Jon B. Applequist -- Maxwell-Cartesian Spherical Harmonics

Let r be the distance of a point from the origin in a Cartesian coordinate system. A Maxwell-Cartesian spherical harmonic of degree n is a partial derivative of 1/r of order n with respect to x,y,z multiplied by (-1)nr2n+1. These functions occur in the theory of multipole potentials and many other physical problems involving central forces.

The values of the functions on the unit sphere show the dependence of the functions on the direction in space. Such forms, known as surface spherical harmonics, are familiar as the angular part of atomic orbital wave functions. However, the functions normally chosen for atomic orbitals, known as tesseral harmonics, are distinct from the Maxwell-Cartesian harmonics in important respects.

The following "tricorn" arrays show some important mathematical properties of the Maxwell-Cartesian surface spherical harmonics.

Each function is defined by three indices which are the degrees of the function (a homogeneous polynomial) in x,y,z, respectively. The sum of the indices is the total degree n of the function. Each tricorn contains all of the possible functions of a given total degree, which ranges from 1 to 4 in the figures shown. The absolute value of the function in any direction in space is given by the distance of the colored surface from its center. Gold surfaces correspond to positive values; blue surfaces correspond to negative values.

The reader will recognize the functions of degree 1 as the angular part of p orbitals. Likewise, the functions of degree 2 bear a close resemblance to d orbitals, and so on. However, one will notice that the number of functions of degree 2 or higher is not the value 2n + 1 expected from quantum mechanics. Furthermore, the symmetry of each set of functions with respect to x,y,z is not found in the usual treatments of atomic orbitals in terms of tesseral harmonics.

A tesseral harmonic of degree n can always be calculated as a linear combination of the Marxwell-Cartesian functions in one horizontal row of the tricorn of degree n. This is one example of the utility of the tricorn for displaying the mathematical properties of the functions.

If you would like to read more about the Maxwell-Cartesian spherical harmonics, click here.

If you would like to generate formulas and graphics of the Maxwell-Cartesian and tesseral harmonics on your computer, click here to obtain a Mathematica package that will do this. You will need to have Mathematica installed on your computer to use the package.

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