An Attempt to Construct an Isosurface Having Symmetry Elements
Sumio TOKITA, Takao SUGIYAMA, Fumio NOGUCHI, Hidehiko FUJII and Hidehiko KOBAYASHI
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1 Introduction
An isosurface is a surface that represents points of a constant value, for example, temperature, pressure, density, and so on. Isosurfaces are displayed by computer graphics, and are used as visualization methods in many fields of chemistry. Orbital wavefunction, probability density, electron density and electrostatic potential have been visualized as isosurfaces [1  11]. The most promising method to draw isosurfaces has been reported as "marching cubes algorithm" [12]. Although the usefulness of the above algorithm has been established by many websites [13  15] and original papers [11, 12, 16], another method for the drawing of an isosurface of an atomic orbital is pursued in this paper with a novel attempt to use symmetry operations.
2 Volume visualization based on symmetries in atomic orbitals
The Schrodinger equation of an electron of a hydrogen atom is represented as follows.
where, h is Planck's constant, m is a mass of an electron, k_{0} is the electrostatic constant (Coulomb force constant), e is an elementary charge, and x, y, z are coordinates of an electron. The solutions, c of the equation (1) are called atomic orbital wavefunctions.
It turns out that each orbital is defined by three quantum numbers that are restricted to certain discrete values.

The principal quantum number, n = 1, 2, ..., determines the energy E of the electron in the atom.

The azimuthal quantum number, l = 0, 1, 2, ..., n1, determines the orbital angular momentum of the electron.

The magnetic quantum number, m = 0, ±1, ..., ± l determines the amount of angular momentum of the electron around a particular axis.
Since the wavefunction c_{nlm} is a complex quantity where m <> 0, it is transformed by the following linear combinations (2)(3).
The orbitals thus obtained are called 1s, 2s, 2p, 3s, 3p, 3d, ..., where the numbers 1, 2, 3, ..., correspond to the values of n, and the letters s, p, d, ..., correspond to the values of l.
l = 0, 1, 2, 3, 4, ...
s, p, d, f, g, ...
The letters then run alphabetically, with the omission of i.
The shape of the isosurfaces of hydrogen atomic orbitals is shown in Figure 1. As the shapes of the orbitals represented by the equation (2) or (3) are identical, one of them is shown in Figure 1. These atomic orbitals have the following symmetry.
Figure 1. Isosurfaces of hydrogen atomic orbitals
The 1s, 2s, 3s,... atomic orbitals, namely, the orbitals having the value of azimuthal quantum number l = 0, have spherical symmetry, in other words, have symmetry of C_{¥} with respect to every axis including the origin.
The shortest way to draw the isosurface of 1s orbital,
where,, is to calculate the function value of c_{1s} from r = 0 to +¥ until the value reaches an appropriate constant a, for example,
and to give a sphere centered on the origin and having radius r that satisfies the equation (5).
In the case of c_{2s} orbital, the maximum number of such spheres having the constant value of ±a is 4. The isosurfaces having the constant value of a = ±0.027 au^{3/2} are drawn as concentric spheres (a small sphere of a = +0.027 au^{3/2}, and a large sphere of a = 0.027 au^{3/2}) as are shown in Figure 2.
A vertical line of r = 3.5 au in Figure 2 left represents the distance from the origin to one of the 6 faces of the cube in Figure 2 right.
Figure 2. Concentric spheres representing the isosurfaces of the c_{2s} atomic orbital (a = ±0.027 au^{3/2}) [3].
The atomic orbitals having the value of magnetic quantum number m = 0 have cylindrical symmetry of C_{¥} with respect to the Z axis (C_{¥}^{z}). When the value of the azimuthal quantum number l is even, the atomic orbital is symmetric with respect to s(xy), on the other hand, when l is odd, the orbital is antisymmetric with respect to s(xy).
For example, in order to draw the isosurfaces of c_{4f5z33zr2} atomic orbital (n = 4, l = 3, m = 0), we must obtain only two red lines in the first quadrant A and B in Figure 3. Antisymmetric operation with respect to XY plane and succeeding rotational operation with respect to Z axis gave a framework shown in Figure 4. Shading and smoothing procedure gave the isosurfaces represented in Figure 1.
Figure 3. Contour map of hydrogen 4f_{5z33zr2} atomic orbital which has the function values of +0.005 (solid line), and 0.005au ^{3/2} (broken line) [17].
Figure 4. Wireframe representation of hydrogen 4f_{5z33zr2} atomic orbital
When the value of the azimuthal quantum number l equals n1, and the value of magnetic quantum number m equals +1 or 1, the shape of the resulting orbitals c_{npx} or c_{npy} obtained by equation (2) or (3) is identical with the c_{npz} atomic orbital ( l = n  1, m = 0 ) because c_{npx} orbital has a symmetry axis C_{¥}^{x} and c_{npy} has C_{¥}^{y}.
When the absolute value of the magnetic quantum number equals or is larger than 1, the hydrogen atomic orbital has an antisymmetric axis C_{m}^{2}. The hydrogen atomic orbital having an even value of l + m is symmetric and an odd value of l + m is antisymmetric with respect to s(xy).
The total number of the nodal surface of hydrogen atomic orbital is n1. The c_{1s} orbital has no nodal surfaces. When the azimuthal quantum number equals to 0, namely, 1s, 2s, ..., the orbital has n1 spherical nodes. When the azimuthal quantum number equals 1, the atomic orbital has a single plenary node and n2 spherical node(s).
When the azimuthal quantum number l is larger than 1 and the magnetic quantum number m equals 0, the atomic orbital has two conical nodes and n  3 spherical nodes ( l: even) or n  4 spherical node and a single planar nodes ( l: odd). For example, c_{5d3z2r2} orbital ( n = 5, l = 2, m = 0 ) has two conical nodes of q = 54.7° and q = 125.3°(Figure 5 and Figure 6) and two spherical nodes (Figure 5 (b) and (c)).
Figure 5. Schematic representation [4] of hydrogen c_{5d3z2r2} atomic orbital: (a) in the cube of 100×100×100; (b) in the cube of 64×64×64; (c) in the cube of 20×20×20 au^{3}
Figure 6. Two conical nodes of q = 54.7°(the upper part) and q = 125.3°(the lower part)
When the absolute value of the magnetic quantum number m is larger than 0, the atomic orbital has m + 1 planar node(s) including Z axis and 0 ( l: odd) or 1 ( l: even) planar nodes of XY plane, and n  m  2 ( l: odd) or n  m  3 ( l: even) spherical node(s).
The above discussion about the symmetry operation and the information about the shape and number of nodes facilitates to minimize the drawing duration to obtain isosurfaces. Several symmetry operations are already included in our recent software. The software [18] will be distributed on request.
3 Conclusion
Consideration of symmetry elements in hydrogen atomic orbitals was shown to facilitate efficient drawing of their isosurfaces.
The authors acknowledge a GrantinAid for Scientific Research Category B, No. 15300267 from Japan Society for the Promotion of Science.
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