Redundant condition

Let us now extend the results to dealing with systems where the estimability and redundancy conditions are satisfied. A measurement is considered redundant if its removal causes no loss of estimability. If we consider that the rank of M = g and (m +1) > g, that is, more information is available than is necessary for a unique determination, the following can be stated (Stanley and Mah, 1981a). 

In this chapter, similar arguments to that of dynamic observability were extended to establish the conditions for estimability in steady-state processes when the redundancy condition is satisfied. This concept allows decomposition of the general estimation... 

This is called a redundant condition. In a planar XY3 molecule (f), the number of internal coordinates is seven when the coordinate A0, which represents the deviation from planarity, is considered. Since the number of vibrations is six, one redundant... 

The redundant conditions are more complex in ring compounds. For example, the number of internal coordinates in a triangular X3 molecule (h) exceeds the number of vibrations by three. One of these redundant conditions (A j species) is... 

The other two redundant conditions ( species) involve bond stretching and angle deformation coordinates such as... 

Comparison of this result with that obtained in Sec. 1.8 immediately suggests that three redundant conditions are included in these bending vibrations (one in Aig and one in Eg). Therefore, % (R) for genuine vibrations becomes... 

In general the potential may be written as a function of 3N-6 internal coordinates. In some cases, e.g., CH4, there are more than 3N-6 valence coordinates." One may always delete sufficient coordinates from the list to obtain an independent set. In some cases, however, either to take advantage of symmetry or obtain or use transferable and physically meaningful force constants, it is convenient to use more than 3N-6 internal coordinates. In such a case one or more coordinates are redundant. Another possibility is that a redundancy condition is satisfied only for a restricted range of geometries, including the reference geometry of the force field this is sometimes called a const raint. If redundant coordinates are retained or there is a con-... 

Let S be the subset of features to be selected by MRMR. Then the minimum redundancy condition between two features Fi and Fj is ... 

The CO2 solid contains four distinctly oriented molecules per primitive unit cell and therefore at q = 0 there are eight angular degrees of freedom and twelve displacement coordinates j(lk), or there are four redundant coordinates. The redundancy conditions are obtained from the conditions of normalization, i.e., the sum of the squares of the direction cosines must be unity both before and after displacement... 

If now the are independent, it follows that = 0 for all i. If, on the other hand, we use m variables, of which n are independent, then there exist m — n constraints in the form of redundancy conditions connecting the coordinates u, ... 

In the present case (for q = 0) we use twelve displacement coordinates Aj(0A ) of which eight are independent, as already discussed, and the four relations of (2.53) represent the redundancy conditions R of (2.56). In this case the 4>j, for the four redundant coordinates will not be zero in the potential expansion (2.54). In fact, Walmsley and Pople (1964) construct symmetry coordinates... 

Equation (2.59) represents the redundancy conditions (2.53) after suitable transformations (2.53) relates linear displacements to second-order terms in the displacements. Substitution of the four equations (2.59) into (2.58) results in an expansion of the potential in second-order terms of the independent coordinates only ... 

Geometrical considerations show that these are not all independent their sum always equals zero. These coordinates are said to form a redundant set, Avith one redundancy condition connecting them. If the four obviously independent CII extensions are also used as internal cooi dinates, the total is ten, or one more than 3A — 6 = 9, thus indicating the presence of redundancy. 

It is possible to exclude one of the angles arbitrarily and to modify the treatment of the previous subsection to accommodate situations in which not all the members of a symmetrically equivalent set are used, but it is usually more convenient to retain complete sets and cope with the redundancy later. The procedure then follows exactly that of the previous subsection, except that the total number of combinations of internal coordinates will exceed the number of normal coordinates by the number of redundancy conditions. These redundancy conditions, or suitable linear combinations of them, ivill fall into the various symmetry species, and Avhen these species are identified, the number of combinations of internal coordinates of each species can be suitably diminished to give the number of normal coordinates of each species. More formal methods of treating this problem Avill be given later (Sec. 6-8), but the problem can often be handled by inspection. 

The case of methane is tabulated in 3 able 6-3. The arrangement is exactly the. same as that of Table 6-2, which should be con.sulted for an explanation. Notv, hotvever, n is not always the sum of n f and but the sum minus the number of sets of redundancy conditions of the given species. Since here the redundancy condition, i.e., the sura of the angle increments equals zero, is by inspect ion completely symmetrical (species Ai), the unity under opposite /li represents this condition and is rejected. 

However, when the second-order terms are retained, the redundancy condition must be written in the formb... 

It is now necessary to review the argument which was used to prove that no linear terms occurred in the expansion of the potential energy. Essentially, the argument was that at equilibrium the potential energy must be at a minimum hence (5F/3redundant coordinates are included, the argument must be revised as follows. Let ( ) = 0 be the redundancy condition then the condition for a minimum in the potential energy, with unrestricted variation of all the coordinates, is that... 

It is nevertheless still possible to formulate the potential energy as a pure quadratic form. For the linear term may be transformed to the quadratic terms ndth the aid of the explicit redundancy condition (7) given to the second order. Thus if the potential energy of bending of... 

Since the redundancy condition in this case is known to belong to tlic totally symmetric species /li, it follows that only the totally symmetric quadratic terms Si + S etc., appear. 

As a matter of fact, some authors have even introduced additional parameters in similar cases. To the first order in the small angular displacements, the redundancy condition for methane is... 

Here / and /" are the strain constants which occur since the four XY stretches ( ) arc not independent of the six YY stretches (90). In using this type of potential function, it is necessary eventually to eliminate the redundant coordinates, of which there are seven in the present example. Moreover, the redundancy conditions must be obtained to the second order to permit proper accounting for the linear potential energy constants f and f. When this is done, the potential energy may ultimately be written in the form... 

In the unmodified UBFF, elaborated by Shimanouchi [10], off-diagonal terms in V are represented by atom pair 1,3 non-bonded interactions. Because this introduces inherent redundancies in the coordinates, the equilibrium configuration of the molecule represents a state with internal tensions, and therefore linear terms must be included in the force field. By introducing the redundancy condition, the linear terms can be eliminated from V, although the coefficients of the quadratic terms then contain intramolecular tension as well as the valence and non-bonded force constants. The important characteristic of such a UBFF is that only a limited... 

In the GVFF there is no inherent limit to the number of FyS that are included. There is only the restraint that the number of Fs should not exceed the number of observable frequencies in fact, it should generally be much smaller so as to overdetermine the force field. Since an independent set of coordinates can be chosen, e.g., local symmetry coordinates, and the redundancy conditions explicitly given, there is no need to include linear terms in F, and Eq. (5-1) is the most general representation of the force field. Our discussion will focus on applications of the GVFF. 

The redundancy condition implies that changes in the sugar units or occurrence of side chains do not necessarily destroy the conformational preference of the molecule. Redundancy is to be associated with repetitive and similar three dimensional structures over the range of... 

This evolutionary trend is an ill lustration of both the redundancy condition and the autocatalytic assembly concept. The latter concept is associated with a matrix role of the cellulose crystal lattice in relation to biosynthesis. Cellulose and mannan in algae are more highly... 

The quantity (v), an effective atomic charge, is determined by purely vibrational contributions to the dipole moment changes. The elements of Px(v) are not interconnected by die equations resulting from relation (4.19). Still, nine redundancy conditions as defined by Eq. (4.18) are present. The removal of these remaining redundancies can be accomplished by defining the problem in bond Cartesian displacement coordinate space [Eqs. (4.96) and (4.97)]. 

There are three redundant conditioning units, each one works with only one of the three redundant trip units. Hence, there is a direct one-to-one correlation between each conditioning unit and each trip unit. 

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