Transformation operators Og

Let be a set of functions belonging to the irreducible representation P and H an operator which commutes with all the transformation operators Og, then... 

To find the symmetry adapted combinations we first csonsider the application of the transformation operators Og to the 33 atomic orbitals. We see at once that for all symmetry operations Jt of the Pt, point group, the central atom M is left unchanged and consequently any will only transform metal orbitals into metal orbitals (or combinations of metal orbitals) and ligand orbitals into ligand orbitals (or combinations of ligand orbitals). Thus, we can immediately reduce (the reducible representation using aU 33 atomio orbitals) to the... 

The table below gives the effects of the transformation operators Og for the symmetry operations H of tho point group on four functions /j, /j, /j,... 

In the first place, we consider those representations which are produced by the same transformation operators Og and the same fimction space but with different choices of basis functions describing that space to be equivalent. We will see that any pair of such equivalent representations have corresponding matrices which are linked by a similarity transformation (see 4-6). As it will always be possible to find a set of basis functions which produce unitary matrices (a unitary repreaentotion), convenience dictates that we choose such a set for producing a representation which is typical of the other equivalent ones. 

Now let us consider the transformation operator Og (corresponding to the symmetry operation R) which is defined by the equation which We have had before ... 

Ik this chapter we introduce the Schrodinger equation this equation is fundamental to all applications of quantum mechanics to chemical problems. For molecules of chemical interest it is an equation which is exceedingly difficult to solve and any possible simplifications due to the symmetry of the system concerned are very welcome. We are able to introduce symmetry, and thereby the results of the previous chapters, by proving one single but immensely valuable fact the transformation operators Og commute with the Hamiltonian operator, It is by this subtle thread that we can then deduce some of the properties of the solutions of the Schrodinger equation without even solving it. 

In this ohapter 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 Og which are homomorphic with the symmetry operations and that from these we can construct... 

From our point of view the most significant thing about the Hamiltonian operators H9l and f/nno is that they both commute with the operators Og, we say that i/el and f/nuc are invariant under all symmetry transformation operators of the point group of the molecular framework... 

We now consider transformations of a function of position, /(r). Following Wigner [1] and most later authors, we define the effect of a symmetry operation G on the function by introducing an operator Og satisfying... 

Suppose that P is an n-dimensional representation of a group of transformation operators acting on the functions of an n-dimensional function space and that we have basis functions /, f/ with the property that the first m (m < ) are transformed among themselves for all O, (e.g. in 6-3, the p-orbitals Pi and p, were transformed among themselves by all Og and so m = 2 for this case) ... 

These three terms remove the Og, go and 00 terms respectively from (3.229) and repetition of the transformation ultimately yields a Hamiltonian which, with p = p2 = I, operates only on the i//ul functions, as desired (to order c-2). However, as (3.233) shows, this transformation is unacceptable if the two particles have equal masses (e.g. two electrons). It was realised subsequently that there exists a family of related transformations, of which (3.233) is just one member. The transformation using (3.233) in fact goes further than we require, in that it leads to complete separation of all four types of function. We would be satisfied with the more limited objective of separating... 

Nonetheless, by using the Fourier transform and two other mathematical operations, convolution and correlation, we can obtain several important structural parameters fiom the experimentally measured scattered intensity. The usual assumption is that the system can be represented by two-phases, with either sharp or difiuse boundaries between the phases. Some structural parametas of interest include average separation distance between the phases specific surface, Og average thicknesses cf the two phases and width of the boundary between phases, if diffose. These parameters are obtained from the "auto-correlation" function of p(x), which is a specific type of convolution. 

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