7. Dry Lab III: Molecular Symmetry

Transcription

1 0 7. Dry Lab III: Moecuar Symmetry Topics: 1. Motivation. Symmetry Eements and Operations. Symmetry Groups 4. Physica Impications of Symmetry 1. Motivation Finite symmetries are usefu in the study of moecues. They are used in the cassification of moecues, simpifying quantum mechanica cacuations on moecues, determining the presence of certain moecuar properties such as moecuar poarity and chiraity. In the next two dry abs, we wi focus on finite symmetries and groups of finite symmetry operators. We wi use P.W. tkins, Physica hemistry, hapter 15, as our principe reference. nother book by F.. otton, entited hemica ppications of Group Theory, either the first or second edition, is an exceent source book too. In hemistry 1/, you have had some exposure to finding symmetry eements and symmetry operations. This Dry Lab is meant to be a either a practica review or an introduction to symmetry eements and symmetry operations. The set of a symmetry operations for a moecue form a mathematica structure caed a group. ere, we wi ook at group structure, casses of symmetry operations, and the naming of moecuar point groups. We wi examine how to use the group structure to predict when a moecue is poar or chira. You wi appy these ideas to severa moecues or moecuar ions.. Symmetry eements and operations symmetry operation wi transform a moecue into itsef so that the transformed moecue wi be indistinguishabe from the origina structure. so, at east one point in the moecue wi be eft undisturbed by the transformation. ence the origin of the name moecuar point group. Often, two or more atoms are permuted during the course of the moecuar transformation. Since atoms of the same type are indistinguishabe, the transformed moecue is indistinguishabe from the starting moecue. So, a moecuar point group consists of a those symmetry operations that eave a point in the moecue invariant and permutes identica atoms. 1 Symmetry operations come in severa favours: inversion, refections, rotations, and improper rotations, and of course no operation at a. The ast operation is caed the identity operation and is present in a moecues. It is normay denoted by the symbo E. For each symmetry operation there corresponds a symmetry eement. More than one operation may correspond to the same symmetry eement rotation operation takes pace about a rotation axis. rotation takes pace through an ange of π/n. (y convention, a rotation by a positive ange is considered to be countercockwise, a rotation through a negative ange woud then be cockwise.) Such a rotation is said to be an n-fod rotation. The rotation is about some axis in the moecue caed an n-fod rotation axis. If a moecue has more than one rotation axis, the one with the highest vaue of n is caed the principa axis of rotation. The n- fod rotation operation is denoted by n (pain text) and the axis of rotation is identified by the symbo n (itaics). For exampe, the water moecue has one principe axis of rotation, a - fod axis,, through the O atom and bisecting the O ange. This a rotation by 180, transforming the moecue into itsef. In contrast, the rf 5 moecue has a 4-fod axis, 4, of rotation containing the axia rf bond. Notice that a 4-fod axis aways contains a -fod axis. For exampe, the square panar structure XeF 4 has a 4 axis through the Xe atom and orthogona to the pane of the moecue. arrying out two consecutive 4-fod rotations, i.e. 4 = 4 4, about a 4 axis is equivaent to conducting a rotation about the same axis. Therefore, the axis is coincident with the 4 axis. In O, the axis is the principa axis of rotation. In XeF 4, the 4 axis is the principa axis of rotation. For a diatomic moecue, a rotation by any arbitrary ange can be performed about the internucear axis. Such an axis is caed a axis and is the principa axis of rotation. 1 Note that not a identica atoms need be permuted amongst each other. We sha see an exampe of this ater. The principa axis of rotation need not be unique. Dry Lab III: Moecuar Symmetry hemistry 7

2 1 refection takes pace in a pane of symmetry, sometimes caed a mirror pane. The refection operation is denoted by σ (pain) and the mirror pane are both denoted by σ (itaics). Normay, refection panes contain rotation axes or are orthogona to an axis of rotation. If a pane is orthogona to a principa axis of rotation it is designated by the symbo σ h. If a pane contains a principa axis of rotation it is usuay denoted by σ v. If the pane bisects the ange between two axes it is caed a σ d and such a pane is caed a dihedra pane of symmetry. Often, other criteria are needed to distinguish between σ v panes and σ d panes. Water has two refection panes, both of them σ v panes. In XeF 4, the refection pane orthogona to the 4 axis (the principa axis) is a σ h pane. n inversion operation, I (sometimes written as i), through the centre of inversion (I or i) takes the point (x,y,z) in the moecue and transforms it to (-x,-y,-z). The chemica environment at the points (x,y,z) and (x,-y,-z) are identica. Water does not possess an inversion centre whereas XeF 4 does. n n-fod improper rotation, S n, about an n- fod improper rotation axis (same symbo) is composed of two successive transformations: The first component is an n-fod rotation about S n foowed by the second component, a refection in a pane orthogona to the S n axis. Note that the n-fod rotation need not correspond to an actua n-fod rotation axis in the moecue. Simiary, the refection pane orthogona to the S n axis need not be an actua refection pane. The water moecue does not possess an improper axis of rotation. 4 moecue has three S 4 axes but no 4 axis.. Symmetry Groups We use the phrases symmetry groups and moecuar point groups synonymousy. mathematica group, G = {G, }, consists of a set of eements 4 G = {E,,,,D,...} and a binary reation, caed group mutipication or group product or simpy mutipication or product, denoted by, which satisfies the foowing properties: (a) The product of any two eements and in the group is another eement in the group, i.e., we write G. (b) If,, are any three eements in the group then ( ) = ( ). Therefore, group mutipication is associative, and frequenty, we omit the brackets. (c) There is a unique eement E in G such that E = E=, for every eement in G. The eement E is caed the identity eement. (d) For every eement in G, there is a unique eement X in G, such that X = X = E. The eement X is referred to as the inverse of and is denoted -1. The identity is its own inverse. The number of eements in a group is caed the order of the group. Frequenty, it is denoted by the symbo h. Frequenty, when no confusion can arise the symbo for the product is omitted. so, when there can be no confusion, we wi use the symbo G for the group rather than G. If we think of the group eements as symmetry operations of a moecue and if by we mean first we perform the symmetry operation on the moecue foowed by symmetry operation. The net resut of such consecutive action on a moecue is another symmetry operation. Take dichoromethane as our exampe. The moecuar structure and artesian axis system are shown beow. x z y Frequenty, the σ h pane is caed a horizonta pane and the σ v pane is referred to as a vertica refection pane. There is some danger in doing this since not a σ h panes need be horizonta and not a σ v panes need be vertica. 4 Do not confuse symmetry eements discussed in section with group eements. The set of symmetry eements do not form a group, ony the symmetry operations form a group. The term eement used in this definition is standard usage in set theory. hemistry 7 Dry Lab III: Moecuar Symmetry

3 Think of the artesian axes as being centred at the atom with the z-axis bisecting the and the anges. The x-axis ies in the pane, whie the y-axis is in the pane. The identity operation, E, eaves the moecue unchanged. The axis ies aong the z-axis. The operation transforms the dichoromethane moecue as arrying out two consecutive operations is equivaent to the identity transformation. There are two refection panes in the moecue; both contain the rotation axes. One pane is the pane of the page; containing the pane. We denote this pane by σ(yz). The second pane is perpendicuar to the pane of the page; denote it by σ(xz). The action of σ(yz) is to give the arrangement of atoms shown beow, where the two hydrogen atoms have been interchanged, whie the two chorine atoms and carbon are σ(yz) unchanged. The σ(xz) permutes the chorine atoms but eaves carbon and the two hydrogen atoms fixed. This is represented in the foowing diagram σ(xz) ppying the pane σ(yz) twice, i.e., (σ(yz)) = σ(yz)σ(yz) = E, we get the identity. This means that σ(yz) is its own inverse. Simiary, we find that (σ(xz)) = σ(xz)σ(xz) = E, and σ(xz) is its own inverse. Now, if we carry out a σ(xz) refection first and foow it by a σ(yz) refection, we get σ(yz)σ(xz) omparing this diagram to that of a rotation we see that the resut is identica. Therefore, we say that σ(yz)σ(xz) = (1) You can show that performing the refections in reverse order yieds the same resut. Note that the symmetry eements remain fixed and are not transformed to new positions when the atoms in the moecue are moved to new positions. What about carrying out a rotation foowed by the refection σ(yz)? Performing these symmetry operations yieds σ(yz) which is equivaent to a σ(xz) operation. Show that carrying out these operations in reverse order affords the same resut. Next, we compute the product σ(xz) : σ(xz) and this is identica with a σ(yz) operation. gain check that the reverse sequence of operations yieds the same resut. Using the definition of the group and the products of symmetry operations that we have just uncovered, we can construct a group mutipication tabe: v E σ(xz) σ(yz) E E σ(xz) σ(yz) E σ(yz) σ(xz) σ(xz) σ(xz) σ(yz) E σ(yz) σ(yz) σ(xz) E This tabe contains a the information about the group and its structure. The name of this moecuar point group is v. There are some observations to make about this tabe. (1) Notice the inner four-by-four box. In each row and each coumn, each operation appears once and ony once. In other words, each row and each coumn is a permutation of the others. This is a feature possessed by a group mutipication tabes. () We can identify smaer groups within the arger one. For exampe, {E, } is a group. There are two others; what are they? These smaer groups are caed subgroups of v. Dry Lab III: Moecuar Symmetry hemistry 7

4 () In this particuar tabe, we observe that the group product is commutative. This is not necessariy true for other groups. In the ammonia moecue, N the nitrogen atom is the fixed point. The moecue has a axis of rotation. Note that the both the and operations occur about the same axis. There are three refection panes, each pane containing an N bond and bisecting the opposing N ange. Denote each pane by the number on the hydrogen atom it contains; thus σ 1 is the pane containing 1. This refection interchanges atoms and, eaving fixed N σ 1 N 1 1 Note that the numbers remain fixed to their origina positions. The symmetry eements must not shift with the atoms when they are transformed to new positions. In the previous exampe, we used abes for the refection panes that were expressed in terms of the fixed axes externa to the moecue. In this exampe, it is not obvious how to do that. So keep in mind here that the numbers stay fixed and the etters move. You wi want to use the doube abes whenever it is inconvenient to abe axes with references outside the moecue. It is irreevant whether the numbers stay fixed and the etters move or vice versa. Just be consistent within a given appication. Since σ 1 = σ = σ = E, each refection is its own inverse. Since = = = E, is the inverse of a rotation (or is the inverse of ). Reca that a rotation is a 10 rotation in an countercockwise direction about the rotation axis, whie is a 40 countercockwise rotation. so, we can interpret a rotation as a -10 rotation (cockwise). Given these definitions and considerations, here is the compete group mutipication tabe v E σ 1 σ σ E E σ 1 σ σ E σ σ 1 σ E σ σ σ 1 σ 1 σ 1 σ σ E σ σ σ σ 1 E σ σ σ 1 σ E Exercise 1: Prove the tabe for v using N as the sampe moecue. (If you use another moecue, the tabe wi basicay be the same except some rows and coumns may be interchanged.) You wi not have to prove a the products. Use the fact that each row (coumn) is a permutation of another row (coumn) and that no eement can occur more than once in a row or coumn to assist you. Observe that the group is not commutative. Exercise : The point group of the rf 5 moecue is 4v. There are eight symmetry operations in the group. What are they? onstruct the group mutipication tabe. You may use the textbook to assist you. It is not necessary to derive a group mutipication tabe each time you want to find the point group of a moecue or moecuar ion. It is sufficient to determine the presence of ony a particuar subset of symmetry eements. Once you have discovered the moecuar point group, the character tabe for the point group wi give you the remaining group operations. In tkins, hapter 15, page 4, foow the Exampe 15.1 and refer to fow chart in Figure s another exampe, consider ammonia. Is the moecue inear? The answer is no. It has ony one axis of rotation, a axis, which is the principa axis of rotation. re there axes orthogona to the principa axis? The answer is no. Is there a refection pane orthogona to the principa axis? gain, the answer is no. re there three σ v panes containing the axis and the answer is yes. Therefore, N must have v symmetry. hemistry 7 Dry Lab III: Moecuar Symmetry

5 4 Exercise : Determine the moecuar point groups for the foowing species: (a) SO, (b) O, (c) =, (d) XeF 4, (e) XeO F, (f) F 4, (g) Ni(O) 4, (h) Fe (O) 9, (i) cis-1,-dicoroethene, (j) trans-1,-dichoroethene, (k) cis-o(n ) 4, () trans-o(n ) 4 (m) S 4 N 4, (n) [o(n) 6 ] + When you examine group character tabes, you wi notice that some of the eements are grouped together. These tabes are given on page 951 in tkins. For exampe, in the character tabe for v, we see that the eements are umped together as {E,,σ v }. That is because the two operations and beong to the same cass. Simiary, the three refections beong to the same cass. In a group G={E,,,,...}, we say that two eements and are conjugate to each other if -1 =, for some eement in G. n eement and a its conjugates form a cass. So, for instance, in v, if =, we have E (E) -1 = E E =, since E is its own inverse, ( ) -1 = =, every eement is conjugate to itsef, ( ) ( ) -1 = ( ) =, and the remaining three can be read from the group mutipication tabe: σ 1 (σ 1 ) -1 = σ 1 σ 1 = σ σ 1 =, σ (σ ) -1 = σ σ = σ σ =, σ (σ ) -1 = σ σ = σ 1 σ =, This means that ony and are conjugate to each other and beong to the same cass. For the refections, we have Eσ 1 (E) -1 = E σ 1 E = σ 1, σ 1 ( ) -1 = σ 1 = σ, σ 1 ( ) -1 = σ 1 = σ, σ 1 σ 1 (σ 1 ) -1 = σ 1 σ 1 σ 1 = Eσ 1 = σ 1, σ σ 1 (σ ) -1 = σ σ 1 σ = σ = σ 1, σ σ 1 (σ ) -1 = σ σ 1 σ = σ = σ 1. Therefore, the three refection operations beong in the same cass. Exercise 4: From the group mutipication tabe that you have worked out for the 4v group in Exercise, determine a the casses. You may confirm your answers by ooking in the group character tabes on page 951 in tkins. simpify cacuations of certain matrix eements or expectation vaues. Often, it is possibe to decide when matrix eements or expectation vaues are zero just on the basis of symmetry aone. In vibrationa spectroscopy, frequenty, the vibrationa modes of a moecue are cassified according to their behaviour under the symmetry operations of the moecuar point group. We sha expore some of these ideas in Dry Lab IV. This week we sha examine the connection between symmetry and moecuar poarity and moecuar chiraity. Poarity: The idea here is very simpe. poar moecue has a permanent eectric dipoe moment, v µ, which is a vector quantity. The dipoe moment has a specific orientation in the moecue. If a moecue has a rotation axis, then the dipoe moment must ie aong the rotation axis, since no dipoe moment can change under a rotation. If a moecue possesses noncoincidenta rotation axes, then it can have no dipoe moment since v µ woud have to ie aong a such rotation axes. This is physicay impossibe. Thus, for such moecues v µ = 0. If a moecue has a centre of inversion, then it can have no permanent dipoe moment. dipoe moment woud change sign under inversion and that is not possibe physicay. nother way of ooking at this is that the dipoe moment woud have to be a point, and that is not possibe for a vector quantity. Exercise 5: Which of the foowing moecues in Exercise woud have permanent eectric dipoe moments: (a), (b), (i), (j), (k), (), (c), (f)? hiraity: If a moecue is chira, it is opticay active, i.e., it exists in enantiomeric forms. For a moecue to exhibit chiraity it must have no improper axes of rotation. Note that S = i and S 1 = σ, a refection. So if a moecue possesses refection panes or an inversion centre, it cannot be chira. Exercise 6: Which of the foowing species is chira? (a) [o(en) ] +, (b) cis-[o(en) ] +, (c) trans-[o(en) ] + 4. Physica Impications of Symmetry Symmetry is used in a wide variety of ways in chemistry. In quantum chemistry, symmetry is used to cassify moecuar orbitas and state wave functions. With this cassification, we can Dry Lab III: Moecuar Symmetry hemistry 7