Tesseral harmonics

Let ) ( ) represents the (21 + l)th derivative of the (n + l)th Laguerre polynomial (20) and P7 (cos ) is Ferrers associated Legendre function of the first kind, of degree l and order m. Yim Zm thus constitutes a tesseral harmonic (21). The p s are in this form orthogonal and normalized, so that they fulfill the conditions... 

On application of the ordinary methods of perturbation theory, it is seen that the first-order perturbed wave function for a normal hydrogen atom with perturbation function f r)T, , perturbation function. The statements in the text can be verified by an extension of this argument. 

The rest are called tesseral harmonics, which have both vertical and horizontal nodal lines on the unit sphere ... 

TABLE 1 Relevant tesseral harmonics for intensity calculations before and after rotations x —> y —> z —> x and x y z x in C4V symmetry (Prather, 1961)... 

These are called tesseral harmonics since they divide the surface of a sphere into tesserae—four-sided figures bounded by nodal lines of latitude and longitude. (Nodes are places where the wavefunction equals zero.) When m = 0, the spherical harmonics are real functions ... 

The expansion coefficient is real, but 7 and Y are complex quantities. It is more convenient to replace the spherical harmonics yO Y and Y by the tesseral harmonics Z g, Z and Z, which are real. The relations between the spherical and tesseral harmonics are ... 

The superscripts c and s stand for the presence of the factors cos qq>) and sin( p) in the tesseral harmonics. The expressions for the tesseral harmonics in polar coordinates can be calculated easily with eqs. (14)-(16) from the expressions of the spherical harmonics. 

The tesseral harmonics in cartesian coordinates for A = 0 to A = 6 have been tabulated by Prather (1961). His results together with the tesseral harmonics for k=l are given in Appendix 2. Equation (10) can be rewritten as ... 

We will illustrate the development of the even part of the crystal-field potential for the symmetries C4 and D3, First of all we determine which tesseral harmonics have the same symmetry elements as the point group C4. In C4, only a fourfold rotation axis is present. This axis can generate the symmetry operations C, C (= C2), C and the identity. The tesseral harmonics Z q, and Zgf, have cylindrical symmetry and are present in the even part of every crystal-field potential. The C4 axis is a symmetry element for the harmonics Z44, Z44, Zg4 and Zg4. We have thus the following coefficients in the even part of the crystal-field potential for C4 Bl, Bq, B, B l, 5 and B. The potential can be written as... 

We want to give here also some comments on the definitions of the crystal-field parameters by Wybourne (1965). His crystal-field formalism is sometimes a little bit confusing. First, he introduces the parameters which have the same transformation properties as spherieal harmonies. These parameters are in general complex numbers. Both fij and oeeur. Moreover, his parameters transform as tesseral harmonics, like our parameters do. It is therefore easy to confuse and S. Prather (1961) always uses tesseral harmonies for his parameters and also for his operators. 

These ratios are the cubic ratios and are also reported by Morrison and Leavitt (1982) and by Caro (1976). The ratios reported by Prather (1961) are a factor /2 larger, because his operators are also tesseral harmonics. Both 5q and Bq will be positive. 

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