Formulas, electronic creation

In Chapter 15, for the CFP with a detached electrons, we obtained a relationship (15.27) whose right side has the form of a vacuum average of a certain product of second-quantized operators q>. To obtain algebraic formulas for CFP, it is necessary to compute this vacuum average by transposing all the annihilation operators to the right side of the creation operators. So, for N = 3, we take into account (for non-repeating terms) the explicit form of operators (15.2) and (15.5), which produce pertinent wave functions out of vacuum, and find (cf. [107])... 

Formula (17.16) is the most general form of the two-electron matrix element in which all four one-electron wave functions have different quantum numbers. We shall put it into general formula (13.23), whereupon the creation and annihilation operators will be rearranged to place side by side those second-quantization operators whose rank projections enter into the same Clebsch-Gordan coefficient. Summing over the projections then gives... 

The theory of symmetry-preserving Kramers pair creation operators is reviewed and formulas for applying these operators to configuration interaction calculations are derived. A new and more general type of symmetry-preserving pair creation operator is proposed and shown to commute with the total spin operator and with all of the symmetry operations which leave the core Hamiltonian of a many-electron system invariant. The theory is extended to cases where orthonormality of orbitals of different configurations cannot be assumed. 

The above considerations may be put in mathematical form by referring the following formulas 100). Let a be the creation operator for an electron in the single-particle state j>) and Uj the corresponding annihilation operator. Then the Hamiltonian operator of N electrons can be written ... 

These mixed-valenee systems have been discussed by several authors with respect to final state effects in their core level spectra [12,35]. A distinction is generally made between the cases where the extra electrons occur as itinerant conduction band electrons (metallic case) or whether they are completely localized to single sites. In the former case, the core ionization of one site will in itself lead to the creation of localized levels (by pulling down from the conduction band), whose occupancy in the final state of the ionization process will depend statistically on the conduction electron density. Thus, the final state localized level may be either filled or empty. The net result is a loss of direct correlation between observed relative peak intensities and the number of inserted electrons per formula unit (x value), since the population of the two possible core-hole states is considered to be entirely a final state phenomenon. On the other hand, for the case of complete single-site localization (non-metaUic case) the relative intensities are expected to truly represent the relative number of the two possible valence states before ionization and will not be affected by effects due to the final state of the core ionization process. 

The formal similarities between the above treatment and the second quantized approach are obvious. The last result of Eq. (8.14) resembles very much to the second quantized representation of a one-electron operator, cf. Eq. (4.27), and the second quantized counterparts of all previous formulae can easily be identified. The correspondences that have been obtained so far are collected in Table 8.1. This shows that creation operators are analogs of ket functions, while annihilation operators correspond to fera-functions. The eigenprojector i>particle number operator Nj = aj does. The resolution of identity is analogous to the operator of the total number of particles. The... 

We have already seen in Sect. 3.4 that the homogeneous halfwidth of ZPL is due to two-phonon Raman-like processes one phonon is created and the other phonon is annihilated. It is obvious that the quadratic interaction AC C yields two-tunnelon processes of such a type in the electron-tunnelon system and the probability of a simultaneous creation and annihilation of a tiumdon is proportional to (1 — f)f = (2ch(E/2kT)). Therefore, the effect of the quadratic electron-tunnelon interaction on the homogeneous halfwidth of ZPL is given by the following formula [89, 90]... 

We believe this behavior to be indicative of a calculation with open-shell character. The swapping of the HOMO positions between pairs of subsystems is actually equivalent to the exchange of electrons between these subsystems. On average, the situation probably amounts to the creation of unpaired electrons in two or more subsystems. The pseudo closed-shell density matrix formula in equation (14) is not equipped to deal with unpaired electrons, so convergence is difficult to achieve, and any results are nonphysical. 

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