Axis systems

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 ... 

Another distinction we make concerning synnnetry operations involves the active and passive pictures. Below we consider translational and rotational symmetry operations. We describe these operations in a space-fixed axis system (X,Y,Z) with axes parallel to the X, Y, Z) axes, but with the origin fixed in space. In the active picture, which we adopt here, a translational symmetry operation displaces all nuclei and electrons in the molecule along a vector, say. 

In the active picture adopted here the (X,Y,Z) axis system remains fixed in space and a translational synnnetry operation changes the (X,Y,Z) coordinates of all nuclei and electrons in the molecule by constant amounts, (A X, A Y, A Z) say,... 

Figure Al.4.4. The definition of the Euler angles (0, ( ), x) that relate the orientation of the molecule fixed (x, y, z) axes to the (X, Y, Z) axes. The origin of both axis systems is at the nuclear centre of mass O, and the node line ON is directed so that a right handed screw is driven along ON in its positive direction by twisting it from Z to z through 9 where 0 < 9 < n. ( ) and x have the ranges 0 to In. x is measured from the node line.

We consider rotations of the molecule about space-fixed axes in the active picture. Such a rotation causes the (x, y, z) axis system to rotate so that the Euler angles change... 

Initially, we neglect tenns depending on the electron spin and the nuclear spin / in the molecular Hamiltonian //. In this approximation, we can take the total angular momentum to be N(see (equation Al.4.1)) which results from the rotational motion of the nuclei and the orbital motion of the electrons. The components of. m the (X, Y, Z) axis system are given by ... 

The axis labels (p,q,r) are chosen In order not to confuse this axis system with other systems, such as the molecule fixed axes (x,y,z) discussed below, used to describe molecular motion. 

The electric field gradient is again a tensor interaction that, in its principal axis system (PAS), is described by the tluee components F Kand V, where indicates that the axes are not necessarily coincident with the laboratory axes defined by the magnetic field. Although the tensor is completely defined by these components it is conventional to recast these into the electric field gradient eq = the largest component,... 

There are 3M-6 vibrations of a non-linear molecule containing M atoms a linear molecule has 3M-5 vibrations. The linear molecule requires two angular coordinates to describe its orientation with respect to a laboratory-fixed axis system a non-linear molecule requires three angles. 

For a three-axis system, three triangles in mutually perpendicular planes may be used. For navigational purposes, the output of the laser gyroscope may be integrated to determine the heading of an aircraft. Laser-based navigation systems have been in use on commercial and military aircraft since the early 1980s. 

For estimating concentrations from more than one source, it is convenient to use map coordinates for locaHons. Gifford (4) has pointed out that the resulting calculated concentration is the same whether the preceding axis system is used or whether an origin is placed at the ground beneath the receptor, with the x-axis oriented upwind, the z-axis remaining vertical, and the y-axis crosswind. 

This latter axis system is convenient in assessing the total concentration at a receptor from more than one source provided that the wind direction can be assumed to be the same over the area containing the receptor and the sources of interest. 

Potential energy surfaces are also central to our quantum-mechanical studies, and we are going to meet them again and again in subsequent chapters. Let s start then with Figure 3.1, which shows H2+. We are not going to be concerned with the overall translational motion of the molecule. For simphcity, I have drawn a local axis system with the centre of mass as the origin. By convention, we label the intemuclear axis the z-axis. 

Fig. 9.7 (A) Structures of the two possible diastereomers of the spiroindene 1. The RDCs determined in a PDMS/CDCf gel are color coded to the structures (red, negative RDCs blue, positive RDCs green, no RDCs or RDCs not used in analysis). The axis systems to the...

Values of ty larger than 1 indicate that the octahedral axes do no represent a proper axis system for the EFG in the sense of (4.32). To achieve 0 < rj < 1, the axes may be just interchanged... 

Rotational constants G = A, B or C are inversely proportional to principal moments of inertia Ia through the expressions G = h/Sn2Ia, where a refers to one of the three principal inertia axis directions a, b or c. The Ia are related to the coordinates of the atoms i in the principal axis system via the... 

The classification of critical points in one dimension is based on the curvature or second derivative of the function evaluated at the critical point. The concept of local curvature can be extended to more than one dimension by considering partial second derivatives. d2f/dqidqj, where qt and qj are x or y in two dimensions, or x, y, or z in three dimensions. These partial curvatures are dependent on the choice of the local axis system. There is a mathematical procedure called matrix diagonalization that enables us to extract local intrinsic curvatures independent of the axis system (Popelier 1999). These local intrinsic curvatures are called eigenvalues. In three dimensions we have three eigenvalues, conventionally ranked as A < A2 < A3. Each eigenvalue corresponds to an eigenvector, which yields the direction in which the curvature is measured. 

Sediment-free, on-axis systems at tectonic spreading centres, which have high temperature gradients (as they lie directly above magma chambers). The mean temperature of the ejected water is 620-640 K, the throughput 24 km3/year. 

Off-axis systems, on the flanks of spreading centres, driven by free convection due to cooling of the ocean crust, where the mean water temperature is around 420 K. 

Figure 2 Principal axis system of a shielding tensor assumed to be of axial symmetry.

Figure 5 Definition of the polar angles in the principal axis system of the diffusion-rotation tensor.

Figure 17 The a,a,2,6-tetrachlorotoluene. Small letters refer to the rotation-diffusion principal axis system.

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