PSI - Issue 83

Alla V. Balueva et al. / Procedia Structural Integrity 83 (2026) 196–207

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is reduced the minimization of the quadratic functional to determine the true energy minimum, E min : ሬ ⃖ ௘௟௘௖ Ĥ ௘௟௘௖ ௘௟௘௖ ሺ ,⃗ݎ ሬ⃗ ሻ ൌ ௘௟௘௖ (1.9) The next step, DFT introduces the electron density function, ρ (r), which helps describe the energy distribution of electrons (1.9): ( r ) = ∑ ௜ ሺ ሻ ௝ ሺ ሻಿ మ ௜ (1.10) This allows to reduce 3N dimensional problem about the minimization of a quadratic functional (1.9), where N is the number of particles, to 3 dimensional problem of the minimization of the functional with respect to the density function, ρ (r), which is itself is pretty sophisticated problem, where as a 3N problem is actually the task, which is not available for modern computers. Thus, the Hamiltonian equation (1.9), which governs the total energy of electrons, can be rewritten in terms of ρ (r) this way: ሾρሿൌܶ ሾρሿ ൅ ׬ ௘௫௧ ሺ ሻρሺ ሻd ൅ ଵ ଶ ׬ ׬ ᇱ ஡ሺ ሻ஡൫ ᇲ ൯ ⌈ ି ᇲ ⌉ ൅ ௫௖ [ ] (1.11) Now, this leads us to the Hohenberg - Kohn theorems, which state that the density ρ (r), that delivers the minimum of a quadratic functional (1.11), gives us a true position of the nuclei in the ground state and a true ground state energy of the system. These ground state atom’s configurations and ground state energies are unique and contain all the information of the system. Specifically, the electron density uniquely determines the external potential, the external potential being the positions and charges of the nuclei. Furthermore, in Density Functional Theory (DFT), reactants and products exist at the most stable points—called local minima on the potential energy surface, which represents how energy changes based on atomic positions (see Fig. 2). The reaction follows the path that requires the least energy, passing through a saddle point known as a transition state (Fig. 2). The energy needed to overcome this barrier is called the activation energy (E act ), which allows the reaction to proceed. The energy difference between reactants and products, known as the binding energy ( Δ G), represents the energy stored in the final product (Fig. 2). 3. Results: Chemical reactions of Tricalcium Phosphate with Titanium and chemical reaction of Tricalcium Silicon Trioxide with Titanium and Comparative characterizes of Binding Energy 3.1. Density Functional Theory Optimizations calculations were performed between the coatings, Ca 3 (PO 4 ) 2 and Ca 3 (SiO 3 ) 2 , with the Ti substrate to be able to determine the ground state of the compounds. Then, the ground state energy is used to calculate the binding energy of the compounds [Balueva et al., 2025]. Binding energy allows us to compare coating effectiveness. The optimizations calculations were performed with Gaussian 09W, with calculation setup being Ground state, DFT, default spin, the B3LYP hybrid-exchange-correlation functional (Becke three-parameter Lee-Yang-Parr functional) for the methods and the basis set: 6-31G + (d, p). In order to determine the binding energy of the compounds, the ground state energy of each molecule or reactant needs to be calculated separately. Then, the reactants are used to form a complete compound, the compound is optimized as well to calculate the ground state energy. Once all the ground state energies are collected, the binding