Accurate Energy Levels Research Articles

Determination of the energy bands of metallic lithium by the variational cellular method

The recently proposed variational cellular method (VCM) is extended to periodic structures with one atom per unit cell. The method as outlined in this paper is applied to the calculation of the band structure of metallic lithium. The results show that the VCM leads to accurate energy levels for Li requiring a small computational effort. It is concluded that the results are sufficiently encouraging to suggest the application of the method to more complex crystals.

Accurate He-Ag(110) interaction potential determination by selective adsorption

The first accurate energy level measurements of a He-metal interaction potential are reported for the Ag(110) surface as obtained from selective adsorption data. The fitting of the experimental data with a suitable phenomenological potential leads to a well depth of 6.0 ± 0.1 meV. The potential curve is considerably softer than those obtained for He interaction with either insulators or semiconductors and is in good agreement with theoretical predictions.

Accurate He-Ag(110) interaction potential determination by selective adsorption

The first accurate energy level measurements of a He-metal interaction potential are reported for the Ag(110) surface as obtained from selective adsorption data. The fitting of the experimental data with a suitable phenomenological potential leads to a well depth of 6.0 ± 0.1 meV. The potential curve is considerably softer than those obtained for He interaction with either insulators or semiconductors and is in good agreement with theoretical predictions.

Simple procedure to compute accurate energy levels of an anharmonic oscillator

A procedure based on the piecewise perturbation treatment of the Schr\"odinger equation is developed. The procedure is shown to be convergent for any bounded perturbation. A simple way to construct the algorithm is also developed, which can be used for any potential of polynomial form. Our procedure is further particularized for the anharmonic-oscillator potential $\mathcal{U}(x)={m}^{2}{x}^{2}+2\ensuremath{\lambda}{x}^{2\ensuremath{\alpha}}$ ($\ensuremath{\lambda}>0$ and integer $\ensuremath{\alpha}\ensuremath{\ge}2$) and applied to produce highly accurate numerical values for its energy levels. The experimental results also show that this procedure exhibits almost equal efficiency for all relevant values of the principal quantum number $n$, mass parameter ${m}^{2}$, coupling $\ensuremath{\lambda}$, and anharmonicity $\ensuremath{\alpha}$.

Far infrared laser lines in CH<inf>3</inf>OH, optically pumped to CO stretch states and to strongly interacting vibrational states

A commented table is presented, containing 90 assigned FIR emission lines of the CH 3 OH laser, which can be dixectly or indirectly related to the torsional ground state of the CO stretch. Based on a model, which empirically takes into account interactions with other vibrational modes, the table compares measured and calculated pump and emission frequencies. For 14 pump transitions with J \leq 26 , the rms deviation is 0.005 cm-1. For 26 FIR emission transitions involving J \leq 26 , where the accuracy of the model can be checked against frequency measurements, the rms deviation is 0.004 cm-1. Pump transitions to a strongly perturbing mode are identified, and accurate energy level parameters for this mode are given.

Commission 14: Atomic and Molecular Data

The rapid expansion during the past few years of the spectral region accessible to astronomical observers, from gamma rays to the radio region, has resulted in a corresponding expansion of the need for a wide variety of atomic and molecular data. Included are needs for accurate wavelengths, atomic and molecular energy levels, and transition probabilities. The continually improving resolution that has been attainable has resulted in the requirement of improved insight into line broadening mechanisms of various types. This expansion has placed an increasing premium on data compilation and dissemination, so that available information can be made readily available to potential users. Among the numerous compilations that have appeared might be mentioned the important National Bureau of Standards Bibliography on Atomic Levels and Spectra, which is up-dated periodically via successive supplements, and the NBS compilation and bibliography by the Data Center on Atomic Transition Probabilities. Several compilations or bibliographies on collision cross-sections are now available, such as that published by the Information Center at JILA. In the field of molecular spectra there has appeared the very comprehensive “Constants of Diatomic Molecules” by Huber and Herzberg. Other useful compilations are referred to in the reports of the five Working Groups that appear below.

Solitons and magnon bound states in ferromagnetic Heisenberg chains

We investigate the relation between magnon, magnon bound states, and the classical soliton solutions in the isotropic, the anisotropic-exchange, and the easy-axis ferromagnetic Heisenberg chains. The Dyson-Maleev boson representation is used to study the interaction between magnons, and bound states are investigated in terms of two-spin Green's functions. For easy-axis and anisotropic-exchange magnets, a mapping to the Bose gas with attractive $\ensuremath{\delta}$-function interactions is established, and it yields the eigenvalues of the magnon and the $m$th bound state in the weak-coupling and continuum limits. The classical limit of the two-spin Green's function reveals that, to leading order in temperature, the bound-state resonance and the associated effects on the two-magnon continuum survives. An important result is a ${T}^{2}$ dependence of the "binding" energy of the bound-state resonance. As a consequence, in a classical description the bound states enter only in order ${T}^{2}$. Finally, we quantize the soliton solutions according to the Bohr-Sommerfeld and de Broglie rules. This approach is found to be exact for the $s=\frac{1}{2}$ isotropic Heisenberg chain and for sufficiently small wave numbers for all $s$ values. In the anisotropic-exchange and easy-axis models, it agrees with the results obtained from the mapping on the Bose gas, for large quantum numbers and in the easy-axis case for $s\ensuremath{\gg}\frac{1}{2}$ in addition. On this basis, we conclude that the envelope solitons considered here lead to bound-state resonances and associated effects in a classical treatment. Moreover, their semiclassical quantization gives remarkably accurate energy levels.

Tests of approximation schemes for vibrational energy levels and partition functions for triatomics: H2O and SO2

Various approximate schemes are used to obtain vibrational energy levels and partition functions for H2O and SO2. These results are then compared with accurate quantum mechanical values computed for the same original quartic force fields. The neglect of interaction force constants in internal coordinates and the use of a Morse model to describe stretching anharmonicity are both shown to provide reasonably accurate perturbation theory energy levels and partition functions, while the neglect of interaction force constants in normal coordinates is found to yield much less accurate results. In addition, the method of Pitzer and Gwinn to calculate partition functions without first calculating energy levels is shown to provide a good approximation to the accurate partition functions.

The [formula omitted] single vibronic level fluorescence spectrum of styrene vapor

Single vibronic level (SVL) fluorescence spectra following excitation into the 0 0 0 band and many sequence and cross-sequence bands in the A ̃ 1A′- X ̃ 1A′ system of styrene have been recorded and assigned. The technique is shown to be very powerful in obtaining accurate energy levels for low-wavenumber vibrations, particularly in the X ̃ state, where it is capable of giving information regarding vibrational levels inaccessible by conventional infrared or Raman spectroscopy. Interpretation of the spectra leads to many new assignments in the absorption spectrum and to an accurate knowledge of many vibrational energy levels. An important reassignment is that of the 41 0 142 0 1 band previously assigned to 40 0 2 ( ν 40, ν 41, and ν 42 are all a″ vibrations). The two most important pieces of information which derive from analysis of the SVL spectra are (a) the first five vibrational levels of ν 42, the C(1)- C( α) torsional vibration, in the X ̃ state, each accurate to about ±0.2 cm −1, and (b) the fact that the normal coordinates of ν 42 and ν 41, an out-of-plane substituent vibration, are very heavily mixed in the A ̃ , relative to the X ̃ , state—an extreme case of the Duschinsky effect. As a result of analysis of the SVL spectra, analysis of the absorption spectrum is now so complete that we can be confident that the Hui and Rice proposal that the CH 2 group of the substituent is perpendicular to the plane of the rest of the molecule in the A ̃ state has no evidence to support it.

Accurate energy levels for the anharmonic oscillator and a summable series for the double-well potential in perturbation theory

We introduce a generalization of Wick-ordering which maps the anharmonic oscillator (AO) Hamiltonian for mass m and coupling λ exactly into a “Wick-ordered” Hamiltonian with an effective mass M which is a simple analytic function of λ and m. The effective coupling Λ = λ M 3 is bounded. We transform the AO perturbation series in λ into one in Λ. This series may then be summed using Borel summation methods. We also introduce a new summation method for the AO series (which is a practical necessity to obtain accurate energy levels of the excited states). We obtain a numerical accuracy for (E PT − E exact) E exact of at least 10 −7 (using 20 orders of perturbation theory) and 10 −3 (using only 2 orders of perturbation theory) for all couplings and all energy levels of the anharmonic oscillator. The methods are applicable also to the double-well potential (DWP, the AO with a negative mass-squared). The only change is that now the effective coupling is unbounded as λ → 0. The series in Λ is, however, still summable. The relative accuracy in the energy levels for 20 orders of perturbation theory varies from 10 −7 for large coupling to 1% at λ = 0.1 and to 10% at λ = .05. We also present results for the sextic oscillator.

Accurate energy levels for the anharmonic oscillator and a summable series for the double-well potential in perturbation theory: William E. Caswell. Department of Physics, Brown University, Providence, Rhode Island 02912

Accurate energy levels for the anharmonic oscillator and a summable series for the double-well potential in perturbation theory: William E. Caswell. Department of Physics, Brown University, Providence, Rhode Island 02912

Core charge polarisation effects in atomic lithium

The semiclassical, consistent formulation of core charge polarisation effects presented in an earlier paper has been applied to atomic lithium. When the core polarisation potential is added to a Hartree-Fock calculation one obtains accurate atomic energy levels for the valence and excited states, reasonable values for the hyperfine splitting constant of the valence S state and oscillator strengths in excellent agreement with the tabulated values. This suggests that this procedure accurately describes the non-relativistic aspects of atomic lithium. The spin-orbit splitting and atomic g shift are not given well by the standard one-particle expressions. This suggests that the one-particle expressions are invalid for relativistic corrections in many-body systems.

Accurate Energy Levels and Calculated Wavelengths of ^86Kr i*

A total of about 240 spectral lines of 86Kr i has been measured interferometrically during the past decade. Some have been measured in only one laboratory while others have been measured in as many as 12. On the whole, the agreement is good, but some discrepancies appear. For this work, almost all of the wavelengths have been averaged and used for the determination of a best set of energy-level values for this isotope of krypton. Forty-five even and 66 odd levels have been calculated by the method of least squares, with uncertainties ranging from ±0.0001 to ±0.0016 cm−1 These interferometrically determined levels have, in turn, been used to calculate a completely consistent set of wavelengths that are preferable to any single set of measurements or averages of measurements of 86Kr i. The list of 530 lines contains only those calculable transitions that have been observed from krypton sources in the spectral region 3300–40 700 Å. With the exception of a number of weak lines and some close pairs, the final list of wavelengths is recommended for consideration for adoption as secondary standards.

Quartic Oscillator as a Basis for Energy Level Calculations of Some Anharmonic Oscillators

A variational treatment facilitating accurate energy level calculations for the mixed harmonic—quartic oscillator within the frameworks of both the harmonic oscillator and the quartic oscillator representations is formulated. The usefulness of the quartic oscillator as a basis for such calculations is examined in detail. It is shown that the quartic oscillator representation yields significantly better energies than the conventional harmonic oscillator representation even for moderate quartic anharmonicities. The first 10 energy levels, expectation values of x2, x4, and some transition moments are presented over the range of anharmonicities from the pure harmonic oscillator to the pure quartic oscillator.

Energy Levels and Thermodynamic Properties of the Internal Rotator

To facilitate the calculation of the thermodynamic properties of internal rotators lying outside the limits of available tabulations, methods of determining the energy levels are studied for the purpose of finding the most expeditious route to accurate results. It proves to be unnecessary in any case to calculate many levels from the wave mechanics. A partition function is set up in terms of the correct ground state and a series of approximate levels derived from the old quantum theory, and this is corrected to constancy by the successive substitution of correct levels, starting with the first excited state. A set of levels obtained in this way which gives the correct partition function always leads to correct values of all the thermodynamic properties. To obtain accurate energy levels, the continued fractions of Koehler and Dennison are recommended as most convenient.