Molecular symmetry

Symmetry groups

Molecular symmetry groups are sets of the operations listed above together with the binary operation of consecutive application. If A, B and C are group operations and the application of operation A to the molecule "followed by" operation B results in an equivalent result to operation C the process may be shown as C = BA or BA = C. For example, a C₂ rotation followed by a σ_v reflection is seen to be a σ_v′ symmetry operation: σ_vC₂ = σ_v′. The binary combination consists of applying first one symmetry operation and then the other. Operations are applied from right to left. Another example is the sequence of a C₄ rotation about the z-axis and a reflection in the xy-plane, denoted σ(xy)C₄. The composition (repeated application) of symmetry operations is sometimes shown as B ∗ A to emphasise the fact that binary operation is being performed but the most common practice in modern literature is just right-to-left placement. Molecular symmetry groups are also called point groups because the set of symmetry operations leave at least one point fixed (though for some symmetries an entire axis or an entire plane remains fixed). In other words, a point group is a group that summarises all symmetry operations that all molecules in that category have. The symmetry of a crystal, by contrast, is described by a space group of symmetry operations, which includes translations in space.

The symmetry operations of a molecule provide a specific example of a mathematical group, a concept from linear algebra combining a set of operations with a binary operation that obeys the following three axioms

1. there exists an identity element that in a binary operation with another element simply reproduces that element: for all x ∈ G there exists E ∈ G such that xE = Ex = x
2. there exists an inverse element for every element in the group: for all x ∈ G there exists y ∈ G such that xy = yx = E
3. combinations of elements are associative; for all x, y, z ∈ G, x(yz) = (xy)z. ie the order in which binary operations are applied does not matter so long as the overall order is maintained.

Assigning each molecule to a point group classifies molecules into categories with similar symmetry properties. Schoenflies symbols for point groups evolved during the 19th century from the detailed examination of crystals and minerals but are used now only in atomic and molecular work. Hydrogen peroxide (H₂O₂) has the structure shown with hydrogen atoms attached to two oxygen atoms in a 'V' shape relative to the oxygen–oxygen bond. It provides an example of the simplest non-trivial point group C₂, containing just the identity element E and a 2-fold rotational symmetry about the vertical axis, the set {E, c}. Clearly, Ec = c, cE = c, EE = E and cc = c² = E. This molecule has two enantiomers that cannot be superimposed on each other by rotational motions in three dimensional space. Each enantiomer has hydrogen atoms placed on the opposite side of the V shape. Any molecule described completely by a rotational group has enantiomers but it is possible that the two might be interconverted by rotations about bonds. A molecule which has no improper axis for any value of n is a chiral molecule. So, molecules belonging to rotational groups C_n, D_n, T and O are potentially enantiomeric

Formaldehyde is a planar molecule with two hydrogen atoms and an oxygen atom attached to a carbon atom as shown in the diagram. It has a very clear 2-fold rotational axis through the carbon and oxygen atoms and so exhibits C₂ symmetry. There are also vertical mirror symmetry planes labelled σ_v and σ_v' at right angles to each other so the molecule exhibits the higher order C_2v group symmetry. Binary combinations of these four operations are shown in the C_2v operation table where a slightly different notation is used for the reflections because in low symmetry molecules of this kind the mirror planes can be identified with the x, y and z axes. Operation σ_xz in the top row "followed by" operation C₂ in the left hand column is equivalent to σ_yz read from the body of the table. In this point group the order in which operations are applied makes no difference to the outcome but for the vast majority of groups it does make a difference. Water (H₂O) and hydrogen sulfide (H₂S) molecules share identical symmetry operations with formaldehyde and therefore belong to the same symmetry group and are subject to the same four symmetry operations with order 4.

Although simple, the procedure above is typical of Schoenflies point group symbols. A rotational group C₂ of order 2 is extended to a non-rotational group C_2v of order 4 through a combination of cyclic group elements with mirror reflection. Most all of these symbols have the form C_nx or D_nx in which C_n and D_n are cyclic and dihedral rotational groups and x is a mirror reflection. A cyclic group C_n contains n elements (its order) while a dihedral group D_n has 2n elements. Some molecules have symmetry operations limited to just rotational operations and the rotational group symbol is as far as the analysis can proceed. It is worth noting that the number of proper and improper operations is always equal whenever improper symmetry is present at all.

Trigonal molecules

Phosphoric acid only has 3-fold cyclic rotational symmetry and the point group C₃ contains three symmetry operations: rotation through 120°, 240° and 360°, which is equivalent to the identity E. A glance at the gallery image on the right shows a central phosphorus atom with four oxygen atoms and three hydrogen atoms. It is easy to imagine a rotational axis through the central phosphorus atom and the lone oxygen atom about which rotations occur. If the 120° rotation is labelled c the set of three operations can be shown as the set {c, c², c³ = E} and binary operations on these elements amount to simply counting indices modulo n, so c²c² = c.

Phosphoryl fluoride POF₃ belongs to the same symmetry group as molecules PCl₃, XeO₃, and NH₃ and therefore shares identical symmetry operations. This molecule obviously has the same three symmetry elements {c, c², c³ = E} as the preceding phosphoric acid example and so provides an example of a C₃ symmetry molecule. Closer examination shows that this molecule also has a vertical mirror plane of symmetry that did not exist in the previous example. In addition to the three rotational operations there are three combinations with the vertical mirror plane operation σ_v, these being {cσ_v, c²σ_v, σ_v} so they are placed in one point group, C_3v, with order 6. Almost all Schoenflies point group symbols are constructed in this way. First find the rotational group to which the molecule belongs then double the number of elements by combining rotational elements with the reflection operation. This classification system helps scientists to study molecules more efficiently, since chemically related molecules in the same point group tend to exhibit similar bonding schemes, molecular bonding diagrams, and spectroscopic properties.

The relationship between rotational group C₃ and its non-rotational super-group C_3v is just one of the relationships shown in the table. A cyclic group C_n has n elements but every element in the group is a product of a generating rotation c of ⁠360°/n⁠ about the z axis and the elements are c, c², c³,... , cⁿ = E. Group C_nv has 2n elements: the same n elements as the rotational group together with the product of these n elements with the vertical mirror plane σ_v . Notice that all 2n operations of the group C_nv can be obtained as products of the two generating elements c and σ_v.

Tetragonal molecules

Two low-symmetry groups of order 2 have symbols C_s and C_i, and are respectively combinations of group C₁ with mirror reflection and spatial inversion. One remaining series of Schoenflies symbols does not follow the pattern of a rotational group combined with a mirror reflection. Rotation-reflection point groups S_2n (where n is a simple integer) have a single generating operation that combines a 2n-fold rotation with a horizontal mirror reflection. Consider as an example the propadiene molecule C₃H₄ with three carbon atoms on an imaginary z axis and two hydrogen atoms in the zy plane and two other hydrogen atoms in the xz plane. Imagine a 90° rotation about the z axis combined with a reflection in a horizontal plane through the central carbon atom. This combination brings the atoms back into coincidence with their original positions but neither the rotation or the reflection by itself would do so. Two applications of the combined improper operation are equivalent to a single application of a 2-fold rotation operation so point group S₄ contains group C₂ as a subgroup and generally rotation-reflection groups S_2n contain C_n as a subgroup. It follows from this relationship that the subscript n in symbol S_n is always an even number.

Hexagonal molecules

A benzene molecule has a 6-fold axis with carbon and hydrogen atoms positioned at apices of a hexagon imagined to lie in the xy plane and is therefore an example of a C₆ molecule of order 6. It has 6 additional 2-fold axes at right angles to the principal 6-fold axis and is also an example of a D₆ molecule of order 12. Finally, it has vertical, diagonal and horizontal mirror symmetry resulting from the fact that it has a centre of symmetry. Only one of these improper operations is necessary so Schoenflies describes this as point group D_6h of order 24.

Spherical molecules

All of the molecular group examples above have a principal rotational axis and might also have a 2-fold rotational axis at right angles to the principal axis or mirror reflection symmetry at right angles to either of these axes. There is a small number of symmetry groups that have multiple rotational symmetry axes that are not at right angles to each other, these being the tetrahedral, octahedral and icosahedral systems.

Molecular image gallery

The following table shows a large number of molecules belonging to point groups labelled using Schoenflies notation. Each row contains example molecules a group belonging to the Schoenflies symbol on the left of the table In each row, the descriptions and examples have no higher symmetries, meaning that the named point group captures all of the point symmetries and is the highest order group applicable to that molecule. This table is excellent for an overall view of molecular forms but greater detail and the ability to move images is provided by the Otterbein site.

Laue classes

All of the group operations described above and the symbols for crystallographic point groups themselves were first published by Arthur Schoenflies in 1891. Later Max von Laue published the results of experiments using X-ray diffraction to elucidate the internal structures of crystals, producing a limited version of the table of "Laue classes" shown below which are sometimes described as "Laue groups" or "Friedel classes" (after Georges Friedel). Hermann-Mauguin notation is almost invariably used now to describe crystallographic groups in Laue classes but this system provides no advantage for atomic and molecular work.

When adapted for molecular work this table first divides point groups into three kinds: asymmetric, symmetric and spherical tops. These are categories based on the angular momentum of molecules, having respectively 3, 2 and 1 distinct values of angular momentum, becoming more symmetrical down the table. A further sub-division into systems is defined by the rotational group G in the leftmost column then into rows of Laue classes that take the form of cyclic and dihedral groups in the first two categories and tetrahedral and octahedral classes in the third.

Rotational groups contain only pure rotational operations, sometimes called proper operations and occur only in the first column of the table. Von Laue showed that x-ray diffraction can not distinguish between groups in a row of the table and shows each one to be the centred point group on the right hand column of the table. Groups in this column contain the inversion operation itself as a member. The middle two columns contain non-rotational groups belonging to the same abstract group as that in the first column -this is why they result in the same diffraction pattern

For example, seven groups in the hexagonal system all contain the C₆ cyclic system, mostly as physical rotational group but in the third column of the table as an abstract group. So, C₆ and C_3h are distinct manifestations of the same group while C_6h is simply C₆ × i. Groups D₆, C_6v and D_3h are also example of the same abstract group and D_6h is the direct product D₆ × i. Tetrahedral and octahedral point groups have a relationship similar to that between cyclic and dihedral groups and the tetrahedral group occurs in all cubic groups.

Representations and their characters

A set of matrices that multiply together in a way that mimics the multiplication table of the elements of a group is called a representation of the group. The simplest method of obtaining a representation of molecular group transformations is to trace the movements of atoms in a molecule when symmetry operations are applied. For example, a water molecule belonging to the C_2v point group might have an oxygen atom labelled 1 and two hydrogen atoms labelled 2 and 3 as shown in the right hand column vector below. If the hydrogen atoms are imagined to rotate by 180 degrees about an axis passing through the oxygen atom we have the familiar C₂ operation of this point group. The oxygen atom in position number 1 stays in position but the atoms in positions 2 and 3 are moved to positions 3 and 2 in the resulting column vector. The matrix connecting the two provides a 3 × 3 representation for this operation.

${\displaystyle \left[\begin{array}{c}1\\ 3\\ 2\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]\times {\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]}_{}}$

This point group only contains four operations and matrices for the other three operations are obtained similarly, including the identity matrix which just contains 1s on the leading diagonal (top left to bottom right) and 0s elsewhere. Having obtained the representation matrices in this way it is not difficult to show that they multiply out in exactly the same way as the operations themselves.

${\displaystyle \underset{C_2}{\underbrace{\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]}}\times \underset{{\sigma }_{\text{v}}}{\underbrace{\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}}=\underset{{\sigma }_{\text{v}}^{\prime }}{\underbrace{\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]}}}$

Although an infinite number of such representations exist, the irreducible representations (or "irreps") of the group are all that are needed as all other representations of the group can be described as a direct sum of the irreducible representations. The first step in finding the irreps making up a given representation is to sum up the values of the leading diagonals for each matrix so, taking the identity matrix first then the matrices in the order above, one obtains (3, 1, 3, 1). These values are the traces or characters of the four matrices. Asymmetric point groups such as C_2v only have 1-dimensional irreps so the character of an irrep is exactly the same is the irrep itself and the following table can be interpreted as irreps or characters.

Looking again at the characters obtained for the 3D representation above (3, 1, 3, 1), we only need simple arithmetic to break this down into irreps. Clearly, E = 3 means there are three irreps and a C₂ representation sum of 1 means there must be two A and one B irreps so the only combination that adds up to the characters derived is 2A₁ + B₁.

Robert Mulliken was the first to publish character tables in English and so the notation used to label irreps in the above table is called Mulliken notation. For asymmetric groups it consists of letters A and B with subscripts 1 and 2 as above and subscripts g and u as in the C_2h example below. (Subscript 3 also appears in D₂.) The irreducible representations are those matrix representations in which the matrices are in their most diagonal form possible and for asymmetric groups this means totally diagonal. One further thing to note about the irrep/character table above is the appearance of polar and axial base vector symbols on the right hand side. This tells us that, for example, cartesian base vector × transforms as irrep B₁ under the operations of this group. The same collection of product base vectors is used for all asymmetric groups but symmetric and spherical groups use different sets of product base vectors.

Point group C_2h has the operations {E, C₂, i, σ_h} and the 1,5-dibromonapthalene (C₁₀H₆Br₂) shown in the figure belongs to this symmetry group. It is possible to construct four 18 × 18 matrices representing the transformations of atoms during its symmetry operations in the style of the water molecule example above and reduce it to 18 single-dimensional irreps. Notice however that carbon atom number 1 either stays in place or it is exchanged with carbon atom number 5 and these two atoms can be analysed separately from all the other atoms in the molecule. The transformation matrix for these two atoms alone during the molecular C₂ rotation is

${\displaystyle \left[\begin{array}{c}5\\ 1\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\times {\left[\begin{array}{c}1\\ 5\end{array}\right]}_{}}$

with character 0. When this computation is carried out for each of the operations above the characters obtained are (2, 0, 0, 2) because two operations leave the atoms in place and two move them. The irrep table for this group is below. The first column tells us there are two single-dimensiona; irreps, the second column (C₂) that there is one A and one B while columns 3 and 4 reveal that one irrep has subscript g the other has to have subscript u. This means that the irreps resulting from the two atoms are A_g + B_u. In fact, the 18 atoms in this molecule are paired off in exactly the same way as carbon atoms 1 and 5 so that, from a symmetry perspective, the atom consists of 9 pairs of equivalent atoms related through symmetry. It follows that each pair contributes the same irreps as the pair examined above to give a total 18 dimensional irrep result of 9(A_g + B_u).

Symmetric point group representations

Symmetric point groups are divided into systems based on the increasing order of the main rotational axis from three to infinity. Systems are in turn divided into cyclic and dihedral groups and within a system the order of the dihedral group is twice that of the cyclic group. Cyclic groups only have one dimensional representations as shown in the table of irreps and the number of irreps is equal to the order of the group. The irreps shown use standard notation for the rotational group of a class but Mulliken sometimes gave different symbols to other members of the same class even though they belong to the same abstract group and therefore have the same irreps.

Dihedral point groups contain a cyclic group of the same rotational order: so group D_n always contains group C_n as an index-2 subgroup. It follows that dihedral irreps are superimposed on cyclic irreps because the cyclic group within a dihedral one does not cease to be a cyclic group. A dihedral group also contains a 2-fold rotational axis at right angles to the main cyclic axis and this has two consequences. Firstly, the A and B cyclic irreps are split into pairs of one dimensional irreps identified by subscripts 1 and 2. Secondly, pairs of 1D E_−x and E_+x cyclic irreps combine to form single E_x 2D irreps in the dihedral group because the 2-fold horizontal rotation makes pairs of rotations equivalent. For example, a 60° rotation about the main axis becomes equivalent to a 300° (i.e. −60°) rotation because the 2-fold horizontal rotation makes them equivalent. Combinations of this kind are said to form a class. Infinite order dihedral group irreps sometimes use Greek symbol descriptions, σ, Π, Δ that follow from early linear molecule calculations.

Taking benzene as a simple example, we have a molecule that belongs to the series of point groups C₆, D₆ and D_6h with increasing orders 6, 12 and 24. The six carbon atoms may be represented by a 6 × 6 matrices which in group C₆ have irreps A, B, E₊₁, E₋₁, E₊₂ and E₋₂, because n objects in an n-fold cyclic group always produce one of each irrep. If these cyclic irreps are promoted to group D₆ we obtain A₁, B_x, E₁ and E₂ when subscripts are added to the 1D A and B irreps and the others merge to form 2D irreps. A₁ is there because the most symmetric irrep has to occur once and only once in the result leaving only the B irrep subscript to be deduced. The character of the 2-fold horizontal rotation operation is 2 because 2 carbon atoms stay in place during the rotation, Char(C₂) = 2 telling us that there are two more 1 subscripts than 2 subscripts so the result is A₁, B₁, E₁ and E₂. Finally, promotion to D_6h requires the addition of g and u subscripts. Since Char(i) = 0 there are an equal number of g and u subscripts, and A_1g has to be present as the most symmetric group so B_1u is mandatory. Furthermore, odd and even 2D irreps take u and g subscripts so the final result for the carbon atoms is (A_1g, B_1u, E_1u, E_2g), but with the hydrogen atoms we get 2(A_1g, B_1u, E_1u, E_2g).

Boric acid and boron trifluoride provide further hexagonal examples in spite of their slightly misleading Schoenflies symbols C_3h and D_3h. Taking boric acid first, we have three sets of equivalent atoms: 1 boron, 3 oxygen and 3 hydrogen. Obviously the oxygen and hydrogen atoms produce the same irreps so only one has to be deduced. Applying the 6-fold cyclic group to (say) the hydrogen atoms produces characters (3,0,0,3,0,0) yielding irreps A + E₊₂ + E₋₂. Doubling up and adding an irrep for the central boron atom produces A + 2(A + E₊₂ + E₋₂) in standard Laue class notation. Unfortunately, Mulliken used a different notation for C_3h and D_3h irreps to that used for other groups and a conversion table would be needed if that was important.

Boron trifluoride has a central boron atom with 3 fluorine atoms and belongs to cyclic subgroup C_3h and the larger dihedral group D_3h. Following the above reasoning, the irreps in C_3h are A + (A + E₊₂ + E₋₂) and when promoted to the dihedral group this becomes A₁ + (A₁ + E₂). Again, conversion to Mulliken notation is required if that is important.

Spherical point group representations

Spherical classes are defined by the tetrahedral, octahedral and icosahedral rotational groups T, O and I. The first two of these, T and O, are related in much the same way as cyclic and dihedral groups are related in symmetric groups. Both tetrahedral and octahedral molecules are often shown with their atoms inscribed in the apices or faces of cubes and might be considered as a single "cubic" system. The first Laue class of this system contains only the tetrahedral rotational group T of order 12 and the direct product of this group with space inversion T_h of order 24. Every point group in the following octahedral class contains the tetrahedral rotational group as a subgroup. Irreps of tetrahedral and octahedral groups are also related similarly to cyclic and dihedral groups and the table below shows how tetrahedral irreps are incorporated in octahedral irreps

Tetrahedral symmetry has 3 one-dimensional irreps (A, E₊, E₋) and one 3-dimensional irrep T then the A and T irreps split into two irreps with subscripts 1 and 2 while the two 1D E irreps combine into a single 2D irrep. Notice that the T irrep is always 3 dimensional but the E irrep only becomes 2 dimensional in the higher order group.

Methane (CH₄) is often used as an example and, although often described as a tetrahedral molecule because of the very visible rotational symmetry, it really belongs to the octahedral symmetry class. Considering methane first as a tetrahedral molecule the 12 operations of group T are {E, 3 × c, 4 × b, 4 × b³} where c is a 180° rotation along the x, y and z axes and b is a 120° rotation about the apices of a cube. It is not difficult to convert 5 × 5 symmetry operation transformation matrices to reducible matrices and thence to molecular irreps but this not necessary.

Methane has two sets of equivalent atoms: a single carbon atom and 4 hydrogen atoms. The atoms of each set are transformed into each other during operations. A single atom can only ever be transformed into itself and therefore always contributes the most symmetrical irrep to the end total irrep count. Additionally, there is a rule of group theory that the most symmetrical irrep must occur once and only once in the irreps of any equivalent atom set so the five dimensions of irreps being sought contain two A and three others. Although E₁ and E₂ are 1 dimensional they have to occur together in the irreps of any equivalent set of atoms. it follows that he only way of filling the remaining three dimensions is to adopt 3D irrep T so the irreps are 2A + T. (E irreps have to be taken in pairs in physical molecular applications.)

Extending this treatment to the octahedral group T_d requires six 4-fold roto-inversion operations (f) about the main axes and six 2-fold roto-inversions (a), appearing as mirror reflections through opposite edges of the imaginary cube in which methane is placed. So half the operations of this group are rotational and half non-rotational. Rotational group T exists within the non-rotational group T_d = {E, 3 × c, 8 × b, 8 × b³, 6 × f, 6 × a} so the irreps in group T in the expansion to T_d. Again we have 2 sets of equivalent atoms and each set must contribute one and only one of the most symmetrical irrep, in this case A₁. Reasoning as above, we know that the irreps in T_d must be 2A₁ + T_x so the last step is to find the 3D subscript. A brief look at the 4 × 4 transformation matrix for the 4-fold rotation operation f shows character Ch(f) = 0 and the × subscript has to be 2 to balance the 1 on the A irrep. so the final result is 2A₁ + T₂

Sulfur hexafluoride (SF₆) can also be treated first as a tetrahedral molecule T, then as octahedral O and finally as centred molecule O_i. There are two sets of equivalent atoms consisting of a single sulfur atom and six fluorine atoms. Transformations of the fluorine atoms generate a six dimensional representation that can only reduce into the direct sum of tetrahedral irreps A, E₊₁, E₋₁ and T because the direct sum must include the most symmetrical irrep once and only once, leaving five dimensions that can only be satisfied in the way shown—a direct sum of 5 can only be made up from a 2 and a 3—no other combination is possible. These irreps are "promoted" to 2A₁ + E₁ + T_x in group O. To get the × subscript observe that the 4-fold rotation in SF₆ has character Ch(f) = 2 because two atoms stay in position and a glance at this column of the table suggests A₁ + E₁ + T₁. Finally the inversion operation (i) applied go the fluorine atoms has character Ch(i) = 0 indicating equal numbers of g and u subscripts (because none of the atoms remains in position). Since the most symmetrical irrep must occur once the only possible result is A_1g + E_1g + T_1u. The single sulfur atom always has the most symmetric irrep to the final reduction of the seven dimensional matrices to a direct sum is 2A_1g + E_1g + T_1u.

Representations are labeled according to a set of conventions:

• A is a one-dimensional representation if the character under the principal rotation is +1
• B is a one-dimensional representation if the character under the principal rotation is −1
• when higher dimensions are permitted 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.

A summary of possible point group irreducible representations

Cyclic groups have Schoenflies symbols C_n, S_n, C_nh, C_s and C_i positioned together in rows of the Laue class table above. These point groups are represented by the one dimensional symbols A, B, E_+x and E_−x. If the group has a centre of symmetry the number if irreps doubles and subscripts g and u have to be added to each of the simple cyclic symbols. For example, the eight possible irreps of C_4h are A_g, B_g, E_+1g, E_−1g, A_u, B_u, E_+1u and E_−1u. Cotton (page 96) shows how cyclic group irreps are derived for groups of any order.

Dihedral groups have Schoenflies symbols D_n, C_nv, D_nd and D_nh also positioned together in rows of the Laue class table. They are represented by one and two dimensional irreps derived from the cyclic irreps above. Cyclic A and B irreps split into separate one dimensional irreps A₁, A₂, B₁ and B₂ while pairs of E_+x and E_−x irreps merge to form single two dimensional degenerate irreps. For example, the possible irreps of D₄ (also C_4v and D_2d) are A₁, A₂, B₁, B₂ and E. (E₁ is normally shown simply as E.) A group with a centre of symmetry also has irreps with g and u subscripts. Once the irreps of a cyclic group C_n are known it is a simple task to deduce the irreps of the corresponding dihedral group D_n.

In addition to the one and two dimensional irreps so far described, tetrahedral groups can a three dimensional degenerate irrep T that expands in octahedral groups to yield irreps T₁ and T₂.

The following reference of character tables uses symbols Z_x for abstract cyclic groups C_x with A₄ and S₄ (alternating and symmetric permutations of 4 objects) for T and O. Many authors just use C, T and O in two senses, making it clear which is intended.

Historical background

The article above provides an insight into the development of symmetry theory in the context of crystallographic investigations during the 19th century. Point groups were derived from observations of the macroscopic forms of crystals, leading up to the Schoenflies system used to describe them. Max von Laue's invention and use of X-ray diffraction to elucidate the internal structure provided an insight not possible from an examination of external crystal shapes. Laue showed that the 32 crystallographic point groups can be collected into 11 classes each containing 2, 3 or 4 point groups that all produce the same diffraction pattern. Crystallographic Laue classes are now very familiar and, although usually shown in international notation, would probably originally have been published in Schoenflies notation. (International notation was developed 10 years after Laue's work). Each class contains one rotational group, one non-rotational group that is the direct product of the rotational group with space inversion and 0, 1 or 2 more groups obtained from combinations of rotational group operations with space inversion. Of course, the crystallographic restriction applies only to crystals and in the wider field of molecular symmetry an infinite number of Laue classes become possible.

While crystallographic research continued apace in the first part of the 20th century there were also major discoveries at the atomic level. Rutherford showed that a single atom consists of a massive nucleus surrounded mainly by empty space containing electrons, leading directly to the planetary Bohr model of the atom. Planck discovered that black body radiation could be explained if he allocated particles to discrete energy levels then Einstein showed that the energy levels were characteristic of particles themselves. De Broglie extended this reasoning to all particles, providing a relationship between the allowed energy levels and the wave nature of the particle. All of this work on the combination of Newtonian particle particle behaviour combined with the wave nature of the particles led up to Schrodinger's three dimensional wave equation, perhaps the most fundamental mathematical expression in modern chemistry. Solutions to this equation are defined by 3 quantum numbers that label the allowed energy levels when it is applied to particles, The equation might be applied to any quantized state.

A particle in energy level E₁ might be stimulated to rise to level E₂ by the absorption of a photon with energy ε provided that

E₂ − E₁ = ε

or a particle might fall from the higher energy level to the lower one and emit a photon with the same energy. Like all particles the photon has an energy related to its frequency ε = hν where h is the Planck constant (6.626×10⁻³⁴ J Hz⁻¹) and ν is the frequency. Higher frequencies have greater energies. This is the basis of "spectroscopy", a procedure in which the energy levels in atoms and molecules probed by photons of varying frequency until the frequency matches an energy level difference. These techniques may be applied in the following three areas

• electronic transitions in atoms and molecules that have frequencies in the range 430 to 770 THz
• vibrational transitions in molecules with typical frequencies 10 to 100 THz
• rotational transitions of molecules with common frequencies in the range 8 to 10 GHz

Clearly, these frequencies are in the order electronic > vibrational > rotational so the energy differences also change in this order. Suppose an atom or molecule emits or absorbs at 500 THz, a quick calculation shows the energy involved to be

ε = hν = 6.626×10⁻³⁴ × 500×10¹² = 3.313×10⁻¹⁹ J

and it becomes clear that at the atomic level levels involved here are incredibly small in relation to day to day experience.

Soon after the Schrodinger equation became familiar to early researchers Hans Bethe showed how atomic orbitals could be modified by symmetric "crystal fields" resulting from surrounding charges and this work was extended to his study of ligand field theory in 1929. Eugene Wigner used group theory to explain the selection rules of atomic spectroscopy.

E. Bright Wilson used character tables in 1934 to predict the symmetry of vibrational normal modes.

Electronic and vibrational spectra provide enormous detail about molecular structures and, although rotational spectra cannot be linked to individual point groups, it does often supply useful information.

Symmetry of molecular orbitals

When Schrodinger's 3D wave equation is applied to a one-electron atom it provides a number of solutions called wave functions that are then used to label the allowed energy levels in that atom. Exact solutions of this kind are usually described by three quantum numbers, n, l and m from which the probable radial and angular distribution of the electron around the atom can be computed. This type of deduction leads to the familiar s, p, d, f,… description of atomic orbitals based on the l and m quantum numbers. Each solution is a base vector from which more complex structures may be constructed. Descriptions of many-electron atoms use the one-electron model to build models that are sometimes pictured as multiple electrons in the simple structure. Molecular orbitals then take linear combinations of atomic orbitals (LCAOs) to explain the distribution of electrons over multiple atoms within a molecule. Atomic orbital symmetry follows from the angular part of the wave function which increases in complexity in the series s, p, d, f,… so that s orbitals only have radial symmetry while p orbital base vectors have a symmetry identical to that of the Cartesian polar base vectors.

Consider the example of water (H₂O), which has the C_2v symmetry described above. The 2p_x orbital of oxygen has, like the × base vector, B₁ symmetry. It 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. Likewise, the 2p_z orbital is seen to have the symmetry of the A₁ irreducible representation (i.e., none of the symmetry operations change it), 2p_y B₂, and the 3d_xy orbital A₂. These assignments are noted in the rightmost columns of the table.

Each molecular orbital also has the symmetry of one irreducible representation. For example, ethylene (C₂H₄) has symmetry group D_2h, and its highest occupied molecular orbital (HOMO) is the bonding π orbital which forms a basis for its irreducible representation B_1u.

Symmetry of vibrational modes

Electronic bonds between atoms in molecules can be imagined to be equivalent to springs that extend, contract and bend, absorbing energies appropriate for the strength of the bond. Each of the N atoms in a molecule can move to a limited extent in three dimensions so that a total of 3N distinct motions become possible but 3 of these correspond to the overall translational motion of the molecule and 3 to overall rotation. It follows that there are 3N − 6 genuine motions relative to the whole semi-rigid molecule (or 3N − 5 in linear molecules). Vibrational motions of this kind may be resolved into normal modes of molecular vibration in which each mode has a symmetry which forms a basis for one irreducible representation of the molecular symmetry group. This relationship is the basis of vibrational spectroscopy. Irreps for the normal modes of a molecule are derived through a fairly mechanical procedure

1. find the point group of the molecule
2. deduce the irreps for that molecule
3. look up the irreps corresponding to x, y and z base vectors for an atom in the point group
4. take the direct product of the molecular irreps and the three base vector irreps
5. subtract the six irreps that represent overall rotations and translations

Vibrational modes for a water molecule

A water molecule (H₂O) has three atoms and so has 3 × 3 − 6 = 3 normal modes of vibration. The molecular symmetry of water is C_2v and the molecular irreps 2A₁ + B₂. A glance at the irrep/character table shows that the x, y and z base vectors for C_2v are A₁, B₁ and B₂. The direct product is then

(2A₁ + B₂) × (A₁ + B₁ + B₂) = 2(A₁ + B₁+ B₂) + (B₂ + A₂ + A₁) = 3A₁ + A₂ + 2B₁ + 3B₂

There are three atoms in the molecule so the resulting representation has 3N = 3 × 3 = 9 dimensions. Subtracting the three polar base vectors A₁, B₁ and B₂ and the axial base vectors B₂, B₁ and A₁ a final result is obtained

(3A₁ + A₂ + 2B₁ + 3B₂) − ((A₁, B₁, B₂) + (B₂, B₁, A₁)) = 2A₁ + B₂

The overall symmetry of the three vibrational modes is therefore 2A₁ + B₂.

The three modes represents a symmetric stretch in which the two O–H bond lengths vary in phase with each other, an asymmetric stretch in which they vary out of phase, and a bending mode in which the bond angle varies. Symmetric stretching and the bending modes have symmetry A₁, while the asymmetric mode has symmetry B₂.

Diborane (B₂H₆) has D_2h molecular symmetry. The terminal B–H stretching vibrations which are active in IR are B_2u and B_3u.

Vibrational modes for an ammonia molecule

An ammonia molecule (NH₃) has 4 atoms and therefore 3N − 6 = 6 normal modes of vibration. It has a trigonal pyramidal shape with 3 hydrogen atoms equidistant from a nitrogen atom and belongs to point group C_3v, with symmetry operations E, C₃ and σ_v. Both of the examples above are asymmetric molecules in which only 1-dimensional irreps are possible and in which each irrep corresponds to a single vibrational mode related to a single frequency. Ammonia is a symmetric molecule and therefore its vibrational modes may include 2-dimensional irreps representing two modes of the same frequency (degenerate modes). Symmetry group C_3v for has the three symmetry species A₁, A₂ and E₁ although the last of these is usually written simply as E.

A brief glance at the molecule shows that the nitrogen atom contributes irrep A₁ while the three hydrogen atoms contribute A₁ + E so the molecular irreps are 2A₁ + E. The translational modes for this point group are A₁ + E and so the direct product of these three vectors with the molecular irreps produces a 12 dimensional representation

(2A₁ + E) × (A₁ + E) = 3A₁ + A₂ + 4E

Now the 3 translational modes A₁ + E and 3 rotational modes A₂ + E have to be subtracted from this representation

(3A₁ + A₂ + 4E) − (A₂ + E) − (A₁ + E) = 2A₁ + 2E

All three hydrogen atoms travel symmetrically along the N–H bonds, either in the direction of the nitrogen atom or away from it. This mode is known as symmetric stretch (ν₁) and reflects the symmetry in the N–H bond stretching. Of the three vibrational modes, this one has the highest frequency.

In the bending (ν₂) vibration, the nitrogen atom stays on the axis of symmetry, while the three hydrogen atoms move in different directions from one another, leading to changes in the bond angles. The hydrogen atoms move like an umbrella, so this mode is often referred to as the "umbrella mode".

There is also an asymmetric stretch mode (ν₃) in which one hydrogen atom approaches the nitrogen atom while the other two hydrogens move away.

Vibrational modes for spherical molecules

Molecules such as methane and carbon tetrachloride with a general formula AB₄ have a very obvious tetrahedral appearance and are usually described as such even though they belong to the octahedral Laue class. They have five atoms: a central atom A surrounded by four equivalent B atoms so there might be expected to be (5 × 3) − 6 = 9 vibrational modes. Molecular irreps for this shape were earlier found to be 2A₁ + T₂ and a direct product of this five dimensional expression with the irrep representing x, y and z base vector transformations (T₂) produces a 15-dimensional result

(2A₁ + T₂) × T₂ = (2T₂ + (A₁ + E + T₁ + T₂)) = A₁ + E + T₁ + 3T₂

A subtraction of the polar and axial base vectors (T₂ and T₁) produces a final 9-dimensional result

A₁ + E + T₁ + 3T₂ − (T₁ + T₂) = A₁ + E + 2T₂

All three of these modes are Raman active but only T₂ is IR active.

While there are simple rules to deduce the results of direct products for asymmetric and symmetric groups the results for spherical molecules are best found from tables. For example the product above used the identity T₂ × T₂ = A₁ + E + T₁ + T₂. This is equally applicable to other octahedral groups because O and T_d are distinct examples of the same abstract group.

W(CO)₆ has octahedral geometry. The irreducible representations contributing to the six dimensional representation for the C–O stretching vibration are A_1g + E_g + T_1u. Of these, only T_1u is IR active.

Symmetry and molecular rotation

Molecular rotation is quantized but the relationship between molecular symmetry and allowed energy levels is not as detailed as that for electronic and vibrational motions. Rotational transitions depend only on the kinetic energy of the rotating molecule while electronic and vibrational levels involve both kinetic and potential energies. In spite of this difference the general procedure for relating molecular structure to observed spectra is similar to that of the earlier systems. First of all an expression is derived for the energy of the rotating molecule, taking into account the Laue partition to which it belongs because rotational motion is not related to individual point groups or even to their class. Asymmetric molecules have three different moments of inertia from which three different angular momenta can be derived: L_x, L_y and L_z. The fact that there are three distinct values of angular momentum means that solutions to the Schrodinger for this kind of molecule are exceedingly difficult. This does not stop asymmetric molecules exhibiting clear rotational spectra: water vapour provides clear transition lines.

Symmetric molecules have two equal moments of inertia and so two equal angular momenta, making their energy expressions more easily soluble. Oblate symmetric molecules have a discus shape (e.g. benzene) and prolate examples are cigar shaped (e.g. methyl chloride). Prolate symmetric molecules become linear when the main axial order reaches infinity at which point there is only one moment of inertia and one angular momentum. This makes the Schrodinger equation easily soluble and it is possible to relate (say) diatomic molecule structure to their structure very accurately. Spherical molecules also only have one moment of inertia so their energy levels are easily calculated but not always observed.

Point group symmetry describes the symmetry of a molecule when fixed at its equilibrium configuration in a particular electronic state. It does not allow for tunneling between minima nor for the change in shape that can come about from the centrifugal distortion effects of molecular rotation.

The molecular symmetry group

One can determine the symmetry operations of the point group for a particular molecule by considering the geometrical symmetry of its molecular model. However, when one uses a point group to classify molecular states, the operations in it are not to be interpreted in the same way. Instead the operations are interpreted as rotating and/or reflecting the vibronic (vibration-electronic) coordinates and these operations commute with the vibronic Hamiltonian. They are "symmetry operations" for that vibronic Hamiltonian. The point group is used to classify by symmetry the vibronic eigenstates of a rigid molecule. The symmetry classification of the rotational levels, the eigenstates of the full (rotation-vibration-electronic) Hamiltonian, can be achieved through the use of the appropriate permutation-inversion group (called the molecular symmetry group), as introduced by Longuet-Higgins.

Molecular nonrigidity

As discussed above in § The molecular symmetry group, point groups are useful for classifying the vibrational and electronic states of rigid molecules (sometimes called semi-rigid molecules) which undergo only small oscillations about a single equilibrium geometry. Longuet-Higgins introduced the molecular symmetry group (a more general type of symmetry group) suitable not only for classifying the vibrational and electronic states of rigid molecules but also for classifying their rotational and nuclear spin states. Further, such groups can be used to classify the states of non-rigid (or fluxional) molecules that tunnel between equivalent geometries and to allow for the distorting effects of molecular rotation. The symmetry operations in the molecular symmetry group are so-called 'feasible' permutations of identical nuclei, or inversion with respect to the center of mass (the parity operation), or a combination of the two, so that the group is sometimes called a "permutation-inversion group".

Examples of molecular nonrigidity abound. For example, ethane (C₂H₆) has three equivalent staggered conformations. Tunneling between the conformations occurs at ordinary temperatures by internal rotation of one methyl group relative to the other. This is not a rotation of the entire molecule about the C₃ axis, although each conformation has D_3d symmetry, as in the table above. The molecule 2-butyne (dimethylacetylene) has the same molecular symmetry group (G₃₆) as ethane but a very much lower torsional barrier. Similarly, ammonia (NH₃) has two equivalent pyramidal (C_3v) conformations which are interconverted by the process known as nitrogen inversion.

Additionally, the methane molecule (CH₄) and trihydrogen cation (H+3) have highly symmetric equilibrium structures with T_d and D_3h point group symmetries respectively; they lack permanent electric dipole moments but they do have very weak pure rotation spectra because of rotational centrifugal distortion.

Sometimes it is necessary to consider together electronic states having different point group symmetries at equilibrium. For example, in its ground (N) electronic state the ethylene molecule C₂H₄ has D_2h point group symmetry whereas in the excited (V) state it has D_2d symmetry. To treat these two states together it is necessary to allow torsion and to use the double group of the molecular symmetry group G₁₆.