Ideas of Quantum Chemistry P97 shows how quantum mechanics is applied to chemistry to give it a theoretical foundation. The structure of the book (a TREE-form) emphasizes the logical relationships between various topics, facts and methods. It shows the reader which parts of the text are needed for understanding specific aspects of the subject matter. Interspersed throughout the text are short biographies of key scientists and their contributions to the development of the field. | 926 C. GROUP THEORY IN SPECTROSCOPY If we obtain another orbital v2 then we may begin to play with it by applying all the symmetry operations. Some operations will lead to the same new orbital sometimes with the opposite sign. After other operations we will obtain the old orbital v1 sometimes with the opposite sign and sometimes these operations will lead to a third orbital v3. Then we apply the symmetry operations to the third orbital etc. until the final set of orbitals is obtained which transform into themselves when subject to symmetry operations. The set of such linearly independent orbitals Vi i 1 . n may be treated as the basis set in a vector space. All the results of the application of operation on the orbitals vi are collected in a transformation matrix Ri R i V RI V where v V1 Vn The matrices Ri i 1 2 . g form the n-dimensional representation in general reducible of the symmetry group of the molecule. Indeed let us see what happens if we apply operation T tR 1 tR2 to the function vi T Vi tR . Vi R1 Rl v R R1v R R v R1R2 t v This means that all the matrices Ri form a representation. BASIS OF A REPRESENTATION The set of linearly independent functions yi which served to create the representation forms the basis of the representation. The basis need not have been composed of the orbitals it could be expressions like x y z or x2 y2 z2 xy xz yz or any linearly independent functions provided they transform into themselves under symmetry operations. We may begin from an atomic orbital and after applying symmetry operations will soon obtain a basis set which contains this orbital and all the other equivalent orbitals. Decomposition of a function into irreducible representation components Let us take a function f belonging to a Hilbert space. Since see eq. S P a 1 where a goes over all the irreducible representations of the group f can be written as the sum of its components fa each component belonging to the corresponding subspace of the .
