Here we summarize the basic concepts of the spherical-tensor formalism. This is not intended as an explanatory account, but merely as a summary of the principal formulae and definitions that we need. A fuller account may be found in one of the many textbooks on angular momentum theory, such as Brink and Satchler (1993) or Zare (1988).
B.1 Spherical harmonics
Recall that the spherical harmonics, usually denoted Ylm(θ, φ), are the functions of θ and φ that satisfy the eigenvalue equations for the angular momentum operators:
They are normalized so that ∫|Ylm|2 sin θ dr dθ dφ = 1, and because they are eigenfunctions of the hermitian operators Lˆ2 and Lˆz, they are orthogonal:
The angular momentum operator Lˆ = r × pˆ has components defined by
In spherical polar coordinates, Lˆz = –iℏ∂/∂φ, while
The operators Lˆ± = Lˆx ± Lˆy are important in the theory. The spherical harmonics satisfy
so that Lˆ+ and Lˆ− have the effect of shifting the eigenvalue m up or down by one unit. For this reason they are called shift or ladder operators.
These evidently satisfy the same eigenvalue equations, but their normalization is different: they satisfy C l0(0, 0) = 1. The use of these functions avoids the factors of that otherwise clutter up the equations. The explicit form of the spherical harmonics is derived in some elementary texts on quantum mechanics, and is
where is an associated Legendre function (Abramowitz and Stegun 1965, Zare 1988), and the phase factor ϵ m, important for maintaining the phase relationships required by eqn (B.1.2), is (−1)m for m 〉 0 and 1 for m ≤ 0.
We can also define related functions called the regular and irregular spherical harmonics:
where r, θ and φ form the spherical polar representation of the vector argument r. These functions satisfy Laplace’s equation: ∇2 Rlm = 0 everywhere, while ∇2 Ilm = 0 except at the origin. This is easily demonstrated using the fact that
The first few of the regular spherical harmonics are
Here x, y and z are the cartesian components of r. As the functions with non-zero m are complex, it is helpful to define real functions Rlmc and Rlms, for m 〉 0, by
This relationship between the real and complex components can be written in the form
Here we use the Greek label κ generically for the real components; for a given value of l ittakes the values 0, 1c, 1s, …, lc, ls. The transformation coefficients X κm are zero except for X 00 = 1 and
The matrix X is unitary.
It is sometimes more convenient to express the relationship between real and complex components differently:
and m¯ ≡ –m. If this is done, there is no need to restrict m in eqn (B.1.8) to be positive, and the definitions ensure that Rlm¯c = Rlmc and Rlm¯s = –Rlms. We also see that
for all m, and that R l0c = R l0 and R l0s = 0.
The labels c and s stand for ‘cosine’ and ‘sine’ respectively, since Rlmc and Rlms are proportional to cos mφ and sin mφ respectively. Some of these functions are tabulated in Table E.1. Real irregular spherical harmonics are defined similarly, and can be obtained from the corresponding regular harmonics by dividing by r 2l+1. We shall also use real forms Clmc and Clms of the ordinary spherical harmonics.∗
B.2 Rotations of the coordinate system
The spherical harmonics Ylm and Clm and the regular and irregular solid harmonics Rlm and Ilm all depend on the angular coordinates in the same way. Accordingly they all transform under (p.274) rotations in the same way. A rotation is described by the Euler angles defined in Fig. 1.2, and the effect of such a rotation on a spherical harmonic is
where the are the Wigner rotation matrices. That is, the rotated function R(α,β, γ)Ylm can be expressed as a linear combination of the original set of functions, with coefficients that are elements of the Wigner rotation matrix. Here we are using the active convention for rotations; that is, the l.h.s. of eqn (B.2.1) describes the function that we obtain by rotating the spherical harmonic Ylm according to the Euler angles α, β and γ, while the r.h.s. is a linear combination of the original set of unrotated functions. The alternative passive convention involves rotating the axes rather than the functions; see Brink and Satchler or Zare. The Wigner rotation matrices are defined by eqn (B.2.1); since is the coefficient of Ylm′ in the rotated function R(α,β, γ)Ylm, we can use the orthogonality of the spherical harmonics to write
Explicit tables of the Wigner rotation matrices are given in Brink and Satchler for l ≤ 2.
B.3 Spherical tensors
A spherical tensor of rank l is now defined as any set of 2l + 1 quantities, labelled by m = l, l − 1,…, −l like the spherical harmonics, for which the same transformation law holds as for the spherical harmonics. That is, if
where the are the Wigner rotation matrices as in eqn (B.2.1), then the set Tlm is a spherical tensor of rank l.
For our purposes, the main importance of this is that the multipole moments, when expressed in spherical-tensor form, do satisfy this requirement. However, any quantum-mechanical operator can be expressed in spherical-tensor form. For example, the spherical-tensor components of a vector operator with cartesian components vx, vy and vz are
Instead of regarding the as matrices, defined for particular values of α, β and γ, it is possible to regard them as functions of α, β and γ for particular values of l, m and k. In this context they are called Wigner functions. In fact it turns out that is the normalized rotational wavefunction of a symmetric top whose orientation is described (p.275) by the Euler angles α, β and γ and which has angular momentum quantum number l, with component m in the global z direction and component k in the molecule-fixed z direction. From this it follows that the set of , for any fixed k, is a spherical tensor of rank l.
B.4 Coupling of wavefunctions and spherical tensors
Given a set of eigenfunctions φ l 1 m 1 with angular momentum eigenvalues l 1 and m 1, and another set ψ l 2 m 2 with eigenvalues l 2 and m 2, it is possible to construct from the products φ l 1 m 1 ψ l 2 m 2 a set of angular momentum functions ΨLM for each value of the eigenvalue L from l 1 + l 2 by integer steps down to |l 1 −l 2| (the Clebsch–Gordan series). For example, from the 15 products Y 1m 1 Y 2m 2 of the rank 1 and rank 2 spherical harmonics, such as arise in the wavefunction for an atom with one p electron and one d electron outside closed shells, we can construct functions with L = 3, 2 and 1. The formula for the new functions is
where the coefficient 〈l 1 l 2 m 1 m 2|LM〉 is called a Wigner or Clebsch–Gordan or vector coupling coefficient. It is zero unless m = m 1 + m 2 and L is one of the values l 1 + l 2, l 1 + l 2 − 1,…, |l 1 − l 2|. The latter condition is called the triangle condition, and can be expressed symmetrically: l 1 + l 2 + L must be an integer, and none of the three may exceed the sum of the other two. There is a formula for the Wigner coefficient (Brink and Satchler 1993, Zare 1988), but it is quite complicated and values are usually obtained from tables, for example in Varshalovich et al. (1988, pp. 271–278).
The same procedure can be applied when the factors in the products are not wavefunctions φ l 1 m 1 and ψ l 2 m 2 but spherical-tensor operators R l 1 m 1 and S l 2 m 2. In this case it often happens that l 1 = l 2 and we require L = 0. In this case the coupled function is a scalar: it is invariant under rotations. The Wigner coefficients take a particularly simple form in this case, and the coupled operator is
The reason for the importance of this case is that the Hamiltonian for an isolated system is invariant under rotations, so these invariant combinations of operators are the only ones that can occur in such a Hamiltonian. It occurs so often, in fact, that it is usual to define the ‘scalar product’ of two sets of spherical-tensor operators as
in agreement with the conventional cartesian definition. Indeed, this is the main reason for the use of the definition (B.4.2), the scalar product of two vectors being then the same whether the cartesian or the spherical tensor formulation is used.
(p.276) Where we have a spherical-tensor ‘scalar product’ of the form ∑m(–1)m A l,–m Blm, it can always be replaced by a scalar product in terms of the real components. For any m 〉 0 we can use eqn (B.1.5) to show that
When we sum over m from 1 to l and add the m = 0 term we get
B.4.1 Wigner 3j symbols
A similar situation sometimes arises when we have three sets of spherical tensor quantities, R l 1 m 1 S l 2 m 2 and T l 3 m 3 and we wish to construct a scalar from them. In this case we can begin by constructing a rank l 3 tensor from R and S (this is possible only if l 1, l 2 and l 3 satisfy the triangle condition) and then form the scalar product of this with T:
where the quantity in parentheses is a Wigner 3j symbol, defined by
The phase factor here is chosen to make the 3j symbol as symmetric as possible: it is invariant under even permutations of its columns, while odd permutations multiply it by a factor (–1)l 1+l 2+l 3
One example is the S function introduced in §3.3. It takes the form
Here θ and φ are the polar angles describing the direction of the intermolecular vector R. We see that apart from some additional numerical constants, this has the form of eqn (B.4.4), with j and m taking the place of l 3 and m 3. (Remember that is the m 1 component of a spherical tensor of rank l 1.)
When three or more sets of spherical tensor quantities are coupled together, there are usually several ways of doing it. The relationship between the different ‘coupling schemes’ is expressed by Wigner 6 j and 9 j symbols. For an explanation of these quantities, consult Brink and Satchler (1993) or Zare (1988).
(∗) Some authors use a different notation, writing, for example, Zlm for Clmc and Zlm¯ for Clms. This notation has several disadvantages: it requires a new letter, Z instead of C, so that the link with the complex form is less apparent; the form Zlm is not obviously a real rather than complex component; and the regular and irregular harmonics either require two new, less mnemonic, letters than R and I, or must be expressed less compactly as rlZlm and r –l–1 Zlm. The only compensating advantage, if indeed it is an advantage, is the loss of the c or s suffix.