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A Novel Representation of the Angular Part of Atomic Orbitals

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Sumio TOKITA^{a}*, Fuyuko KIDO^{a}, Takao SUGIYAMA^{a} and Haruo HOSOYA^{b}

^{a}Department of Applied Chemistry, Faculty of Engineering, Saitama University

255 Shimo-Ohkubo, Urawa, Saitama, 338-8570 Japan

^{b}Department of Information Sciences, Ochanomizu University

2-1-1 Otsuka, Bunkyo-ku, Tokyo, 112-8610 Japan

**e-mail: *

(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 *R*_{n,l}(*r*) and angular part *Y*_{l,m}(q, j) using *r*, q, and j as polar coordinates, *i.e.*, c(*r*, q, j)=*R*_{n,l}(*r*)*Y*_{l,m}(q, j), where *n*, *l*, *m* are principal, angular momentum, and magnetic quantum number, respectively. The angular part *Y*_{l,m}(q, j), defined as *Y*_{l,m}(q, j)= Q_{l,m}(q)j*m*(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)=|*Y*_{l,m}(q, j)| on this vector for all q and j. If we suppose the equation |*Y*_{l,m}(q, j)|/*r*, the point on the above vector where *r* equals to **r**(q, j) will provide the value of this equation |*Y*_{l,m}(q, j)|/*r* as unity (Figure 3). The product of *r* and |*Y*_{l,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 *r**Y*_{l,m}(q, j)/*r*^{2}(*x*, *y*, *z*) equal unity should coincide with the conventional representation of the angular part. In this study, we calculated the function values of *r**Y*_{l,m}(q, j)/*r*^{2}(*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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