A GROUP-THEORETICAL TREATMENT OF VIBRATIONAL, TORSIONAL, AND ROTATIONAL MOTIONS IN CH3—C≡C—SiH3

1966 ◽  
Vol 44 (5) ◽  
pp. 1169-1182 ◽  
Author(s):  
Jon T. Hougen
Keyword(s):  
Normal Modes ◽  
Energy Operator ◽  
Wave Functions ◽  
Theoretical Treatment ◽  
Symmetry Coordinates ◽  
Rotational Wave ◽  
Symmetry Properties ◽  
Symmetry Species ◽  
Molecular Wave Function ◽  
Rotational Part

A set of coordinates is described for methylsilylacetylene which allows a separation of the molecular wave function into a vibrational part, a torsional part, and a rotational part. The transformation properties of these coordinates, and vibrational, torsional, or rotational wave functions containing them, are discussed in terms of a double group of the molecular symmetry group appropriate for the classification of the overall wave functions of the molecule. Particular attention is given to the symmetry properties of vibrational symmetry coordinates and normal modes, since these symmetry properties are rather different from those encountered in molecules without free internal rotation. An exact and an approximate rotational–torsional–vibrational kinetic-energy operator is derived. Selection rules on symmetry species and also on angular-momentum quantum numbers are presented for electric dipole transitions.

A GROUP-THEORETICAL TREATMENT OF ELECTRONIC, VIBRATIONAL, TORSIONAL, AND ROTATIONAL MOTIONS IN THE DIMETHYLACETYLENE MOLECULE

1964 ◽  
Vol 42 (10) ◽  
pp. 1920-1937 ◽  
Author(s):  
Jon T. Hougen
Keyword(s):  
Wave Function ◽  
Optical Transition ◽  
Total Wave Function ◽  
Theoretical Treatment ◽  
Linear Portion ◽  
Selection Rules ◽  
Symmetry Species ◽  
Molecular Wave Function ◽  
Rotational Part ◽  
Rotational Motions

The Hamiltonian for the dimethylacetylene molecule is expressed in terms of a set of coordinates which allow a separation of the molecular wave function into an electronic part, a vibrational part, a torsional part, and a rotational part. Symmetry species are introduced which can be used to classify separately any one of the parts of the molecular wave function as well as the total wave function itself. These symmetry species correspond not only to the single-valued representations of the group proposed by Longuet-Higgins for the dimethylacetylene molecule, but also to double-valued representations of that group. The use of these symmetry species and of the corresponding selection rules is illustrated by the example of an optical transition which would correspond to a [Formula: see text] transition in the linear portion of the dimethylacetylene molecule.

Symmetry properties of the normal modes of vibration of calcite and α-corundum

1969 ◽  
Vol 47 (13) ◽  
pp. 1381-1391 ◽  
Author(s):  
E. R. Cowley
Keyword(s):  
Theoretical Analysis ◽  
Normal Modes ◽  
Dispersion Curves ◽  
Dynamical Matrix ◽  
Space Group ◽  
Symmetry Properties ◽  
Modes Of Vibration ◽  
Symmetry Species ◽  
Group Theoretical Analysis

A group theoretical analysis is carried out on the symmetry properties of the normal modes of vibration of crystals with the calcite and α-corundum structures. Both of these structures have the space group [Formula: see text]. The symmetry species present in the dispersion curves are determined for important wave vectors, and a transformation to block-diagonalize the dynamical matrix at the zone center is given.

Quantum Boxes, Stringed Instruments

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.

Symmetry Properties of Wave Functions in Magnetic Crystals

1962 ◽  
Vol 127 (2) ◽  
pp. 391-404 ◽  
Author(s):  
J. O. Dimmock ◽  
R. G. Wheeler
Keyword(s):  
Wave Functions ◽  
Symmetry Properties

Vibrations of a Cage-like Molecule, P4S3: Some Theoretical Aspects

1981 ◽  
Vol 36 (7) ◽  
pp. 774-777 ◽  
Author(s):  
S. J. Cyvin ◽  
B. N. Cyvin ◽  
M. Somer ◽  
W. Brockner
Keyword(s):  
Potential Energy ◽  
Energy Distribution ◽  
Force Field ◽  
Normal Modes ◽  
Potential Energy Distribution ◽  
Symmetry Coordinates ◽  
Method Of Fragments ◽  
Compliance Matrices

Abstract Two independent symmetry coordinate sets for P4S3 are developed, starting from the "method of fragments". A simple, approximate force field is expressed in terms of the two sets of symmetry coordinates, and the corresponding compliance matrices are given. The invariance of compliants is demonstrated. The potential energy distribution (PED) is discussed. An example is shown where the PED terms are clearly inadequate for the description of normal modes. A general warning against the interpretation of the PED in terms of such descriptions for cage-like structures seems to be warranted. V ib r a tio n s o f a C a g e -lik e M o le c u le , P 4S3 : S o m e T h e o r e tic a l A sp e c ts

The molecular orbital theory of chemical valency. XII. The excited states of diatomic molecules

1953 ◽  
Vol 216 (1124) ◽  
pp. 1-10 ◽  
Keyword(s):  
Wave Equation ◽  
Excited States ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Wave Functions ◽  
Triplet States ◽  
Orbital Theory ◽  
Lennard Jones ◽  
Singlet States ◽  
Symmetry Properties

A method is developed for the solution of the wave equation for two electrons in the presence of two centres. The work of Lennard-Jones & Pople (1951) on the ground state of such a system is generalized so as to apply to all the excited states. Full advantage is taken of the symmetry properties of the wave functions, both in three-dimensional and six-dimensional space, to reduce the wave equation to a number of component parts, each of a particular symmetry type. This leads to sets of equations with characteristic symmetry properties appropriate to singlet states and triplet states, whether even or odd, positive or negative in the standard notation ( 1 ∑ - g ).

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

2021 ◽  
Vol 75 (3) ◽  
pp. 229-233
Author(s):  
Faig Pashaev ◽  
Arzuman Gasanov ◽  
Musaver Musaev ◽  
Ibrahim Abbasov
Keyword(s):  
Group Theory ◽  
Energy Levels ◽  
Wave Functions ◽  
Electron Wave ◽  
Hamilton Operator ◽  
Atomic Orbitals ◽  
Symmetry Properties ◽  
Symmetry Transformations ◽  
Space Symmetry ◽  
Polyatomic Systems

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.

scholarly journals ON A METHOD OF CALCULATION OF NON-AXIAL ROTATIONAL WAVE FUNCTIONS AND ENERGY LEVELS OF ODD-A NUCLEI

1963 ◽  
Vol 19 (2) ◽  
pp. 99
Author(s):  
HAN WEN-SHU ◽  
SOU YOU-LIANG ◽  
YAN ING-CHOU
Keyword(s):  
Energy Levels ◽  
Wave Functions ◽  
Method Of Calculation ◽  
Rotational Wave

Ab initio Calculations of the Rotation-Vibration Spectrum of Na3+

1993 ◽  
Vol 58 (1) ◽  
pp. 24-28 ◽  
Author(s):  
Ladislav Češpiva ◽  
Vlasta Bonačič-Koutecký ◽  
Jaroslav Koutecký ◽  
Per Jensen ◽  
Vojtěch Hrouda ◽  
...  
Keyword(s):  
Potential Surface ◽  
Vibration Spectrum ◽  
Wave Functions ◽  
Basis Set ◽  
Morse Oscillator ◽  
Fundamental Frequencies ◽  
Rotational Wave ◽  
Rigid Molecule ◽  
Full Ci ◽  
Ci Calculations

SCF, 6C-SCF, MP4 and valence-electron full CI calculations were performed in order to determine the potential surface of Na3+. A power series in the variables yi = 1 - exp (-a∆ri), where ∆ri are bond length displacements from equilibrium, has been fitted through the surface obtained and used in a variational rotation-vibration calculations with a basis set of products of Morse-oscillator eigenfunctions and symmetric top rotational wave functions. In contrast to H3+, Na3+ behaves as a very rigid molecule and does not exhibit any anomalous anharmonicity. With our best potential surface, MP4, the predicted E' and A1' fundamental frequencies are 105.1 and 146.7 cm-1, and the harmonic E' and A1' frequencies are 106.5 and 148.3 cm-1.

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