Sign vectors orthogonality

Next, it is necessary to define the concept of orthogonality of sign vectors. Two sign vectors a and b are said to be orthogonal (a T b) if either (1) the supports of a and b have no indices in common, or (2) there is an index i for which a, and bi have the same signs and there is another index j (j i) for which aj and bj have opposite signs. Given these definitions, the thermodynamic constraint may be stated as ... 

The vector orthogonal to the vector given by Eq. (244) is formed by interchanging its elements and placing a negative sign before one of them we obtain... 

Figure 9.3 Cluster of unit cells of the cesium chloride crystal structure. This figure shows that ions of the same sign in this structure line up along the 100 directions. Thus the three rows are orthogonal to one another. Translation of a (100) plane of ions over its nearest (100) neighboring plane keeps ions of opposite sign adjacent to one another. This is also the case on the (110) planes, but the translation vector is V2 larger than for the the (100) planes.

Therefore the above sign pattern is not thermodynamically feasible because it is not orthogonal to the sign pattern of a vector from the right null space of the stoichiometric matrix of internal reactions. 

A vector yi orthogonal to one of the known vectors, say Xo, is written down. It can be obtained, for example, by making all elements of Xo zero except two, interchanging their position and placing a negative sign before either one of the two—for instance... 

The base vector Vi is therefore the same as that for Vi except for a change of sign, which makes them orthogonal. 

The sign of the rotatory strength for the /-th mode is thus determined by the angle between the vector gi, o(0 of the EDTM and the vector M i, o(0 of the MDTM, which can be seen from Eq. 15.3. If two vectors are orthogonal with respect to each other, as in achiral molecules, the rotatory strength is zero. 

The assertion of the lemma is implied by the fact that any matrix can be represented as the product of an orthogonal matrix by an upper-triangular matrix. If the columns of the matric B are interpreted as a frame in R, then the unkown rotation of R2 turns the frame so that the first vector is collinear to the first basis vector and the second lies in the coordinate plane spanned by the first two basis vectors. This reasoning implies that if the eigenvalues of the matrix A are distinct, then the elements of the upper-triangular matrix B are uniquely determined up to sign. 

Equation (6.63) gives the first of two degenerate vibrations to obtain the second we need a new generating vector that is likely to lead to an orthogonal result. In < i(Ei ), M2 and M3 occur with opposite sign, so that H2 and H3 move out of phase with one another. Accordingly, a possible choice of generating vector is one which forces these two atoms to move in phase with one another, the simplest form of which is... 

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