Variational calculations of the ground-state energy of hard-sphere, helium-4, Coulomb, and nuclear Yukawa fluids are carried out. The long-range part $v(r>d)\mathrm{}$ and a constant part $\lambda$ at $r<d$ of the bare interaction are assumed to contribute to the average field. The correlation function $f\left(r\right)\mathrm{}$ is obtained by minimizing the two-body cluster contribution due to the remainder interaction, and the distance $d$ is determined by minimizing the energy as calculated with the hierarchy of hypernetted-chain-type equations. The convergence of two- and three-particle distribution functions given by these equations is discussed. The method is shown to be exact in the low-density limit, and the comparison of the present results with the available exact (standard) results in the high-density limit (region) verifies its accuracy over the entire density range. It is shown that the variationally obtained value of $d$ in the high-density Coulomb fluid can be understood with conventional meanfield theory. A solidification criterion is obtained by studying the energies in liquid and solid phases with the present $f\left(r\right)\mathrm{}$. This criterion explains the solidification of the Coulomb, hard-sphere, and helium systems, and the inability of the nuclear Yukawa system to solidify. The variational method in the form presented here could be very useful for studying complicated quantum systems.

Specific-volume measurements on methoxybenzylidene butylaniline (MBBA) and $p$-azoxyanisole (PAA) using the differential-dilatometer method are reported. The nematic-isotropic pretransition region over a range ± 15°K about the transition point is examined. The data, interpreted in terms of the de Gennes-Landau theory, are consistent with an $\alpha =0.5\mathrm{}$ critical exponent. The pseudocritical temperature derived from the volume data is significantly different from that reported for optical experiments.

The usual formulas for stimulated bremsstrahlung and inverse bremsstrahlung are modified to take into account the effect of electron-electron collisions in highly ionized plasmas. We consider only limiting cases in which the collision time is much larger or much smaller than the period of oscillation of the intense electromagnetic field.

We present computationally convenient theoretical methods for calculating the structural and thermodynamic properties of a fluid in the semiclassical limit where quantum effects are small. The concept of the effective two-body Mayer functions has been used to develop two approximations, one for the pair correlation function and the other for the Helmholtz free energy. The theory has been applied to the calculation of quantum effects on the radial distribution function and Helmholtz free energy of a fluid of hard spheres. It is shown that because of quantum effects a significant change in the structure of a hard-sphere fluid takes place in the neighborhood of the point of contact.

Measurements were made of the small-angle x-ray scattering intensity from argon near the critical point. After the scattering curves are corrected for all known effects except those resulting from irrelevant variables, a value of the critical exponent $\eta =0.09\mathrm{}\pm 0.02$ is obtained, while if a correction for the estimated effect of irrelevant variables is also made, $\eta =0.03\mathrm{}\pm 0.03$. The scattering data for argon therefore show that if irrelevant variables can be neglected, the critical exponent $\eta$ for argon is in good agreement with the value obtained by Warkulwiz, Mozer, and Green from small-angle neutron scattering data for neon. These values of $\eta$, however, are clearly greater than those obtained by calculations using high-temperature expansions and renormalizationgroup techniques. On the other hand, if the effects of irrelevant variables are given by the estimate used in correcting the scattering data, the $\eta$ value computed from the scattering curves agrees with the theoretical results.

The expressions for the fourth and sixth frequency moments of current correlation functions are simplified by performing the angular integrations associated with the three-body contributions to these moments. The resultant expressions are applicable both in the long-wavelength limit as well as for any momentum transfer. These expressions are then numerically evaluated for a liquid-argon-like system near its triple point using the molecular dynamics data of Verlet for the static pair-correlation function. We have used Kirkwood superposition approximation for the triplet correlation function and a low-order decoupling approximation for the quadruplet correlation function. The Maxwell relaxation time for the longitudinal mode is calculated using our computed numbers for the fourth and sixth moments. The results for the relaxation time are compared with those of other authors as well as with the experimental observations. It is concluded that the approach of Machida and Murase is comparatively adequate and has to be modified in order to get a better wave-number dependence of the relaxation time.

The continued-fraction representation of the generalized Langevin equation developed by Mori is used to calculated the coherent and incoherent scattering functions of simple classical liquids. A three-pole approximation, together with the assumption that the decay of the correlation function of the third random force is a function of $t^2$, is used to derive a microscopic expression for the Maxwell relaxation time for the longitudinal mode. No adjustable parameters are introduced, and the corresponding expression for $S(q, \omega )$ has the correct large wave-vector dependence at zero energy transfer. The method is extended to calculate the incoherent scattering function $S_s(q, \omega )$. Numerical predictions for the relaxation time in liquid argon are compared with other existing estimates, the present results are in very good agreement with the experimental observations. It is also found that our results give a better description of the behavior of $S(q, \omega )$ and $S_s(q, \omega )$ in liquid argon as revealed by the neutron inelastic scattering measurements of Sköld et al.

Reflection spectra of homogeneous fluid xenon are presented for the wavelength region $115<\mathrm{}\lambda <165\mathrm{}$ nm and densities ranging from that near the critical point to that at the triple point. For each atomic line in this spectral region two bands are observed: one broadened and somewhat shifted band corresponding to the atomic transition, and another distinct band corresponding in position to that of the exciton observed in the triple-point liquid. Dispersion analysis of the two (atomic and excitonic) ${}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ bands yielded the optical constants ${\epsilon }^{\prime }$, ${\epsilon }^{\prime \prime }$, $n$, and $k$ as a function of photon energy as well as parameters of the absorption bands. The results are interpreted in terms of exciton transitions appearing in momentary clusters of the fluid. The analysis of the parameters of the absorption bands was compared with the statistical fluctuations of the fluid density. It was estimated from this comparison that the ${}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ exciton can be formed in a momentary cluster of atoms provided two conditions are fulfilled: the cluster has at least the volume of 1.5×${10}^{-21\mathrm{}}$ ${\mathrm{cm}}^3$ and contains at least ten atoms.

The transport coefficients of hard-sphere, soft-sphere, Lennard-Jones, and one-component-plasma systems are expressed in terms of the corresponding internal entropies, to reveal some universal characteristics. The transport of all the inverse $n\mathrm{th}$-power repulsive potentials appears to be related, and can be numerically estimated by the hard-sphere model, although the hard-sphere transport is different. The entropy dependence of the coefficients of self-diffusion and (to a lesser extent) shear viscosity, for systems with purely repulsive potentials, is only slightly affected by the addition of an attractive potential.

A molecular theory of cholesteric liquid crystals is presented. By means of symmetry considerations we obtain a general form of the intermolecular potential containing chiral contributions, which are then shown to be responsible for forming the cholesteric phase. The temperature dependence of the cholesteric pitch is calculated. It is found that the pitch depends on temperature through a ratio of orientational order parameters in the form ${\left[\frac{{\sigma }_4\mathrm{}\left(T\right)\mathrm{}}{{\sigma }_2\mathrm{}\left(T\right)\mathrm{}}\right]}^2$. A general mean-field theory for binary cholesteric mixtures is also presented, along with a formula showing how the pitch depends on temperature and concentration. Under certain conditions, the formula reduces at fixed $T-T_c$ to a simple quadratic rational fraction in the concentration. This is consistent with experimental observation.

A model is presented to explain the observed high surface compressibility and low surface prsssure in the liquid-gas critical region of fatty-acid monolayers on water. The molecules are assumed to form tightly bound clumps, and the clumps then act as the basic particles of the gas. With several assumptions, the model is reduced to a three-parameter equation of state. With parameters chosen to give correct critical temperature, area, and surface pressure, the equation approximates the remaining data only qualitatively; but it is a distinct improvement over past theories, and much of the remaining discrepancy can be attributed to the assumptions made.

A multiple scattering theory, using the Ornstein-Zernike approximate for the correlation functions, previously obtained for thermal-neutron critical scattering from ferromagnets and fluids, is applied to x-ray critical scattering from fluids. These results are compared to the single-scattering theory using Fisher's approximate for the correlation function in order to obtain a multiple-scattering-induced $\eta$ for critical x-ray scattering. Comparison is made with experiment.

An exact density-functional theory is developed for nonuniform classical systems, expressing the free energy in terms of direct correlation functions. It is shown how this theory may be used in a practical fashion to remove several unjustified, although physically plausible, approximations made in earlier work. Further understanding of the approximations is obtained.

The Madelung rules for the order of the filling of successive electron shells in the ground states of neutral atoms (and hence the properties of the resulting periodic table) suggest that the sum $n+l\equiv k$ is in the nature of an "energy-ordering" quantum number. Thus for neutral atoms, the levels of the outermost electron are filled in the order of increasing $k$ values as the atomic number $Z$ is increased. For a constant $k$, the shells are filled in the order of increasing $n$ or decreasing $l$ values. The general applicability of these ordering rules suggests that they may also be relevant to the structure of the excited-state energy spectrum of the valence electron of the alkali-metal atoms. With the exception of lithium, it was found that the excited-state levels are energy ordered according to the increasing value of $k$ ("$k$ ordering"). For states with the same value of $k$, the levels with increasing energy $E_{\mathrm{nl}}$ exhibit a sequence of $l$ values (the "$l$ pattern") which is characteristic of the alkali-metal atom considered (Na, K, Rb, and Cs), and is independent of $k$. Important additional evidence for the $k$ ordering and the constant $l$ patterns has been obtained from the energy spectra of the singly ionized alkaline-earth atoms, i.e., the ${\mathrm{Mg}}^{+}$, ${\mathrm{Ca}}^{+}$, ${\mathrm{Sr}}^{+}$, ${\mathrm{Ba}}^{+}$, and ${\mathrm{Ra}}^{+}$ ions. Several possible implications of the present results are discussed.

A computational procedure for calculating correlated wave functions is proposed for ground and excited states of three-particle systems with Coulomb interaction. Calculations are performed for the muonic helium atom as an example. Hylleraas-type trial functions multiplied by appropriate angular factors are used as the basis of the correlated wave functions. The expansion coefficients and two nonlinear parameters are obtained variationally. Results for the states ${\mathrm{}\left(1s\right)\mathrm{}}_e{\mathrm{}\left(1s\right)\mathrm{}}_{\mu } {}_{}{}^{1,3}{}_{}{}^{}S$, ${\mathrm{}\left(1s\right)\mathrm{}}_e{\mathrm{}\left(2p\right)\mathrm{}}_{\mu } {}_{}{}^{1,3}{}_{}{}^{}P$, ${\mathrm{}\left(1s\right)\mathrm{}}_e{\mathrm{}\left(3d\right)\mathrm{}}_{\mu } {}_{}{}^{1,3}{}_{}{}^{}D$, ${\mathrm{}\left(1s\right)\mathrm{}}_e{\mathrm{}\left(4f\right)\mathrm{}}_{\mu } {}_{}{}^{1,3}{}_{}{}^{}F$, and ${\mathrm{}\left(1s\right)\mathrm{}}_e{\mathrm{}\left(5g\right)\mathrm{}}_{\mu } {}_{}{}^{1,3}{}_{}{}^{}G$ are listed.

The spin-density functional formalism is investigated with the objective of producing practical methods for calculating the spin (magnetic moment) density, $m\left(\overrightarrow{\mathrm{r}}\right)$, of atoms, molecules, and metals. Calculations are performed for the atoms Li, Na, K, N, P, Mn, and As in their spherically symmetric ground states using two approximations for the exchange-correlation functional: (i) the exact treatment of exchange with a local approximation for correlation (referred to as "SUHF-C"), (ii) the local spin-density (LSD) approximation for both exchange and correlation. The "SUHF-C" values of $m\left(0\right)\mathrm{}$ for Li, Na, K, N, P, Mn, and As are 108, 99, 91, 200, -30, 108, and 93% of experiment, respectively. For Li, Na, and K the LSD approximation yields 98, 109, and 106%, respectively. Away from the nucleus, in all the cases studied, the LSD and "SUHF-C" results for $m\left(\overrightarrow{\mathrm{r}}\right)$ are in close agreement. We conclude that for systems where there is a large direct contribution the LSD approximation should give better than 90% accuracy, even at the origin. The results using "SUHF-C" suggest that it has the capability of yielding accurate values of $m\left(\overrightarrow{\mathrm{r}}\right)$ for atoms and molecules.

Drift velocities of electrons in helium at 293 K have been measured in the range $2.5\le \frac{E}{N}\le 7.0$ Td (1 Td = ${10}^{-17\mathrm{}}$ V/${\mathrm{cm}}^2$), and analyzed to determine the momentum-transfer cross section between 4 and 12 eV. The estimated error bounds on the cross section are ±3% in the range $4\le \epsilon \le 7$ eV and ±5% in the range $7<\mathrm{}\epsilon \le 12$ eV. The implications of the new result to the problem of determining standard cross sections for electron-helium scattering below the first excitation threshold are discussed.

The $K$-shell ionization probability for silver has been measured at large angles up to 110°. A large increase has been found for the smallest impact parameters which can be explained by a dependence on the scattering angle calculated some years ago by Ciocchetti and Molinari. These theoretical results have been introduced in the binary-encounter approximation (BEA) as an additional correction together with the well-known relativistic, trajectory and binding corrections. In the range of impact parameter considered here, the agreement of this fully corrected BEA with the experimental data is satisfying. New results of the ionization probability for 2-MeV protons on Cu are given; as expected from the theory, the ionization probability does not depend on the scattering angle in this particular case.

The Magnus approximation (or sudden approximation) is applied to derive the transition amplitude and the cross section for $K$-shell ionization of atoms by heavy-ion impact. The target electron is described by a hydrogenic wave function and the projectile as a point charge moving along a straight-line trajectory. The transition amplitude for each partial wave of the ejected electron is expressed as an infinite (but rapidly converging) sum over hypergeometric functions. To obtain the total cross section, only integrals over impact parameter and the final electron momentum have to be evaluated numerically. The approach, because it is nonperturbative, should be particularly useful for treating collisions of light atoms with much heavier projectile ions. It also allows the study of the impact-parameter dependence of the ionization process. The connection with the Glauber approximation is pointed out.