Electron-hole symmetry

The interpretation of thermoelectric power data in most materials is a delicate job and this is particularly true for the case of carbons and graphites. In the case of SWCNTs the data are not consistent with those calculated from the known band structure which leads to much smaller values than observed. Hone et al. [11] suggest from their data that they may indicate that the predicted electron-hole symmetry of metallic CNTs is broken when they are assembled into bundles (ropes). 

As for the case of the two-leg ladder, we select the Local Configurations (LC) which are the much important for the ground state wave function. This selection of the relevant LC is made by combining energetic and symmetry considerations. For that purpose, we use the electron-hole symmetry operator, J, to classify the LC. On the Fock space of a single site n, the action of this operator are summarized as follows [40]... 

The electron-hole symmetry operators for several site is just given by the direct product of such single site operator J = H jn. 

The MC are the LC completely localized on the monomers. They give an appropriated description of the system in the strong dimerized limit. There is two kinds of MC in the (+) electron-hole symmetry sector [36, 37],... 

Notes that the LC where a. monoexcitation take place in a double bond, let s call it. S -LC, lower in energy than the D-LC, is in the (-) electron-hole symmetry sector therefore, it will be a local constituent for excited states [36],... 

There is more LC extended over two n.n. double-bonds within the (+) electron-hole symmetry class - and, of course, there is much more LC more extended - but we believe the two selected ones are the more important [37]. For instance, we saw that the S-LC is in the (-)class of symmetry, however, with the simple product form of the electron-hole symmetry operator, a LC with two S-LC - or more generally with an even number of S-LC - is in the (+) class of symmetry and should be considered here [37], However, this kind ofLC are sufficiently high in energy to be reasonably neglected. 

Table 2. Exact Pariser-Parr-Pople results[21] with standard parameters for the first three singlet excitations (in eV) of pyrene (Fig. 2) in >2a and electron-hole symmetry. In parenthesis, experimental excitations and assignments, from[52]. The...

In this figure, small thick arrows indicate electron spins symbolizing the electron occupation of the levels, and dashed vertical arrows indicate the possible absorption transitions. Electron-hole symmetry is preserved, so the electron or hole levels are at the same distance from their corresponding band edge, and the various transition energies do not depend on the sign... 

In practice, however, the electron-hole symmetry relation (16) is never perfectly satisfied. There are deviations due to small corrections neglected in the spectrum (2). Following a lattice compression, the electron spectrum is altered and these deviations magnify under pressure and tend to suppress the logarithmic singularity (17) T, thus rapidly decreases and even vanishes above some critical pressure (Fig. 5). 

The Hiickel model as applied to polyenes possesses a symmetry known as alternancy symmetry, since the polyene system can be subdivided into two sublattices such that the Hiickel resonance integral involves sites on different sublattices. In such systems, the Hamiltonian remains invariant when the creation and annihilation operators at each site are interchanged with a phase of +1 for sites on one sublattice and a phase of -1 on sites of the other. Even in interacting models this symmetry exists when the system is half-filled. The alternancy symmetry is known variously as electron-hole symmetry or charge-conjugation symmetry [16]. 

Hubbard chains at half-filling also possess electron-hole symmetry. This allows labelling of the states as either corresponding to covalent subspace, or corresponding to ionic subspace. The dipole-allowed excited states are found in the subspace while the... 

The electron-hole symmetry operator interchanges, with a phase, the creation and annihilation operators at a site. 

For this molecule, as only carbon atoms make up Ji-conjugation, electron hole symmetry appears 12-14 jjj which the coefficients of the i-th highest-occupied orbital are equal to those of the i-th lowest-unoccupied one after reversing the signs of the coefficients of alternate carbon atoms. This symmetry is obviously observed in the calculation result between HOMO and LUMO for phenylene vinylenes, and is also observed between HOMO-1 and LUMO+1, as is seen in Fig. 3. The electron distributions are approximately the same between an occupied orbital and the corresponding unoccupied one. As such, the transition moment from HOMO-1 to HOMO is approximately the same as that from LUMO to LUMO+1, and the transition energies too. 

The shapes of the orbitals for the other molecules of Fig. 2 are shown in Fig. 5. Molecules B and C of Fig. 2 have donors at the both ends. For B and C in Fig. 5, HOMO-1 spreads somewhat into extremities, and electron-hole symmetry does not hold either between HOMO and LUMO nor between HOMO-1 and LUMO-i-1. These shapes are beneficial for %(3). However, the orbital HOMO does not spread into the extremities. This means that transition from HOMO-1 to HOMO does not have so much moment as that of SBA. The orbital shape for molecules B and C is not so good as SBA. With respect to the orbitals of the molecules E and F, both HOMO and HOMO-1 spread, and the electron-hole symmetry does not hold. These are beneficial for Molecules D, E and F, which have amino at both ends and have nitrogen atoms at the X positions of Fig. 1, are expected to have large third order susceptibilities. These results correspond well with the experimental fact that their measured X is large and as good as conjugated polymers. The x values of SBA and SBAC, which is dichloro SBA, are 1.0 X 10 esu and 1.3 x lO" esu. The X( ) of PU-STAD is 1.5 X 10 esu, which is a derivative of molecule F in Fig. 2. 

M. Baitoul, J. W6ry, S. Lefrant, E. Faulques, J.-P. Buisson, and O. Chau-vet. Evidence of electron-hole symmetry breaking in poly(p-phenylene vinylene). Phys. Rev. B Condens. Matter, 68(19) 195203, Nov 2003. 

TEP experiments on YbCuAl were performed down to low temperatures. By analogy with transition metals, flie low-temperature TEP of 4f instability compounds was interpreted in a crude two-band model of a heavy band and a broad d band according to the Mott formula (Mott 1936). Analysis of the YbCuAl data above 1K in the two-band model clearly reveals the electron-hole symmetry in comparison with cerium compounds (Jaccard et al. 1985, Jaccard and Sierro 1982). 

ELECTRON-HOLE SYMMETRY (QUALITATIVE) Metallic <-Insulating- Metallic... 

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