Principle axis system

Table 23.1. Euler angles of the C and N CSA principle axis system (PAS) relative to fiber axis system (FAS) for [l- C]Gly, [l- C]Ala, [ N]Gly and [ NjAla sites of B. mori silk fibroin samples and that relative to molecular symmetry axis (MSA) for [l- C]-[ N]double labeled sites of Boc-Gly-Ala- [l- C]Gly-[ N]Ala-Gly-Ala-OPac and Boc-Ala-Gly-[l- C]Ala-[ N]Gly-Ala-Gly-OPac...

From these equations it is obvious that it is possible, at least in principle, to determine the electric dipole moment of a molecule from the rotational Zeeman effect data of two isotopic species (see Fig. II,1). If the isotopic substitution causes a parallel shift of the principle axis system a p=av + Aa, b p = bp- - Ab, c p—Cp->r Ac, etc., where Aa is the a-coordinate of the center of mass of the daughter molecule referred to the principal axes of the parent molecule, the corresponding equations are given by... 

Is the velocity of the t-th particle with respect to the principle axis system of the nuclear moment of inertia tensor. 

The dependence of the dipolar couplings on the orientation of the internuclear vector in the principle axis system of the electric field gradient has Inherent information about the molecular geometry of the relevant and sites. For example, in a peptide bond, where two carbons are bound to the same nitrogen, the dipolar coupling of each depends on the orientation of its respective internuclear vector with respect to the common coordinate system of the nitrogen electric field gradient principle axis system. Thus, the NMR spectrum is analyzed in terms of the relative orientation of the internuclear vectors, In principle, this can lead to an independent determination of the conformation of the peptide bond. 

Two of the three constants eqQaa, eqQbb id eqC c were determined from microwave optical double resonance (MODR) spectra of NH2 [1 to 4], microwave absorption spectra of ND2 [5], and by ab initio (UHF) calculations [6]. The third constant follows from eqQaa+ eqQbb + QQcc = 0 [3]- The nuclear quadrupole coupling constants listed in the following table are given in terms of the principle axis system (b C2, c 1 molecular plane). 

Anisotropic constants are given in terms of the principle axis system (b C2, c molecular plane). 

Another way of rationalizing these results is to treatx, y, z as px, py, p- orbitals of central atom A in an AH molecule. A molecule with C2v symmetry is H2S. A convention to set up the Cartesian coordinates for this molecule is as follows. Take the principle axis (C2 in this case) as the z axis. Since H2S is planar, we take the x axis to be perpendicular to the molecular plane. Finally, the y axis is taken so as to form a right-handed system. Following this convention, the px, py, p- orbitals on sulfur in a H2S molecule are shown in Fig. 6.3.2. When the orientations of these orbitals are examined, it is obvious that the p- orbital is symmetric with respect to all four operations of the C2v point group E, C2, other hand, the px orbital is symmetric with respect to E and arv(xz), but antisymmetric with respect to C2 and [Pg.182]

Another method of characterizing the asymmetry of the molecular shape is related to the choice of the main direction in the molecule. In the principle axis Z in Fig. 11 tangential to the worraJike chain at its middle point can be chosen as this direction. However, in this system of coordinates the asymmetry of the shape of the molecule is adequately expressed only in the range of conformations close to a rod whereas for the Gaussian coil conformation (x — < ), according to Eq. (29), p is equal to unity and the coil is herical (see also Pig. 13). Hence, the asymmetry of the shape of the Gaussian coil cannot be adequately expressed by a system of coordinates determined by the middle element of a worm-like chain. 

The four-fold axis is the highest order (n is largest) axis and is considered the principle axis. The highest order axis plays an important role in distinguishing one crystal system from another. 

If only a single interaction tensor is present, it is always possible to employ the principle axis (PAS) system of the tensor as the coordinate system, resulting in the following expression ... 

The averages are just the microscopic order parameters. There are a maximum of L independent order parameters in the principal axis system, but the principal axes may be defined only if symmetry allows. In particular, there are generally 25 order parameters for L = 2. In principle, the distribution function may be obtained from x-ray and neutron scattering studies. However, this is difficult in practice. If /(f2) is supposed to originate from an orientational pseudo-potential V(fi), then... 

The rotational g tensor is a tensor of rank two with in principle nine independent elements. The rotational magnetic moment is only well defined in the principal rotational axis system, where the moment of inertia tensor is diagonal, and only the diagonal elements of g in this principal axes system contribute to the rotational Zeeman effect (Flygare 1974 Gauss et al. 1996). 

CAS uses an extension to the CIP system to denote the configuration of mononuclear coordination compounds. The stereodescriptors are composed of a system indicator (e.g., OC-6 for octahedral), a geometry label, and potentially a chirality symbol. The geometry label is assigned on the basis of CIP priorities and the chirality symbol describes the handedness of the ligand arrangement with respect to some principle axis through the center. The DARC system also uses priorities (not based on CIP) and a method has been proposed to apply them to the stereochemistry of coordination compounds, ... 

For a removal attempt a molecule is selected irrespective of its orientation. To enhance the efficiency of addition attempts in cases where the system possesses a high degree of orientational order, the orientation of the molecule to be added is selected in a biased way from a distribution function. For a system of linear molecules this distribution, say, g u n ), depends on the unit vector u parallel to the molecule s symmetry axis (the so-called microscopic director [70,71]) and on the macroscopic director h which is a measure of the average orientation in the entire sample [72]. The distribution g can be chosen in various ways, depending on the physical nature of the fluid (see below). However, g u n ) must be normalized to one [73,74]. In other words, an addition is attempted with a preferred orientation of the molecule determined by the macroscopic director n of the entire simulation cell. The position of the center of mass of the molecule is again chosen randomly. According to the principle of detailed balance the probability for a realization of an addition attempt is given by [73]... 

As was pointed out earlier, when we have considered the physical principles of the ballistic gravimeter and the pendulum an influence of the Coriolis force was ignored. Now we will try to take into account this factor and consider the motion of a particle near the earth s surface. With this purpose in mind let us choose a non-inertial frame of reference, shown in Fig. 3.5a its origin 0 is located near the earth s surface and it rotates together with the earth with angular velocity a>. The unit vectors i, j, and k of this system are fixed relative to the earth and directed as follows i is horizontal, that is, tangential to the earth s surface and points south, j is also horizontal and points east, k is vertical and points upward. As is shown in Fig. 3.5a SN is the earth s axis, drawn from south to north, I is the unit vector along OiO, and K is a unit vector parallel to SN. 

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