TRIPLET-STATE ROTATIONAL ENERGY LEVELS FOR NEAR-SYMMETRIC ROTOR MOLECULES OF SYMMETRY C2v, D2, AND D2h

1967 ◽  
Vol 45 (3) ◽  
pp. 1363-1387 ◽  
Author(s):  
F. Creutzberg ◽  
J. T. Hougen
Keyword(s):  
Energy Levels ◽  
Rotational Energy ◽  
Polyatomic Molecules ◽  
Spin Splitting ◽  
Triplet States ◽  
Quantum Numbers ◽  
Symmetric Rotor ◽  
Symmetry Species ◽  
Definition Of ◽  
Second Order Perturbation Theory

The ideas involved in Hund's coupling cases for diatomic molecules are extended to allow the definition of three coupling cases, (a), (ab), and (b), for the triplet states of orthorhombic polyatomic molecules. Case (a) is dealt with by second-order perturbation theory. Case (b) has recently been discussed by Raynes. Case (ab), which can be subdivided into types I, II, and III, is studied by examining the results of numerical calculations for selected values of the rotational constants, rotational quantum numbers, and spin-splitting parameters for a near-prolate rotor. The assignment of symmetry species to the various spin-rotational functions under consideration is described.

On the vibration–rotational energy levels of the hydrogen molecular ion HD+

1985 ◽  
Vol 63 (9) ◽  
pp. 1201-1204 ◽  
Author(s):  
L. Wolniewicz ◽  
J. D. Poll
Keyword(s):  
Diatomic Molecules ◽  
Energy Levels ◽  
Rotational Energy ◽  
New Method ◽  
Dissociation Limit ◽  
Radiative Effects ◽  
Vibrational States ◽  
Hydrogen Molecular Ion ◽  
Molecular Ion ◽  
Quantum Numbers

A new method for calculating vibration–rotational energies of diatomic molecules is discussed and applied to the case of HD+. This method is designed to obtain accurate results for all vibrational states including those close to the dissociation limit. Nonadiabatic, relativistic, and radiative effects are taken into account for all the bound vibrational states with rotational quantum numbers J ≤ 5; the estimated accuracy is of the order of 0.001 cm−1.

The influence of environment upon the spectra of molecules in liquids and crystals

1960 ◽  
Vol 255 (1280) ◽  
pp. 5-21 ◽  
Keyword(s):  
Ground State ◽  
Electronic Excitation ◽  
Energy Levels ◽  
Rotational Energy ◽  
Intermolecular Distance ◽  
Pressure Broadening ◽  
Basic Difference ◽  
Solid States ◽  
Quantum Numbers ◽  
Electronic Ground

The effect on the spectrum of a molecule of the environment in which it is located depends upon the changes which the surroundings produce in the electronic, vibrational, rotational and nuclear energies of the upper and lower states of the molecule. Studies of the influence of environment in the gaseous, liquid or solid states can thus be made by any of the appropriate techniques listed in table 1, and it is clearly desirable in studying any one system to use as many different techniques as possible. A basic difference between the effect of environment on electronic and vibra­tional energy levels arises from the very much greater overlap of electron density with the environment that results from electronic excitation. Hence while for the consideration of changes which arise in the vibrational spectrum it is adequate to consider only the distortion of the curve relating the interaction energy to the intermolecular distance in the electronic ground state, for electronic spectra, how­ever, the changes in the potential curves in both upper and lower states must clearly be taken into account. Collisions between molecules in gases lead to the broadening of rotational energy levels, and much useful information on inter­molecular force fields has resulted from observations on pressure broadening of pure rotational lines in the microwave region. Both self-broadening and broadening by different foreign gases have been studied as well as the dependence of line half­width Δ v 1/2 on the rotational quantum numbers J and K (Townes & Schawlow 1955).

A Perturbation Theory for the Population of Atomic Energy Levels in Non-Lte Plasmas

1988 ◽  
Vol 102 ◽  
pp. 343-347
Author(s):  
M. Klapisch
Keyword(s):  
Perturbation Theory ◽  
Small Parameter ◽  
Classification Scheme ◽  
Sum Rules ◽  
Energy Levels ◽  
Natural Classification ◽  
Quantum Numbers ◽  
Formal Expansion ◽  
Atomic Energy Levels ◽  
As Effects

AbstractA formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum rules can be formulated, allowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.

scholarly journals Zur exakten Behandlung von kollektiven Rotationen Teil A: Theorie / On the Exact Treatment of Collective Rotations Part A: Theory

1973 ◽  
Vol 28 (2) ◽  
pp. 206-215
Author(s):  
Hanns Ruder
Keyword(s):  
Coordinate System ◽  
Perturbation Expansion ◽  
Rotational Energy ◽  
The Body ◽  
System A ◽  
N Body Problem ◽  
Definition Of ◽  
Fixed System ◽  
Body Fixed Coordinate System ◽  
Groundstate Energy

Basic in the treatment of collective rotations is the definition of a body-fixed coordinate system. A kinematical method is derived to obtain the Hamiltonian of a n-body problem for a given definition of the body-fixed system. From this exact Hamiltonian, a consequent perturbation expansion in terms of the total angular momentum leads to two exact expressions: one for the collective rotational energy which has to be added to the groundstate energy in this order of perturbation and a second one for the effective inertia tensor in the groundstate. The discussion of these results leads to two criteria how to define the best body-fixed coordinate system, namely a differential equation and a variational principle. The equivalence of both is shown.

scholarly journals Semiclassical calculations of the quadruple excited four-electron systems

2007 ◽  
pp. 33-44
Author(s):  
N. Simonovic ◽  
M. Predojevic ◽  
V. Pankovic ◽  
P. Grujic
Keyword(s):  
Energy Levels ◽  
Point Of View ◽  
Interstellar Space ◽  
Vibrational Modes ◽  
Electron Systems ◽  
Atomic Systems ◽  
Excited Atoms ◽  
Relative Contribution ◽  
Quantum Numbers ◽  
Semiclassical States

Highly excited atoms acquire very large dimensions and can be present only in a very rarified gas medium, such as the interstellar space. Multiply excited beryllium-like systems, when excited to large principal quantum numbers, have a radius of r ? 10 ?. We examine the semiclassical spectrum of quadruple highly excited four-electron atomic systems for the plane model of equivalent electrons. The energy of the system consists of rotational and vibrational modes within the almost circular orbit approximation, as used in a previous calculation for the triply excited three-electron systems. Here we present numerical results for the beryllium atom. The lifetimes of the semiclassical states are estimated via the corresponding Lyapunov exponents. The vibrational modes relative contribution to the energy levels rises with the degree of the Coulombic excitation. The relevance of the results is discussed both from the observational and heuristic point of view.

On Bose-Einstein Condensation

1954 ◽  
Vol 50 (1) ◽  
pp. 65-76 ◽  
Author(s):  
P. T. Landsberg
Keyword(s):  
Closed System ◽  
Canonical Ensemble ◽  
Energy Levels ◽  
Volume Density ◽  
Einstein Condensation ◽  
Bose Einstein Condensation ◽  
Quantum Numbers ◽  
Finite Integer ◽  
Bose Einstein ◽  
Grand Canonical

ABSTRACTThis paper contains a proof that the description of the phenomenon of Bose-Einstein condensation is the same whether (1) an open system is contemplated and treated on the basis of the grand canonical ensemble, or (2) a closed system is contemplated and treated on the basis of the canonical ensemble without recourse to the method of steepest descents, or (3) a closed system is contemplated and treated on the basis of the canonical ensemble using the method of steepest descents. Contrary to what is usually believed, it is shown that the crucial factor governing the incidence of the condensation phenomenon of a system (open or closed) having an infinity of energy levels is the density of states N(E) ∝ En for high quantum numbers, a condition for condensation being n > 0. These results are obtained on the basis of the following assumptions: (i) For large volumes V (a) all energy levels behave like V−θ, and (b) there exists a finite integer M such that it is justifiable to put for the jth energy level Ej= c V−θand to use the continuous spectrum approximation, whenever j ≥ M c θ τ are positive constants, (ii) All results are evaluated in the limit in which the volume of the gas is allowed to tend to infinity, keeping the volume density of particles a finite and non-zero constant. The present paper also serves to coordinate much of previously published work, and corrects a current misconception regarding the method of steepest descents.

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