Yves Jean - Molecular Orbitals of Transition Metal Complexes (793957), страница 44
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This idea can be expressed as follows: σ 2 =E, where E represents the identity operation. A reflection plane is thereforeassociated with a single symmetry operation that leads to a configurationthat is equivalent to but not identical with the initial configuration.6.1.2. Inversion centreIf the change of the coordinates (x, y, z) for every atom to (−x, −y, −z)leads to a molecular configuration that is equivalent to the initial one,the point at the origin of the frame of reference is an inversion centre (orcentre of symmetry) for the molecule.
The symbol i is used both for thissymmetry element and for the associated operation. Two exampleswill illustrate the fact that the inversion centre may be located onone of the atoms of the molecule, but this is not necessary. In theoctahedral complex [FeCl6 ]4− , the iron atom is situated at the inversioncentre: the inversion operation leaves its position unchanged, whereasthe chlorine atoms are exchanged in pairs: Cl1,2 , Cl3,4 , and Cl5,6 (6-4).But in the ethylene molecule, the inversion centre is situated in themiddle of the carbon–carbon bond, and all the atoms are exchanged bythe inversion operation (6-5).Cl1Cl3Cl 6Cl 5Fe Cl 4Cl 2iCl4Cl5Cl 26-4FeCl 1Cl 6Cl 3Symmetry elements and symmetry operationsH2H1C1C2H4H4H3iC1C2H2H3H16-5Notice that a molecule may not contain more than one inversioncentre, and that there is only one symmetry operation associated withthis element.
If the inversion operation is performed twice (i2 ), theresulting molecular configuration is identical to the initial one (i2 = E).6.1.3. Rotation axesWe return to the example of dimethylether (6-1). A rotation of 180◦ (π)around the z-axis leads to a molecular configuration that is equivalent tothe initial one (6-6), with the following pairs of atoms being exchanged:(C1 , C2 ), (H1 , H4 ), (H2 , H6 ), and (H3 , H5 ). The z-axis is therefore asymmetry element of the molecule. The ratio between 360◦ and therotation angle associated with the symmetry operation (180◦ in thisexample) defines the order of the axis. In this case it is therefore a twofold axis (or of order two), written C2 . A single symmetry operation isassociated with this axis, shown in 6-6 and written C21 or simply C2 .
Ifthis operation is applied twice (a rotation of 2 × 180◦ = 360◦ , writtenC22 ), we return to the initial molecular configuration (C22 = E). Note thatby convention, the rotation is performed in a clockwise sense.H1H4OC1H2H3C2C2H6H4H5(180°)H1OC2H6H5C1H2H36-6xNH1zH3H26-7Ammonia (NH3 ) is an example of a molecule that possesses a threefold axis (C3 ). This is the z-axis that passes through the N atom and isperpendicular to the equilateral triangle defined by the three hydrogenatoms (6-7).A rotation of 120◦ (or 1×2π/3) around this axis leads to an equivalentconfiguration in which H1 , H2 , and H3 have been replaced by H2 , H3 ,and H1 , respectively (6-8).
This operation is written C31 or C3 . A rotationof 240◦ (or 2 × 2π/3) also leads to a configuration that is equivalent tothe initial one ((H1 , H2 , H3 ), replaced by (H3 , H1 , and H2 )), (6-9). Thisnew operation is written C32 . A rotation of 360◦ (or 3 × 2π/3) brings usElements of group theory and applicationsback to the initial configuration, that is, C33 = E. There are thereforetwo symmetry operations, different from the identity operation, that areassociated with a C3 -axis.1NH1C3H3H2(120°)NH2H1H36-8NH1H3H2C 32(240°)NH3H2H16-9z C4 and C2Cl3Cl2PtCl4Cl1xC⬙2C⬘26-10More generally, an axis of order n (Cn ) is present if a rotation of360◦ /n (or 2π/n) leads to an equivalent configuration for the molecule.This axis generates (n − 1) symmetry operations, C1n , C2n , . .
. , Cn−1n ,nthe operation Cn being equivalent to the identity operation(Cnn = E).Some molecules have several rotation axes. The axis of highest orderis called the principal axis. The complex [PtCl4 ]2− , in which the platinum atom is located in the centre of the square defined by the fourchlorine atoms (6-10), possesses a C4 -axis, perpendicular to the planeof the square, and four C2 -axes in the plane of the square: two of these−Pt−−Cl3 and Cl2 −−Pt−−Cl4 (C2′ ), andare co-linear with the bonds Cl1 −−Pt−−Cl (C2′′ ). The principal axis isthe two others bisect the angles Cl−therefore four-fold (of order four).
The existence of a four-fold rotationaxis implies the presence of a co-linear two-fold axis, as the operation C42(a rotation of 2 × 2π/4) is identical to the operation C21 (a rotation of1 × 2π/2).The set of operations associated with a rotation around the axisthat is perpendicular to the molecular plane can thus be written: C41 ,C42 (=C21 ), C43 and C44 (=E). Usually, the notation used for these is C41 ,C2 , C43 , and E, or sometimes 2C4 , C2 , and E, where the two operations C41 and C43 are grouped together.
This type of situation occursevery time that there is an axis whose order is even and greaterthan 2 (in practice, 4, 6, or 8). For example, if a molecule possesses a C6 -axis, a C3 -axis and a C2 -axis are necessarily co-linear withthis axis. The symmetry operations are thus: C61 , C62 (=C31 ), C63 (=C21 ),C64 (=C32 ), C65 and C66 (=E), which can also be written 2C6 , 2C3 , C2 ,and E.Symmetry elements and symmetry operationsCommentThe [PtCl4 ]2− example (6-10) allows us to make several ideas more preciseabout planes of symmetry. The z-axis defines the ‘vertical’ direction.
Themolecular plane (xy), which is perpendicular to the principal (z) axis, iswritten σh (‘h’ for ‘horizontal’). The two planes which contain the principal−Cl bonds are written σv (‘v’ for ‘vertical’). And the twoaxis and two Pt−planes xz and yz, which contain the principal axis and which bisect the−Pt−−Cl, are written σd (‘d’ for ‘dihedral’).angles Cl−6.1.4. Improper rotation axesThe symmetry operation associated with an improper axis of order n(written Sn ) is performed in two steps: a rotation of 2π/n around thisaxis, followed by a reflection in a plane that is perpendicular to this axis.It is important to realize that the axis used is not necessarily a Cn -axis,nor is the plane that is involved necessarily a symmetry element of themolecule.H4H3H1Crot.
90°H3CH2H3H1H2H4refl.H1H2CH46-11z (C3 and S3)F4xy (h) F1PF2F3F56-12As a first example, we consider the methane molecule (CH4 ), whosegeometry is tetrahedral. If we perform a rotation of 90◦ (2π/4) around−C−−H2 and H3 −−C−−H4 , followed by athe axis bisecting the angles H1 −reflection in the plane perpendicular to this axis passing through the carbon atom, we obtain a configuration that is equivalent (but not identical)to the initial one (6-11). The axis which we used is therefore an improperfour-fold axis (S4 ), and the symmetry operation described in 6-11 is written S41 , or simply S4 .
Notice that this axis is not a C4 -axis in the molecule(but a C2 -axis), and that the plane used is not a symmetry element in themolecule. As in the cases of Cn axes, the S4 operation can be performedm times in succession, and these are written Smn . In the example above,21it is easy to verify that S4 = C2 (a two-fold axis, co-linear with S4 ) andthat S44 = E (see Exercise 6.4). The only two operations that are uniqueto the S4 -axis are therefore S41 and S43 .The hypervalent molecule PF5 adopts a trigonal-bipyramidal (TBP)geometry (6-12). The fluorine atoms F1 , F2 , and F3 define the base of thebipyramid (the xy plane, written σh ) and are situated at the vertices ofan equilateral triangle (the angles between bonds are 120◦ ). The atomsElements of group theory and applications−P−−F5 axis (theF4 and F5 define the vertices of the bipyramid, the F4 −z-axis) being perpendicular to the plane σh .A rotation of 120◦ (2π/3) around the z-axis, followed by a reflectionin the xy plane, leads to an equivalent configuration for the molecule(6-13).
The z-axis is therefore an improper three-fold axis (S3 ). Noticethat in this example both the axis and the plane that were used toperform the S3 operation are themselves symmetry elements in themolecule: the z-axis is also a C3 -axis, and the σh plane is a reflectionplane. A characteristic property of the S3 -axis (and, more generally, ofimproper axes whose order is odd) is that the S33 operation does notlead to a configuration that is identical to the initial one (S33 is differentfrom E).
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