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On the Summations Involving Wigner Rotation Matrix Elements Journal of Mathematical Chemistry 24 (1998) 123±132 123 On the summations involving Wigner rotation matrix elements Shan-Tao Lai a, Pancracio Palting b, Ying-Nan Chiu b and Harris J. Silverstone c a Vitreous State Laboratory, The Catholic University of America, Washington, DC 20064, USA b Center for Molecular Dynamics and Energy Transfer, Department of Chemistry, The Catholic University of America, Washington, DC 20064, USA c Department of Chemistry, The Johns Hopkins University, Baltimore, MD 21218, USA Received 3 June 1997 Two new methods to evaluate the sums over magnetic quantum numbers, together with Wigner rotation matrix elements, are formulated. The ®rst is the coupling method which makes use of the coupling of Wigner rotation matrix elements. This method gives rise to a closed form for any kind of summation that involves a product of two Wigner rotation matrix elements. The second method is the equivalent operator method, for which a closed form is also obtained and easily implemented on the computer. A few examples are presented, and possible extensions are indicated. The formulae obtained are useful for the study of the angular distribution of the photofragments of diatomic and symmetric-top molecules caused by electric-dipole, electric-quadrupole and two-photon radiative transitions. 1. Introduction Recently, we discussed [5,6] a number of summations involving Wigner rota- tion matrix elements which are needed for the angular distribution of photofragments caused by dissociative electric-dipole, electric-quadrupole and two-photon radiative transitions. The basic method previously employed involved the use of the recurrence relations of the Wigner rotation matrix elements. In the following, we report two alternative methods: (i) the coupling operator method [4,7], and (ii) the equivalent operator method. As illustrations, we shall use these methods to perform three kinds of summations. 2. The coupling method From the coupling rule of the Wigner rotation matrix elements, we obtain the ®rst kind of summation [4,7]: X m jjj3 j 2 m0 jjj3 (−) d 0 (β) = (−) P (cos β), (1) m m 0 m m m m 0 j3 m J.C. Baltzer AG, Science Publishers 124 S.T. Lai et al. / Summations involving Wigner rotation matrix elements ::: − where ::: are Wigner 3 j symbols, Pj(cos β) is a Legendre polynomial, and j3 = 0, 1, 2, :::,2j: This closed form is useful in dealing with the following prototypical summation: X k j 2 m dm0m(β) ,(2) m for which k can assume the values 0, 1, 2, :::,2j. From equation (1) it is found that X 2 j 2 1 1 0 2 m d 0 (β) = j(j + 1) − j(j + 1) − 3m P (cos β), (2a) m m 3 3 2 m X 3 j 2 1 2 0 1 2 0 2 0 m d 0 (β) = (3j + 3j − 1)m cos β − (3j + 3j − 1) − 5m m P (cos β), m m 5 5 3 m (2b) X 4 j 2 1 2 m d 0 (β) = j(j + 1)(3j + 3j − 1) m m 15 m 1 − (6j2 + 6j − 5) j(j + 1) − 3m0 2 P (cos β) 21 2 1 + 3j(j + 1)(j + 2)(j − 1) − 5(6j2 + 6j − 5)m0 2 + 35m0 4 35 × P4(cos β), (2c) X 15j2(j + 1)2 − 50j(j + 1) − 70j(j + 1)m2 + 63m4 + 105m2 + 12 m j 2 2 2 0 2 × m d 0 (β) = 15j (j + 1) − 50j(j + 1) − 70j(j + 1)m m m 0 4 0 2 0 + 63m + 105m + 12 m P5(cos β) (2d) and X −5j3(j + 1)3 + 105j2(j + 1)2m2 − 315j(j + 1)m4 + 231m6 + 40j2(j + 1)2 m 2 4 2 j 2 − 525j(j + 1)m + 735m − 60j(j + 1) + 294m d 0 (β) m m = −5j3(j + 1)3 + 105j2(j + 1)2m0 2 − 315j(j + 1)m0 4 + 231m0 6 + 40j2(j + 1)2 0 2 0 4 0 2 − 525j(j + 1)m + 735m − 60j(j + 1) + 294m P6(cos β): (2e) Equations (2d) and (2e) can be reduced to the following closed forms: X 5 j 2 1 2 2 m d 0 (β) = 15j (j + 1) − 50j(j + 1) + 12 m m 63 m 0 2 0 4 0 − 35 2j(j + 1) − 3 m + 63m m P5(cos β) 1 + 2j(j + 1) − 3 (3j2 + 3j − 1)m0P (cos β) 9 1 S.T. Lai et al. / Summations involving Wigner rotation matrix elements 125 2 0 2 0 − (3j + 3j − 1) − 5m m P3(cos β) 1 − 15j2(j + 1)2 − 50j(j + 1) + 12 m0P (cos β)(3a) 63 1 and X 6 j 2 1 4 3 m d 0 (β) = j(j + 1)(3j + 6j − 3j + 1) m m 21 m 1 − j(j + 1) − 3m0 2 5j(j + 1)(j − 1)(j + 2) + 7 P (cos β) 21 2 1 + 3j(j + 1) − 7 3j(j + 1)(j − 1)(j + 2) 77 2 0 2 0 4 − 5(6j + 6j − 5)m + 35m P4(cos β) 1 + −5(j − 2)(j − 1)j(j + 1)(j + 2)(j + 3) 231 + 21 5j2(j + 1)2 − 25j(j + 1) + 14 m0 2 0 4 0 6 − 105 3j(j + 1) − 7 m + 231m P6(cos β), (3b) respectively. The coupling rule may also be used to obtain the second kind of summation: X − m j + kj kj3 j+k j−k (−) d 0 (β)d 0 (β) mm0 m m m m m 0 j + kj− kj3 = (−)m P (cos β), (4) m0 m0 0 j3 where j3 = 2k,2k + 1, :::,2j and k = 0, 1=2, 1, :::, j. This equation can be reduced to [4,7] X X q 2k + j3 2j − j3 (−)L−m (j + k)2 − m2 (j − k)2 − m2 L j + k + m − L m L j − 2k × 3 j+k j−k dm0m(β)dm0m(β) j3 − L X q 0 2k + j3 2j − j3 = (−)L−m (j + k)2 − m0 2 (j − k)2 − m0 2 L j + k + m0 − L L j − 2k × 3 Pj3 (cos β), (5) j3 − L ::: where ::: are binomial coef®cients. 126 S.T. Lai et al. / Summations involving Wigner rotation matrix elements A third kind of summation that can be obtained by this coupling method is X m j + kjj3 j+k j (−) d 0 (β)d 0 (β) mm m m m m m 0 m0 j + kjj3 = (−) Pj (cos β): (6) m0 m0 0 3 In summary, the general coupling form is given by [4,2] X X j j j 1 2 3 j1 j2 Dm0 m (αβγ)Dm0 m (αβγ) m1 m2 m3 1 1 2 2 m1 m2 j j j ∗ 1 2 3 j3 = D 0 (αβγ), (7) 0 0 0 m m3 m1 m2 m3 3 where j ≡ im0α j imγ Dm0m(αβγ) e dm0m(β)e : (8) 3. The equivalent operator method We shall deal with the ®rst kind of summation by using the so-called equivalent operator method. From the de®nition [3,5] 0 j ≡ iβJy dm0m(β) jm e jm ,(9) we begin by deriving an equality. For two noncommuting operators A and B, we have, by using the Baker±Campbell±Hausdorff identity, followed by the Taylor expansion, 1 X (iλ)k B0 = eiλABe−iλA = eiλAB = AkB, (10) k! k=0 in which AkB denotes the kth degree nested commutator, AkB = A, A,[A, :::[A, B], :::] : (11) Under a similarity transformation, A = Ay, and the operators B and B0 are said to be equivalent. For example, the Cartesian components of a rotational angular momentum obey the commutation relations [Jx, Jy] = iJz,[Jy, Jz] = iJx,[Jz, Jx] = iJy: (12) Using equation (10), we have the equivalent-operator equality iαJx −iαJx e Jze = Jz cos α + Jy sin α, (13) where α is the angle of rotation with respect to the x-axis. S.T. Lai et al. / Summations involving Wigner rotation matrix elements 127 Similarly, we may obtain a useful equivalent-operator equality iβJy −iβJy e Jze = Jz cos β − Jx sin β. (14) An alternative geometrical interpretation of equations (10), (13) and (14) is given in the appendix. Here, the similarity transformation of Jz has been reduced to explicit forms which are more convenient for operating upon rotational wave functions. We now give examples of the usefulness of equation (14) for the evaluation of summations of the ®rst kind. Example (1) X X j 2 0 iβJy −iβJy 0 m dm0m(β) = jm e Jz jm jm e jm m m 0 iβJy −iβJy 0 = jm e Jze jm 0 0 0 = jm Jz cos β − Jx sin β jm = m cos β. (15) Here, we have used the resolution of the identity for the jth irreducible vector space of O+(3), viz. X jjmihjmj = 1, (16a) m along with the identities 1 0 0 J = (J + J−)andhjm jJjjm i = 0: (16b) x 2 + Example (2) X 2 j 2 0 − 2 0 m dm0m(β) = jm (Jz cos β Jx sin β) jm m 1 m0 2 = j(j + 1) sin2 β + (3 cos β − 1): (17) 2 2 Example (3) X 3 j 2 0 − 3 0 m dm0m(β) = jm (Jz cos β Jx sin β) jm m m0 = m0 3 cos β + sin2 β cos β 3j(j + 1) − 5m0 2 − 1 : (18) 2 128 S.T. Lai et al. / Summations involving Wigner rotation matrix elements Equations (15), (17) and (18) agree with the results obtained by using the recurrence relations [4,5,7]. We note that for the equivalent operator method the powers of the expression (Jz cos β − Jx sin β) are expressible as a symmetric sum, i.e., Xn n k n−k k (Jz cos β − Jx sin β) = (−) (Jz cos β) (Jx sin β) , (19) k=0 where the brackets occurring in the right-hand side indicate the symmetric sum of the product contained therein.
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