Visualization of Orbitals (2)Probability Density Distribution in the Three DimensionalRepresentation of a Hydrogen 1s Orbital in a Glass Block

In contrast to Bohr's concept of a simple circular "orbit" with a fixed radius, a quantum chemical "orbital" is represented by the probability density of finding an electron. A three dimensional (3-D) representation of the probability density of a hydrogen 1s orbital in a glass block (Figure 1 (a)) was developed. It was compared with other models of the atomic orbital, such as several 3-D isosurface models (Figure 1 (b-d)), a 2-D probability density model (Figure 2 (a)), 2-D contour maps (Figure 2 (b, c)), and a pseudo 3-D contour map (Figure 2 (d)). Isosurface models shown in Figure 1 (b), (c), and (d) represent the boundary or contour surface where probability density χ1s2 has a constant value, chosen so that there is a certain probability − for example 90% − of finding the electron within the contour. These diagrams cannot show the entire region where an electron can be found. On the other hand, in the diagram of Figure 1 (a), an electron is found everywhere around the nucleus. The density of the dots sculptured in the glass block shows the probability density of finding an electron. The distribution of the probability density χ1s2 on the x-y plane is well described by the contour map in Figure 2 (b), or by the solid line in Figure 3 (the plot of χ1s2 versus distance from the nucleus r). In this Figure, the electron probability density is greatest at r = 0 (the nucleus) and falls off with increasing distance r. The 1s atomic orbital is spherically symmetric, so the probability of finding a 1s electron is the same at every point on a spherical shell having a constant radius r. The surface area of each spherical shell, 4πr², increases very rapidly with increasing r(the dashed line in Figure 3). The product of the probability χ1s2 and the surface area of the spherical shell 4πr² is illustrated in Figure 4 as the radial distribution function 4πr2χ1s2 . Where r is very small, the surface area 4πr² is so small that the total probability of finding an electron on that surface is very low. It has a maximum at a particular distance of r = 1 au (atomic unit), namely, the Bohr radius, 52.92 p.m. In the classical Bohr model of hydrogen, an electron is found only at the radius of 52.92 p.m., whereas in the quantum chemical refined orbital model, an electron can potentially be found at any distance from the nucleus, but, depending on the square of the hydrogen 1s wavefunction, exists more frequently on the spherical shell at the Bohr radius. If we divide the hydrogen atom into very thin (thickness: dr) concentric shells, much like the layers of an onion, we can calculate the probability of finding an electron on each spherical shell. The delicacy of each shell, namely, the electron probability density is greatest at r = 0. In contrast, the volume of the spherical shell 4πr²dr at r = 0 is 0, and the total delicacy at the center of the atom is 0, but the volume increases rapidly with increasing r, the total delicacy of each shell has a maximum at the Bohr radius.