One-Electron Eigenvalues

In recent years density-functional methods32 have made it possible to obtain orbitals that mimic correlated natural orbitals directly from one-electron eigenvalue equations such as Eq. (1.13a), bypassing the calculation of multi-configurational MP or Cl wavefunctions. These methods are based on a modified Kohn-Sham33 form (Tks) of the one-electron effective Hamiltonian in Eq. (1.13a), differing from the HF operator (1.13b) by inclusion of a correlation potential (as well as other possible modifications of (Fee(av))-... 

By introducing eqn (10-2.3) into eqn (10-2.4) it is possible to arrive at a simple set of n one-electron eigenvalue equations, called the Hartree-Fock equations ... 

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. 

The only variational degree of freedom concerns the orbital set, which is therefore chosen to minimize the energy expectation value with the constraint that the orbitals remain orthonormal. This leads to a set of Euler equations that in turn lead to the Hartree-Fock equations, finally giving the (f)/ set although in an iterative way because the Hartree-Fock equations depend on the orbitals themselves. This dependency arises from the fact that the HF equations are effective one-electron eigenvalue equations... 

First, consider Fermi statistics. Slater (1974) has shown for Xa the proper interpretation of the one-electron eigenvalues... 

The problem of two valence electrons that are singlet-coupled is richer than the problem of two electrons triplet-coupled because the two orbitals need not be orthogonal, i.e. for the triplet state the one-electron eigenvalue problem (Eq. (6)) returns two orthonormal orbitals. Consider the singlet ground state of the hydrogen molecule. In the approximate two-determinantal wavefunction of Heitler and London (1927),... 

In the case shown in Fig. 2, although/ (/0 >IJ K), we have (7 )jj < (/ ) . This crossing-over of the one-electron eigenvalues and ionization energies will occur for any pair of eigenvalues for which the difference is smaller than... 

The Hellmann-Feynman theorem holds for the density-functional expression for Eu, and it is instructive to work through this. Writing the one-electron eigenvalues in terms of the Hamiltonian h in (9), (11) becomes ... 

Fortunately, this issue has so far not appeared as a limitation in practical applications of COHP, and the reason is simple. The successes of one-electron theories in quantum chemistry (see Section 2.11.1) is due in no small part to the intimate connection between the sum of one-electron eigenvalues (COHP terms) and the total energy, and this connection is one of the pillars of qualitative molecular orbital theory. Experience also reveals that the most important chemical information contained in the COHP (and also COOP) analyses results from the shape of these bonding indicators. 

The first term on the right-hand side of (11.8) is the band structure energy which is equal to the sum of the one-electron eigenvalues Ei of the occupied states given by a tight-binding Hamiltonian Htb,... 

This is shown in Figure 4.7, where the curves for the four individual cages have been shifted by a constant value to yield the coincidence depicted. We show in Table 4.10 both the ground-state energy E and the sum of the one-electron eigenvalues (in atomic units). The equilibrium radii R (in angstroms) are also recorded in this table. 

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