Electronic state occupancy

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

Specific electronic states may also be specified using the Gue s=Alter keyword, which allows you to explicitly designate orbital occupancies. See the Gaussian User s Reference for details. 

Fig. 1. Electronic states [or iron-group atoms, showing number of states as qualitative [unction of electronic energy. Electrons in band A are paired with similar electrons of neighboring atoms to form bonds. Electrons in band B are d electrons with small interatomic interaction they remain unpaired until the band is half-filled. The shaded area represents occupancy of the states by electrons in nickel, with 0.6 electron lacking from a completely filled B band. (States corresponding to occupancy of bond orbitals by unshared electron pairs are not shown in the diagram.)...

XPS has typically been regarded primarily as a surface characterization technique. Indeed, angle-resolved XPS studies can be very informative in revealing the surface structure of solids, as demonstrated for the oxidation of Hf(Sio.sAso.5)As. However, with proper sample preparation, the electronic structure of the bulk solid can be obtained. A useful adjunct to XPS is X-ray absorption spectroscopy, which probes the bulk of the solid. If trends in the XPS BEs parallel those in absorption energies, then we can be reasonably confident that they represent the intrinsic properties of the solid. Features in XANES spectra such as pre-edge and absorption edge intensities can also provide qualitative information about the occupation of electronic states. 

Thus far, I have mainly discussed neutral impurities. From the treatment of the electronic states, however, it should be clear that occupation of the defect level with exactly one electron is by no means required. In principle, zero, one, or two electrons can be accommodated. To alter the charge state, electrons are taken from or removed to a reservoir the Fermi level determines the energy of electrons in this reservoir. In a self-consistent calculation, the position of the defect levels in the band structure changes as a function of charge state. For H in Si, it was found that with H fixed at a particular site, the defect level shifted only by 0.1 eV as a function of charge state (Van de Walle et al., 1989). 

A defining feature for carbenes is the existence of two non-bonding orbitals on one carbon atom. There are two electrons to distribute among these two orbitals and their placement defines the electronic state of the molecule. A simple representation showing the electron occupancy of the non-bonding orbitals is displayed in Fig. 1. The orbital perpendicular to the... 

Electron occupation in the frontier bands of metal crystals varies with different metals as shown in Fig. 2-7. For metallic iron the frontier bands consist of hybridized 4s-3d-4p orbitals, in which 4s and 3d are partially occupied by electrons but 4p is vacant for electrons. Figure 2-8 shows the electron state density curve of metallic iron, where the 3d and 4s bands are partially filled with electrons. Electrons in metals occupy the energy states in a frontier band successively fix>m the lower band edge level to the Fermi level, leaving the higher levels vacant. 

Fig. 2-20. Electron state density and ranges of Fermi energy where electron occupation probability in the conduction band of an electron ensemble of low electron density (e.g., semiconductor) follows Boltzmann function (Y i)or Fermi function (y > 1) y = electron activity coeffident ET =transition level from Y 4= 1 to Y > 1 0(t) = electron energy state density CB = conduction band. [From Rosenberg, I960.]...

Figure 3.5. Probability of occupation of electron states versus energy of states E (a)T= OK (/ )T>0K.

One important feature of carbenes is the presence of two nonbonding electrons and two available orbitals, which are nominally located on the carbon atom. In bent carbenes, the two orbitals have different energies and are often denoted as a and -ir, with a being the in-plane orbital and -ir the out-of-plane orbital. Within this simple picture, four electronic states can be envisioned, which are depicted in Fig. la. Singlets and S3 have the two electrons in the same orbital (a and -IT, respectively) and are often characterized as closed-shell singlets. On the other hand, singlet S2 has the same open-shell electronic occupation (arr) as the triplet (T) state. 

EKS approach, based on the more formal Janak s type of argumentation, doesn t put these restrictions on the ground-state occupation numbers below HOMO. In what follows in this chapter we will use the fractional occupation of HOMO only, in which case both approaches agree. Using Eqn (4) for HOMO energy with respect to (A — 1 + /) electrons, and for HOMO energy Ar+i with respect to (N + /) electrons, one can express I and A in the following form ... 

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