This paper presents a rigorous mathematical framework for the molecular orbital (MO) method in quantum mechanics. The MO method is a quantum-mechanical approach to describe the electronic structure of molecules, where each electron is assigned to a one-electron wave function, or molecular orbital. The paper discusses the general principles of the MO method, including the construction of molecular wave functions from molecular spin orbitals (MSOs), which are products of spatial orbitals and spin functions. The total N-electron wave function is built as an antisymmetrized product of MSOs, ensuring compliance with the Pauli exclusion principle. The paper also introduces the Hartree-Fock self-consistent field method for closed-shell ground states, where the best molecular orbitals are determined by minimizing the energy of the system. The method involves solving a set of equations derived from the variational principle, leading to the determination of molecular orbitals and their corresponding energies. The paper further discusses the LCAO (linear combination of atomic orbitals) self-consistent field method, which approximates molecular orbitals by combining atomic orbitals. The paper concludes with a discussion of ionization and excitation energies, where the energy required to remove an electron from a molecular orbital is calculated based on the difference between the energy of the ionized state and the ground state. The paper emphasizes the importance of mathematical rigor in the MO method and highlights the practical implications of the Hartree-Fock and LCAO methods in molecular quantum mechanics.This paper presents a rigorous mathematical framework for the molecular orbital (MO) method in quantum mechanics. The MO method is a quantum-mechanical approach to describe the electronic structure of molecules, where each electron is assigned to a one-electron wave function, or molecular orbital. The paper discusses the general principles of the MO method, including the construction of molecular wave functions from molecular spin orbitals (MSOs), which are products of spatial orbitals and spin functions. The total N-electron wave function is built as an antisymmetrized product of MSOs, ensuring compliance with the Pauli exclusion principle. The paper also introduces the Hartree-Fock self-consistent field method for closed-shell ground states, where the best molecular orbitals are determined by minimizing the energy of the system. The method involves solving a set of equations derived from the variational principle, leading to the determination of molecular orbitals and their corresponding energies. The paper further discusses the LCAO (linear combination of atomic orbitals) self-consistent field method, which approximates molecular orbitals by combining atomic orbitals. The paper concludes with a discussion of ionization and excitation energies, where the energy required to remove an electron from a molecular orbital is calculated based on the difference between the energy of the ionized state and the ground state. The paper emphasizes the importance of mathematical rigor in the MO method and highlights the practical implications of the Hartree-Fock and LCAO methods in molecular quantum mechanics.