Three-dimensional Deformed Klein–Gordon Equation in the Presence of the Improved Deformed Generalized Deng–Fan with Improved Yukawa Potential Class in 3D-(R/NR)NCQs Symmetries: Eigensolution, HLM, and Composite Systems

In this paper, we examine the bound state solutions of the three-dimensional deformed Klein–Gordon equation (3D-DKGE) for an improved deformed Deng–Fan potential in generalized form along with an improved class Yukawa potential (I(DGDF-YCP)). The standard perturbation theory and Bopp’s shifts method are employed, utilizing a proper correspondence to the centrifugal potential term. We obtain new energy eigenvalues for all atomic quantum numbers ([Formula: see text] and [Formula: see text]) quantum states. In addition, we found that the new energy is related to the potential depths ([Formula: see text]), the adjustable constants ([Formula: see text]), the deformation parameter [Formula: see text] and the noncommutativity parameters ([Formula: see text], [Formula: see text], and [Formula: see text]). We noticed that the energy value reflects its known value in the literature (absence of deformation), in addition to an additional correction resulting from the effect of space properties, which clearly revealed the effect of spin–orbit and intrinsic magnetic effects resulting from the movement of electrons, as these effects appeared automatically. The analysis is performed on the nonrelativistic limit of new energy spectra under I(DGDF-YCP). By appropriately adjusting the combined potential parameters ([Formula: see text]), we analyze the obtained new bound state eigenvalues of the 3D-DKGE and three-dimensional deformed Schrödinger equation (3D-DSE) with the I(DGDF-YCP) in 3D-(R/NR)NCQs symmetries and obtain the modified Yukawa potential, the modified class of Yukawa potential, the modified Deng–Fan Yukawa potential model and the modified Deng–Fan potential model. The new analytical energy expression was used to predict the spectral masses ([Formula: see text] and [Formula: see text]) of the heavy-light mesons ([Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]) and heavy quarkonia such as charmonium [Formula: see text] and bottomonium [Formula: see text] in 3D-(NR)NCQs regimes. We found that the spectral masses consist of their fundamental terms ([Formula: see text] and [Formula: see text]) in the framework of 3D-(NR)Qm regimes plus the impact of spatial deformation. Within the framework of the 3D-(NR)NCQs regime, the homogenous diatomic molecules (H 2 , I 2 ); the heterogeneous diatomic molecules (CO, HCl, LiH); the neutral transition metal hydrides (ScH, TiH, VH, CrH); the transition-metal lithide CuLi; the transition-metal carbides (TiC, NiC); the transition metal nitrite ScN and the transition metal fluoride ScF under I(DGDF-YCP) under I(DGDF-YCP) models are examined.