Molecular Symmetry The symmetry elements of mebela/chm3411_chapter15.pdf · PDF file An...
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The symmetry elements of objects Some objects are ‘more symmetrical’ than others. A sphere is more symmetrical than a cube because it looks the same after it has been rotated through any angle about any diameter. A cube looks the same only if it is rotated through certain angles about specific angles (90°, 180°, or 270° about an axis passing through the centers of any of its opposite faces or by 120° or 240° about an axis passing through any of its opposite corners.
An NH3 molecule is more symmetrical than H2O because NH3 looks the same after rotations of 120° or 240°, whereas H2O looks the same only after a rotation of 180°. An action that leaves an object looking the same after it has been carried out is called a symmetry operation. Typical symmetry operations; rotations, reflections, and inversions. Symmetry element for each symmetry operation – the point, line, or plane with respect to which the symmetry operation is performed. A rotation (a symmetry operation) is carried out around an axis (the corresponding symmetry element). We shall see that we can classify molecules by identifying all their symmetry elements, and
grouping together molecules that possess the same set of symmetry elements.
Operations and symmetry elements The classification of objects according to symmetry elements that leave at least one
common point unchanged gives rise to the point groups. There are five kinds of symmetry operation (and five kinds of symmetry element) of this kind.
The identity, E – doing nothing (or rotation by 360°around any axis); the corresponding symmetry element – the entire object. Because every molecule is indistinguishable from itself if nothing is done to it, every object possesses at least the identity element. An n-fold rotation (the operation) about an n-fold axis of symmetry, Cn – a
rotation through 360°/n. The operation C1 is equivalent to the identity operator E. H2O has one twofold axis, C2. NH3 has one threefold axis, C3, and there are two symmetry operations associated with it: clockwise and anticlockwise 120° rotations. A pentagon has a C5 axis, a cube has three C4 axes, four C3 axes, and six C5 axes. A sphere possesses an infinite number of symmetry axes (along any diameter) of all possible integral values of n. If a molecule possesses several rotational axes, then the one (or more) with the greatest value of n is called the principal axis. The principal axis of a benzene molecule is C6 perpendicular to the hexagonal ring. A reflection (the operation) in a mirror plane, σ, (the element) may contain the
principal axis or be perpendicular to it. If the plane is parallel to the principal axis, it is
called ‘vertical’, σv. H2O has two σv planes, NH3 has three.
A vertical mirror plane that bisects the angle between two C2 axis is called a ‘dihedral plane’, σd. When the plane is perpendicular to the principal axis it is called ‘horizontal’, σh. A C6H6 molecule has a C6 principal axis and a horizontal mirror plane.
In an inversion (the operation) through a center of symmetry, i (the element), we imagine taking each point in a molecule, moving it to the center of the molecule, and then moving it out the same distance on the other side: the point (x,y,z) is taken to the point (-x, -y,-z). H2O and NH3 do not have a center of inversion, but a sphere or a cube do. C6H6 or a regular octahedron do have a center of inversion; a regular tetrahedron (CH4) does not.
A n n-fold improper rotation (the operation) about an n-fold improper rotation axis, Sn, (the symmetry element) – two successive transformations: 1) a rotation through 360°/n; 2) a reflection through a plane perpendicular to the axis of this rotation; neither of these operations alone needs to be a
symmetry operation. CH4 has three S4 axis and the staggered form of ethane a S6 composed of 60° rotation followed by a reflection.
The symmetry classification of molecules A molecule belongs to the group C1 if it has no element other than the identity.
Ci – a molecule has the identity and the inversion alone (structure 3). Cs – a molecule has the identity and a mirror plane alone (4). Cn – a molecule possesses an n-fold axis. An H2O2 molecule (5) has the elements E and C2, so it belongs to the group C2.
If, in addition to the identity and a Cn axis, a molecule has n vertical mirror planes σv, then it belongs to the group Cnv. H2O: the elements are E, C2, and 2σv, so it belongs to the group C2v. NH3: the elements are E , C3, and 3σv, so it belongs to the group C 3v. A heteronuclear diatomic molecule (HCl) belongs to the group C∞v because all rotations around the axis and reflections across the axis are symmetry operations. Other members of the group C∞v include the linear OCS molecule and a cone.
Objects that, in addition to the identity and an n- fold principal axis, also have a horizontal mirror plane σ h belong to the groups Cnh: trans- CHCl=CHCl (6) – C2h, B(OH)3 (7) – C3h. In C2h the operations C2 and σh jointly imply the presence of a center of inversion.
A molecule that has an n-fold principal axis and n twofold axes perpendicular to Cn belongs to the group Dn. A molecule belongs to Dnh if it also possesses a horizontal mirror plane. The planar trigonal BF3 molecule (8) has the elements E, C3, 3C2, and σ h (with one C 2 axis along each B-F bonds) – D3h. C6H6: elements are E, C6, 3C2, 3C2’, and σh – D6h. All homonuclear diatomic molecules (N2) belong to the group D∞h because all rotations
around the axis are symmetry operations, as are end-to-end rotation and end-to-end reflection.
D∞h is also a group of the liner OCO and HCCH molecules and of a uniform cylinder. Other examples of Dnh molecules are C2H4 (9) – D2h, PCl5 (10) – D3h, [AuCl4]
- (11) – D4h. A molecule belongs to the group Dnd if in addition to elements of Dn it possesses n
dihedral mirror planes σd. Examples: the twisted, 90° allene (12) – D2d; the staggered conformation of ethane (13) – D3d.
Molecules which possess one Sn axis belong to the group S n: tetraphenylmethane (14) – S4. Molecules belonging to Sn with n > 4 are rare. The group S2 is the same as Ci.
A number of very important molecules (for example, CH4 and SF6) possess more than one principal axis. Most belong to the cubic groups, and in particular to the tetrahedral groups T, Td and T h or to the octahedral groups O and Oh. A few icosahedral (20-faced) molecules belonging to icosahedral group, I, are also known: they include some of the boranes (B12H12
2-) and buckminsterfullerene C60 (15).
The groups Td and Oh are the groups of the regular tetrahedron (CH4) and the regular octahedron (SF6), respectively. If the object possesses the rotational symmetry of the tetrahedron or the octahedron, but none of their planes of reflection, then it belongs to the simpler groups T or O. The group Th is based on T but also contains a center of inversion.
We can use the flow chart to determine symmetry group for any molecule. For example, 16 has D 5h symmetry and 17 belongs to the D5d group.
The full rotation group, R3 (the 3 refers to rotation in three dimensions), consists of an infinite number of rotation axes with all possible values of n. A sphere and an atom belong to R3, but no molecule does. One can use the R3 group to apply symmetry arguments to atoms, and this is an alternative approach to the theory of orbital angular momentum.
A summary of the shapes corresponding to different point groups
Some immediate consequences of symmetry
Polarity A polar molecule has a permanent dipole moment. If the molecule has
Cn symmetry (n > 1), it cannot possess a charge distribution with a dipole moment perpendicular to the symmetry axis because the symmetry of the molecule implies that any dipole moment that exists in one direction perpendicular to the axis is cancelled by an opposing dipole. Any dipole moment in such molecules must be parallel to the rotational axis. Example: H2O has a dipole moment parallel to its twofold axis. The same applies generally to the group Cnv, so molecules belonging to any of the Cnv groups
may be polar. In all other groups, such as C3h, D, etc., there are symmetry operations that take one end of the molecule into the other. Therefore, such molecules may not have dipole moment both perpendicular and along the symmetry axis. Only molecules belonging to the groups Cn, Cnv, and Cs may have a permanent dipole moment. For Cn and Cnv, that dipole moment must lie along the symmetry axis. Examples: ozone (O3) is angular and belongs to the group C2v and therefore may be polar (and is), but CO2 is linear and belongs to the group D∞h and hence is not polar.
Chirality A chiral molecule (from the Greek word ‘hand’) is a molecule that cannot be
superimposed on its mirror image. An achiral molecule can be superimposed on its mirror image. Chiral molecules are optically active in the sense that they rotate the plane of
polarized light. A chiral molecule and its mirror image partner constitute an enantiometric pair of isomers and rotate the plane of polarization in equal but opposite directions. A molecule may be chiral, and therefore optically active, only if it does not
possess an axis of improper rotation Sn. Molecules belonging to the groups Cnh possess a Sn axis implicitly because
they possess both Cn and σh