Clebsch Gordan coefficients

Clebsch-Gordan coefficients have already occurred several times in our considerations in the Introduction (formula (2)) while generalizing the quasispin concept for complex electronic configurations, while defining a relativistic wave function (formulas (2.15) and (2.16)), in the addition theorem of spherical functions (5.5) and in the definition of tensorial product of two tensors (5.12). Let us discuss briefly their definition and properties. There are a number of algebraic expressions for the Clebsch-Gordan coefficients [9, 11], but here we shall present only one  

The Clebsch-Gordan coefficient may be expressed in terms of the Wigner coefficient (symbol in round brackets)  

Many special cases of the Clebsch-Gordan coefficients have very simple algebraic expressions, usually without sums. Thus, a general formula for the Clebsch-Gordan coefficient with all projection parameters equal to zero may be easily found from Eqs. (5.8) and (5.6). If one of the parameters ji = 0, then 

As we can see from Eq. (6.8), the Wigner coefficients are much more symmetrical than the Clebsch-Gordan coefficients. Therefore, usually their tables are presented (see, for example, in [11] the tables for 71+72+73 16). 

In angular momentum theory a very important role is played by the invariants obtained while summing the products of the Wigner (or Clebsch-Gordan) coefficients over all projection parameters. Such quantities are called 7-coefficients or 3ny-coefficients. They are invariant under rotations of the coordinate system. A j-coefficient has 3n parameters (n = 1,2,3.), that is why the notation 3nj-coefficient is widely used. The value n = 1 leads to the trivial case of the triangular condition abc, defined in Chapter 5 after formula (5.25). For n = 2,3,4. we have 67 -, 9j-, 12j-,. .. coefficients, respectively. 3nj-coefficients (n 2) may be also defined as sums of 67-coefficients. There are also algebraic expressions for 3nj-coefficients. Thus, 6j-coefficient may be defined by the formula 

The extension of a given determinantal wavefunction (called the parent wave-function, which in the simplest case can be just a one-electron spin-orbital) to include another non-equivalent electron (or even a group of non-equivalent electrons) is made with the help of vector-coupling or Clebsch-Gordan coefficients 

For the coupling of two angular momenta a and b to the resulting value c with the corresponding magnetic quantum numbers a, P and y, the Clebsch-Gordan coefficients (atxbp cy) are defined as expansion coefficients in the relation 

If this condition is not fulfilled, the Clebsch-Gordan coefficients vanish, otherwise they have certain numerical values (see Table 7.1). The Clebsch-Gordan coefficients are related to the Wigner coefficients [Wig51], also called 3j symbols jt = a, h = b, j3 = c,m1 = a, m2 = P,m3 = y), defined by 

These and other j symbols can be found as program packages in many computer libraries. 

It is important to note that different authors use different phase conventions. Those of Condon and Shortley [CSh35] will be employed here, requiring [STa63] 

This is straightforward matrix multiplication but a useful exercise nevertheless to confirm that the matrix Z is correctly given by k. 

The inner direct product (DP) (or inner Kronecker product) 

lJ are called Clebsch-Gordan (CG) coefficients. They have the property 

The generalization of eq. (3) to magnetic groups (Bradley and Davis (1968) Karavaev (1965)) is 

Exercise 14.3-1 Write down the non-zero CG coefficients for the inner DPs of the point group mm2 (C2V). [Hints See Table 14.4. Recall that %3 (R) means x(r3) and that for this group T3 = T4.] Using Table 14.6 derive expressions for the non-zero CG d coefficients of the magnetic point group 4mm in terms of the cijk and evaluate these. Hence write down the CG decomposition for the Kronecker products of the IRs T of 4mm. 

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Clebsch-Gordan coefficients coupling ground and excited levels that = transitions coupled by linearly... 

The coefficients 0 are variously called angular momentum addition coefficients, or Wigner coefficients, or Clebsch-Gordan coefficients. Their importance for quantum mechanics was" first recognized by Wigner,6 who also provided a formula and a complete theory of them. The notation varies among different authors who deal with them7 ours follows most closely that of Rose. 

Secondly, due to the smallness of the rotational temperature for the majority of molecules (only hydrogen and some of its derivatives being out of consideration), under temperatures higher than, say, 100 K, we replace further on the corresponding summation over rotational quantum numbers by an integration. We also exploit the asymptotic expansion for the Clebsch-Gordan coefficients and 6j symbol [23] (JJ1J2, L > v,<0... 

The conditions determined by the selection rules for Clebsch-Gordan coefficients and 6j symbols provide the following block-diagonal structure of the operator... 

The asymptotic expressions for the corresponding Clebsch-Gordan coefficients take the form... 

In Eq. (12), l,m are the photoelectron partial wave angular momentum and its projection in the molecular frame and v is the projection of the photon angular momentum on the molecular frame. The presence of an alternative primed set l, m, v signifies interference terms between the primed and unprimed partial waves. The parameter ct is the Coulomb phase shift (see Appendix A). The fi are dipole transition amplitudes to the final-state partial wave I, m and contain dynamical information on the photoionization process. In contrast, the Clebsch-Gordan coefficients (CGC) provide geometric constraints that are consequent upon angular momentum considerations. 

M-sum unitarity of the first two Clebsch-Gordan coefficients now means that the sum over p reduces to a simple delta function ... 

From (4.56) and Table 4.3, we derive the relative intensity ratios 3 2 1 1 2 3 for the hyperfine components of a Zeeman pattern of a powder sample. The transition probability for the case of the polar angle 6 = Oq can readly be calculated by integrating (4.56) only over the azimuthal angle (j). One obtains a factor (1 + cos 0o)/2 and sin 0o for m = 1 and m = 0, respectively, which are multiplied by the square of the Clebsch-Gordan coefficients. As a consequence of the angular correlation of the transition probabilities the second and fifth hyperfine components (Fig. 4.17) disappear if the direction k of the y-rays and the magnetic field H are parallel (0q = 0). 

Where appropriate the spin components are then coupled via the Clebsch-Gordan coefficients. In this way therefore the wave functions for a given dx configuration may be obtained from the dx 1 results. 

The products of second-rank tensor components, such as A1-, j ( P/ JA1-, j (0Fi), can be expressed in coupled form using the Clebsch-Gordan coefficients, yielding tensor terms, of spatial rank / = 0, 2, and 4 ... 

The expansion coefficients on the right hand side of Eq. (1.25) are the Clebsch-Gordan coefficients.2 The eigenfunctions of the angular momentum, which can be written, abstractly, using Dirac notation 11, m >, satisfy the equations (h=1)... 

Inserting the appropriate values of the Clebsch-Gordan coefficients, one obtains... 

The result (2.167) is particularly important, since it is used to analyze experimental data. It is merely a consequence of the fact that the quadrupole operator is a tensor of rank 2. Sj is just the square of the Clebsch-Gordan coefficients in... 

In this expression the coefficients in brackets < > are the isoscalar factors (Clebsch-Gordan coefficients) for coupling two 0(4) and two 0(3) representations, respectively. They can be evaluated either analytically using Racah s factorization lemma (Section B.14) or numerically using subroutines explicitly written for this purpose.2... 

We define the Clebsch-Gordan coefficients with the usual (Condon and Short-ley, 1967) phase convention... 

The Clebsch-Gordan coefficients satisfy the orthogonality relations... 

Instead of Clebsch-Gordan coefficients it is often convenient to use the Wigner 3 — j symbols... 

Sharp, R. T. (1960), Simple Derivation of the Clebsch-Gordan Coefficients, Am. J. Phys. 28, 116. 

We can pass from tree a to b using the suitable Clebsch-Gordan coeficient (eq. 12). The tree (c) illustrates the hyperspherical parametrization that leads to the hyperspherical harmonics Yn- Xm(, W 9) They are related to the harmonics of tree a through the Z coeficient defined in eg. (15). The connection between (b) and (c) requires a Clebsch Gordan coefficient and a phase change related to a (see eq. (14)). 

Here, the subscript (c) is short for the set of expansion parameters (c) = (2i, 22, A, L, oi, u2) r, is the vibrational coordinate of the molecule i R is the separation between the centers of mass of the molecules the Q, are the orientations (Euler angles a, jS y,) of molecule i Q specifies the direction of the separation / the C(2i22A M[M2Ma), etc., are Clebsch-Gordan coefficients the DxMt) are Wigner rotation matrices. The expansion coefficients A(C) = A2i22Al u1u2(ri,r2, R) are independent of the coordinate system these will be referred to as multipole-induced or overlap-induced dipole components - whichever the case may be. 

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