The Hartree-Fock (HF) method is a vibrational, wave function-based approach used to determine the wave function of a system by employing a single Slater determinant or a single permanent Nspin orbital . The core principle of HF involves calculating a vibrational wave function for single-particle orbitals. However, as the number of electrons in the system increases, the computational complexity of determining the wave function also grows exponentially, making direct calculations increasingly challenging . To manage this complexity, the Born-Oppenheimer Approximation is often applied, simplifying the problem by decoupling nuclear motion from electronic motion. This approximation reduces the computational burden by allowing the electronic degrees of freedom to be treated separately, thus enabling more manageable calculations even as the number of electrons increases . Consequently, Hartree-Fock theory, as a mean-field approach, provides a solution for many cases, making it the most straightforward wave function-based method.