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Spectroscopy of XY3Z (C3v) molecules: A tensorial formalism adapted to the O (3) ⊃ C∞v ⊃ C3v group chain
Journal of Molecular Spectroscopy 234 (2005) 113–121 www.elsevier.com/locate/jms
Spectroscopy of XY3Z (C3v) molecules: A tensorial formalism adapted to the O (3) C1v C3v group chain A. El Hilali, V. Boudon *, M. Loe¨te Laboratoire de Physique de lUniversite´ de Bourgogne, CNRS UMR 5027, 9, Avenue Alain Savary, BP 47870, F-21078 Dijon Cedex, France Received 7 June 2005; in revised form 26 August 2005 Available online 18 October 2005
Abstract A tensorial formalism adapted to the case of XY3Z symmetric tops has been developed. We use the O (3) C1v C3v group chain. All the coupling coefficients and formulas for the computation of the matrix elements are given for this chain. Such relations are also deduced in C3v group itself. 2005 Elsevier Inc. All rights reserved. Keywords: Group theory; Tensorial formalism; Symmetric tops; Rovibrational spectroscopy
1. Introduction In molecular spectroscopy, it is often considered that sophisticated group theoretical and tensorial formalism methods are only really useful for spherical top (i.e., highly symmetrical) molecules, for which they have proven their high efficiency [1,2]. Consequently, it is usually admitted that symmetric and asymmetric tops (i.e., lower symmetry) species should be treated using more ‘‘conventional’’ methods [3]. However, some key elements of the formalism developed in our group for tetrahedral or octahedral molecules [1,4] can be used with great profit even for less symmetrical systems: the ability of performing systematic developments of all rovibrational interactions in case of complex polyads and the so-called ‘‘vibrational extrapolation’’ which makes global analyses much easier. Recently, we have presented several extensions of the tensorial formalism to some quasi-spherical top molecules with C4v [5–7] or C2v symmetry [8–11], and also to D2h species [12,13], this last case deriving itself from an original idea of Sartakov et al. [14]. The idea of the present paper is to derive similar tools for XY3Z molecules with C3v symme*
Corresponding author. Fax: +33 3 80 39 59 71. E-mail address: [email protected] (V. Boudon).
0022-2852/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2005.09.002
try. In fact, this case has already been treated and used by Roche [15,16] and Nikitin et al. [17,18]. The formalism presented here is relatively similar but with some differences. It intends (i) to be fully consistent with preceding works on Td, Oh, C4v, C2v, and D2h and (ii) to serve as a basis for the extension to the C3v radicals with an odd number of electrons (similarly to what has been done for octahedral open-shell molecules [19–22]). The basic idea of this approach can be explained as follows: in molecular spectroscopy problems, if G is the point group of the molecule, we usually have to consider a larger group, denoted G00 (G is a subgroup of G00 ). In this case, we consider the G00 G group chain. In some cases, however, we are brought to consider an intermediate group G0 which corresponds to an approximate symmetry of the system, and we have then a chain G00 G0 G. For XY3Z molecules, with an equilibrium configuration belonging to the C3v group, and among the various possibilities, let us only mention two of them. One possibility consists in describing a ‘‘quasi-spherical’’ C3v symmetric top like CH3D or C35Cl337Cl using a chain like O (3) Td C3v (i.e., to perform a reorientation of the O (3) Td formalism [1] into the C3v subgroup). However, this is not as simple as for the similar cases of quasi-spherical symmetric C4v and asymmetric C2v molecules treated recently [2] for the follow-
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ing reason: in the usual O (3) Td formalism for tetrahedral spherical tops, the quantization axis (z) is taken as one of the S4 axes which lies in between two X–Y bonds. For an XY3Z molecule with C3v symmetry, the z axis is taken, of course, as the C3 axis along the X–Z bond. We have thus two different axis orientations (unless we rewrite the O (3) Td formalism with the z axis along one of the X– Y bonds which would be quite meaningless for tetrahedral molecules themselves). This is why, for C3v molecules, in general, it is simpler to use a second scheme, that is the O (3) C1v C3v chain. In this case, the usual projection quantum number K corresponds to a C1v irreducible representations. C1v accounts for the approximate ‘‘cylindrical’’ (or ‘‘conical’’) symmetry of the molecule. We propose, in this theoretical work, the development of a tensorial formalism adapted to the study of XY3Z type molecules which possess integer angular momenta (i.e., in a singlet electronic state) by using the O (3) C1v C3v chain. This slightly differs from Roches [15,16] and Nikitins [17,18] works, who used the O (3) C3v direct chain. For this, we recall the bases of the spherical formalism for the SO (3) group (the extension to O (3) = SO (3) CI being trivial). Then we build a formalism adapted to the O (3) C1v chain that could be incidentally used for linear molecules such as CO, HF, HCN, . . . Then we finally carry out the C1v C3v orientation. Concerning the orientation of tensors in the G subgroup (C1v or C3v), we define a general method of calculation of the transformation matrix which realizes this symmetry adaptation. The main coupling coefficients can be calculated to obtain the necessary tools for the development of the rovibrational model. We intend, on the one hand, to use this formalism for development of the Hamiltonian and transition moment operators (dipole moment and polarizability), and, on the other hand, to generalize it to the SU ð2Þ C I C S1v C S3v group chain (the S superscript indicating the use of spinorial representations) [23,24] to study XY3Z type open-shell species which possess half-integer momenta such as CH3O and CH3S radicals. The present article, although rather technical, will serve as a starting point for all these extensions, which will be described in forthcoming papers. 2. Formalism in the O (3) group The ideas presented in this section will be used throughout all this work. We first consider proper symmetry operations (rotations) only, i.e., the SO (3) group, before finally extending expressions to the improper symmetry operations of the O (3) group. SO (3) is a Lie group with three parameters. All its elements are proper symmetry operations which leave invariant a point O. Any rotation R can be decomposed in a single way as a succession of three rotations characterized by Euler angles. From a passive point of view, we can write a representation of R as
Dða; b; cÞ ¼ exp
iaJ z ibJ y icJ z exp exp ; h h h
ð1Þ
where a, b, and c are the Euler angles, and Jy, Jz are two components of the angular momentum operator0 J. The m matrix elements of D denoted by ½DðjÞ ða; b; cÞm can be written in the standard basis {|j,mæ} as ðjÞ m0 D ða; b; cÞ m ¼ hj; m0 jDða; b; cÞjj; mi m0 0 ð2Þ ¼ eim a d ðjÞ ðbÞ m eimc ; d(j) (b) being the well-known Wigners function evaluated as [25] m0 m0 ½d ðjÞ ðbÞm ¼ DðjÞ ð0; b; 0Þ m 1 ðj þ m0 Þ!ðj m0 Þ! 2 X jm0 r r ¼ C jþm C jm ðj þ mÞ!ðj mÞ! r 2rþm0 þm 2j2rm0 m jm0 r ð1Þ cosðb2Þ sinðb2Þ ; ð3Þ where max (0, m m ) 6 r 6 min(j m , j + m). We can notice that the quantum number j also represents an irreducible representation (irrep) of SO (3) (m being an irrep of SO (2)). In a covariant formalism (see below) the transformation of an irreducible tensor T ðjÞ m of rank j and component m (j 6 m 6 j) can be expressed as X m0 ðjÞ DðjÞ ðRÞ m T m0 . ð4Þ 8R 2 SOð3Þ P R T ðjÞ m P R1 ¼ 0
0
m0
In the following, states of type |j, mæ will be considered as standard and contravariant in the molecular fixed frame, and states of type Æj, m| will be considered as standard and covariant. We write jj; k i WkðjÞ () contravariant set;
ð5Þ
ðjÞ
hj; k j ¼ Wk () covariant set. In a general way, the contra-covariant change is made using the metric tensor [26] j 0 ðjÞ T m ¼ ej ð6Þ T mðjÞ ; 0 mm 0 mm ðjÞ T mðjÞ ¼ e0j ð7Þ T m0 . j We adopt here Einsteins summation convention. ej and e0j are phase factors to be chosen. For involutive transformations, we have ej ¼ e0j and e2j ¼ 1. Let us note, for example, that with the choice of variance (5) E E E m m ðj j Þm m ð8Þ Wðj1 j2 jÞ ¼ F m11m22 ðjÞ Wðj11 Þ Wðj22 Þ ; where F are the Clebsch–Gordan coefficients of the SO (3) group. In all the following, symbols like [C] represent the dimension of the irrep C. In the case of SO (3), ½j ¼ 2j þ 1.
ð9Þ
A. El Hilali et al. / Journal of Molecular Spectroscopy 234 (2005) 113–121
As explained in [26–28], the various coupling coefficients for two angular momenta can be related by general relations : 0 0 j m1 m1 m2 m2 m1 m2 ðjÞ ðj j Þm0 F ðj1 j2 Þm ¼ Kðj1 j2 jÞ F m01m20 ðjÞ 0 1 2 mm j1 j2 ð10Þ 0 0 0 m1 m1 m2 m2 mm ðj j jÞ F ðjm11mj22jÞm ¼ K 0 ðj1 j2 jÞ F m01m20 m0 ; 1 2 j j1 j2 ðj j mÞ
F m11m22 ðjÞ m m ðjÞ
F ðj11 j22mÞ
mm ðj j jÞ iUðj1 j2 jÞ 12 ¼ ej e ½j F m11m22 m0 ; j j 1 0 ¼ e0j eiWðj1 j2 jÞ ½j2 F mðj11mj22jÞm . mm0
ð11Þ
0
ð12Þ ð13Þ
For our work, we take the same choices as those made by Michelot [26] for SO (3), ej ¼ e0j ¼ 1; 2j
Kðj1 j2 jÞ ¼ ð1Þ ; K 0 ðj1 j2 jÞ ¼ 1; 2j
2j
eiUðj1 j2 jÞ ¼ ð1Þ 1 ; eiWðj1 j2 jÞ ¼ ð1Þ 2 .
ð14Þ
This implies that j j mm0 2j jm ¼ ¼ ð1Þ ¼ ð1Þ dm0 ;m . m0 m mm0 j One can then write in this case m m ðjÞ
ðj j mÞ
F ðjm11mj22jÞm ¼ F mðj11mj22jÞm ; ðj j Þm
F m11m22 ðjÞ m m ðjÞ
F ðj11 j22mÞ
ð16Þ ð17Þ
mm 1 ðj j jÞ 2j ¼ ð1Þ 1 ½j2 F m11m22 m0 ; j j 1 0 2j2 2 ¼ ð1Þ ½j F mðj11mj22jÞm 0 mm 0
ð18Þ ð19Þ
and all these coefficients are real. In SO (3), the Wigner–Eckart theorem, for covariant quantities, can be written D E
0 1 ðkjÞm0 m ¼ ð1Þkjþj ½j0 2 F qmðj0 Þ j0 T ðkÞ j ; j0 ; m0 T ðkÞ ð20Þ j; q where Æ . . . i .. i . . . æ are the reduced matrix elements. Relation (2) gives SO (3) irreps. By carrying out an extension to O (3) = SO (3) CI (where CI = {E, I} and I is the space inversion) we get: 8 ðj s Þ ðjÞ > < D ðRÞ ¼ D ðRÞ; ð21Þ R 2 SOð3Þ ) Dðjg Þ ðIRÞ ¼ DðjÞ ðRÞ; > : ðj Þ ðjÞ u D ðIRÞ ¼ D ðRÞ; where s = g or u is the parity index. Let us note that each ji has its parity index and Clebsch–Gordan coefficients of O (3) have the additional non-zero condition: s1 s2 s3 ¼ g.
3. Orientation in the O(3) C‘v group chain The first part of our tensorial study consists in performing the reduction of O (3) into C1v. This operation results in a simple basis change, and allows us to express the C1v components of the oriented spherical tensor according to the standard components. A first basic step makes use of Wang type functions : 1 ð23Þ jjs ; Ki ¼ pffiffiffi ½jjs ; þK i jjs ; K i ðK 6¼ 0Þ. 2 In the special case K = 0, we have ( j ; 0þ ¼ j ; 0 if j is even; g g j ; 0 ¼ j ; 0 if j is odd; g
ð24Þ
g
and similarly but with 0+ and 0 inverted for ju. In the following, the parity index is omitted for simplicity but is implicitly considered. ðjÞ Thus, for covariants quantities, the passage from T k ðjÞ spherical components to the T Kd symmetrized spherical components (d = ±) is expressed as ðjÞ X ðjÞ ðjÞ W kKd T k . ð25Þ T Kd ¼ k¼K;þK
ð15Þ
F ðj11 j22mÞ ¼ F m11m22 ðjÞ ;
115
ð22Þ
In fact, K = |k| is an irreducible representation (irrep) of C1v and d corresponds to its component. ðjÞ W kKd is the Wang matrix. Table 1 gives reduction rules in O (3) C1v. The C1v character table and the multiplication rules for the C1v irreps are given in Appendix A. We can notice that: [K] = 2 for K > 0 and [0] = [0+] = 1. In the O (3) C1v group chain, we can write relations similar to those used in the spherical formalism (the phases being taken equal to 1, as in Eq. (14)), 0 0 j K d Kd ðjÞ ðjÞ K 0 d0 Kd T K 0 d0 . ð26Þ and T ¼ T Kd ¼ T ðjÞ ðjÞ KdK 0 d0 j Our oriented metric tensor is then X j j ðjÞ W kKd ¼ 0 kk KdK 0 d0 k;k 0 X jk ðjÞ ¼ ð1Þ W kKd
0
ðjÞ
W kK 0 d0
ðjÞ
W k K 0 d0 .
ð27Þ
k
This gives, j j 2j jþK / ¼ ð1Þ ¼ ð1Þ ð1Þ d dK;K 0 dd;d0 . K 0 d0 Kd KdK 0 d0 ð28Þ Table 1 Reduction of the Dðjg Þ representations of O(3) in C1v jg of O(3)
Irreps of C1v
0 1 2 3. ..
R+ ” 0+ R ¯ P ” 0 ¯ 1 R+ ¯ P ¯ D ” 0+ ¯ 1 ¯ 2 R .. ¯ P ¯ D ¯ U ” 0 ¯ 1 ¯ 2 ¯ 3 .
Rules for Dðju Þ are obtained by exchanging 0+(R+) and 0(R).
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A. El Hilali et al. / Journal of Molecular Spectroscopy 234 (2005) 113–121
where, for K > 0, þ1 if d ¼ þ; /d ð1Þ ¼ 1 if d ¼ . For K = 0, we have j j j ¼ ¼ ð1Þ . 0 0 0þ 0þ
ð29Þ
eK 1 K 2 K is a phase factor to be chosen. In fact, the choices in Eq. (28) for O (3) C1v and similar ones that we made for C1v (see below) imply a relation between isoscalar factors and their complex conjugates that leads to the following condition:
ð30Þ
e2K 1 K 2 K ¼
ð1Þ
¼ 1.
ð31Þ
Contrary to what is usually adopted for such formalisms (i.e., SO (3) O [23], O (3) Td [26,1] and O (3) C3v [15,17]) and for reasons of coherence with our forthcoming work on radicals [32], we prefer to use Clebsch–Gordan coefficients instead of Racahs V or Wigners 3j. Thus, we define oriented Clebsch–Gordan coefficients in the O (3) C1v chain by X ðj j ÞKd ðj j Þk ðj1 Þ W kK11 d1 ðj2 Þ W kK22 d2 ðjÞ W Kd F k11k22ðjÞ . F K 11 d21 K 2 d2 ðjÞ ¼ k k1 k2 k
ð32Þ In this O (3) C1v chain, the Wigner–Eckart theorem, for contravariant quantities, can be written D 0 E
1 ðkjÞK 0 d0 ðj Þ ðkÞ ðjÞ WK 0 d0 T K 0 d0 WKd ¼ ½j0 2 F K 0 d0 Kdðj0 Þ ei/E j0 T ðkÞ j . ð33Þ As Michelot for SO (3) [26], we take ei/E ¼ 1.
ð34Þ
We are now able to deduce Clebsch–Gordan coefficients for the C1v group itself. We follow a method given by Lulek [29,30] and also used in [23]. This method is based on the use of Racahs factorization lemma, which can be written here as: ðj j ÞKd
ðj j ÞK
F K 11 d21 K 2 d2 ðjÞ ¼ K K 11 K22 ðjÞ K d K d ðjÞ
2 2 F ðj11 j21ÞKd
K K ðjÞ
¼ K ðj11 j22ÞK
2 F d1 d12 ðKÞ ;
ðK K Þd
ð35Þ
d d ðKÞ
ð36Þ
F ðK1 12K 2 Þd ;
where the K 0 s are called isoscalar factors. The Clebsch–Gordan matrix being unitary, we can define X ðj j ÞKd 2 N ðj1 K 1 ; j2 K 2 ; jKÞ ¼ F K 11 d21 K 2 d2 ðjÞ d1 d2 d
ðj j ÞK 2 ¼ ½K K K 11 K22 ðjÞ ;
e0þ 0þ 0þ ¼ 1; e110þ ¼ 1; e0 0 0þ ¼ 1; e110 ¼ i; e0þ 0 0 ¼ 1;
e10 1 ¼ i; e112 ¼ 1;
ð41Þ
e10þ 1 ¼ 1.
As before, the contra-covariant change in C1v itself is made means of a metric tensor (the phases are again taken equal to 1, as in Eq. (14)) 0 K dd ðKÞ ðKÞ d0 d Td ¼ T d0 . ð42Þ T ðKÞ and T ðKÞ ¼ dd0 K With the phases choices realized in [26], we can derive relations similar to Eqs. (10), (12), and (13) d d ðKÞ
F ðK1 12K 2 Þd ¼ ð1Þ2K
d01 d1
d02 d2
K ðK K 2 ÞK ; F d0 d10 ðKÞ 1 2 dd0
K1 K2 0 0 0 d2 d2 dd d1 d1 ðK K KÞ F dðK1 d12Kd2 KÞ ¼ F d0 d10 d20 ; 1 2 K K1 K2 0 1 dd ðK K 2 Þd ðK K KÞ 2K F d1 d12 ðKÞ F d1 d12 d20 ; ¼ ð1Þ 1 ½K 2 K K 1 0 d1 d2 ðKÞ 2K 2 2 F ðK 1 K 2 Þd ¼ ð1Þ ½K F dðK1 d12Kd2 KÞ . dd0
We choose, 0 K K dd 2K / ¼ ¼ ð1Þ d dd;d0 . ¼ ð1Þ dd0 d0 d K
ð43Þ ð44Þ ð45Þ ð46Þ
ð47Þ
The Table 2 give the exact values of some oriented symbols as well as our Clebsch–Gordan coefficients for the C1v group calculated by the method exposed above. We have limited the table to the C1v irreducible representations necessary to carry out the C1v C3v orientation described in the next section. 4. Formalism in the C‘v C3v chain
1
½K 2 ðj j ÞKd ¼ eK 1 K 2 K pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F K 11 d21 K 2 d2 ðjÞ ; N ðj1 K 1 ; j2 K 2 ; jKÞ
ð38Þ ðj1 Þ
ðj2 Þ
where j1 (j2) takes the minimum values such that D ðD Þ contains K1 (K2) (see Table 7 in Section 5.1). j is such that DðjÞ contains K and is compatible with the condition: j j1 j2 j 6 j 6 j 1 þ j2 .
For the lowest K values, we take the following phase factors:
ð37Þ
and get: ðK K 2 Þd F d1 d12 ðKÞ
ð40Þ
i¼1
This also implies that for K = 0, we define /d
3 Y j þK ð1Þ i i .
ð39Þ
We choose the C3v generators as: • The C3 operation is a 2p/3 rotation around the molecular axis z. We denote it C3z. • The mirror rv = C2 I considered as a product of a C2 rotation of the D3 group by space inversion I (the C2 axis was taken parallel to (Ox) axis of the molecular fixed frame).
A. El Hilali et al. / Journal of Molecular Spectroscopy 234 (2005) 113–121 Table 2 Exact values of some oriented Clebsch–Gordan coefficients for O (3) C1v and Clebsch–Gordan coefficients for the C1v group ðj j ÞKd
ðK K Þd
j 1 j2 j3
K1 K2 K3
d1 d2 d3
F K 11 d21 K 2 d2ðjÞ
2 F d1 d12 ðKÞ
0 1 0 1
0 0 0 1 1
0 0 0 0 0
+++ + + +++ +
1 p1ffiffi3 1 p1ffiffi 3 p1ffiffi
1 1 1
110
+
p1ffiffi2 p1ffiffi
0 1 1 1
0 0 1 0
111
0 0 0 1 1
3
+ 101
101
111
101
+++ + +
112
112
+
2
1 1 p1ffiffi 2 p1ffiffi 2 p1ffiffi 2 p1ffiffi 2 p1ffiffi 2 p1ffiffi 2
+ +++ + +
(
We have then: Dðjg Þ ðC 3z Þ ¼ DðjÞ ðC 3z Þ Dðjg Þ ðrv Þ ¼ DðjÞ ðC 2 Þ
( and
117
Table 4 C1v C3v correlation table C of C3v
K of C1v +
+
R ”0 R ” 0 P”1 D”2 U .. ” 3 .
p1ffiffi 2 p1ffiffi2 piffiffi
A1 A2 E E A .. 1 ¯ A2 .
2
piffiffi 2
X
1 1 i
T Cr ¼
i
or for contravariant quantities, X ðKÞ Cr d V d T ðKÞ T Cr ðKÞ ¼
ðKÞ
ðKÞ
ðKÞ
V dCr T d ;
ð50Þ
d
p1ffiffi 2 p1ffiffi 2 p1ffiffi 2 p1ffiffi 2
ð51Þ
d
Dðju Þ ðC 3z Þ ¼ DðjÞ ðC 3z Þ Dðju Þ ðrv Þ ¼ DðjÞ ðC 2 Þ
.
ð48Þ We thus obtain the O (3) matrices for the generators (and thus for all the symmetry elements) of the C3v group. The C3v matrices of theses generators are themselves chosen as in Table 3. The Wang transformation does not carry out the complete orientation, because, for K = 1 + 3p,2 + 3p,3 + 3p where p is an integer, it appears that the matrix blocs which match with the E irreps differ from a phase factor. A second transformation is thus used to orient C1v into C3v by making all the C1v irreps matrices for the C3v generators equivalent to those that we have chosen in Table 3. This orientation is carried out thanks to a V-unitary transformation, that perform the C1v C3v reduction recalled in Table 4. The oriented F coupling coefficients of C1v C3v can now be calculated in terms of C1v Clebsch–Gordan coefficients as: X ðK K ÞCr ðK K 2 Þd ðK 1 Þ d1 V C1 r1 ðK 2 Þ V dC22 r2 ðKÞ V Cr F d1 d12 ðKÞ . F C11r1 C2 2 r2 ðKÞ ¼ d
In theses two equations, C indicates an irrep of C3v and r is a component of this irrep. For example, in the case of the E irrep, we look for a unit matrix ½ðKÞ V dCr such that : 8 pffiffi ! 3 > ðKÞ d ðKÞ d 1 12 > 2 ðKÞ > V ðC Þ V ¼ D p ffiffi > 3z Cr Cr < 23 12 ð52Þ ; > > 1 0 1 > ðKÞ d > V Cr DðKÞ ðrv Þ ðKÞ V dCr ¼ : 0 1 where DðKÞ ðRi Þ ¼
ðjÞ
W kKd
DðjÞ ðRi Þ
ðjÞ
W kKd
1
;
and D ðRi Þ is the matrix of the j irreps of O (3). As shown below, the metric tensor of C3v depends on (1)K, that is on the parity of K. That is the reason why we need to define phase factors for all the values of K modulo 6 (although the C1v C3v reduction is the same for all K modulo 3 values, see Table 4). Table 5 gives the numerical values of the V-matrix elements. For the C1v C3v chain, we can write analog relations to those used before (the phase remains again the same, as before): 0 0 K C r Cr ðkÞ ðKÞ C 0 r0 Cr T Cr ¼ and T ¼ T C0 r0 . T ðkÞ ðKÞ CrC 0 r0 K
d1 d2 d
ð54Þ ð49Þ
We can write for covariant quantities Table 3 Matrices for the irreducible representations of C3v. A1 A2 E
C3z
rv
Basis set in terms of jj,mæ
1 1
1 1
|A1æ = |0,0æ |A2æ = |1,0æ
pffiffi ! p12ffiffi 23 23 12
ð53Þ
ðjÞ
1 0
0 1
jE1 i ¼ p1ffiffi2 j1; 1i þ i p1ffiffi2 j1; 1i jE2 i ¼ p1ffiffi2 j1; 1i þ i p1ffiffi2 j1; 1i
Considering phases chosen (see before) it follows that the oriented metric tensor is X K K 0 ðKÞ d V Cr ðKÞ V dC0 r0 ¼ 0 0 0 CrC r dd d0 d X / ¼ ð1Þ d ðKÞ V dCr ðKÞ V dC0 r0 . ð55Þ d
This gives, K CrC 0 r0
¼ ð1Þ
2K
K C 0 r0 Cr
K
¼ ð1Þ dC;C0 dr;r0 .
ð56Þ
118
A. El Hilali et al. / Journal of Molecular Spectroscopy 234 (2005) 113–121
Table 5 Numerical values of all V-matrix for jg elements 2m
6 + 6p 5 + 6p 4 + 6p 3 + 6p 2 + 6p 1 + 6p 0
2m + 1
ðj;KÞ
ðKÞ
ðj;6þ6p;A1 Þ
T T ðj;6þ6p;A2 Þ ðj;5þ6p;EÞ T1 ðj;5þ6p;EÞ T2 ðj;4þ6p;EÞ T1 ðj;4þ6p;EÞ T2 T ðj;3þ6p;A1 Þ T ðj;3þ6p;A2 Þ ðj;2þ6p;EÞ T1 ðj;2þ6p;EÞ T2 ðj;1þ6p;EÞ T1 ðj;1þ6p;EÞ T2 þ T ðj;0 ;A1 Þ
1 0 i 0 1 0 i 0 1 0 i 0 —
0 i 0 1 0 i 0 1 0 -i 0 1 —
— — — — — — — — — — — — 1
T ðj;6þ6p;A2 Þ T ðj;6þ6p;A1 Þ ðj;5þ6p;EÞ T1 ðj;5þ6p;EÞ T2 ðj;4þ6p;EÞ T1 ðj;4þ6p;EÞ T2 T ðj;3þ6p;A2 Þ T ðj;3þ6p;A1 Þ ðj;2þ6p;EÞ T1 ðj;2þ6p;EÞ T2 ðj;1þ6p;EÞ T1 ðj;1þ6p;EÞ T2 T ðj;0 ;A2 Þ
1 0 0 i 0 1 i 0 0 1 0 i —
0 i 1 0 i 0 0 1 i 0 1 0 —
— — — — — — — — — — — — 1
T Cr T ðj;K;CÞ r
K
jg
6 + 6p 5 + 6p 4 + 6p 3 + 6p 2 + 6p 1 + 6p 0
Vþ Cr
ðKÞ
V Cr
ðKÞ
V 0Cr
Those for ju are obtained by exchanging the blocks corresponding to j = 2m and j = 2m + 1.
In the C1v C3v chain, the Wigner–Eckart theorem, for contravariant quantities, can be written D E
1 ðK 0 KÞk 0 i/E 0 ðk Þ ðK 0 Þ ðK Þ ðKÞ Wk0 T k0 0 Wk K T K . ð57Þ ¼ ½K 0 2 F k0 kðK 0Þ e As before, we take ei/E ¼ 1.
ð58Þ
5. Tensorial algebra in the C3v group
It is again possible to calculate Clebsch–Gordan coefficients for C3v thanks to Racahs factorization lemma: ðK K ÞCr
ðK K ÞC
ðC C Þr
2 F r1 r1 2 ðCÞ ;
C r C 2 r2 ðKÞ F ðK11 K1 2 ÞCr
r r ðCÞ F ðC1 1 C2 2 Þr .
¼
C C ðKÞ K ðK11 K2 2 ÞC
ð59Þ
3 Y ð1ÞK i ;
ð62Þ
i¼1
here and in (61) K1(K2) takes the minimum values such that DðK 1 Þ ðDðK 2 Þ Þ contain C1(C2) only once (see Table 7). These minimum values of K, say Kmin, are listed for each irreducible representation of C3v in Table 7, while K is such that DðKÞ DðCÞ and: DðK 1 Þ DðK 2 Þ DðKÞ .
ð63Þ
We take the following phase factors:
The Clebsch–Gordan matrix being unitary, we can define X ðK K ÞCr 2 N ðK 1 C 1 ; K 2 C 2 ; KCÞ ¼ F C11r1 C2 2 r2 ðKÞ r1 r2 r
ðK K ÞC 2 ¼ ½C K C11C2 2ðKÞ .
ð61Þ
where eC1 C2 C3 is a phase factor to be chosen. In fact, the choices in Eq. (56) for C1v C3v, and similar ones that we made for C3v (see below), imply a relation between isoscalar factors and their conjugates that leads to the following condition: e2C1 C2 C ¼
5.1. Clebsch–Gordan coefficients and the Wigner–Eckart theorem in C3v
F C11r1 C2 2 r2 ðKÞ ¼ K C11C2 2ðKÞ
1
½C 2 ðC C 2 Þr ðK K ÞCr ¼ eC1 C2 C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F C11r1 C2 2 r2 ðKÞ ; F r1 r1 2 ðCÞ N ðK 1 C 1 ; K 2 C 2 ; KCÞ
ð60Þ
This leads to the desired Clebsch–Gordan coefficients :
eA1 A1 A1 ¼ 1;
eA2 A2 A1 ¼ 1;
eA1 A2 A2 ¼ 1;
eEEA1 ¼ 1; eEA2 E ¼ 1;
eEEA2 ¼ 1; eEEE ¼ 1;
eEA1 E ¼ 1.
ð64Þ
We impose that our 3C r coefficients (see below) are left unchanged by any even permutation of their columns and that for an odd permutation they are multiplied by the factor.
A. El Hilali et al. / Journal of Molecular Spectroscopy 234 (2005) 113–121
ð1ÞC1 þC2 þC .
ð65Þ
The (1)C phases are obtained in multiplying each C3v representation by itself. As a result, the symmetric part is associated to (1)C = 1 and the antisymmetric one to (1)C = 1, so that we obtain (see Table 10 in Appendix B): A1
¼ ð1Þ ¼ 1;
ð66Þ
A2
¼ 1.
ð67Þ
ð1Þ ð1Þ
E
Table 6 Exact values of some oriented Clebsch–Gordan coefficients for C1v C3v and Clebsch–Gordan coefficients for the C3v group r1 r2 r
F C 1 1r1 C2 2 r2 ðKÞ
2 F r1 r1 2 ðCÞ
0+ 0+ 0+ 0 0 0+ 0+ 0 0 1 1 0+
A1 A1 A1 A2 A2 A1 A1 A2 A2 E E A1
Æ Æ Æ Æ Æ Æ Æ Æ Æ 11.
1 1 1 p1ffiffi2 p1ffiffi
1 1 1 p1ffiffi 2 p1ffiffi 2
1 1 0
E E A2
p1ffiffi2 p1ffiffi2 1 1 1 1 p1ffiffi
p1ffiffi2 þ p1ffiffi2 1 1 1 1 p1ffiffi p1ffiffi 2 p1ffiffi 2 p1ffiffi 2
22.
E A1 E
1 0 1
E A2 E
r r ðCÞ F ðC1 1 C2 2 Þr
112
EEE
r r ðCÞ
F ðC1 1 C2 2 Þr
C 0 r0
1. .1 .2 .2 .1 11
ð68Þ
¼
1 C 2 CÞ F ðC r1 r2 r ;
ð69Þ
221 122
p1ffiffi2 p1ffiffi
ð70Þ
212
p1ffiffi2
pffiffiffiffiffiffi ðC C CÞ ½C F r1 r1 2 r2 ; pffiffiffiffiffiffi r rr ¼ ½C F ðC1 1 C2 CÞ .
ðC C Þr0
r r ðCÞ
ð72Þ
ð74Þ
The Wigner–Eckart theorem in C3v can be expressed (for covariant sets) by: D E
1 ðC 0 CÞr0 i/E 0
ðC 0 Þ Wr0 T rðC0 0 Þ WðCÞ C T ðC 0 Þ C ; ð75Þ ¼ ½C 0 2 F r0 rðC 0 e r Þ
kmin jmin
A1
A2
E
0 0
0 1
1 1
5.2. Recoupling coefficients for C3v We will now consider briefly the recoupling coefficients for C3v. By analogy with what has been done by Fano and Racah [31] for O (3), we can define 6C coefficients from the recoupling matrix for three irreducible representations by the relation: jjj C 1 j C 2 i; C 12 i j C 3 i; C; ri C1 C2 1 X 1 2 2 ½C 23 ¼ ½C 12 C3 C C 23
C 12 C 23
kC 1 kC 2 i C 3 i; C 23 i; C; ri.
again with ei/E ¼ 1.
2
2
Table 7 Minimum values of j and K such that DðjÞ DðKÞ DðCÞ
1 2
2 1 2 F r1 r1 2 ðC 0 Þ F ðC C Þr ¼ dC 0 C dr0 r . 1 2
2
ð71Þ
and the Clebsch–Gordan coefficients are real. The values of our Clebsch–Gordan symbols are given in Table 6. We can also introduce the orthogonality relations which will be useful for the calculation of the 6C factors (see next paragraph): X ðC C Þr r r ðCÞ 2 F r0 r1 0 ðCÞ F ðC1 1 C2 2 Þr ¼ dr01 r1 dr02 r2 ; ð73Þ X
2 1 2 1 2 1
¼
We have chosen C ¼ dr;r0 ; rr0
r01 r02
2
12.
ðC C 2 Þr F r1 r1 2 ðCÞ ;
2 ¼ F r1 r1 2 ðCÞ
ðC C Þr
C1 C2 C
1 0+ 1
ðC C Þr
ðK K ÞCr
K1 K2 K
Likewise, the 3C r and Clebsch-Gordan coefficients can be related through expressions similar to Eqs. ()(10)–(13) and with the same phase factors choice as in the preceding section :
r1 r2 r F ðC 1 C 2 CÞ
119
ð76Þ
We can also note a useful formula which relates the reduced matrix elements in C3v to those in C1v. sffiffiffiffiffiffiffiffi
0 0 ðK ;C Þ ½C 0 ðK 0 KÞC0 0
ð K 0 Þ K ; ð77Þ K C T 0 0 KC ¼ 0 K C 0 CðK 0 Þ K T ½K as well as the formula relating reduced matrix elements in O(3) and C3v for an O (3) C1v C3v oriented tensor: sffiffiffiffiffiffiffiffi
0 0 0 ðj ;K ;C Þ
½C 0 ðj0 jÞK 0 ðK 0 KÞC0 0
ðj0 Þ j . j K C T 0 0 0 jKC ¼ 0 K K 0 Kðj0 Þ K C 0 CðK 0 Þ j T ½j ð78Þ
ð79Þ
Expanding the two members of expression (79) and taking our convention as well as (73) and (74) into account, we obtain C 1 C 2 C 12 C 3 C C 23 C þC þC C ð1Þ 1 2 3 X ðC1 C2 Þr12 ðC12 C3 Þr r1 r23 ðCÞ r2 r3 ðC23 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F r r ðC Þ F r r ðCÞ F ðC1 C23 Þr F ðC2 C3 Þr23 ½C 23 ½C 12 ½C allr 1 2 12 12 3 ð80Þ The nonvanishing conditions for a 6C are:
120
C1 C3
A. El Hilali et al. / Journal of Molecular Spectroscopy 234 (2005) 113–121
C2 C
8 C 1 C 23 C; > > > < C 2 C 3 C 23 ; C 12 6¼ 0 ) > C 12 C 3 C; C 23 > > : C 1 C 2 C 12 .
Table 8 Characters of the irreducible representations of C1v
ð81Þ +
It is possible to demonstrate the following very useful formula, which gives the reduced matrix elements of two coupled tensor operators of C3v acting on the same space. D E ðC0 Þ
C 0 T ðC1 Þ T ðC2 Þ
C X 1 C1 C2 C0 C 1 þC 2 þCþC 0 2 ½C 0 ¼ ð1Þ C C 0 C 00 C 00 0 ðC Þ 00 00 ðC Þ C T 1 C C T 2 C ð82Þ
ð83Þ
We merely say that these 9C symbols can be calculated from the 6C coefficients though the relation 8 9 < C 1 C 2 C 12 = C C 4 C 45 : 3 ; C 13 C 24 C X C 2 C 4 C 24 C 1 C 3 C 13 0 ¼ ½C C 24 C C 0 C 3 C 0 C 34 C0 C 12 C 34 C . ð84Þ C0 C1 C2 The nonvanishing conditions for a 9C are: 8 C 1 C 2 C 12 ; > > > > 9 8 > C 3 C 4 C 34 ; > > > > < C C C; = < C 1 C 2 C 12 > 13 4 C 3 C 4 C 34 6¼ 0 ) > > > C C 3 C 13 ; > ; : > 1 C 13 C 24 C > > > C 2 C 4 C 24 ; > > : C 12 C 34 C.
R ”0 R ” 0 P”1 D”2 U .. ” 3 .
ð85Þ
This enable us to express the reduced matrix elements of two coupled tensor operators acting on two different spaces:
D E ðCÞ
C 01 C 02 C 0 T ðC1 Þ U ðC2 Þ C 1 C 2 C 9 8 0 C 1 C 02 C 0 > > = < 1 ¼ ð½C ½C 0 ½CÞ2 C1 C2 C > > ; : C1 C2 C
C 01 T ðC1 Þ C 1 C 02 T ðC2 Þ C 2 . ð86Þ We can also finally introduce the 12C coefficients given in terms of 9C and 6C symbols [33]:
E
2C(u)
1rv
1 1 2 2 ..2 .
1 1 2 cosu 2 cos(2u) ..2 cos(3u) .
1 1 0 0 0. ..
Table 9 Multiplication table for the irreducible representations of C1v +
R R P D .. .
In the same way, we can define the 9C symbols by jjjC 1 jC 2 i; jC 12 i jjC 3 jC 4 i; C 34 i; C; ri 8 9 < C 1 C 2 C 12 = 1 ¼ ð½C 12 ½C 12 ½C 12 ½C 12 Þ2 C 3 C 4 C 45 : ; C 13 C 24 C jjjC 1 jC 3 i; jC 13 i jjC 2 jC 4 i; C 24 iC; ri
+
R+
R
+
R R P D .. .
R R+ P D .. .
P
D
...
P P R+¯R ¯ D P¯U .. .
D D P¯U + R .. ¯R ¯ C .
... ... ... ... ...
Table 10 Multiplication table for the irreducible representation of C3v A1 A2 E
A1
A2
E
[A1] A2 E
A2 [A1] E
E E [A1 + E] + {A2}
[. . .] symmetric part, {. . .} antisymmetric part.
8 > < C1 C5 > : C9
C2
C3
C6 C 10
C7 C 11
9 C4 > = C8 > ; C 12
9 8 > = < C1 C2 C3 > X C þC þC þC ¼ ð1Þ 3 7 9 10 ½X C 5 C 6 C 7 > > ; : X C 9 C 10 X C 11 C 3 C 4 C 11 C 9 C 8 . C 7 C 12 X C 10 C 12 X
ð87Þ
6. Conclusion To study C3v XY3Z molecules, we have developed a tensorial formalism adapted to the chain O(3) C1v C3v. We have determined coupling and recoupling coefficients in this chain but also for the C3v group itself. The Wigner–Eckart theorem with useful formulas for the computation of reduced matrix element have been given. Special attention has been given to the various phases choices. We intend, on the one hand, to use this formalism for the development of the Hamiltonian and transition moment operators (dipole moment and polarizability), and, on the other hand, to generalize it, to the SU ð2Þ C I C S1v C S3v chain (where C S3v is the C3v point
A. El Hilali et al. / Journal of Molecular Spectroscopy 234 (2005) 113–121
group with its spinorial representations) and to apply it to the spectroscopy of methoxy (CH3O) and thiomethyl (CH3S) radicals having a half integer angular momentum. This will be the subject of forthcoming papers. Appendix A see Tables 8 and 9. Appendix B see Table 10. References [1] J.-P. Champion, M. Loe¨te, G. Pierre, Spherical top spectra, in: K.N. Rao, A. Weber (Eds.), Spectroscopy of the Earths atmosphere and interstellar medium, Academic Press, San Diego, 1992, pp. 339–422. [2] V. Boudon, J.-P. Champion, T. Gabard, M. Loe¨te, F. Michelot, G. Pierre, M. Rotger, Ch. Wenger, M. Rey, J. Mol. Spectrosc. 228 (2004) 620–634. [3] D. Papousˇek, M. Aliev, Molecular Vibrational-Rotational Spectra, Elsevier, New York, 1982. [4] N. Cheblal, M. Loe¨te, V. Boudon, J. Mol. Spectrosc. 197 (1999) 222– 231. [5] M. Rotger, V. Boudon, M. Loe¨te, J. Mol. Spectrosc. 200 (2000) 123– 130. [6] M. Rotger, V. Boudon, M. Loe¨te, J. Mol. Spectrosc. 200 (2000) 131– 137. [7] Ch. Wenger, M. Rotger, V. Boudon, J. Quant. Spectrosc. Radiat. Transfer 74 (2002) 621–636. [8] M. Rotger, V. Boudon, M. Loe¨te, J. Mol. Spectrosc. 216 (2002) 297– 307. [9] M. Rotger, V. Boudon, M. Loe¨te, L. Margule`s, J. Demaison, H. Ma¨der, G. Winnewisser, H.S.P. Mu¨ller, J. Mol. Spectrosc. 222 (2003) 172–179.
121
[10] V. Boudon, M. Rotger, N. Zvereva-Loe¨te, M. Loe¨te, J. Mol. Struct. (in press) (2005). [11] Ch. Wenger, M. Rotger, V. Boudon, J. Quant. Spectrosc. Radiat. Transfer 93 (2005) 429–446. [12] W. Raballand, M. Rotger, V. Boudon, M. Loe¨te, J. Mol. Spectrosc. 217 (2003) 239–248. [13] W. Raballand, M. Rotger, V. Boudon, M. Loe¨te, J. Breidung, W. Thiel, J. Mol. Struct. (in press) (2005). [14] B. Sartakov, J. Oomens, J. Reuss, A. Fayt, J. Mol. Spectrosc. 185 (1997) 31–47. [15] Ch. Roche, PhD Thesis, Dijon (France), 1992. [16] Ch. Roche, J.-P. Champion, A. Valentin, J. Mol. Spectrosc. 160 (1993) 517–523. [17] A. Nikitin, J.-P. Champion, V.G. Tyuterev, L.R. Brown, G. Mellau, M. Lock, J. Mol. Struct. 517 (2000) 1–24. [18] A. Nikitin, J.-P. Champion VI, G. Tyuterev, J. Quant. Spectrosc. Radiat. Transfer 82 (2003) 239–249. [19] M. Rey, V. Boudon, M. Loe¨te, F. Michelot, J. Mol. Spectrosc. 204 (2000) 106–119. [20] M. Rey, V. Boudon, M. Loe¨te, J. Mol. Struct. 599 (2001) 125–127. [21] M. Rey, V. Boudon, M. Loe¨te, P. Asselin, P. Soulard, L. Manceron, J. Chem. Phys. 114 (2001) 10773–10779. [22] M. Rey, V. Boudon, C. Wenger, G. Pierre, Sartakov, J. Mol. Spectrosc. 219 (2003) 313–325. [23] V. Boudon, F. Michelot, J. Mol. Spectrosc. 165 (1994) 554–579. [24] V. Boudon, F. Michelot, J. Moret-Bailly, J. Mol. Spectrosc. 166 (1994) 449–470. [25] A. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, Princeton, NJ, 1957. [26] F. Michelot, The`se de´tat, Dijon (France), 1980. [27] V. Boudon, PhD thesis, Dijon (France), 1995. [28] M. Rey, PhD Thesis, Dijon (France), 2002. [29] B. Lulek, T. Lulek, B. Szczepaniak, Acta Phys. Pol. A 54 (5) (1978) 545–559. [30] B. Lulek, T. Lulek, Acta Phys. Pol. A 54 (5) (1978) 561–572. [31] U. Fano, G. Racah, Irreducible Tensorial Sets, Academic Press, New York, 1959. [32] A. El Hilali, V. Boudon, M. Loe¨te, in preparation. [33] M. Rotenberg, N. Metropolis, R. Bivins, J.K. Wooten Jr., The 3 j and 6 j Symbols, The Technology Press, Cambridge, MA, 1959.
Spectroscopy of XY3Z (C3v) molecules: A tensorial formalism adapted to the O (3) C1v C3v group chain A. El Hilali, V. Boudon *, M. Loe¨te Laboratoire de Physique de lUniversite´ de Bourgogne, CNRS UMR 5027, 9, Avenue Alain Savary, BP 47870, F-21078 Dijon Cedex, France Received 7 June 2005; in revised form 26 August 2005 Available online 18 October 2005
Abstract A tensorial formalism adapted to the case of XY3Z symmetric tops has been developed. We use the O (3) C1v C3v group chain. All the coupling coefficients and formulas for the computation of the matrix elements are given for this chain. Such relations are also deduced in C3v group itself. 2005 Elsevier Inc. All rights reserved. Keywords: Group theory; Tensorial formalism; Symmetric tops; Rovibrational spectroscopy
1. Introduction In molecular spectroscopy, it is often considered that sophisticated group theoretical and tensorial formalism methods are only really useful for spherical top (i.e., highly symmetrical) molecules, for which they have proven their high efficiency [1,2]. Consequently, it is usually admitted that symmetric and asymmetric tops (i.e., lower symmetry) species should be treated using more ‘‘conventional’’ methods [3]. However, some key elements of the formalism developed in our group for tetrahedral or octahedral molecules [1,4] can be used with great profit even for less symmetrical systems: the ability of performing systematic developments of all rovibrational interactions in case of complex polyads and the so-called ‘‘vibrational extrapolation’’ which makes global analyses much easier. Recently, we have presented several extensions of the tensorial formalism to some quasi-spherical top molecules with C4v [5–7] or C2v symmetry [8–11], and also to D2h species [12,13], this last case deriving itself from an original idea of Sartakov et al. [14]. The idea of the present paper is to derive similar tools for XY3Z molecules with C3v symme*
Corresponding author. Fax: +33 3 80 39 59 71. E-mail address: [email protected] (V. Boudon).
0022-2852/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2005.09.002
try. In fact, this case has already been treated and used by Roche [15,16] and Nikitin et al. [17,18]. The formalism presented here is relatively similar but with some differences. It intends (i) to be fully consistent with preceding works on Td, Oh, C4v, C2v, and D2h and (ii) to serve as a basis for the extension to the C3v radicals with an odd number of electrons (similarly to what has been done for octahedral open-shell molecules [19–22]). The basic idea of this approach can be explained as follows: in molecular spectroscopy problems, if G is the point group of the molecule, we usually have to consider a larger group, denoted G00 (G is a subgroup of G00 ). In this case, we consider the G00 G group chain. In some cases, however, we are brought to consider an intermediate group G0 which corresponds to an approximate symmetry of the system, and we have then a chain G00 G0 G. For XY3Z molecules, with an equilibrium configuration belonging to the C3v group, and among the various possibilities, let us only mention two of them. One possibility consists in describing a ‘‘quasi-spherical’’ C3v symmetric top like CH3D or C35Cl337Cl using a chain like O (3) Td C3v (i.e., to perform a reorientation of the O (3) Td formalism [1] into the C3v subgroup). However, this is not as simple as for the similar cases of quasi-spherical symmetric C4v and asymmetric C2v molecules treated recently [2] for the follow-
114
A. El Hilali et al. / Journal of Molecular Spectroscopy 234 (2005) 113–121
ing reason: in the usual O (3) Td formalism for tetrahedral spherical tops, the quantization axis (z) is taken as one of the S4 axes which lies in between two X–Y bonds. For an XY3Z molecule with C3v symmetry, the z axis is taken, of course, as the C3 axis along the X–Z bond. We have thus two different axis orientations (unless we rewrite the O (3) Td formalism with the z axis along one of the X– Y bonds which would be quite meaningless for tetrahedral molecules themselves). This is why, for C3v molecules, in general, it is simpler to use a second scheme, that is the O (3) C1v C3v chain. In this case, the usual projection quantum number K corresponds to a C1v irreducible representations. C1v accounts for the approximate ‘‘cylindrical’’ (or ‘‘conical’’) symmetry of the molecule. We propose, in this theoretical work, the development of a tensorial formalism adapted to the study of XY3Z type molecules which possess integer angular momenta (i.e., in a singlet electronic state) by using the O (3) C1v C3v chain. This slightly differs from Roches [15,16] and Nikitins [17,18] works, who used the O (3) C3v direct chain. For this, we recall the bases of the spherical formalism for the SO (3) group (the extension to O (3) = SO (3) CI being trivial). Then we build a formalism adapted to the O (3) C1v chain that could be incidentally used for linear molecules such as CO, HF, HCN, . . . Then we finally carry out the C1v C3v orientation. Concerning the orientation of tensors in the G subgroup (C1v or C3v), we define a general method of calculation of the transformation matrix which realizes this symmetry adaptation. The main coupling coefficients can be calculated to obtain the necessary tools for the development of the rovibrational model. We intend, on the one hand, to use this formalism for development of the Hamiltonian and transition moment operators (dipole moment and polarizability), and, on the other hand, to generalize it to the SU ð2Þ C I C S1v C S3v group chain (the S superscript indicating the use of spinorial representations) [23,24] to study XY3Z type open-shell species which possess half-integer momenta such as CH3O and CH3S radicals. The present article, although rather technical, will serve as a starting point for all these extensions, which will be described in forthcoming papers. 2. Formalism in the O (3) group The ideas presented in this section will be used throughout all this work. We first consider proper symmetry operations (rotations) only, i.e., the SO (3) group, before finally extending expressions to the improper symmetry operations of the O (3) group. SO (3) is a Lie group with three parameters. All its elements are proper symmetry operations which leave invariant a point O. Any rotation R can be decomposed in a single way as a succession of three rotations characterized by Euler angles. From a passive point of view, we can write a representation of R as
Dða; b; cÞ ¼ exp
iaJ z ibJ y icJ z exp exp ; h h h
ð1Þ
where a, b, and c are the Euler angles, and Jy, Jz are two components of the angular momentum operator0 J. The m matrix elements of D denoted by ½DðjÞ ða; b; cÞm can be written in the standard basis {|j,mæ} as ðjÞ m0 D ða; b; cÞ m ¼ hj; m0 jDða; b; cÞjj; mi m0 0 ð2Þ ¼ eim a d ðjÞ ðbÞ m eimc ; d(j) (b) being the well-known Wigners function evaluated as [25] m0 m0 ½d ðjÞ ðbÞm ¼ DðjÞ ð0; b; 0Þ m 1 ðj þ m0 Þ!ðj m0 Þ! 2 X jm0 r r ¼ C jþm C jm ðj þ mÞ!ðj mÞ! r 2rþm0 þm 2j2rm0 m jm0 r ð1Þ cosðb2Þ sinðb2Þ ; ð3Þ where max (0, m m ) 6 r 6 min(j m , j + m). We can notice that the quantum number j also represents an irreducible representation (irrep) of SO (3) (m being an irrep of SO (2)). In a covariant formalism (see below) the transformation of an irreducible tensor T ðjÞ m of rank j and component m (j 6 m 6 j) can be expressed as X m0 ðjÞ DðjÞ ðRÞ m T m0 . ð4Þ 8R 2 SOð3Þ P R T ðjÞ m P R1 ¼ 0
0
m0
In the following, states of type |j, mæ will be considered as standard and contravariant in the molecular fixed frame, and states of type Æj, m| will be considered as standard and covariant. We write jj; k i WkðjÞ () contravariant set;
ð5Þ
ðjÞ
hj; k j ¼ Wk () covariant set. In a general way, the contra-covariant change is made using the metric tensor [26] j 0 ðjÞ T m ¼ ej ð6Þ T mðjÞ ; 0 mm 0 mm ðjÞ T mðjÞ ¼ e0j ð7Þ T m0 . j We adopt here Einsteins summation convention. ej and e0j are phase factors to be chosen. For involutive transformations, we have ej ¼ e0j and e2j ¼ 1. Let us note, for example, that with the choice of variance (5) E E E m m ðj j Þm m ð8Þ Wðj1 j2 jÞ ¼ F m11m22 ðjÞ Wðj11 Þ Wðj22 Þ ; where F are the Clebsch–Gordan coefficients of the SO (3) group. In all the following, symbols like [C] represent the dimension of the irrep C. In the case of SO (3), ½j ¼ 2j þ 1.
ð9Þ
A. El Hilali et al. / Journal of Molecular Spectroscopy 234 (2005) 113–121
As explained in [26–28], the various coupling coefficients for two angular momenta can be related by general relations : 0 0 j m1 m1 m2 m2 m1 m2 ðjÞ ðj j Þm0 F ðj1 j2 Þm ¼ Kðj1 j2 jÞ F m01m20 ðjÞ 0 1 2 mm j1 j2 ð10Þ 0 0 0 m1 m1 m2 m2 mm ðj j jÞ F ðjm11mj22jÞm ¼ K 0 ðj1 j2 jÞ F m01m20 m0 ; 1 2 j j1 j2 ðj j mÞ
F m11m22 ðjÞ m m ðjÞ
F ðj11 j22mÞ
mm ðj j jÞ iUðj1 j2 jÞ 12 ¼ ej e ½j F m11m22 m0 ; j j 1 0 ¼ e0j eiWðj1 j2 jÞ ½j2 F mðj11mj22jÞm . mm0
ð11Þ
0
ð12Þ ð13Þ
For our work, we take the same choices as those made by Michelot [26] for SO (3), ej ¼ e0j ¼ 1; 2j
Kðj1 j2 jÞ ¼ ð1Þ ; K 0 ðj1 j2 jÞ ¼ 1; 2j
2j
eiUðj1 j2 jÞ ¼ ð1Þ 1 ; eiWðj1 j2 jÞ ¼ ð1Þ 2 .
ð14Þ
This implies that j j mm0 2j jm ¼ ¼ ð1Þ ¼ ð1Þ dm0 ;m . m0 m mm0 j One can then write in this case m m ðjÞ
ðj j mÞ
F ðjm11mj22jÞm ¼ F mðj11mj22jÞm ; ðj j Þm
F m11m22 ðjÞ m m ðjÞ
F ðj11 j22mÞ
ð16Þ ð17Þ
mm 1 ðj j jÞ 2j ¼ ð1Þ 1 ½j2 F m11m22 m0 ; j j 1 0 2j2 2 ¼ ð1Þ ½j F mðj11mj22jÞm 0 mm 0
ð18Þ ð19Þ
and all these coefficients are real. In SO (3), the Wigner–Eckart theorem, for covariant quantities, can be written D E
0 1 ðkjÞm0 m ¼ ð1Þkjþj ½j0 2 F qmðj0 Þ j0 T ðkÞ j ; j0 ; m0 T ðkÞ ð20Þ j; q where Æ . . . i .. i . . . æ are the reduced matrix elements. Relation (2) gives SO (3) irreps. By carrying out an extension to O (3) = SO (3) CI (where CI = {E, I} and I is the space inversion) we get: 8 ðj s Þ ðjÞ > < D ðRÞ ¼ D ðRÞ; ð21Þ R 2 SOð3Þ ) Dðjg Þ ðIRÞ ¼ DðjÞ ðRÞ; > : ðj Þ ðjÞ u D ðIRÞ ¼ D ðRÞ; where s = g or u is the parity index. Let us note that each ji has its parity index and Clebsch–Gordan coefficients of O (3) have the additional non-zero condition: s1 s2 s3 ¼ g.
3. Orientation in the O(3) C‘v group chain The first part of our tensorial study consists in performing the reduction of O (3) into C1v. This operation results in a simple basis change, and allows us to express the C1v components of the oriented spherical tensor according to the standard components. A first basic step makes use of Wang type functions : 1 ð23Þ jjs ; Ki ¼ pffiffiffi ½jjs ; þK i jjs ; K i ðK 6¼ 0Þ. 2 In the special case K = 0, we have ( j ; 0þ ¼ j ; 0 if j is even; g g j ; 0 ¼ j ; 0 if j is odd; g
ð24Þ
g
and similarly but with 0+ and 0 inverted for ju. In the following, the parity index is omitted for simplicity but is implicitly considered. ðjÞ Thus, for covariants quantities, the passage from T k ðjÞ spherical components to the T Kd symmetrized spherical components (d = ±) is expressed as ðjÞ X ðjÞ ðjÞ W kKd T k . ð25Þ T Kd ¼ k¼K;þK
ð15Þ
F ðj11 j22mÞ ¼ F m11m22 ðjÞ ;
115
ð22Þ
In fact, K = |k| is an irreducible representation (irrep) of C1v and d corresponds to its component. ðjÞ W kKd is the Wang matrix. Table 1 gives reduction rules in O (3) C1v. The C1v character table and the multiplication rules for the C1v irreps are given in Appendix A. We can notice that: [K] = 2 for K > 0 and [0] = [0+] = 1. In the O (3) C1v group chain, we can write relations similar to those used in the spherical formalism (the phases being taken equal to 1, as in Eq. (14)), 0 0 j K d Kd ðjÞ ðjÞ K 0 d0 Kd T K 0 d0 . ð26Þ and T ¼ T Kd ¼ T ðjÞ ðjÞ KdK 0 d0 j Our oriented metric tensor is then X j j ðjÞ W kKd ¼ 0 kk KdK 0 d0 k;k 0 X jk ðjÞ ¼ ð1Þ W kKd
0
ðjÞ
W kK 0 d0
ðjÞ
W k K 0 d0 .
ð27Þ
k
This gives, j j 2j jþK / ¼ ð1Þ ¼ ð1Þ ð1Þ d dK;K 0 dd;d0 . K 0 d0 Kd KdK 0 d0 ð28Þ Table 1 Reduction of the Dðjg Þ representations of O(3) in C1v jg of O(3)
Irreps of C1v
0 1 2 3. ..
R+ ” 0+ R ¯ P ” 0 ¯ 1 R+ ¯ P ¯ D ” 0+ ¯ 1 ¯ 2 R .. ¯ P ¯ D ¯ U ” 0 ¯ 1 ¯ 2 ¯ 3 .
Rules for Dðju Þ are obtained by exchanging 0+(R+) and 0(R).
116
A. El Hilali et al. / Journal of Molecular Spectroscopy 234 (2005) 113–121
where, for K > 0, þ1 if d ¼ þ; /d ð1Þ ¼ 1 if d ¼ . For K = 0, we have j j j ¼ ¼ ð1Þ . 0 0 0þ 0þ
ð29Þ
eK 1 K 2 K is a phase factor to be chosen. In fact, the choices in Eq. (28) for O (3) C1v and similar ones that we made for C1v (see below) imply a relation between isoscalar factors and their complex conjugates that leads to the following condition:
ð30Þ
e2K 1 K 2 K ¼
ð1Þ
¼ 1.
ð31Þ
Contrary to what is usually adopted for such formalisms (i.e., SO (3) O [23], O (3) Td [26,1] and O (3) C3v [15,17]) and for reasons of coherence with our forthcoming work on radicals [32], we prefer to use Clebsch–Gordan coefficients instead of Racahs V or Wigners 3j. Thus, we define oriented Clebsch–Gordan coefficients in the O (3) C1v chain by X ðj j ÞKd ðj j Þk ðj1 Þ W kK11 d1 ðj2 Þ W kK22 d2 ðjÞ W Kd F k11k22ðjÞ . F K 11 d21 K 2 d2 ðjÞ ¼ k k1 k2 k
ð32Þ In this O (3) C1v chain, the Wigner–Eckart theorem, for contravariant quantities, can be written D 0 E
1 ðkjÞK 0 d0 ðj Þ ðkÞ ðjÞ WK 0 d0 T K 0 d0 WKd ¼ ½j0 2 F K 0 d0 Kdðj0 Þ ei/E j0 T ðkÞ j . ð33Þ As Michelot for SO (3) [26], we take ei/E ¼ 1.
ð34Þ
We are now able to deduce Clebsch–Gordan coefficients for the C1v group itself. We follow a method given by Lulek [29,30] and also used in [23]. This method is based on the use of Racahs factorization lemma, which can be written here as: ðj j ÞKd
ðj j ÞK
F K 11 d21 K 2 d2 ðjÞ ¼ K K 11 K22 ðjÞ K d K d ðjÞ
2 2 F ðj11 j21ÞKd
K K ðjÞ
¼ K ðj11 j22ÞK
2 F d1 d12 ðKÞ ;
ðK K Þd
ð35Þ
d d ðKÞ
ð36Þ
F ðK1 12K 2 Þd ;
where the K 0 s are called isoscalar factors. The Clebsch–Gordan matrix being unitary, we can define X ðj j ÞKd 2 N ðj1 K 1 ; j2 K 2 ; jKÞ ¼ F K 11 d21 K 2 d2 ðjÞ d1 d2 d
ðj j ÞK 2 ¼ ½K K K 11 K22 ðjÞ ;
e0þ 0þ 0þ ¼ 1; e110þ ¼ 1; e0 0 0þ ¼ 1; e110 ¼ i; e0þ 0 0 ¼ 1;
e10 1 ¼ i; e112 ¼ 1;
ð41Þ
e10þ 1 ¼ 1.
As before, the contra-covariant change in C1v itself is made means of a metric tensor (the phases are again taken equal to 1, as in Eq. (14)) 0 K dd ðKÞ ðKÞ d0 d Td ¼ T d0 . ð42Þ T ðKÞ and T ðKÞ ¼ dd0 K With the phases choices realized in [26], we can derive relations similar to Eqs. (10), (12), and (13) d d ðKÞ
F ðK1 12K 2 Þd ¼ ð1Þ2K
d01 d1
d02 d2
K ðK K 2 ÞK ; F d0 d10 ðKÞ 1 2 dd0
K1 K2 0 0 0 d2 d2 dd d1 d1 ðK K KÞ F dðK1 d12Kd2 KÞ ¼ F d0 d10 d20 ; 1 2 K K1 K2 0 1 dd ðK K 2 Þd ðK K KÞ 2K F d1 d12 ðKÞ F d1 d12 d20 ; ¼ ð1Þ 1 ½K 2 K K 1 0 d1 d2 ðKÞ 2K 2 2 F ðK 1 K 2 Þd ¼ ð1Þ ½K F dðK1 d12Kd2 KÞ . dd0
We choose, 0 K K dd 2K / ¼ ¼ ð1Þ d dd;d0 . ¼ ð1Þ dd0 d0 d K
ð43Þ ð44Þ ð45Þ ð46Þ
ð47Þ
The Table 2 give the exact values of some oriented symbols as well as our Clebsch–Gordan coefficients for the C1v group calculated by the method exposed above. We have limited the table to the C1v irreducible representations necessary to carry out the C1v C3v orientation described in the next section. 4. Formalism in the C‘v C3v chain
1
½K 2 ðj j ÞKd ¼ eK 1 K 2 K pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F K 11 d21 K 2 d2 ðjÞ ; N ðj1 K 1 ; j2 K 2 ; jKÞ
ð38Þ ðj1 Þ
ðj2 Þ
where j1 (j2) takes the minimum values such that D ðD Þ contains K1 (K2) (see Table 7 in Section 5.1). j is such that DðjÞ contains K and is compatible with the condition: j j1 j2 j 6 j 6 j 1 þ j2 .
For the lowest K values, we take the following phase factors:
ð37Þ
and get: ðK K 2 Þd F d1 d12 ðKÞ
ð40Þ
i¼1
This also implies that for K = 0, we define /d
3 Y j þK ð1Þ i i .
ð39Þ
We choose the C3v generators as: • The C3 operation is a 2p/3 rotation around the molecular axis z. We denote it C3z. • The mirror rv = C2 I considered as a product of a C2 rotation of the D3 group by space inversion I (the C2 axis was taken parallel to (Ox) axis of the molecular fixed frame).
A. El Hilali et al. / Journal of Molecular Spectroscopy 234 (2005) 113–121 Table 2 Exact values of some oriented Clebsch–Gordan coefficients for O (3) C1v and Clebsch–Gordan coefficients for the C1v group ðj j ÞKd
ðK K Þd
j 1 j2 j3
K1 K2 K3
d1 d2 d3
F K 11 d21 K 2 d2ðjÞ
2 F d1 d12 ðKÞ
0 1 0 1
0 0 0 1 1
0 0 0 0 0
+++ + + +++ +
1 p1ffiffi3 1 p1ffiffi 3 p1ffiffi
1 1 1
110
+
p1ffiffi2 p1ffiffi
0 1 1 1
0 0 1 0
111
0 0 0 1 1
3
+ 101
101
111
101
+++ + +
112
112
+
2
1 1 p1ffiffi 2 p1ffiffi 2 p1ffiffi 2 p1ffiffi 2 p1ffiffi 2 p1ffiffi 2
+ +++ + +
(
We have then: Dðjg Þ ðC 3z Þ ¼ DðjÞ ðC 3z Þ Dðjg Þ ðrv Þ ¼ DðjÞ ðC 2 Þ
( and
117
Table 4 C1v C3v correlation table C of C3v
K of C1v +
+
R ”0 R ” 0 P”1 D”2 U .. ” 3 .
p1ffiffi 2 p1ffiffi2 piffiffi
A1 A2 E E A .. 1 ¯ A2 .
2
piffiffi 2
X
1 1 i
T Cr ¼
i
or for contravariant quantities, X ðKÞ Cr d V d T ðKÞ T Cr ðKÞ ¼
ðKÞ
ðKÞ
ðKÞ
V dCr T d ;
ð50Þ
d
p1ffiffi 2 p1ffiffi 2 p1ffiffi 2 p1ffiffi 2
ð51Þ
d
Dðju Þ ðC 3z Þ ¼ DðjÞ ðC 3z Þ Dðju Þ ðrv Þ ¼ DðjÞ ðC 2 Þ
.
ð48Þ We thus obtain the O (3) matrices for the generators (and thus for all the symmetry elements) of the C3v group. The C3v matrices of theses generators are themselves chosen as in Table 3. The Wang transformation does not carry out the complete orientation, because, for K = 1 + 3p,2 + 3p,3 + 3p where p is an integer, it appears that the matrix blocs which match with the E irreps differ from a phase factor. A second transformation is thus used to orient C1v into C3v by making all the C1v irreps matrices for the C3v generators equivalent to those that we have chosen in Table 3. This orientation is carried out thanks to a V-unitary transformation, that perform the C1v C3v reduction recalled in Table 4. The oriented F coupling coefficients of C1v C3v can now be calculated in terms of C1v Clebsch–Gordan coefficients as: X ðK K ÞCr ðK K 2 Þd ðK 1 Þ d1 V C1 r1 ðK 2 Þ V dC22 r2 ðKÞ V Cr F d1 d12 ðKÞ . F C11r1 C2 2 r2 ðKÞ ¼ d
In theses two equations, C indicates an irrep of C3v and r is a component of this irrep. For example, in the case of the E irrep, we look for a unit matrix ½ðKÞ V dCr such that : 8 pffiffi ! 3 > ðKÞ d ðKÞ d 1 12 > 2 ðKÞ > V ðC Þ V ¼ D p ffiffi > 3z Cr Cr < 23 12 ð52Þ ; > > 1 0 1 > ðKÞ d > V Cr DðKÞ ðrv Þ ðKÞ V dCr ¼ : 0 1 where DðKÞ ðRi Þ ¼
ðjÞ
W kKd
DðjÞ ðRi Þ
ðjÞ
W kKd
1
;
and D ðRi Þ is the matrix of the j irreps of O (3). As shown below, the metric tensor of C3v depends on (1)K, that is on the parity of K. That is the reason why we need to define phase factors for all the values of K modulo 6 (although the C1v C3v reduction is the same for all K modulo 3 values, see Table 4). Table 5 gives the numerical values of the V-matrix elements. For the C1v C3v chain, we can write analog relations to those used before (the phase remains again the same, as before): 0 0 K C r Cr ðkÞ ðKÞ C 0 r0 Cr T Cr ¼ and T ¼ T C0 r0 . T ðkÞ ðKÞ CrC 0 r0 K
d1 d2 d
ð54Þ ð49Þ
We can write for covariant quantities Table 3 Matrices for the irreducible representations of C3v. A1 A2 E
C3z
rv
Basis set in terms of jj,mæ
1 1
1 1
|A1æ = |0,0æ |A2æ = |1,0æ
pffiffi ! p12ffiffi 23 23 12
ð53Þ
ðjÞ
1 0
0 1
jE1 i ¼ p1ffiffi2 j1; 1i þ i p1ffiffi2 j1; 1i jE2 i ¼ p1ffiffi2 j1; 1i þ i p1ffiffi2 j1; 1i
Considering phases chosen (see before) it follows that the oriented metric tensor is X K K 0 ðKÞ d V Cr ðKÞ V dC0 r0 ¼ 0 0 0 CrC r dd d0 d X / ¼ ð1Þ d ðKÞ V dCr ðKÞ V dC0 r0 . ð55Þ d
This gives, K CrC 0 r0
¼ ð1Þ
2K
K C 0 r0 Cr
K
¼ ð1Þ dC;C0 dr;r0 .
ð56Þ
118
A. El Hilali et al. / Journal of Molecular Spectroscopy 234 (2005) 113–121
Table 5 Numerical values of all V-matrix for jg elements 2m
6 + 6p 5 + 6p 4 + 6p 3 + 6p 2 + 6p 1 + 6p 0
2m + 1
ðj;KÞ
ðKÞ
ðj;6þ6p;A1 Þ
T T ðj;6þ6p;A2 Þ ðj;5þ6p;EÞ T1 ðj;5þ6p;EÞ T2 ðj;4þ6p;EÞ T1 ðj;4þ6p;EÞ T2 T ðj;3þ6p;A1 Þ T ðj;3þ6p;A2 Þ ðj;2þ6p;EÞ T1 ðj;2þ6p;EÞ T2 ðj;1þ6p;EÞ T1 ðj;1þ6p;EÞ T2 þ T ðj;0 ;A1 Þ
1 0 i 0 1 0 i 0 1 0 i 0 —
0 i 0 1 0 i 0 1 0 -i 0 1 —
— — — — — — — — — — — — 1
T ðj;6þ6p;A2 Þ T ðj;6þ6p;A1 Þ ðj;5þ6p;EÞ T1 ðj;5þ6p;EÞ T2 ðj;4þ6p;EÞ T1 ðj;4þ6p;EÞ T2 T ðj;3þ6p;A2 Þ T ðj;3þ6p;A1 Þ ðj;2þ6p;EÞ T1 ðj;2þ6p;EÞ T2 ðj;1þ6p;EÞ T1 ðj;1þ6p;EÞ T2 T ðj;0 ;A2 Þ
1 0 0 i 0 1 i 0 0 1 0 i —
0 i 1 0 i 0 0 1 i 0 1 0 —
— — — — — — — — — — — — 1
T Cr T ðj;K;CÞ r
K
jg
6 + 6p 5 + 6p 4 + 6p 3 + 6p 2 + 6p 1 + 6p 0
Vþ Cr
ðKÞ
V Cr
ðKÞ
V 0Cr
Those for ju are obtained by exchanging the blocks corresponding to j = 2m and j = 2m + 1.
In the C1v C3v chain, the Wigner–Eckart theorem, for contravariant quantities, can be written D E
1 ðK 0 KÞk 0 i/E 0 ðk Þ ðK 0 Þ ðK Þ ðKÞ Wk0 T k0 0 Wk K T K . ð57Þ ¼ ½K 0 2 F k0 kðK 0Þ e As before, we take ei/E ¼ 1.
ð58Þ
5. Tensorial algebra in the C3v group
It is again possible to calculate Clebsch–Gordan coefficients for C3v thanks to Racahs factorization lemma: ðK K ÞCr
ðK K ÞC
ðC C Þr
2 F r1 r1 2 ðCÞ ;
C r C 2 r2 ðKÞ F ðK11 K1 2 ÞCr
r r ðCÞ F ðC1 1 C2 2 Þr .
¼
C C ðKÞ K ðK11 K2 2 ÞC
ð59Þ
3 Y ð1ÞK i ;
ð62Þ
i¼1
here and in (61) K1(K2) takes the minimum values such that DðK 1 Þ ðDðK 2 Þ Þ contain C1(C2) only once (see Table 7). These minimum values of K, say Kmin, are listed for each irreducible representation of C3v in Table 7, while K is such that DðKÞ DðCÞ and: DðK 1 Þ DðK 2 Þ DðKÞ .
ð63Þ
We take the following phase factors:
The Clebsch–Gordan matrix being unitary, we can define X ðK K ÞCr 2 N ðK 1 C 1 ; K 2 C 2 ; KCÞ ¼ F C11r1 C2 2 r2 ðKÞ r1 r2 r
ðK K ÞC 2 ¼ ½C K C11C2 2ðKÞ .
ð61Þ
where eC1 C2 C3 is a phase factor to be chosen. In fact, the choices in Eq. (56) for C1v C3v, and similar ones that we made for C3v (see below), imply a relation between isoscalar factors and their conjugates that leads to the following condition: e2C1 C2 C ¼
5.1. Clebsch–Gordan coefficients and the Wigner–Eckart theorem in C3v
F C11r1 C2 2 r2 ðKÞ ¼ K C11C2 2ðKÞ
1
½C 2 ðC C 2 Þr ðK K ÞCr ¼ eC1 C2 C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F C11r1 C2 2 r2 ðKÞ ; F r1 r1 2 ðCÞ N ðK 1 C 1 ; K 2 C 2 ; KCÞ
ð60Þ
This leads to the desired Clebsch–Gordan coefficients :
eA1 A1 A1 ¼ 1;
eA2 A2 A1 ¼ 1;
eA1 A2 A2 ¼ 1;
eEEA1 ¼ 1; eEA2 E ¼ 1;
eEEA2 ¼ 1; eEEE ¼ 1;
eEA1 E ¼ 1.
ð64Þ
We impose that our 3C r coefficients (see below) are left unchanged by any even permutation of their columns and that for an odd permutation they are multiplied by the factor.
A. El Hilali et al. / Journal of Molecular Spectroscopy 234 (2005) 113–121
ð1ÞC1 þC2 þC .
ð65Þ
The (1)C phases are obtained in multiplying each C3v representation by itself. As a result, the symmetric part is associated to (1)C = 1 and the antisymmetric one to (1)C = 1, so that we obtain (see Table 10 in Appendix B): A1
¼ ð1Þ ¼ 1;
ð66Þ
A2
¼ 1.
ð67Þ
ð1Þ ð1Þ
E
Table 6 Exact values of some oriented Clebsch–Gordan coefficients for C1v C3v and Clebsch–Gordan coefficients for the C3v group r1 r2 r
F C 1 1r1 C2 2 r2 ðKÞ
2 F r1 r1 2 ðCÞ
0+ 0+ 0+ 0 0 0+ 0+ 0 0 1 1 0+
A1 A1 A1 A2 A2 A1 A1 A2 A2 E E A1
Æ Æ Æ Æ Æ Æ Æ Æ Æ 11.
1 1 1 p1ffiffi2 p1ffiffi
1 1 1 p1ffiffi 2 p1ffiffi 2
1 1 0
E E A2
p1ffiffi2 p1ffiffi2 1 1 1 1 p1ffiffi
p1ffiffi2 þ p1ffiffi2 1 1 1 1 p1ffiffi p1ffiffi 2 p1ffiffi 2 p1ffiffi 2
22.
E A1 E
1 0 1
E A2 E
r r ðCÞ F ðC1 1 C2 2 Þr
112
EEE
r r ðCÞ
F ðC1 1 C2 2 Þr
C 0 r0
1. .1 .2 .2 .1 11
ð68Þ
¼
1 C 2 CÞ F ðC r1 r2 r ;
ð69Þ
221 122
p1ffiffi2 p1ffiffi
ð70Þ
212
p1ffiffi2
pffiffiffiffiffiffi ðC C CÞ ½C F r1 r1 2 r2 ; pffiffiffiffiffiffi r rr ¼ ½C F ðC1 1 C2 CÞ .
ðC C Þr0
r r ðCÞ
ð72Þ
ð74Þ
The Wigner–Eckart theorem in C3v can be expressed (for covariant sets) by: D E
1 ðC 0 CÞr0 i/E 0
ðC 0 Þ Wr0 T rðC0 0 Þ WðCÞ C T ðC 0 Þ C ; ð75Þ ¼ ½C 0 2 F r0 rðC 0 e r Þ
kmin jmin
A1
A2
E
0 0
0 1
1 1
5.2. Recoupling coefficients for C3v We will now consider briefly the recoupling coefficients for C3v. By analogy with what has been done by Fano and Racah [31] for O (3), we can define 6C coefficients from the recoupling matrix for three irreducible representations by the relation: jjj C 1 j C 2 i; C 12 i j C 3 i; C; ri C1 C2 1 X 1 2 2 ½C 23 ¼ ½C 12 C3 C C 23
C 12 C 23
kC 1 kC 2 i C 3 i; C 23 i; C; ri.
again with ei/E ¼ 1.
2
2
Table 7 Minimum values of j and K such that DðjÞ DðKÞ DðCÞ
1 2
2 1 2 F r1 r1 2 ðC 0 Þ F ðC C Þr ¼ dC 0 C dr0 r . 1 2
2
ð71Þ
and the Clebsch–Gordan coefficients are real. The values of our Clebsch–Gordan symbols are given in Table 6. We can also introduce the orthogonality relations which will be useful for the calculation of the 6C factors (see next paragraph): X ðC C Þr r r ðCÞ 2 F r0 r1 0 ðCÞ F ðC1 1 C2 2 Þr ¼ dr01 r1 dr02 r2 ; ð73Þ X
2 1 2 1 2 1
¼
We have chosen C ¼ dr;r0 ; rr0
r01 r02
2
12.
ðC C 2 Þr F r1 r1 2 ðCÞ ;
2 ¼ F r1 r1 2 ðCÞ
ðC C Þr
C1 C2 C
1 0+ 1
ðC C Þr
ðK K ÞCr
K1 K2 K
Likewise, the 3C r and Clebsch-Gordan coefficients can be related through expressions similar to Eqs. ()(10)–(13) and with the same phase factors choice as in the preceding section :
r1 r2 r F ðC 1 C 2 CÞ
119
ð76Þ
We can also note a useful formula which relates the reduced matrix elements in C3v to those in C1v. sffiffiffiffiffiffiffiffi
0 0 ðK ;C Þ ½C 0 ðK 0 KÞC0 0
ð K 0 Þ K ; ð77Þ K C T 0 0 KC ¼ 0 K C 0 CðK 0 Þ K T ½K as well as the formula relating reduced matrix elements in O(3) and C3v for an O (3) C1v C3v oriented tensor: sffiffiffiffiffiffiffiffi
0 0 0 ðj ;K ;C Þ
½C 0 ðj0 jÞK 0 ðK 0 KÞC0 0
ðj0 Þ j . j K C T 0 0 0 jKC ¼ 0 K K 0 Kðj0 Þ K C 0 CðK 0 Þ j T ½j ð78Þ
ð79Þ
Expanding the two members of expression (79) and taking our convention as well as (73) and (74) into account, we obtain C 1 C 2 C 12 C 3 C C 23 C þC þC C ð1Þ 1 2 3 X ðC1 C2 Þr12 ðC12 C3 Þr r1 r23 ðCÞ r2 r3 ðC23 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F r r ðC Þ F r r ðCÞ F ðC1 C23 Þr F ðC2 C3 Þr23 ½C 23 ½C 12 ½C allr 1 2 12 12 3 ð80Þ The nonvanishing conditions for a 6C are:
120
C1 C3
A. El Hilali et al. / Journal of Molecular Spectroscopy 234 (2005) 113–121
C2 C
8 C 1 C 23 C; > > > < C 2 C 3 C 23 ; C 12 6¼ 0 ) > C 12 C 3 C; C 23 > > : C 1 C 2 C 12 .
Table 8 Characters of the irreducible representations of C1v
ð81Þ +
It is possible to demonstrate the following very useful formula, which gives the reduced matrix elements of two coupled tensor operators of C3v acting on the same space. D E ðC0 Þ
C 0 T ðC1 Þ T ðC2 Þ
C X 1 C1 C2 C0 C 1 þC 2 þCþC 0 2 ½C 0 ¼ ð1Þ C C 0 C 00 C 00 0 ðC Þ 00 00 ðC Þ C T 1 C C T 2 C ð82Þ
ð83Þ
We merely say that these 9C symbols can be calculated from the 6C coefficients though the relation 8 9 < C 1 C 2 C 12 = C C 4 C 45 : 3 ; C 13 C 24 C X C 2 C 4 C 24 C 1 C 3 C 13 0 ¼ ½C C 24 C C 0 C 3 C 0 C 34 C0 C 12 C 34 C . ð84Þ C0 C1 C2 The nonvanishing conditions for a 9C are: 8 C 1 C 2 C 12 ; > > > > 9 8 > C 3 C 4 C 34 ; > > > > < C C C; = < C 1 C 2 C 12 > 13 4 C 3 C 4 C 34 6¼ 0 ) > > > C C 3 C 13 ; > ; : > 1 C 13 C 24 C > > > C 2 C 4 C 24 ; > > : C 12 C 34 C.
R ”0 R ” 0 P”1 D”2 U .. ” 3 .
ð85Þ
This enable us to express the reduced matrix elements of two coupled tensor operators acting on two different spaces:
D E ðCÞ
C 01 C 02 C 0 T ðC1 Þ U ðC2 Þ C 1 C 2 C 9 8 0 C 1 C 02 C 0 > > = < 1 ¼ ð½C ½C 0 ½CÞ2 C1 C2 C > > ; : C1 C2 C
C 01 T ðC1 Þ C 1 C 02 T ðC2 Þ C 2 . ð86Þ We can also finally introduce the 12C coefficients given in terms of 9C and 6C symbols [33]:
E
2C(u)
1rv
1 1 2 2 ..2 .
1 1 2 cosu 2 cos(2u) ..2 cos(3u) .
1 1 0 0 0. ..
Table 9 Multiplication table for the irreducible representations of C1v +
R R P D .. .
In the same way, we can define the 9C symbols by jjjC 1 jC 2 i; jC 12 i jjC 3 jC 4 i; C 34 i; C; ri 8 9 < C 1 C 2 C 12 = 1 ¼ ð½C 12 ½C 12 ½C 12 ½C 12 Þ2 C 3 C 4 C 45 : ; C 13 C 24 C jjjC 1 jC 3 i; jC 13 i jjC 2 jC 4 i; C 24 iC; ri
+
R+
R
+
R R P D .. .
R R+ P D .. .
P
D
...
P P R+¯R ¯ D P¯U .. .
D D P¯U + R .. ¯R ¯ C .
... ... ... ... ...
Table 10 Multiplication table for the irreducible representation of C3v A1 A2 E
A1
A2
E
[A1] A2 E
A2 [A1] E
E E [A1 + E] + {A2}
[. . .] symmetric part, {. . .} antisymmetric part.
8 > < C1 C5 > : C9
C2
C3
C6 C 10
C7 C 11
9 C4 > = C8 > ; C 12
9 8 > = < C1 C2 C3 > X C þC þC þC ¼ ð1Þ 3 7 9 10 ½X C 5 C 6 C 7 > > ; : X C 9 C 10 X C 11 C 3 C 4 C 11 C 9 C 8 . C 7 C 12 X C 10 C 12 X
ð87Þ
6. Conclusion To study C3v XY3Z molecules, we have developed a tensorial formalism adapted to the chain O(3) C1v C3v. We have determined coupling and recoupling coefficients in this chain but also for the C3v group itself. The Wigner–Eckart theorem with useful formulas for the computation of reduced matrix element have been given. Special attention has been given to the various phases choices. We intend, on the one hand, to use this formalism for the development of the Hamiltonian and transition moment operators (dipole moment and polarizability), and, on the other hand, to generalize it, to the SU ð2Þ C I C S1v C S3v chain (where C S3v is the C3v point
A. El Hilali et al. / Journal of Molecular Spectroscopy 234 (2005) 113–121
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