The paper by C. C. J. Roothaan introduces a rigorous mathematical framework for the molecular orbital (MO) method, which is an extension of the Bohr theory of electron configurations from atoms to molecules. The MO method is contrasted with the valence bond (VB) method, which originates from a chemical perspective. Both methods aim to describe molecular wave functions, but differ in their approach and applications. The VB method is more suitable for chemical valence, while the MO method is better for describing excitation and ionization processes. Roothaan's paper focuses on the electronic part of molecular wave functions, neglecting nuclear motion and magnetic effects. The total wave function is constructed as a product of molecular spin orbitals (MSOs), which are one-electron wave functions that extend over the entire molecule. The MSOs are antisymmetrized to form the molecular wave function, ensuring that no more than two electrons occupy each MSO. The paper discusses the Hartree-Fock self-consistent field method and the linear combination of atomic orbitals (LCAO) self-consistent field method for closed-shell ground states. These methods involve solving a set of equations to find the best MSOs or LCAOs that minimize the energy of the system. The LCAO method is particularly useful for molecules without central symmetry, as it allows for a more straightforward calculation compared to the Hartree-Fock method. Finally, the paper touches on the calculation of ionization and excitation energies, noting that these processes are more complex due to the lack of closed-shell structure in excited states and the need to maintain orthogonality with lower-energy states. A less accurate but simpler method is proposed to address these complications.The paper by C. C. J. Roothaan introduces a rigorous mathematical framework for the molecular orbital (MO) method, which is an extension of the Bohr theory of electron configurations from atoms to molecules. The MO method is contrasted with the valence bond (VB) method, which originates from a chemical perspective. Both methods aim to describe molecular wave functions, but differ in their approach and applications. The VB method is more suitable for chemical valence, while the MO method is better for describing excitation and ionization processes. Roothaan's paper focuses on the electronic part of molecular wave functions, neglecting nuclear motion and magnetic effects. The total wave function is constructed as a product of molecular spin orbitals (MSOs), which are one-electron wave functions that extend over the entire molecule. The MSOs are antisymmetrized to form the molecular wave function, ensuring that no more than two electrons occupy each MSO. The paper discusses the Hartree-Fock self-consistent field method and the linear combination of atomic orbitals (LCAO) self-consistent field method for closed-shell ground states. These methods involve solving a set of equations to find the best MSOs or LCAOs that minimize the energy of the system. The LCAO method is particularly useful for molecules without central symmetry, as it allows for a more straightforward calculation compared to the Hartree-Fock method. Finally, the paper touches on the calculation of ionization and excitation energies, noting that these processes are more complex due to the lack of closed-shell structure in excited states and the need to maintain orthogonality with lower-energy states. A less accurate but simpler method is proposed to address these complications.