Orbital energies and Koopmans theorem

Comparison with equation (2.23) shows that the first ionization energy of He is equal to minus the orbital energy. 

The general statement that the energy required to remove an electron from an atom or a molecule, is equal to minus the orbital energy of the electron that is removed, is known as Koopman s theorem. The general theorem can be proven if we use an atomic or molecular orbital model and assume that ionization is not followed by relaxation. Comparison with experiments shows that predictions based on Koopman s theorem usually are accurate to about 10%. The accuracy would be even poorer if there was not a fortunate cancellation of errors  

Problem 2.2 Use equation (2.25) and Koopman s theorem to predict the second ionization energy of Li and the electron affinity of the H atom. (The efectron affinity of H is equal to the energy 

Since the orbital energy includes the energy of repulsion to the other electrons in the molecule, the orbital energy will change if we move one of the other electrons to another orbital or if we remove it completely (ionization). The orbital energies are not constant for a given atom but must be calculated separately for each electron configuration. 

What is the relationship between the total energy of an atom and the sum of the orbital energies of the constituent electrons According to equation (2.19) the energy of the atom is given by 

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