The reason we cannot use single electron equations to accurately describe multi-electron atoms and molecules is due to the complex interactions and correlations that occur among the electrons.
In quantum mechanics, the behavior of electrons is described by wave functions, which are mathematical functions that represent the probability distribution of finding an electron in a particular state. The Schrödinger equation is used to determine the wave function of an electron in the presence of a potential, such as the electric field generated by the atomic nucleus.
For a single-electron system, such as the hydrogen atom, the Schrödinger equation can be solved exactly to obtain the wave function and energy levels of the electron. However, when multiple electrons are present, their interactions with each other become significant, and solving the Schrödinger equation analytically becomes mathematically intractable.
The interactions between electrons in multi-electron systems give rise to phenomena such as electron-electron repulsion, electron shielding, and electron exchange effects. These interactions cause the energy levels and behavior of the electrons to become interdependent, making it impossible to treat each electron independently using single-electron equations.
To address these complexities, approximations and computational methods have been developed, such as the Hartree-Fock method and density functional theory (DFT). These methods attempt to approximate the behavior of the entire electron system by considering the average interactions between electrons. They provide a more accurate description of multi-electron systems and are widely used in quantum chemistry to study atoms, molecules, and solids.
In summary, the interactions and correlations among electrons in multi-electron atoms and molecules prevent the use of single-electron equations. The accurate description of these systems requires the consideration of the collective behavior of all electrons using more sophisticated theoretical and computational methods.