The solid harmonics

In this normalization, Coo(, p) is equal to I - compare with the spherical harmonics in Table 6.1. 6.4.2 THE SOLID HARMONICS 

Henceforth, we shall work almost exclusively with radial functions of the form 

Associating the radial factor in (6.4.11) with the spherical-harmonic part of the one-electron functions (6.3.13), we may write the one-centre functions in the form 

In Section 6.4.3, we shall demonstrate that the solid harmonics may be expressed as 

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Vi = Y so that Ytm is the value of the solid harmonic at points on the surface of the unit sphere defined by the coordinates 8 and (p, and hence Y is called a surface harmonic of degree l. Surface harmonics are orthogonal on the surface of the unit sphere and not at r = 0, as commonly assumed in the definition of atomic orbitals. 

It is useful to define in general differential operators from the solid harmonics by performing in Eqs. (15 a) or (15b) the substitutions... 

When the differential operator operates on a solid harmonic Eq. (5a) will yield the operator in a particularly simple form. Since the solid harmonic is subject to the Laplace equation V 2 (r 3 ) = 0, all but the first term can be excluded from the expression of the operator. Having decided on the analytical form of the reduced matrix will be deduced. With the sohd harmonic in the form Eq. (5 b), one has... 

The reduced matrix can be used to write down the result of the action of the differential operator on the solid harmonic ri 3t,... 

For r = 1, V = y so that y is the value of the solid harmonic in the surface of the unit sphere at points defined by the coordinates 6 and surface harmonics of degree /. The associated Legendre polynomials Pf(cosd) have l — m roots. Each of them defines a nodal cone that intersects a constant sphere in a circle. These nodes, as shown in Figure 20-5, are in the surface of the sphere and not at r = 0 as assumed in the definition of atomic orbitals. Surface harmonics are obviously undefined for r = 0. The linear combinations i/ i i/i i define one real and one imaginary function directed along the X and Y Cartesian axes respectively, but these functions (denoted and ipy) are no longer eigenfunctions of L, but of or Ly instead. 

The potential within the volume of the solute molecule is a smooth function and can be expanded in an appropriate set of expansion functions gifr). This is a generalization of the usual multipole expansion. The latter uses Cartesian monomials 1, x, y, z, xy, y, xz, yz, x, ..., or the corresponding solid spherical harmonics to expand the potential. However, for reasons explained in [17], the origin-centered multipole expansion is unsuitable for most systems. We experimented with several expansion sets. One important requirement is that the expansion functions should not diverge to infinity like the solid harmonics do. We finally settled on a sine function expansion of the potential. It is conveniently defined in an outer box that is larger than the extent of the molecular electronic density, to avoid problems with the periodic nature of the sine expansion. Other... 

To remove the constant fraction in front of each term, the solid harmonics in Racah s normalization are used ... 

The complex solid harmonics are eigenfiinctirxis of the total angular-momentum operator, which in (6.3.4) is given in polar coordinates. The form of this operator is rather complicated, however, making the solution of the associated eigenvalue problem (in order to arrive at the solid harmonics)... 

The solid harmonics are easily seen to be solutions to Laplace s equation [14]. Thus, from the expressions (6.3.3) and (6.4.1), we find... 

Let us determine, in Cartesian coordinates, the solutions to Laplace s equation that correspond to the solid harmonics for a given angular momentum 1. AccortUng to the preceding discussion, the solution must be a polynomial in x, y and z of degree / ... 

We have thus established that the solutions (6.4.37) to Laplace s equation represent products of r with the spherical harmonics - that is, they correspond to the solid harmonics 9Jm-It remains to determine the expansion coefficients (6.4.34). Expressing p and q in terms of t and m and substituting the results into (6.4.30), we obtain the recurrence relation... 

Except for the normalization constant (see Exercises 6.3 and 6.4), this step completes our derivation of the solid harmonics (6.4.33) from Laplace s equation. Our results are summarized in (6.4.14)-(6.4.16). 

Note that all terms in the solid harmonics of order / in (6.4.47) are of degree / exactly. The reader may wish to verify that (6.4.47) reproduces the real solid harmonics in Table 6.3. 

In Section 6.4.4, we derived an explicit expression for the real solid harmonics. Often, it is more convenient to calculate the solid harmonics by recursion, in particular when the full set of solid harmonics up to a given angular momentum / is needed. In the present subsection, we shall derive a set of recurrence relations for the real solid harmonics of the form... 

The factor has here been introduced since, in our discussion of multipole expansions of the Coulomb integrals in Section 9.13, we shall work with different sets of scaled solid harmonics. Of course, when the solid harmonics are calculated from an explicit expression such as (6.4.47), any scaling factor is trivially incorporated and requires no special attention. 

To simplify matters, we shall consider the generation of the solid harmonics Sim in two steps. First, we evaluate all functions of the type 5 , / by diagonal recursion, where the quantum numbers I and m are changed simultaneously. Next, the remaining solid harmonics Sim are generated by vertical recursion, where m is kept fixed and I is incremented. While the diagonal recurrence relations are easily established, the derivation of the vertical recurrences requires more work. 

To obtain the diagonal recurrences, we first note from (6.4.19)-(6.4.21) that the solid harmonics Si, i may be written in the simple form... 

To obtain the desired vertical recurrence relations for the solid harmonics, it is sufficient to determine a recurrence for the expansion coefficients of the form... 

Box 6.1 One-electron basis functions expressed in terms of the generalized L uerre polynomials (6.5.1) and the solid harmonics (6.4.13). The functions are not normalized... 

At this point, it would be possible to express this expansion in terms of the solid harmonics (6.4.22)... 

It is often convenient to employ a common notation for the cosine and sine components of the solid harmonics. In such cases, we shall indicate real components by the use of Greek letters, adopting the following convention to distinguish between the cosine and sine components... 

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