Scalar triple

In the nonrelativistic case, at a given the quantity x was shown to be invariant under the hansformation in Eq. (16), for a = y = 0. This invariant, whose value depends on was used to systematically locate confluences, [18-21], intersection points at which two distinct branches of the conical intersection seam intersect. Here, we show that the scalar triple product, gij X is the invariant for q = 3. Since the g t, and h cannot be... 

Some of the common manipulations that are performed with vectors include the scalar product, vector product and scalar triple product, which we will illustrate using vectors ri, T2 and r3 that are defined in a rectangular Cartesian coordinate system ... 

The vector product and the scalar triple product can be conveniently written as matrix leterminants. Thus ... 

Figure 1.11. The vector product [b x c] defines a new vector whose magnitude is given by the area of the parallelogram forming the base of the parallelepiped and whose direction is perpendicular to the plane of b and c. The scalar triple product is thus the area of the parallelogram multiplied by the projection of the slant height of a on the vector [f> x c].

Hint From Figure 1.8, we can see that sin /3 is equal to the cosine of the angle, 6, between the cross product vector and a. Thus the scalar triple product... 

In Eqs. 4.25-4.27, the numerators contain cross (vector) products and the denominators are scalar triple products equal to the volume of the real-space unit cell. It may be computed from its determinant (see practice problem 7). 

Show how to compute the scalar triple products in Eqs. 4.21-4.23 through the determinant. 

If we define three vectors a, b and c, as in Problem 5.9, the expression a (6 X c), known as the scalar triple product, yields a scalar quantity, the magnitude of which provides the formula for the volume, F, of a parallelepiped with adjacent edges defined by vectors a, b and c (an example in chemistry being a crystalline unit cell). If the determinantal representation of A x c is used, then, on expanding the determinant from the first row, and evaluating the three scalar products, we obtain ... 

We explore the application of the scalar triple product for evaluating the volume of a crystallographic unit cell in the final two problems of this chapter. 

The scalar triple products, involving three vectors. 

This product is called the scalar triple product and is written as [V1V2V3]. 

It also follows that cyclic permutation of the subscripts does not change the value of the scalar triple product so that... 

The scalar triple product ti (r2 x rs) equals the scalar product of ri with the vector product of r2 and r3. The result is a scalar. The scalar triple product has a useful geometrical interpretation it is the volume of the parallelepiped whose sides correspond to the three vectors (Figure 1.9). 

During the optimisation of the structure against the distance constraints it is usual to incorporate chiral consiraints. These are used to ensure that the final conformation is the desired stereoisomer. Chiral constraints are necessary because the interatomic distances in two enantiomeric conformations are identical and as a consequence the wrong isomer may quite legitimately be generated. Chiral constraints are usually incorporated into the error function as a chiral volume, calculated as a scalar triple product. For example, to maintain the correct stereochemistry about the tetrahedral atom number 4 in Figure 9,16, the following scalar triple product must be positive ... 

Where 6 is the angle between V2 and V3 and 0 is the angle between V and the normal to the plane of V2 and V3. This product is called the scalar triple product and is written as [V1V2V3]. 

The reason is that, contrary to a line or a surface, a volume has no orientation in space and does not require a vector to represent it. The operator expressing a volume is the mixed product of three vectors, or scalar triple product, which gives a scalar equal to the volume of the parallelepiped defined by the three vectors, as shown in Figure 5.3c. 

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