A Simple Group-Theoretical Derivation of the Selection
P. K. Chattaraj and A. 6. Sannigrahi Indian Institute of Technology, Kharagpur 721 302, India
It is well known that an electric dipole-induced transition between two stationary states, 11.1 and 11.2 of a system can occur only when at least one component of the~transition moment, (&111.2) . . - . . - is nonzero. The conditions that must be met in order to obtain nonzero transition moments are called selection rules. For almost every type of transition there are two kinds of selection rules-moss and soecific. The moss selection rule for pure rotationil transitions is that the molecule under consideration must oossess a permanent dipole moment, whereas for Raman rotkional tr&itions the presence of anisotro~ic - -~olarizabilitvis the necessarv condition (now p is replaced by wind = at,where a is the polarizability tensor and r is the electric field vector of the radiation). These selection rules can be obtained in a straightforward manner as bas been shown in many textbooks on molecular spectroscopy (1-5) and quantum chemistry (6-9). When the stationary states are characterized by a set of quantum numbers corresponding to each constant of the motion of a system, the specific selection rules are expressed by stating how the pertinent quantum numbers should change for a nonvanishing transition moment. In the case of pure and Raman rotational transitions these selection rules are AJ = i 1 and AJ = 0, *2, respectively, where J = 0, 1, 2.. .denotes the rotational auantum number. The former selection rule has been derived in a numher of textbooks (69) usinn rather complicated methods that require a thorough knowlidge of the a&ociated Legendre polynomials and various recursion relations among them. Recently Sannigrahi and Das (10) . nresented a simpler derivation of this selection rule following an idea of irk (11) that is based on certain commutation relations between aneular momentum operators and cartesian coordinates. ~ o i e v e ran , attempt (i2)to extend this approach toobtain the s~ecificselectionrules for Raman rotaiional transitions did n o t meet with success. Most of the standard textbooks either do not include the derivation of this selection rule or suhstantiate it by advancine theaualitative argument that the polarizability ellipsoid ofa molicule comes back twice to its original position during one complete rotation. In the present article we have arrived at the specific selection rules (henceforth the word "specific" is dropped) for both pure and Raman rotational transitions wine simnle m o u ~theoretical areuments. A treatment of this genre can be'found elsewhere (8, 9, 13). The present a ~ o r o a c his. however. of didactical nature and thus expected tk be intelligible to a'wide cross section of readers. In the studv of electronic and vibrational spectra of a molecule the rkevant point group is used to characterize its different stationary states. In contrast, a rotating molecule has the spherical symmetry and its energy eigenstates are described by the full rotation group R(3). It is an infinite group and not usually discussed in standard texts on group theory (14-16). Since the rotational Hamiltonian RBcommutes with any symmetry operation &of R(3) (its symmetry elements are E and an infinite number of proper rotations., C(a)). . ... the eieenfunctions of Ra. namelv. the soherical harmonics YJM(8,-+) will form the bases fortbe irrdducible reoresentations (irren) of R(3). An irrep correspondinn to a &en value of j wit be (2J 1)-dimensionai sincefor a
.
u
. "
+
+
particular value of J , M can take 25 1values (J,J - 1, J 2, . , -J 1,-4. For example, we have one s orbital (J = 0). three D orbitals ( J = 1). and so on. On rotation, anv one of thk +'I)spheridal h&konics for a given valueof j will he transformed into a function that is a linear combination ofthe basis functions corresponding to same J only. Tbus p, or p, orbital can be generated by rotating a p, orbital but not from s or d or f orbitals. Once the basis functions are known, the application of group theory to derive the rotational selection rules can be accomplished as described in the following paragraphs. For the sake of clarity we will first illustrate the group theoretical principle of deriving selection rules taking a finite group as an example. s Let the stationam states $1 and $I belone to the i r r e ~D(') and D(2), respectively, of & arbitrary fiiite point group. Further, let the dipole moment vector p (equivalently x , y, or z) and the polarizability tensor a (equivalently, x2,y2, 9,xy, vz.. or z x ) belone to the irreps D(*)and D(a),respectively, of . the same pointgroup. ~ h 'necessary condition for an allowed transition between 11.1 and 11.2 is that the transition moment integral,
..
+
(w
TMI=
(hi&)
(1)
is nonzero. For Raman transitions p in eq 1 has to be replaced by a. The TMI will be zero if the direct product (reducible) representations (d.p. rep), D(') @ D(r)@ D@)and D(') @ D(=)@ D(2)do not contain the totally symmetric irrep D(" (here 0 stands for Alg or Ag or A1 or A1 depending upon the nature of the point group). It is worth noting that even if the d.p. rep contains D(O)the TMI may be zero, but, if it does not contain D(O),it will never be nonzero. In case of finite point groups the d.p. can be reduced using the reduction formula
where a0 is the number of times D(0)will occur in the d.p.rep, h is the order (number of symmetry elements) of the group, andx(0)(R) and ~ ( ~ . p . ~ ~ are p ) (thecharadersfor R) the symmetry operation R in D(O) and d.p.rep, respectively. Since the characters for D(0)for all the symmetry operations are +1, eq 2 reduces to
Thus, whether the d.p.rep contains D(O)or not can be ascertained from eq 3 in a straightforward manner. For the same purpose we can also make use of an important property of the point groups according to which the totally symmetric representation D(O) occurs in the d.p.rep of two irreps of a given point group iff the irreps are the same. Tbus, if we know the irrep to which D(r) or D(")belongs, we can at once infer that the TMIwill be nonzero if D(') @ D@)c D(p)or D(=). The reduction formula (eq 3) cannot be used for R(3) since its order iu infinity. It is therefore necessary to resort to some other technique. Since a knowledge of the characters of the Volume 67 Number 8 August 1990
653
irreps of R(3) is essential for any group theoretical discussion of rotational dynamics, we will first discuss how to obtain them from the spherical harmonics bases. Let us consider the effect of rotation on spherical harmonics, YJM(~,@) which can be written as YJM(~,+) = py(cos 9)eiM\ where py(cos 8) is the associated Legendre polynomial. Considering the C, axis to be coincident with the z axis, any anticlockwise rotation by an angle X will transform YJM(B,+) into py(:os 8) eN(4-"). Thus, the entire (2J 1) dimensional baas, for a particular J, would transform as follows
But from eq 8 E e CID(') gives
+
Therefore,
2 2 J,
which implies that the transformation matrix representative of can be written as /e-""
...
0
J*
e-"M,+M>'L
MI=-J, Ms=-Js
=
2C, 2 P
+
e-'""
(11)
m=-!
+
The maximum value of [MI Mz1 is (JI Jz) because -J15 MI C J1and -JzC MZ5 5 2 . Accordingly, ImlC (J1+ Jz) and hence .? 5 (J1+Jz), i.e., Ce = 0 for .? > (JI Jz). Now, the maximumvalue of m, viz., (J1+ Jz) will come only if Ml = J1 and Mz = Jz, i.e., Cj,+j2= 1.The next value of m, viz., (JI J z - 1) can be obtained in two different ways: (1) MI = JI a n d M 2 = J z - l a n d ( 2 ) M l = J 1 - l a n d M z = Jz.Butoneof them would be obtained form from t = JI J z (since .? = JI J z implies m = (JI Jz), (JI + J z - 1) . . .). Thus, CJItJ3-1 is also unity. Continuing this way one can show that ICel=1,1J,-J21~t5Jl+J,
+
o\
+
Therefore, the diagonal matrix O(Ci) having the following property = O(C;)IYJM(~, (6) C:IYJM(O,~)I can be used t o obtain information regarding the symmetry operation Ci. From eq 5 we have
+
Therefore, from eqs 9-11
d
X(C:)=
e i M h = 1 + 2 ~ o s A + Z ~ o s 2 h +2cosJh +_.. (7) x(CJ =
In eq 7 the first term comes from M = 0, and the relation elnn e-'"8 = 2 cos n 3 has been used in deriving the other terms. Let us now go back to the rotational seiection rules. The of a rotator are the spherical harmonics wave functions (YJMJspanning the irreducible representations ID("] of the full rotation group R(3). The dipole moment vector (p), like any other vector, will span the D(') representation. In order to know why it is so let us take a simplistic approach. Any wave function with quantum number J will span the D(J) irrep. The s orbitals being spherically symmetric span the D(0) irrep., p orbitals or any vectors (functions linear in coordinates) will belong to D(l) irrep., d orbitals or any second-rank tensor (functions quadratic in coordinates, e.g., the components of the polarizability tensor x2, y2, xy, yz, .. .) will be the bases for D@)irrep and so on. Therefore, (TMl)dipole= ($11P1$2) for a possible transition between the states and $2 would not necessarily be zero if the direct product representation D(J~)@ D(') @ D(J2) contains the totally symmetric representation D(O). Similarly for rotational Raman transition
+
will be zero if D(Jd @ D(2)@ DcJ2) does not contain D(". Thus one should now search for D(') in case of dinole transition -~~ and D'2' in case of Raman transition in the direct product renresentation D(J#'@D'J inorder that the resoective tran&ions are allowed. forms the basis for the i r r e ~Dm: the As we know IY.rd , product basis (Y&M, YJ~M~! will span the d.p.rep D(J~)@ D(JJ. This can be reduced if we recall that any arbitrary function can be written as a linear combination of the members of a complete orthonormal basis set, say (YJM)in this case. Therefore, we have ~~~~
~
~
~~~
'
.
@
Dud =
- c@!PI
(8)
P=O
where the summation index t is a quantum number that determines the resultant angular momentum. The righthand side of eq 8 can be expanded if we consider the character ~ ( C Ain)d.p.rep as follows: 654
Journal of Chemical Education
7
IJ,+JJ
"I ... .. -.,
~~~~~
+
+
X'~'(CI)
(12)
P= J-J:
Consequently, the d.p.rep D(J1) @ D(J2) reduces to the following Clebsch-Gordan series. DIJl
DIJ:l
= D!J+J2)erDIJt+J:-l)
@
,, ,
DIIJ-J:~'
(13)
Now our next job is to find out for what values of J1and J z do D(1)andD@)appear in the r.h.s. of eq 13. In the series IJ1 - Jzl, . . ( J I Jz) where each successive term differs by unity (due to quantization) the minimum (also positive) is IJ1- Jzl. Thus, for IJ1 - Jz1 = 0 the Clebsch-Gordan series ,J1+J z = 2J1, and, if lJ1contain terms with J = 0,1,2, Jzl= 1, there will be terms having J = 1,2, . . ., (J1 Jz). Therefore,D(') will come f ~ o m D (@~D(J2) ~ ) only if IJ1- J z=~ 0 or 1, when pure rotational transitions are considered. In case IJ1- J z l > 1, D(') will not appear in the Clebsch-Gordan series. I t is important to note that, if we consider (J1+J z ) = 0 or 1or even try to get IJ1- Jzl= 0 or 1from any possibility other than these two discussed above, we will land up with the unphysical situation of negative values for the quantum number t. In the case of Raman rotational transitions D@)will occur on reduction of D"]) @D(Js)only if IJ1- J 2 1 = 0,1, or 2 due to the same reason as above. Thus, we have deduced that the specific selection rules for pure rotational spectroscopy is A J = 0, *1 (cf. IJI- Jzl = 0, 1) and the same for rotationalRaman spectroscopy is A J = 0, f 1, f2. Now, from the simple symmetry argument we can infer that since p is an odd function of coordinates the product $;,$J? should also be odd such that the TMI should not vanish. But we know (J.JJfunctions are alternately even and odd for even and odd values of J , respectively, ex., 8, d, g, . . are even functions while p, f, h, . . . are odd functions. Therefore, A J = 0 will give rise to an even function whose product with p will make the integrand odd and TMI will vanish. Hence, the "exact" selection rule for this type of transition is AJ = f 1. Using similar argument and remembering that a is an even function, the selection rule for rotationalRaman transition becomes A J = 0, *2. But A J = 0 indicates the case of the Rayleigh (elastic) scattering where there is no energy transfer between the molecule and the
.
+
...
.
+
incoming photon, So the specific selection rule for rotational Raman transition is A J = i 2 .
B A&w, P W MdecYLy Qvantum Mmhanicr 2nd 4.;Oxford Uniwmity:
Orfmd.
1983. 9. Flurry, R. L., Jr. Quantum Chsmislrr An Introduction: Prentic-Hell: New Jersey, >am 10. Ssnnigrahi, A. B.: Dab, R. J Chom. Edur 1980,57,786. 11. Dirac. P.A. M. Plinciples ofBvonturn Mechanics; Oxford University: Orfmd, 1958. 12. Bandyopadhyay, T . M.Se. project lvarkauhmittod to I L T . Kharsgpur, 1987. 13. Flurry, R. L., Jr. Symmetry Croups: Theory and Chemical Applieofiow; PrentiHall: New Jersey, 1969. 14. C o t w ~F. , A. Chemicd Applicotiow of OIOUP Theory; Wiley: New York,1971. 15. Bi8hop.D. M. Cmvp Theory and Chemktry: Clarendon:Orford, 1973. 16. Kettle, S. F. A. Symmetry and Sfruefue:Wiley: New York. 1985.
Volume 67 Number 8 August 1990
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