Atomic orbitals beyond the hydrogen atom are not exactly "approximations" in the sense that they are incorrect or inaccurate descriptions. Rather, they are theoretical models that are developed to describe the behavior of electrons in atoms more complex than hydrogen.
In the hydrogen atom, the electron is governed by a single proton in the nucleus, resulting in a simple and analytically solvable system. The atomic orbitals in hydrogen are described by mathematical functions called wave functions, which give the probability distribution of finding the electron in a particular region around the nucleus.
However, for atoms with more than one electron, the problem becomes significantly more complex due to the mutual repulsion between electrons. The behavior of electrons in such atoms cannot be solved analytically, and we have to resort to approximate methods to describe their behavior.
The most commonly used approximation method is the Hartree-Fock method, which treats each electron as moving in an average electrostatic field created by all the other electrons. This method assumes that electrons move independently in average fields and neglects the electron-electron correlation effects. While this approximation works reasonably well for many atoms, it becomes less accurate for highly correlated systems or when considering properties involving electron-electron interactions.
To overcome these limitations, more advanced theoretical methods, such as density functional theory (DFT), configuration interaction (CI), and coupled-cluster theory, are employed. These methods provide better accuracy by incorporating more sophisticated treatments of electron-electron interactions. However, they also involve more computational complexity.
Therefore, atomic orbitals beyond the hydrogen atom are not mere approximations in the sense of being incorrect. They are sophisticated models that take into account the complexities of electron-electron interactions to describe the behavior of electrons in multi-electron atoms to varying degrees of accuracy.