Highly correlated electron gas

It is worth pointing out here, however, that 1/ (-T curves of this form are also predicted for a highly correlated electron gas (Chapter 4, Section 9). 

Fig. 4.6 The highly correlated electron gas of Brinkman and Rice. The doubly occupied and unoccupied states are shown. On the singly occupied sites the moment can change direction each time an electron or hole passes (Mott 1972b).

Brinkman and Rice (1970b) were the first to show that the highly correlated electron gas has these properties. They started with the Hamiltonian... 

We think that all these observations are to be explained by the assumption that metallic V203 is a highly correlated electron gas, as first suggested by Brinkman and Rice (1970b) and described in Chapter 4. The very low degeneracy temperature suggests that there may also be some mass enhancement of the carriers by polaron formation. Two electrons per atom would just half fill an ej band, so that the number of electron-like and hole-like carriers would be... 

It turns out that in this class of material we are in the presence of a highly correlated electronic gas, and a simple perturbative approach (weak U regime) does not work. This result is confirmed by the analysis of the density of states at the Fermi level which is, on the one hand, calculated from the tight binding model and on the other hand deduced from the experimental data (magnetism and low temperature electronic specific-heat experiments) (see Table 3.15). 

The free energy of activation at the QCISD(T)/6-31 H-- -G(d,p) level amounts to 21.1 kcal/mol. According to the authors, the large electron density redistribution arising upon cyclization makes it necessary to use extended basis sets and high-order electron correlation methods to describe the gas-phase thermodynamics, which indicates clearly the gas-phase preference of the azido species. However, the equilibrium is shifted toward the tetrazole as the polarity of a solvent is increased. For instance, SCRF calculations (e = 78.4) yield a relative free energy of solvation with respect to the cw-azido isomer of —2.4 kcal/mol for the tmns-zziAo compound and of —6.8 kcal/mol for the tetrazole isomer. At a much lower level, the... 

Gell-Mann, M., and Brueckner, K. A., Phys. Rev. 106, 364, Correlation energy of an electron gas at high density/ ... 

Inserting equation (6-14) into equation (6-12) retrieves the p4/3 dependence of the exchange energy indicated in equation (3-5). This exchange functional is frequently called Slater exchange and is abbreviated by S. No such explicit expression is known for the correlation part, ec. However, highly accurate numerical quantum Monte-Carlo simulations of the homogeneous electron gas are available from the work of Ceperly and Alder, 1980. 

The amount of high precision experimental structural data on conjugated polyenes is limited. Some structure results are presented in Table 5. In gas electron diffraction studies it is difficult to determine closely spaced bond distances accurately, because these parameters are highly correlated with the corresponding vibrational amplitudes. Today it is possible to calculate the vibrational amplitudes accurately, if the vibrational frequencies are known. This was, however, not the case when the GED studies presented in Table 5 were carried out. The observed differences between the terminal and central C=C bonds in the GED studies of traw.s-l,3,5-hexatriene and c/s-l,3,5-hexatricne are probably too large29. A very accurate X-ray study of traw.s-l,3,5-hexatriene has, however, been carried out also in connection with the preparation of this chapter4. Figure 4 shows the molecular structures of trans-1,3-butadiene and trans-l,3,5-hexatriene as found in the crystal lattice. 

As mentioned above, LSD yields a reasonable description of the exchange-correlation hole, because it satisfies several exact conditions. However, since the correlation hole satisfies a zero sum rule, the scale of the hole must be set by its value at some value of . The local approximation is most accurate at points near the electron. In fact, while not exact at m = 0, LSD is highly accurate there. Thus the on-top hole provides the missing link between the uniform electron gas and real atoms and molecules [18]. 

Table 2. Correlation energy of the uniform electron gas, comparing the highly-accurate parametrization of Perdew and Wang [4] with the LYP functional [10]...

The exchange-correlation functional for the uniform electron gas is known to high precision for all values of the electron density, n. For some regimes, these results were determined from careful quantum Monte Carlo calculations, a computationally intensive technique that can converge to the exact solution of the Schrodinger equation. Practical LDA functionals use a continuous function that accurately fits the known values of gas(/i). Several different... 

The electron gas thus obtained, though nearly antiferromagnetic , has no antiferromagnetic order. It is called highly correlated" because only some 10% or less of the atomic sites contain two (or no) electrons the others each contain one electron, of which the spin is resonating between the two possible positions. This behaviour is illustrated in Fig. 4.6. 

Fig. 4.13 Expected susceptibility as a function of temperature of a highly correlated antiferromagnetic electron gas near the value of BjV where moments disappear.

Apart from the drop in the susceptibility as we approach the transition, the increase in the Hall-coefficient ratio (Fig. 10.18) is further evidence for the model as we have seen, evidence from doped semiconductors shows that RH does not deviate from the free-electron form when EF lies in the pseudogap for current carriers in a highly correlated gas (unless antiferromagnetic order sets in). Also, the susceptibility increases with temperature this may be because the molecular dimers dissociate. 

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