Abstract: We describe several harmonic functions; the solid harmonics (SH), the mixed SH, the reducing SH and the reducing mixed SH. We do the solid harmonic gradient (SHG) operator. Using the SHG, we derive a new Gauss-transform and a new Fourier-transform for real-type general Slater-type orbitals (STO). Several formulas can be derived for the calculus of harmonic functions. They will be useful especially for evaluating the angular part of molecular integrals over STOs and over Gaussian-type orbitals with higher-angular-momenta.

Keywords: Several harmonic functions; Molecular integrals; Solid harmonics; Higher-angular-momenta; Slater-type orbital; Solid harmonic gradient.

We have succeeded in improving the gain of the base quantum cascade laser (QCL) through experiments and device simulations using the nonequilibrium Green’s function (NEGF) method, and we investigate the factor that increases the gain by reducing the thickness of the barrier by 10%. Specifically, we analyzed the minibands (Wannier–Stark states), density of states (DOS), and electron density that contribute to the emission. The results show that the gain enhancement is due to the increase in the electron density of the quantum wells in the active region and the increase in the oscillator strength between the minibands that contribute to the emission. Quantum mechanical calculation like the NEGF method is very effective for mesoscopic systems such as the active layer of QCL.

The NMR chemical shifts for metal ions in aqueous solutions of inorganic salts often change with the solution concentration. These changes are referred to herein as dilution–concentration (dc) shifts. For main-group elements, the dc shifts generally increase with increasing atomic number. We calculated the dc shifts for various metal and nonmetal ions in aqueous nitrate solutions of elements in periods 5 and 6 by the DFT method based on the spin–orbit ZORA Hamiltonian. The calculation results revealed that the dc shifts of metal and nonmetal ions with the outermost shell electron configuration of 6s² are primarily determined by the changes in the spin–orbit interaction terms of the shielding constants. Furthermore, the spin densities of the Pb(II) ions in aqueous lead nitrate solutions under an external magnetic field were calculated using the matrix/modified Dirac–Kohn–Sham Hamiltonian, which confirmed the presence of Fermi contacts between the spin-polarized electrons and the lead nucleus. When a suitable threshold for visualizing the spin density was set, there was no Fermi contact in the infinite dilution state, whereas the upward-spin electron was in Fermi contact with the lead nucleus in the saturation state. There have been few reports of the NMR chemical shifts of heavy atoms in terms of Fermi contacts with relativistically spin-polarized electrons. We succeeded in clarifying how the dc shifts occur on the hydrated ions of heavy atoms by visualizing the Fermi contact of spin-polarized electrons with heavy-metal nuclei.

In this work, we performed reactive molecular dynamics-based sliding simulations of diamond-like carbon/Fe in the presence of H₂O and O₂ molecules to analyze tribochemical reaction processes and the atomic-scale wear mechanism. The atomic-scale wear amount in the O₂ environment model is smaller than that in the H₂O environment model. H₂O molecules adsorbed on the surface prevent adhesion between surfaces. However, they are ejected from the contact surface when the high contact pressure was applied, allowing the direct convex-to-convex contact. On the other hand, O₂ molecules reacted with the Fe surface, forming a chemically inert oxide layer, thereby leading to preventing atomic-scale adhesive wear.

A new method is proposed to translate molecule Lewis structures to machine-readable format data and a software based on Lewis structures has been developed to simulate the practice of chemists writing tautomerization mechanisms. For a tautomeric compound, from one Lewis formula entered by users, the software can generate its other equivalent tautomer (s), which share a common tautomerization intermediate stabilized by resonance. Not only atom balance but also electron balance are guaranteed in the tautomerism representation. The stereochemistry changes caused by tautomeric equilibrium can be tracked too. All the stereoisomers for the compounds containing tetrahedral carbon stereocenters and double bond stereocenters can be identified.