Find more information on the Altmetric Attention Score and how the score is calculated. Images of isosurfaces and contour lines for a large set of hydrogen atomic orbitals were generated with new and expanded sets of parametric equations. Parametric equations for 4f and 5g orbitals are derived for the first time. Atomic orbitals are a theme throughout the undergraduate chemistry curriculum, and visualizing them has been a theme in this journal.
Contour plots as isosurfaces or contour lines in a plane are the most familiar representations of the hydrogen wave functions. The easiest method to generate an isosurface is to use an equation for the surface. Ramachandran and Kong derived parametric equations to generate atomic orbital isosurfaces for 2s, 2p, 3p, 3d x 2 — y 2 , and 3d z 2 orbitals in terms of r , the distance from the origin.
The alternative to these equations is to generate the isosurfaces or contour lines implicitly through an algorithm that may require programming or software with the ability to implement them. The marching cubes algorithm is a common algorithm for isosurfaces, 25 and other algorithms have been used for computerized plotting of orbitals. Here we generate vector graphic images for a large set of orbitals using parametric equations, many of which are newly described.
These representations are generated with gnuplot, a readily available, free, open-source program. The orbitals were generated using three sets of parametric equations: 1. The 2s, 2p, 3d, 4f, and 5g orbitals were generated with a new set of parametric equations. These parametric equations use the Lambert W function. More details about the Lambert W function and its implementation are found in the Supporting Information.
These are the first parametric equations for f and g orbitals in the literature, although they cannot be extended to higher principal quantum numbers. In this work, the surface is generated using the cylindrical symmetry of these orbitals around the z -axis; in addition, expressions for the additional orbitals are derived. The p x and p y orbitals can be generated by exchanging the variables.
The d x 2 — y 2 orbitals have a similar form as an ellipse in plane geometry for x and z around the y -axis and for y and z around the x -axis at a specific value of r. The other orbitals are generated through rotations of the d x 2 — y 2 orbitals around axes.
The derivations and complete details for the parametric equations are given in the Supporting Information. In all, the parametric equations represent 52 orbitals.
The parametric equations were used to generate image files for the isosurfaces using gnuplot. Previous implementation of isosurfaces in gnuplot 19 and Winplot, 24 a similar program, was limited to the angular wave function. Both formats are vector graphics, consisting of curved and straight lines and text; such file types can be enlarged without pixelation, and SVG images can be edited for example, with Inkscape or Gimp.
For these reasons, SVG images are the preferred image format for line drawings at Wikipedia. A sample isosurface of the 4f xyz orbital is provided in Figure 1. Figure 1. The distance units are atomic units. Gnuplot input files and image files for several contour lines are provided in the Supporting Information , as well as in a macro-enabled Microsoft Excel spreadsheet. The image files and gnuplot input files can be used for pedagogical purposes in several ways.
Most textbooks still use artist renditions of orbitals, often with incorrect features such as lobes that touch each other and distorted shapes, which can create misconceptions. Hence, an instructor can use the image files directly to describe atomic orbitals, or generate his or her own image files in different orientations.
Students can also load the input files into gnuplot and manipulate the orbitals interactively. In addition, the equations can be used in any mathematical package that plots surfaces parametrically, thus expanding the available options for orbital contour plotting.
All the representations show features of atomic orbitals, such as the location and number of nodes, orbital shape, and, through the use of color, the phases and symmetry for the orbitals.
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Propylene Glycol. Note that each p orbital has just one nodal plane. In each case, the phase of the wave function for each of the 2 p orbitals is positive for the lobe that points along the positive axis and negative for the lobe that points along the negative axis.
It is important to emphasize that these signs correspond to the phase of the wave that describes the electron motion, not to positive or negative charges. Each orbital is oriented along the axis indicated by the subscript and a nodal plane that is perpendicular to that axis bisects each 2 p orbital. The phase of the wave function is positive orange in the region of space where x , y , or z is positive and negative blue where x , y , or z is negative.
Just as with the s orbitals, the size and complexity of the p orbitals for any atom increase as the principal quantum number n increases. The electron configuration of an atom is the representation of the arrangement of electrons distributed among the orbital shells and subshells. Commonly, the electron configuration is used to describe the orbitals of an atom in its ground state, but it can also be used to represent an atom that has ionized into a cation or anion by compensating with the loss of or gain of electrons in their subsequent orbitals.
Many of the physical and chemical properties of elements can be correlated to their unique electron configurations. The valence electrons, electrons in the outermost shell, are the determining factor for the unique chemistry of the element. Steven Farmer Sonoma State University. Objectives After completing this section, you should be able to describe the physical significance of an orbital.
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