Rotational angle space

It is obvious that the volume of the rotation angle space is 2n. However, the cosine function is symmetric in this space and, therefore, the solution of Eq. 4.138 does not yield... 

T is a rotational angle, which determines the spatial orientation of the adiabatic electronic functions v / and )/ . In triatomic molecules, this orientation follows directly from symmetry considerations. So, for example, in a II state one of the elecbonic wave functions has its maximum in the molecular plane and the other one is perpendicular to it. If a treatment of the R-T effect is carried out employing the space-fixed coordinate system, the angle t appearing in Eqs. (53)... 

As discussed in Section II. A, the adiabatic electronic wave functions, a and / 1,ad depend on the nuclear coordinates R> only through the subset (which in the triatomic case consists of a nuclear coordinate hyperradius p and a set of two internal hyperangles this permits one to relate the 6D vector W(1)ad(Rx) to another one w(1 ad(q J that is 3D. For a triatomic system, let aIX = (a1 -. blk, crx) be the Euler angles that rotate the space-fixed Cartesian frame into the body-fixed principal axis of inertia frame IX, and let be the 6D gradient vector in this rotated frame. The relation between the space-fixed VRi and is given by... 

Figure 4 A third-generation CT measurement exposure geometry where the disk rotation angle, is 105° (see Figure 3). Typically, from 200 to 1000 such measurements are made, depending on the design of the scanner, at equally spaced angular intervals in order to generate one CT slice.

Another idea is to use different runs of the same program with slightly different models it is most aptly described as an application of a consistency principle, namely it is required that the solution should appear consistently in all runs, even with a rather low score. Special algorithms have been developed to cluster similar solutions in eulerian angles space and convincing results have shown that it is indeed possible to increase the signal-to-noise ratio of the rotation function in this way (Urzhumtsev and Urzhumtseva, 2002). 

As we mentioned already, the problem of the rotation function is its score, leading to a difficult energy landscape to be searched we can now describe another way to tackle this problem. Since the Translation Function score is much more sensitive, one might try to run a translation for every possible rotation angle, therefore exploring the 6D space exhaustively. The space to be searched in eule-rian angles depends on the space group of the crystal and can be found in Rao et al. (1980). It turns out that it is doable in most cases within reasonable cpu time with a normal workstation. 

Microwave 1 to 100 Rotational excitation Spacings of rotational energy levels bond distances and bond angles (Section 9-6)... 

For the description of crystal structures one adds to the preceding molecular parameters an appropriate combination of rotation angles and translations 6 parameters or less according the space group symmetry. 

Figure 3. Contour diagram of the base stacking potential energy Kg of sequential adenine bases and the hydrogen bonding potential energy VHB of the complementary A T base pairs as a function of the phosphodiester rotation angles

Rotation in Space Angle Close to Spherical Angle... 

To do a full rotation in three-dimensional space, Eulerian rotation angles can be defined, as three successive rotations, each in two dimensions. 

In chapter 2 we introduced the molecule-fixed axis system (x, y, z) which rotates in space with the molecule. This axis system is related to the non-rotating (but translating) axis system (X, Y,Z) by the three Euler angles (0,0,/). The coordinates of a point i in the molecule-fixed axis system are related to its coordinates expressed in the space-fixed axis system by the transformation... 

Figure 1 Three line-position and line intensity roadmaps as a function of rotation angle (deg), for rotations about three orthogonal rotation axes, obtained from solid-state 31P cross-polarization (CP) NMR spectra at fixed magnetic fields of 4.7 T, for single-crystals (space group C2/m) of TMPS (tetramethyldiphosphine sulfide). One sees the spectra from two site types occurring in abundance ratio 1 2, each with a spin-spin doublet structure. Taken from Ref. 31.

The repeating unit is a distinct sequence of choices from (1) and (2) in the molecule this sequence corresponds to a row of successive atoms, the relative positions of which are fixed if bond and dihedral angles are kept constant. Such a repeating sequence of atoms is called a motif (or module). In general, a motif will coincide with its successor in the molecular string after a translation combined with a rotation in space. This implies that in general these molecular conformations are... 

Fig. 24. Dependence of the resolution of BR-24 spectra on the pulse spacing t. The rotation angle of the sample was different from that used previously and was chosen such that the dipolar coupling of the methylene protons was particularly strong. Note that for t = 5 /is the Nyquist frequency which is marked by the sharp edge, is within the spectral range shown in the figure.

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