scholarly journals IV. The triplets of helium

1929 ◽  
Vol 228 (659-669) ◽  
pp. 151-196 ◽  
Keyword(s):  
Perturbation Theory ◽  
Fine Structure ◽  
Energy Levels ◽  
Wave Functions ◽  
The Other ◽  
Recent Theory ◽  
Spin Effects ◽  
Electronic Spin ◽  
Fundamental Equations ◽  
Good Agreement

Heisenberg’s theory of the helium atom, following his famous principle of “resonance,” accounted satisfactorily for the ortho- and para- states. His more detailed treatment of the triplets, besides giving ground for minor criticisms, necessarily had to be derived from Schrödinger’s equation , with a somewhat incongruous electronic “spin” grafted upon it. The present work uses Dirac’s recent theory, in which the spin effects grow more naturally from the fundamental equations. Calculations which are in dependent of the spin, such as the approximate energy-levels and the ortho-para separations, are the same on either theory. We deal here with spin effects, such as the fine structure of the triplets and intercombinations between ortho- and para- states. The fundamental equations, the perturbation theory which is used, and the perturbing spin energy, have been discussed fully elsewhere. The present paper, after a short recapitulation of the general theory, gives in detail the calculations whose results alone were quoted in the other. §§ 4–7 work out the structure of the helium triplets with one excited election. The resulting spin separations are different from Heisenberg’s, but in equally good agreement with experiment. At the same time, the correct first approximations to the wave-functions are found, and they are used in § 8 to verify the ordinary classification and selection rules.

The Polarizability of Molecular Hydrogen H2

1936 ◽  
Vol 32 (2) ◽  
pp. 260-264 ◽  
Author(s):  
C. E. Easthope
Keyword(s):  
Wave Function ◽  
Perturbation Theory ◽  
Molecular Hydrogen ◽  
Applied Field ◽  
Minimum Energy ◽  
Wave Functions ◽  
The Other ◽  
System A ◽  
The One ◽  
Two Parameter

1. The problem of calculating the polarizability of molecular hydrogen has recently been considered by a number of investigators. Steensholt and Hirschfelder use the variational method developed by Hylleras and Hassé. For ψ0, the wave function of the unperturbed molecule when no external field is present, they take either the Rosent or the Wang wave function, while the wave functions of the perturbed molecule were considered in both the one-parameter form, ψ0 [1+A(q1 + q2)] and the two-parameter form, ψ0 [1+A(q1 + q2) + B(r1q1 + r2q2)], where A and B are parameters to be varied so as to give the system a minimum energy, q1 and q2 are the coordinates of the electrons 1 and 2 in the direction of the applied field as measured from the centre of the molecule, and r1 and r2 are their respective distances from the same point. Mrowka, on the other hand, employs a method based on the usual perturbation theory. Their numerical results are given in the following table.

Zur paramagnetischen Elektronenresonanz metallorganischer Sandwich-Verbindungen, Teil I

1965 ◽  
Vol 20 (11) ◽  
pp. 1411-1416
Author(s):  
Manfred Nussbaum ◽  
Jürgen Voitländer
Keyword(s):  
Magnetic Resonance ◽  
Energy Levels ◽  
Total Spin ◽  
Wave Functions ◽  
Spin Hamiltonian ◽  
Short Treatment ◽  
Ionic Model ◽  
Quantitative Calculation ◽  
Resonance Properties ◽  
Good Agreement

After a short treatment of the ionic model of ROBERTSON and McCONNELL, two methods for the calculation of the coefficients of the spin Hamiltonian are given and discussed in connection with HUNDS rule and the magnitude of the effective total spin S. In addition, the dependence of the effective magnetic moment on those coefficients in shown. A quantitative calculation of the energy levels, wave functions and magnetic resonance properties of V (C5H5) 2, the latter being in good agreement with the experiments of McCONNELL, concludes this section.In Part II the magnetic-resonance behaviour of Ni (C5H5)2 and Co (C5H5)2 is investigated theoretically under various assumptions and as far as possible compared with experiment.

Electronic levels in simple conjugated systems, I. Configuration interaction in cyclobutadiene

1950 ◽  
Vol 202 (1071) ◽  
pp. 498-506 ◽  
Keyword(s):  
Molecular Orbital ◽  
Configuration Interaction ◽  
Energy Levels ◽  
Valence Bond ◽  
Wave Functions ◽  
The Other ◽  
Orbital Theory ◽  
Valence Bond Theory ◽  
Symmetry Properties ◽  
Bond Theory

In the simplest cyclic system of π-electrons, cyclobutadiene, a non-empirical calculation has been made of the effects of configuration interaction within a complete basis of antisymmetric molecular orbital configurations. The molecular orbitals are made up from atomic wave functions and all the interelectron repulsion integrals which arise are included, although those of them which are three- and four-centre integrals are only known approximately. In this system configuration interaction is a large effect with a strongly differential action between states of different symmetry properties. Thus the 1 A 1g state is several electron-volts lower than the lowest configuration of that symmetry, whereas for 1 B 1g the comparable figure is about one-tenth of an electron-volt. The other two states examined, 1 B 2g and 3 A 2g are affected by intermediate amounts. The result is a drastic change in the energy-level scheme compared with that based on configuration wave functions. Neither the valence-bond theory nor the molecular orbital theory (in which the four states have the same energy) gives a satisfactory account of the energy levels according to these results. One conclusion from the valence-bond theory which is, however, confirmed, is the somewhat unexpected one that the non-totally symmetrical 1 B 2g state is more stable than the totally symmetrical 1 A 1g . On the other hand, it is clear that the valence-bond theory, with the usual value for its exchange integral, grossly exaggerates the resonance splitting of the states, giving separations between them several times too great. Thus the valence-bond theory leads to large values of the resonance energy (larger, per π-electron, than in benzene) and so associates with the molecule a considerable π-electron stabilization. This expectation has no support in the present more detailed and non-empirical calculations.

A COMMENT ON THE "PEIERLS SUBSTITUTION"

2000 ◽  
Vol 14 (05) ◽  
pp. 187-194
Author(s):  
M. K. FUNG ◽  
Y. F. WANG
Keyword(s):  
Perturbation Theory ◽  
Energy Levels ◽  
The Other ◽  
Raising And Lowering Operators ◽  
Lowering Operators

We present a new way of doing perturbation theory of the Landau problem. We introduce two sets of raising and lowering operators. One set is for shifting the coordinates of the center of the cyclotron orbit. The other set is for shifting the energy levels. This method is compared to that of the Peierls substitution.

scholarly journals The effect of a nuclear spin on the optical spectra

1929 ◽  
Vol 124 (795) ◽  
pp. 568-591 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Hyperfine Structure ◽  
Nuclear Spin ◽  
Energy Levels ◽  
Principal Series ◽  
Wave Functions ◽  
Central Field ◽  
Similar Work ◽  
Electronic Spin ◽  
Working Out

Some recent work has been done by Back and Goudsmidt on the “hyperfine” structure of the optical spectrum of bismuth,* and more recently similar work has been carried out for cæsium by Jackson. In each of these investigations the line structure was examined closely with a view to revealing a still finer structure, and it was found in both cases that the lines attributed to electronic spin were themselves composed of several distinct lines. In fact, for cæsium, each of the fine (electron spin) lines of the principal series was found to split up into two ; for bismuth the hyperfine structure was more complicated. Back and Goudsmidt attributed the structure to a nuclear spin, and working out the consequences of this on the lines of the old quantum mechanics they found that a nuclear spin of 41/2 quanta is necessary to account for the facts ; a spin of a 1/2 quantum is similarly attributed by Jackson to the nucleus of cæsium. The hypothesis explains very satisfactorily in a qualitative way the results of observation. In the work described in the present paper the methods of the new quantum mechanics have been applied to the problem. More precisely, we consider the motion of a single electron in a Coulombian field due to a nucleus possessing a 1/2 quantum of spin. It will be seen that the results can easily be extended to the case of any central field, and the principle could also be extended to the case of an atom with a nuclear spin of 1/2( nh /2π), but the detailed working out would be very heavy for n > I (at any rate, using the methods explained in this paper), owing to the large number of wave functions which would be necessary to specify any state of the atom. It will be seen that the results we obtain are substantially the same as Jackson’s so far as the energy levels are concerned, but the calculated intensities are not consistent with the observed transitions, and we deduce a combination rule which is radically different from Jackson’s.

scholarly journals Energy States of Some Diatomaic Molecules: The Exact Quantisation Rule Approach

2015 ◽  
Vol 70 (2) ◽  
pp. 85-90 ◽  
Author(s):  
Babatunde J. Falaye ◽  
Sameer M. Ikhdair ◽  
Majid Hamzavi
Keyword(s):  
Hypergeometric Functions ◽  
Vibrational Energy ◽  
Energy Levels ◽  
Wave Functions ◽  
Energy Eigenvalues ◽  
Molecular Potential ◽  
Approximate Analytical Solutions ◽  
Rule Approach ◽  
Radial Schrödinger Equation ◽  
Good Agreement

AbstractIn this study, we obtain the approximate analytical solutions of the radial Schrödinger equation for the Deng–Fan diatomic molecular potential by using the exact quantisation rule approach. The wave functions were expressed by hypergeometric functions via the functional analysis approach. An extension to the rotational–vibrational energy eigenvalues of some diatomic molecules is also presented. It is shown that the calculated energy levels are in good agreement with those obtained previously (Enℓ–D; shifted Deng–Fan).

scholarly journals Self-consistent field, with exchange, for beryllium

1935 ◽  
Vol 150 (869) ◽  
pp. 9-33 ◽  
Keyword(s):  
Wave Functions ◽  
The Other ◽  
Numerical Work ◽  
Spin Effects ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Average Distribution ◽  
Approximate Wave ◽  
The Individual

Except for the lightest atoms, most calculations of approximate wave functions and fields for many-electron atoms have been carried out by the method of the “self-consistent field,” of which the principle is, shortly, the determination of a set of one-electron wave functions such that each represents a stationary state of an electron in the field of the nucleus and the Schrödinger charge distribution of the electrons occupying the other wave functions of the set. This method has been found quite practicable for numerical work, even for the heaviest atoms. As so far applied, it involves three main approximations, namely, ( a ) neglect of relativity and spin effects, ( b ) neglect of exchange effects, and ( c ) treatment of the wave function of the whole atom as built up of functions of the co-ordinates of the individual electrons only, its depen­dence on the mutual distances between every pair of electrons being neglected; or, in other words, each electron is replaced by a statistical average distribution, in calculating its effect on the other electrons on the atom.

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.

Ultrastructure and Staining of Developing Elastic Tissue

1971 ◽  
Vol 29 ◽  
pp. 244-245
Author(s):  
E. N. Albert
Keyword(s):  
Fine Structure ◽  
Phosphotungstic Acid ◽  
Staining Method ◽  
Rat Aorta ◽  
Uranyl Acetate ◽  
The Other ◽  
Elastic Fibers ◽  
Elastic Tissue ◽  
Lead Citrate ◽  
New Born

Silver tetraphenylporphine sulfonate (Ag-TPPS) was synthesized in this laboratory and used as an electron dense stain for elastic tissue (Fig 1). The procedures for the synthesis of tetraphenylporphine sulfonate and the staining method for mature elastic tissue have been described previously.The fine structure of developing elastic tissue was observed in fetal and new born rat aorta using tetraphenylporphine sulfonate, phosphotungstic acid, uranyl acetate and lead citrate. The newly forming elastica consisted of two morphologically distinct components. These were a central amorphous and a peripheral fibrous. The ratio of the central amorphous and the peripheral fibrillar portion changed in favor of the former with increasing age.It was also observed that the staining properties of the two components were entirely different. The peripheral fibrous component stained with uranyl acetate and/or lead citrate while the central amorphous portion demonstrated no affinity for these stains. On the other hand, the central amorphous portion of developing elastic fibers stained vigorously with silver tetraphenylporphine sulfonate, while the fibrillar part did not (compare figs 2, 3, 4). Based upon the above observations it is proposed that developing elastica consists of two components that are morphologically and chemically different.

Further Studies on the Ultrastructure of Harderian Gland in Adult Rats

1974 ◽  
Vol 32 ◽  
pp. 288-289
Author(s):  
Alfredo Feria-Velasco ◽  
Guadalupe Tapia-Arizmendi
Keyword(s):  
Fine Structure ◽  
Domestic Fowl ◽  
Animal Species ◽  
Acinar Cells ◽  
Harderian Gland ◽  
Myoepithelial Cells ◽  
The Other ◽  
Albino Rats ◽  
Adult Rats ◽  
Cell Components

The fine structure of the Harderian gland has been described in some animal species (hamster, rabbit, mouse, domestic fowl and albino rats). There are only two reports in the literature dealing on the ultrastructure of rat Harderian gland in adult animals. In one of them the author describes the myoepithelial cells in methacrylate-embbeded tissue, and the other deals with the maturation of the acinar cells and the formation of the secretory droplets. The aim of the present work is to analize the relationships among the acinar cell components and to describe the two types of cells located at the perifery of the acini.

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