Invariance from coordinates

The elements with such an electronic configuration known as luminescent centers are Cr, Mn and They are capable of substituting in a wide variety of metal oxide host systems. They are invariably oxygen-coordinated with six nearest neighbors, and may be in a pure octahedral or a distorted octahedral symmetry site. These luminescent centers exists in a d configuration (Fig. 5.23), and the electronic repulsion, which results from placing three electrons in the same set of d-orbitals yields several states identified as free... 

The form of equations (16) does not depend on the parameter presentation of the paths, as is easily demonstrated. Likewise one shows on the basis of equations (10a) of chap. Ill, that equations (16) are also invariant against coordinate transformation. It finally follows from (10c) that they are invariant against gauge transformation as well. 

Stresses transform from one coordinate-axis system to another according to well-defined transformation laws that utilize direction cosines of the angles of rotation between the final and initial coordinate-axis systems. Matrixes that obey such transformation laws are referred to as tensors (McClintock and Argon 1966). There are three sets of stress relations that are scalar and invariant in coordinate-axis transformations. The first such stress invariant of particular interest is the mean normal stress o- , defined as. 

Both sides of eq.(18) are contravariant vectors and change according to the same rule. Mathematically, a solution of eq.(18) yields a curve invariant from the actual coordinate system. 

Again, it follows from the tensor character of this equation that any solution curve is invariant under coordinate transformations. 

Now the Lagrangean associated with the nuclear motion is not invariant under a local gauge transformation. Eor this to be the case, the Lagrangean needs to include also an interaction field. This field can be represented either as a vector field (actually a four-vector, familiar from electromagnetism), or as a tensorial, YM type field. Whatever the form of the field, there are always two parts to it. First, the field induced by the nuclear motion itself and second, an externally induced field, actually produced by some other particles E, R, which are not part of the original formalism. (At our convenience, we could include these and then these would be part of the extended coordinates r, R. The procedure would then result in the appearance of a potential interaction, but not having the field. ) At a first glance, the field (whether induced internally... 

In summary, the groups of Espenson and Loh observe catalysis of Diels-Alder reactions involving monodentate reactants by Lewis acids in water. If their observations reflect Lewis-acid catalysis, involvirg coordination and concomitant activation of the dienophile, we would conclude that Lewis-acid catalysis in water need not suffer from a limitation to chelating reactants. This conclusion contradicts our observations which have invariably stressed the importance of a chelating potential of the dienophile. Hence it was decided to investigate the effect of indium trichloride and methylrhenium trioxide under homogeneous conditions. 

The objectivity of the spatial stress rate relation (5.154) may be investigated by applying the coordinate transformation (A.50) representing a rotation and translation of the coordinate frame. The spatial strain and its convected rate are indifferent by (A.58) and (A.62). So are the stress and its Truesdell rate. It is readily verified from (5.151), (5.152), and the fact that K has been assumed to be invariant, that k and its Truesdell rate are also indifferent. Using these facts together with (A.53) in (5.154) with c and b given by (5.155)... 

The metric matrix is the matrix of all scalar products of position vectors of the atoms when the geometric center is placed in the origin. By application of the law of cosines, this matrix can be obtained from distance information only. Because it is invariant against rotation but not translation, the distances to the geometric center have to be calculated from the interatomic distances (see Fig. 3). The matrix allows the calculation of coordinates from distances in a single step, provided that all A atom(A atom l)/2 interatomic distances are known. 

The practical way of calculating 2 is different from that used in the derivation of (4.18). Since 2 is invariant with respect to canonical transformations, it is preferable to seek it in the initial coordinate system. Writing the linearized equation for deviations from the instanton solution 6Q,... 

To look ahead a little, there are properties that depend on the choice of coordinate system the electric dipole moment of a charged species is origin-dependent in a well-understood way. But not the charge density or the electronic energy Quantities that have the same value in any coordinate system are sometimes referred to as invariants, a term borrowed from the theory of relativity. 

Finally, I should tell you that structural databases invariably contain Cartesian coordinates. A typical paper from the early 1990s addresses the problem. 

From the invariance of the coupling constant 2J(SiP) between HMPA-P and Si over a wide temperature range (compounds 4-7, 9-11) a rigid coordination of the HMPA to silicon can be deduced. Only in the case of the methyl complex 6 above 25 °C is the beginning of exchange of HMPA observed. However, a fast exchange of the coordinated acetonitrile at room temperature has been found for 12. 

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