Koopmans' theorem
Encyclopedia
Koopmans' theorem states that in closed-shell Hartree-Fock
theory, the first ionization energy
of a molecular system is equal to the negative of the orbital energy of the highest occupied molecular orbital (HOMO
). This theorem is named after Tjalling Koopmans
, who published this result in 1934.
Koopmans became a Nobel laureate in 1975, though neither in physics nor chemistry, but in economics
.
Koopmans' theorem is exact in the context of restricted Hartree-Fock theory if it is assumed that the orbitals of the ion are identical to those of the neutral molecule (the frozen orbital approximation). Ionization energies calculated this way are in qualitative agreement with experiment – the first ionization energy of small molecules is often calculated with an error of less than two electron volts. Therefore, the validity of Koopmans' theorem is intimately tied to the accuracy of the underlying Hartree-Fock
wavefunction. The two main sources of error are:
Empirical comparisons with experimental values and higher-quality ab initio
calculations suggest that in many cases, but not all, the energetic corrections due to relaxation effects nearly cancel the corrections due to electron correlation.
A similar theorem exists in density functional theory
(DFT) for relating the exact first vertical ionization energy and electron affinity to the HOMO and LUMO
energies, although both the derivation and the precise statement differ from that of Koopmans' theorem. Ionization energies calculated from DFT orbital energies are usually poorer than those of Koopmans' theorem, with errors much larger than two electron volts possible depending on the exchange-correlation approximation employed. The LUMO energy shows little correlation with the electron affinity with typical approximations. The error in the DFT counterpart of Koopmans' theorem is a result of the approximation employed for the exchange correlation energy functional so that, unlike in HF theory, there is the possibility of improved results with the development of better approximations.
For example the electronic configuration of the H2O molecule is (1a1)2 (2a1)2 (1b2)2 (3a1)2 (1b1)2, where the symbols a1, b2 and b1 are orbital labels based on molecular symmetry
. From Koopmans’ theorem the energy of the 1b1 HOMO corresponds to the ionization energy to form the H2O+ ion in its ground state (1a1)2 (2a1)2 (1b2)2 (3a1)2 (1b1)1. The energy of the second-highest MO 3a1 refers to the ion in the excited state (1a1)2 (2a1)2 (1b2)2 (3a1)1 (1b1)2, and so on. In this case the order of the ion electronic states corresponds to the order of the orbital energies. Excited-state ionization energies can be measured by photoelectron spectroscopy
.
For H2O, minus the near-Hartree-Fock orbital energies of these orbitals in eV
, are 1a1 559.5, 2a1 36.7 1b2 19.5, 3a1 15.9 and 1b1 13.8. The corresponding ionization energies are 539.7, 32.2, 18.5, 14.7 and 12.6 eV. As explained above, the deviations are due to the effects of orbital relaxation as well as differences in electron correlation energy between the molecular and the various ionized states.
as the energy of the lowest unoccupied molecular orbitals (LUMO
) of the respective systems. However, Koopmans' original paper makes no claim with regard to the significance of eigenvalues of the Fock operator
other than that corresponding to the HOMO
. Nevertheless, it is straightforward to generalize the original statement of Koopmans' to calculate the electron affinity
in this sense.
Calculations of electron affinities using this statement of Koopmans' theorem have been criticized on the grounds that virtual (unoccupied) orbitals do not have well-founded physical interpretations, and that their orbital energies are very sensitive to the choice of basis set used in the calculation. As the basis set becomes
more complete; more and more "molecular" orbitals that are not really on the molecule of interest will appear, and care must be taken not to use
these orbitals for estimating electron affinities.
Comparisons with experiment and higher-quality calculations show that electron affinities predicted in this manner are generally quite poor.
the validity of Koopmans’ theorem for ROHF in general has been proven provided that the correct orbital energies are used.
The spin up (alpha) and spin down (beta) orbital energies do not necessarily have to be the same.
(DFT) a similar theorem exists that relates the first ionization energy and electron affinity to the HOMO and LUMO energies. This is sometimes called the DFT-Koopmans' theorem. More generally, for a fixed geometry defined by the N-electron system, the HOMO energy is equal to the ionization energy of the N-electron system when the total number of electrons is in the range N − δN for 0 < δN < 1, and is equal to the electron affinity when the total number of electrons is in the range N + δN for 0 < δN < 1 (with the δN occupying what is the LUMO of the N-electron state). While these are exact statements in the formalism of DFT, the use of approximate exchange-correlation potentials makes the calculated energies approximate. As in HF theory, the electron affinity calculated in this way is less accurate than the ionization energy.
A proof of the DFT counterpart of Koopmans' theorem usually employs Janak's theorem: that the derivative of the total DFT energy, E, with respect to the occupation of a given orbital, ni is equal to the corresponding orbital energy, εi:
Hartree-Fock
In computational physics and chemistry, the Hartree–Fock method is an approximate method for the determination of the ground-state wave function and ground-state energy of a quantum many-body system....
theory, the first ionization energy
Ionization energy
The ionization energy of a chemical species, i.e. an atom or molecule, is the energy required to remove an electron from the species to a practically infinite distance. Large atoms or molecules have a low ionization energy, while small molecules tend to have higher ionization energies.The property...
of a molecular system is equal to the negative of the orbital energy of the highest occupied molecular orbital (HOMO
Homo
Homo may refer to:*the Greek prefix ὅμο-, meaning "the same"*the Latin for man, human being*Homo, the taxonomical genus including modern humans...
). This theorem is named after Tjalling Koopmans
Tjalling Koopmans
Tjalling Charles Koopmans was the joint winner, with Leonid Kantorovich, of the 1975 Nobel Memorial Prize in Economic Sciences....
, who published this result in 1934.
Koopmans became a Nobel laureate in 1975, though neither in physics nor chemistry, but in economics
Nobel Memorial Prize in Economic Sciences
The Nobel Memorial Prize in Economic Sciences, commonly referred to as the Nobel Prize in Economics, but officially the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel , is an award for outstanding contributions to the field of economics, generally regarded as one of the...
.
Koopmans' theorem is exact in the context of restricted Hartree-Fock theory if it is assumed that the orbitals of the ion are identical to those of the neutral molecule (the frozen orbital approximation). Ionization energies calculated this way are in qualitative agreement with experiment – the first ionization energy of small molecules is often calculated with an error of less than two electron volts. Therefore, the validity of Koopmans' theorem is intimately tied to the accuracy of the underlying Hartree-Fock
Hartree-Fock
In computational physics and chemistry, the Hartree–Fock method is an approximate method for the determination of the ground-state wave function and ground-state energy of a quantum many-body system....
wavefunction. The two main sources of error are:
- orbital relaxation, which refers to the changes in the Fock operator and Hartree-FockHartree-FockIn computational physics and chemistry, the Hartree–Fock method is an approximate method for the determination of the ground-state wave function and ground-state energy of a quantum many-body system....
orbitals when changing the number of electrons in the system, and
- electron correlation, referring to the validity of representing the entire many-body wavefunction using the Hartree-FockHartree-FockIn computational physics and chemistry, the Hartree–Fock method is an approximate method for the determination of the ground-state wave function and ground-state energy of a quantum many-body system....
wavefunction, i.e. a single Slater determinantSlater determinantIn quantum mechanics, a Slater determinant is an expression that describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements and consequently the Pauli exclusion principle by changing sign upon exchange of fermions . It is named for its discoverer, John C...
composed of orbitals that are the eigenfunctions of the corresponding self-consistent Fock operator.
Empirical comparisons with experimental values and higher-quality ab initio
Ab initio
ab initio is a Latin term used in English, meaning from the beginning.ab initio may also refer to:* Ab Initio , a leading ETL Tool Software Company in the field of Data Warehousing.* ab initio quantum chemistry methods...
calculations suggest that in many cases, but not all, the energetic corrections due to relaxation effects nearly cancel the corrections due to electron correlation.
A similar theorem exists in density functional theory
Density functional theory
Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
(DFT) for relating the exact first vertical ionization energy and electron affinity to the HOMO and LUMO
HOMO/LUMO
HOMO and LUMO are acronyms for highest occupied molecular orbital and lowest unoccupied molecular orbital, respectively. The energy difference between the HOMO and LUMO is termed the HOMO-LUMO gap...
energies, although both the derivation and the precise statement differ from that of Koopmans' theorem. Ionization energies calculated from DFT orbital energies are usually poorer than those of Koopmans' theorem, with errors much larger than two electron volts possible depending on the exchange-correlation approximation employed. The LUMO energy shows little correlation with the electron affinity with typical approximations. The error in the DFT counterpart of Koopmans' theorem is a result of the approximation employed for the exchange correlation energy functional so that, unlike in HF theory, there is the possibility of improved results with the development of better approximations.
Generalizations of Koopmans' theorem
While Koopmans' theorem was originally stated for calculating ionization energies from restricted (closed-shell) Hartree-Fock wavefunctions, the term has since taken on a more generalized meaning as a way of using orbital energies to calculate energy changes due to changes in the number of electrons in a system.Ground-state and excited-state ions
Koopmans’ theorem applies to the removal of an electron from any occupied molecular orbital to form a positive ion. Removal of the electron from different occupied molecular orbitals leads to the ion in different electronic states. The lowest of these states is the ground state and this often, but not always, arises from removal of the electron from the HOMO. The other states are excited electronic states.For example the electronic configuration of the H2O molecule is (1a1)2 (2a1)2 (1b2)2 (3a1)2 (1b1)2, where the symbols a1, b2 and b1 are orbital labels based on molecular symmetry
Molecular symmetry
Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can predict or explain many of a molecule's chemical properties, such as its dipole moment...
. From Koopmans’ theorem the energy of the 1b1 HOMO corresponds to the ionization energy to form the H2O+ ion in its ground state (1a1)2 (2a1)2 (1b2)2 (3a1)2 (1b1)1. The energy of the second-highest MO 3a1 refers to the ion in the excited state (1a1)2 (2a1)2 (1b2)2 (3a1)1 (1b1)2, and so on. In this case the order of the ion electronic states corresponds to the order of the orbital energies. Excited-state ionization energies can be measured by photoelectron spectroscopy
Ultraviolet photoelectron spectroscopy
Ultraviolet photoelectron spectroscopy refers to the measurement of kinetic energy spectra of photoelectrons emitted by molecules which have absorbed ultraviolet photons, in order to determine molecular energy levels in the valence region.-Basic Theory:...
.
For H2O, minus the near-Hartree-Fock orbital energies of these orbitals in eV
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...
, are 1a1 559.5, 2a1 36.7 1b2 19.5, 3a1 15.9 and 1b1 13.8. The corresponding ionization energies are 539.7, 32.2, 18.5, 14.7 and 12.6 eV. As explained above, the deviations are due to the effects of orbital relaxation as well as differences in electron correlation energy between the molecular and the various ionized states.
Koopmans' theorem for electron affinities
It is sometimes claimed that Koopmans' theorem also allows the calculation of electron affinitiesElectron affinity
The Electron affinity of an atom or molecule is defined as the amount of energy released when an electron is added to a neutral atom or molecule to form a negative ion....
as the energy of the lowest unoccupied molecular orbitals (LUMO
Lumo
Lumo is a 2007 documentary film about twenty-year-old Lumo Sinai, a woman who fell victim to "Africa's First World War." While returning home one day, Lumo and another woman were gang-raped by a group of soldiers fighting for control of the Democratic Republic of the Congo during the 1994 Rwandan...
) of the respective systems. However, Koopmans' original paper makes no claim with regard to the significance of eigenvalues of the Fock operator
Fock matrix
In the Hartree-Fock method of quantum mechanics, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of basis vectors....
other than that corresponding to the HOMO
Homo
Homo may refer to:*the Greek prefix ὅμο-, meaning "the same"*the Latin for man, human being*Homo, the taxonomical genus including modern humans...
. Nevertheless, it is straightforward to generalize the original statement of Koopmans' to calculate the electron affinity
Electron affinity
The Electron affinity of an atom or molecule is defined as the amount of energy released when an electron is added to a neutral atom or molecule to form a negative ion....
in this sense.
Calculations of electron affinities using this statement of Koopmans' theorem have been criticized on the grounds that virtual (unoccupied) orbitals do not have well-founded physical interpretations, and that their orbital energies are very sensitive to the choice of basis set used in the calculation. As the basis set becomes
more complete; more and more "molecular" orbitals that are not really on the molecule of interest will appear, and care must be taken not to use
these orbitals for estimating electron affinities.
Comparisons with experiment and higher-quality calculations show that electron affinities predicted in this manner are generally quite poor.
Koopmans' theorem for open-shell systems
Koopmans' theorem is also applicable to open-shell systems. It was previously believed that this was only in the case for removing the unpaired electron, butthe validity of Koopmans’ theorem for ROHF in general has been proven provided that the correct orbital energies are used.
The spin up (alpha) and spin down (beta) orbital energies do not necessarily have to be the same.
Counterpart in density functional theory
In density functional theoryDensity functional theory
Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
(DFT) a similar theorem exists that relates the first ionization energy and electron affinity to the HOMO and LUMO energies. This is sometimes called the DFT-Koopmans' theorem. More generally, for a fixed geometry defined by the N-electron system, the HOMO energy is equal to the ionization energy of the N-electron system when the total number of electrons is in the range N − δN for 0 < δN < 1, and is equal to the electron affinity when the total number of electrons is in the range N + δN for 0 < δN < 1 (with the δN occupying what is the LUMO of the N-electron state). While these are exact statements in the formalism of DFT, the use of approximate exchange-correlation potentials makes the calculated energies approximate. As in HF theory, the electron affinity calculated in this way is less accurate than the ionization energy.
A proof of the DFT counterpart of Koopmans' theorem usually employs Janak's theorem: that the derivative of the total DFT energy, E, with respect to the occupation of a given orbital, ni is equal to the corresponding orbital energy, εi: