The Spherical Harmonics

if we transform the function according to the matrix above, 

It s worth noting the weirdness of this example. Normally if you take a second derivative of a polynomial of a certain degree, that degree goes down by 2. In other words, this trick of using polynomials won t work for the regular Laplace operator, so why should it work for the spherical one The key lies in the finite dimensional hypothesis in Schur s lemma. For the lemma to work, the group G and the operator A must commute on this 

As a matter of fact, if we look at what happens to those polynomials when we substitute Euler angles and set the radius to 1, we find something quite different. The polynomial z, for example, becomes 

If we pass to polynomials of degree 2, we have to be careful. They do not always look homogeneous when we finish setting x2 + y2 + z2 = 1. Let us look at z2, for example, which is not harmonic and so should not be an eigenfunction of A. Passing to spherical coordinates, and setting r = 1, this gives the function 

G is an eigenfunction of A with eigenvalue —6. Now, if we were to create a bunch of new functions by applying our rotation group to this one, we would indeed create the irreducible representation described by Schur s lemma. Does this correspond to a harmonic function in x,y,z Well, you can check that 

We begin in Section 6.4.1 by reviewing, without proofs or derivations, the standard properties of the spherical harmonics and their relationship to the associated Legendre polynomials. The closely related solid harmonics are next introduced in Section 6.4.2. In Sections 6.4.3 and 6.4.4, we derive explicit Cartesian expressions for the complex and real solid harmonics, respectively. Finally, in Section 6.4.5, we derive a set of recurrence relations for the real solid harmonics. 

The spherical harmonic functions [10,11] are characterized by two quantum numbers 

Let us consider the general form of the spherical harmonics. In the standard treatment, the spherical harmonics are formed by taking products of the associated Legendre polynomials in cos and exponentials in mup-. 

The associated Legendre polynomuds [11,12] may be obtained from the Rodrigues expression (in the phase convention of Condon and Shortley) 

The associated Legendre polynomials are orthogonal on the interval — 1 x 1 in the sense that 

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In summary, separation of variables has been used to solve the full r,0,( ) Schrodinger equation for one electron moving about a nucleus of charge Z. The 0 and (j) solutions are the spherical harmonics YL,m (0,(1>)- The bound-state radial solutions... 

The spherical harmonics are defined in terms of the associated Legendre polynomials, of variable cos 6, and exponential functions in... 

The model of non-correlated potential fluctuations is of special interest. First, it can be solved analytically, second, the assumption that subsequent values of orienting field are non-correlated is less constrained from the physical point of view. The theory allows for consideration of a rather general orienting field. When the spherical shape of the cell is distorted and its symmetry becomes axial, the anisotropic potential is characterized by the only given axis e. However, all the spherical harmonics built on this vector contribute to its expansion, not only the term of lowest order... 

The orbitals d. and dy can be expressed in terms of the complex forms di and whose angular parts are given by the spherical harmonics and respectively. The matrix of orbital angular momentum about the z axis in the complex basis is... 

By reexpressing the spherical harmonics in the form of D rotation matrices they may be effectively substituted by a Clebsch-Gordan series yielding... 

We may readily derive a general expression for the spherical harmonic Yim(9, (p) which results from the repeated application of L+ to Yi-i(9, tp). We begin with equation (5.38a) with m set equal to — /... 

Because both and Lz are hermitian, the spherical harmonics Yim(0, q>) form an orthogonal set, so that... 

Thus, the operators H and have the same eigenfunctions, namely, the spherical harmonics Yj iO, q>) as given in equation (5.50). It is customary in discussions of the rigid rotor to replace the quantum number I by the index J m the eigenfunctions and eigenvalues. 

All the s orbitals have the spherical harmonic Too(0, q>) as a factor. This spherical harmonic is independent of the angles 0 and spherically symmetric about the origin. Likewise, the electronic probability density is spherically symmetric for s orbitals. 

Since the spherical harmonics are normalized, the value of the double integral is unity. 

Transitions between states are subject to certain restrictions called selection rules. The conservation of angular momentum and the parity of the spherical harmonics limit transitions for hydrogen-like atoms to those for which A/ = 1 and for which Am = 0, 1. Thus, an observed spectral line vq in the absence of the magnetic field, given by equation (6.83), is split into three lines with wave numbers vq + (/ bB/he), vq, and vq — (HbB/he). 

When atoms occupy highly symmetrical sites, a further limitation of the current multipolar expansions is the limited order of the spherical harmonics employed, that do not usually extend past the hexadecapolar level (/ = 4). Only two multipolar studies published to date used spherical harmonics to orders higher than 1 = 4 graphite [15] and crystalline beryllium [16]. In the latter work, the most significant contribution to the valence density was indeed shown to be given by a pole of order 1 = 6. 

The expansion coefficients of the spherical harmonics for a deliberate distribution g((p, y/) are easily computed from... 

In Equation 1.3, the radial function Rnl (r) is defined by the quantum numbers n and l and the spherical harmonics YJ" depend on the quantum numbers l and W . When the spin of the electron is taken into account, the normalized antisymmetric function is written as a Slater determinant. The corresponding eigenvalues depend only on n and l of each single electron, which determine the electronic configuration of the system. 

Taking into account that Bq parameters represent the coefficient of an operator related to the spherical harmonic ykq then the ranges of k and q are limited to a maximum of 27 parameters (26 independent) Bq with k = 2,4,6 and q = 0,1,. .., k. The B°k values are real and the rest are complex. Due to the invariance of the CF Hamiltonian under the operations of the symmetry groups, the number of parameters is also limited by the point symmetry of the lanthanide site. Notice that for some groups, the number of parameters will depend on the choice of axes. In Table 2.1, the effect of site symmetry is illustrated for some common ion site symmetries. 

The spherical harmonics in real form therefore exhibit a directional dependence and behave like simple functions of Cartesian coordinates. Orbitals using real spherical harmonics for their angular part are therefore particularly convenient to discuss properties such as the directed valencies of chemical bonds. The linear combinations still have the quantum numbers n and l, but they are no longer eigenfunctions for the z component of the angular momentum, so that this quantum number is lost. 

The eigenfunctions of the L operators will be denoted by the spherical harmonics Y(m. 

The solutions of the angular dependent part are the spherical harmonics, Y, known to most chemists as the mathematical expressions describing shapes of (hydrogenic) atomic orbitals. It is noted that Y is defined only in terms of a central field and not for atoms in molecules. 

One can see then that the states l,m> are the spherical harmonics Ylm. It is convenient to give also the realization of the operators Z+ and / ... 

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