Yves Jean - Molecular Orbitals of Transition Metal Complexes (793957), страница 46

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The action of the symmetry operations (E, C31 , C32 , σv (1),σv (2), σv (3)) on these orbitals gives the results presented in Table 6.3.Each (3 × 3) matrix associated with a symmetry operation(Figure 6.1) can easily be constructed from this table. Notice in particularthat the matrix associated with the identity operation (E) is diagonal; allits non-zero terms lie on the diagonal and moreover, they are all equalto 1.Each of these matrices can be characterized by its trace, that is thesum of its diagonal elements.

In group theory, this trace is called thecharacter of a matrix, written χ . Notice that the character in Figure 6.1Elements of group theory and applicationsTable 6.3. Transformation of the atomic orbitals1sH1 , 1sH2 , and 1sH3 of the hydrogen atoms of theNH3 molecule by the action of the symmetryoperations of the C3v point groupC3vEC31C32σv (1)σv (2)σv (3)1sH11sH21sH31sH11sH21sH31sH31sH11sH21sH21sH31sH11sH11sH31sH21sH31sH21sH11sH21sH11sH3100010 01sH11sH10 1sH2  = 1sH2 χ (E) = 3010001 1sH31sH110 1sH2  = 1sH1 χ (C31 ) = 0001100 01sH21sH11 1sH2  = 1sH3 χ (C32 ) = 01σv (1) 00001 1sH11sH101 1sH2  = 1sH3 χ (σv (1)) = 10σv (2) 01010 1sH111sH30 1sH2  = 1sH2 χ (σv (2)) = 10σv (3) 10100 1sH201sH10 1sH2  = 1sH1 1sH31sH31χ (σv (3)) = 1EC31C32100001sH31sH31sH31sH31sH31sH31sH21sH11sH21sH1Figure 6.1. Matrix representation of the action of the symmetry operations of the C3vpoint group on the 1sH1 , 1sH2 , and 1sH3 orbitals of the hydrogen atoms of the NH3molecule.associated with the identity operation (three in this example) is equalto the dimension of the basis; this result is general.

Notice also that thecharacters associated with the symmetry operations that belong to agiven class (see § 6.2.1) are equal (to 0 for the operations C31 and C32 , andto 1 for the operations σv (1), σv (2), and σv (3)).Symmetry groupsFrom now on, the action of the symmetry operations on a basis willbe represented by the set of characters associated with each symmetryoperation; the operations may be grouped into classes where appropriate. The symmetry properties of the basis (ŴH ) formed by the 1sH1 ,1sH2 , and 1sH3 orbitals of the hydrogen atoms of the NH3 molecule canbe summarized as follows (Table 6.4):Table 6.4.

Characters associated with thebasis ŴH formed by the orbitals 1sH1 , 1sH2 ,and 1sH3 of the hydrogen atoms in the NH3molecule (C3v point group)C3vEC31C32σv (1)σv (2)σv (3)ŴH300111Or alternativelyC3vE2C33σvŴH3016.2.5. Character tablesA character table provides a complete list of the irreducible representations associated with a given point group, as well as other usefulinformation.

The appearance of these tables, which is identical forall point groups, is illustrated below for the C2v (e.g. H2 O) and C3v(e.g. NH3 ) point groups.6.2.5.1. Character table for the C2v point groupThis table can be described in the following ways (Table 6.5):1. The symbol of the point group appears in the top left-hand corner.2. The symmetry operations, grouped in classes, appear on the firstline. However, in this point group, there is only one operation perclass.3.

The symbols given to the different irreducible representations of thepoint group appear underneath the point-group symbol, in the firstcolumn. These are known as Mulliken symbols.4. The characters associated with each irreducible representation arefound in the central part of the table. For example, the charactersfor the A1 representation are (1, 1, 1, 1), for the B2 representationElements of group theory and applicationsTable 6.5. Character table for the C2vpoint groupC2vEC2zσxzσyzA1A2B1B2111111−1−11−11−11−1−11zxyx 2 , y2 , z2xyxzyz(1, −1, −1, 1), etc. Every irreducible representation of this group isone-dimensional, since χ (E) = 1 for each of them.5.

The last two columns on the right contain algebraic functions whichare bases for the irreducible representations on the same line. Forexample, z is a basis for the A1 irreducible representation, and xz abasis for the B1 irreducible representation.These last two points need further comment. The A1 representation,for which every character is 1, is also known as the totally symmetricrepresentation, since any of the possible symmetry operations of thegroup transforms a basis function into itself.

There is a totally symmetric representation in each point group. The characters for the otherrepresentations are either 1 or −1, depending on whether the functionis transformed into itself or into its opposite. The algebraic functionsgiven in the last two columns on the right allow us to find immediatelythe symmetry of the AO on the ‘central’ atom, that is, the one whoseposition is unchanged by any of the symmetry operations, or the onelocated at the intersection of all the symmetry elements (the oxygenatom in the case of the water molecule). So the px , py , and pz orbitalstransform in the same way as x, y, and z, respectively, and the s orbital(with spherical symmetry) like (x 2 + y2 + z2 ). The symmetries of theseorbitals are therefore A1 (s, pz ), B1 (px ), and B2 (py ).

The character tablefor the C2v point group therefore allows us to reduce the basis ŴO formedby the valence orbitals of oxygen:ŴO = 2A 1 ⊕ B1 ⊕ B2(6.3)Notice that the last column of the character table contains functionsthat are squares or second-order products of x, y, and z. The symmetriesof these functions indicate the symmetries of the d orbitals on the centralatom, which is clearly very useful for the study of transition metalcomplexes. Thus, in a complex with C2v symmetry, the dx 2 −y2 and dz2orbitals have A1 symmetry, whereas the dxy , dxz , and dyz orbitals have A2 ,B1 , and B2 symmetries, respectively (Table 6.5). The basis of dimension5 formed by the d orbitals of the central atom (Ŵd ) can therefore beSymmetry groupsreduced as follows:Ŵd = 2A1 ⊕ A2 ⊕ B1 ⊕ B2(6.4)6.2.5.2.

Character table for the C3v point groupWe shall now examine a second character table, that associated with theC3v point group (Table 6.6), the point group for the NH3 molecule. Itis presented in the same format as the previous one, but it illustratesseveral characteristics of the point groups in which a rotation axis ispresent whose order is higher than 2 (here, a C3 -axis).Notice first that on the first line, the symmetry operations aregrouped in classes.

Thus, the two rotation operations around the C3 -axis(C31 and C32 , 6-8 and 6-9) are written 2C3 , and the reflections in the threeplanes of symmetry, σv (1), σv (2), and σv (3) (6-21) are written 3σv . Ingeneral, the notation that is used indicates the number of symmetryoperations in the class, followed by the symmetry element that is concerned.

One of the irreducible representations in this point group has acharacter of two associated with the identity operation. This is thereforea two-dimensional irreducible representation, written E (be careful notto confuse this with the identity operation, E).If we consult the last two columns of the character table, we canestablish the symmetry properties of the orbitals on the central atom.The s orbital is a basis for the A1 representation (or, more simply, it hasA1 symmetry), like the pz orbital. The px and py orbitals form a basisfor the two-dimensional representation (E). This indicates that from thesymmetry point of view, these orbitals cannot be separated.

Neither ofthem taken separately, nor any linear combination of them, forms a setthat is stable to the action of the symmetry operations C31 and C32 . Ifwe consider the d orbitals on the central atom, the final column in thecharacter table shows us that dz2 has A1 symmetry, whereas the orbitalpairs (dx 2 −y2 , dxy ) and (dxy , dyz ) have E symmetry.To finish this section, we provide a few comments about theMulliken symbols that are used for irreducible representations, thoughwithout full details. One-dimensional representations are indicated bythe letters A or B, whereas the letters E and T are used for two- andTable 6.6.

Character table for the C3v point groupC3vE2C33σvA1A2E11211−11−10zx 2 + y2 , z2(x, y)(x 2 − y2 , xy), (xz, yz)Elements of group theory and applicationsthree-dimensional representations, respectively. In the groups whichcontain an inversion centre, the subscript g (from the German gerade,which means even) is added to the representations that are symmetricwith respect to inversion (a positive character in the column headed byi), and the subscript u (from the German ungerade, for odd) to the representations that are antisymmetric with respect to inversion (a negativecharacter). Thus in the Ci point group, in which the inversion centreis the only symmetry element, the two (one-dimensional) irreduciblerepresentations are written Ag and Au , and the functions that are basesof these representations transform into themselves or their opposites,respectively.6.3.

The reduction formulaAs we have already seen (§ 6.2.5), the character table gives us informationon orbital symmetry properties. If the molecule contains a central atom,the symmetries of the orbitals of this atom are indicated in the last twocolumns of the table. However, the orbitals on non-central atoms, forexample the 1sH orbitals in H2 O or NH3 , are not individually bases for anirreducible representation (Tables 6.1 and 6.3). These AO form a basisfor a reducible representation that can be decomposed into a sum ofirreducible representations of the point group. Although the charactertable does not give the result immediately, it does enable us to find it byusing the reduction formula.6.3.1. The reduction formulaIf the characters χŴ associated with a reducible representation Ŵ areknown, it can be decomposed into a sum of irreducible representations(Ŵ = i ai Ŵi ) of the point group by using the reduction formula:ai =1χŴ (Rk ) × χi (Rk ) × n(Rk )h(6.5)kwhere the summation (k) is carried out over the classes, h is the numberof symmetry operations in the point group, also known as its order,χŴ (Rk ) is the character of the reducible representation for a given class,χi (Rk ) is the character of the irreducible representation for that class, andn(Rk ) is the number of symmetry operations in that class.Before being able to apply the reduction formula, it is thereforenecessary to determine the characters of the reducible representationbeing studied.The reduction formula6.3.2.

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