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where

Although a vast number of methods of orbital visualization have already been reported [4 - 13], most of them are two-dimensional (2-D) projections of 3-D pictures. Shading or 3-D rotation technique helps us to image 3-D pictures from 2-D projections. Real 3-D representations such as solid models also exist [4]. Even if they accurately describe the shape of an orbital, they are not sufficient by themselves. Their sharp boundaries often mislead beginners that the electron moves along the surface. They are far from being suitable images of "electron clouds" or "probability density". A laser sculpture method in a glass block has been employed for reproducing the boundary surface features of the atomic orbitals [5]. This paper deals with a novel application of the laser technique to visualize the "real" 3-D probability densities of the hydrogen atomic orbitals.

(i) Let

(ii) The variables

(iii) A parameter

(iv) By utilizing uniform random numbers

(v) If > M ×

the point

(vi) Procedures (iv and v) are repeated to get a desired number of probable points.

In the

Figure 1. Characteristic features of the hydrogen 2s orbital; (**a**): function value of c_{2s} orbital against distance *r* from atomic nucleus. Points A and C correspond to the radii of two concentric sphere shells having the same absolute value of 0.027 au^{-3/2}. Point B (*r* = 2.00 au) corresponds to the radius of the nodal sphere; (**b**)-(**e**): snapshots of an animation of hydrogen c_{2s} isosurfaces in a cube of *R* × *R* × *R* au^{3}. By changing the display region *R*, the inner shell is shown.

For a representation of the inner information of the probability density of an atomic orbital, the cross section [6, 7, 11] of the electron clouds is useful (Figure 2(**a**-**c**)). Circular cross sections of spherical nodes (strictly, nodes of the original orbital, not of the square of it) are clearly shown as dark circles in Figure 2(**b**) and Figure 2(**c**). Figure 2(**d**) shows the images of the 3-D probability densities of the hydrogen 1s to 3s orbitals in the present study. At a glance, they look like the cross sections in Figure 2(**a**), Figure 2(**b**), and Figure 2(**c**), respectively, because they are 2-D projected photographs of the original pictures in 3-D glass blocks. The original ones are basically different from the cross sections from the following two standpoints. First, the present model is a "real" 3-D orbital in 3-D space; on the other hand, the cross section representation is a 2-D picture of the 3-D orbital. Second, one can take this model in one's hands and examine it closely from every direction. This is the most advantageous feature of this model. One can easily image the 3-D spherical shapes of the s orbitals, and not the 2-D circles or disks; one can recognize the inner information including the spherical nodes, which are concentric spherical shells instead of circular lines. It must be noted that the present visualization method is the first one to display both the shape of an orbital, which is characteristic of the isosurface technique, and the inner information which is characteristic of the cross section approach.

Figure 2. Representations of the probability density of hydrogen s orbitals. Among the probability densities of the cross section of hydrogen 1s (**a**), 2s (**b**), and 3s (**c**) orbitals [7], the inner nodes are represented in (**b**) and (**c**). Images of the 3-D probability densities of hydrogen 1s, 2s, and 3s orbitals in a glass block show the spherical shape of the orbitals together with the inner nodes (**d**).

Figure 3(**a**) shows the images of the 3-D probability densities of the hydrogen 2p_{z} to 4p_{z} orbitals in glass blocks. One can recognize the planar node (Figure 3(**a**)) and the spherical node (Figure 3(**a**), center and right) at the same time. The similarity of the shape of each p_{z} orbital is also recognized. In the conventional isosurface representations (Figure 3(**b**)-Figure 3(**d**)), the spherical nodes are hard to recognize from the view along the *z* axis (Figure 3(**c**), Figure 3(**d**), bottom), because the outer surface in 3p_{z} or 4p_{z} completely hides the inner surfaces.

Figure 3. Comparison of probability density representations with the isosurface technique of hydrogen p_{z} orbitals. Images of the 3-D probability densities of hydrogen 2p_{z}, 3p_{z}, and 4p_{z} orbitals in a glass block are shown in (**a**). Isosurfaces of hydrogen 2p_{z}, 3p_{z}, and 4p_{z} atomic orbitals are represented in (**b**), (**c**) and (**d**), respectively. The views from the *z* axis ((**b**)-(**d**) bottom) are essentially identical.

The 3-D probability density of the hydrogen 3d_{3z2-r2} orbital in the present study (Figure 4(**a**)) seems to have no torus (doughnut), which is the characteristic shape of the isosurface representation (Figure 4(**b**)). This difference in the orbital shape is based on the fact that the plot of probability density is obtained for the whole space in the cube, whereas the isosurface technique represents only surfaces having the same absolute function value. Some applications of orbital visualization are freely available on the Internet: "Atom in a box" [12], "Hydrogen Atom Viewer" [13], and so on. For example, "Atom in a box" is a piece of public domain software, presented by Dauger Research Inc., to visualize hydrogen atomic orbitals. This program traces a ray of light through a 3-D cloud density that represents the probability density of wavefunctions and presents the results in real-time. The representation of 3d_{3z2-r2} by this program gave a picture that was almost the same as Figure 4(**a**). The familiar isosurface representation of 3d_{3z2-r2} (Figure 4(**b**)) is only a "mental model". The actually existing shape is that of "Atom in a box" or the present study (Figure 4(**a**)). "Atom in a box" is an excellent program to represent a 3-D probability density with high speed (up to 48 frames per second [12]). The rotation with high speed of electron clouds compensates for the 2-D output on a cathode ray tube or on a liquid crystal display; nevertheless, it is essentially a virtual method to image a 3-D object from a 2-D output. It is complicated to image a shape by rotating the picture on the display with a mouse and it is impossible to observe the whole shape at a glance.

Figure 4. Representations of the hydrogen 3d_{3z2-r2} orbital. The probability density in a glass block is shown in (**a**). Conventional isosurface representations, (**b-1**) and (**b-2**), differ from the real 3-D picture in (**a**).

Exactly speaking, "clouds" are not an appropriate expression for a single electron. Schrodinger interpreted that the square of a wavefunction represents the electron density [15]. This description, which sometimes appears even in recent papers [16 - 18], considers a mass of "clouds" having mass and electric charge. If an electron is described as "clouds", the Schrodinger equation (1) for hydrogen atomic orbitals must contain an expression of electron repulsion. But it does not include any such term. If we could detect a single electron, it would give us only a single particle image. We can never detect a fragment of a cloud. The representation of electron density has proved to be very useful in practice. However, it cannot be justified so rigorously as the probability density interpretation [3]. The present study agrees with the above discussion because the probability density defined by Born [2] is represented as an accumulation of detected particles.

We would like to thank Professor Kozo Kuchitsu for his helpful comments and discussion. This work was partly supported by Grants-in-Aid for Scientific Research from Japan Society for the Promotion of Science.

[ 2] M. Born,

[ 3] E. Cartmell and G. W. A. Fowles,

[ 4] N. N. Greenwood and A. Earnshaw,

[ 5] T. Nagao, http://www.juen.ac.jp/scien/cssj/cejrnl.html (2003).

[ 6] S. Tokita,

[ 7] S. Tokita,

[ 8] C. F. Matta and R. J. Gllespie,

[ 9] G. P. Shusterman and A. J. Shusterman,

[10] S. Tokita and T. Sugiyama,

[11] J. E. Douglas,

[12] D. E. Dauger, http://daugerresearch.com/orbitals/ (1998).

[13] http://www.falstad.com/qmatom.

[14] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetlerling,

[15] E. Schrodinger,

[16] J. M. Zou, M. Kim, M. O'Keeffe, and J. C. H. Spence,

[17] E. R. Scerri,

[18] E. R. Scerri,

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