J. Edmund White' Southern Illinois University Edwardsville, 62025
An Introduction to Group Theory for Chemists
Developments over the last ten to fifteen years, especially in the areas dealing with transition metal complexes, have made clear the importance of symmetry considerations in chemistry. Since group theory is the most powerful method for handling symmetry relationships, a working knowledge of group theory is essential for most inorganic chemists, and some understanding is desirable for all chemists. Symmetry and group theory may be applied in solving secular equations in molecular orbital calculations, in determining the proper atomic orbitals to combine into hybrid orbitals, in predicting infrared spectra from modes of vibration and obtaining wave functions for the normal modes, and in x-ray crystallography. Most of the recent books on molecular structure, molecular spectroscopy, and quantum chemistry contain some introduction to symmetry and group theory, perhaps as an appendix or perhaps as a short section in a chapter. Some of these are mentioned later in the section, "Suggestions for Further Reading." Although these books probably are available in most scientific libraries, their specialized nature may not suggest to teachers that they do contain a presentation of group theory. The main purpose here is to present the fundamentals and to define clearly the basic terms of group theory in the hope that the reader will be prepared to get the maximum understanding when he proceeds to the more advanced articles or books. Even if he goes no further into group theory than this introduction, he should recognize in his other chemical reading when it is being used and that terms such as "tz, orbital" come directly from group theory terminology. Another purpose is to describe a method of presenting group theory to stndents for the first time. The meaning of “trio orbitals" and "e, orbitals" might be considered a secondary theme of this article. (Sometimes the symbols d, and d. are used) ( I ) . Quite often in writing not intended for the specialist in inorganic chemistry, these odd-looking terms are used without identification of their origin, much less of their true meaning. They are presented as convenient symbols for symmetry species, molecular wave functions, or types of orbitals. The reasons for the choice of these particular symbols will appear later. The author wishes to express his appreciation to the Donors of the Petroleum Research Fund for a Faculty Award for Advanced Scientific Study and to the Massachusetts Institute of Technology and Professor F. Albert Cotton far an appointment as Guest of the Institute for the year 196344. Present address through July, 1967: Department of Chemistry, University College London, Gower Street, London WC1, England.
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It should be acknowledged that group theory is not essential for most of the situations where it is usefill. I n most cases in quantum mechanics, for example, "the use of group theory could be circumvented by detailed algebraic considerations" (g). Recent articles on crystal field theory and splitting in THIS JOURNAL patterns (3, 4) and the molecular orbital and ligand field theories (5-7) have illustrated the importance of symmetry in chemistry but have not really used group theory. Manch and Fernelius (3) briefly describe the use of a character table to determine the symmetry species of the d orbitals, whereas Cotton (7) uses pictorial arguments for the same purpose in order to show that group theory is not essential. He does give a short explanation of the &, and e, symbols. The main purpose of Companion and Komarynsky (4) is to present an approach alternative to group theory for obtaining splitting diagrams. A recent article by Kettle (8) describes a specific application of group theory and symmetry in finding the correct linear combination of ligand orbitals for octahedral complexes and does use and describe briefly several of the terms which are elaborated in the following. Those articles and books to which the chemist or chemistry teacher is likely to go for an introduction to group theory usually present it in conjunction with symmetry concepts. This intimate mixing may leave the reader with the impression that symmetry and group theory are more or less the same thing, that symmetry operations are part of the terminology of group theory. This is an erroneous impression; group theory was a respectable area of mathematics long before Bethe (9) used it to develop crystal field theory. The approach used here avoids this confusion by introducing the basic concepts of group theory with a nonscientific example. The particular example used has two features which recommend it. I t requires no props or models other than the body of the instructor (or reader) and no previous discussion of symmetry and symmetry elements, which is the starting place in many of the brief introductions mentioned above. The use of rotation about an axis which is not an axis of symmetry provides a natural stepping-stone to an explanation of what is required to make it an axis of symmetry; thus one introduces symmetry elements and operations after the mathematical techniques for dealing with them are developed. I n six or eight years, such an elementary introduction should not be necessary for undergraduate students. The new approaches to mathematics in the elementary grades teach the concept of the group, perhaps called a "closed set"; thus the chemistry professor of the
future should be able to launch into applications of group theory with legitimate chemical illustrations. The Group
Comparisons of the positions of a soldier which result when he executes certain combinations of the facing movements willlead us to the identifying features, or defining properties, of the mathematical group. This illustration was previously used in briefer form by Higman (10) but here is expanded and rephrased in the United States military terminology rather than the British. Those readers who have not served in the Armed Forces should not have difficulty visualizing the operations: right face (90' clockwise), left face (90" counterclockwise), and about face (180" clockwise), designated R, L, and A respectively, and a complete rotation of 360'. The veterans, however, will pause before trying to carry out an order to rotate 360". One's feet would get rather tangled.= During that pause, the veteran should realize that such a rotation would return him to his starting position and that he can achieve the desired result simply by standing still. The operation of "doing nothing" or rotating 360°,720°, . . . is designated E. I n the following, after each military situation has been presented, it will be restated in group theoretical terminology to make specific the generalization being illustrated. Finally these properties will be listed in a compact formal definition of a group. ( I ) Right face followed by about face. The same final position could have been achieved simply by ordering left face; in other words R combined with A equals L. Thus the performance in succession of two operations gives the same result as the performance of a single third operation. An operation may combine with itself, following the same rule: R combined with R = A, which means that two successive right faces bring the soldier to the same position as if he had executed one about face. I n general, the combination of two operations must be a third operation which is a member of the group. The words "combined with" are written out here to emphasize that the combination process may be different for different groups, perhaps addition, multiplication, or simply "followed by" as in this case. The convention is to use the terminology of multiplication, i.e., one speaks of "multiplying" two "elements" of a group and obtaining a "product." The convention for writing down the equation for a comhination process will seem backward: the operation to be performed first is placed to the right. Thus "right face combined with about face" is written AR, and LAR means "first do R, then A, and finally L." (This is no more "backward" than the mathematical operation imtruction: b2/dx~.) (2) Right face. About face. At ease. Attention. Left face. Compare this to Right face. At ease. Attention. About face. Left face. The fact that the final positions are the same may be represented symbolically by L(AR) = (LA)R. I n short, the associe tive law holds. "he teacher is advised to practice the facing movements hefore attempting them before a class! The safest approach may he to let an R.O.T.C. student demonstrate.
This seems an appropriate place to emphasize that the commutative law does not necessarily hold. I n the present simple example it does; but, in a highly symmetrical molecule or complex ion, there will be symmetry operations for which X Y does not equal YX. (3) Right face. About face. Right face. This s e quence returns the soldier to his starting position. It is as if he had done nothing or had been left alone: RAR = E. We can see now why the operation E (doing nothing) was introduced above and why i t is necessary in every group. Situation (3) is a case where the product of two (or more) elements amounts to no change, but situation (1) requires that such a product be an element of the group. Thus we are forced to accept "doing nothing" as an element. The operation E is called the "identity operation" and is the "identity element," or "unit element," of the group. Another aspect of E is shown in the next military situation. It should be clear that the product of the identity element with a second element is merely the second element: ER = R; AE = A = EA; etc. (4) Right face. Left face. This sequence also returns the man to his starting position: LR = E. When one operation cancels the effect of the other, one is called the "reciprocal" or "inverse" of the other. The inverse of right face, which is written R-I, is left face in this example. By definition, an element multiplied by its inverse equals theidentity element: RR-' = RL = E. This combination is always commutative: RR-' = R-'R = E. Conversely, if the product of two elements is the identity element, the two must be inverses of each other. An element may be its own inverse. This clearly is true of about face: since AA = E, A-' = A. A comment on the use of the terms "reciprocal" and "inverse" may dispel confusion and aid in understanding the concept to which they refer. Most of us automatically think of the reciprocal of a number as that number divided into the number one; thus a number multiplied by its reciprocal equals one. We now must appreciate that this is a restriction on the reciprocal concept, applying for the special case of the set of rational numbers. I t does not apply, for example, in the second example below. The general definition is that used in (4) : the reciprocal of a given element is that element of the group which combines with it to give the unit element. I n an effort to avoid confusion due to the well-established limited usage of "reciprocal," the term "inverse" is often used and will be used in this article. Formal Definition of a Group
A set of elements for which a method of comhination has been defined and which satisfies the following requirements : (1) The product of any two elements is an element of the group. (2) The associative law holds. (3) There is a unit element, E, such that EX = X E = X. (4) For each element, X , there must be another element which is its inverse, Y = X-I, such that X Y = E. Volume 44, Number 3, March 1967
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Examples. Greater confidence that the properties stated for a group really "work" may come from considering numerical examples where the combination procedures are familiar. One should ask: "What are the method of combination, the identity element, and the inverse relationship?" For the set of rational numbers ( l / m to a), the method of combination is multiplication, the identity element is the number one, and the inverse of a number is one divided by the number. To show that the four defining properties are satisfied: (1) 2 x 3 = 6; 6 is an element of the grow (2) (2 X 3) X 4 = 2(3 X 4) D X 4 = 2 X 1 2 24 = 24; associative law holds (3) 2 X 1 = 2; combination with identity element does not,
change the element. = I ; cornhination with inverse gives t,he identity elemenb
(4) 2 X
The reader is strongly urged to test himself by carrying out the same analysis of another group-the set of all positive and negative integers and zero-before looking at the answers in the f ~ o t n o t e . ~He might decide also whether the set of all positive and negative integers without zero is a group. Special T e r m i n o l o-. a~
Multiplication Table. The results of all binary combinations of the elements of a group may he collected conveniently in a "table of combinationsn-in conventional terminology, a "multiplication tnble." Once the table is worked out, it is used to find quiclcly the element which is the product of the combination of any two elements. An example is Table 1 which mas prepared by listing all of the elements of the group of facing operations across the top and down the left side, thus labeling four columns and four rows. The sixteen Toble 1.
A
Multiplication Toble for the Group of Facing Movements
A
It
spaces were filled in by combining the element at the top of each column with the element at the side of each row. To illustrate: the second row is given by Rfl = R, RR = A, EL = E , RA = L. Remember that the commutation law in general does not hold; therefore, specification of the order of combination is necessary. The conventional order is the column element followed by the row element. Ovder. The "order" of a group is simply the number of elements in the group and may be finite or infinite. Except for the case of linear molecules, only finite groups will concern chemists. The group of facing Method of combination is addition, ident,ity element is zero, and inveme of a. number is its negative: J
Answer: Because t,he product of R with it,self is A , which is not an element of E,R.
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operations obviously is of order 4 (h = 4). Similarly, the order of a sub-group or a class (see below) is t,he numbcr of clement,^ in that particular grouping. Sub-group. A "sub-group" is any selection of somc of the elements of a zroun which amone themselves satisfy t,he definition of a group. By definition any sub-group must contain E, so that the only possible sub-group of order one is fl itself. I n the military example, there is one sub-group of order two: E,A. Before looking at the answer in the footnote-why is E,R not a sub-group?& Conjugate IClemenls and Class. A lass" is another subdivision of t,he elenlents of a group, one which is particularly valuable because it simplifies the "character tables" to be described later. A class includes all of the group elements which are "conjugate" to each other. TWOelements, X and I', are said to b? "conjugate" if t,hey satisfy the relationship: u
-
.
P'XZ
= 1-
(1)
where Z is not neoessarily in t,he same class as X and Y hut i s in the same group. The overworked but useful "facing" group consists of the identity operation and three different rotations about the same axis. Let us see if these operations belong to the mme class by applying the ahove defining expression. If the identity operation, E , is chosen as X in eqn. (I), it should be clear from the group properties numbered (3) and (4) that I' must always be E . I n other words, E is conjugate only wit,h itself and therefore, is always in a class by it,self. Substituting X = R and Z = A in eqn. ( 1 ) gives ARA = Y, since A is its own inverse. From Table 1, we find (column followed by row) that RA = L. Then AL = Y but, again referring to Table 1, AL = R. Thus Y = R = X, and R seems to be its own conjugate. To prove this conclusion, each clement should be substituted for Z, keeping X = R:
Since the conjugate of R turns out to be R for every possible choice of X, we conclude that R has no conjugate, or is "self-conjugate," and must be in a class by itself. The reader should prove to himself that the same is true for L and A and that, in fact, this particular group contains four classes. It becomes awkward to say more about conjugation without specific examples of group elements which may be conjugate, such as are found among the symmetry operations of highly symmetrical molecules. The inadequate group of military facing operations now will be replaced by a very similar group of true symmetry operations. This will bring the discussion closer to chemical applicat,ions of group theory and will provide better illustrations in the following. Symmetry Elements and Operations. The four positions which a soldier achieves by the facing movements R, A, L, and E are at 90° intervals around a circle, and all could be reached by successive left faces or right faces. The axis of the soldier's body, perpendicular to the ground, is then a fourfold axis of rotation, meaning that four movements of 3G0°/4 (90" each) are
rcquircd to bring t,he soldier back to the starting position. Now, however, the analogy of the soldier must be completely discarded, because an axis of rotation can be a "symmetry element" and a rotation about the axis can be a "symmetry operation" only if, to a fixed observer, the rotated object looks exactly the same after the rotation, which the human body obviously does not. I n the last sentence, notice that a "symmetry operation" is a movement of the object and a "symmetry element" is a line (or plane, or point) with respect to which the object moves. A chemical illustration of a fourfold symmetry axis is found in thc square plauar complex ion [PtC14]-2. Upon a 360' rotation about the axis perpendicular to the plane of the ion and passing through the P t atom, thc chlorine atoms pass through three positions which are equivalent to the starting position and stop at a fourth which is identical to the starting position. Successive rotations of 90" eaeh would take a marked chlorine atom to eaeh of these positions, or a single rotatiou of 180' mould take it to the second position, or 270" to the third, or 360" to the fourth. Thus the 90' rotation, designated CP, generates three more operations: C2, C2, and Cd4(and an infinite number more, but each of thcse is equivalent to one of the basic four: C8 = CP, etc.). Since Ca2 = 180' = C2 and Ca4= 330' = E, the proper designation of the set of four symmetry operations associated with a four-fold rotation axis is: E, C2, Cz, C2. This set does constitute a group and has a multiplicatiou table like Table 1. Further details concerning symmetry are found in marly of the refercuces listed (5, 6, 11-15). Only the rotation operations just described will he used here. Move on Conjugation. In symmetry groups of higher orders, conjugate group elements do appear, and the consolidation into classes is possiblc. Several generalizations permit the first steps in classitication to be made quickly: (1) The identity operatiou always constitutes a class of order one (as we saw earlier). (2) Symmetry operations of distinctly d8erent types such as rotation about an axis and rpflection through a plane will never he in the same class. (3) A rotation operation of one angle (e.g., 120') will never be in thc same class as a rotation of another angle (lSOo). (The beginner may want to eorivince himself of this.) The ultimate refinement, when inspection must be replaced by eqn. (I), occurs when several operations of the same type are present. 111 an octahedral figure, for example, there are niue twofold axes of rotation, and it turns out that six of the nine corresponding C2 operations constitute one class and three another. The three will be thc a, y, and z axes of a Cartesian ooordinate system if the system is oriented so that the axes pass through the apices of the octahedron. Letting X and Y of eqn. (1) represent C2rotations about the x and y axes (Cz.zand Cz,,) and choosing for Z the C4 rotation about the z axis, we have C2 Cz., C2 for the left-hand side of eqn. (1) since C2 = (Cdl)-l. The product of the combination of these symmetry operations may be worked out by referring to Figure
1. A point at a, above the xy plane, is moved by C41,,to b. The next operation, CZ.,, moves the point to c, below the plane. Finally C45 moves thc point to d, below the plane. The same result, a to d, is obtained by the single operatiou CZ,,. Therefore C4a.. Cz,zC41,1 = C*.Vwhich is to say that CZ.=and C Z . ~ are conjugate and C41,sis the "conjugating element." Similarly, Ca.. can be shown to belong in this class.
Figure 1.
Movement of o point by rototion operations.
The "conjugate" relatiouship has a more fundamental significance than merely sorting group elements into classes. Consider the last example from the different viewpoint pictured in Figure 2. I n 2 A , the operatiou Cz.=moves the point from a to b; in 2B, the operatiou C2,,achieves the same result. The difference between 2A and 2B is that the coordinate syst,em has been reoriented with respect to points a and b by a rotatiou of 90' or Gal,,. We can say that two
Figure 2. Movement of the coordinate system combined with movement 0 point by rotation operotionr
of
operations are conjugate when they give the same actual change although this change is described differently because a different coordinate system is used for the two operations. "Conjugation" is the transformation of one operator into another by a change of coordinate system. Thus Cz., is the conjugate group element of C2.=because of the presence in the group of the conjugating, or transforming, element C,I.,. Point Groups. The symmetry operations which may be performed on a finite body such as a molecule are restricted to those which leave fixed at least one point in the body, othenvise the operation would not "send the body into itself." Thus the set of operations permissible for a given symmetrical body is called a point group." Rotation about an axis, reflection through a plane, inversion through a center of symmetry, and the identity operation are the only symmetry operations which leave one point fixed. Usually the combination of a rotation followed by reflcctiou in a plane perpendicular to the rotation axis is defined as a single operation called an "improper rotation." An infinite number of point groups exists, and every molecule fits one of them; even for the completely 'C
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unsymmetrical molecule there is the trivial group consisting of a one-fold rotation. I n crystals, only the 2-, 3-, 4-, and 6- fold rotation axes are permitted by the requirement that the unit cells fill all space, leading to the 32 "crystallographic point groups." Crystal lattices, however, are considered infinite, and symmetry operations are ~ermittedwhich do not leave onepoiut fixed, such as a glide plane, leading to 230 "space groups." Two systems of symbols for groups are commonly used. The Schoenflies system is convenient for dealing with molecular symmetry and is favored by inorganic chemists and spectroscopists. The HermannMauguin symbols are more informative when dealing with space groups and are preferred by crystallographers. For example, the symmetry elements of [PtCL]-= include a 4-fold rotation axis; the four operations which accompany it (E, C2, CZ,C2) constitute a subgroup which is the axial point group designated C4 by Schoenflies and 4 in the Herman-Mauguin system. The ion also has four twofold axes perpendicular to the principal axis and a horizontal plane of symmetry, putting it in the Schoenflies Dlh point group. B u m (16) gives a table of the 32 point groups, the 230 space groups, and the corresponding symbol in each system. Representations of Groups: "The Heart of the Matter"5
Definition. A representation of a group usually is defined as a set of elements which satisfies two conditions:
(1) There are elements in the set which can be associated with every element of the group. (2) The multiplication table of the set is equivalent to that of the group. The representation probably will he a set of numhers or matrices but could he a set of wave functions. Higman (10) uses the term "realization of a group" for the general case and restricts "representation" to matrix realizations.
Tables 2 and 3 are the multiplication tables for the C4 point group and the set of numhers +1, -1, f 1, -1. Associating E with +1, CI1 with -1, C2 with +1, and C43with -1, we see that the two tables are equivalent; thus both conditions are met and the set of numbers +1, -1, +1, -1 is a representation of the C4 point group. It should be obvious that the set of
numbers +1, +1, +1, +1 could he used and thus is also a representation of this group. I n fact, the latter set (all +1) is called the "totally symmetric representation" and fits the multiplication table of every group. The usual format for tabulating representations is to list the elements of the representation on a line, each falling in a column beneath the corresponding element of the group. Letting r stand for a representation, we have: C4
rl rp
Multiplication Table for the Point Group C4 E
Caa
Table 3. Multiplication Table for the Set of Numbers + 1
-1,
+1,
-1
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Ca 1 1
C43 1 -1
-
The matrix which transforms
1 .
[:I
into
[PC] is
This may be considered simply another
-2
symbol representing the operation also described as a "90" rotation" or a "C2 operation." The set of transformation matrices for the symmetry operations of point group C4, written as a third repre sentation of the group, are:
For any square matrix, the sum of the elements on the principal diagonal (upper left to lower right corners) is called the "trace" or the "character" of the matrix. It is invariant under similarity transformations, i.e., it is independent of the coordinate system, and it. may be 6
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1
CJ 1 -1
To find other representations of this group, we must consider matrices. A set of matrices will be a group representation if the above two conditions are met plus the additional condition that the matrices are square. A particular set of matrices which meets these conditions and is useful as a representation consists of the matrices of the transformations of coordinates associated with each mouu operation. Matrix Representatim. To avoid lengthening this article or straying unnecessarily into matters which are not part of group theory, no matrix algebra will he used or explained. The reader must appreciate that matrices are essential in applying group theory and that he should be sure he understands the statements about matrices in the following. Consider Figure 1 again, but now let a and b lie in the zy plane. The symmetry operation C4' which moves the point from a to b is said to "transform" the point. This can be treated as a rotation of the coordinate system similar to that illustrated in Figure 2. The mathematical method for calculating the values of the new coordinates of a point in terms of its coordinates in the original coordinate system is to r e p resent any point as a vector from the origin to the point. The coordinates then represent the components of the vector and are written as a single-column matrix. The matrix which will multiply the vector for a to give the new vector for b transforms the coordinates and is called the "transformation matrix." If the coordinates of a and b are x.,y. and xb,yb, it is clear from Figure 1 that xa = -y. and ya = x,,.
L.-
Table 2.
E 1
Higman (lo), title of Chap. V.
used to represent the matrix to avoid writing out the entire array of symbols. For example, the set of matrices r3above can be written 2,0, -2,0, where each number is the character of the respective matrix and the set of numbers is the "character of the representation." This double usage of the word "character" is not always made clear. Confusion would be avoided if the term "trace" were universally used for the "character of a matrix." Irreducible Representations. Although an infinite number of group representations is possible, it turns out that there are a few fundamental ones from which all others can be built up. These are called the "irreducible representations." Any other representation is called "reducible" because it can be expressed as, or "reduced" to, the summation of some grouping of the irreducible representations. More precisely, a representation is said to be reducible if there is a transformation which will simultaneously convert every matrix in the representation to the same block form. This statement must be left without further explanation: the reader is referred to Cotton (Is), p. 60-63. To complete the description of the C4 group, an additional complicating feature must be considered, with which we will conclude discussion of ways of obtaining irreducible representations. So far we have three irreducible representations of the C, group, but a theorem of group theory states that there must always be as many irreducible representations as there are classes in the group. The fact that the representation r3has a 2 under E (E is said to be of "dimension 2") indicates that this representation is degenerate and that it may be considered to be a pair of representations. These necessarily will contain imaginary characters and are written separately but are bracketed and given a single symbol. The complete list of irreducible representations for the C4point group is given in Table 4. For certain applications, the two bracketed representations are added together to give again the real, twodimensional representation 2,0, -2, 0. I n other point groups, e.g., Car C5, Can,or 8 6 , this complication is resolved by the use of complex characters of the type: 6 = exp' (2ai/n). Degenerate representations occur only in groups containing a rotation axis for which n is greater than 2. There are several other theorems or rules coucerning the characters of irreducible representations which would be needed in some applications. One of these which does not require extensive explanation will be mentioned here: the sum of the squares of the individual characters in an irreducible representation will equal the order of the group. The reader can check that this rule works for the four irreducible representations in Table 4, if each imaginary number is not squared strictly but is multiplied by its complex conjugate. Character Tables. In Table 4, new symbols have Table 4. Abbreviated Character Table for the Point Group C4
been introduced for the rJs. These "Mullikeu symbols" consist of A and B for one-dimensional, E for two-dimensional, and T for three-dimensional representation~.~An "A" representation is symmetric about the principal rotation axis whereas "B" means antisymmetric. Difference in behavior with respect to other axes is indicated by subscripts 1 and 2, with respect to a center of inversion by subscripts g and u,and with respect to planes perpendicular to the principal axis by primes. It is not necessary to memorize these rules. The symbols primarily serve as labels for the various representations, and normally one would have to look up a given point group to find the actual character of the representation indicated. In referring to atomic orbitals, lower case letters are used. Thus the symbols h, and e,, mentioned in the introduction, should now be recognized as the labels for one triply degenerate and one doubly degenerate irreducible representation. The subscript 2 indicates that there is a t least one other triply degenerate representation (t,) in the group. The subscript g means that this tz and e are symmetric with respect to a center of symmetry and shows that there is another tz and another e representation which are unsymmetric with respect to the center of symmetry (tZu and e,). If there were not, the g would be omitted. The complete table for the point group C, as it is usually found is reproduced as Table 5, which is known as a "character table." Character tables are included in several books (Id, 14, 81, W )and can be looked up when needed. A great deal of information is included in these tables. The first column on the left gives the MuUiken symbols for each irreducible representation. The next group of columns lists at the top all of the symmetry operations of the group collected into classes and, below, the characters of all of the irreducible representations of the group, where each number is the trace (or "character") of the transformation matrix which corresponds to each symmetry operation. This part of the table, of course, is identical to Table 4. The irreducible representations are sometimes called the "symmetry species" of the point group. Those molecular properties which are required to have the symmetry of the molecule, such as vibrational and rotational motions and electronic orbitals, can be associated with one of the symmetry species of the point group of the molecule. Such information is included in columns to the right. The T,, T,, T, terms refer to translation along the x,y, z axes. Let us deduce why T, is written beside the A sym8 A similar set of symbols was used by Placzek (17). The principal difference in Mulliken's set is the use of T instead of F for three-dimensional representrations(I8,19). Slater (20, p. 362) gives a table comparing six systems of symbols for irreducible representations.
Toble 5. Complete Character Table for the Point Group C4
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metry species, by referring to Figure 3, where the C4 axis corresponds to the z axis and translations in the x, y, and z directions are represented by short vectors. We see that application of the C4 operation leaves the z vector (T,) pointing in the same direction. Algebraically, we can write C4(T,) = T,. Thus the transformation matrix for this operation, Q in the expression Q (T,) = T,, must have a trace of +1 and therefore must be a one-by-one matrix with element I. he z vector is not affected by E, CZ, or Cla; therefore, each matrix in the set of four representing the behavior of T, under the symmetry operations of Figure 3. Trondaiionol vectors of a the group has a trace point. of +l. This is identical to the A irreducible representation of the point group, and we say that T, "belongs to the A representation" or "transforms as the A representation." When this information is included in the complete character table, the analysis just described is unnecessary. The table tells us that T, and T, together belong to the doubly degenerate E representation. I n some tables the symbols T,, T,, T, are replaced by the briefer x, y, z. These symbols also represent the three atomic p orbitals, p,, p,, p,, and the components of the dipole moment vector, M,, M,, M,. Since the orbitals and the components lie along the coordinate axes, they transform in the same way as the corresponding translation vectors. Knowledge of the symmetry behavior of orbitals is needed in molecular orbital theory and of the dipole moment in deducing selection rules for spectral transitions. The symbols R,, R,, R, in Table 5 represent the axial components of the angular momentum vector which expresses a rotation about an arbitrary axis. The symmetry species for each component could be determined in a manner similar to that used above for T,. I n the last column on the right, the symbols are the subscripts used to distinguish the five d orbitals, d,,, ,,d,, etc. Thus the table states that the d,, orbital transforms as the B irreducible representation of the group. To verify this for only one operation, picture the four lobes of the orbital with alternating positive and negative sign. A 90" rotation, Ca, will move a positive lobe to a spot previously occupied by a negative lobe. It is as if the sign of the corresponding wave function were changed or the function were n~ultipliedby -1, which is the character under C4 in the B representation. This last observation illustrates another interpretation of the character under a given operation in the character table. A negative character means that, when that symmetry operation is applied to a molecular property which has a positive or negative value, the property will be transformed into its opposite sign. The absolute value remains the same, as required if the change is to be a symmetry operation. For an illustration, refer to the T, vector in Figure 3. If the
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Journal o f Chemical Education
Ca operation about the z axis is applicd, the T, vector will move to the opposite side of the yz plane but will he pointing in the -x direction. This result is predicted by the -1 under C'? for the T, or E representation in Table 5. Conclusion
Returning to the "secondary theme" mentioned in the introduction, it is now possible to state the meaning of "e," and "l?," orbitals whirh appear so often in articles concerned with ligand field theory and transition metal complexes. These are the symbols for the irreducible representations which transform under the symmetry operations of t,he point group in the same way as do the orbitals, or, more simply, these are the symmetry species to which the d orbitals "belong." I n the free t.ransition-metal atom or ion, the five d orbitals have the same energy, and this energy level, or "subshell," is described as being "degenerate." When the atom or ion is placed in a symmetrical, but not spherical, arrangement of ligands, t,he electrostatic field will cause the fivefold degenerate energy level to split into several levels of different energies. If the arrangement of ligands is tetrahedral or octahedral, the splitting is into two levels of three- and twofold degeneracy. Those d orbitals with lobes lying between the axes (d,,, d,,, d,,) are equivalent in every way; thus they remain together as a triply-degenerate set and have the same symmetry behavior. Those d orbitals having lobes lying on the axes (da>,d+vs) are also equivalent and have the same symmetry7; thus they remain together as a doubly-degenerate set. For the octahedral case, in the character table for the 0,point group (not reproduced here), one will find in the right-hand column, on the same line as E, in the left-hand column, the ent.ry (2z2- x 2 - y2, x2 - y2)and, on the same line as Tz,, the entry (xz, yz, xy). This have E , symmetry and are means that d,. and d,.-,. "e, orbitals," and d,,, d,,, and d,, have Tz0 symmetry and are "k0 orbitals." Remember that different ligand arrangements affect d orbitals differently. The main purpose of this article has been to present to teachers of chemistry the essential concepts of group theory. I t is hoped that the reader is convinced that these fundamentals are simple and involve no complicated mathematics and that group theory is important for chemists. It has been said that group theory at present is "a tool not at t,he fingertips of most chemists" (4). Probably the same could be said for quantum mechanics. Most chemist,^, however, believe that t,hey have some understanding of what quantum mechanics is and what it can do in chemistry. In describing the basic concepts and terminology of group theory, this article has attempted to increase the general understanding of what. group theory is and what it can do in chemistry. The terminology and applications of symmetry and group theory should become at least as familiar to chemistasas are the terminology and applications of quantum mechanics. Suggestions for Further Reading
For additional study, two articles written 20 and 30 7 This will he more obvious if it is remembered that d , ~ may be looked a t as a linear eomhinatio~of two orbitals: d,*-,r and d ~ ~which 3 , is properly written d9,?.,a.,,? (7, 1s).
ycars ago (23, $4) are reconnncndcd as starters, taking the most recent first ($4). The original article in this field by Bethe in 1929 (9) is available in English. For the formal mathematical theory, the first English booli (1st edition in 1897) by Burnside (85) is still available. Examples of modern books are those of Weyl (26) and Littlewood (27). A small hook by Alexandroff (f8), designed for "n~athcmatically-inclined pupils in senior classes of grammar school," is a good starting placc for one interested in the "pure mathcmatics." Recent hooks which are rather thornugh and rigorous in group theory and emphasize physical applications include those of Higman (lo), Lomont (29), Hammermesh (50), and McWeeny (31). Applications in quantum mechanics are presented by Heiue (2) and Slater (ZO), the latter including a thorough treatment of group theory. Other current hooks which contain chapters or appendices on matrix algebra, symmetry operations, and group theory- includc those of Barrow (I I ) , Bauman (Sf), Jaffe and Orchin (55), and Sandosfy (34) on various chemical aspects of spectroscopy; that of Streitwieser (55) on molecular orbital calcul* tions; and that of Brand and Speakman (36) on molecular structure determination. Dealing primarily with symmetry are t,wo recent paperbacks by Dorain (IS) and by J d e and Orchin (14). For a detailed, practical, and rcadahle presentation of the fundamentals of group theory and of symmetry in molecules plus illustrations and explanations of the more imporhnt applications in chemistry, thc booli by Cotton (12) is highly recommended. Literature Cited (1) S ~ r r r aL. ~ ,E., J. CHEM.EDUC.,37, 498 (l!I60). (2) IJEINE, V., "Uro~ip Theory in Q~lant,wnhlechmies," Perg~lmonPress, I w . , Now l'o'urk, 1960. (3) MANCH, W., A N D FERNEMUS, W. C., J . CREM.EDUC.,38, 1W (1961). (4) COMPANION, A. L., .AND KOMARYNYKY, 1\I. A,, J. CIIEM. JCDUC., 41, 257 (1!)64). (5) LIEHR,A. D., J . CHEM.EDUO.,39, 135 (1062). (.6.) G R ~,YJI.. B.. J . CHEM.EDUC.. . 41.. 2 (19641. . , (7) C O ~ O NF. , A,, J . CHFM.EDUC.,41, 466 (1964). ( 8 ) KETTLE,S. F.A., J . CIIEM.EOUC.,43, 21 (1966). (9) RETHE,H., A m . P h y ~ i k 3, , 133 (1029). English Twrrsls, ~~
tion: Consollants Bureau, Inc., New York. (10) IIIGMAN,B., "Applied GmupTheuretic and Matrix
hfethods," Oxford Univemity Pres~, Lwldon, 1955. (Dover, New Yark, 1964.) (11) BARROW, G. M., "Intmduct,ion to Molecular Spectroscopy," NIcGraw-Hill Book Co., New York, 1962. (12) COTTON, F. A., "Chemical Applications of Gmup Theory," Inberscienee Publishers (division of John Wilev & Sons. Ine.), New York, 1963. (13) DORAIN,P. B., "Symmet,ry in Inorganic Chemistry," Addison-Wesley Publishing Co., Reading, Mass., 1965. (14) JAFFE,1%.H., AND ORCHIN,M., "Symmetry in Chemistry," John Wiley & Sons, Inc., New York, 1965. (15) ZELUIN,M., J . CHEM.EDUC.,43, 17 (1966). (16) BUNN,C. W., "Chemical Crystallography," Oxford University Press, London, 1945. (171 PLACZEK. G.. "Handbuch der Radiolaeie. - , 2 Auflare. Band T'I, ~ e iI? l Akad. 'I'erlagsges, Leipzig, 1934, p.205. MULLIKEN, R. S., Phys. Rev., 43, 279 (1933). MULLIREN, 11. S., J. Phys. Chem., 41, 159 (1937). SLATER, J. C., "Quantum Theory of Molecules and Solids," McGraw-Hill Book Co., New York, 1963, vol. I. EYEING,H., WALTER,J., AND KIMBALL,G. E., "Quantum Chemistry," John Wiley & Sons, Ine., New York, 1944. WILSON,E. B., DECIU~, J. C., AND CROSS,P. C., "n~olecuiar Vibrations." MeGraw-Hill Bonk Co.. New Yo*. 1955. R O S E N T HJ. A E., ~ AND MURPHY, G. M., R m Mod. Phvs., 8, 217
~----,.
(24) MEISTER,A. G., CLEVELAND, F. F., AND hfURRhY, M. J., Amer. J. Phys., 11, 239 (1943). (25) BURNSIDE,W., "Theory of Groups of Finite Order," 2nd
ed., Cambridge University Press, London, 1911. (Dover, New York, 1955.) 126) , , WEYL. ' 13.., "The Classical Grouus.,' , rev. ed.. P k c e t a n University Press, Princeton, N. J., 1946. (27) LITTLEWOOD, D. E., "The Theory of Gronp Characters," Oxford Univessi(.y Press, London, 1940. (28) ALEXANDROFF, P. S., "An Introduction t,o bhe Theory of Groups," Hafner Publishing Go., New York, 1959. (29) LOMONT, J. S., "Applications of Finite Groups," Academic Press, Inc., New York, 1959. (30) HAMERMESH, hl., "Group Theory and Its Application to physical Problems," Addison-Wesley Publishing Co., Reading, Mass., 1962. (31) MCWEENY,R., "Symmetry," Pergemon Pmss, Inc., New Ynrk - *. .., 106.7 - .-- . (32) BAUMAN, R. P., "Absorpt,ion Speetroseopy," John Wiley & Sons, Inc., New York, 1962. (33) JAFFE,H. R., AND ORCHIN,M., "Then~yand Applicatiom of Ultraviolet S~ectroscouv," .. . John Wilsv & Sans, Inc., New York, 196; (34) SANDORFY. C., "Electronic Spectra. and Qumtnm Chemist,ry," Prent,ice-Hall, Inc., Englewood cliff^, N. J., 1964. (35) STREITWIESER, A,, JR., ''M~lecula~. Orhital Theory for Organic Chemists," John Wiley & SOLS,Ine., New York,
.
1961. AND SPEAKMAN, J. C., "Mole~ularSt1.u~. tnre," Edward Arnold, London, 1960.
(36) BRAND, J . C. D.,
Volume 44, Number 3, March 1967
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