The Coulomb interaction of electron pair in $\textbf{r}_i$ of neighbour ions at $\textbf{R}_j$ within the tight-binding representation comes either to the covalent-bond energy ($\Gamma$) or to the hopping integral ($t$) that is calculated for two-dimensional systems such as fullerenes (FUL) and carbon nanotubes (CNT). The theory of semiconductor (s/c) systems leads to the dependence of bond energy $\Gamma(T)$ on temperature $T$, which is caused by the chemical (covalent) bond fluctuations (CBF). The calculated s/c characteristics (forbidden-band width $E_g(T)$, effective mass of electrons $m_e^*$ and holes ($|m_h^*| > m_e^*$), electrical resistance (ER), etc.) depend on the set of parameters – $\Gamma$ and CBF. The catalytic properties of FUL and CNT, the hydrogen accumulation into them, the work function are expressed through $\Gamma(T)$. The calculation precision is conditioned by the definiteness of introduction of the many-electron operator spinors (MEOS) in the Fock spaces. Fe atomic and magnetic diagrams are caused by the competition of band and covalent (within the MEOS representation) $3d$−$t_{2g}$- and $e_g$-electrons. Antibonding electrons of neighbour sites in $\gamma$-Fe form antiferromagnetic (AFM) order. Its Neel temperature $T_N < 10^2$ K is determined by the AFM exchange ($A_{ex}^* \sim \Gamma^e$) and rises when particle-size is decreased (in nanoparticles, $T_N > 10^2$ K) that is explained by the $|\Gamma_e|$ increase on a particle surface. At $T < T_M$, the $t_{2g}$-electrons’ localization in the covalent state with $S_r$ spin at the $r$ site leads to the ferromagnetic (FM) order of $\alpha$-Fe when $T < T_c \sim A_{ex}^t(T)$. The dependence $T_M(B)$ on magnetic field $B$ (even in the presence of FM phase) is strengthened by the competition of band and covalent energies of $t_{2g}$-electrons. The observed nonlinearities for magnetization $M^2(T)$ and susceptibility $\chi^{−1}(T)$ are interpreted as the effect of $A_{ex}^t(T)$ renormalization by CBF spectrum. The theory of ferroelectric (FE) phase of dielectrics within the deformation FE model interprets the square-law dependence of FE polarization $P(T)$ and its jump $Р(Т_с) \sim Р(0)$ in the first-kind transition points $T_c$ for BaTiO$_3$ and other FE crystals.