Atomic vector contributions

This brings us to the last factor to consider—the size and direction of the atomic vector contributions - The size can only be significant if the coefficients of the 2p orbitals located on atom n in at least one of the most localized orbitals Xa nd Xh ( nd indeed, in the first approxi-... 

This brings us to the last factor to consider—the size and direction of the atomic vector contributions size can only be significant if... 

Figure 4.25. Sum-over-atoms factor in the spin-orbit coupling vector // a) in orthogonally twisted ethylene and b) in (0, 90°) twisted trimethylene biradical, using Equation (4.12) and (4.13) most localized orbitals x - Xh and nonvanishing atomic vectorial contributions from Xh (white through-space, black through-bond).

To take into account the disparity in electrons, and hence scatting power, for the various atoms in the unit cell, the atomic number Zj has been introduced as a means of defining amplitudes for the component waves. The total diffracted wave for the entire crystal will be the product of the equation above with the total number of unit cells in the crystal.1 For a single unit cell, like that in Figures 5.12a and 5.12b, an atom s contribution to the total structure factor of one unit cell can also be illustrated in vector terms as in Figure 5.13. To put the expression in the correct units, it is necessary to multiply the summation by a constant, the volume of the unit cell. Here, that is simply V = a x b x c. Thus... 

Figure 6a. A vector diagram illustrating the native protein (Fp) and heavy atom (F ) contributions to the structure factor (Fp,d for the heavy atom derivative of the protein. Sp, and are the phases for the native protein, the heavy atom, and the heavy atom derivative of the protein, respectively, b. The Marker construction for the phase calculation by the method of single isomorphous replacement corresponding to the situation shown in Figure 6a. The scale has been reduced slightly. The vector AB represents the amplitude (Fh) and phase (an) of the heavy atom. With centre A a circle radius Fp is drawn. Similarly, with centre B a circle radius Fp is drawn. The intersections of the circles at O and O represent the two possibilities for ap. Only one (O) is the correct solution.

Figure 2.14 For the SIRAS case a vector diagram can be drawn from hk( and hki labelled + and — respectively for protein P and heavy atom H contributions to the structure factor. From this, equations (2.19), (2.20) and (2.21) can be derived. This is an Argand diagram in the complex plane where S is the imaginary and is the real axis.

On the other terminal carbon, the roles of A and B are interchanged. The direction of the atomic vectorial contribution is along the direction perpendicular to the plane of the carbon atoms. If these out-of-plane vectors add, condition (3) will be satisfied if they cancel, SOC will vanish. Working out the directions from the detailed expression for the vectorial contribution shows that the two contributions add, and SOC will not vanish. This conclusion can also be reached quite easily using symmetry arguments. (See Section 4.2). 

From the vector properties of orbitals, the bond orbitals oah> and of Figure 7.6 may be decmnposed into atomic orbital contributions as shown in 7 Jl. So we can easily see that the MOs of bent AHj are approxiinated by the bond orbitals as shown in Figure 7.7. Consider for instance the linear combinations of the two bonding orbitals oah The positive combination of the two leads to the Os MO, and the s character of A is retained. The negative combination leads to the Ox, and removes the s character of A. Therefore, the two linear combinations of the degenerate bonding orbitals oah become different in energy. Similarly, linear combinations of the two a orbitals or the two orbitals lift the de-... 

Thus, the through-space term makes a positive contribution to the z component of the atomic vector and the two through-bond terms make... 

From the vector properties of orbitals, the bond orbitals (Toh. nd (T oh of Figure 7.8 may be decomposed into atomic orbital contributions as shown on the left-hand side of 7.29 for the two n bond orbitals. One can force the bond orbitals to have the full symmetry properties associated with the molecule by taking linear combinations of each degenerate set. Consider for instance the linear combinations of the two nonbonding orbitals no- The positive combination of the two, 7.29, leads to a molecular orbital we shall call n r- By decomposing each bond... 

The mentioned peaklike structure is due to the layered atomic structure of the sample vertical to the surface. Figure 3.2.1.10a displays a simple example whereby atom number j is in layer j, so that with the layer vectors Cj, the atomic vectors can be written as pj = Cj + pi (note that cj = 0). The atoms are the lesser shaded the deeper they reside within the surface, indicating that they contribute less and less to the structure factor because of electron attenuation. Certainly, we need an analytic tool to describe that attenuation in Eq. (3.2.1.17). Before introducing it, we simplify... 

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