Planes defined

Robb, Bemaidi, and Olivucci (RBO) [37] developed a method based on the idea that a conical intersection can be found if one moves in a plane defined by two vectors xi and X2, defined in the adiabatic basis of the molecular Hamiltonian H. The direction of Xi corresponds to the gradient difference... 

To ensure that the arrangement of four atoms in a trigonal planar environment (e.g., a sp -hybridized carbon atom) remains essentially planar, a quadratic term like V(0) = (fe/2) is used to achieve the desired geometry. By calculating the angle 9 between a bond from the central atom and the plane defined by the central... 

Here i is now a double bond up to the seventh sphere (D,7 ) of non-rotatablc bonds centered on the proton, Td is the distance of the proton from the center of the double bond, and is the angle, in radians, between the plane defined by the bond and the distance q, (Figure 10.2-6a). 

Various other ways to incorporate the out-of-plane bending contribution are possible. For e3plane bend involves a cakulation of the angle between a bond from the central atom and the plane defined by I he central atom and the other two atoms (Figure 4.10). A value of 0° corresponds to all four atoms being coplanar. A third approach is to calculate the height of the central atom above a plane defined by the other three atoms (Figure 4.10). With these two definitions the deviation of the out-of-plane coordinate (be it an angle or a distance) can be modelled Lt ing a harmonic potential of the form... 

Crystal stmcture analyses of cyanine and related dyes are reviewed in Ref. 32. Most typical sensitizers are nearly planar, with angles of less than 15° between planes defined by heterocycHc rings. Distinct solvent of crystallization is present in most of the cationic dyes. X-ray crystal analyses also provide intermolecular data. Because of photographic use of cyanine and carbocyanine dyes, the cation-cation arrangements of most interest have been those for l,l -dieth5l-2,2 -quinocyanine chloride [2402-42-8] 5,5, 6,6 -tetrachloro-l,l, 3,3 -tetraethylbenzimidazolocarbocyanineiodide [3520-43-2] and 5,5 -dichloro-3,3, 9-triethylthiacarbocyanine bromide [18426-56-7] (32) (see Fig. 8). 

A metal bar of width w is compressed between two hard anvils as shown in Fig. Al.l. The third dimension of the bar, L, is much greater than zu. Plastic deformation takes place as a result of shearing along planes, defined by the dashed lines in the figure, at a shear stress k. Find an upper bound for the load F when (a) there is no friction between anvils and bar, and (b) there is sufficient friction to effectively weld the anvils to the bar. Show that the solution to case (b) satisfies the general formula... 

Ethanol is an achiral molecule. The plane defined by atoms C-1, C-2, and O is a plane of symmetry. Ary carbon atom with two identical ligands contains a plane of symmetry feat... 

Occupied zone The volume of air confined by horizontal and vertical planes defined to include space occupied by persons. 

The gravity tool face angle, or angle between the plane defined by the borehole axis and the vertical and the plane defined by the borehole axis and the BHA axis below the bent sub, can also be calculated. Figure 4-226 shows the gravity tool face angle. It is readily calculated using the equation... 

Figure 4-258. Solid geometry sketch of the planes defining the azimuth angie.

Since the only constraint we have placed on Qi is that it be less than 9, the second isotherm can be as arbitrarily close to the first as we wish. The conclusion that states exist on this second isotherm that cannot be reached from a point on the first isotherm by any adiabatic path is therefore general. Thus, we can argue that there are states located in the plane defined by 6 and xi that are inaccessible from state 1. 

Now we can consider the effect of variations of 9 with a second variable, x2. Since we have been general about the nature of the variable, xj, we can expect to obtain similar behavior for the variable X2. We construct isotherms using the same B and 62, but this time in the direction of x2. Our initial point will be the same state 1 as earlier. The value of x2 in state 1 will fix the location of this state on the isotherm in the new direction. A reversible adiabatic path can be constructed that connects state 1 with a state on the second isothermal in the x2 direction. Irreversible states located on one side of this point will be inaccessible from state 1 by adiabatic paths, while states located on the other side of that point will be accessible. Thus, there exist states located on the plane defined by 9 and X2 that are inaccessible from point 1. Similar conclusions can be drawn by considering isotherms localized on the planes formed by 9 and each of the x,. 

First, the VB part of the description of benzene. Each C atom is sp2 hybridized, with one electron in each hybrid orbital. Each C atom has a p.-orbital perpendicular to the plane defined by the hybrid orbitals, and it contains one electron. Two sp2 hybrid orbitals on each C atom overlap and form cr-bonds with similar orbitals on the two neighboring C atoms, forming the 120° internal angle of the benzene hexagon. The third, outward-pointing sp2 hybrid orbital on each C atom forms a hydrogen atom. The resulting cr-framework is the same as that illustrated in Fig. 3.20. 

The variants are distinguished by the locants of those ring atoms that lie outside a reference plane (defined below) and are listed for some examples in Table 1. The locants of ring atoms that lie on the side of the reference plane from which numbering appears clockwise (i.e. the upper side in the normal Haworth representation of furanoses and pyranoses) are written as superscripts and precede the letter those that lie on the other side are written as subscripts and follow the letter. Heteroatoms (e.g. O, S) are indicated by their subscript or superscript atomic symbols. Table 1 gives the notations and Chart III some examples. 

In [Tl 18-crown-6] [CIO4] (Fig. 2.1), the thallium(I) cation resides 75 pm above the plane defined by the oxygen atoms of the crown ether and 66 pm in [ri 18-crown-6] [TII4] [39] whereas, for example, in the analogous potassium compound [K 18-crown-6] [CIO4] [40] the cation is found right in the middle of the crown ether (Fig. 2.2). 

In order to establish conditions for the isolation of the image of point D under the EM map, the projection is performed by first taking a section through the surface ct = 0 at fixed J, an example of which is shown in the right-hand panel of Fig. 15, for the critical value = S — N. The shaded area of the (K, plane defines the classically allowed range for the specified value. The lines indicate energy contours for y = 0.5. Those that touch the section correspond to relative equilibria of the Hamiltonian, whose values... 

All the atoms of butadiene lie in a plane defined by the s p hybrid orbitals. Each carbon atom has one remaining p orbital that points perpendicular to the plane, in perfect position for side-by-side overlap. Figure 10-42 shows that all four p orbitals interact to form four delocalized molecular orbitals two are bonding MOs and two are antibonding. The four remaining valence electrons fill the orbitals, leaving the two p orbitals empty. 

Figure 1.42 A90° pulse brings the equilibrium magnetization to the )i -plane. Itsorien-tation in the x y-plane depends on the direction of the pulse. Applying the pulse along one axis causes the magnetization to rotate in a plane defined by the other two axes.

In Fig. 4 we present the final induced decision tree, as well as the partition of the (xi,X4) plane defined by its leaves, together with a projection of all the available (x, y) pairs on the same plane. These two decision variables are clearly influencing the current performance of the refinery unit, and the decision tree leaves perform a reasonable partition of the plane. To achieve better performance, we must look for operating zones that will result in obtaining mostly y = 3 values. Terminal nodes 2... 

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