Calculation of One-Electron Wave Functions and Energy Levels of N-Butane Molecule on the Basis of Slater Atomic Orbitals

Abstract It is known that the application of the group theory greatly simplifies the problems of polyatomic systems possessing to any space symmetry. The symmetry properties of such systems are their most important characteristics. In such systems, the Hamilton operator is invariant under unitary symmetry transformations and rearrangements of identical particles in the coordinate system. This allows to obtain information about the character of one-electron wave functions — molecular orbitals — the considered system, i.e. to symmetrise the original wave functions without solving the Schrödinger equation.

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Energy levels and electron wave functions in semiconductor quantum wells having superlattice alloylike material (0.9 nm GaAs/0.9 nm AlGaAs) as barrier layers

Applied Physics Letters ◽

10.1063/1.96197 ◽

1985 ◽

Vol 47 (3) ◽

pp. 295-297 ◽

Cited By ~ 30

Author(s):

H. Sakaki ◽

M. Tsuchiya ◽

J. Yoshino

Keyword(s):

Quantum Wells ◽

Energy Levels ◽

Wave Functions ◽

Electron Wave ◽

Barrier Layers ◽

Semiconductor Quantum Wells

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

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1950.0115 ◽

1950 ◽

Vol 202 (1071) ◽

pp. 498-506 ◽

Cited By ~ 28

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.

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Low-temperature transport properties of the alkali metals. I. The electron-phonon interaction

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1961.0178 ◽

1961 ◽

Vol 263 (1315) ◽

pp. 531-544 ◽

Cited By ~ 13

Keyword(s):

Fermi Surface ◽

Alkali Metals ◽

Plane Waves ◽

Wave Functions ◽

Electron Wave ◽

Thermo Electric ◽

Fermi Surfaces ◽

Electron Phonon Interaction ◽

Symmetry Properties ◽

Umklapp Scattering

The form of the electron-phonon matrix element is calculated for metals with non-spherical Fermi surfaces by using electron wave functions, which are linear combinations of two plane waves, as in the model of nearly free electrons. Numerical calculations are made with the use of the ‘twelve cone’ approximation to the Brillouin zone. It is shown that, if the Fermi surface bulges towards the zone faces, there is a significant increase in the probability of Umklapp scattering of electrons, the increase depending on the amount of distortion of the Fermi surface, and on the symmetry properties of the electron wave functions. The increase in Umklapp scattering has important consequences for calculations of the resistivities of metals, and particularly for calculations of the ‘phonon drag’ contribution to the thermo-electric power.

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Imaging Electron Wave Functions of Quantized Energy Levels in Carbon Nanotubes

Science ◽

10.1126/science.283.5398.52 ◽

1999 ◽

Vol 283 (5398) ◽

pp. 52-55 ◽

Cited By ~ 245

Author(s):

L. C. Venema

Keyword(s):

Carbon Nanotubes ◽

Energy Levels ◽

Wave Functions ◽

Electron Wave

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Magnetotunneling spectroscopy imaging of electron wave functions in self-assembled InAs quantum dots

Uspekhi Fizicheskih Nauk ◽

10.3367/ufnr.0171.200112f.1365 ◽

2001 ◽

Vol 171 (12) ◽

pp. 1365

Author(s):

E.E. Vdovin ◽

Yu.N. Khanin ◽

Yu.V. Dubrovskii ◽

A. Veretennikov ◽

A. Levin ◽

...

Keyword(s):

Quantum Dots ◽

Wave Functions ◽

Electron Wave ◽

Inas Quantum Dots ◽

Self Assembled

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Quantum Boxes, Stringed Instruments

10.1093/oso/9780198808275.003.0008 ◽

2017 ◽

Author(s):

Frank S. Levin

Keyword(s):

Energy Level ◽

Schrödinger Equation ◽

Quantum System ◽

Schrodinger Equation ◽

Probability Distributions ◽

Wave Functions ◽

Energy Level Diagram ◽

One Dimensional ◽

Stringed Instruments ◽

Symmetry Properties

Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.

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A free-electron theory of conjugated molecules

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100028851 ◽

1953 ◽

Vol 49 (4) ◽

pp. 650-658 ◽

Cited By ~ 11

Author(s):

J. Stanley Griffith

Keyword(s):

Free Electron ◽

Wave Functions ◽

Electron Charge ◽

Algebraic Equations ◽

Electron Wave ◽

Electron Theory ◽

Conjugated Molecules ◽

Electron Charge Density ◽

Linear Polyenes ◽

Alternant Hydrocarbons

ABSTRACTThe values of a free-electron eigenfunotion at the carbon nuclei of a conjugated hydrocarbon are found to satisfy a system of algebraic equations. These equations are similar in form to those obtained in the method known as the linear combination of atomic orbitale but only coincide with them for linear polyenes and benzene. The symmetry, degeneracy and energy of the eigenvectors of these free-electron equations correspond exactly to those of the free-electron wave functions found by the usual methods. From this correspondence, a theorem is deduced about the free-electron charge density in alternant hydrocarbons.

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