Virtual bound resonance

The visible and near ultra-violet (1-6 eV), and vacuum ultra-violet (4-8.5 eV) spectroscopic studies focus on transitions from the occupied states at the top of the valence band, primarily O 2p tt nonbonding states to the conduction band states, primarily O 2p ir and cr antibonding states that are mixed TM atomic states in the context of SLAC s. These spectra also include intra-d-state d-d transitions between occupied ground states and empty excited states of band edge defects these are not be confused with d-d transitions that terminate in virtual bound resonance antibonding states within the vacuum continuum. In the context of many-electron theory as applied to X-ray measurements, these states are referred to respectively as shake-up and shake-off states. The SE instruments used in these studies were developed by D.E. Aspnes during his research studies at Bell Labs, and more recently at NCSU [4,5]. 

Fig. 12 OK edge XAS 2nd derivative spectra for Ti02 (a) conduction and valence band edge defects, (b) virtual bound resonance states defect states and shallow Ti and O core states...

We studied in the previous section several types of phase transition, namely, bound-virtual, bound-resonance, and so on. A characteristic of a phase transition is that two different solutions merge (a / 1), or coexist at the critical point (a = 1). Many-body and multiparameter Hamiltonians could present more complicated transitions, and we will call them multicritical points. 

If the localized electron tunnels out through the barrier (state 1 in Fig. 12 b) a certain amount of f-f overlapping is present. States like 1 in Fig. 12 b are called sometimes resonant states or "virtually bound" states. In contrast with case 2 in Fig. 12b, which we may call of full localization , the wave function of a resonant state does not die out rapidly, but keeps a finite amplitude in the crystal, even far away from the core. For this reason, overlapping may take place with adjacent atoms and a band may be built as in ii. (If the band formed is a very narrow band, sometimes the names of localized state or of resonance band are employed, too. Attention is drawn, however, that in this case one refers to a many-electron, many-atoms wave function of itinerant character in the sense of band theory whereas in the case of resonant states one refers to a one-electron state, bound to the central potential of the core (see Chap. F)). 

If the active metal becomes highly diluted the minimum polarity model leads to the virtual bound-state model (127, 128, 129). This model has also been applied to highly diluted Ni-Cu alloys (121a). The nickel d-states are then found to form not a common band with the copper d-states but narrow virtual levels between the copper d-states and the Fermi level. The levels are in resonance with the s,p-band of the metal. 

Additionally and equally significant, the spectral features assigned to the antibonding state of Hf 5f electrons display seven features indicating a completely removal of the Hf 4fs/2 and Hf 4f7/2 degeneracies of three and four, respectively. This is consistent with the local field induced symmetries of Hf 4f orbitals that are mixed with O 2p, and possibly O 2s states as well. This is the same mechanism that activated the Ti 3p and O 2s virtual bound state resonance absorptions in Fig. 12. The spectral widths of the Hf 5d" features (4 states) and Hf 4f features (7 states)... 

Fig. 22 Derivative OK edge XAS spectra (a) empty band edge d empty Mn " " states for hexagonal perovskite, h-HoMn03 and (b) virtual bound d resonance states of Mn for hexagonal...

In general, expansions involving resonant states may be divided in two broad classes a first class that involves bound, resonant, and continuum states, where usually resonant states are considered in the interval (0,oo) [13, 17, 21, 24], and a second class that refers fo expansions defined in the interval (0, a), where a stands for fhe radius of the finite range potential, that involves the full set of bound, antibound (virtual), and resonant states. Here, the expansions are purely discrete and follow using Gauchy s infe-gral theorem [15, 16, 18, 25]. Some of these expansions are discussed in Ref. [26]. 

The KS potential of conventional functionals doesn t show the correct — 1/r asymptotic decay and thus anions are often unbound and few virtual bound KS orbitals are present. On the other hand in EXX methods the asymptotic decay is correctly reproduced and Rydberg series of virtual orbitals are present in the KS spectrum, which allows a correct description of Time-Dependent DFT (TD-DFT) excitation energies and a better evaluation nuclear magnetic resonance shielding constants ... 

The addition of a third element to an ordered binary alloy can modify considerably the point-defect properties for example, the behavior of FeCo-2% V is very complex, because vanadium forms a resonant virtual bound state at the Fermi level, the partial filling of which depends strongly on the state of order (Riviire et al., 1983b). Solutes (specifically Mn, Re, and Ti) have also been shown to have complicated effects in CoAl (Fleischer, 1993). 

Another possibility concerns the resonance integrals /Sab which appear in the Klopman-Salera equation. In a Hiickel picture, these are independent of the orbital energy, but in a double-zeta or better description we would expect the more tightly-bound electrons to have more contracted orbitals, and the higher virtual orbitals to be more diffuse.131 It may be that the HOMO and LUMO have the optimum spatial distribution for strong interaction, and that interactions involving more contracted and more diffuse orbitals are weaker.122... 

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