Conical correction

Figure 8.6 Distribution of 1509 N—H O = C hydrogen bonds before and after application of a conic correction factor.8...

Elements of the matrix —are usually small in the vicinity of a conical intersection and can be added to to give a corrected diabatic energy matrix. As can be seen, whereas in Eq. (15) contains both the singular... 

This analysis is heuristic in the sense that the Hilbert spaces in question are in general of large, if not infinite, dimension while we have focused on spaces of dimension four or two. A form of degenerate perturbation theory [3] can be used to demonstrate that the preceding analysis is essentially correct and, to provide the means for locating and characterizing conical intersections. 

In most rotational viscometers the rate of shear varies with the distance from a wall or the axis of rotation. However, in a cone—plate viscometer the rate of shear across the conical gap is essentially constant because the linear velocity and the gap between the cone and the plate both increase with increasing distance from the axis. No tedious correction calculations are required for non-Newtonian fluids. The relevant equations for viscosity, shear stress, and shear rate at small angles a of Newtonian fluids are equations 29, 30, and 31, respectively, where M is the torque, R the radius of the cone, v the linear velocity, and rthe distance from the axis. 

Procedure. To determine the purity of a sample of boric acid, weigh accurately about 0.8 g of the acid, transfer quantitatively to a 250 mL graduated flask and make up to the mark. Pipette 25 mL of the solution into a 250 mL conical flask, add an equal volume of distilled water, 2.5-3 g of mannitol or sorbitol, and titrate with standard 0.1 M sodium hydroxide solution using phenolphthalein as indicator. It is advisable to check whether any blank correction must be made dissolve a similar weight of mannitol (sorbitol) in 50 mL of distilled water, add phenolphthalein, and ascertain how much sodium hydroxide solution must be added to produce the characteristic end point colour. 

Pipette 25 mL of the standard 0.1 M silver nitrate into a 250 mL conical flask, add 5mL of 6M nitric acid and 1 mL of the iron(III) indicator solution. Run in the potassium or ammonium thiocyanate solution from a burette. At first a white precipitate is produced, rendering the liquid of a milky appearance, and as each drop of thiocyanate falls in, it produces a reddish-brown cloud, which quickly disappears on shaking. As the end point approaches, the precipitate becomes flocculent and settles easily finally one drop of the thiocyanate solution produces a faint brown colour, which no longer disappears upon shaking. This is the end point. The indicator blank amounts to 0.01 mL ofO.lM silver nitrate. It is essential to shake vigorously during the titration in order to obtain correct results. ... 

Furthermore, for any two-mirror telescope in which each mirror is individually corrected for spherical aberration (i,e. composed of two conics) one can solve for the third order aberrations in closed form. 

With these closed form aberration equations it is possible to do something interesting Rather than use the mirrors at their individual conic conjugates which would correct for spherical aberration in each mirror independently, we can set the overall spherical and comatic aberration to zero and solve for a different set of conics. This is called the aplanatic condition, when 3 order... 

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