Lab-fixed coordinates

For symmetric top species, Pave hes along the symmetry axis of the molecule, so the orientation of Pave can again be described in terms of 0 and (j), the angles used to locate the orientation of the molecule s symmetry axis relative to the lab-fixed coordinate system. As a result, the El integral again can be decomposed into three pieces ... 

The result of all of the vibrational modes contributions to la (3 J-/3Ra) is a vector p-trans that is termed the vibrational "transition dipole" moment. This is a vector with components along, in principle, all three of the internal axes of the molecule. For each particular vibrational transition (i.e., each particular X and Xf) its orientation in space depends only on the orientation of the molecule it is thus said to be locked to the molecule s coordinate frame. As such, its orientation relative to the lab-fixed coordinates (which is needed to effect a derivation of rotational selection rules as was done earlier in this Chapter) can be described much as was done above for the vibrationally averaged dipole moment that arises in purely rotational transitions. There are, however, important differences in detail. In particular. 

For linear molecules, the vibrationally averaged dipole moment (iave lies along the molecular axis hence its orientation in the lab-fixed coordinate system can be specified in terms of the same angles (0 and (f>) that are used to describe the rotational functions Yl,m (0,( )). Therefore, the three components of the (% I 4,ave I fr> integral can be written as ... 

Here m is the mass of the molecules and Vx, Vy, and Vz label the velocities along the lab-fixed cartesian coordinates. 

The derivation of these selection rules proceeds as before, with the following additional considerations. The transition dipole moment s (itrans components along the lab-fixed axes must be related to its molecule-fixed coordinates (that are determined by the nature of the vibrational transition as discussed above). This transformation, as given in Zare s text, reads as follows ... 

Figure 26. Scheme of the a = H2O and fo = H2 coordinates. x,x ,y are lab-fixed frames and the dashed line is the intermolecular axis, with R being the intermolecular distance. The dipole of H2O is along and the H-H internuclear axis is oriented along rj. Angles as shown. 

Here V, is the particle velocity of species i and y, the mole fraction of species i By particle velocities, we mean the vector-average velocities of millions of A molecules at a point. For a binary mixture of species A and B, we let Va and Vg be the particle velocities of species A and B, respectively. The flux, of A with respect to a fixed coordinate system (e.g., the lab bench), Wa, is just the product of the concentration of A and the particle velocity of A ... 

Now let Co, et, and be the basis (unit) vectors of the molecular coordinate system and ex, ey and ez the basis vectors (unit vectors) of the space fixed coordinate system (lab system) which leads to (compare Fig. IV. 1) ... 

For motion of the whole molecule, however, it is the molecule s orientation and location in the laboratory that matters, and so we use the lab-fixed or space-fixed coordinates 0 and O to describe the orientation of the nuclei in the lab, and the coordinates X, Y, and Z, which pinpoint the center of mass of the molecule. 

A FIGURE 5.5 The distinct lab-fixed and molecule-fixed coordinate systems. As the... 

Classical trajectory simulations are usually done by integrating Hamilton s equations of motion. Hamilton s equations are used for convenience since they are first-order. They are normally written in lab-fixed Cartesian coordinates. For an A -atom system Hamilton s equations are ... 

The method presented in this paper is based on the existing curriculum of an autonomous institution where flexibility is given to the program coordinator and the board of studies to modify the course content and a new course can be introduced without removing the fundamental courses. PO attainment of a course is the sum of PO attainment obtained from the direct assessment tools and the indirect assessment tools. The direct assessment tool assumed in this method is Continuous Internal Evaluation (CIE) analysis which is derived from the test paper, quiz paper, and lab performance analysis of a class of students. The indirect assessment tools assumed here are the Semester End Exam (SEE) result analysis, exit survey report, and faculty feedback report. The program coordinator in consultation with the Academic Audit Committee fixes the weightage for the assessment tools, fii this paper, it is assumed that 70 % come from the CIE analysis, 10 % from the SEE analysis, 10 % from the exit survey, and 10 % from the faculty feedback report to calculate the PO attainment of a course through the CO attainment. The components of PO attainment are shown in Pig. 4. 

This book uses the symbol E sys for the energy of the system measured in an earth-fixed lab frame. If during a process the system as a whole undergoes motion or rotation relative to this lab frame, then its energy in the lab frame depends in part on coordinates that are not properties of the system. In such situations E sys is not a state function, and we need the concept of internal energy. 

In the preceding sections of Appendix G, we assumed that a lab frame whose coordinate axes are fixed relative to the earth s surface is an inertial frame. This is not exactly true, because the earth spins about its axis and circles the sun. Small correction terms, a centrifugal force and a Coriolis force, are needed to obtain the effective net force acting on particle i that allows Newton s second law to be obeyed exactly in the lab frame. ... 

Reactive-scattering formalism. We start out by choosing a system of coordinates that spans the 6-MD configuration space in which the motion of the triatomic system, after removal of the motion of the center of mass, takes place. Let Oxyz be a system of coordinates whose origin 0 is the center of mass of the system and whose axes are parallel to a system of laboratory-fixed axes (the laboratory frame ). The space-fixed (SF) coordinate system Oxyz is also called laboratory-fixed (LAB) system. The spherical polar coordinates of the scaled Ra in this system are Let us also define a... 

Figure 5.1 The laboratory (LAB) frame, space-fixed (SF) frame, and body-fixed (BF) frame for the A+BC system in Jacobi coordinates. All are right handed axis systems. The SF frame originates on the center of mass (CM) of the A-fBC system, and is parallel with the LAB frame, which originates on the experimental apparatus. The BF frame also originates on the center of mass of the A+BC system, but rotates in space with the system so that the BF z-axis lies on the Jacobi scattering coordinate R, and the BF x-axis lies in the plane of the three paxticles. The transformation from the SF frame to the BF frame is a three dimensional rotation symbolized by 7, which may be specified in terms of the Euler angles (, 0, ). [The two versions of A+BC in this Figure are identical in every way. The axis systems, however, axe different in the two pictures.] The details and conventions axe discussed in the main text.

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