Rotational analysis

The unique factors are usually removed as they are considered to represent irrelevant information specific to the particular descriptors alone (such as experimental error). This leaves the common factors, which like the principal components are orthogonal. It sometimes happens that several descriptors may be loaded onto one or more factors. The factors are often rotated in order to try to arrange for each factor to be largely associated with as few variables as possible. Thus if any given factor is largely associated with only one or two variables then it can be much easier to interpret any subsequent analysis. Rotation can also be applied to a set of principal components. 

Extraction Method Principal Component Analysis, Rotation Method Oblimin Oblique Rotation with Kaiser Normalization. Correlation of Components Asian to Caucasian 0.307, Asian to African -0.615, Caucasian to African -0.119... 

Figure 1. shows the measured phase differenee derived using equation (6). A close match between the three sets of data points can be seen. Small jumps in the phase delay at 5tt, 3tt and most noticeably at tt are the result of the mathematical analysis used. As the cell is rotated such that tlie optical axis of the crystal structure runs parallel to the angle of polarisation, the cell acts as a phase-only modulator, and the voltage induced refractive index change no longer provides rotation of polarisation. This is desirable as ultimately the device is to be introduced to an interferometer, and any differing polarisations induced in the beams of such a device results in lower intensity modulation. 

The external reflection of infrared radiation can be used to characterize the thickness and orientation of adsorbates on metal surfaces. Buontempo and Rice [153-155] have recently extended this technique to molecules at dielectric surfaces, including Langmuir monolayers at the air-water interface. Analysis of the dichroic ratio, the ratio of reflectivity parallel to the plane of incidence (p-polarization) to that perpendicular to it (.r-polarization) allows evaluation of the molecular orientation in terms of a tilt angle and rotation around the backbone [153]. An example of the p-polarized reflection spectrum for stearyl alcohol is shown in Fig. IV-13. Unfortunately, quantitative analysis of the experimental measurements of the antisymmetric CH2 stretch for heneicosanol [153,155] stearly alcohol [154] and tetracosanoic [156] monolayers is made difflcult by the scatter in the IR peak heights. 

To calculate N (E-Eq), the non-torsional transitional modes have been treated as vibrations as well as rotations [26]. The fomier approach is invalid when the transitional mode s barrier for rotation is low, while the latter is inappropriate when the transitional mode is a vibration. Hamionic frequencies for the transitional modes may be obtained from a semi-empirical model [23] or by perfomiing an appropriate nomial mode analysis as a fiinction of the reaction path for the reaction s potential energy surface [26]. Semiclassical quantization may be used to detemiine anliamionic energy levels for die transitional modes [27]. 

If the experunental technique has sufficient resolution, and if the molecule is fairly light, the vibronic bands discussed above will be found to have a fine structure due to transitions among rotational levels in the two states. Even when the individual rotational lines caimot be resolved, the overall shape of the vibronic band will be related to the rotational structure and its analysis may help in identifying the vibronic symmetry. The analysis of the band appearance depends on calculation of the rotational energy levels and on the selection rules and relative intensity of different rotational transitions. These both come from the fonn of the rotational wavefunctions and are treated by angnlar momentum theory. It is not possible to do more than mention a simple example here. 

Figure Bl.5.7 Rotational anisotropy of the SH intensity from oxidized Si(l 11) surfaees. The samples have either ideal orientation or small offset angles of 3° and 5° toward tire [Hi] direetion. Top panel illustrates the step stnieture. The points eorrespond to experimental data and tlie fiill lines to the predietion of a symmetry analysis. (From [65].)...

Figure B2.3.3. Crossed-moleciilar beam apparatus employed for die study of the F + D2 —> DF + D reaetion. Indieated in the figure are (1) the effusive F atom soiiree (2) slotted-disk veloeity seleetor (3) liquid-nitrogen-eooled trap (4) D2 beam souree (7) skimmer (8) ehopper (9) eross-eorrelation ehopper for produet veloeity analysis and (11) rotatable, ultralrigh-vaeuum, triply differentially pumped, mass speetrometer deteetor ehamber. Reprinted with pemrission from Lee [29], Copyright 1987 Ameriean Assoeiation for the Advaneement of Seienee.

The homonuclear rare gas pairs are of special interest as models for intennolecular forces, but they are quite difficult to study spectroscopically. They have no microwave or infrared spectmm. However, their vibration-rotation energy levels can be detennined from their electronic absorjDtion spectra, which he in the vacuum ultraviolet (VUV) region of the spectmm. In the most recent work, Hennan et al [24] have measured vibrational and rotational frequencies to great precision. In the case of Ar-Ar, the results have been incoriDorated into a multiproperty analysis by Aziz [25] to develop a highly accurate pair potential. 

Kreutz T G and Flynn G W 1990 Analysis of translational, rotational, and vibrational energy transfer in collisions between COj and hot hydrogen atoms the three dimensional breathing ellipse model J. Chem. Phys. 93 452-65... 

Also, rotational state resolution of cross-sections can be obtained by employing a coherent state analysis [51] for the situation of weak coupling between rotational and vibrational degrees of freedom. A suitable rotational coherent state can be expressed as... 

Resonant timesteps can be estimated on the basis of one-dimensional analysis [65, 62] from the propagating rotation matrices in phase-space for... 

---