Canonical orbitals Koopmans’ theorem

The next five chapters are each devoted to the study of one particular computational model of ab initio electronic-structure theory Chapter 10 is devoted to the Hartree-Fock model. Important topics discussed are the parametrization of the wave function, stationary conditions, the calculation of the electronic gradient, first- and second-order methods of optimization, the self-consistent field method, direct (integral-driven) techniques, canonical orbitals, Koopmans theorem, and size-extensivity. Also discussed is the direct optimization of the one-electron density, in which the construction of molecular orbitals is avoided, as required for calculations whose cost scales linearly with the size of the system. 

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. 

The photoelectron spectrum is frequently discussed in terms of Koopmans theorem, which states that the ionization potentials (IPs) are approximately related to the energies of the canonical orbital found in molecular orbital calculations.106. The relationship is approximate because two factors are neglected the change in the correlation energy, and the reorganization energy, which is a consequence of the movement of electrons in response to the formation of a cation. The two quantities are approximately equal and opposite. 

Similar habits are reinforced by Hartree-Fock theory, where Koopmans s theorem [1] enables one to use canonical orbital energies as estimates of ionization energies and electron affinities. Here, orbitals that are variationally optimized for an N-electron state are used to describe final states with N 1 electrons. Energetic consequences of orbital relaxation in the final states are ignored, as is electron correlation. 

In Sections 10.3 and 10.4, we introduced the Fock operator as an effective Hamiltonian for the calculation of Hartree-Fock orbitals (the canonical orbitals) and orbital energies by the repeated diagonalization of the Fock matrix. In the present section, we consider in more detail the properties of the canonical orbitals and the associated orbitals energies - the eigenfunctions and eigenvalues, respectively, of the Fock operator. In particular, we shall introduce Koopmans theorem and identify the ionization potentials and electron affinities of a closed-shell system with the negative energies of the canonical orbitals. 

We conclude that the occupied canonical c bitals are the MOs best suited to describe ionization processes in the sense that they are the solutions to the small Cl problem (10.5.7). Furthermore, we identify the orbital energies with the negative IPs according to (10.5.8). This result is known as Koopmans theorem. 

These identifications are based on Koopmans theorem, which states that the canonical orbitals are optimal for the description of ionized states, yielding the same results as a Cl calculation in the configurations generated by the application of the full set of annihilation or creation operators to the neutral state. 

Thus the spectrum which arises when Eq. (8) is Fourier transformed consists of a set of -functions at the energies corresponding to the stationary states of the ion (which via the theorem of Koopmans) are the one-electron eigenvalues of the Hartree-Fock equations). The valence bond description of photoelectron spectroscopy provides a novel perspective of the origin of the canonical molecular orbitals of a molecule. Tlie CMOs are seen to arise as a linear combination of LMOs (which can be considered as imcorrelated VB pairs) and coefficients in this combination are the probability amplitudes for a hole to be found in the various LMOs of the molecule. 

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