Reduced-Mass Calculations

In the interpretation of the spectral data it is usually the constants A and B in Eq. (3.27) or some other set of reduced potential constants that are evaluated. The barrier height, AB2/4 for B 0, is thus determined directly. However, if one wants to relate the value of the dimensionless parameter Z I at the minimum of a double-minimum potential function to the absolute geometry of the puckered ring, the reduced mass must be known in order to find x from IZI. One is then required to introduce assumptions about the dynamical model of the ring-puckering motion. 

For cyclobutane, the motion of the four carbon atoms is relatively unambiguous. 

Model calculations to reproduce the variation of rotational constants with vibrational state are sensitive to the value of to. Thus an approximate value may be determined from them that yields experimental information about the dynamics of the vibration. 

If the Hamiltonian with variable reduced mass is used, the dependence of the reduced mass on coordinate will be a function of to. On the other hand, if the constant effective mass model is used, the reduced mass can be evaluated from Eqs. (3.1) to (3.10) for an infinitesimal displacement from the planar conformation. The constant effective reduced mass derived in this fashion is independent of to, and thus, no knowledge of to is needed to use the Hamiltonian Eq. (3.22) conversely, no information about the value of to can be determined from the vibrational data and Eq. (3.22). 

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This study is one of the earliest attempts to calculate equilibrium fractionation factors using measured vibrational spectra and simple reduced-mass calculations for diatomic molecules. For the sake of consistency I have converted reported single-molecule partition function ratios to units. 

Once /z-and thus a value for the puckering coordinate at the minimum of the well-is obtained, it is possible to extract some structural information about the molecule in the ground state, such as the ring dihedral angle. A number of papers have appeared in which dihedral angles are reported where neither geometric information nor the details of the reduced mass calculation are given. In some cases the reduced mass has been assumed by comparison with p s calculated for other... 

Bands in the IR spectrum - ° correspond to vibrational modes of a molecule. The position of the band, v, depends (Eq. 10.15) on the strength of the bond(s) involved as measured by a force constant k, and on the reduced mass of the system, m,. Equation 10.16. shows the reduced mass calculated for a simple diatomic molecule, where m and mz are the atomic weights of the two atoms ... 

E10.6 For the diatomic molecule Na2, 5 = 230.476 J-K-1-mol" at T= 300 K, and 256.876 J-K-,-mol-1 at T= 600 K. Assume the rigid rotator and harmonic oscillator approximations and calculate u, the fundamental vibrational frequency and r, the interatomic separation between the atoms in the molecule. For a diatomic molecule, the moment of inertia is given by l pr2, where p is the reduced mass given by... 

The isotopic difference between the mean squares of the displacements in equation (7) can be calculated if the carbon-hydrogen oscillator is treated as a diatomic molecule. It is easily shown that for constant potential the mean square of the displacement from the equilibrium position of the harmonic oscillator will be inversely proportional to the square root of the reduced mass, /x, and hence... 

In other models, the confined exciton was treated as one electron with the reduced mass p = (1 /m -I- 1/nih) which moves in the field of the walls, and a positive hole h fixed at the centre of the sphere. Curve a in Fig. 34 gives the lowest energy level (wavelength of transition to this level) as calculated in the semi-classical approxima-... 

Atomic isotopes help detection so that the shift for 13 CO is perfectly predictable and isotope ratios are known from various sources, as we shall see later on. The change in the rotational constant with reduced mass is easily calculated and hence the change in the frequency of the J = J = 0 can be calculated, as we saw in Section 3.3. 

Microwave spectra Calculation of reduced mass, rotational constant, bond... 

A solution of CO in tetrachloromethane absorbs infrared radiation of frequency 6.42 x 1013 Hz. If this is interpreted as a vibration quantum both the force constant of the C-0 bond and the spacing of vibrational levels can be calculated directly. The reduced mass yco = 1-14 x 10-26 kg. The force constant k = A-n2vly = (47r2)(6.42 x 1013)(1.14 x 10-26) = 1.86 x 103 Nm 1. The separation between the vibrational levels of CO is... 

Capital, volume 3. Grossmann s claim to have faithfully represented Marx s theory rests on passages in section 3 of chapter 15, Excess Capital and Excess Population .7 The breakdown scenario, in which the mass of surplus value dries up in year 35 of the Bauer schema, is interpreted by Grossmann (1992 76) as a case of overaccumulated capital . Quoting from Marx, there would be a steep and sudden fall in the general rate of profit (Marx 1959 246). Moreover, The fall in the rate of profit would then be accompanied by an absolute decrease in the mass of profit. And the reduced mass of profit would have to be calculated on an increased total capital (ibid. 247). 

However, it is pertinent to restate the underlying principles embodied in these calculations, that is the vibrational frequency of a bond is expected to increase when the bond strength increases, and also when the reduced mass of the system decreases . 

It should be noted in passing that Mulliken also examined the isotope effect on the quadratic terms in the equations for the band heads. These ratios should theoretically show an isotope effect proportional to the reduced masses of the diatomic molecules (rather than the square root of the reduced masses). While Mulliken concludes that these ratios also confirm that the molecule is BO rather than BN, the four experimental ratios show a fairly large scatter so that the case for identifying the molecule is not as strong as that from the experimental a and b ratios. He also measured some of the rotational lines in the spectra of BO and considered the measured and theoretical isotope effects. Here one experimental isotope ratio checks the theoretically calculated ratio quite well, but for the other two the result was unsatisfactory. However, Mulliken judged the error to be within the experimental uncertainty. 

As mentioned, most calculations we have done so far have concerned molecular systems. However, prior to development of the non-BO method for the diatomic systems, we performed some very accurate non-BO calculations of the electron affinities of H, D, and T [43]. The difference in the electron affinities of the three systems is a purely nonadiabatic effect resulting from different reduce masses of the pseudoelectron. The pseudoelectrons are the heaviest in the T/T system and the lightest in the H/H system. The calculated results and their comparison with the experimental results of Lineberger and coworkers [44] are shown in Table 1. The calculated results include the relativistic, relativistic recoil. Lamb shift, and finite nuclear size corrections labeled AEcorr calculated by Drake [45]. The agreement with the experiment for H and D is excellent. The 3.7-cm increase of the electron affinity in going from H to D is very well reproduced by the calculations. No experimental EA value is available for T. 

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