemical reactivity.
Isotope separation is important in industry, such as uranium enrichment. It is also very important in geochemistry and cosmo-chemistry because heavy and light isotopes are gradually fractionated in global equilibrium processes. For isotopes of heavy elements such as uranium and gadolinium, this fractionation occurs mainly owing to the difference of nuclear charge radii between the isotopes. This is because electronic states of isotopologues are different because of the difference of nuclear charge radii. Nuclear volume terms (lnKnv) in the equilibrium constants of isotope fractionations can be calculated from the electronic energy difference of isotopologues, obtained by relativistic quantum chemical calculations with finite-size nuclear models. In this review, we summarize our theoretical results in uranium isotope separations, using U (III)-U (IV) and U (IV)-U (VI) redox reaction systems. Calculated nuclear volume term (lnKnv) is reasonably close to the ones estimated from experimental analysis of temperature dependence in equilibrium constants. We used the four-component Dirac-Coulomb Hartree-Fock method with the Fermi and Gaussian-type nuclear models in our calculations. We also briefly mention how the nuclear volume effect is important in geochemistry or cosmo-chemistry referring to recent relevant works.
The thermodynamic properties and electronic structure of hydrated Ra2+ have been investigated using ab initio quantum chemical calculations that apply the relativistic model core potential method and compared with those of the other hydrated divalent alkaline earth metal ions (Mg2+, Ca2+, Sr2+, and Ba2+). The solvation free energies calculated for [Ra (H2O)n]2+ (n = 1–9) in a continuum dielectric media (semi-continuum model) showed that the hydration number of Ra2+ is in the range of 7–9. Natural population analysis (NPA), natural bond orbital (NBO) analysis and localized molecular orbital energy decomposition analysis (LMO-EDA) showed that the dominant interaction between hydrated Ra2+ ions and solvent water molecules is electrostatic interaction to form coordination bonds which have a strong ionic bond character. On the other hand, not only electrostatic interaction but also covalent interaction accompanying charge-transfer from solvent water molecules to the central ion are important in the interaction between hydrated Mg2+ or Ca2+ (lighter divalent alkaline earth metal ions) and solvent water molecules.