The "atomic limit" (kinetic energy zero) of the Hubbard Hamiltonian has extreme degeneracy under very general circumstances. The kinetic energy or hopping term removes this degeneracy. It has been shown by Harris and Lange that there are important first-order manifestations of this fact. For instance, nonlocal correlations exist which invalidate usual Green's function truncation schemes. We present a mass operator perturbation scheme devised on the principle that, first of all, the role of kinetic energy as a breaker of degeneracy must be accounted for. A sort of Taylor's expansion of the mass operator in terms of the kinetic energy is found, the coefficients of which are functionals of the exact Green's functions at zero kinetic energy. The ambiguity due to the degeneracy is removed from these functions self consistently and thereby the degeneracy properly treated. We present a Green's function which accounts for all first-order effects of the kinetic energy.

The Mott-insulator and paramagnetic metal ranges of the half-filled energy band, governed by the Hubbard Hamiltonian, are discussed in terms of an electron-hole picture.

The electronic band structures of CsAu and LiAg have been determined by a combination of the Green's function (KKR) and the quantum-defect (QDM) methods. The construction of the phase shifts is based on the preliminary determination of the appropriate quantum defects. The latter are adjusted to take into account an estimated charge transfer of $0.55e$ from alkali to the noble metal. Our results show that despite the similarily of their constituents and of their crystal structures the two alloys exhibit markedly different features: CsAu is an extrinsic $n$-type semiconductor with a 2.7-eV minimum gap in the (111) direction, while LiAg has the band structure of a metallic alloy, with the (cubic) Brillouin zone full except for small empty regions at the corners. Our results for CsAu are in agreement with available data and with one previous calculation. For LiAg, there are no experimental data available for comparison.

The energy band structure of face-centered cubic metals Ca, Sr, and Ba was investigated by a pseudopotential-like approximation, using the screened model potential of Abarenkov, Heine, and Animalu. All three of the metals exhibit a negative energy gap at atomospheric pressure, the magnitude of which tends to zero with increasing pressure. According to a symmetry demanded degeneracy along the line WL of the Brillouin zone, the transition to semiconductors with positive energy gap seems to be unlikely. The Fermi surface consists of holes in the first zone around the points $W$ and electrons in the second zone around the points $L$. The dimensions of these surfaces are also decreasing under pressure. The Fermi surface of Ca was studied in detail. Numerical results for the de Haas-van Alphen periods cyclotron masses, are given. The pressure dependence of electrical resistivity of Ca was calculated and found to be in qualitative agreement with the experiments.

We report experimental information which elucidates important features of the ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ alloy system. By means of observations of the Shubnikov-de Haas effect on undoped and doped samples we have followed the transition from a band structure near the Fermi level which is essentially that of pure Bi to one which resembles that of pure Sb. Supplementary information has been obtained from transport measurements. In both Bi and Sb, the conduction band is at the $L$ point in the Brillouin zone. A mirror-image band lies not far below. The upper $L$-point band is the conduction band throughout the alloy system. Our model of the transition involves two processes: (1) The descent of the $T$-point band (the valence band in Bi) relative to the $L$-point bands, a transition to six valence maxima at the nearby $H$ points, and their subsequent rise past the Fermi level to become the valence bands in Sb; (2) the approach, crossing, and separation of the $L$-point bands. At 4.2°K, the descent of the $T$-point maximum below the $T$-point minimum and the $L$-point crossing both occur near $x=8\%$; consequently, there is a transition from semimetallic to direct-gap ($L$-point) semiconducting behavior. The most direct determination of the $L$-point gap is obtained from the nonparabolic behavior of the very small transverse mass components. Strong corroborative evidence is obtained from transport measurements in the temperature range 4°-30°K.

Groves and Paul have, on the basis of experimental evidence, given strong arguments that in grey tin the valence and conduction bands touch at k=0. This suggests that at sufficiently low temperatures this phase may undergo an excitonic transition. The critical temperature may be estimated from the binding energy $E_B$ of an exciton: $k{T}_c\approx E_B=\frac{\left(\frac{\mu }{m}\right)}{{\epsilon }^2}$ where $\mu$ is an effective reduced mass and $\epsilon$ is an effective dielectric constant. From the work of Groves and Paul and of Ewald and co-workers, we estimate $\left(\frac{\mu }{m}\right)\approx 0.02$. If one accepts the estimate $\epsilon =24\mathrm{}$ given by Lindquist and Ewald one finds $T_c\approx 5°$K. From group theoretical considerations one finds that the new phase would belong to one of three space groups of lower symmetry and in one case would be weakly ferromagnetic. These conclusions are, however, highly tentative because of (1) uncertainty about the dielectric constant $\epsilon$ and (2) the suppression of the transition by small concentrations of impurities. Experimental studies of the dielectric constant would be highly desirable and interesting.

An indirect band-gap semiconductor may be converted to a semimetal, or vice versa, by application of pressure. At low temperature, an excitonic phase or some other anomaly must occur in the neighborhood of the transition pressure. We also discuss the direct-band-gap case and the case where a band gap is zero by symmetry, as in gray tin.

Recent electrical resistivity measurements have shown that the hexagonal form of stoichiometric NiS exhibits an abrupt metal-to-semiconductor transition at 264°K. Neutron diffraction studies have shown that a first-order paramagnetic-to-antiferromagnetic transition also occurs at 264°K. No crystal lattice distortion is observed at the transition nor detected at 4.2°K, suggesting that this may be a transition of the kind considered by Adler and Brooks. The measured magnetic moment at 4.2°K is 1.66±0.08 μβ and at 260°K it is 150.±0.10 μβ indicating that the sublattice magnetization is within 10% of saturation immediately upon ordering. The neutron data also shown that no more than about 1% of the Ni atoms migrate to tetragonal interstitial sites on warming from 4.2°K to room temperature. Thus, Ni atom migration apparently plays no part in this transition. The powder magnetic susceptibility is 2.24×${10}^{-6\mathrm{}}$ emu/g at 300°K and is virtually temperature-independent above the transition. $\chi$ increases abruptly at 264°K by about 15% and exhibits some field-cooling effects. Studies on the compounds ${\mathrm{Ni}}_X\mathrm{S}$, for $X=1.01, 0.99, 0.97, \mathrm{and} 0.94$ show that excess sulfur lowers the transition temperature.

Data on the electrical, optical, magnetic, and structural properties of ${\mathrm{Ti}}_2$${\mathrm{O}}_3$ are reviewed; the differences between the transition encountered in ${\mathrm{Ti}}_2$${\mathrm{O}}_3$ and ${\mathrm{V}}_2$${\mathrm{O}}_3$ are particularly stressed. Perusal of the available experimental and theoretical work leads to the conclusion that the transition in ${\mathrm{Ti}}_3$${\mathrm{O}}_3$ can be explained without invoking antiferromagnetic ordering. This is of particular relevance in light of recent experimental work which casts serious doubt on the commonly accepted hypothesis that ${\mathrm{Ti}}_2$${\mathrm{O}}_3$ is antiferromagnetic. An alternative interpretation of the transition is offered, according to which a gradual change in the position and width of two close-lying bands results in a change of ${\mathrm{Ti}}_2$${\mathrm{O}}_3$ from a narrow-gap semiconductor to a semimetal. It is pointed out that a simple theory based on this model is in essential accord with the available experimental data, and that it satisfies the requirements imposed by crystal symmetry.

As shown by Adler and Brooks, the lattice distortion induces the change of electronic states in V${\mathrm{O}}_2$ and ${\mathrm{V}}_2$${\mathrm{O}}_3$. As a consequence, the dipole-dipole interaction of lattice vibration is modified. Taking into account this electron-lattice interaction and solving motion of lattice vibration self-consistently, the following results were obtained. The instability of the lattice-vibration mode of the vanadium ions occurs at $T_c$ on decreasing temperature, on increasing the temperature from the semiconductor region, the instability occurs at a higher temperature $T_0>T_c$. The temperature dependence of the displacement and the dielectric constant of the lattice system is given near the first-order phase transition temperature, and this explains the hysteresis of NMR and conductivity measurements. The parameters obtained are reasonable. This theory also predicts the anomalous behavior of diffuse scattering of x ray and temperature dependence of the frequency spectrum of the lattice vibration.

The explanations given by Brooks and Adler for the electronic properties of V${\mathrm{O}}_2$ and similar compounds is compared to that of the author, which was based on a model due to Fröhlich in which the electrons are localized.

Photoemission quantum yield and energy distribution curves have been measured for V${\mathrm{O}}_2$ in the tetragonal (100°C) and monoclinic (25°C) phases over the photon energy range from 7 to 11.5 eV. The V${\mathrm{O}}_2$ studied was a polycrystalline film of several microns thickness formed by oxidizing vanadium. The experimental data, although distorted by strong energy-dependent inelastic scattering and by work-function variations due to the polycrystallinity, shows most of the major features expected on the basis of a simple model of V${\mathrm{O}}_2$ and supports the interpretation of optical data given by Barker ${\mathit{et}\mathit{al}.\mathrm{}}^1$ In both the semiconductor and metallic phases, there is evidence of a high density-of-states band starting approximately 2.5 eV below the Fermi level. This band is believed to be due primarily to the oxygen $2p$ orbitals in V${\mathrm{O}}_2$. The photoelectric yield is considerably higher at 100°C than at 25°C consistent with the formation of a forbidden gap at temperatures below 67°C. This is accompanied by expected changes in the energy distribution curves particularly for these electrons photoexcited from states near the Fermi level.

The optical constants of V${\mathrm{O}}_2$ have been determined between 0.25 and 5 eV, both below and above the semiconductormetal transition temperature $T_t=340\mathrm{}°$ K. Reflectivity and transmission spectra have been measured on both single crystals and thin films. The reflectivity spectra of the bulk crystals were measured with $E\perp c$ axis in the tetragonal phase (or $\perp a$ axis in the monoclinic phase) and $E\parallel$ to these axes. While there are some differences in magnitude between the dielectric constants obtained from thin-film and single-crystal measurements, the structural features are in good agreement. Below $T_t$ there are four prominent absorption peaks centered near photon energies of 0.85, 1.3, 2.8, and 3.6 eV. Above $T_t$ metallic free carrier absorption is observed below 2.0 eV, but the same two absorption peaks near 3 and 4 eV are present. The energy location and polarization dependence of these two higher energy peaks can be related to similar absorption peaks in rutile, and are interpreted using the rutile band structure. The results are consistent with a picture in which filled bands arising primarily from oxygen $2p$ orbitals are separated by approximately 2.5 eV from partially filled bands arising primarily from vanadium $3d$ orbitals. Transitions from the filled $2p$ bands are responsible for the high-energy peaks in the optical absorption in both the high- and low-temperature phases. In the high-temperature metallic phase there is evidence that there is overlap among the $3d$ bands such that at least two bands are partially occupied by the extra $d$-electron per vanadium ion. In the low-temperature semiconductor phase a band gap of approximately 0.6 eV opens up within the $3d$ bands separating two filled bands from higher lying empty bands. The two absorption peaks at 0.85 and 1.3 eV are due to transitions from these two filled bands.

Transition-metal oxides and sulfides can be insulators, semiconductors, or metals, or undergo metal-nonmetal transitions. The experimental situation in these materials is briefly reviewed. Pure Bloch-Wilson theory predicts that many of the insulators should be metallic, so this class cannot be explained in terms of pure band theory. However, it now appears likely that band theory can be adapted to describe most of the materials that exhibit metal-nonmetal transitions.

Measurements have been made of the complex dielectric constant ${\epsilon }^{\prime }-i{\epsilon }^{\prime \prime }$ of sodium-ammonia solutions at 10 GHz. The real part of the dielectric constant ${\epsilon }^{\prime }$ was a very dramatic function of solution concentration. At very low concentrations ${\epsilon }^{\prime }$ was close to the pure ammonia value, then increased slowly with concentration in the range $\chi =2\mathrm{}\times {10}^{-3\mathrm{}}$ to $\chi =2\mathrm{}\times {10}^{-2\mathrm{}}$. This increase has been attributed to the presence of permanent dipoles, and a dipole moment of approximately ${10}^{-16\mathrm{}}$ esu at room temperature was determined. At $\chi =2\mathrm{}\times {10}^{-2\mathrm{}}$ the value of ${\epsilon }^{\prime }$ decreased very abruptly to large negative values. This decrease is interpreted as the nonmetal to metal transition, indicating a very rapid decrease in the mass of the negatively charged carriers as the concentration is increased. In this same concentration range ${\epsilon }^{\prime \prime }$, the imaginary part of the dielectric constant, increased rapidly as would be expected from the increasing dc conductivity. Temperature-variation studies of the dielectric constant have also been made and are consistent with the known temperature behavior of the solution's conductivity, i.e., the lower the temperature the higher the concentration required to reach the metallic concentration region and hence produce a negative real part to the dielectric constant.

A review is given of the properties of metal-ammonia solutions together with a summary of the evidence for the existence of a metal-nonmetal transition.

This article presents the results of conductivity and density measurements of liquid, gaseous, and supercritical mercury up to 1700°C and 2100 bar. Beyond the critical temperature of 1490°C the conductivity varies continuously from ${10}^{-4\mathrm{}}$ to ${10}^2$ ${\mathrm{\Omega }}^{-1\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$ if the density is increased from 2 to 6 g/${\mathrm{cm}}^3$. This variation is an example of nonmetal-metal transition and is discussed accordingly. The activation energy of conductivity in the supercritical fluid decreases markedly with increasing density and approaches very small values around 5 g/${\mathrm{cm}}^3$. Conductivity data for dense gaseous mercurycesium mixtures are given.

Various anomalous optical, magnetic, and transport phenomena have been found in Eu-chalcogenide alloys. Anomalous optical properties are explained by the magnetic exciton in which a hole in the $4f$ conduction band and an electron (for optically active magnetic exciton) in the $5d$ conduction band make a bound state. Strong $d-f$ exchange interaction between the magnetic-exciton electron of $5d$ character and the $4f$ electrons in the surrounding ${\mathrm{Eu}}^{2+}$ ions is the origin of the anomaly. Anomalous magnetic properties in dilute ${\mathrm{R}}^{3+}$ alloys are explained by the properties of the giant spin molecule which comes from the strong $s-f$ exchange interaction between the impurity electron and the $4f$ electrons at the central $R^{3+}$ and the surrounding ${\mathrm{Eu}}^{2+}$. The anomalous transition from the hopping type to the metallic-impurity band conduction in dilute ${\mathrm{R}}^{3+}$ alloys is also explained by the temperature dependence of the activation energy due to the $s-f$ exchange interaction in the magnetic-impurity state.

A general survey of the idea of the metal-nonmetal transition is given with particular emphasis to the following points: (a) is the transition of first order; (b) can one observe the transition; and (c) how is the transition affected when it occurs in ionic lattices or polar liquids.