and
R. 1. Ellis' H. H. Jaffb
University of Cincinnati Cincinnati, Ohio 45221
I
I
The Symmetries and Multiplicities of Electronic States in Polyatomic Molecules
The problem of determining the number of clrrtronic st:ites oi 3n tltom rrrising f n m n given electronic configurarion, their inultipliciries, and qmmetry types arises in both graduate and undergraduate courses in chemistry. Solution of this problem is straightforward if the total orbital angular momentum and the total spin angular momentum can be considered as constants of motion for the system. This allows the possible states of the distribution in question to be classified by their various L and S values. A knowledge of L values leads to the symmetries of the statesS , P, D, etc.; and the S values lead to the multiplicities (1).
In nonlinear polyatomic molecules, the orbital angular momentum is not a constant of motion, and thus no values of L exist with which to classify the symmetries of electronic states. The general approach to the solution of the problem in molecules rests in the theory of groups. Several authors have approached this problem at the introductoiy level (B, S), but the cases treated are in general restricted to molecules of low symmetry. As a consequence, systems with both equivalent and nonequivalent electrons are not treated in a general fashion. Benzene has sufficiently high symmetry to present the problem of the treatment of both equivalent and nonequivalent electrons. In this context, equivalent electrons are defined as electrons occupying a set of degenerate orbitals, and the interesting problems arise where such a set is not. completely filled. Consequently, we have chosen the (a?,)2(e1,)2(e2.)2configure tion of benzene as an example. This configuration is a doubly excited configuration of the molecule, accessible from the ground state by simultaneous excitation of two electrons, and all the states corresponding to this configuration are doubly excited states. Such states are not normally observed by standard spectroscopic techniques; however, recent advances in laser technology have made the study of such states feasible (4). The configuration is illustrated in Figure 1in terms of a molecular orbital energy diagram; it gives rise to several states of different symmetry and multiplicity. We shall show how one can determine what states arise from this configuration. Electronic Stater in Organic Molecules
In organic molecules the orbital and spin angular momenta of the individual electrons are not strongly coupled. Thus the wave function (spin orbital), Supported in part by NSF Grant # GP 15944. 1 Procter & Gamble Research Fellow, 1969-70. 2 This approximation breaks down when spin-orbiting coupling is large, i s . in compounds of h e w atoms.
92
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Journal o f Chemical Education
Figure 1.
Energy level diagram of the lol.121etolaledz cofiguration of
benzene.
+P(j) of an individual electron, j, may be considered as a product. * P ( j ) = *i(nj).i (1) where +' is a function of the space coordinates, q,, of the electron j, and is called a space orbital, and o< is a function of the spin coordinate only. The spin function u1is necessary in eqn. (1) to. take into account the fact that each electron has a spin s = which can orient itself parallel or anti-parallel to any preferred direction. This means that the spin of an electron can be treated as a vector of length unit. In a many electron syst,em, such as described by eqn. (4) (vide infra) the spins of the individual electrons can be added vectorially to form a resultant S. This is exactly the same situation which arises in many electron atoms. The resultant spin S is a characteristic of each electronic state. As in atoms, the coupling of the spin S to the orbital motion may lead to a splitting of the molecular electronic states into 25 1 components, where S is the magnitude of S . This multiplicity is written as a superscript in front of the symbol representing the symmetry type of the state. Thus, for S = 0, we have singlet states, indicated by a preceding superscript 1,and for S = 1we have triplet states. In all cases under consideration here, the spin functions o, transform as the totally symmetric species of the point group of the mole~ule.~Consequently each spin orbital transforms in the same irreducible repre
+
sentation as the space orbital it contains. Thus, the spin orbitals form a basis of the appropriate representation. As a first approximation to the wave function of a many electron system, a product of spin orbitals might be chosen. Then the wave function 9 for the many electron system is %, = ~(q,)ux*~(q*).*. . .*&")." (2) However, this function violates the requirement that any wave function describing a physically real system of particles must be either symmetric or antisymmetric with respect to the exchange of the position coordinates of each particle. Since we are dealing here with electronic functions, we are only interested in antisymmetric functions (5). The exchange of ql and qz in eqn. (2) leads to which is equal to neither the 9 of eqn. (2), nor its negative, and hence the functions are neither symmetric nor antisymmetric. A suitable function can be constructed by linear combination, and in general is expressed in the form of a Slater determinant.
Such a function as this guarantees antisymmetry since the exchange of any two rows of a determinant changes its sign. The exchange of two rows of the Slater determinant given by eqn. (4) has the same meaning as the exchange of the coordinate of any two particles of our n particle system. Since the Pauli Principle allows two electrons with different spin functions to be described by the same space function, we shall treat the problem at hand by subdividing the total electron population of a molecule into a set of "building blocks" (6) each of which is the set of electrons occupying one space orbital, or a set of degenerate space orbitals. For each "building block" we can then determine a multiplicity and symmetry species, and each will transform as an irreducible or as a reducible representation of the molecular point group in question. Finally by properly multiplying the "building blocks" together we can obtain a description of the states of the molecule. Since each space orbital accommodates a maximum of 2 electrons, each "building block" accommodates a maximum of 2n electrons, where n is the degree of degeneracy of the orbital(s) making up the "building block." Since we are dealing here with an excited configuration, not all "building blocks" have their maximum occupation. As previously stated, the $' are suitable functions with which to form the basis of the irreducible representations of the point group of the molecule. Consequently, it will be profitable to examine the transformation behavior of a general set of functions which form the basis of an irreducible representation of the operations of the point group. Dired Product of Irreducible Representation
Suppose there exist two sets of functions fl", fZa,
. . . fnmand gf, g2@,. . .g"flwhich form the basis of the two
irreducible representations a and 8, respectively. Examples of such sets might be the molecular orbit,als of benzene; e.g., for a = a2,, the whole set is the single
orbital ql with (n = I), while for a = el, with n = 2, the MO's $S and $3 are such a set. Forming the yroduct of these two sets of functions, faagB, a system of n.n8 new functions is obtained which can serve as a basis for a new representation of the group. This new representation will have the dimension n,na, and is called the direct product of a and 8. It is irreducible only if a or 8 is unity. The characters of the representation of the direct product are equal to the product of the characters of the two components: let be any operation of the group, then =
&Gz,*fp
(5.)
where the GI,are constants which depend on the nature of 6, i.e., the elements of the transformation matrix of 6. Thus, under the operation Cs, the orbital $2 (a = el,) transforms into (7)
G@ = 1/2
GII. = 4 %
Similarly
and
In eqn. (7),if fl" andfk are hand $8 and glBand g 8 are h and $5 of benzene, &f?gSp) represents the transformation of the product $&. Letting the characters of the direct product of the a and p representations be denoted as [x" X xsl(G), we have from eqn. (7) for the sum of the diagonal elements of the matrix G
There isno reason why the two irreduciblerepresentations a and 8 in this discussion cannot be the same (e.g. both el,). For such a situation the two sets of functions jI.. .f, and g1. . .g, both form the basis of the same representation. In fact, as we shall see, this is the exact situation which arises when two electrons occupy the same space orbital. The direct product is given by the n2functionsf;g, and has the characters [ x X xl(G) = [(G)12
(9)
Here, the [x(G)IZare the squares of the characters of the irreducible representation for which the functions f, and gn form the basis. This representation can be reduced into two representations, which in general may themselves be reduced. These two representations are given by the 'lZn(n 1) functionsj;ga jagr, known as the symmetric product, and the '/zn(n - 1) funct.ions f,g, - fag,, known as the antisymmetric produd. The two new sets of functions are suitable to use as basis functions since the members of each set transform only into linear combinations of themselves under the 3 operations. The characters of the symmetric product are denoted as [X~](G), and the characters of the antisymmetric product as (X2)(G). The characters of the symmetric product are formed in the following way. From eqn. (7) .
+
+
Volume 48, Number 2, February 1971
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93
a(f~gr+fw4
=
C G~iG,s(fig~+fngd t,m
C ( G ~ ~ G ~ ~ + G , @ ~jng,) ) ( ~ , (~l o~) +
=
1% .,...
The characters are then given by [xzl (G) =
C (GiiGa + G o G d
(11)
i.n
them to be identical to the characters of the al, representation of the Dohpoint group. The character table for the Dm point group may be found in Table 2. From this we mav conclude that nlacinz two electrons in a, leads to a "building block," Br, of symmetry al., and further since the total spin must be S = 0, the spin multiplicity must be 1.- Therefore, this 'first "building block" may be symbolized as
-
But since
BI
C GiiG=
=
[x(G)IZ
i.n
and
C GdIhcli = x(GP)
(I2) (13)
The characters of the antisymmetric product are found in a similar manner. (14)
lx'l(G) = '/aIIx(G)IP - x(Ga)l
In the special case where n = 1 only the symmetric product has meaning. The characters of such a product are given directly by eqn. (9). The (02.)"e,,)P(e,.)*
Configuration of Benzene
Using the results of the previous section, we can now proceed to determine the symmetries and multiplicities which arise from the electron configuration of benzene given in Figure 1. Benzene belongs to the point group Deh, and in Figure 1 the individual molecular orbitals $i have been labeled by the irreducible representations they form the basis. If we put twoelectrons for in a,, the Pauli Principle requires that the spins must be paired. The total spin contribution from (a,)%is, therefore, S = 0. The symmetry is given by the product a~.
x
a~,
The characters for this product as given by eqn. (9) are listed in Table 1. Inspection of these characters shows Table 1.
=
'A,,
(15)
It can readily be shown that any "building block" consisting of a doubly occupied nondegenerate orbital is equivalent to lA1, in Do), or a singlet of the totally symmetric species in any point group. The same holds for any n-fold degenerate set of orbitals occupied by 2n electrons, i.e., for any closed shell. The next "building block" to he formed is one in which two electron' occupy a two-fold degenerate molecular orbital. The Pauli Principle now permits two possible spins S = 0 and S = 1. If two electrons occunv ' " two orbitals. each in a different "building block," there arise states of each symmetry contained in the direct product of the symmetries of the two orbitals, and for each symmetry there arises a state of each of the predicted multiplicities. However, when the two electrons occupy degenerate orbitals, this is not the case. The auestion is, then. to which svm0 metry components are'the spin functions with S and S = 1 to be assigned. The spin function u = a(1)8(2) - 8(l)or(2) is antisymmetric with respect to = 0. e x c h a w of electrons and corresponds to On the other hand, the spin functions u = a(1)8(2) P(l)a(2), a(l)or(2), and 8(1)8(2) are symmetric with respect to exchange of electronic coordinates, and correspond to the values M , = -1,0, +1, respectively, of S = 1. Recalling that the total wave functions given in eqns. '(1) and (2) must always be antisymmetric, we must conclude that the symmetric spin functions be associated with an antisymmetric space function, and the antisymmetric spin function be associated with a symmetric space function (8). We are left, therefore, with the task of finding the symmetric and antisym-
=
+
The Characters of the Symmetric and Antisymmetric Direct Products of Representations a,, and el, of Point Group DM
These characters are based on the el, representation of D6i. Table 2.
94 / Journal o f Chemical Education
The Character Table of the Point Group
bn
metric portions of the direct product of the space function But these are given by just the symmetric and antisymmetric products discussed in the preceding section. I n order to compute the characters of the symmetric and antisymmetric direct product, the values of [x(G)I2 and x(G2) are required. The first term is the square of the characters of the el, representation found in Table 2. The characters, x(G2), are obtained from the trace of the square of each of the matrices forming the el, representation. Fortunately it is unnecessary to know the exact form of these matrices, since we can use the symmetry operation itself to determine the desired characters. As an example, consider the C6 operation of Dm. If we apply this operation twice, the result is the same as having applied C3 once. The character of the square of the operation in the el, representation will be the same as the characters of the C3 operation. Thus we see that all two-fold opere tions, that is operations which, when applied twice, yield the original conformation back, will have the character of the identity operation. The operation C6 and S6,when squared, yield C8,while C3 and S3yield CS2. The formation of the characters x(G2) is summarized in Table 3. Using the result of this table and Table 3.
Result of Applying the Symmetry Operations Twice For a DonMolecule
Substituting eqns. (15), (It?), and (17) into eqn. (18) we have But multiplication of any set of characters by the characters of A,, (in &), or of the totally symmetric representation in any point group produce no change in the characters so multiplied. Neither does multiplication of a term of any multiplicity by a singlet change the multiplicity, since multiplication by a singlet corresponds to addition of a vector S = 0 to the other S vector. Consequently, any closed shells, which were shown to be 'A,,, produce nothing new in the multiplication, and can be neglected. This fact, of course, explains why we have been able to ignore the c-electrons completely, since they form a closed shell. Expanding the product given in eqn. (19) results in nine terms.
++ +
++ +
('A,, X 'A,,) ('Au X 'A*) ('AI. X 'Ew) (8A2, X 'A,.) PA%,X ('A,. X 'E9,) ('Ez. X 'A,,) ('En. X =Axp) ('E20 X 'E*,) (20)
In order to evaluate eqn. (20) we require a knowledge of the spin which can arise when species of different spins are multiplied. Two spin vectors S, and Skmay be added in only certain ways. The length of the resultant S,% is given by sir = Si Sk,Si S* - 1, . . ., pi - SbI (21) Thus if the product of two triplet species is formed the resulting allowed spins are
+
+
sir These characters are based on the e,, representationof DL.
the characters of the el, representation, we can now compute the quantities necessary to solve eqns. (13) and (14) for the symmetric and antisymmetric direct product of el,. These quantities are given in Table 1 along with the characters of the symmetric and antisymmetric direct product. Inspecting the characters of the antisymmetric direct product, we see that they are identical to the characters of the a,representation of Dm. Since, as has been stated previously, the spin function which must be associated with the space function of this symmetry have S = 1, this portion of the "building block" is given as 3Az,. The characters of the symmetric direct product are not equivalent to any of the irreducible representations of the group. This representation can, however, be reduced using standard techniques. The results show that this representation is given by Associating these symmetries with the spin function with S = 0, the total "building block" BII is given by BII
=
'A,,
+ aA1, + 'Enr
(16)
The "building block" B ~ IisI formed in the same manner as BII and the results turn out to be identical to BTI, thus BIII = 'A,,
+ $Ax. + lE%E1,
(17)
Using these results, the total manifold of symmetries and multiplicities may be found by forming the product of all of the "building blocks"
++
=
2, 1, 0
The multiplicities of the species are 5, 3, 0. The results of forming the product of species with different spins is summarized in Table 4. Using these results, Table
4.
Multiplicities Arising from of Two Terms
-Separate TermsSinglet Singlet Singlet Doublet Doublet Triplet
+ Singlet + Doublet + Triplet, Doublet +++ Triplet Triplet
the
---Combined Singlet Doublet Triplet Singlet Doublet Singlet
Combination
Terms----
,+ Triplet
Quartet ++ Triplet + Quintet,
eqn. (20) may be reduced to yield 'A,.
+ aA,, + 'Exg++ OAsg + (6A1D+ SAl. + + .Ezs 'E2. + 3E2. + ('A,. + IA2. + ' E P ~ )(22) lA1.)
Thus the possible st,ates that can be formed from the electron configuration (a2,)2(e1,)2(ez,)zin benzene (or any Dm case) are 'A,,, 'A,,, 'Ad3h 'A&),
'Am 'E&),
'Ed3)
Spectroscopic States of XeFa
I n most introductory courses in inorganic chemistry the student is introduced to ligand and cryst,al field theory and the construction of Orgel diagrams. While any detailed discussion of these subjects is beyond t,he scope of the present paper, it seems logical to discuss the construction of the strong field side of an Orgel diagram in light of the results of t.he above section on Direct Product of Irreducible Representation. Volume 48, Number 2, February 7 977
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95
The compound XeF4is an example of Dm symmetry. The splitting of the Xe p orbitals in the field of the four flourine atoms leads to an an, and an e, molecular orbital (9). This splitting is shown schematically in Figure 2. These orbitals are the highest occupied and the lowest unoccupied molecular orbitals of the XeF4 molecule. The question is what states of what symmetries may arise from a two electron occupancy of these orbitals. The possible configuration are shown in Figure 2b. From the above results we know that the first configuration can give rise only to an 'A1, state. The second configuration is the product of two "building blocks." The result of this product is BI X Blr = 'Am X ?EsE1. = 'Ea, 'EsEne I n order to determine the states which may arise from the third configuration it is necessary to construct the
+
atomic 2p orbitals
\\,
symmetric and antisymmetric direct products of e. X e.. The quantities necessary to solve eqns. (13) and (14), along with the symmetric and antisymmetric direct product are given in Table 5. Comparison of the characters of the antisymmetric direct product with the characters of the various irreducible representations of D4h,given in Table 6, reveals them to be identical to those of a,. Therefore the term with az, symmetry will have a multiplicity of three. The symmetric direct product does not correspond to any of the irreducible representation of the D,,group. This product can be reduced to yield states of symmetry a,,, bl, and bz,. Each of these terms must have a multiplicity of one. Therefore, the configuration (e2,)%gives rise to one triplet, 3Ao, and three singlets, 'A1,, 'BI,and 'BaP Extension to Triply Degenerate Sets
The problem of the (azu)Z(e133(e2Y) and (az,J2(elo)(en,)aconfigurations of benzene is trivial. These can give rise to "building blocks" with either one or three electrons. I n the three electron "building block" two electrons must be paired. The resultant spin in either case is I / , , and the multiplicity of the resulting "bnilding block" is 2. Further, since there are only two ways in which one or three electrons may be distributed over a doubly degenerate orbital, the symmetry must be E. Therefore, either one or three electron occupancy of a doubly degenerate orbital will lead to a ZE "building block." The finding of the preceding paragraph is partly combined in the general result that the termsarisingfrom i electrons being placed in an n-fold degenerate "building block" are identical to those which arise when (2n - i) electrons are placed in the same "building block." As another example, consider the "building block" of a tetrahedral ( T d )molecule which forms the basis of the T I representation. There are only three ways of distributing one or five electrons in this "building block"; therefore, the term which arises is 2 T ~ .Where two or four elect,ronsoccupy a T I "building block," the terms which arise are found in the same manner as discussed for two electrons in a doubly degenerate orbital. The symmetric product may be reduced to al e t,. The autisymmetric product is tl. Tberefore, the terms are 'Al, 'E, 'Tz,and 'TI.
'.,.--
Qzu 2p orbitals in a field
Deb
(a
Figure 2.
(01
Splitting of o p orbital in a D4nfleld.
+ +
Ib) The configurations
I O Z ~ ) m~d ,(ed'. ~ ~ ~ ~ ~ ~ , Table 5.
The Characters of the Symmetric and Antisymmetric Direct Products of Representation e, of Point Group D,h
n..
1
,,. , \ - ,
,
2Cde)
C 2 = C".
2C9
2C'.
2s"
m
2m6
2gs
Sl = i
-
These are the characters of e, in the Do group. 6 These characters are based on those of e.. Table 6.
Da
I
2C(z)
AU A," -4%. Azu
+I
B*"
+I +1 +1 +1 +1 +1 +1 +1
3, E..
f,",
B1.
B1"
Bs
96
/
+1 +1 +1 -1 -1 -1 -1 0 0
C?= Cut +I +1 +1 +l +1 +1 +1 +1 -2 -2
Journal of Chemical Education
The Character Table of the Point Group D,P,
2C1
2Crn
c,,
+i
+I +1 -1 -1 -1 -1 +1 +1
+I -1 +1 -1 +1 -1 +1 -1 -2 +2
+1 -1 -1 +1 +1 -1 -1 0 0
0
0
2s" +I -1 -1 +1 +1 - 11 +1 0 0
2sa
+I -1
-1 +1 -1 +1 +1 -1
0
0
2& +I -1 +l -1 1 +1
-1
+1 0 0
--
& i +I -1 +1 1 +1 -1 +1 -1 +2 -2
While the problem of three electrons in a triply degenerate "building block" can be handled in a way analogous to the problem of two electrons in a doubly degenerate "building block," the solution is in fact much more difficult. To overcome this difficulty the (t)3 configuration is analyzed with respect to its relation to the ( P ) ~configuration of an atom (6). The (71)~configuration in an atom can be resolved into the terms 4S, ZD,zP. For a molecule of Td symmetry the 4Sgoes over to the 4Az. The %D splits into two components. These are %Eand 2T1.The zP goes over into % T 2 . Thus the terms that may arise from a tz "building block," in a molecule of Td symmetry, occupied by three electrons are 4 A ~%E, , % T Iand 2Tz. Summary
A simple approach has been presented for determining the symmetries and multiplicities of electronic states arising from configurations containing both equivalent and noneqnivalent electrons. The method is based on theuse of different "building blocks" for each set of electrons described by the same space function. The
first step is the determination of the symmetry and multiplicity of each "building block." If a degenerate "building block" is encountered and it contains two electrons the results of the third section (The (az,)%(e1,)2(e1,)2Configuration of Benzene) are used to determine the symmetries and multiplicities. If more or less than two electrons are present the results of the fourth section (Spectroscopic States of XeF4) are used. The second step is to form the direct product of the various "building bloclts." This procedure leads to all states that may arise from a given electronic distribution consistent with the assumed model. Literature Cited Atomic Spectra and Atomic Structure,'' Dover Publioa14,Chapter 11. (21 J A ~ P H.~15, . AND O n c n r ~ .M.. "Symmetrv in Chemistrr." John Wilev &Sons, New York, 1965. (3) COTTON.F. A,. " C h e m i d Applications of Group Theory: Interscience Publishers (division of John Wiley &Sons, Ino.), New York. 1963. (4) J o m s ,W. J., Vuorteily Renicws. 23.73 (1969). (5) W = n , H.,"The Tlmw of Groups and Quantum Mechanics," E. P. Duttonand Co.. 1931,pp. 238-246. (6) H E n z ~ ~ n H., a , "Eleotronia Speotra of Polyatomic Molecules." D. van Nostrand Co., New York. 1966, Chapter 111. (7) J A F PH. ~ ,H..AND OROHIN,M., "Symmetry in Chemistry.'' John Wiley & 80118, New York, 1965,p. 85. (8) JAPPB, H. H., AND ORCHIN.M., "Theory and Appiiaations of Ultraviolet S~eotrosoopy,"Johnmiley &Sons. New York. 1962,p. 47. (9) Reference (8).Ch. 18.
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