Axis convention

Table Al.4.2 The multiplication table of the point group using the space-fixed axis convention (see text).

If we were to define the operations of the point group as also rotating and reflecting the (p.q.r) axis system (in which case the axes would be tied to the positions of the nuclei), we would obtain a different multiplication table. We could call this the nuclear-fixed axis convention. To implement this the protons in the o, O2 and planes in figure Al.4.2 would be numbered H, H2 and respectively. With this convention the operation would move the a plane to the position in space originally occupied by the 02 plane. If we follow such a C3 operation by the reflection (in the plane containing Ft ) we find that, in the nuclear-fixed axis convention ... 

Similarly, with the nuclear-fixed axis convention, we detennine that... 

Nuclear pemuitations in the N-convention (which convention we always use for nuclear pemuitations) and rotation operations relative to a nuclear-fixed or molecule-fixed reference frame, are defined to transfomi wavefunctions according to (equation Al.4.56). These synnnetry operations involve a moving reference frame. Nuclear pemuitations in the S-convention, point group operations in the space-fixed axis convention (which is the convention that is always used for point group operations see section Al.4.2,2 and rotation operations relative to a space-fixed frame are defined to transfomi wavefiinctions according to (equation Al.4.57). These operations involve a fixed reference frame. 

Figure 27. Molecular frame axis convention for the C2V NO dimer. The y axis is along the N—N bond. Both pump and probe laser polarizations are parallel to the y axis.

The 6D fully correlated TRCIS data set may be cut, projected, or filtered to reveal both scalar and vector correlations as a function of time. We restrict our discussion here to angular correlations. The molecular frame axis convention for the NO dimer is shown in Fig. 27. Note that the pump and probe laser polarizations were parallel to each other in these experiments. 

It will be noted that the above derivation was based on the assumption that dTIdy was negative and that duldy was positive and that the buoyancy forces acted in the -direction, i.e., they applied to assisting flow. In opposing flow, the buoyancy forces act in the opposite direction to the x-axis, this -axis conventionally being taken in the direction of the forced flow. Thus, in general, Eqs. (9.95), (9.96), and (9.97) can be written as ... 

Following a minimum aging period of 48 hr, specimens were cut so that the fibers were parallel to the long or tensile axis. Conventional 1.5-inch dogbone tensile specimens (1 inch gage length) were tested at 0.2 inch/min on an Instron testing machine. 

The vibration-rotation interaction is the effect arising from coupling terms between angular and vibrational momenta as well as from the dependence of the rotational G-matrix elements (the /u-tensor) on the internal coordinates. The importance of this effect may to some extent be reduced provided an appropriate axis convention is used. The axis convention is the set of rules defining the orientation of the molecular axes, eg, g = x,y, z, relative to an arbitrary configuration as given by the position vectors, Ra, a. = 1, 2,... N. These rules can be expressed in three relations between the rag components, similar to the center of mass conditions(2.4). We shall refer to these relations as the axial constraints . Usually Eckart-condi-tions39 are imposed, but other possibilities may be considered. 

All components of these vectors can in principle be explicitly written in terms of the generalized coordinates, provided that all atomic coordinate functions, ragU7i > < 2> <73Ar-6). appearing in Eq. (2.6) have been formulated. If the vibrational coordinates are purely geometrically defined, however, the vibrational s-vectors and their LS-components, sk> aF, are independent of the axis convention used in formulating -functions, contrary, of course, to their MS-components, sk> ag. Applying Eq. (2.36) we realize that the vibrational part of the G-matrix is also independent of the axis convention under these special conditions. 

Exploring the possibilities of this idea we consider the three rotational constraint relations on the atomic coordinates that follow from the axis convention. It is assumed that these relations can be cast in the form... 

First we must formulate the axis convention in accordance with Eq. (2.45). 

Eckart System (ES). It is well-known how the axis convention proposed by Eckart39 enters the standard vibration-rotation theory2 49 52) in the alternative method of deriving the kinetic energy the Eckart conditions are used in formulating rotational s-vectors. For this purpose we may proceed exactly as in the PAS example above. 

The Eckart conditions play an important role in this connection. We shall discuss this in more detail below, since the arguments presented apply equally well to the treatment of nonrigid molecules. Hence, to study the basis of introducing Eckart conditions, let us for a moment go back to an earlier stage where axis conventions were not yet formulated. We recapitulate that we are looking for the conditions required in order that the atomic position coordinates, rag, can be given as unique functions of 3 N-6 internal coordinates, or equivalently stated, in order that the expansion [Eq. (3.6)] can be determined as a unique inverse of Eq. (3.5). 

We adopt here an axis convention in which the ground state of the phenolate anion (PhO ) is described by the following basic orbital configuration . .. (13ai). .. (8 2) (3i>i) The reference configurations for the A2, 5i and... 

Figure 5.13 The axis convention for determining the sign of the unit vector n directed per-pendicuiar to the piane containing the vectors a and b...

In the monoclinic system, the point group symbols refer to the unique axis, conventionally taken as the b-axis. The point group symbol... 

Mirror plane normal to the unique b-axis (conventional) Mirror plane normal to the unique c-axis... 

Figure 2a. Histogram of the distribution of the concentration of zinc in serum in 27 samples. Concentration in mg/L on x-axis. Conventional histogram (after Herber and Pieters. 1982).

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