Koopmans operator

The connection with the Fock-like equations (8.2.8) becomes clear when we evaluate the expectation value of dH. On inserting (3.6.9) in the Koopmans operator (8.2.27), and making use of the anticommutation relations, (3.6.7), we easily find first... 

The significance of the Koopmans operator, whose expectation value is given expUcitly in (8.2.29) for any kind of wavefimction (exact or approximate), is now clear. According to (8.2.30), the elements of K coincide with those of —c, the matrix of Lagrangian multipliers in the MC SCF equations and they in turn are matrix elements of a 1-electron Hamiltonian containing the effective potential felt by any electron in the presence of the others. In full,... 

A Search Problem.—An example of an operations research problem that gives rise to an isoperimetric model is a search problem, first given by B. Koopman,40 that we only formulate here. Suppose that an object is distributed in a region of space with ... 

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 operator T in this equation is called the Fock operator, and E,- is the energy of orbital / . According to Koopmans theorem, —e, is approximately equal to the energy required to ionize a molecule by removing an electron from / . 

In the Fock operator, the core Hamiltonian h( 1) does not depend on the orbitals, but the Coulomb and exchange operators (1) and ( 1) depend on ( 1). If (1,2,3,..., Ne) is constructed from the lowest energy Ne orbitals, one has the lowest possible total electronic energy. By Koopmans theorem, the negative of the orbital energy is equal to one of the ionization potentials of the molecule or atom. 

The UHF formalism becomes inconvenient for open-shell configurations of atoms or molecules with point-group symmetry. Unless specific restrictions are imposed, the self-consistent occupied orbitals fall into sets that are nearly but not quite transformable into each other by operations of the symmetry group. By imposing equivalence and symmetry restrictions, these sets become symmetry-adapted basis states for irreducible representations of the symmetry group. This makes it possible to construct symmetry-adapted /V-clcctron functions, as described in Section 4.4. The constraints in general invalidate the theorems of Brillouin and Koopmans. This restricted theory (RHF) is described in detail for atoms by Hartree [163] and by Froese Fischer [130],... 

In practice, approximations in the self-energy operator are needed. Efficient, perturbative improvements to Koopmans results may be produced by the neglect of off-diagonal matrix elements of the self-energy operator in the canonical, Hartree-Fock basis. Such diagonal approximations in the selfenergy, which are also known as quasiparticle methods, yield an especially simple form of the Dyson equation. 

Formally, Koopmans theorem applies to eigenfunctions of the Fock operator of a closed shell restricted HF wavefunction or an open shell unrestricted HF wavefunction... 

Certain algorithms can sometimes be used to force convergence to this hole state (that is how the lo-g level is obtained in Tables 16 and 17), but it would be better to avoid the SCF hole state calculation altogether. In particular, it is conceptually and operationally appealing if all ionized states are described in terms of the same set of reference orbitals, namely those for the neutral, while occupying them as needed for the various possible principal ionizations. This means that we will describe the 2energetically optimum determinant. This choice follows the spirit of Koopmans theorem and does not allow the ionized state orbitals to relax. 

In between Koopmans and ASCF calculations, a method was developed termed the transition operator method [12], in which the Fock operator involved in the calculation of the electronic structure of the ionized species is modified so as to adjust an occupation of 1/2 in the ionized core level. Fairly good results were obtained with this approach, as well as with improved versions involving third-order perturbation corrections to the transition operator method, followed by extrapolation using a geometric approximation [13]. 

All transition amplitudes corresponding to primary ionization events thus become equal to unity at this level of approximation. The above result expresses the EP analog of Koopmans theorem. To go beyond Koopmans theorem, better choices must be made for the reference state and operator manifold. 

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