A Novel Representation of the Angular Part of Atomic Orbitals

Sumio TOKITAa*, Fuyuko KIDOa, Takao SUGIYAMAa and Haruo HOSOYAb

aDepartment of Applied Chemistry, Faculty of Engineering, Saitama University
255 Shimo-Ohkubo, Urawa, Saitama, 338-8570 Japan
bDepartment of Information Sciences, Ochanomizu University
2-1-1 Otsuka, Bunkyo-ku, Tokyo, 112-8610 Japan

(Received: April 7, 2000; Accepted for publication: June 20, 2000; Published on Web: November 20, 2000)

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 c(r, q, j) is represented by the product of radial part Rn,l(r) and angular part Yl,m(q, j) using r, q, and j as polar coordinates, i.e., c(r, q, j)=Rn,l(r)Yl,m(q, j), where n, l, m are principal, angular momentum, and magnetic quantum number, respectively. The angular part Yl,m(q, j), defined as Yl,m(q, j)= Ql,m(q)jm(j), is expressed in terms of spherical harmonics. A vector OP is decided by angles q and j (Figure 2). The angular part has been visualized by plotting distance r(q, j)=|Yl,m(q, j)| on this vector for all q and j. If we suppose the equation |Yl,m(q, j)|/r, the point on the above vector where r equals to r(q, j) will provide the value of this equation |Yl,m(q, j)|/r as unity (Figure 3). The product of r and |Yl,m(q, j)| is transformed to Descartes coordinates by the following expression: x = r sinqcosj, y = r sinqsinj, and z = r cosq. Namely, the isosurface where the function values of the following equation rYl,m(q, j)/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 rYl,m(q, j)/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).

Keywords: Atomic Orbital, Angular Part, Four-Dimensional Atomic Orbital, Isosurface, AVS

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