The circle

The equation of a circle with coordinates of the center as (a,b) and radius r is 

Expanding Equation 1.57a, the general equation of a circle can be developed as 

---

Fig. XV-16. A schematic drawing of the arrangement of polyglutamates having alkyl side chains in an LB film. The circles represent the rodlike polyglutamate backbone oriented perpendicular to the page the wiggly lines are the alkyl sidechains. (From Ref. 182.)...

In the graphical representation of the integral shown above, a line represents the Mayer function f r.p between two particles and j. The coordinates are represented by open circles that are labelled, unless it is integrated over the volume of the system, when the circle representing it is blackened and the label erased. The black circle in the above graph represents an integration over the coordinates of particle 3, and is not labelled. The coefficient of is the sum of tln-ee tenns represented graphically as... 

Figure A2.5.28. The coexistence curve and the heat capacity of the binary mixture 3-methylpentane + nitroethane. The circles are the experimental points, and the lines are calculated from the two-tenn crossover model. Reproduced from [28], 2000 Supercritical Fluids—Fundamentals and Applications ed E Kiran, P G Debenedetti and C J Peters (Dordrecht Kluwer) Anisimov M A and Sengers J V Critical and crossover phenomena in fluids and fluid mixtures, p 16, figure 3, by kind pemiission from Kluwer Academic Publishers.

BC CA Physically allowed combinations lie inside the circle, with the conical... 

One can trace the continuous evolution of 0 (or of 0/2) as <() describes the circle q = constant. This will yield the topological phase (as well as intermediate, open-path phase during the circling). We illustrate this in the next two figures for the case q > 1 (encircling the ci s). 

Figure 5. Phase tracing for the case of trigonal degeneracies when the circle encompasses all four ci s and the Berry phase is 271.

In what follows, we discuss the H2D system. For this purpose Eq. (186) is employed for which it is obtained that the straight line seam is defined for the following values of 9j and with constant p and 0 encircle the seam. The fact that 0j is no longer zero implies that not all the circles with constant p and 0 encircle the seam thus, circles for which 0 > 0s will encircle the seam and those with 0 < 9j will not. 

In this series of results, we encounter a somewhat unexpected result, namely, when the circle surrounds two conical intersections the value of the line integral is zero. This does not contradict any statements made regarding the general theory (which asserts that in such a case the value of the line integral is either a multiple of 2tu or zero) but it is still somewhat unexpected, because it implies that the two conical intersections behave like vectors and that they arrange themselves in such a way as to reduce the effect of the non-adiabatic coupling terms. This result has important consequences regarding the cases where a pair of electronic states are coupled by more than one conical intersection. 

Figure 12, Results for the C2H molecule as calculated along a circle surrounding Che 2 A -3 A conical intersection, The conical intersection is located on the C2v line at a distance of 1,813 A from the CC axis, where ri (=CC distance) 1.2515 A. The center of the circle is located at the point of the conical intersection and defined in terms of a radius < . Shown are the non-adiabatic coupling matrix elements tcp((p ) and the adiabatic-to-diabatic transformation angles y((p i ) as calculated for (ii) and (b) where q = 0.2 A (c) and (d) where q = 0.3 A (e) and (/) where q = 0.4 A. Also shown are the positions of the two close-by (3,4) conical intersections (designated as X34).

Going back to our case and recalling that x(

conjugate functions, namely, iTn((p) where nr((p) = V T 2 + Tjj + T 3- In Figure 13a and b we present tn(conical intersections and they occur at points where the circles cross their axis line. 

The values due to the two separate calculations are of the same quality we usually get from (pure) two-state calculations, that is, veiy close to 1.0 but two comments have to be made in this respect (1) The quality of the numbers are different in the two calculations The reason might be connected with the fact that in the second case the circle surrounds an area about three times larger than in the first case. This fact seems to indicate that the deviations are due noise caused by CIs belonging to neighbor states [e.g., the (1,2) and the (4,5) CIs]. (2) We would like to remind the reader that the diagonal element in case of the two-state system was only (—)0.39 [73] [instead of (—)1.0] so that incorporating the third state led, indeed, to a significant improvement. 

Fig. 8.2 Simple Monte Carlo integration, (a) The shaded area under the irregular curve equals the ratio of the number of random points under the curve to the total number of points, multiplied by the area of the bounding area, (b) An estimate of tt can be obtained by generating random numbers within the square, v then equals the number of points within the circle divided by the total number of points within the square, multiplied by 4.

FIGURE 3.2 Possible results of increasing the order of Moller-Plesset calculations. The circles show monotonic convergence. The squares show oscillating convergence. The triangles show a diverging series. 

The circle m a hexagon symbol was first suggested by the British chemist Sir Robert Robinson to represent what he called the aromatic sextet —the six delocalized TT electrons of the three double bonds Robinson s symbol is a convenient time saving shorthand device but Kekule type formulas are better for counting and keeping track of electrons especially m chemical reactions... 

The circle mnemonic was de vised by Arthur A Frost a theoretical chemist at North western University... 

For qualitative purposes the circle itself isn t even necessary We could locate the Huckel MOs by simply working with the polygons themselves The circle is needed only when Frost s method is used quantitatively In those cases the radius of the circle has a prescribed value allowing each MO to be assigned a specific energy... 

In the preceding section we observed that both the rate of polymerization and the degree of polymerization under stationary-state conditions can be interpreted to yield some cluster of the constants kp, kj, and k j. The situation is summarized diagramatically in Fig. 6.4. The circles at the two bottom corners... 

Fig. 8. Rephcation. The amino adenosine X and the pentafluorophenyl ester Y form a hydrogen-bonded dimer XY, prior to reaction between the amine and the activated ester groups (shown in the circle). The reaction product is a <7 -amide conformer cis-Z that isomeri2es to the more stable trans- acnide Z. The rephcative process is cataly2ed by the reaction product Z (also referred to as the template). First, a termolecular complex XYZ is formed from X, Y, and Z.

The amide formation reaction (highlighted by the circle) leads to the production of a hydrogen-bonded dimer (ZZ) of the reaction product Z with the template Z. The dimer is in thermodynamic equilibrium with free template in the reaction medium. 

The branching is continued until all of the safety functions are considered. At this point a conclusion is reached about the result. For the flat tire example, only two results are possible the driver is either stranded or back on the road. The circle used to terminate the stranded result is given an X to denote it as an unfavorable outcome. 

Fig. 12. Phase diagram of the CaO—H2O—P20 (calcium orthophosphate) system where the circle represents the variable hydroxylapatite composition and...

Fig. 4. Example of a halftone illustration. The circle in (a) indicates the enlarged area shown in (b), which reveals the image to be composed of halftone...

Fig. 6. Schematic illustration of stmctural relationships in quart2 where the circles represent siUcon centers only, projected on the basal plane (oxygen atoms are not shown) (Q) represent the highest level, ( ) represent the intermediate level, and (O) represent the lowest level. The lines are an aid to visuali2ation...

Fig. 1. Electrode combinations for alkaline storage batteries where the substance within the circle comprises the negative electrode and the combinations...

Fig. 2. A representation of the cellulose chain ia solution, projected against three two-dimensional surfaces. The circles represent the oxygen atoms that link the iadividual glucose residues, and the lines take the place of the sugar residues. This result of a modeling study (39) iadicated a molecule somewhat more...

---