Molecular symmetry facts for kids

Point Group Chart

Molecular symmetry is a basic idea in chemistry. It is about the symmetry of molecules. It puts molecules into groups according to their symmetry. It can predict or explain many of a molecule's chemical properties.

Chemists study symmetry to explain how crystals are made up and how chemicals react. The molecular symmetry of the reactants help predict how the product of the reaction is made up and the energy needed for the reaction.

Molecular symmetry can be studied several different ways. Group theory is the most popular idea. Group theory is also useful in studying the symmetry of molecular orbitals. This is used in the Hückel method, ligand field theory, and the Woodward–Hoffmann rules. Another idea on a larger scale is the use of crystal systems to describe crystallographic symmetry in bulk materials.

Scientists find molecular symmetry by using X-ray crystallography and other forms of spectroscopy. Spectroscopic notation is based on facts taken from molecular symmetry.

Historical background
Symmetry concepts
- Elements
- Operations
Point groups
Character tables
Images for kids
See also

Historical background

Physicist Hans Bethe used characters of point group operations in his study of ligand field theory in 1929. Eugene Wigner used group theory to explain the selection rules of atomic spectroscopy. The first character tables were compiled by László Tisza (1933), in connection to vibrational spectra. Robert Mulliken was the first to publish character tables in English (1933). E. Bright Wilson used them in 1934 to predict the symmetry of vibrational normal modes. The complete set of 32 crystallographic point groups was published in 1936 by Rosenthal and Murphy.

Symmetry concepts

Mathematical group theory has been adapted to study of symmetry in molecules.

Elements

The symmetry of a molecule can be described by 5 types of symmetry elements.

Water molecule is symmetrical

Symmetry axis: an axis around which a rotation by $\tfrac{360^\circ} {n}$ results in a molecule that appears identical to the molecule before rotation. This is also called an n-fold rotational axis and is shortened to C_n. Examples are the C₂ in water and the C₃ in ammonia. A molecule can have more than one symmetry axis; the one with the highest n is called the principal axis, and by convention is given the z-axis in a Cartesian coordinate system.
Plane of symmetry: a plane of reflection through which an identical copy of the original molecule is given. This is also called a mirror plane and abbreviated σ. Water has two of them: one in the plane of the molecule itself and one perpendicular (at right angles) to it. A symmetry plane parallel with the principal axis is dubbed vertical (σ_v) and one perpendicular to it horizontal (σ_h). A third type of symmetry plane exists: if a vertical symmetry plane additionally bisects the angle between two 2-fold rotation axes perpendicular to the principal axis, the plane is dubbed dihedral (σ_d). A symmetry plane can also be identified by its Cartesian orientation, e.g., (xz) or (yz).

Benzene

Center of symmetry or inversion center, shortened to i. A molecule has a center of symmetry when, for any atom in the molecule, an identical atom exists diametrically opposite this center an equal distance from it. There may or may not be an atom at the center. Examples are xenon tetrafluoride (XeF₄) where the inversion center is at the Xe atom, and benzene (C₆H₆) where the inversion center is at the center of the ring.
Rotation-reflection axis: an axis around which a rotation by $\tfrac{360^\circ} {n}$ , followed by a reflection in a plane perpendicular to it, leaves the molecule unchanged. Also called an n-fold improper rotation axis, it is shortened to S_n, with n necessarily even. Examples are present in tetrahedral silicon tetrafluoride, with three S₄ axes, and the staggered conformation of ethane with one S₆ axis.
Identity (also E), from the German 'Einheit' meaning Unity. It is called "Identity" because it is like the number one (unity) in multiplication. (When a number is multiplied by one, the answer is the original number.) This symmetry element means no change. Every molecule has this element. The identity symmetry element helps chemists use mathematical group theory.

Operations

Each of the five symmetry elements has a symmetry operation. People use a caret symbol (^) to talk about the operation rather than the symmetry element. So, Ĉ_n is the rotation of a molecule around an axis and Ê is the identity operation. A symmetry element can have more than one symmetry operation associated with it. Since C₁ is equivalent to E, S₁ to σ and S₂ to i, all symmetry operations can be classified as either proper or improper rotations.

Point groups

A point group is a set of symmetry operations forming a mathematical group, for which at least one point remains fixed under all operations of the group. A crystallographic point group is a point group which will work with translational symmetry in three dimensions. There are a total of 32 crystallographic point groups, 30 of which are relevant to chemistry. Scientists use Schoenflies notation to classify point groups.

Group theory

Mathematics define a group. A set of symmetry operations form a group when:

the result of consecutive application (composition) of any two operations is also a member of the group (closure).
the application of the operations is associative: A(BC) = (AB)C
the group contains the identity operation, denoted E, such that AE = EA = A for any operation A in the group.
For every operation A in the group, there is an inverse element A⁻¹ in the group, for which AA⁻¹ = A⁻¹A = E

The order of a group is the number of symmetry operations for that group.

For example, the point group for the water molecule is C_2v, with symmetry operations E, C₂, σ_v and σ_v'. Its order is thus 4. Each operation is its own inverse. As an example of closure, a C₂ rotation followed by a σ_v reflection is seen to be a σ_v' symmetry operation: σ_v*C₂ = σ_v'. (Note that "Operation A followed by B to form C" is written BA = C).

Another example is the ammonia molecule, which is pyramidal and contains a three-fold rotation axis as well as three mirror planes at an angle of 120° to each other. Each mirror plane contains an N-H bond and bisects the H-N-H bond angle opposite to that bond. Thus ammonia molecule belongs to the C_3v point group which has order 6: an identity element E, two rotation operations C₃ and C₃², and three mirror reflections σ_v, σ_v' and σ_v".

Common point groups

The following table contains a list of point groups labelled using the Schoenflies notation, which is common in chemistry and molecular spectroscopy. The description of structure includes common shapes of molecules, which can be explained by the VSEPR model.

Point group	Symmetry operations	Simple description of typical geometry	Example 1	Example 2	Example 3
C₁	E	no symmetry, chiral	bromochlorofluoromethane (both enantiomers shown)	lysergic acid	L-leucine and most other α-amino acids except glycine
C_s	E σ_h	mirror plane, no other symmetry	thionyl chloride	hypochlorous acid	chloroiodomethane
C_i	E i	inversion center	meso-tartaric acid	mucic acid (meso-galactaric acid)	(S,R) 1,2-dibromo-1,2-dichloroethane (anti conformer)
C_∞v	E 2C_∞ ∞σ_v	linear	hydrogen fluoride (and all other heteronuclear diatomic molecules)	nitrous oxide (dinitrogen monoxide)	hydrocyanic acid (hydrogen cyanide)
D_∞h	E 2C_∞ ∞σ_i i 2S_∞ ∞C₂	linear with inversion center	oxygen (and all other homonuclear diatomic molecules)	carbon dioxide	acetylene (ethyne)
C₂	E C₂	"open book geometry," chiral	hydrogen peroxide	hydrazine	tetrahydrofuran (twist conformation)
C₃	E C₃	propeller, chiral	triphenylphosphine	triethylamine	phosphoric acid
C_2h	E C₂ i σ_h	planar with inversion center	trans-1,2-dichloroethylene	trans-dinitrogen difluoride	trans-azobenzene
C_3h	E C₃ C₃² σ_h S₃ S₃⁵	propeller	boric acid	phloroglucinol (1,3,5-trihydroxybenzene)
C_2v	E C₂ σ_v(xz) σ_v'(yz)	angular (H₂O) or see-saw (SF₄)	water	sulfur tetrafluoride	pyrrole
C_3v	E 2C₃ 3σ_v	trigonal pyramidal or tetrahedral	ammonia	phosphorus oxychloride	cobalt tetracarbonyl hydride, HCo(CO)₄
C_4v	E 2C₄ C₂ 2σ_v 2σ_d	square pyramidal	xenon oxytetrafluoride	pentaborane(9), B₅H₉	nitroprusside anion [Fe(CN)₅(NO)]²⁻
C_5v	E 2C₅ 2C₅² 5σ_v	'milking stool' complex	Ni(C₅H₅)(NO)	corannulene
D₂	E C₂(x) C₂(y) C₂(z)	twist, chiral	biphenyl (skew conformation)	twistane (C₁₀H₁₆)	cyclohexane twist conformation
D₃	E C₃(z) 3C₂	triple helix, chiral	Tris(ethylenediamine)cobalt(III) cation	tris(oxalato)iron(III) anion
D_2h	E C₂(z) C₂(y) C₂(x) i σ(xy) σ(xz) σ(yz)	planar with inversion center	ethylene	pyrazine	diborane
D_3h	E C₃ 3C₂ σ_h 2S₃ 3σ_v	trigonal planar or trigonal bipyramidal	boron trifluoride	phosphorus pentachloride	cyclopropane
D_4h	E 2C₄ C₂ 2C₂' 2C₂ i 2S₄ σ_h 2σ_v 2σ_d	square planar	xenon tetrafluoride	octachlorodimolybdate(II) anion	Trans-[Co^III(NH₃)₄Cl₂]⁺ (excluding H atoms)
D_5h	E 2C₅ 2C₅² 5C₂ σ_h 2S₅ 2S₅³ 5σ_v	pentagonal	cyclopentadienyl anion	ruthenocene	C₇₀
D_6h	E 2C₆ 2C₃ C₂ 3C₂' 3C₂‘’ i 2S₃ 2S₆ σ_h 3σ_d 3σ_v	hexagonal	benzene	bis(benzene)chromium	coronene (C₂₄H₁₂)
D_7h	E C₇ S₇ 7C₂ σ_h 7σ_v	heptagonal	tropylium (C₇H₇⁺) cation
D_8h	E C₈ C₄ C₂ S₈ i 8C₂ σ_h 4σ_v 4σ_d	octagonal	cyclooctatetraenide (C₈H₈²⁻) anion	uranocene
D_2d	E 2S₄ C₂ 2C₂' 2σ_d	90° twist	allene	tetrasulfur tetranitride	diborane(4) (excited state)
D_3d	E 2C₃ 3C₂ i 2S₆ 3σ_d	60° twist	ethane (staggered rotamer)	dicobalt octacarbonyl (non-bridged isomer)	cyclohexane chair conformation
D_4d	E 2S₈ 2C₄ 2S₈³ C₂ 4C₂' 4σ_d	45° twist	sulfur (crown conformation of S₈)	dimanganese decacarbonyl (staggered rotamer)	octafluoroxenate ion (idealised geometry)
D_5d	E 2C₅ 2C₅² 5C₂ i 3S₁₀³ 2S₁₀ 5σ_d	36° twist	ferrocene (staggered rotamer)
S₄	E 2S₄ C₂		tetraphenylborate anion
T_d	E 8C₃ 3C₂ 6S₄ 6σ_d	tetrahedral	methane	phosphorus pentoxide	adamantane
O_h	E 8C₃ 6C₂ 6C₄ 3C₂ i 6S₄ 8S₆ 3σ_h 6σ_d	octahedral or cubic	sulfur hexafluoride	molybdenum hexacarbonyl	cubane
I_h	E 12C₅ 12C₅² 20C₃ 15C₂ i 12S₁₀ 12S₁₀³ 20S₆ 15σ	icosahedral or dodecahedral	Buckminsterfullerene	dodecaborate anion	dodecahedrane

Representations

Symmetry operations can be written in many ways. A good way to write them is by using matrices. For any vector representing a point in Cartesian coordinates, left-multiplying it gives the new place of the point transformed by the symmetry operation. Composition of operations is done by matrix multiplication. In the C_2v example this is:

\underbrace{ \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} }_{C_{2}} \times \underbrace{ \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} }_{\sigma_{v}} = \underbrace{ \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} }_{\sigma'_{v}}

Although an infinite (going on forever) number of such representations (ways of showing things) exist, the irreducible representations (or "irreps") of the group are commonly used, as all other representations of the group can be described as a linear combination of the irreducible representations. (The irreps span the vector space of the symmetry operations.) Chemists use the irreps to sort the symmetry groups and to talk about their properties.

Character tables

For each point group, a character table summarizes information on its symmetry operations and on its irreducible representations. The tables are square because there are always equal numbers of irreducible representations and groups of symmetry operations.

The table itself is made of characters which show how a particular irreducible representation changes when a particular symmetry operation is applied (put to it). Any symmetry operation in a molecule's point group acting on the molecule itself will leave it unchanged. But for acting on a general entity (thing), such as a vector or an orbital, this does not have to be what happens. The vector could change sign or direction, and the orbital could change type. For simple point groups, the values are either 1 or -1: 1 means that the sign or phase (of the vector or orbital) is unchanged by the symmetry operation (symmetric) and -1 denotes a sign change (asymmetric).

The representations are labeled according to a set of conventions:

A, when rotation around the principal axis is symmetrical
B, when rotation around the principal axis is asymmetrical
E and T are doubly and triply degenerate representations, respectively
when the point group has an inversion center, the subscript g (German: gerade or even) signals no change in sign, and the subscript u (ungerade or uneven) a change in sign, with respect to inversion.
with point groups C_∞v and D_∞h the symbols are borrowed from angular momentum description: Σ, Π, Δ.

The tables also tell the Cartesian basis vectors, rotations about them, and quadratic functions of them transformed by the symmetry operations of the group. The table also shows which irreducible representation transforms in the same way (on the right hand side of the tables). Chemists use this because chemically important orbitals (in particular p and d orbitals) have the same symmetries as these entities.

The character table for the C_2v symmetry point group is given below:

C_2v	E	C₂	σ_v(xz)	σ_v'(yz)
A₁	1	1	1	1	z	x², y², z²
A₂	1	1	-1	-1	R_z	xy
B₁	1	-1	1	-1	x, R_y	xz
B₂	1	-1	-1	1	y, R_x	yz

For example, water (H₂O) which has the C_2v symmetry described above. The 2p_x orbital of oxygen is oriented perpendicular to the plane of the molecule and switches sign with a C₂ and a σ_v'(yz) operation, but remains unchanged with the other two operations (obviously, the character for the identity operation is always +1). This orbital's character set is thus {1, -1, 1, -1}, corresponding to the B₁ irreducible representation. Similarly, the 2p_z orbital is seen to have the symmetry of the A₁ irreducible representation, 2p_y B₂, and the 3d_xy orbital A₂. These assignments and others are in the rightmost two columns of the table.

Images for kids

XeF4, with square planar geometry, has 1 C4 axis and 4 C2 axes orthogonal to C4. These five axes plus the mirror plane perpendicular to the C4 axis define the D4h symmetry group of the molecule.