Orthogonal -systems

As is well known, Laplace s equation in an orthogonal system of coordinates has the form ... 

HPLC and CE may also be coupled to give an on-line multidimensional set-up [960] CZE and HPLC are highly orthogonal systems. On-line HPLC-GC has been reviewed repeatedly [49,906,961-963], as well as MDHPLC [964] various textbooks are available [965,966]. 

Equations 4.22 1.24 are the 2D equivalents of Equations 4.17 1.20. The comparison of the two sets of equations shows a surprising consequence. If the peak capacities of the 1D and 2D separation systems were identical, the 2D separation would lead to more severe overlap. In order to have the same number of components isolated as singlets with a 1D and a 2D separation system, the peak capacity of the 2D system ( 2d) should be double of that of the ID system (nw). Ideally, in an orthogonal system 112D = n j D, but part of the gain in peak capacity is lost due to the increased probability of peak overlap provided the 2D chromatogram is disordered. 

The vector spherical harmonics YjtM form an orthogonal system. The state of the photon with definite values of j and M is described by a wave function which in general is a linear combination of three vector spherical harmonics... 

Chemistry of the dithiasuccinoyl (Dts) function, (orthogonal systems) J Am Chem... 

FIGURE 5.13 Protecting group combinations employed in solid-phase synthesis. The protector written in italics is removed after each residue is incorporated into the chain. (A) Boc/Bzl3 is not an orthogonal system — all substituents are removed by acidolysis. (B) Fmoc/tBu27 [Atherton et al., 1978] is an orthogonal system - Fmoc is removed by P-elimi-nation, other substituents by acidolysis. More suitable variants of the side-chain protectors are also used. 

An unknown mixture can be screened on a set of orthogonal systems as a first step in the method development procedure. The chromatographic and/or electrophoretic system, on which the best separation was achieved, can then be retained for further method optimization. Sequentially, the pH and the organic modifier composition of the mobile phase can be adjusted to improve the separation on the CS. If necessary, also the temperature can be modified, while for gradient methods the gradient slope can be considered. For CE methods, the optimization steps will be different from RP chromatography methods. Other factors will be optimized depending on the type of CE method, e.g., CZE and MEKC. However, for the development of CE methods, we would like to refer to Chapter 4 of this book. 

The separation selectivities in the hrst dimension shonld largely differ from that in the second dimension. Best results are achieved in so-called orthogonal systems with non-correlated retention times in both dimensions [172,173]. Mobile phase, flow rate, and in some cases, temperature should be optimized in each dimension to increase the number of resolved compounds in a single rnn. 

This looks complicated, but for orthogonal systems (a,, y nicely to the expected expression ... 

The AIM electron-population displacements, d/V, are strongly coupled through the olf-diagonal hardness matrix elements //y y>,i Thus, a given displacement d/Vk strongly affects the chemical potentials of all AIM. This representation considers all AIM populational parameters as independent variables, which can be interpreted as projections of the populational vector (/V, d, + N2 2 + dm) onto the orthogonal system of populational axes associated with the constituent atoms, i.e., the AIM populational basis vectors ... 

Fig. 45 h. Optically polished gla>< showing orthogonal system of cracks... 

In a Cartesian coordinate system, by applying the definition of a cross product in this orthogonal system, the unit vectors ex, ey, ez are related as follows ... 

The strain component S12 is usually the deformation of the body along axis 1, due to a force along axis 2 the strain tensor s is usually symmetrical, = s and thus, of the nine terms of s, at most six are unique. Both P and s can be represented as ellipsoids of stress and strain, respectively, and can be reduced to a diagonal form (e.g., P j along some preferred orthogonal system of axes, oblique to the laboratory frame or to the frame of the crystal, but characteristic for the solid the transformation to this diagonal form is a... 

As already discussed in Section 2.4, in 3-D any point (denoted by the vector r) can be described by its three coordinate projections x, y, z (in emits such as m, nm, A, or pm) using an orthogonal coordinate system with emit vectors eX/ ex, ex hence r = xex + yeY + zez- In noncrystallographic textbooks, the position vector r is usually given in a Cartesian (orthogonal) system. 

---