We evaluate trace formulas for various perturbations of two-dimensional harmonic oscillators. Such systems arise naturally in the expansion of generic potentials about local minima. For large enough perturbations, the usual theory for isolated orbits applies and we can reproduce the long and medium-range oscillations in the density of states in terms of the shortest periodic orbits. For small perturbations, or low energies, the Gutzwiller amplitudes diverge due to the approaching degeneracy of the harmonic oscillator. We employ a perturbative analysis of the classical dynamics to give a treatment of the trace formula that is valid near the degenerate harmonic regime. First-order perturbation theory works for generic cases. For certain potentials, such as Hénon-Heiles, discrete symmetries lead to a null result at first order and second-order calculations are necessary to capture the dominant features.

Quantum dissipation in a thermal environment is investigated, using the path-integral approach. The reduced density matrix of the harmonic oscillator system coupled to a thermal bath of oscillators is derived for an arbitrary spectrum of bath oscillators. Time evolution and the end point of the two-body decay of unstable particles are then elucidated: After early transient times unstable particles undergo the exponential decay, followed by the power-law decay and finally ending in a mixed state of residual particles containing contributions from both on and off the mass shell, whose abundance does not suffer from the Boltzmann suppression.

A divergence property of approximating sequences $\{{\psi }_n={\sum }_{m=1\mathrm{}}^n{c}_{\mathrm{nm}}{\phi }_m\}$ that converge in the norm of the Hilbert space $L_2\left(R^N\right)$ to a fast-decay function $\psi$ is studied. The expansion ${\psi }_n$ can be a Fourier one or obtained by solving an eigenproblem by the Ritz method and the basis set $\{{\phi }_m\}$ need not be orthogonal in $L_2\left(R^N\right)$. The notion of uniform boundedness is used to show that if $\{{\psi }_n\}$ is nonuniformly bounded, then it diverges from its correct limit $\psi$ in such a way that there is an increasing separation between the asymptotic tails of ${\psi }_n$ and $\psi$ as $n$ increases. The analytical and numerical examples show that the rate of this divergence may be exponential, hence the divergence of expectation-value sequences $\{S\left({\psi }_n\right)\}$ is proved for some operators $S$ whose correct expectation value $S\left(\psi \right)$ depends mainly on the long-range behavior of $\psi$. The compatibility between several convergence properties of the approximating sequence $\{{\psi }_n\}$ and basis set properties with the nonuniform boundedness property is shown. We show that the well-known property of some trial wave functions to generate correct expectation values of some operators and incorrect values for other operators is connected with the property of nonuniformly bounded sequences $\{{\psi }_n\}$ to converge correctly on a finite region and diverge on its complementary one, hence it is proved that correct expectation values can be obtained from a nonuniformly bounded sequence by using a suitable limiting procedure. As model examples, Fourier and Ritz expansions of the ground state $\psi {=2Z}^{3/2}{e}^{-\mathrm{Zr}}$ of a hydrogenlike atom are considered.

The relation between quantum interference and classical interference is discussed in terms of the Wigner function and an analogous classical expression. For two displaced coherent states and two displaced electric-field Gaussian pulses, both the quantum-mechanical and classical Wigner functions exhibit oscillatory behavior. Classical analogs of squeezing, photon-number oscillation, and the $Q$ function are presented.

We generalize the procedure of entanglement swapping to obtain a scheme for manipulating entanglement in multiparticle systems. We describe how this scheme allows one to establish multiparticle entanglement between particles belonging to distant users in a communication network through a prior distribution of singlets followed by only local measurements. We show that this scheme can be regarded as a method of generating entangled states of many particles and compare it with existing schemes using simple quantum computational networks. We highlight the practical advantages of using a series of entanglement swappings during the distribution of entangled particles between two parties. Applications of multiparticle entangled states in cryptographic conferencing and in reading messages from more than one source through a single measurement are also described.

We present a family of additive quantum error-correcting codes whose capacities exceed those of quantum random coding (hashing) for very noisy channels. These codes provide nonzero capacity in a depolarizing channel for fidelity parameters $f$ when $f>\mathrm{}0.809\mathrm{}44$. Random coding has nonzero capacity only for $f>\mathrm{}0.810\mathrm{}71$; by analogy to the classical Shannon coding limit, this value had previously been conjectured to be a lower bound. We use the method introduced by Shor and Smolin of concatenating a nonrandom repetition (cat) code within a random code to obtain good codes. The cat code with block size five is shown to be optimal for single concatenation. The best known multiple-concatenated code we found has a block size of 25. We derive a general relation between the capacity attainable by these concatenation schemes and the coherent information of the inner code states.

When a confined system interacts with its walls (treated quantum mechanically), there is an intertwining of degrees of freedom. We show that this need not lead to entanglement, hence decoherence. However, it will generally lead to error. The wave-function optimization required to avoid decoherence is also examined.

The formation of excited projectile states via Coulomb excitation is investigated for hydrogenlike and heliumlike bismuth projectiles $\mathrm{}(Z=83\mathrm{})\mathrm{}$ in relativistic ion-atom collisions. The excitation process was unambiguously identified by observing the radiative decay of the excited levels to the vacant $1s$ shell in coincidence with ions that did not undergo charge exchange in the reaction target. In particular, owing to the large fine-structure splitting of Bi, the excitation cross sections to the various $L$-shell sublevels are determined separately. The results are compared with detailed relativistic calculations, showing that both the relativistic character of the bound-state wave functions and the magnetic interaction are of considerable importance for the $K$-shell excitation process in high-$Z$ ions such as Bi. The experimental data confirm the result of the complete relativistic calculations, namely, that the magnetic part of the Liénard-Wiechert interaction leads to a significant reduction of the $K$-shell excitation cross section.

We present a self-interaction-free relativistic density-functional theory (DFT). The theory is based on the extension of our recent nonrelativistic DFT treatment with optimized effective potential (OEP) and self-interaction correction (SIC) [Phys. Rev. A 55, 3406 (1997)] to the relativistic domain. Such a relativistic OEP-SIC procedure yields an orbital-independent single-particle local potential with proper long-range Coulombic $(-1/r)$ behavior. The method is applied to the ground-state energy calculations for atoms with $Z=2\mathrm{}$–106. A comparison with the corresponding nonrelativistic OEP-SIC calculations and other relativistic calculations is made. It is shown that the ionization potentials (obtained from the highest occupied orbital energies) and individual orbital binding energies determined by the present relativistic OEP-SIC method agree well with the experimental data across the Periodic Table.

The possibility to symmetry assign x-ray absorption spectra using x-ray emission spectroscopy is demonstrated. The oxygen $K$ spectra of ${\mathrm{CO}}_2$ are used to illustrate the possibility to assign core excitations through excitation-energy and angular dependence of resonantly excited x-ray emission spectra. The Rydberg-associated structures just below the ionization threshold in the O $K$ x-ray absorption spectrum are assigned to be mainly due to excitations to orbitals of ${\sigma }_g$ character. The results, which contradict some previous literature assignments, are supported by quantum-chemical calculations.

The x-ray-absorption cross section above the $L$ edges in xenon gas has been measured using synchrotron radiation. Multielectron excitation effects are investigated over a few hundred eV region from the Xe $L_{\mathrm{I}},$ $L_{\mathrm{II}},$ and $L_{\mathrm{III}}$ edges and the contributions from the effects of shake-up and shake-off to the photoabsorption spectrum are elucidated. Several multielectron excitation features were detected and analyzed by the theoretical energy of the shake-up process. Previous observations of the $\left[2\right(s,p\left)5\right(s,p\left)\mathrm{}\right]$ and $\left[2\right(s,p\left)4d\right]$ transitions have been confirmed. $\left[2\right(s,p\left)4p\right]$ transitions are clearly identified in the present study. It is very difficult to detect the $\left[2s3s\right]$ multielectron transition with the sensitivity of $\sim 2.0\times {10}^{-4}$ at the $L$ edge jump. The $\left[2s3d\right]$ edge was distinguished using the first derivative of the absorption curve. A new transition is found between $\left[2\right(s,p\left)4p\right]$ and $\left[2\right(s,p\left)4d\right]$ transitions and identified as a three-electron transition $\left[2\right(s,p\left)4d5p\right].$ Moreover, the difference in the behavior of the multielectron excitation as a function of x-ray energy among the $L_{\mathrm{I}}$ and $L_{\mathrm{II}},$ or $L_{\mathrm{III}}$ edges is confirmed and investigated.

The $F=3\mathrm{}$ to $F=2\mathrm{}$ hyperfine transitions in the $1s$ ground state of the two isotopes ${}^{185}{\mathrm{Re}}^{74+}$ and ${}^{187}{\mathrm{Re}}^{74+}$ were measured to be $(4560.5\mathrm{}\pm 3)\AA$ and $(4516.9\mathrm{}\pm 3)\AA ,$ respectively, using emission spectroscopy in an electron beam ion trap. After applying appropriate corrections for the nuclear charge distribution and QED effects, a Bohr-Weisskopf effect of ɛ=2.23(9)% and 2.30(9)% are found for ${}^{185}\mathrm{Re}$ and ${}^{187}\mathrm{R}\mathrm{e},$ respectively. This value is almost twice that of a previous theoretical estimate, and indicates a distribution of the nuclear magnetization far more extended than that of the nuclear charge. A radius of the magnetization distribution of $〈r_m^2{〉}^{1/2}=7.57\mathrm{}\left(32\right)\mathrm{}$ fm and $〈r_m^2{〉}^{1/2}$=7.69(32) fm for ${}^{185}\mathrm{Re}$ and ${}^{187}\mathrm{R}\mathrm{e},$ respectively, is inferred from the data. These radii are larger than the nuclear charge distribution radius $[〈r_c^2{〉}^{1/2}=5.39\mathrm{}(1)\mathrm{}$ fm] for both isotopes by factors 1.40(6) and 1.43(6), respectively. We find that the Bohr-Weisskopf effect in H-like ions is a sensitive probe of nuclear magnetization distribution, especially for cases where the charge distribution and magnetic moments are accurately known.

We have studied the polarization properties of dielectronic satellite lines in Be-like Fe ions excited through resonant electron capture by an electron beam. Using the photon density-matrix formalism, we have calculated the degree of polarization and polarization-dependent spectra of dielectronic satellite lines, i.e., the spectral intensity distribution of lines associated with a given polarization state. Theoretical results have been compared with experiments performed at the Lawrence Livermore National Laboratory electron-beam ion trap where dielectronic satellite line emission from Fe ions produced at different energies of the electron beam was simultaneously recorded with two crystal spectrometers. These spectrometers had different polarization sensitivities. The experimental spectra recorded by the two spectrometers are reproduced by the theory. This ability to model the polarization dependence of x-ray line spectra is important for the diagnosis of electron beams in plasmas.

Two computationally efficient implementations of first-principles molecular dynamics are examined using the reactive scattering of two linear ozone molecules to yield three molecules of oxygen. These methods are based on Kohn-Sham density-functional theory in which the charge density is fitted variationally. The linear combination of Gaussian-type orbitals approach is used to construct the orbital and local-potential basis sets. The first method is a completely general approach requiring variational fitting of the exchange-correlation energy density on a grid of points. The second method avoids the grid of points, but is restricted to the $X\alpha$ functional. Key elements of the dynamics calculation include self-consistent-field convergence at each time step, use of fractional occupation numbers to model the configurational mixing that occurs upon bond formation or dissociation, and accurate analytic evaluation of the appropriate forces acting upon the nuclei. Trajectories computed using modern local-density and gradient-corrected functionals, as well as $X\alpha$ $(\alpha$ = 0.6667) treated analytically, are compared. It is found that reactive trajectories obtained using analytic $X\alpha$ and the generalized gradient approximation are very similar when the initial kinetic energy is well above threshold for reaction. Under similar conditions, trajectories resulting from use of modern local-density functionals exhibit somewhat distinctive behavior. These results suggest that the particularly efficient analytic $X\alpha$ approach is suitable for preliminary investigations of chemical reactions using first-principles molecular dynamics based on density-functional theory.

Primitive semiclassical quantization of H${}_2^{+}$ and D${}_2^{+}$ without the restriction of the separation of electronic and nuclear motions, reveals the existence of a small (10%) classical contribution to the one-electron chemical bond. It is found that in the electronic ground state the electron is exchanged classically between the two nuclei and that this exchange is correlated with the molecular vibration.

Ionization potentials for most atoms are described accurately by total-energy differences calculated within density-functional theory (DFT) in the local-spin-density approximation (LSD). Electron affinities cannot be obtained in the same way, because the long-range behavior of the LSD potential is incorrect and negative ions are not bound. Here I show how electron affinities can be accurately estimated taking advantage of the nearly quadratic dependence of the LSD energy on the occupation numbers around the neutral atomic configuration. The procedure yields electron affinities, for plain LSD atoms, in better agreement with experiments than other self-interaction corrected DFT-based results.

The theory of $\mathrm{PT}$-odd interactions relevant to existing experimental measurements of the hyperfine structure of TlF is reviewed. We outline a relativistic electronic structure theory based on single-particle four-component Dirac spinors, and implemented using methods borrowed from ab initio quantum chemistry. Numerical calculations are reported of the electronic structure of TlF, some of its chemical properties, and of its $\mathrm{PT}$-odd electronic matrix elements. From these results, and from published experimental data, we derive bounds on the value of the electric dipole moment of the proton, $d_p,$ the tensor coupling constant $C_T,$ and the Schiff moment of the ${}^{205}$Tl nucleus, $Q$, which are now the tightest available for these quantities. General issues regarding the calculation of the electronic structures of molecules containing heavy elements are also addressed.

The wavelengths of the ${1s}_{1/2}{-2p}_{1/2}$ and ${1s}_{1/2}{-2p}_{3/2}$ Lyman-α transitions have been measured in hydrogenic ${\mathrm{Mg}}^{11+}$ with an accuracy as high as 24 ppm. The measurement was carried out on an electron-beam ion trap and utilized a quasimonolithic crystal setup absolutely calibrated relative to optical standards. The resulting values for the two transitions were $0.84250\pm 0.00004$ and $0.84190\pm 0.00002\mathrm{nm},$ respectively. The measurement confirms calculations of the $1s-2p$ wavelengths and tests the size of the $1s$ Lamb shift to within 13%.

We have systematically studied the line shapes and dynamics of several dissociative resonances near the second dissociation limits of ${\mathrm{H}}_2$, ${\mathrm{D}}_2$, and HD. Spectral overlap is eliminated using laser double resonance through the $\mathrm{EF}$ ${\mathrm{}}^1{\Sigma }_{\mathrm{g}}^{+}$state, and state-selective detection is used to distinguish dissociation to the $\mathrm{H}\mathrm{}\left(1s\right)+\mathrm{H}\mathrm{}\left(2s\right)\mathrm{}$ and $\mathrm{H}\mathrm{}\left(1s\right)+\mathrm{H}\mathrm{}\left(2p\right)\mathrm{}$ channels. The threshold region contains both dissociative Rydberg states belonging to the $3p$ complex and shape resonances from the $B$, $B^{\prime }$, and $C$ states. Line shapes range from nearly symmetric to highly asymmetric, and most fit well with Beutler-Fano profiles. The observed linewidths of near-threshold shape resonances are in strong disagreement with nonadiabatic theoretical calculations. In some instances, broad resonances coincide almost exactly with the dissociation thresholds, greatly affecting the cross sections and dissociation dynamics. An unusual resonance in HD is observed, in which the $\mathrm{H}\mathrm{}\left(2s\right)\mathrm{}$ cross section exhibits a strong interference line shape at the position of the threshold for $\mathrm{D}\mathrm{}\left(2s\right)\mathrm{}$ production.

A theoretical model for the reaction $\mathrm{Na}\mathrm{}\left(3p\right)+\mathrm{Na}\mathrm{}\left(3p\right)\mathrm{}\to {\mathrm{Na}}_2^{+}+e$ is compared to a large number of experimental data. This model is considering population of long-range adiabatic ${\mathrm{Na}}_2$ molecular states from two colliding excited atoms, and transfer of this population to short-range doubly excited autoionizing states of ${}^3{\Sigma }_u^{+},$ ${}^1{\Sigma }_g^{+},$ ${}^3{\Pi }_u,$ ${}^1{\Pi }_u,$ ${}^1{\Delta }_g,$ and ${}^3{\Delta }_u$ symmetries. Good agreement is obtained with experimental results for total cross sections and polarization dependence of the ion signal at collision velocities $v_c>~400\mathrm{m}/\mathrm{s}.$ At lower velocities the sudden approximation for spin uncoupling is not valid. The computed velocity dependence of the ion signal does not reproduce the maximum in the ion signal observed in some experiments.