Bond direction cosines

We can simplify this expression by decomposing the vectors along the three axes, with unit vectors i, j and k and introducing the direction cosines of the bond (1,2), namely cosri2, cosyi2 and coszi2- The z-components of V12 and V34, cancel, while V23 is directed along x. This yields the equation... 

Again, using the direction cosines and the coordinates of the midpoint of the bond, v,y, yij and lij, and observing that (yi 2 coszn — zi2cosyi2 = 0, we obtain the equation... 

In the foregoing treatment of the water molecule, which we shall use as an example, each of the two bond orbitals of the oxygen atom has been calculated to have 6 percent s character and 94 percent p character. Each of the two unshared-pair orbitals then has 44 percent s character and 56 percent p character. The maxima for the unshared-pair orbitals lie in directions making an angle of 142° with one another and such that their resultant is opposed to that for the two bond orbitals, which have their maxima at 93.5° with one another. The component for the four unshared-pair electrons is determined by the direction cosine —0.34, and that of the two bonding electrons of the oxygen atom by the direction cosine 0.68 hence the contribution of the four unshared-pair electrons to the dipole moment is just balanced by that of the two bonding electrons.18... 

Second, an alternative hat-curved-cosine-squared potential (HC-CS) model is also considered, which, as it seems, is more adeuate than the HC-HO model. The CS potential is assumed to govern angular deflections of H-bonded rigid dipole from equilibrium H-bond direction. The HC-CS model agrees very well with the experimental spectra of water. 

The matrix elements Hkk follow directly from (1) and correspond to directional cosines in a vector space. Transfers between adjacent sites are proportional to tpq. The local structure of VB diagrams limits the outcome to possibilities for Hkki that can readily be enumerated [13]. Spin problems in the covalent basis have even simpler[28] Hkki. The matrix H is not symmetric when the basis is not orthogonal, but it is extremely sparse. This follows because N sites yield about N bonds and each transfer integral gives at most two diagrams. There are consequently 2N off-diagonal Hkk> in matrices of order Ps N, Ne)/4 for systems with inversion and... 

The determination of the asymmetry parameters and the direction cosines, together with x-ray investigations of the crystal structure, will shed some light on the crystal field effect in solids. NQR powder spectra can only permit very rough and qualitative conclusions on the intermolecular forces. The results on the Mentschukin complexes with AsCl3 show that the metal-chlorine bond is virtually unaffected by the formation of molecular compounds. From singlecrystal NQR spectroscopy, particularly on the As nucleus, some geometrical information about the molecular compounds can be expected. 

The hyperfine coupling associated with Line A is characteristic of an a-proton bonded to C2 with ir-spin density at the carbon. The fit between the direct cosine associated with Amin and the computed direction of C2-H is excellent, they differ... 

It is important to note that the proportional relationship between Amax, Amid, and Amin for these couplings is the same for 100% spin density, and for the present case with approximately 50% spin density. When this is so it indicates that there is no rocking motion at the radical site. This is good evidence therefore that the radical site is essentially planar. The best evidence for radical planarity comes from the analysis of the direction cosines associated with each principal values of the hyperfine coupling tensor. The direction of Amin (Table 18-2) is known to be associated with the direction of the >C-H bond, while the direction associated with the Amid indicates the direction of the n-clcctron orbital. These directions are easily calculated from the crystal structure, and are included in Table 18-2. One sees that the direction associated with Amid deviates only 2.0° from the computed perpendicular to the ring plane, while the direction of Amin, deviates only 2.8° from the computed direction of the C6-H bond. The errors listed on these values are at the 95% confidence level. This is very clear evidence that the radical shown here is planar in the solid-state. Any torsional motion of the C6-H would lead to asymmetries of the hyperfine coupling tensor, and would not produce the observed agreement between the direction cosines and the known directions obtained from the crystal structure. 

Normalized dipole moment (direction cosine) Covalent-bond length (see Fig. 1)... 

The equation for RDC, Eq. (2), could be rewritten in the form of the expectation value for a vector in a Saupe order matrix, i.e., Eqs. (3,4) [9]. The x,y,z in Eq. (3) are directional cosines of a bond vector in an arbitrary molecular frame (Eig. la), the most convenient would be an existing PDB coordinate frame Saupe element Sjj is the averaged projection of axes of the molecular frame onto the direction of Bq with jSx.v.z (Fig. la) specifying the projection angle for each axis 8,y is Kronecker delta. The Saupe matrix is symmetric and traceless and contains five unknown variables. 

Let the bond make the polar and azimuthal angles 6 and 0 with respect to the axes OX1X2X2 of the unit and choose the Oxi axis so that it lies in the 0X1X2 plane. This can always be done because of the cylindrical symmetry of the tensor. The direction-cosine matrix linking the sets of axes OV1V2X3 and OX 1X2X2 is then... 

Similarly, by picking d, e, and/proportional to the direction cosines of the other CH lines, we form localized orbitals along these bonds. From (15.49) the direction cosines of the CH lines are... 

The joint three-site problem has been treated by P. A. Hart and C. F. Anderson (unpublished) using an approach for each P—O bond of the phosphodiester (or any other bond pairs of the phosphodiester) that is analogous to that of Tsutsumi (1979). To see how the analog of Eq. (2) takes shape, I focus on the phosphodiester model of Fig. 4 and die potential-energy profile of Fig. 5. The model of internal motion allows rotation to occur about each P—O bond randomly, independendy, and discretely. For each bond the potential energy profile of Fig. 5 is required in which /, Fj 3 and 1 21 = IFj, = W 2 Overall motion is required to be isotropic. As usual, direction cosines and P—H distances are computed using bond angjes and distances, determined by X-ray analysis or estimated by other means, for the nine joint conformations. 

A schematic diagram of the hybrids making up a bond, and the calculation of the bonding antibonding matrix clement for a field in a [100] direction. For this choice of field-direction, all bonds make the same angle (with cosine = 3 ) with the field. 

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