Resource Papers-lX Prepared under the sponsorship of
The Advisory Council on College Chemistry
Milton Orchin and H. H. Jaff-5 University of Cincinnati Cincinnati, Ohio 45221
I
I
Symmetry, Point Groups, and Character Tables Part 111, Character tables and their significance
In preceding sections we have discussed the symmetry properties of molecules in terms of their geometry and have analyzed the dipole moments of molecules. The geometry and the dipole moment are static properties of the molecule. I n order to analyze the dynamic properties of molecules, we must consider whether the particular property is symmetric (unchanged) or antisymmetric under the symmetry operations appropriate to the molecule. All properties must be (or must be decomposable into components which are) either symmetric or antisymmetric with respect to each symmetry operation that can be p e r f ~ r m e d . ~I n order to determine whether a property is symmetric or antisymmetric, we must have a clear understanding of these terms. We will first examine motions, specifically translations, of objects or molecules. If one stood in front of a mirror and threw a ball up in the air parallel with the mirror, one sees that, when the hall moves upward, so does its reflection in the mirror, and with the same speed. When the ball reaches its apogee and starts to descend, the image likewise changes its direction and descends with the same speed as the ball. The up and down motion of the hall (molecule) is said to be symmetric with respect to reflection in the mirror parallel to the motion because the actual motion and its reflection always travel in the same direction with the same speed (or magnitude). Now let us throw the ball directly at the mirror, and analyze this motion which is now perpendicular to the mirror. The reflection or image travels a t the same speed as the ball, but the ball is moving toward the mirror and away from the thrower, while the reflection is moving in t,he opposite direction, so that the ball and its image collide a t the mirror. The motion of the image has the same magnitude but is now opposite in direction or sign to the real motion. Thus, motion (or translation) perpendic-
This threcyart paper represents the last in n series of "Resource Pi~pe1.s"prepared under the spo~isor~hip of tho Advisory Coonril on College Chemistry (AC3) which is supporlcd by the National Science Foundation. Professor L. Cnl.roll King of Northwestern Uuivelaity is tho chairman. Single copy reprints of this paper are being sent to chemistry department chairme11 of every U S . Institution off e r i q college chemistry courses and to others on the ACa mailing list,. This is Serial Pahlieation No. 49 of t,heAdvisory Council.
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ular to the mirror is said to be antisymmetric with respect to reflection in the mirror. If we placed these motions in a coordinate system with the z coordinate vertical and in the plane of the paper, the x coordinate perpendicular to the paper, and the y direction horizontal and in the plane of the paper, we see that, with respect to reflection in a vertical plane, translational motion in the z and y directions, which are parallel with the mirror plane, are symmetric, while translational motion in the x direction, perpendicular to the mirror plane, is antisymmetric. When a molecule has a center of symmetry and we wish to discuss the symmetry behavior of some dynamic property of the molecule, we use the special terms gerade (German for even) and ungerade (German for odd) for symmetric and antisymmetric behavior, respectively. With respect to motion in the x, y, or z direction, reflection through the center of symmetry always reverses the direction of the motion, and hence such motion is always ungerade. Since orbitals or wave functions describe electronic motion, the molecular orbitals of molecules with a center of symmetry must be either gerade or ungerade. Thus, the a orbital in ethylene, Figure 5A, is ungerade, a", while the a*
Figure 5. Gerade and ungerode orbital%
orbital in Figure 5B is gerade, ,:a as are all d atomic orbitals, Figure 5C. I n Figure 5B, for example, if we look a t a point in the upper left positive lobe and draw a line from it through the center of the molecule and continue an equal distance in the same direction, we encounter a point with the same sign; hence, this orbital is gerade.
-
Parts I and I1 of this series appeared in the April and May issues, respectively, and dealt with the topics of symmetry operations and classification of molecules into point aroups. All footnotes, figures, tables, and stmctmes are numbered Eonsecutively throughout the series. The material in fine print indicated by a. 5 adds rigor to some of the arguments hut is not considered essential for the introductory course in organic chemistry 8s it is now generally strootwed. 'An exception are properties of molecules with C, or S, with p > 2 (i.e., molecules belonging to degenerate point groups), which are treated in B later section.
8 Symmetric and Antisymmetric Character of Singlet and Triplet States In dealing with electronic absorption spectroscopy and photochemistry, we are generally cur~cernedwith promoting one electron from an orbital with a certain energy into a vacant (or virtual) orbital with higher energy. Although the two electrons which share the original orbital must, of necessity, possess opposite spin, when the two electrons occupy different orbitals, this restriction no longer applies. The spin of the two electrons in the excited state can be eit,her the same (i.e., both or both -'/%), or the two elect,ronx can have opposed spin (i.e., if one is +xil the other is -'/% or vice versa). The multiplicity of a state, J , is equal t o 2 8 ) 1 where S is the sum of tho spin numbers of either f'/2.When both electrons have the same spin, 21S1 1 = 3 and we have the so-called t,riplet state. When 1 = 1 and we have the the elect,rons have opposed spin, 281 so-called singlet state. I n a. common convention, the two possible spins of an electron, and -I/%, are denoted as o and 8, respectively. Since two electrons are involved. we mav call these electrons 1 and 2. thouzh of course we cannot distinguish them. The possible spin combinations of the two elect,rons in different orbitals are
+
+
~
+
-
~~~
Thus, in (a) the two electrons both have spin of ' / z ,in (h) both I n either (a) or (b), it doesn't make any difference which of the two electrons ( I ) or ( 2 ) is being used, since in both cases both electrons have the same spin. Thus, in (a), o(2)ar(l) is identical with a(l)a(2). Both (a) and (b) are therefore symmetric with respect t o exchange of electrons! However, in (c) and (d) we have two equivalent expressions and neither can he considered done; s. combination of the two is necessary lo describe the situation. Thus. if we exchsneed electrons in (c) we - 1 .
Character Tables
I n the study of the properties of molecules we are frequently interested in the motion of the molecule itself and in the motion of the atoms relative to one another (vibrations, ir absorption) as well as the motion of the electrons in the molecule (molecular orbital theory and electronic spectra). When we are dealing with a molecule like water which belongs to point group C2"(no degeneracies) all motions of the molecnle must be either symmetric or antisymmetric with respect to each of the four symmetry operations of the group. We may for our purposes here agree to characterize symmetric behavior as +I and antisymmetric behavior as -1 and to call the +1 and -1 the "character" of the motion with respect to the symmetry operation. For example, let us examine the behavior of the p, orbital6in water, Figure 6 , under some of the symmetry
Figure 6. The p* orbit01 in woter.
operations of point group Cza. Rotation around the z-axis changes the signs of the two lobes, hence under C2' the p, orbital is antisymmetric and has the character -1. Similarly, reflection on a%,gives a change in 'sign and hence is -1, while reflection in av, transforms the orbital into itself and hence has the character +l. The other symmetry operation in point group Czar heside CSz, as2,and a Y ~ is, the trivial identity operation I,which, antisymmetric with respect to exchange of electrons. I n such cases, we must take linear combinations of the two functions, since it leaves the molecule unchanged, obviously has i.e., we add them together to give one combination and then subthe character +l. The total number of symmetry tract them t o give the second linear cambinstion operations in s. point group is called the order of the point gronp; in Cz,the order is four. (symmet~ic,triplet) (c dl: o(1)8(2) a(2)8(1) (antisymmetric, singlet) We stated earlier that one of the properties of a (c - dl: a(l)P(2) - o(2)8(1) group is that the product of any two elements in the i ~,~utir?mnertvic~1111ravter Kow let u r iwcitlg:ue t h e s y m m ~ t rur gronp is also an element of the group. I n Cz,we have of t l w r t w o spit) fuuvtioni with I C ~ X Y I 10 e s < l . a ~ ~ g ctlnr two three non-trivial elements, and since the third is the elwtroni. \\'c see t l m r if we e*ch:ilqe the r l t u r o ~ i 1 k l . J ( 2 product of any of the other two, we need only be conin the first combination (c d) we get no change in sign and hence this function is also symmetric. But now let us exchange cerned with the character of two elements or operations. the electrons (1) and (2) in (c - d), whereuponweget o(2)@(1)Since we have two possible independent operations and o(l)p(2), which is precisely the result obtained by multiplying each can have the character of + 1, we have a total of (C - d) by -1; hence (e - d ) is antisymmetric with respect t o four possible combinations. Let us arbitrarily consider exchange of electrons. The antisymmetric spin function characterizes the singlet state (the total wave function must be antiCzSand a=, as the two independent operations; both can symmetric, and, in the singlet state, the orbit81 part of the wave be +1, and they can also be +1, - 1; - 1 1 or function is symmetric and the spin part of the wave function is - 1, -1. We can put this information into a table antisymmetric) and the symmetric spin function characterizes called a character table, and for point group Cz,, Table 3 the triplet state. As a point of fact, the energies of the total wave is such a table. This tahle shows that under the idenfunction, of which the spin functions (s); (b), and ( e + d) are a. part, are all equal (degenerate) (in the absence of a magnetic tity operation, I, every property is symmetric, as exfield) and together constitute the triplet state. Accordingly, we pected, since the I operation does nothing. The four may define the triplet state a3 the state whose spin function is possible combinations of Cz' and us,are shown, and it is symmetric with respect to exchange of electrons. readily verified that in each case the character of u,, is the product of CS' X a,,.
+
+
+
-
'The symmetry discussed here is of a somewhat different nature than the symmetry discussed in the rest of this paper. The operators used in the remaining parts are rotation, reflection, inversion, and rotation-reflection. In this section, the operat,or is a more subtle one, one which exchanges the coordinates of electron 1 and those of electron 2. Otherwire, the treatments are equivalent. When the p orbitals are shown as two circles of opposite sign touching a t a. point, the angular dependence of the wave funct,ion is being illustrated.
Symmetry Species of Irreducible Representations
We earlier examined the behavior of the p, orbital in the water molecule, (Fig. G ) , and showed that under the symmetry operations of point gronp C2", Table 3, the orbital is +I with respect to I, -1 with respect to CE',- 1 with respect to u,,, and 1with respect to u,,. This +1, -1, -1, +l behavior is one of the four possible ways t,hat every property of the molecule can be
+
Volume 47, Number 7, July 1970
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described. These four distinct behavior patterns are called symmetry species or irreducible representations, and their number is equal to the order of the group. The symmetry species have, for convenience, been given shorthand symbols which are shown in the case of C2,in the first column of Table 3. All symmetry species that are symmetric with respect to the highest rotational axis are designated by A, and those antisymmetric are designated by B. In the case of point groups like Dl,, where there are three two-fold axes, and therefore no rotational axis that is of highest order, only the symmetry species that is symmetric to all three Cpaxes is designated A . Where more than one species or representation is symmetric with respect to the highest rotational axis, as in C 2 , ,they are distinguished in the subscripts (or sometimes by primes) and the totallysymm~tricspecies, i.e., the species as in C2"which is plus with respect to every operation, is always the A, species. Subscripting of the R species is more arbitrary. I n the case of molecules belonging to CZ,,the rules for orientation given earlier is usually unambiguous and after setting up the coordinate system, the B species that is symmetric to a,, is called B1. I n all point groups with a center of symmetry, the subscripts u and g are used for all species; the assignment depends upon the behavior with respect to the center of inversion, i. Thus, consider the character table of point group Cnn,Table 4. I n this point group there are four operations: I; CZe;u,, (or a,,); i; and again four irreducible representations. The two A species are here designated A, and A,; the former is totally symmetric, i.e., symmetric with respect to all operations, Let us examine some property of a molecule in point group CZ*. We choose s-trans-butadiene and analyze the lowest energy a bonding molecular orbital (Fig. 7). (The molecule is shown in the plane of the paper and the p a system is perpendicular to the paper.) Under the symmetry operations I, C2', an,i, this orbital transforms as +1, +1, -1, -1, and hence belongs to symmetry species A.. -
Toble 3. Chorocter Table for Point Group - --- -~-
Cz,
I
A, Aa
+1 1 1 1 -
-
C2(4
+++
HI H,
Table 4. -
-1
Gv(~d
+-11
+-11
-1
-
+1 -1 -1 1 -
+
-
z
Rz
2, R r
Y, Rz
I
C2m
+1 +1 +1 +1
+- 1 1 +l
-1
-
- -
-
--
i
q(=~)
+-11 +-11
+-11
R, B
RZ,Rv
+1 -1
2, Y
-
I n a point group like Dzh,Table 5, where there are six B species, both numbers and the g and u letter designation are required. Now let us examine two of the possible ir vibrations of ethylene, a molecule which belongs to point group Dz,. An out of plane deformation mode shown in Figure SA transforms as species BZc. The plus signs indicate
Figure 8.
Symmetry species of two vibrational modes in ethylene, D z ~ .
motion out of the plane of the paper toward the observer, and the minus signs indicate motion behind the plane away from the observer. Thus, C1 is moving toward, C2moving away, HI moving away, H, toward, Ha away, H, toward the observer. Reflection of these motions through the xy-plane gives motions a t each of the atoms which are exactly the reverse of the original a t that atom, and hence this out of plane bending mode is antisymmetric with respect to reflection on uZ, and has the character of - 1. Testing this bending mode under each of the operations listed in character table D,, shows that it transforms as species Bz,. The vibrational mode shown in Figure SB can be similarly demonstrated to belong to species Bz,.
5 Degenerate Point Groups
Chorader Table for Point Group C2* --
Cm A, A, R, H" --.
J ,=d
++11
C,,
Figure 7. The mrnt bonding molecular orbital in s-trans-butodiene, C,lu
-
The discussion of character tables up to thk point can he a g plied to point groups CI, C,, C;, C Z ,Csh, Ds,and D l h . AU of these point groups do not involve a symmetry axis C or S greater than two-fold (DZahas an & axis). As soon as an axis greater than two-fold arises, the problems of symmetry species become much more difficult. Let us use, as an example, the ion PtCLS- and examine how the three p orbitals of the platinum atom transform under the sym-
-
Table 5. Character Table for Point Group Dxh -
DZL E VL A. A. Bu
+1 +1 1 +1
HI" B1. B2"
+
B,,
+I
B1"
5 12
I
+1 +1
/
+1
+Y) +l -1 +1 -1 -1 +1 -1 1
+
Journal of Chemical Education
.-
c( X Z 1
~YZ?
i
+1 -1 -1 +1 +1 -1 -1 1
+1 -1 -1 fl -1 +l +I -1
+1 -1 +1 -1 +1 -1 +I -1
+
Vh
c*(~,
C@
c+'
+1 +1 +1 1 1 1 -1 -1
+1 1 -1 -1 +1 +l -1 -1
+l 1 -1 -1 -1 -1 +I +l
+-
+
+
. ..
... R,
e
R. Y
R, 2
If the multiplication of the 2 X 2 metrix
ti)
if1
(dl
Figure 9. Thep orbitals of PI in[PfClrlz,
metry operations appropriate to the point group Dl*, Table 6, to which this square planar ion, Figure 9A, belongs. The z- and y-axes (coordinate system Fig. 9B) are CSaxes; the C9axes which bisect the angles between bonds are designated Cn' axes. The two vertical planes of symmetry that include the z, y, axes a t r e o. and those that include the C9' axes are called ad in the character table. The p, orbital, Figure 9C, presents no particular problem.' If we apply all the symmetry operations on the p, orbital in turn as listed in the chracter table, we get, starting wit,b I, +1, +l, +1, -1, -1, -1, +1, +1, -1, a n d f o r i , -1, which tells us that p, belongs to species Al,. Now let us examine the behavior of the p, orbital, Figure 9D. (For convenience, the coordinate sysbem in B has been rotated 90' around the y-axis to give E, which places t,he zy plane in the plane of the paper.) If we perform a 90" clockwise rotation, C,: we see that p, is transformed to p, and hence p, is neither symmetric nor antisymmetric under the operation. However, the C4' operation simultaneously trsnsforms p, into -p,; obviously, the transformations are related and the two orbitals transform together with t,he result shown in Figure SF. Although the above transformations can he discussed in general terms in vector notation, we shall develop the trsnsfarmations ming the familiar orbitals. When we perform ZI clockwise rotation of 90' on the p, orbital, we get a new orbital which has none of the old orbital in it and is exactly equal to the original pp, orbital. If we call the new orbital p',, we may state the fact mathematically
+
IP, p'. = OP, (1) Similarly, t,he transformation of t,he old p, before the C,- into the new p', may be written P ' ~= -Ips
+ OP,
(2)
A special met,hod of writing these equations is possible
Table 6. Ddn
dl. At,
As
A%"
%
RI"
&
Bs" En E"
I +1 +1 +1 1
++1 +1 +l +l
+2 +2
C,¶ = C*"
ZC2
2C2'
+1 +1 +1 +1 -1 -1 -1 1 0 0
+1 +1
+1
++l 1 -1
-
+1 1 +1 +1 -2 -2
+
>
-
")bytbe
2 X 1 matrix or column vector were carried out according t o the rules of mstrix multiplicetian, the two equations (1) and (2) would be obtained. The sum of the numbers appearing in the diagonal from upper left t o lower right of the transformation matrix is called the trace (German spur) of the matrix and the actual number found by the addition is called the character of the transformation matrix. I n the present example the character is 0. Accordingly, if we refer to the charscter table of Dl*, we must look for a symmetry species which under the C4operation has a character of 0. Here we can use a short-cut to determine the correct symmetry species. When we are dealing with s. pair of orbit& of equal energy that transform together we have a so-called degenerate set. A set of two degenerate orbitals always belongs to an E species (a set of three degenerate orbitals to n l'species). Hence, our orbitals belong to one of the E species in Dlh, and it remains only to specify the behavior under the operation i. A p orbital is always antisymmetric t o a center of symmetry, snd hence our p,, p, arbitdls together belong to species E, in point group D,&, the characters of which are found in the last row of the character table. As an exercise let us determine the transformation matrix of the p,, p, orbitals under the operation it o confirm that the character is -2 as shown in the character table. Referring to Figure SF we see that under i P'= = -IPS P'.
=
+ OP,
OP, - lp,
and the character of the transformat,ion mstrix
Let us, instead of rotating the p,, p, orbitals 90' in the clockwise direction as we did earlier, rotate in the cor~nterclackwise direction, i.e., instead of C4do C',. The transformation may be written P'. = OP, - IP, transformation matrix P'" = lP, OP,
+
(
-3
The character of the t,ransformation matrix is again zero even though the off-diagonal elements are now different, than those in the transformation matrix obt,ained from the C, operat,ion. We now see why C, and C', belong to the same "class"; they have the sitme character. I n the mathematics of group theory, a "clad' includes all the group elements which are "conjugate" t o each other. Any two elements X and Y of a group are "conjugate" t o each other if there exists some element of the group, say Z, such that I n our example C4and C', are conjugate to each other and belong 'When the p orbit,sl is drawn dumbbell shaped, the square of the angular dependence of the wave function is being illustrated. This representation is appealing because the square of the wave function, being a probability, has physical significance. However, the two parts of the dumhbell should then both he positive. I n keeping with the common but incorrect practice of mixing the representation of the angolar part of the wave function and its square, we draw the dumbbell with different signs in the two halves.
Character Tab le for Point Group Dan
ZC,(z)
++11
(--I O
+1
-1
-1 +1 +1 -1 -1 0 0
-1 -1
-1 +1 +1 0 0 Volume 47, Number 7, July 1970
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51 3
to the same class; hence there are 2C4 because C4 and C', both have the same character. The Ca and C'. or 90' clockwise and counterclockwise ratations give, respectively, the transformat,ion matrices
These numbers correspond to the values of the sin and eos of &90' and indeed that is their origin. We may generalize the 2 X 2 transformation metrix for any degree of rotat,ion by using the matrices eos 8 sin 8 cos 8 -sin 8 cw 8 (-sin 8 cos 8) and (sin 8 and in either ease 2 cos 8 corresponds to the character. Thus, in any doubly degenerate species, E, the character for the appropriate rotation of 8' either clockwise or counter-clockwise around the C, axis is 2 cos 8. In these examples we have rotated the orbitals (or more generally, the vectors), hut frequently one rotates the coordinate system instead. The choice affects the sign of the off-diagonal elements but is immaterial, since the diagonal elements which determine the character remain the same. I n the non-degenerate point groups discussed earlier, the transformation matrices are all 1 X 1 matrices and so we could immediately assign +1 to symmetric and -1 to antisymmetric behavior. Symmetry Properties of Tmnslational and Rotational Motion
It will be noted that character tables such as those shown in Tabks 3-6 all include in a last column the notations x, y, z and R,, R,, R,. These symbols are assigned to particular symmetry species in each point group. They inform us to which symmetry species the translations (of a molecule, for example) along the x-, y-, z-axes belong, and to which symmetry species the rotations R,, R,, R, around the x-, y-, and z-axes, respectively, belong. Let us examine the motion of the in the direction of each water molecule (point group CZa) of the Cartesian axes. First we will ascertain how a translation along the positive y-axis (Fig. 10A) trans-
R1. Motion in the x and z directions can be similarly analyzed, and we see that such motion belongs to the representations BIand A1,respectively. Since the p,, p,, and p, orbitals behave like translations in these directions, this notation also tells us how these orbitals transform. Similar transformations of rotational motion around the three Cartesian coordinates can be assigned to symmetry species. Let us analyze how rotation around the z-axis transforms (Fig. 11). Again, to help us analyze the situation, we employ arrows and show them going in and out of the plane of the paper a t each hydrogen. The @ sign indicates motion upward out of the plane; the Q sign indicates motion downward away from the plane. The molecule is now rotating around the zaxis in a clockwise direction so that Hais moving toward, and H. moving away from the observer. I n performing the operations I, C2%, U Z Z , uyl, we can focus on the behavior of the arrows; if they change direction after the operation the character is - 1 and if the arrows remain pointing in the same direction as the original, the rotation is + 1 with respect to the operation. The behavior of the rotational motions after each of the operations is shown in Figure 11. On the CZzoperation, we exchange H. and Ha, but the atom on the left is still going away, and the atom on the right is still coming toward the observer; thus, although we exchanged atoms, the atoms are indistinguishable, and the direction of motions of the atoms on the left and right are identical after the operation to that before the operation. Hence, the Cz2operation is + 1 with respect to rotational motion around the z-axis. The results of reflections in the two planes of symmetry are shown in Figure 11, and the characters belong to symmetry species AD If we look a t the CZ,character table, we see R , in the last column of the AZ species. The species assignments for R, and R, are also indicated in the table. $ Some Features of Character Tables and Irreducible Representations We have mentioned explicitly some properties of character tables and symmetry species and implied others, and it is perhaps
( 0 )
(bl
forms under the symmetry operations of Cz,. To simplify the analysis, we draw an arrow from the center of the molecule along the y-axis. Under I the molecule is unchanged, so the motion is + 1 with respect to I. Under C*'we transform Figure 1 0 B to Figure 10C and we see that now the arrow is pointing in a direction opposite to the original, i.e., the vector has changed sign and the motion in the y direction is thus antisymmetric with respect to Cziand hence has the character -1. On reflection in the xz-plane, the arrow would again be pointing in the direction opposite to the original, as in CzZ, and again the character would be - 1 . Finally, reflection in the yz-plane leaves the arrow unchanged .and hence under this operation, the character is + l . The behavior under the four operations is thus +1, - 1, - 1 , f l , which belongs to symmetry species R,. In the character table for CZu, we see that in the last column of the row of characters in the representation B1 we have the symbol y, which tells us that motion in the y direction of the water molecule transforms as 514
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Journal of Chemical Education
Table 7.
( = )
Figure 10. Behovior of tranrloiional motion in the y direction on CP.
Character Table for Point Group Td
Ts
I
8C3
6-6
684
A,
+l
+1
+1
+1
0
0 -1 +1
+l
E
T, Tz
+2
+3
i-3
0
c: itll
Figure 11.
0
-1
-*. e 11
3S2 = 3Ca
++11
+2 -1 -1
R*, R,, R, 5111.2
1-11
Behovior of rotational motion around the r-oxis under the symmetry operations of C s..
useful t o summarize in one place some characteristics of these tables. As a typical example we can choose the character table of point group Td,Table 7. The symmetry operations spanning this point group can be appreciated by reference to a Td molecule such as methane, XLVIb. The following generalizations can he verified by reference t o character table Ts, Table 7. 1 ) There are as many irreducible representations (symmetry species) as there are clazses of symmetry operations (five for m
~
1 dl.
2) For each irreducible representation, the square of the character in each class multiplied by the order of the class and summed over all classes is equal to the order of the group. ThusforT.inTd: 3 % X1 + 1 2 X 6 + ( - 1 I P X 6 + (-1YX 3 = 24. This i?, in effect, a normdimtion condition. 3) The sum of the squares of the characters in any class, multiplied by the order of the class is equal t o the total number of symmetry operations. For the operation I, this may be stated alternately, as the sum of the square of the degeneracies of the irreducible representations is equal t o the order of the group. For Td: ll(A1) l2(A%) 2YE) 3YT1) 3%X (T2) = 24. This follows from (2) and the Hermetian nature of the transformation matrices. 4) The product obtained by multiplying the character of each class by the order of the class, summed over all clesses is zero for all irreducible representations except the totally sym(-1) X metric one. Thus for specie8 E in Ta: 2 X 1 8 + 0 X 6 + 0 X 6 + 2 X 3 = 0 . 5) The sum of the products of the character of any two symmetry species multiplied by the order of the class is always (0 X [-I] X zero. T h u s i n Td, T* X E = (3 X 2 X 1) 8) (1 X 0 X 6) (-1 X 0 X 6) (-1 X 2 X 3) = 0. In other words, any two symmetry species are orthogonal. 6) The orbital.. p,, p,, p, transform identically as the symmetry species of the translations along the z-,y-, and z-axes, found in the last column of a character table. I n Td the three p orbitals transform jointly as species Tn.
+
+
+
+
+
+
+
+
+
5 Group or Symmetry Orbitals Although the utility of symmetry arguments is enormous and many examples could be cited, we shall restrict our discussion to one application of symmetry arguments tbat has become especially important in transition metal chemistry but can be exemplified by very simple molecules. Again, we use water as a n example. If we look a t the atomic orbitals of the molecule and focus first on the nat,ure of the hydrogen-oxygen bonds, we discover an immediate discrepancy. Let us look a t the H 1s orbital of H. in Figure 12B. Consider only this one atomic arhitsl (Fig. 12C). This single orbit4 does not have the symmetry properties
id!
ibl
of the molecule, i.e., on reflection or rotation it is neither symmetric nor antisymmetric because there is no corresponding orbital on the other side; we simply cannot treat the single atomic orbital alone, hut somehow must comhine the two hydrogen atomic orbitals (Fig. 12D) in such a way tbat the combination will have the symmetry properties of the molecule. In making combinations of atomic orbitals, we combine linearly (LCAO), which means that we simply add and subtract. the two orhit,als. The combinations are shown in Figure 12E and F. These two combinstions are called symmetry or group orbitals and now possess the symmet,ry of the molecule; the sum belongs to species A, and the difference to species B1 in point graup Czs The oxygen atom bas one s and three p orbitals, p,, p,, and p,, assigned to symmetry species A,, At, BI, and Bs, respectively, as shown in Figure 12. I n order to generate the molecular orbitals, we combine the atomic or group orbitals of the appropriate Hlss comhination (A,) can symmetry. We see that the Hls, combine with the oxygen 2p, orbital ( A , ) (Fig. 126) and that the Hls, - Hiss comhination has the same symmetry (Bn) as the oxygen 2p, orbital and combines with i t (Fig. 1217). I n n simple picture we can disregard the participation of the 28 orbital in bonding, i.e., neglect hybridi%ation. We have two electrons available from the hydrogen atoms and six valence electrons from the oxygen. The two bonding molecular orbitals we have described accommodate four electrons and the additional four electrons can be placed in the oxygen 2s and 2p, orbitals as lane pair electrons. If we allow the 2s orbital t o mix with the 2p, orbital snd with Hlss combination (they all belong to AI), one of the the Hls, lone pairs ttcquires some p character, and becomes concentrated on the p, axis, mostly on the side away from the H atoms. According to this picture, the two lone pairs, one in this hybrid orbital and the other in the p, orbital, are nonequivalent, and this is in accord with experimental evidence from spectroscopic and ionization potential dsts. However, when t,hese lone pairs are used to form H-bonds, it appears like they are equivalent,, and describe, together with the other two OH bonds, a slightly distorted tetrahedron. This apparent conBadiction with respect t o the equivalency of the lone pairs has no solution, since we cannot distinguish electrons. Only when the equivalence is tested using some probe, can we get an apparent answer. But of necessity the probe perturbs the system, and depending on the nature of the perturbation we obtain one answer or the other. The description of the lone pair orbitals we have given above involves one as having pure p chart~cter(p,, species B,), the other a hybrid, mostly s with some p character; bath w e occupied by two electrons. It is now Dossible. without itffect,ine the orediet i m of a n y ot,~rrvaldrproperty d the ridccule, t o form lincrr comt,inntions 0: t h r r two lone pnir orhitnls. The re>nlt.
