A second-quantized theory is presented with the aim to study the nature and interactions of well-localizable chemical bonds in molecules. The basis set is partitioned by assigning each basis function to a chemical bond possessing two electrons. The Schrödinger equation within each limited basis subset is solved exactly for each bond, leading to correlated, strictly localized, intrabond wave functions. The total many-electron wave function $\Psi$ is defined as the antisymmetrized product of these strictly localized geminals. The second-quantized Hamiltonian $\hat{H}$ is partitioned as $\hat{H}={\hat{h}}^0+\hat{W}$, where the zeroth-order Hamiltonian ${\hat{h}}^0$ contains all terms which contribute to the energy of the strictly localized wave function $\Psi$, while the expectation value of $\hat{W}$, as calculated by $\Psi$, is zero. Since ${\hat{h}}^0$ is not simply the sum of intrabond Hamiltonians, the strictly localized wave function accounts for certain interbond interactions (inductive effects). The concept of bond creation and annihilation operators is introduced which formally shows Bose-type behavior since they refer to a two-electron composite system.

The existence in dynamical systems of chaotic bands delimited on both sides by period-doubling cascades is a general two-parameter phenomenon. Here we show evidence that, whenever these chaotic regions disappear, the bifurcation convergence rate undergoes a slowing down and asymptotically approaches the square root of the universal number $\delta =4.6692\mathrm{}\dots$. A simple renormalization-group analysis is performed to explain this critical behavior and its scaling properties. In particular a theoretical universal function describing the evolution of the convergence rate from ${\delta }^{\frac{1}{2}}$, to $\delta$ is given and numerically verified.

Taking into account the possibility of particle-antiparticle pair production, we have investigated the onset of Bose-Einstein condensation in an ideal relativistic Bose gas confined to restricted geometries. Through an extensive use of the Poisson summation formula, we have carried out an explicit evaluation of the summations over states appearing in the problem, which enables us to make a rigorous analysis of the temperature dependence of the thermogeometric parameter $y$ of the system in the case of a cubical enclosure under periodic boundary conditions. This, in turn, leads us to determine the growth of the condensate fraction $\frac{{\rho }_0}{\rho }$ as a smooth function of temperature from $T\gtrsim T_c$ down to $T=0\mathrm{}$ K. Finite-size corrections to the standard bulk results are obtained in explicit terms and are shown to be in complete agreement with the Fisher-Barber scaling theory for such effects. In the end, special geometries, such as narrow channels and thin films, are also examined.

A computer simulation based on statistically independent, noninteracting, shielded ions is developed. This simulation procedure differs from the usual molecular-dynamics approach in several respects and, for some problems, provides less-time-consuming and more-accurate results. Simulation results are compared with analytically known plasma functions and the basic limitations of the method are discussed.

The conditional-probability function plays a central role in the development of stochastic models for spectral line shapes in plasmas. We discuss some of the physical properties of this function using various analytic models as well as the results of a computer simulation.

The full Coulomb interaction between radiating ions and perturbing electrons is included in the calculation of Lyman-series spectral line profiles emitted by highly ionized hydrogenic radiators in a dense, hot plasma. Profiles from this calculation and from a similar calculation based on the dipole approximation for the radiator-perturber interaction are compared over a range of plasma densities. The significance of inelastic collisions between the radiator and perturbing electrons is also examined.

Dynamical features of solitary excitations in deoxyribonuclei acid (DNA) double helices are described by a revised theory in which the H-bonding energy between complementary base pairs has a more reasonable form. Four modes of sine-Gordon solitons which predict the existence of the open states in DNA double helices are found. By using the statistical-mechanical formalism which has been established previously, the average soliton number density in DNA double helices is estimated as a function of temperature and compared with the experimental results.

A gauge-invariant formulation of the de Gennes model for the nematic—to—smectic-$A$ transition is presented. In this formulation the energy associated with the gauge field $\overrightarrow{A}$ reduces to the Frank elastic energy with the application of the constraint ${\overrightarrow{n}}_0·\overrightarrow{A}=0\mathrm{}$ where ${\overrightarrow{n}}_0$ is the uniform equilibrium director and $\overrightarrow{A}$ is to be identified with deviations $\delta \overrightarrow{n}$ of the director from equilibrium. It is shown that thermodynamic quantities and renormalization-group recursion relations are gauge invariant. All gauge dependence appears in the exponent $\eta$ describing order-parameter correlations. The gauge invariance of a negative dielectric anisotropy smectic-$A$ in an external electric field is also studied.

Electrostatic stability properties of nonrelativistic non-neutral electron flow in a cylindrical diode with applied magnetic field $B_0{\hat{e}}_z$ are investigated within the framework of the macroscopic cold-fluid-Poisson equations. The electrostatic eigenvalue equation is derived for perturbations about the general class of slow rotational equilibria with angular velocity profile ${\omega }_b^{-}\mathrm{}\left(r\right)=\frac{V_{\theta b}^0\mathrm{}\left(r\right)\mathrm{}}{r}=\left(\frac{{\omega }_c}{2}\right)\{1-{[1-(\frac{4}{{\omega }_c^2{r}^2}\left)\int a^{r}d{r}^{\prime }{r}^{\prime }{\omega }_{\mathrm{pb}}^2(r^{\prime }\right)]}^{\frac{1}{2}}\}$. Here, ${\omega }_c=\frac{e{B}_0}{\mathrm{mc}}$, ${\omega }_{\mathrm{pb}}^2\mathrm{}\left(r\right)=\frac{4\pi n_b^0\mathrm{}\left(r\right)\mathrm{}{e}^2}{m}$ $n_b^0\mathrm{}\left(r\right)\mathrm{}$ is the equilibrium electron density profile, and the cathode is located at $r=a$ and the anode at $r=b$. Space-charge-limited flow is assumed with $E_r^0\mathrm{}(r=a)=0\mathrm{}$ and ${\phi }^0\mathrm{}(r=a)=0\mathrm{}$. The exact eigenvalue equation is simplified for low-frequency flute perturbations with $k_z=0\mathrm{}$ and ${|\omega -l{\omega }_b^{-}\mathrm{}(r\left)\right|}^2\ll {\omega }_c^2-{\omega }_{\mathrm{pb}}^2\mathrm{}\left(r\right)\mathrm{}$, assuming ${\omega }_{\mathrm{pb}}^2\mathrm{}\left(r\right)<{\omega }_c^2$ and a moderate-aspect-ratio diode $(R_0\gg b-a)$. In this regime, it is shown that $\frac{\partial n_b^0}{\partial r}\le 0$ over the interval $a\le r\le b$ is a sufficient condition for stability, and specific examples of stable oscillations (rectangular density profile) and weak resonant diocotron instability (gentle density bump) are analyzed in detail. Finally, the exact eigenvalue equation is solved numerically for a wide range of density profiles $n_0^b\mathrm{}\left(r\right)\mathrm{}$ and values of $\frac{{\omega }_{\mathrm{pb}}^2\mathrm{}\left(r\right)\mathrm{}}{{\omega }_c^2}$ leading to weak and strong instability driven by velocity shear with $\frac{\partial {\omega }_b^{-}\mathrm{}\left(r\right)\mathrm{}}{\partial r}\ne 0$.

In 1972, P. Pechukas [J. Chem. Phys. 57, 5577 (1972)] proposed "classical states" associated with Miller-Good transformations, semiclassical quantization, and the Hamilton-Jacobi equation. In this paper we use some of the concepts established by Pechukas and extend them to define generalized curvilinear coordinates $\overrightarrow{\gamma }$ associated with nonlinear dynamical systems with Hamiltonians of the form $H=T\left(\overrightarrow{\mathrm{p}}\right)+V\left(\overrightarrow{\mathrm{q}}\right)$. For reasons discussed in the paper we call these coordinates "nodal" coordinates. We show that a transformation to nodal coordinates is to a very good approximation dependent only on Cartesian variables. Using this approximation we demonstrate that, within the regular regime of phase space, canonical transformation to $\overrightarrow{\gamma }$ coordinates simplifies the analysis of the classical and quantum mechanics of dynamical systems. Our fundamental conclusions are as follows: (1) The solution of the Hamilton-Jacobi equation is separable in $\overrightarrow{\gamma }$ coordinates; (2) the WKB wave function ${\Psi }^{\mathrm{WKB}}$ is separable in $\overrightarrow{\gamma }$ coordinates; and (3) if the transformation to $\overrightarrow{\gamma }$ coordinates is made conformal (which we show we are allowed to do within the approximation stated above), then the Schrödinger equation and wave functions become separable. Finally, the concepts in this paper are discussed for systems of two degrees of freedom but can be generalized to more degrees of freedom.

Quantum chaos is defined by the following property: Simple dynamical variables (position, momentum, etc.) are represented by pseudorandom matrices, when the Hamiltonian is diagonal. As a consequence, the expectation values of these variables tend to equilibrium values which are insensitive to the initial preparation, for nearly all preparations involving many energy levels. Moreover, the fluctuations around these equilibrium values are, on the average, very small.

The approach proposed in the first part of this paper is illustrated by two examples. The first one involves the Hénon-Heiles model. A small wave packet, initially localized in the classically chaotic region, evolves so that $〈x〉$ and $〈x^2〉$ rapidly reach their equilibrium values. The second example makes use of a pair of rotators with nonlinear coupling. The matrix elements of $J_z$, in the energy representation, are pseudorandom in the energy range which is classically chaotic, but obey selection rules in the classically regular energy range.

A method is developed for improving the trial wave function for a particle in a magnetic field when the Rayleigh-Ritz energy variational principle is used. The energy operator depends on the vector potential, which must be chosen in some gauge. The expectation value of the energy with respect to an arbitrary trial wave function depends on the gauge of the vector potential. When the trial wave function is used to calculate the current, the equation of continuity for charge conservation is not, in general, satisfied. The arbitrary trial wave function can be multiplied by a phase factor which depends on the spatial coordinates. When the energy expectation value is minimized with respect to the phase function, the equation for charge conservation is obtained. This equation can be solved for the phase function, and a solution used in the energy expectation value to obtain a lower energy which is also independent of the choice of the gauge of the vector potential. The method is illustrated by applying it to an anisotropic harmonic oscillator in a constant magnetic field.

For stationary states of quantum charged systems in $\nu$ dimensions, $\nu \ge 2$, it is proven that the reduced-density matrices satisfy a set of sum rules whenever the clustering is faster than ${\mathrm{}\left|x\right|\mathrm{}}^{-(\nu +l)}$. These sum rules, describing the screening properties, are analogous to those previously derived for classical systems. For neutral quantum fluids, it is shown that the clustering cannot be faster than the decay of the force.

We consider the quasistatic displacement of a nonwetting fluid by a wetting one in a porous medium in the presence of buoyancy forces. A simple percolation model of this process is presented and analyzed both theoretically and by Monte Carlo simulation. It is shown that the fact that percolation is a critical phenomenon, with diverging correlation length at the critical point, has a significant effect on the physics of the system, in particular on the dependence of nonwetting phase residual saturation on the density contrast between the phases. An extension of these ideas to the case where the pressure field is generated by viscous rather than buoyancy forces is suggested.

The friction coefficient and the intrinsic viscosity of a linear flexible polymer chain are computed by perturbation theory near four dimensions in the limit of high molecular weights. The resulting series for the dynamical quantities are fitted by exponential functions in the chain length as dictated by the renormalization-group method. The exponents of the corresponding power laws, both in theta and under good-solvent conditions, are obtained to first order in $\epsilon =4-d$ and agree with those expected by dynamical-scaling arguments. The amplitudes are computed to second order in $\epsilon$ and an estimation of the error introduced by the $\epsilon$ expansion for $d=3\mathrm{}$ is given.

The present paper studies the formal aspects of the total-time-ordering-cumulant (TTOC) and the partial-time-ordering-cumulant (PTOC) relaxations of a quantum system, of few degrees of freedom, in weak interaction with a bath, both within and outside the Markovian limit. To this end, the general expressions connecting the matrix elements of the TTOC and PTOC relaxation superoperators with the bath correlation functions are determined. Special attention is paid to two particular cases: a system with a nonequidistant energy spectrum and a system with an equidistant energy spectrum. Discussions revolve mainly around the possibility of applying the secular approximation to the TTOC and PTOC master equations for the off-diagonal matrix elements of the reduced-density operator of the system.

Given a quantum system of a few degrees of freedom in weak interaction with a bath, the expressions which connect its total-time-ordering-cumulant and partial-time-ordering-cumulant relaxation with the corresponding spectral line shapes of dipolar absorption are deduced. For simplicity we consider a system with a nondegenerate and nonequidistant energy spectrum. A special study in the cases of isolated resonances and of a weak interference effect between resonances is made.

We show how quantum-counting quantum nondemolition measurements may be made using standard demolition counting techniques (e.g., photoelectron counting) for two oscillators coupled via a four-wave-mixing interaction. The analysis reveals how the state of one oscillator is reduced to a number eigenstate during the irreversible demolition counting process occurring in another coupled oscillator.

A class of recently discussed time-dependent classical Lagrangians possessing invariants is considered from a quantum-mechanical point of view. Quantum mechanics is introduced directly through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out explicit path integration, that the propagator for these time-dependent problems is related to the propagator of associated time-independent problems. The expansion of the propagator in terms of the eigenfunctions of the invariant operator and the quantum superposition principle follow naturally in our scheme. The theory is applied to obtain explicitly exact propagators for some illustrative examples.