An approximate wave mechanical treatment of the harmonic oscillator and rigid rotator

1963 ◽  
Vol 40 (7) ◽  
pp. 365
Author(s):  
Ronald Rich
Keyword(s):  
Harmonic Oscillator ◽  
Mechanical Treatment ◽  
Rigid Rotator ◽  
Approximate Wave

DERIVATIVES OF MONOGERMANE: PART I. THE VIBRATION–ROTATION SPECTRA OF GERMYL FLUORIDE AND BROMIDE

1962 ◽  
Vol 40 (4) ◽  
pp. 579-589 ◽  
Author(s):  
J. E. Griffiths ◽  
T. N. Srivastava ◽  
M. Onyszchuk
Keyword(s):  
Harmonic Oscillator ◽  
Infrared Absorption ◽  
Thermodynamic Functions ◽  
Low Frequency ◽  
Rotational Constant ◽  
Rigid Rotator ◽  
Harmonic Oscillator Model ◽  
Infrared Absorption Spectra ◽  
Vibration Rotation ◽  
Derivatives Of

The vibration–rotation infrared absorption spectra of germyl fluoride and bromide have been observed. All of the fundamentals in GeH3F were located, and the rotational structure of the E-type bands were resolved and analyzed. The low-frequency band, ν3(a1), in GeH3Br was not observed but an estimate of its position was made from the frequencies of the combination band ν3 + ν6 and of ν6. The rotational constant A″ and the Coriolis constants ζ4, ζ5, and ζ6 were calculated for both molecules, and agreement with microwave A″ values was satisfactory. Thermodynamic functions based upon a rigid-rotator, harmonic-oscillator model have been evaluated for germyl fluoride and bromide.

Schwingungsspektren und Thermodynamische Funktionen von PF3, PCl3, PBr3, P J3, PF2Cl, PF2Br, PF2J, PFCl2, PFBr2, PCl2Br, PCl2J, PClBr2, PClJ2, PBrJ2, PBr2J, PFClBr, AsCl2Br und AsClBr2

1968 ◽  
Vol 23 (5) ◽  
pp. 588-594 ◽  
Author(s):  
A. Müller ◽  
E. Niecke ◽  
B. Krebs ◽  
O. Glemser
Keyword(s):  
Harmonic Oscillator ◽  
Infrared Spectra ◽  
Thermodynamic Functions ◽  
Frequency Data ◽  
Rigid Rotator ◽  
Temperature Range

The complete Raman and infrared spectra of PF2Cl, PF2Br, PFCl2 and PFBr2 are reported. The frequency data for several mixed phosphorus and arsenic halides (see title) given in the literature are revised and completed. Assignments are made for all spectra. Thermodynamic functions in the temperature range 200—2000 °K are calculated for all molecules, assuming the harmonic oscillator — rigid rotator approximation.

Isomeric Higher and Smaller Fullerenes: A Profound Enthalpy/Entropy Interplay

1999 ◽  
Vol 593 ◽  
Author(s):  
X. Zhao ◽  
Z. Slanina ◽  
E. Ōsawa
Keyword(s):  
Harmonic Oscillator ◽  
Ab Initio ◽  
Quantum Chemical ◽  
Temperature Effects ◽  
Rigid Rotator ◽  
Relative Stabilities ◽  
Rotator Model ◽  
Complete Set ◽  
Good Agreement

ABSTRACTComputations of isomeric fullerenes are performed at semiempirical and ab initio quantum-chemical levels: C36, C72, C88. C36 fullerenes and quasi-fullerenes are computed at the SAM1 level, and then at the B3LYP/6-31G* level. Altogether 598 cages are considered. The SAM1 method is also applied to C72, i.e., the solitary isolated-pentagon-rule (IPR) structure and several non-IPR isomers. Finally, the complete set of thirty five topologically different IPR isomers of C88 is computed. In all the cases, energetics is combined with entropy contributions based on the harmonic-oscillator and rigid-rotator model. Considerable temperature effects on the relative stabilities in the systems are found. Relationships to available observed data are discussed throughout and a good agreement is found.

A concise quantum mechanical treatment of the forced damped harmonic oscillator

Open Physics ◽  
2008 ◽  
Vol 6 (4) ◽  
Author(s):  
Ti Li
Keyword(s):  
Harmonic Oscillator ◽  
Mechanical Treatment ◽  
Classical Mechanics ◽  
Quantum Mechanical ◽  
External Influence ◽  
Classical Motion ◽  
Damped Harmonic Oscillator ◽  
Classical Energy ◽  
Quantum Mechanical Treatment ◽  
Generalized Coordinate

AbstractBy selecting a right generalized coordinate X, which contains the general solutions of the classical motion equation of a forced damped harmonic oscillator, we obtain a simple Hamiltonian which does not contain time for the oscillator such that Schrödinger equation and its solutions can be directly written out in X representation. The wave functin in x representation are also given with the help of the eigenfunctions of the operator $$ \hat X $$ in x representation. The evolution of $$ \left\langle {\hat x} \right\rangle $$ is the same as in the classical mechanics, and the uncertainty in position is independent of an external influence; one part of energy mean is quantized and attenuated, and the other is equal to the classical energy.

Normal Coordinate Treatments and Calculated Thermodynamic Properties of Phosphoryl Fluorodichloride, Phosphoryl Chlorodifluoride, Thiophosphoryl Bromide, Thiophosphoryl Fluorodibromide, and Thiophosphoryl Bromodifluoride

1971 ◽  
Vol 25 (2) ◽  
pp. 212-217 ◽  
Author(s):  
Joseph S. Ziomek ◽  
Frank J. Fillwalk ◽  
Edward A. Piotrowski
Keyword(s):  
Thermodynamic Properties ◽  
Harmonic Oscillator ◽  
Matrix Elements ◽  
Normal Coordinate ◽  
Rigid Rotator ◽  
Gaseous State ◽  
Harmonic Oscillator Approximation ◽  
The One ◽  
Raman And Ir Spectra ◽  
The Ideal

The Raman and ir spectra of phosphoryl dichlorofluoride, phosphoryl difluorochloride, thiophosphoryl tribromide, thiophosphoryl dibromofluoride, and thiophosphoiyl difluorobromide were collected and examined for the most probable values for the wavenumbers, intensities, and depolarization factors. The data are as follows: The Raman displacements Δσ in cm−1, the relative intensities I, and the depolarization factors ρ are for POFCl2: Δσ ( I) ρ = 207(3.6)0.55, 254(1.8)6/7, 330(1)0.6, 372(3.4)6/7, 386(5.5)0.3, 547(10)0.05, 620( w)6/7, 894( w)0.45, and 1331 ( m) p; for POF2Cl: 274(10)0.65, 274 (calculated), 410(17)0.23, 424(7)6/7, 424 (calculated), 618(10)0.05, 895( m)0.1, 948( w>)6/7, and 1358( m)0.2; for PSBr3: 115(5.5)6/7, 165(4.1)0.3, 179(5)6/7, 299(10)0.05, 438( w)6/7, and 718 ( m) p; for PSFBr2:129(7)0.6, 159.5(11)6/7, 219(10)0.5, 254(4)6/7, 274(5)0.3, 377(10)0.2, 470( w)6/7, 718( m) p, and 887 ( vw) p; and for PSF2Br: 175(28)0.5, 231(3)6/7, 288(29)0.3, 298( w)6/7, 384(7)0.25, 462( w)0.1, 474(10)0.1, 711(9)0.1, 899( w)0.75, and 930( w)0.75. No published ir spectral data were found for POFCl2, POF2Cl, PSBr3, PSFBr2, and PSF2Br. Also normal coordinate treatments were conducted for POFCl2, POF2Cl, PSFBr2, and PSF2Br on the basis of the Cs model and the one for PSBr3 on the basis of the C3 v model. The results of these treatments established the above listed wavenumbers as fundamentals and lend support for 424 and 274 cm−1 bands for POF2Cl as the missing Raman bands. The F matrix elements obtained for these molecules were determined in such a way that F matrix elements common to POF3, POF2Cl, POFCl2, and POCl3 had nearly the same values and those common to PSF3, PSF2Br, PSFBr2, and PSBr3 also had nearly the same values. Finally, the values of the thermodynamic properties for these substances were computed for the ideal gaseous state using the rigid rotator harmonic oscillator approximation at 1 atm from 200 to 1000 K.

Rotational Cut-Off and Anharmonicity Effects on Thermodynamic Properties of Gases at High Temperatures

1974 ◽  
Vol 25 (4) ◽  
pp. 287-292
Author(s):  
N M Reddy
Keyword(s):  
Partition Function ◽  
Thermodynamic Properties ◽  
Harmonic Oscillator ◽  
Specific Heat ◽  
High Temperatures ◽  
Anharmonic Oscillator ◽  
Rigid Rotator ◽  
Temperature Range ◽  
Vibrational Levels ◽  
Harmonic Oscillator Approximation

SummaryThe exact expressions for the partition function Q and the coefficient of specific heat at constant volume Cv for a rotating-anharmonic oscillator molecule, including coupling and rotational cut-off, have been formulated and values of Q and Cv have been computed in the temperature range of 100°K to 100 000°K for O2, N2 and H2 gases. The exact Q and Cv values are also compared with the corresponding rigid-rotator harmonic-oscillator (infinite rotational and vibrational levels) and rigidrotator anharmonic-oscillator (infinite rotational levels) values. The rigid-rotator harmonic-oscillator approximation can be accepted for temperatures up to about 5000°K for O2 and N2. Beyond these temperatures the error in Cv will be significant, owing to anharmonicity and rotational cut-off effects. For H2, the rigid-rotator harmonic-oscillator approximation becomes unacceptable even for temperatures as low as 2000°K.

Metallurgical Effects of Grain Boundary Ledges

1977 ◽  
Vol 35 ◽  
pp. 126-129
Author(s):  
L.E. Murr
Keyword(s):  
Grain Boundary ◽  
Quantitative Data ◽  
Mechanical Treatment ◽  
Unit Length ◽  
Physical And Mechanical Properties ◽  
Direct Observations ◽  
Transmission Electron ◽  
Properties Of Materials ◽  
Thermo Mechanical Treatment ◽  
Ledge Density

Ledges in grain boundaries can be identified by their characteristic contrast features (straight, black-white lines) distinct from those of lattice dislocations, for example1,2 [see Fig. 1(a) and (b)]. Simple contrast rules as pointed out by Murr and Venkatesh2, can be established so that ledges may be recognized with come confidence, and the number of ledges per unit length of grain boundary (referred to as the ledge density, m) measured by direct observations in the transmission electron microscope. Such measurements can then give rise to quantitative data which can be used to provide evidence for the influence of ledges on the physical and mechanical properties of materials.It has been shown that ledge density can be systematically altered in some metals by thermo-mechanical treatment3,4.

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