Koopmans’ Theorem

Koopmans' Theorem

Koopmans’ theorem provides a fundamental approximation in quantum chemistry: the ionization energy of an atom or molecule is approximately equal to the negative of the orbital energy of the ejected electron. This interpretation arises primarily from photoelectron spectroscopy data.

The theorem is named after the Dutch physicist, mathematician, and economist Charles Koopman (more commonly known as Tjalling C. Koopmans), who formulated the hypothesis within the framework of Hartree–Fock theory, treating electrons as particles in independent orbitals while disregarding electron relaxation and correlation effects.

Speculations and Modern Understanding of Koopmans’ Theorem

Originally developed to estimate ionization energies using confined (closed-shell) Hartree–Fock wave functions, the concept has evolved into a broader and more practical tool. Today, it is widely used as a general approach to approximate energy changes associated with the addition or removal of electrons by directly utilizing computed orbital energies.

          
                In essence, Koopmans’ theorem applies to the expulsion of an electron from any occupied 
                atomic or molecular orbital, resulting in the formation of a positive ion (cation). 
                Removing an electron from different orbitals produces the cation in different electronic states. 
                The lowest-energy state among these is typically (though not always) the one corresponding 
                to ionization from the Highest Occupied Molecular Orbital (HOMO), 
                which represents the ground state of the cation. Higher-energy ionic states arise from 
                ionization of deeper (lower-energy) orbitals.
            
        

This approximation is remarkably useful despite its simplicity, as it allows quick estimation of ionization potentials and electron affinities directly from a single Hartree–Fock calculation, without the need for separate computations on the neutral and ionic species.

MP Assistant Professor Exam 2022

The Koopman's theorem usually deals with the energy of which orbital ?

(A) HOMO (Highest Occupied Molecular Orbital)
(B) LUMO (Lowest Unoccupied Molecular Orbital)
(C) S-orbital
(D) P-orbital

Answer: (A) HOMO (Highest Occupied Molecular Orbital)

Explanation: Koopman's theorem states that the first ionization energy of a molecule is approximately equal to the negative of the orbital energy of the Highest Occupied Molecular Orbital (HOMO). It assumes that the orbitals of the resulting ion remain "frozen" (no electronic relaxation) during the removal of an electron.

Koopmans’ Theorem