As a continuation of Part I, dedicated to the ground state of He-like and Li-like isoelectronic sequences for nuclear charges $Z \leq 20$, and Part II, dedicated to two excited states of He-like sequence, two ultra-compact wave functions in the form of generalized Guevara-Harris-Turbiner functions are constructed for Li-like sequence. They describe accurately the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC) for energies (2-3 significant digits (s.d.)) of the spin-quartet state $1^40^+$ of Li-like ions (in static approximation with point-like, infinitely heavy nuclei). Variational parameters are fitted in $Z$ by 2nd degree polynomials. The most accurate ultra-compact function leads to the absolute accuracy $\sim 10^{-3}$\,a.u. for energy, and $\sim 10^{-4}$ for the normalized electron-nuclear cusp parameter for $Z \leq 20$. Critical charge $Z=Z_B$, where the ultra-compact trial function for the $1^40^+$ state looses its square-integrability, is estimated, $Z_B(1^4\,0^+) \sim 1.26 - 1.30$. As a complement to Part I, square integrability for the compact functions constructed for the {\it ground, spin-doublet state} $1^2\,0^+$ of the Li-like sequence is discussed. The critical charge, for which these functions stop to be normalizable, is estimated as $Z_B( 1^2\,0^+) = 1.62 - 1.65$. It implies that at $Z=2$ - the negative helium ion He$^-$ - both states $1^2\,0^+$ and $1^4\,0^+$ exist as states embedded to continuum.

The Jacobi polynomials $\hat{P}_n^{(\alpha,\beta)}(x)$ conform the canonical family of hypergeometric orthogonal polynomials (HOPs) with the two-parameter weight function $(1-x)^\alpha (1+x)^\beta, \alpha,\beta>-1,$ on the interval $[-1,+1]$. The spreading of its associated probability density (i.e., the Rakhmanov density) over the orthogonality support has been quantified, beyond the dispersion measures (moments around the origin, variance), by the algebraic $\mathfrak{L}_{q}$-norms (Shannon and R\’enyi entropies) and the monotonic complexity-like measures of Cram\’er-Rao, Fisher-Shannon and LMC (L\’opez-Ruiz, Mancini and Calbet) types. These quantities, however, have been often determined in an analytically highbrow, non-handy way; specially when the degree or the parameters $(\alpha,\beta)$ are large. In this work, we determine in a simple, compact form the leading term of the entropic and complexity-like properties of the Jacobi polynomials in the two extreme situations: ($n\rightarrow \infty$; fixed $\alpha,\beta$) and ($\alpha\rightarrow \infty$; fixed $n,\beta$). These two asymptotics are relevant \textit{per se} and because they control the physical entropy and complexity measures of the high energy (Rydberg) and high dimensional (pseudoclassical) states of many exactly, conditional exactly and quasi-exactly solvable quantum-mechanical potentials which model numerous atomic and molecular systems.

Phenylenes network is applied in several fields of chemistry sciences due to its advantages compared to other several columnar networks, recently. This paper aims to introduce a kind of networks which obtained by a family of dicyclobutadieno derivative of linear phenylene chain Ln which is made up of n hexagons and (n+1) quadrangles. Let L2n be the strong prism of the dicyclobutadieno derivative of linear phenylenes Ln. By taking full advantage of the knowleges about the normalized Laplacian spectra, we induce the explicit expressions, with respect to the index n, of the multiplicative degree-Kirchhoff index and the number of spanning tree based on the graph L2n.

Metadynamics simulation has been used to describe the conformational energy landscapes of several helical quinoline oligoamides bearing β-pinene-derived pyridine at either the C or N terminus. Based on the experimental results, helix-sense preference for four types of foldamers with the chiral terminal group has been verified. To compare to the key factors of inducing handedness to helical-sense preference, a terminal group with three hydrogen bond sites is designed and corresponding foldamers are built. The calculated results show the delocalization effect and steric hindrance mainly responsible for a particular helix-sense preference for the investigated foldamers. The more hydrogen bonds between the terminal group and oligoamide units are formed, the more stable foldamers are.

As a continuation of Part I (Int. Journal of Quantum Chem. 2021; 121: qua.26586), dedicated to the ground state of He-like and Li-like isoelectronic sequences for nuclear charges Z ≤ 20, a few ultra-compact wave functions in the form of generalized Hylleraas-Kinoshita functions are constructed, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC) for energies (4-5 significant digits (s.d.)) of two excited states of He-like ions: the spin-singlet (first) excited state 2¹S and for lowest spin-triplet 1³S state. For both states it provides absolute accuracy for energy ∼10−3a.u., exact values for cusp parameters and also for 6 expectation values the relative accuracy ∼10−2. Bressanini-Reynolds observation about the special form of nodal surface of 2¹S state for Helium is confirmed and extended to ions with Z > 2. Critical charges Z = ZB, where ultra-compact trial functions loose their square-integrability, are estimated: ZB(1¹S)≈ZB(2¹S)∼0.905 and ZB(1³S)∼0.902. For both states the Majorana formula - the energy as the second degree polynomial in Z - provides accurately the 4-5 significant digits for Z ≤ 20.

The efficient computation of molecular integrals and their derivatives is a crucial step in molecular property evaluation in modern quantum chemistry. As an integral tensor decomposition technique, the density-fitting (DF) approach becomes a popular tool to reduce the memory and disk requirements for the electron repulsion integrals. In this study, an application programming interface (API) framework, denoted (MFW), for the computation of molecular integrals and their first derivatives, over contracted Gaussian functions, for the density-fitted methods is reported. The MFW is free software and it includes overlap, dipole, kinetic, potential, metric, and 3-index integrals, and their first derivatives. Furthermore, the MFW provides a smooth approach to build the Fock matrix and evaluate analytic gradients for the density-fitted methods. The MFW is a C++/Fortran hybrid code, which can take advantage of shared-memory parallel programming techniques. Our results demonstrate that the MFW is an efficient and user-friendly API for the computation of molecular integrals and their first derivatives.