PSI - Issue 83

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

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energy of the nuclei being treated as a constant, the calculation for the electronic energies and electronic density of the system simplifies. Thus, the Hamiltonian can be written as follows: Ĥൌܶ ௘௟௘௖ ሺ ሻ൅ܸ ௡௨௖ି ௘௟௘௖ ሺ , ሻ ൅ܸ ௡௨௖ି ௡௨௖ ሺ ሻ൅ܸ ௘௟௘௖ି ௘௟௘௖ ሺ ሻ (1.5) or Ĥൌ∑ ቆቀ డ మ డ మ ௫ ቁ൅ቀ డ మ డ మ ௬ ቁ൅ቀ డ మ డ మ ௭ ቁቇെ∑ ∑ ൬ ௓ ೕ หோ ೕି ௥ ೔ ห ൰൅∑ ∑ ൬ ห௥ ଵ ೔ି ௥ ೕ ห ൰൅∑ ∑ ൬ ௓ ೔ ௓ ೕ หோ ೔ି ோ ೕ ห ൰ ூ ௃ழூ ௜ ௝ழ௜ ௜ ௝ ௝ (1.6) The operator is called the Electronic Hamiltonian, and the Schrödinger equation now can be written as: Ĥ ௘௟௘௖ ௘௟௘௖ ሺ ⃗, ሬ⃗ ሻ ൌ ௘௟௘௖ ሺ ሬ⃗ ሻ ௘௟௘௖ ൫ ⃗, ሬ⃗ ൯ (1.7) The eigenvalues (E elec ) correspond to the energy levels of the systems at absolute zero, since the nuclei don’t move, or thus reveal the lowest energy states (Emin), or ground state energy. The ground state energy of the nuclei in the systems is unique to the coordinates of the nuclei ( ሬ⃗ ).

Fig. 2. Potential-energy surface (PES), activation energy and binding energy.

Moreover, the fundamentals of Density Functional Theory (DFT) in relation to electronic optimization can be explained in a mathematical approach for determining the minimum energy of a system by multiplying Schrödinger’s equation by ሬ ⃖ ௘௟௘௖ (1.7) ሬ ⃖ ௘௟௘௖ Ĥ ௘௟௘௖ ௘௟௘௖ ൫ ⃗, ሬ⃗ ൯ ൌ ௘௟௘௖ ൫ ሬ⃗ ൯ ሬ ⃖ ௘௟௘௖ ௘௟௘௖ (1.8) and taking into account the normalization of the wave function (1.8), the solving of the Schrödinger’s equation (1.8)