Cyclic coordinate

In the strictest meaning, the total wave function cannot be separated since there are many kinds of interactions between the nuclear and electronic degrees of freedom (see later). However, for practical purposes, one can separate the total wave function partially or completely, depending on considerations relative to the magnitude of the various interactions. Owing to the uniformity and isotropy of space, the translational and rotational degrees of freedom of an isolated molecule can be described by cyclic coordinates, and can in principle be separated. Note that the separation of the rotational degrees of freedom is not trivial [37]. 

As the angular coordinates x and p enter Eq. (27) only as deri are cyclic coordinates (see Section 6.4.2). Therefore, the three variables can be separated, leading to waveftmetions of the form... 

With the exception of chlorobenzene and 1,2-dichloroethane, halocarbon solvents are unsuitable diluents, as carbon tetrachloride and chloroform may react violently with alkylalumimum derivatives. The hazards of individually mixing 7 alkyla-luminiums with 7 chlorinated solvents have been assessed comparatively. Most of a series of cyclic coordination complexes between triethylaluminium and a-iminoketones decomposed violently when dissolved in halogenated solvents. 

Much attention is currently devoted to the synthesis and properties of shape-persistent macrocycles[l]. Such compounds are interesting for a variety of reasons including formation of columnar stacks potentially capable of performing as nanopores for incorporation into membranes or for the generation of nanowires[2]. Furthermore, in shape-persistent macrocycles incorporating coordination units, enc/o-cyclic metal-ion coordination may be exploited to generate nanowires[3], whereas e.ro-cyclic coordination can be used to construct large arrays of polynuclear metal complexes[4]. Shape-persistent macrocycles with reactive substituents may also be linked to other units to yield multicomponent, hierarchical structures. 

Let us now consider a few examples of probability density pb(0,ip). In cyclic coordinates, as used by us, the z-axis is a selected one see (A.l). It corresponds to the quantization axis in quantum mechanical... 

In physical calculations containing angular momenta, cyclic coordinates are frequently employed [379]. Cyclic unit vectors are introduced according to the following rules ... 

The scalar product of vectors A and B may be expressed in cyclic coordinates as... 

Despite its simplicity this demonstration may be considered of little importance as very few mechanical systems have only cyclic coordinates. What is more important is the fact that any system can be described by more than one set of generalized coordinates, with the possibility of transforming non-cyclic into cyclic coordinates. 

A simple example is provided by the coordinates that may be used to describe the motion of a particle in a plane. As generalized coordinates one can use either Cartesian coordinates qi = x, q2 = y, or plane polar coordinates qi = r, q2 = central force the coordinate p becomes cyclic, while both x and y are non-cyclic. The number of cyclic coordinates thus depends on the choice of generalized coordinates, and for each problem there may be one particular choice for which all coordinates are cyclic. 

Transformation to all-cyclic coordinates is not always feasible and another alternative is to consider the transformation from initial conditions (qo,Po) to the coordinates (q,p) at time t. The transformation equations ... 

Computational methods have been applied to study the conformations of free and metal-complexed oxathiacrown ethers 1-4 shown in Figure 6. The results were compared to variable temperature NMR and i spin-lattice relaxation time measurements <2001JP2988>. Theoretical studies included simulated 111 NMR spectra using PERCH and molecular modeling with PM3 semi-empirical quantum-chemical calculations. The NMR and the computational data both show that Ag+ coordinates equally well to S and O atoms, Bi3+ and Sb3+ prefer O atoms, and that Ptz+ and Pdz+ prefer exo-cyclic coordination only to the S atoms in this maleonitrile macrocycle. 

According to the existing data, sulfate ions may have six different coordination types in relation to the uranyl ions (Fig. 10). Like carbonate and nitrate ions, sulfate ions may have bidentate cyclic coordination to U(VI) atoms. However, the most common coordination is bidentate bridging that has El = 4.25 e (Table... 

The angles x and

know that they enter the wave function in the following manner ... 

The cyclic coordinate search method successively uses search directions that are parallel to the coordinate axes along with line search problems to determine the step sizes. This method is conceptually simple, but the sequence of iterates generated by this method tend to zigzag if the optimal solution lies in a valley. 

To help overcome this problem, the method of Hooke and Jeeves (1961) involves both exploratory moves and pattern moves (acceleration moves) with discrete steps along the search directions. The discrete steps eliminate the need for a line search. In the exploratory move phase, starting at a point X , a modified cyclic coordinate search is performed in which, if possible, the objective function is reduced once along each of the coordinate directions using a prescribed discrete step length. This leads to a new point x +j and establishes a direction of improvement. A pattern move is then performed in the direction x +j - x, leading to an intermediate point y. Now, starting from y, another exploratory move yields the point x +2- If < /(Xa +i) then an improvement has been found... 

Cyclic coordinate seeu ch method, 2549 Cyclic scheduling, 1746-1747 Cylindrical robots, 375... 

We see immediately that y is a cyclic coordinate, as required by symmetry. 

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