The multiplet structure of core $2p$ -vacancy levels for 36 ions belonging to $3d$ transition metals have been calculated and their plots presented. Only the ground-state configuration $2p^{5}3d^n$ ($n=1,\dots ,9\mathrm{}$) is considered for each ion. The spin-orbit interaction has been incorporated exactly, but the crystal-field effect is ignored. While there is general agreement with the available experimental works, higher-energy resolution in x-ray-photoelectron measurements is necessary for detailed comparison of the spectra presented here. On the theoretical side inclusion of crystal field appears to be important. One should go beyond single-configuration approximation to have the observed satellite structure in $2p$ -x-ray-photoelectron spectra appear in the calculations.

The multiplet splittings of core $p$-vacancy levels have recently been observed in various transition-metal complexes and in some rare-earth as well as actinide complexes. This paper presents the calculation of the multiplet structure of core $p$-vacancy levels in the Hartree-Fock free-ion approximation. The spin-orbit and crystal-field effects have been included. The cases studied are ${\mathrm{Mn}}^{2+}$ in Mn${\mathrm{F}}_2$ and the free ion ${\mathrm{Fe}}^{3+}$. For $3p$ vacancies these effects appear to be unimportant from the point of view of photoelectron spectroscopy owing to the limited resolution of the technique. For $2p$ vacancies the spin-orbit interaction is comparable to the electrostatic interaction between the electrons and both the interactions must be considered simultaneously. The conclusion is that the x-ray photoelectron spectra of inner-core electrons are more suitable for chemical analysis than those of outer electrons which, in addition, contain large correlation and many-body effects.

Deep space observations of the JWST have revealed that the structure and masses of very early Universe galaxies at high redshifts ($z\sim15$), existing at $\sim$0.3 Gyr after the Big Bang, may be as evolved as the galaxies in existence for $\sim10$ Gyr. The JWST findings are thus in strong tension with the ${\rm{\Lambda}}$CDM cosmological model. While tired light (TL) models have been shown to comply with the JWST angular galaxy size data, they cannot satisfactorily explain isotropy of the cosmic microwave background (CMB) observations or fit the supernovae distance modulus versus redshift data well. We have developed hybrid models that include the tired light concept in the expanding universe. The hybrid ${\rm{\Lambda}}$CDM model fits the supernovae type 1a data well but not the JWST observations. We present a model with covarying coupling constants (CCC), starting from the modified FLRW metric and resulting Einstein and Friedmann equations, and a CCC + TL hybrid model. They fit the Pantheon + data admirably, and the CCC + TL model is compliant with the JWST observations. It stretches the age of the Universe to 26.7 Gyr with 5.8 Gyr at $z = 10$ and 3.5 Gyr at $z = 20$, giving enough time to form massive galaxies. It thus resolves the ‘impossible early galaxy’ problem without requiring the existence of primordial black hole seeds or modified power spectrum, rapid formation of massive population III stars, and super Eddington accretion rates. One could infer the CCC model as an extension of the ${\rm{\Lambda}}$CDM model with a dynamic cosmological constant.

Sternheimer shielding-antishielding parameters pertinent to the quadrupole perturbations outside an ion ($\lambda$), as well as within the ion owing to an unfilled 4f shell ($R$), have been calculated for rare-earth ions ${\mathrm{La}}^{3+}$, ${\mathrm{Ce}}^{3+}$, ${\mathrm{Pr}}^{3+}$, ${\mathrm{Nd}}^{3+}$, ${\mathrm{Pm}}^{3+}$, ${\mathrm{Sm}}^{3+}$, ${\mathrm{Eu}}^{3+}$, ${\mathrm{Gd}}^{3+}$, ${\mathrm{Tb}}^{3+}$, ${\mathrm{Dy}}^{3+}$, ${\mathrm{Ho}}^{3+}$, ${\mathrm{Er}}^{3+}$, ${\mathrm{Tm}}^{3+}$, ${\mathrm{Yb}}^{3+}$, ${\mathrm{Lu}}^{3+}$, ${\mathrm{Sm}}^{2+}$, and ${\mathrm{Eu}}^{2+}$. Hartree—Fock—Slater-type ionic wave functions have been used. The value of ${\lambda }_0$ varies smoothly from -65.82 for ${\mathrm{La}}^{3+}$ to -58.49 for ${\mathrm{Yb}}^{3+}$, whereas that of $R_0$ (without exchange) varies from 0.081 for ${\mathrm{Ce}}^{3+}$ to 0.128 for ${\mathrm{Yb}}^{3+}$. The calculation of Sternheimer parameters for ${\mathrm{Sm}}^{2+}$ and ${\mathrm{Eu}}^{2+}$ with ${\lambda }_0=-73.28\mathrm{}$ and $R_0=0.124\mathrm{}$ for ${\mathrm{Sm}}^{2+}$ shows that their dependence on the charge of the ions is appreciable. Sternheimer parameters are reported also for the core-electron sites.

The Sternheimer shielding-antishielding parameters are reported for 77 ions in $3d$, $4d$, $5d$, and $5f$ groups, as well as for gold and lithium atoms. The parameters $\lambda$ and $R$, which take into account, respectively, the effect of the core electrons on the quadrupole perturbing potential outside the ion and within the ion (owing to unfilled valence-shell electrons), have been calculated for varying ionic charge and varying atomic number in each group. The electronic wave functions used throughout the work are of the nonrelativistic Hartree-Fock-Slater type. Interesting differences of the results in each group are seen to be the values of Sternheimer parameters and ${\alpha }_q$ for ions with and without the closed-shell configurations. Also, for the lithium atom with $1s^{2}2p^1$ configuration we obtained $1-{\lambda }_0=3.46\mathrm{}$, assuming $2p^1$-shell perturbation to be one-sixth that of the closed $p$ shell. This value, incidentally, is the same as that measured recently by Anderson and Karra for the lithium ion (${\mathrm{Li}}^{+}$) in LiF. For gold atoms the configurations $5d^{10}6p^1$ and $5d^{10}6s^1$ were considered and we obtained the ratios of atomic core $5p_{\frac{3}{2}} \mathrm{to} 4p_{\frac{3}{2}}$ energy-level splittings as 1.6 and 1.7, respectively, against approximately 2 determined experimentally by Novakov and Hollander for Au in gold-cyanide compounds. The significance of these calculations is emphasized by the possibility of experimentally determining the quadrupole splittings of the atomic core by high-resolution electron spectroscopy.

We have shown that the varying physical constant model is consistent with the recently published variational approach wherein Einstein equations are modified to include the variation of the speed of light c, gravitational constant G, and cosmological constant Λ using the Einstein–Hilbert action. The general constraint resulting from satisfying the local conservation laws and contracted Bianchi identities provides the freedom to choose the form of the variation of the constants as well as how their variations are related. When we choose |${\dot{G}}/G = 3\,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{\dot{c}} /c,\,c = {c_0}\,{\rm{exp}}\,[({a^\alpha} - 1)],\,G = {G_0}\,{\rm{exp}}\,[3({a^\alpha} - 1)]$|⁠, and |${\rm{\Lambda }} = {{\rm{\Lambda }}_0}\ \exp [ {( {{a^{ - \alpha }} - 1} )} ]$|⁠, where a is the scale factor and α = 1.8, we are able to show that the resulting model: (a) fits the supernova 1a observational data marginally better than the Lambda cold dark matter (ΛCDM) model; (b) determines the first peak in the power spectrum of the cosmic microwave background temperature anisotropies at a multipole value of |$l = 217.3$|⁠; (c) calculates the age of the Universe as 14.1 Gyr; and (d) finds the BAO acoustic scale to be 145.2 Mpc. These numbers are within less than 3 per cent of the values derived using the ΛCDM model. Surprisingly, we find that the dark-energy density is negative in a Universe that has significant negative curvature and whose expansion is accelerating at a faster rate than that predicted by the ΛCDM model.