Koopman state

Fig. 1. Observable states of a radical cation (left) opposed by the nonobservable Koopmans states and the negative values of the orbital energies (right)...

The reader should be warned that this seeming simplicity of the C2s band system of hydrocarbon PE spectra is spurious, and largely a consequence of the low (inherent and instrumental) resolution of the experiment. Theoretical treatments including the effects of electronic relaxation and electron correlationshow that the observed shapes of the individual C2s bands are the result of the superposition of a whole series of Franck-Condon envelopes, each corresponding to one of many cation states involving multiply excited configurations. However, the positions of these individual bands tend to cluster around the one band (dominantly) associated with the Koopmans state due to simple... 

This effective Hamiltonian proposed by Gadea et for the cations of conjugated molecules is able to give many more eigenvectors of the positive ion than Koopman s theorem. It provides a direct estimate of the spectrum of the positive ion, involving the non-Koopmans states, which appear to occur at quite low energy and are described as two-hole one-particle states in the delocalized MO-CI language. 

Figure 11 VBCI model developed to define electronic relaxation, i.e., the difference between the unrelaxed final state (Koopmans state) and the true relaxed final state/" The left side of the diagram represents the initial state configuration interaction between the d") ground configuration and the charge transfer...

MO calculations of the SCF type for Nj place the state below the X g state. This discrepancy is an example of the breakdown of Koopmans theorem due to deficiencies in fhe calculations. 

The orbitals and orbital energies produced by an atomic HF-Xa calculation differ in several ways from those produced by standard HF calculations. First of all, the Koopmans theorem is not valid and so the orbital energies do not give a direct estimate of the ionization energy. A key difference between standard HF and HF-Xa theories is the way we eoneeive the occupation number u. In standard HF theory, we deal with doubly oecupied, singly occupied and virtual orbitals for which v = 2, 1 and 0 respectively. In solid-state theory, it is eonventional to think about the oecupation number as a continuous variable that can take any value between 0 and 2. 

The matrix elements between the HF and a doubly excited state are given by two-electron integrals over MOs (eq. (4.7)). The difference in total energy between two Slater determinants becomes a difference in MO energies (essentially Koopmans theorem), and the explicit formula for the second-order Mpller-Plesset correction is... 

In the uncorrelated limit, where the many-electron Fock operator replaces the full electronic Hamiltonian, familiar objects of HF theory are recovered as special cases. N) becomes a HF, determinantal wavefunction for N electrons and N 1) states become the frozen-orbital wavefunctions that are invoked in Koopmans s theorem. Poles equal canonical orbital energies and DOs are identical to canonical orbitals. 

When S(E) is neglected, P equals unity for each Koopmans final state. 

Finally, we should note Koopmans theorem (Koopmans, 1934) which provides a physical interpretation of the orbital energies e from equation (1-24) it states that the orbital energy e obtained from Hartree-Fock theory is an approximation of minus the ionization energy associated with the removal of an electron from that particular orbital i. e., 8 = EN - Ey.j = —IE(i). The simple proof of this theorem can be found in any quantum chemistry textbook. 

DFT has come to the fore in molecular calculations as providing a relatively cheap and effective method for including important correlation effects in the initial and final states. ADFT methods have been used, but by far the most popular approach is that based on Slater s half electron transition state theory [73] and its developments. Unlike Hartree-Fock theory, DFT has no Koopmans theorem that relates the orbital energies to an ionisation potential, instead it has been shown that the orbital energy (e,) is related to the gradient of the total energy E(N) of an N-electron system, with respect to the occupation number of the 2th orbital ( , ) [74],... 

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