Visualizing Atomic Orbitals Jing B. Wang, Paul C. Abbott, and Jim F. Williams Department of Physics University of Western Australia Nedlands WA 6907, Australia wang@physics.uwa.edu.au, paul@physics.uwa.edu.au The Mathematica software package provides a range of tools for working with atomic sys- tems. The symbolic tools include orthogonal polynomials and Clebsch-Gordan coefficients whilst the graphical capabilities cover polar plots; spherical plots; density plots; contour plots in two and three dimensions; and animation. These tools are applied to the manipulation and visualization of atomic orbitals. INTRODUCTION The probability distribution of the electron charge cloud in an atom, namely the atomic orbital, often reveals crucial information in atomic physics. Visual representation of such orbitals is invaluable to researchers working in this field and also helps teaching atomic structures to physics or chemistry students. The atomic orbital concepts were discussed in detail by Berry and the cited papers therein, including several visualization devices. However, the newly developed symbolic package Mathematica provides a wider range of graphical tools (including polar plots, spherical plots, density plots, contour plots in two and three dimensions, and animation), which can be applied to help visualise the radial and angular behaviour of multi-dimensional systems. Mathematica also has a wide range of built-in special functions appropriate for the analysis of quantum systems. Orthogonal polynomials — Legendre, Laguerre, Chebyshev, Hermite and Jacobi (corresponding to the Mathemat- ica functions P n x, L n a x, U n x, T n x, H n x, and P n a,b x) — arise when solving a wide class of eigenvalue and boundary value problems. The spherical harmonics, Y l m Θ, Φ, are eigenfunctions of the angular part of the Lapla- cian operator (which is the quantum mechanical angular momentum operator). Clebsch-Gordan coefficients arise in the study of angular momenta in quantum mechanics, and in other applications of the rotation group. These functions, taken together with the orthogonal polynomials and graphical tools, provide a powerful environment for the study of atomic systems. It is worth noting that these methods can be readily utilized in the studies of other multi-dimensional systems. As this article is not intended to be a tutorial on Mathematica, commands and syntax will not usually be explained. However, it is hoped that by showing a well-chosen set of examples, even readers unfamiliar with Mathematica will be able to follow the examples. The graphical output is placed immediately after its corresponding input command as it appears in Mathematica notebooks. Arbitrary units are used in these graphs. 1. NOTATION AND PREREQUISITES The quantity r r Ψ n,l ,m r r pure basis state Ψ n,l ,m r, Ψ n,l r r coherently mixed state Ψ n,l r m c m Ψ n,l ,m r, where Ψ n,l,m r nlm R n,l rY l m Θ, Φ, r r, Θ, Φ, r r 2 r , sinΘ Θ Φ, represents the probability of finding an electron in the volume element r. The spherical harmonics, Y l m Θ, Φ, give the angular distribution of the atomic orbitals and R n,l r is the corresponding radial distribution. In the following, the general spherical harmonic will be denoted using the Mathematica shorthand Y l m Θ, Φ. For example,