Yoshihiro Watanabe

Relativistic X-ray scattering factors are reported for a broad and systematically defined set of chemically relevant atomic anions, including mono- and multivalent species, for elements with atomic numbers Z = 1–85 [Greenwood & Earnshaw (1997). Chemistry of the Elements ]. Dirac–Hartree–Fock wavefunctions were generated using the DFRATOM code [Matsuoka & Watanabe (2001). Comput. Phys. Commun. 139 , 218–234], with soft radial confinement applied where necessary to stabilize diffuse anionic states. The influence of this confinement on the resulting X-ray scattering factors is explicitly and systematically analyzed to ensure that the reported data reflect physically meaningful electronic structure trends rather than artifacts of the stabilization procedure. The quality of the resulting wavefunctions was assessed using previously reported high-quality benchmark energies [Visscher & Dyall (1997). At. Data. Nucl. Tables 67 , 207–224], radial metrics, and direct comparison of X-ray scattering factors with established reference data for neutral atoms and several monovalent anions [Rez et al. (1994). Acta Cryst . A 50 , 481–497; Su & Coppens (1997). Acta Cryst. A 53 , 749–762; Macchi & Coppens (2001). Acta Cryst. A 57 , 656–662]. To ensure physically meaningful results, an X-ray scattering based perturbation metric was introduced to quantify the impact of confinement on the calculated X-ray scattering factors, together with additional high-sin θ/λ consistency checks. The resulting X-ray scattering factors were computed and tabulated in a format consistent with that used in the International Tables for Crystallography [Maslen et al. (2006). Vol. C, Section 6.1.1, pp. 554–589], facilitating direct crystallographic use. Analysis of representative main-group, transition-metal and heavy p -block elements demonstrates that electron attachment leads to a systematic enhancement of the X-ray scattering factors at small momentum transfer, governed primarily by the expansion of the valence electron density, while the high-sin θ/λ region remains dominated by largely invariant core electron contributions. The data presented here provide the first broad, internally consistent reference set of fully relativistic X-ray scattering factors spanning a wide range of multivalent atomic anions of chemical and crystallographic interest, and represent a natural extension of the studies by Olukayode et al. [ Acta Cryst. (2023), A 79 , 59–79; Acta Cryst. (2023), A 79 , 229–245].

Fourier-transform microwave spectroscopy has been applied for the 13C/18O-substituted tropolone to observe tunneling-rotation transitions as well as pure rotational transitions. The tunneling-rotation transitions were observed between the 13C-4 and -6 forms as well as between 13C-3 and -7, between 13C-1 and -2, and between 18O-8 and -9 (we denote these tunneling pairs as 13C-46, etc., below) although they have an asymmetric tunneling potential due to the difference in the zero point energy (ZPE). From the observed tunneling splittings ΔEij (0.9800–1.6824 cm−1), the differences in ZPE Δij for the 13C-46, -37, -12, and 18O-89 species are derived to be −0.1104, 0.5652, −1.3682, and 1.3897 cm−1 to agree well with the DFT calculation. The state mixing ratio of the tunneling states decreases drastically from (44%:56%) to (8.7%:91.3%) for 13C-46 and 18O-89 with an increase in the asymmetry Δij of the tunneling potential function. The observed tunneling-rotation interaction constants Fij decrease from 16.001 to 9.224 cm−1 as the differences in ZPE Δij increase, while the diagonal tunneling-rotation interaction constants Fu increase from 1.767 to 13.70 cm−1, explained well by the mixing ratio of the tunneling states.

In response to Professor Liu’s Comment1 on our recently published paper,2 this article provides our responses to the raised questions. The Comment poses several In response to Professor Liu’s Comment1 on our recently published paper,2 this article provides our responses to the raised questions. The Comment poses several

The generalized Foldy-Wouthuysen (GFW) transformation was proposed as a generic form that unifies four types of transformations in relativistic two-component methods: unnormalized GFW(UN), and normalized form 1, form 2, and form 3 (GFW(N1), GFW(N2), and GFW(N3)). The GFW transformation covers a wide range of transformations beyond the simple unitary transformation of the Dirac Hamiltonian, allowing for the systematic classification of all existing two-component methods. New two-component methods were also systematically derived based on the GFW transformation. These various two-component methods were applied to hydrogen-like and helium-like ions. Numerical errors in energy were evaluated and classified into four types: the one-electron Hamiltonian approximation, the two-electron operator approximation, the newly defined "picture difference error (PDE)," and the error in determining the transformation, and errors in multi-electron systems were discussed based on this classification.

Theoretical discussions are given on issues in relativistic molecular orbital theory to which the quantum electrodynamics (QED) Hamiltonian is applied. First, several QED Hamiltonians previously proposed are sifted by the orbital rotation invariance, the charge conjugation and time reversal invariance, and the nonrelativistic limit. The discussion on orbital rotation invariance shows that orbitals giving a stationary point of total energy should be adopted for QED Hamiltonians that are not orbital rotation invariant. A new total energy expression is then proposed, in which a counter term corresponding to the energy of the polarized vacuum is subtracted from the total energy. This expression prevents the possibility of total energy divergence due to electron correlations, stemming from the fact that the QED Hamiltonian does not conserve the number of particles. Finally, based on the Hamiltonian and energy expression, the Dirac–Hartree–Fock (DHF) and electron correlation methods are reintroduced. The QED-based DHF equation is shown to give information on positrons from negative-energy orbitals while having the same form as the conventional DHF equation. Three electron correlation methods are derived: the QED-based configuration interactions and single- and multireference perturbation methods. Numerical calculations show that the total energy of the QED Hamiltonian indeed diverged and that the counter term is effective in avoiding the divergence. The relativistic molecular orbital theory presented in this article also provides a methodology for dealing with systems containing positrons based on the QED Hamiltonian.

A combined method of the Dirac–Hartree–Fock (DHF) method and the reference interaction‐site model (RISM) theory is reported; this is the initial implementation of the coupling of the four‐component relativistic electronic structure theory and an integral equation theory of molecular liquids. In the method, the DHF and RISM equations are solved self‐consistently, and therefore the electronic structure of the solute, including relativistic effects, and the solvation structure are determined simultaneously. The formulation is constructed based on the variational principle with respect to the Helmholtz energy, and analytic free energy gradients are also derived using the variational property. The method is applied to the iodine ion (I⁻), methyl iodide (CH₃I), and hydrogen chalcogenide (H₂X, where X = O–Po) in aqueous solutions, and the electronic structures of the solutes, as well as the solvation free energies and their component analysis, solvent distributions, and solute–solvent interactions, are discussed.

The applicability of density functional theory (DFT) and wave function theory combined with the three-dimensional reference interaction site model self-consistent field method to the d–d transitions of transition metal aqua complexes was examined using [Cr(H₂O)₆]₃₊ in aqueous solution as an example. DFTs with hybrid functionals, multiconfigurational self-consistent field followed by perturbation theory, and coupled-cluster singles and doubles (CCSD) followed by the equation of motion CCSD, gave reasonable d–d transition energies.

Ferric hexacyanoferrate (FeHCF) or Prussian blue (PB) exhibits selective alkali ion adsorption and has great potential for use in various applications. In the present work, alkali ion (Li⁺, Na⁺, K⁺, and Cs⁺) and water configurations in defective PB (d-PB) were studied by using the statistical mechanics of molecular liquids. The three-dimensional (3D) distribution functions of the ions and water were determined by solving the 3D-reference interaction site model (RISM) equation of systems of a unit lattice of d-PB in electrolyte solutions, i.e., LiCl, NaCl, KCl, and CsCl. The results show the difference in the ion–water configurations and distributions between small (Li⁺ and Na⁺) and large ions (K⁺ and Cs⁺). The adsorption sites of Li⁺ and Na⁺ are located off-center and lie on the diagonal axis. By contrast, the larger ions, K⁺ and Cs⁺, are adsorbed at the center of the unit cell. The degree of dehydration due to the adsorption of alkali ions indicates that there was no water exchange during Li⁺ and Na⁺ adsorption, whereas two and three water molecules were removed after adsorption of K⁺ or Cs⁺ in the unit cell.