A Novel Representation of the Angular Part of Atomic Orbitals

Abstract

Visualization is one of the most suitable methods for understanding atomic orbital wavefunctions. The angular part of an atomic orbital is very important, because it has information on all the symmetry properties of the wavefunctions. The atomic orbital wavefunction χ(r, θ, φ) is represented by the product of radial part R_{n, l}(r) and angular part Y_{l, m}(θ, φ) using r, θ, and φ as polar coordinates, i.e., χ(r, θ, φ)=R_{n, l}(r)Y_{l, m}(θ, φ), where n, l, m are principal, angular momentum, and magnetic quantum number, respectively. The angular part Y_{l, m}(θ, φ), defined as Y_{l, m}(θ, φ)= Θ_{l, m}(θ)φm(φ), is expressed in terms of spherical harmonics. A vector OP is decided by angles θ and φ (Figure 2). The angular part has been visualized by plotting distance r(θ, φ)=|Y_{l, m}(θ, φ)| on this vector for all θ and φ. If we suppose the equation |Y_{l, m}(θ, φ)|/r, the point on the above vector where r equals to r(θ, φ) will provide the value of this equation |Y_{l, m}(θ, φ)|/r as unity (Figure 3). The product of r and |Y_{l, m}(θ, φ)| is transformed to Descartes coordinates by the following expression: x = r sinθcosφ, y = r sinθsinφ, and z = r cosθ. Namely, the isosurface where the function values of the following equation rY_{l, m}(θ, φ)/r²(x, y, z) equal unity should coincide with the conventional representation of the angular part. In this study, we calculated the function values of rY_{l, m}(θ, φ)/r²(x, y, z) at 64, 000 points ((x, y, z) = 40×40×40), and visualized isosurfaces from these data using a software called AVS (Application Visualization System, Figure 1). The present method is applied to visualize not only the angular parts of three-dimensional atomic orbital wavefunctions (Figures 4 -7), but also those of four-dimensional ones (Figure 9).