D. I. Ford
LeTourneau College Longview, Texas 75601
Molecular Term Symbols by Group Theory
Simple molecular orbital theory is currently being applied extensively on the undergraduate level in practically all divisions of chemistry. Even the casual user of MO (molecular orbital) theory is likely to be confronted with the task of determining molecular term symbols. Usually, each type of molecule is treated by special methods. There is a need for a general, yet relatively automatic algorithm for the determination of molecular term symbols. Origin of the Problem and Sources of its Solution
Conceptually, simple LIO theory is closely related to the familiar atomic orbital theory which every freshman chemistry student encounters. Here, the student is asked to build electronic configurations for atoms from single-electron states (orbitals). In M 0 theory, the analogous configuration description is obtained by using molecular orbitals instead of atomic orbitals. Like atomic configurations where one invokes the Russell-Saunders coupling scheme, MO electronic configurations are incomplete and usually each can produce several states differing in properties such as energy, total spin, and symmetry type. Although the configuration specification is adequate for some purposes, the more complete description of the electronic states (which are labelcd by so-called molecular term symbols) are needed to discuss such subjects as uv and visible spectroscopy and correlation diagrams. It is desirable then to know which molecular term symbols arise from a given electronic configuration, and this can be achieved without performing any quantum mechanical calculations. Kilpatrick and Curl (1) give the group-theoretical answer to this question for atoms and Goscinski and Ohrn (2) do likewise for molecules. I n a recent article in THIS JOURNAL (S), Ellis and JaffB give an excellent account of an important portion of the problem but their treatment calls for generalization. In the following sections we attempt to present a general solution in a manner most beneficial to the average user of MO theory. Our discussion is limited to configurations with only one partially filled shell. (A shell is here taken to be a set of degenerate orbitals.) This covers all ground and some excited-state configurations. The way one determines term symbols arising from configurations involving more than one incomplete shell is to first find the term symbols resulting from each shell and then to do a direct-product analysis. The direct-product analysis between incomplete shells is a straightforward problem adequately discussed by others (3, 4). Pertinent Group Concepts
Out of necessity, it will be assumed that the reader has knowledge of elementary group theory on the 336
/ Journal of Chemical Education
level of several texts now available of which Cotton's hook (4) is typical. Only a brief review will he presented here. The elements of a group generally do something such as rotating an object or a vector, or transforming a function into a combination of other functions by a coordinate transformation. We can therefore regard the elements as operators and construct matrix representations of the operators. The objects acted upon are called the basis of the representation. We will be interested in basis-sets which are eigenfunctions of the electronic Hamiltonian of a molecule. It is well known that sets of degenerate eigenfunctions yield irredun%le representations for the group of symmetry elements which leave the classical energy unaltered. Irreducibility means that the matrices cannot be block diagonalized by a change in basis. An important group is the set of rotations and reflections of the nuclear framework of the molecule which carry the framework into itself. Another extremely important group is the set of all possible permutations of the identical electrons. The elements of both of these groups in no way alter the classical energy of the molecule. It is a great asset that the eigenfunctions may be rigorously labeled by the irreducible representations of the associated permutation and point groups. However, it may happen that we choose to employ a basis-set whose members are not necessarily eigenfunctions of the exact Hamiltonian. This representation is in general reducible (block diagonalizable). An important example for us is the hasis-set consisting of products of one-electron eigenfunctions which are derived from either the Hamiltonian of a single electron moving in the field of the bare nuclei, or a t best, in an average field of the nuclei and the other electrons. The representation of the group operations and the exact polyelectronic Hamiltonian afforded by a set of degenerate product-functions is therefore expected to be simultaneously block diagonaliable. This is the origin of the splitting of an electronic configuration into various terms. When electronic repulsion is explicitly taken into account, states corresponding to these various symmetry species differ in energy; however, this calculation is out of the realm of group theory. Group theory tells us that N., the number of times the orirreducible representation occurs in a reducible representation R, is given by
Where h is the order of the group, the sum is over the group elements G, and x,(G) and xn(G) are the traces of the representation of Gin the a and R-representations,
respectively. The asterisk indicates the taking of the complex conjugate. The Pauli Exclusion Principle and Eledron Spin
In the zeroth order approximation to the electronic description of molecules or ions, the electrons are individually assigned to spin-orbitals R(i), which consist of a spatial part +(i) and a spin part u(i). R(i)
=
the correspondence. As an example, for n equal 5, the spin-state partitions correspond to the irreducible representations (51
(4,11
13~21
with Young diagrams
*(Mi)
u(i) can be only one of two states, a(i) or P(i) corresponding to the ith electron having an intrinsic spin respectively. component of or The overall state function is a product of such singleelectron functions and it is specified by the configuration under investigation. Therefore, the total polyelectranic wave junction is a product ofa spin part and a spatial part. Up to now we have done a blasphemous thing by not considering the Pauli exclusion principle. It requires all polyelectronic wave functions to be totally antisymmetric with respect to all pairwise interchanges of the identical electrons. Equivalently stated, they must transform like the (1") irreducible representation of S,, the symmetric group on n-objects. (Readers not familiar with this group should consult Appendix I before proceeding to the next section.) From the configmation standpoint, this excludes those electronic configurations with more than one electron in the same state. Furthermore, nature compels us to take linear combinations of the polyelectronic spin functions and combine them with an appropriate cornbination of spatial functions such that the product between the two is totally antisymmetric. When dealing with two equivalent electrons, we merely use a symmetric spatial function with an antisymmetric spin function or vice-versa. However, for more than two electrons we shall see that other possibilities must be considered.
Each row of the Young diagram represents a number of the partition. By convention, the larger parts of the partition are written first. Filling all of the boxes of the first row with electrons of spin and those of the second row with spin -I/%, we get a suggestive and correct partition to total spin (8) and multiplicity (2S+1) association
(Hextet)
(Gblet)
The Young diagram resulting from an interchange of rows with columns is said to be the canjugate of the original diagram. The conjugates of the above diagrams are, respectively
For instance, the conjugate of ( 4 , l ) is (2,l8]. A partition may be self-conjugate. The two-part partiTable 1.
Characterization of the Spin Functions
We shall have need for the character tables of two groups: the point group of the molecule's symmetry and the symmetric group on the n-electrons in a particular level. The use of the point group will occur in the next section. To classify the spin functions, we need the character table of the symmetric group (6) and some of its properties. Table 1 lists Spthrough S5along with classification of the resultant spin states. We now discuss the method used to make the spin state to irreducible representation correspondence. Each of the various irreducible representations of S, have been labeled by a partition of the number n. There is a one-to-one correspondence between the partitions of n and the irreducible representations of S, (Appendix I). The theory of group representations tells us that this is the most natural label. Having on hand the character tables (which Littlewood (5) gives through Slo) let us now see how one can by inspection of the partition, locate the irreducible representation which belongs to a particular resultant-spin state for electrons. It can be shown that the partitions of n into at most two parts corresponds to a unique spin state (6). Entities called Young diagrams are useful in making
(Quartet)
S. Class Order
[PI
(lPI
Ss
Class Order
Character Tables of the Symmetric Groups 1' 1 1 1
2 1 Spin Basis 1 Triplet -1 Singlet
l8
1,2
1 1
2 1 S,
Class Order r-(41
1 [\Qli,
1' 1 1 3
2
3
1 0 -1
3 2 Spin Bssis 1 Quzrtet -1 Doublet 1
i8,z i,a 6 8 1 1 1 0 0 --1
4 2% 6 3 Soin Basis 1 1 ~uintet -1 -1 Triplet 0 2 Singlet
-11~1
Ss
Class Order
Volume 49, Number 5, Moy 1972
/
337
tions in which we are interested will necessarily have conjugates consisting of only ones and/or twos. In the character tables, we have Connected the conjugate representations by lines. Those with no tie lines are self-conjugate. I n the next paragraph we show that the representations conjugate to the spin representations describe the Pauli required permutation symmetry of the spatial part of the wave functions. Characterization of the Spotiol Functions
An important property of the conjugate pairs of the irreducible representations of S , concerns the characters themselves. The characters of one can be obtained from those of the other by changing the signs of those belonging to classes of odd permutations. From this property of the conjugate, it is easy to show that a direct product representation between conjugate pairs contains (1") once and only once. Any other combination will not contain it. (1") is totally antisymmetric and is the way all wave functions must transform if they are to comply . . with the exclusion principle. Therefore, a r spin function requires a F spatial function where P is the irreduci6le representation of S , cmqjugate to T. The spin-free formalism advocated by Matsen (7) would have us consider only those partitions using ones and/or twos (of the form (2P,1"-2P))and then we would make the spin multiplicity assignment without referring to spin functions by using the relation multiplicity = 2S f 1 = n
the configuration has n-electrons in an orbital of degeneracy d. First choose a possible spin state such as a triplet, singlet, etc., and then with the character table for S,, locate which conjugate representation is needed. Now, imagine that we have made all possible assignments of t h e n electrons to the set of d-degenerate orbitals. Let us suppose that we are able to find the combinations of the resulting product-functions which yield the appropriate symmetry (conjugate to the spin symmetry). If we could determine the trace of all elements G of the point group in this new Pauli-acceptable basis-set, wecould, by inspection or by eqn. (I), resolve the resulting representations against the irreducible representations of the point group and thereby determine which symmetry species are present and how many times they occur. This trace problem has been solved by a theorem proven in Appendix I1 and we need not actually construct a new basis-set. The theorem relates the trace of G in the new basis to the characters of G and its powers. By the closure property of groups, we know that G to any power is also contained in the group, so we need not compute any new characters. The equation for the trace of G in the new basis having permutation symmetry PI is
The sum is over all classes of 8,; h, and @(p:E,) are the order and character of the pth class in the irreducible representation conjugate to that of the spin; x(G) is the trace of G in the irreducible representation of the orbital under inspection. The expression in the brackets is obtained directly from the cycle structure of the pth class. For example, if we were dealing with four electrons, the (12,2) class would yield the term
- 2p + 1
This, as we have seen, amounts to a restatement of the Pauli principle and is applicable when the wave functions are products of a spin and a spatial part (no spin-orbit interactions). Indeed this is a most compact summary of the previous paragraphs, and we could have started at this point if we had wanted to obscure the role of the electron spin.
The problem is solved. Having obtained the traces, we can determine the resulting symmetries by resolution against various irreducible representation of the point group. Let us use eqn. (2) and Table 1to obtain the following useful special cases. Two electrons, n = b.
Solution of the Trace Problem
The following is an outline of the approach taken toward finding the molecular term symbols. Suppose Table 2. On
E
8Ca
6Cz
6C1
3C2
i
AT, A*, Eo
1 1 2 3 3 1 1 2 3 3
1 1
1 -1 0 1 -1 1
0 0
1 -1 0 -1 1 1 -1 0 -1 1
-1
1 1 2 -1 -1 1 1 2 -1 -1
1 1 2 3 3 -1 -1 -2 -3 -3
-1 0 1 -1 -1 1 0 -1 1
1
1
-1
-1
1
1
-1
8
-1
0
TI,
T9,
A>"
A.. E"
T,. T*.
x(G:quartet) eqn. (4b) x(G:doublet) esn. ( 4 ~ )
338
/
The Octahedral Group Character Table
-1
o
0 1 1 -1
Journal of Chemical Education
-1
0 1
0
0 8 ( t ~+ ~ 'Azg, ) ~ %Em=TU, 'Tw
684
8Se
3a.5
6aa
1
1 1 -1
1 1 2 -1 -1 -1 -1 -2 1 1
1 -1 0 -1 1 -1 1 0 1 -1
0
0 0 -1 -1 1 0 0
1 -1
1
0
-1
0
'Aw
'ED,'7'10
¶TxO
Three electrons, n
Other configurations in octahedra symmetry yield the rnolecular term symbols listed in Table 3. Use was made of the well known result that a shell of degeneracy d occupied by n electrons produces the same set of term symbols as the same shell occupied by (2d - n) electrons. This is a great work saver since we need only do a detailed analysis for at most half-filled shells. Also, note that a singly occupied shell is a trivial case. I t always yields a single doublet-term of the same point group symmetry as the orbital, e.g., (e,)'+ aE,.
= 3.
Example II
Example I We now work through an example illustrating the above pracp dure. Suppose we have a molecule or ion with octahedral symmetry which has the electronic configuration (ts)B. That is, it has three electrons in the triply degenerate orbital. We ask for the possible states that this can give rise to when the exclusion principle and electronic repulsions are accounted for. Since n equals 3, we use eqs. (4a) and (4b) for the doublet and the quartet. Ga, G: and their traces must he determined. The computations are best done in tabular form. First, we give the character table of Onand then the scheme for determining the traces and finally, the resulting term symbols (seeTahle2). We note that we could have shortened the computations considerably by observing that the g and u designations can be determined by inspection. They refer to symmetry and antisymmetry with respect to theinversion operator. Since g X g = g and g X u = u and u X u = g, we could have predicted that thestates misingfrom (t~,)~would all beg states since (g X g) X g = g. Therefore, the improper rotations of O h could have been omitted from our table with considerable amount of labor heingsaved.
Table 3.
To illustrate the method further, we determine the molecular term symbols arising from the (g.)' configuration of a molecule - ) icosahedral symmetry (point group or ion (such as B I J I ~ ~with In). Since the product of four u states yields g states, the improper rotations of I*giveus no other usefulinformation. Hence, we examine I, a subgroup of In consisting of the proper rotations of the molecule only. Equations @a), (5b), and (5c) call for G: Ga and G4 to be determined. As with the previous example, the solution is presented in tabular form (Table 4).
Example Ill Consider a linear molecule of D,r, symmetry and having the configuration (T")'. The resulting term symbols will be g states since u X u = g, so we can drop the inversion operation and deal with the subgroup C-,. The computational scheme is given in Table 5 along with the term symbols. For a continnolis group such as we have here, eqn. (1) is not applicable (8)and the resolution is done by inspection. The experienced reader probably used a vector addition scheme to get the same results almost by inspection. This method, however, is only good for linear molecules. This exsmple was chosen to illustrate the generality of the method presented in this article.
Table 5. The Character for
Term Symbols for a Molecule of Octahedral Svmmetw
Confirmration
,C ,
E
2C-+.
z+
1
1 1
A B
1 2 2 2
G G' x(G)¶ in II
E E 4
2-
n
Resultine States
.,
~ W S +
2 cos z+ 2 cos 36
Cm% C, 4 oma+ = 2 2 cos 2+ 2eos24 1 1 2 cos%
+
2 x(G9 in n x(G:triplet), eqn. (3b) 1 x(G:singlet), eqn. (3a) 3 (r..P Y-".%+". 'A"
-
Table 4. I
E
The lcosahedral Group Character Table 12Ca 12C2 20Ca 1
x(G:quintet), eqn. (5e) x(G: triplet), eqn. (56)
1 15
x(G:singlet), eqn. (5a)
20
1
+
C,. m cu
1 -1 0 0 0 *U
E 0 2 -1 1
$2-
4 :
A'
15C1 1
'Ag,'Gg, 3'H. (gd4+ 6A,, 8T~.,BTs.,BGo,W0,
Volume 49, Number 5, May 1972
/ 339
Appendix 1.
The Symmetric Group
Label n of the d components and form the products
The set of all n! permutations of n objects is a. special case of a. permutation group. I t is called the symmetric group and is denoted by S.. This group plays a central role in representation theory and i t finds many physical applications. Since not all chemists are acquainted with its properties and its special not* tion, a short introduction t o this fascinating subject will be given here. neing perrnurations, the el~rnentsof S, ran be represented by an) of severn! ways. Oneia!,~.tlaeaymbol
ai = 1,2, . . d with repetition being allowed. Symmetrize these products with respect t o the Ath irreducible representation of some permutation group P on t h e n labels. Denote the order of P by h. The appropriate group projection-operator (9) is used toperform thesymmetrization.
indicating that the permutation p replaces object 1with object i,, object 2 withobiect i,. etc. For manvmrooses weneed notsnec-ify p so completely and we only indicate its cycle sfruelure. Consider apermutation on eight objectsgiven by
p i s the permutation operator belonging t o P and *,'is its chrtrac ter in the Ath irreducible representation of P. We note that p can operrtte on the labels in parenthesis or on the subscripts. We choose the subscripts. Now, our goal is t o express x(G), the trace of G, not in R but in theset of allfunetionsW"aa a,. . . an{. Using eqs. (1) and (2) weobtain
. -.
".
~A
~~~~
.
It is seen that 2 reolaces 1and 1reolaces 4 while 4 reolaces 2. We say that the has indiced a cycle of lekgth 3. Completing the analysis, p i s seen t o partition the objects into disjoint cycles For sirnplirity, we havedropped thesubsrript o n q'ln,, a,. . . . 0.1 To take the trice of G in t h i ~new spare, we need to rerriwe our symmetrized products. We do this hy using rhciderrtiry
Therefore, p can he represented by a partition which indicates its cycle structure.
Hence
p = (3,2,2,1) or (3,251)
That is, it induces 1cycle of length 3,2 of length 2, and 1of length 1. This description of p is not complete as there are other permutations with the same cycle structure. I t can he shown that the number of permuttttions with cycle structure
For the previous example, this expression has the value =
......
Now, we can take the trace by selecting i = a>,j = az, a. and summing the coefficientsover all a,, a s . . a .
n!/(lb6,!26~bn!.. .na*b.l) 8!/(3.2'.2!.1)
Using this in eqn. (3) we get
P
It happens that all permutrttions with identical cycle structure belong t o the same class. The classes of S. are then conveniently labeled by a partition of n indicating the cycle structure of its members. Furthermore, i t can be shown that even the irreducible represeutations are meaningfully labeled by partitions. S. has the same number of irreducible representations as there are
But
...
*,%
x(G;qk) = l l h
1680
nz
. ..
i
j
L~Z
.
... z =
G~I..PIG~..,P~. . .G...nPI
(I"
GijGjr. . .Gxi = x(Gm)
k
.
where the indices i, j, k . . form a cycle of lnegth m. So, if we . .nb*) we can write represent p by its cycle structure p = (la2*".
nrtrt,itions a... r~~~ . ~ .... .of ..~
The reader s h ~ u l dnow be in B positl~nto appr~viatethe form of the eharacrer tablei presented ~n 'hhle 1. The order of a class is dvfmrd as thenurnher of c l e r n w t ~ i nit. h'otr rhut i n all cases the I I") irreducible representation is totally antisymmetric. That is, the classes of permutations which can be achieved by au odd number of pairwise interchanges (odd classes) have characters of -1 and the other even classes have characters of f l . On the other hand, ( n { is said t o be totally symmetric since all of its characters are f l . The other irreducible represeutations have dimensions greater than 1 and are said t o be of mixed symmetry.
A p ~ e n d i xII. A Theorem Concernina the Trace of a ~ b i r i xin o Certain Symmetrized ~ i r & tProducf Space in Terms of its Trace in the Original Space
.
Consider alinear vector spaceR = (RI, Rn,. . Rd) of dimension d. Let G he a linear operator associated with the space
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/
Journal of Chemical Education
This is eqn. (2) used in the body of this article for apartioulm permutation group, the symmetric group. This elegant and useful result was previously proven by Goscinski and Ohm (3)by using a more involved grouptheoreticd analysis.
Literature Cited (1) CWL. R . F..A N D KILPATRICK. J.E., A m ? . J. Phw.. 28,357 (1960). GOBCINBRI, 0 . . ANY O R R N Y . . .Int. J. Quantum C h e n . , 2,845 (1968). ~.. L.. A N D J ~ r r eH. , H., J. Cmr.Enuo., 48.97 (1971). (3) E G L IR (4) COTTON,F. A,, " C h e m i c d Applications of Group Theory," Intersoience Publishers,NelvYork, 1963, p. 314, (5) LITTLEWOOD. D.. "The Theory of Group Characters," Oxford Press. NervYork. 1950, pp. 264-71. (2)
(6) D*vroov. A . 5.. "Quantum Meohanics." Persamon Preaa. New Y o r k . (7) M * ~ a e n F. , A , , J:Amer. Ckem.Soc., 92,3525 (1970). L..AND C m m , S. J., J. CRE.I Eom., 48,295 (1971). (8) SCX*I.ER, M . . ''Group Theory and I t s Appliaations t o Physical (9) HAMERIESH, Problems." Addison-W&y. Reading, Maaa.. 1064, p. 113.
