Symmetry operations on a position vector

In this chapter we have shown that there are very many different sets of matrices which behave like the symmetry operations of a given point group. We have constructed these so-called representations by considering the action of the symmetry operations on a position vector or on any number of base vectors. Alternatively, we have found that we can find transformation operators Om which are homomorphic with the symmetry operations and that from these we can construct... 

Symmetry operations on a position vector 6-3. Matrix representations for and... 

It is apparent that we can always get a set of 3 x 3 matrices, which form a representation of a given point group, by consideration of the effect that the symmetry operations of the point group have on a position vector. Why this works is shown pictorially in Fig. 5-3.2 for the a = a TCt operation of, T. The symmetry operation C8 on the position veotor p followed by a r on p produces a vector p which is coincidental with the one produced by the operation o on p. The matrices D(Ct), D(a r), X>(o ) then simply mirror what is being done to the point vector. The general mathematical proof that, if symmetry operations R, S, and T obey the relation SR — T, then the matrices D(R), D(S), and D(T), found as above, obey the relation... 

The space group G of a crystal is the set of all symmetry operators that leave the appearance of the crystal pattern unchanged from what it was before the operation. The most general kind of space-group operator (called a Seitz operator) consists of a point operator R (that is, a proper or improper rotation that leaves at least one point invariant) followed by a translation v. For historical reasons the Seitz operator is usually written R v. However, we shall write it as (R ) to simplify the notation for sets of space-group operators. When a space-group operator acts on a position vector r, the vector is transformed into... 

To summarize, we have found that the effect of any symmetry operation Jfl on a position vector p — expressed... 

There are several different methods of obtaining sets of matrices which are homomorphic with a given point group and in this chapter we discuss these methods in some detail. One way is to consider the effect that a symmetry operation has on the Cartesian coordinates of some point (or, equivalently, on some position vector) in the molecule. Another way is to consider the effect that a symmetry operation has on one or more sets of base vectors (coordinate axes) within the molecule. 

Problem 4.1 Figure 4.3 also contains pictures of the 0(2s), 0(2pj) and 0(2p,) orbitals insert the result of each symmetry operation for these in the appropriate position. The paper models of H2O (from Appendix 1) may be useful for this exercise, as they can be used to find the effect of each symmetry operation on the x, y and z vectors at the O atom. A completed version of the figure is included at the end of the chapter. 

In physics and chemistry there are two different forms of spatial symmetry operators the direct and the indirect. In the direct transformation, a rotation by jr/3 radians, e.g., causes all vectors to be rotated around the rotation axis by this angle with respect to the coordinate axes. The indirect transformation, on the other hand, involves rotating the coordinate axes to arrive at new components for the same vector in a new coordinate system. The latter procedure is not appropriate in dealing with the electronic factors of Born-Oppenheimer wave functions, since we do not want to have to express the nuclear positions in a new coordinate system for each operation. 

One more quantum number, that relating to the inversion (i) symmetry operator can be used in atomic cases because the total potential energy V is unchanged when ah of the electrons have their position vectors subjected to inversion (i r = -r). This quantum number is straightforward to determine. Because each L, S, Ml, Ms, H state discussed above consist of a few (or, in the case of configuration interaction several) symmetry adapted combinations of Slater determinant functions, the effect of the inversion operator on such a wavefunction P can be determined by ... 

We now consider the effeot that symmetry operations have on a point or position vector. 

Second, algebraic differences between the equivalent positions for the space group are formed. For each pair of equivalent positions, one coordinate difference will turn out to be a constant, namely 0, 5, 3, 5, depending on the symmetry operator. These define the Harker sections for that space group, which are the planes having one coordinate u,v, or w constant, and that will contain peaks corresponding to vectors between symmetry equivalent atoms. In focusing attention only on Harker sections, the Patterson coordinates u,v,w... 

In addition, the combination of three-dimensional symmetry elements gives rise to a completely new symmetry operator, the screw axis. Screw axes are rototranslational symmetry elements, constituted by a combination of rotation and translation. A screw axis of order n operates on an object by (a) a rotation of 2n/n counter clockwise and then a translation by a vector t parallel to the axis, in a positive direction. The value of n is the order of the screw axis. For example, a screw axis running parallel to the c-axis in an orthorhombic crystal would entail a counter-clockwise rotation in the a - b plane, (001), followed by a translation parallel to +c. This is a right-handed screw rotation. Now if the rotation component of the operator is applied n times, the total rotation is... 

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