Electron Gas Theories

Figure 9. The measured momentum density of an aluminium film. In the left panel we show the measured momentum density near the Fermi level (error bars), the result of the LMTO calculations (dashed line) and the result of these calculations in combination with Monte Carlo simulations taking into account the effects of multiple scattering (full line). In the central panel we show in a similar way the energy spectrum near zero momentum. In the right panel we again show the energy spectrum, but now the theory is that of an electron gas, taking approximately into account the effects of electron-electron correlation (dashed) and this electron gas theory plus Monte Carlo simulations (solid line).

The number of delocalizable electrons can be increased. This is possible because the activation energy of the dark conduction, like the intramolecular excitation energy of the electrons, decreases with increasing number N of delocalizable electrons in agreement with the electron gas theory. Hence, if the relationship between and N for the open [Eq. (48)] and cyclic [Eq. (49)] electron systems is represented graphically, the values obtained from conductivity measurements will generally lie between the two curves see e.g. 13>64>. 

Spackman MA (1986) Atom-atom potentials via electron gas theory. J Chem Phys 85 6579-6586... 

Jackson, M. D., and R. G. Gordon (1988a). Electron-gas theory of some phases of magnesium oxide. Phys. Rev. B38, 5654-60. 

Wolf, G. H., and M. S. T. Bukowinski (1988). Variational stabilization of the ionic charge densities in the electron-gas theory of crystals applications to MgO and CaO. Phys. Chem. Mineral. 15, 209-20. 

The method is based on the free-electron gas theory, where the exchange potential is given by... 

The absorption lines of the low temperature photoreaction products in TS-6 monomer crystals are summarized in the diagram of Fig. 7. The correlation of the A, B, C,. .. photoproduct series to diradical DR intermediates and of the b, c, d,... photoproducts to asymmetric carbene AC intermediates is based on the ESR experiments discussed below. The correlation of the y, 8,6,... series to stable oligomers SO is based on their thermal and optical stability. The correlation of dimer, trimer, tetramer,... molecules follows from the chemical reaction sequences observed in the time resolved optical and ESR measurements as well as from the widths of the one-dimensional potential wells used in the simple electron gas theory , which already has proved successful in its application to dye molecules. Following Exarhos et al. the explicit dependence is given by... 

For a homogeneous electron density in a given volume V, the kinetic energy can be derived form the electron gas theory. For a pair of homogeneous electron densities Pa t) = phA and Pb t) = phg) the analytic form of T ad pA, Pb] obtained using Thomas-Fermi kinetic energy functional applied to Ts[pA + pb], Ts[pa], and Ts[pb] reads ... 

Electrical Conductivity. Metal crystals have already been shown to consist of (a) positive ions arranged on a characteristic ordered lattice, and (b) an equivalent number of free electrons, which we have described as forming the metallic electron cloud. These free electrons behave in many ways like the molecules of a gas, and it is interesting to note that it has been found possible to apply to them the kinetic theory of gases. This has resulted in the building up of an electron gas theory/ but this cannot, however, be further discussed here. 

C. F. Curtiss communicated to the author that Eq. (49) may be deduced from D. Enskog s treatment of gas diffusion. Further applications to the ordinary kinetic gas theory, and to the electron gas theory, were successfully made by Ljimggren, through calculations of on the basis of molecular kinetics. 

On electron gas theory Ae = fe /4ma and on the model a = nd. We see Ae will decrease as the number of conjugated ir electrons increases. While in principle a protein may extend the conjugation path for tt electrons, in practice this can hold for only a few adjacent amino-acid residues if hemoglobin in the dry state is any guide, since the A obtained for the over-all conductivity of this molecule by Cardew is 2.7 e.v., the globin effectively insulating the haem from each other in the crystal. Hydration may in certain instances lower this value, but this is still to be investigated fully. 

A simplified version of magnetic field DFT, namely, an electron gas theory that depends on the sum of the atomic electron densities in the presence of an external field, was used to estimate the FC contribution to the so far unmeasured Xe-H and Xe- Xe couplings as functions of the interatomic distance. Couplings reported are within the order of /xHz to mHz for interatomic distances of physical interest. 

Within the context of Kohn-Sham theory, the assumption underlying the LDA is that each point of the nonuniform electron density is uniform but with a density corresponding to the local value. In the LDA for exchange, the wavefunction is therefore assumed to be a Slater determinant of plane waves at each electron position. The corresponding pair-correlation density g[ r, r p(r) is thus the expectation of Eq. (66) taken with res[ ct to this Slater determinant, with the resulting expression then assumed valid locally. (The superscript (0) indicates the result is derived from uniform electron gas theory.)... 

Muhlhausen, C. and Gordon, R.G. (1981) Electron gas theory of ionic crystals, including many-body effects P/jyr. Rev. B23, 900-923 Muhlhausen, C. and Gordon, R.G. (1981) Density-functional theory for the energy of crystals test of the ionic model Phys. Rev. B24, 2147-2160. 

Hemley, R.J., Jackson, M.D. and Gordon, R.G. (1985) Lattice dynamics and equations of state of high-pressure mineral phases studied with electron gas theory EOS Trans. AGU 66, 357. 

In order to make contact with existing work, we have also performed static lattice calculations in the constrained cubic and the fully relaxed orthorhombic structures. We find that the static energy of cubic perovskite exceeds that of orthorhombic perovskite by 0.017 Hartrees per formula unit at ambient pressure. This value is similar to that found by Hemley et al. [69] with a different electron gas model. Both estimates are roughly three times smaller than those based on independent ab initio electronic structure calculations, all of which give an energy difference of about 0.05 Hartrees [23, 81, 75]. Evidently, the electron gas theories underestimate the energy... 

If the dopant species are monatomic ions, the derivation of their mutual interactions fall into two classes. Using quantum chemical electron gas theory, electronic charge distributions derived from Hartree-Fock calculations on the ions are allowed to interact at various intemuclear distances. " In another method lattice simulation calculations are performed to fit the constants in Eqs. (7) or (8) to measured bulk data, such as elasticity tensor components, dielectric constants and/or lattice structures, thereby providing empirically derived potentials. 

Kinetic theory, non-equilibrium statistical mechanics and non-equilibrium molecular dynamics (NEMD) have proved to be useful in estimating both straight and cross-coefficients such as thermal conductivity, viscosity and electrical conductivity. In a typical case, cross-coefficient in case of electro-osmosis has also been estimated by NEMD. Experimental data on thermo-electric power has been analysed in terms of free electron gas theory and non-equilibrium thermodynamic theory [9]. It is found that phenomenological coefficients are temperature dependent. Free electron gas theory has been used for estimating the coefficients in homogeneous conductors and thermo-couples. 

Onsager relations are satisfied, showing that free electron gas theory is consistent with thermodynamic theory. The free electron theory correctly predicts the temperature dependence of thermo-electric power. Similarly, the interpretation of the phenomenon of thermo-osmosis of gases on the basis of non-equilibrium thermodynamics and kinetic theory of gases is mutually consistent. 

The DFT method has a long history behind it, which began with Thomas, Dirac, Fermi, etc. At the beginning the successes were quite modest (the electron gas theory, known as the Xa method). Real success came after a publication by Jan Andzelm and Erich Wimmer. The DFT method, offering results at a correlated level for a wide spectrum of physical quantities, turned out to be roughly as inexpensive as the Hartree-Fock procedure - this is the most sensational feature of the method. 

A simple electron gas theory is applied on the ootical absorption series of Figure 3-5. The spectral positions of the optical absorptions as a function of the conjugation length 1 is explicite-ly given by... 

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