Mathematical Methods for Physics and Engineering - Matematica.NET

REPRESENTATION THEORY1y42x3Figure 29.4manganese.A molecule consisting of four atoms of iodine and one of29.11.1 Bonding in moleculesWe have just seen that whether chemical bonding can take place in a moleculeis strongly dependent upon whether the wavefunctions of the two atoms forminga bond transform according to the same irrep. Thus it is sometimes useful to beable to find a wavefunction that does transform according to a particular irrepof a group of transformations. This can be done if the characters of the irrep areknown and a sensible starting point can be guessed. We state without proof thatstarting from any n-dimensional basis vector Ψ ≡ (Ψ 1 Ψ 2 ··· Ψ n ) T ,where{Ψ i }is a set of wavefunctions, the new vector Ψ (λ) ≡ (Ψ (λ)1Ψ (λ)2··· Ψ n (λ) ) T generatedbyΨ (λ)i= ∑ Xχ (λ)∗ (X)XΨ i (29.24)will transform according to the λth irrep. If the randomly chosen Ψ happens notto contain any component that transforms in the desired way then the Ψ (λ) sogenerated is found to be a zero vector and it is necessary to select a new startingvector. An illustration of the use of this ‘projection operator’ is given in the nextexample.◮Consider a molecule made up of four iodine atoms lying at the corners of a square in thexy-plane, with a manganese atom at its centre, as shown in figure 29.4. Investigate whetherthe molecular orbital given by the superposition of p-state (angular momentum l = 1)atomic orbitalsΨ 1 =Ψ y (r − R 1 )+Ψ x (r − R 2 ) − Ψ y (r − R 3 ) − Ψ x (r − R 4 )can bond to the d-state atomic orbitals of the manganese atom described by either (i) φ 1 =(3z 2 −r 2 )f(r) or (ii) φ 2 =(x 2 −y 2 )f(r), wheref(r) is a function of r and so is unchanged byany of the symmetry operations of the molecule. Such linear combinations of atomic orbitalsare known as ring orbitals.We have eight basis functions, the atomic orbitals Ψ x (N) andΨ y (N), where N =1, 2, 3, 4and indicates the position of an iodine atom. Since the wavefunctions are those of p-statesthey have the forms xf(r) oryf(r) and lie in the directions of the x- andy-axes shown inthe figure. Since r is not changed by any of the symmetry operations, f(r) can be treated asa constant. The symmetry group of the system is 4mm, whose character table is table 29.4.1106