C 2 '' σ v ' C 2 ' "side on" "in-plane" 2S determine how the MO transforms under each symmetry operation, Figure 3.

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1 Lecture Model nswers to Problems Self-study Problems / Exam Preparation determine the point group of o use VESPR (from st year) to determine that is planar, then use the flow chart. is the molecule linear? O. are there two or more C n axies with n>? O. is there a principle axis? YES. are there nc axies perpendicular to the principle axis (ie C axes)? YES 5. is there are mirror plane? YES o the molecule is D h z x on a molecule of draw the y y symmetry elements, Figure x C '' o the operation consists of σ v '' C rotation around the axis and (z) C '' reflection in the mirror plane. what is the correct axial orientation and why? o the correct orientation has the z- axis coincident with the symmety axis, the position of the x and y axes is in the plane perpendicular to the z-axis however there is no set convention with respect to the relative orientation of x and y although a left handed system is tradional. It can be useful to align one of the - bonds along the x or the y-axis. determine the symmetry label for the molecular orbitals of Figure o (a) has a ' symmetry o (b) has a " symmetry (working shown below). draw a "representation table" for the MO D h E C σ v Γ C ' σ v '. determine how the MO transforms under each symmetry operation, Figure (xy) E (xy) C (y) σ v ' C (y) "side on" "in-plane" Figure diagram depicting symmetry elements (a) σ v '' (b) Figure (a) lies in the plane of the paper, (b) is a po going into the paper. C ' (xy) (z) C (z) σ v C (y) Figure diagram depicting symmetry elements

2 Lecture Model nswers to Problems. compare this representation to the irreducible representations from the character table Γ o MO(b) has a " symmetry D h E C σ v construct a representation table for the "z-axis" and determine which irreducible representation it transforms as o The z-axis has the same phase properties as the p z O of boron and MO(b) above (which is just the p z O of boron, Os can also be MOs!). Thus we need never actually work out the symmetry labels of the p x, p y or p z atomic orbitals because they will always have the same symmetry labels as the x, y and z axes. The representation table is therefor the same as that for the p z orbital and the symmetry label is a ". What is the general procedure to determine the symmetry label of a molecular orbital? (in your words, use bullet points!) This is my process however you should go through the notes and an example and write out the process for yourself in your own words! o determine the point group of the molecule o define the axis system o draw a representation table for the MO o determine how the MO transforms under each symmetry operation o enter + for no phase change, - for a phase change o compare this representation to the irreducible representations from the character table o use a small letter for the symmetry label of a MO What is the general procedure to determine the character of a degenerate set of orbitals? (in your own words, use bullet points!) o take point on tip of each orbital o form the starting matrix o perform the symmetry operation on the orbitals o write coordinates of each point o form the final matrix by combing the coordinates o the character is the TRCE of the final matrix Work out all of the n operations up to 6. o Show for the D h + system that thee are only two unique operations. o Which operation is equivalent to? Draw a diagram clearly illustrating this equality. 6 = E and thus six operations need to be examined o The first operation was given to you in the lecture notes, Figure rotation reflection Figure Components of improper rotation

3 Lecture Model nswers to Problems o The operations up to were also given in the lecture notes (see Figure 5 below) and it was established that =. From this diagram it can also be seen that = = Figure 5 operation of the D h point group the full set of operations is shown in Figure 6 from this diagram it is can be seen that = 6 5 Figure 6 The full set of operations for o thus the only unique operations are and 5 and thus there are only unique operations. S = = S = 5 6 = E Which operation is S equivalent to? Draw a diagram clearly illustrating this equality. o S is equivalent to C, Figure 7 same result = Figure 7 diagram illustrating that is equivalent to

4 Lecture Model nswers to Problems orazine () belongs to the D h point group. On a sketch of borazine illustrate and label the symmetry elements of the D h point group (007/8 exam question, worth marks) o put in the axial definition, note the z axis is coming out of the page in the first diagram because it has to align with the axis, Figure 8 o don't crowd your diagrams, use two or three if that will make things clearer! z y (z), (z) x C ", σ v " C (y), (xy) C ', σ v ' (z), (z) C (y) σ v planes lie perpendicular to the page and pass through the C axes axis is coincident with the axis Figure 8 symmetry elements sketched on a molecule of borazine The Tetrahedral Point Group o discuss and illustrate the symmetry operations of the T d point group (use the notes of Lecture as a guide). o hint: there are three useful ways of thinking about a tetrahedral molecule, Figure 0, each one emphasises a different aspect of symmetry: (a) the C axes (b) the axes (c) the cubic structure o The character table for the T d point group is shown to the left, Figure 0, and it tells us the key symmetry operations in this group are E, 8 C, 6S and 6σ d Figure C axis there are 8 operations o a axis lies along each bond, one axis is shown in Figure, the others are easily predicted because the four atoms are symmetry equivalent, if one has a axis passing through it then they all will, hence there are four axis symmetry elements o the "cube" may be less familiar to you, think of the atoms occupying opposite corners of a cube and the central atom is at the center of the cube o around each axis there are possible operations:, the last operation = E is equivalent to the identity and so is already counted, there are then two symmetry operations associated with each axis and thus there are eight distinct operations in T d : 8 (a) (b) (c) Figure 9 Tetrahedral molecules T d E 8 C 6S 6σ d E T T Figure 0 Td point group h= (T x T y T z )

5 there are C operations o a C axis lies between each pair of - bonds, Figure a, bisecting each pair of atoms and through the center of each pair of faces in the cube, Figure b, as there are pairs of faces to each cube, there will be C axes o as we associate only one operation with each C axis there are C operations in T d there are 6σ d operations Lecture Model nswers to Problems o a σ mirror plane passes through each pair of atoms and contains a C axis, ie two mirror planes cross each pair of faces, Figure, these are dihedral mirror planes σ d. o as there are pairs of faces each with two mirror planes there are 6σ d operations in T d C (a) (b) Figure C axis looking from the side looking from above σ d (a) σ d (b) Figure σd mirror planes there are 6S operations o each C axis has a coincident S axis, consider a rotation of 90º around this axis and then reflection in a plane perpendicular to the axis through the center of the molecule. n example of these elements for the S operation is given in Figure. o notice that I am rotating counter clockwise, I can rotate in any direction I like as long as I am consistent for all operations. If I call counter-clockwise the positive direction, then rotating in the opposite direction becomes the "reverse" operation of C -. o notice that after the C operation the "atom" is now "off" the molecule! owever, once the whole S operation is completed it is back "on" the molecule. This is a consequence of neither the C nor the existing within the T d point group as separate elements!! (c) C S C Figure S operation 5

6 Lecture Model nswers to Problems o S (Figure 5) is the same as C operation and C lies to the left of S and so this operation is not counted with the S operations. In addition the S operation is the same as E and so is not counted here either Figure 5 S operation o thus there are S operations per C axis, and as there are C axes there must be 6S operations in T d Thus we have shown that there are E, 8 C, 6S and 6σ d operations for the T d point group. 6