Koopmans’ theorem

Koopmans’ theorem states that the energy to remove an electron from an orbital of an atom, whose state is described by a single Slater determinant, is approximately the negative value of the corresponding Hartree-Fock orbital energy. In other words, the theorem states that the negative value of the orbital energy of an atom is approximately equal to the ionisation energy IE for the k-th electron in the atom.

To prove the theorem, we begin with the ionisation process, which is generally expressed as

where is an atom, which is irradiated with a photon of energy , and is the corresponding ion with an electron removed.

In terms of energies,

where is the state of the ion, is the state of the atom and is the kinetic energy of the expelled electron.

As ,

Substitute eq95 in eq143,



Show that .


Let . Expanding the equation and eliminating common terms,

Since , we can add it to , giving

Furthermore, and . So,

Relabelling i to j for


Substitute eq145 in eq144,

Relabelling t with j and with in eq109, we have

Multiplying the k-th equation in eq147 by and integrating over , we have


Substitute eq148 in eq146

Eq149 is known as Koopmans’ theorem.

Lastly, let’s study the relation between the relation between the orbital energy  of an atom and the total energy E of the atom. From eq148,

Substitute eq94 in the above equation,



Show that the alternative to eq150 is .


Adding  to both sides of eq148 and summing throughout from ,

Since ,


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