According to Koopman's theorem, how is the ionization energy related to the orbital energy?

Koopman's theorem provides a valuable insight into the relationship between ionization energy and the energy of orbitals in a molecular system. According to this theorem, the ionization energy of an electron removed from a molecular orbital can be approximated as being equal to the negative of the orbital energy of that electron before its removal. This means that if you have an orbital energy given as a positive value, the ionization energy, which reflects the energy required to remove the electron, will be expressed as a negative value.

This negative relationship arises because, in quantum mechanics, energies of bound states (like electrons in orbitals) are conventionally assigned negative values relative to a state of zero energy when the electron is completely removed from the influence of the nucleus. Thus, if an electron has an orbital energy of, say, -10 eV, the ionization energy required to remove it would be +10 eV, which is the negative of the orbital energy.

Therefore, the correct perspective offered by Koopman's theorem directly links the ionization energy to the orbitals in a manner that emphasizes their inverse relationship, highlighting that ionization energy is qualitatively and quantitatively derived from the energy of the orbital from which the electron is being removed.