Electron Probability Maps
- erudite .
- Apr 28, 2022
- 3 min read
By Catherine Qiu
An electron is not stuck at a fixed point in space, nor does it travel in a singular, circular path; Electrons act as particles and waves. You may have also heard of Schrödinger’s uncertainty principle; the more you know about an electron’s position, the less you know about its velocity. Consequently, instead of dealing with absolutes, scientists have settled for knowing the likelihood that an electron will be in a certain location.
Atomic Orbitals
Hydrogen atom orbitals, with only 1 electron, seem to be the least complicated to study. For hydrogen’s spherical-shaped 1S* sublevel, the probability density can be described: the farther away from the nucleus, the less likely we’ll find an electron, with probability approaching zero as the radius increases to infinity.
This figure is from UWEC Lecture Slides and shows 1S sublevel’s electron probability density and the graph of radial probability over distance from the nucleus. Note that electrons are not present super close to the nucleus, but they otherwise follow the trend described above.
*For reference, L is the angular momentum number, where L=0 for S orbitals, L=1 for P orbitals, L=2 for D orbitals, and L=3 for F orbitals. ML is the magnetic quantum number, which specifies the orientation in space. MS is the magnetic spin, which can be positive or negative ½.
Wave functions show how electrons are distributed in space. If you square Schrödinger’s wave function, you’ll get the probability of finding an electron in a defined place. By multiplying this wave function squared by the spherical surface area, you’ll get radial probability function. The peak of this graph is called the most probable radius, where you’re most likely to find electrons. Note that the surface area of a sphere is 4*pi*radius. Each radial probability is the probability of finding an electron at that specific spherical surface (imagine a surface as a hollow ornament ball).
As we move up in energy levels, areas of zero electron probability appear. These regions are called nodes. Since 2S and 3S sublevels have nodes, their radial probability graphs oscillate a bit more to account for the nodes. They have small peaks closer to the nucleus and larger peaks farther away.
The figure is from an “Introduction to Quantum Mechanics” lesson on GitHub, which shows the electron probability, the nodes in 2S and 3S sublevels, and the electron radial probability function.
Molecular Orbitals
Positive nuclei repel each other and negative electrons repel each other. In contrast, nuclei and electrons attract each other. If two nuclei move close enough, the nucleus from atom 1 might attract electrons from atom 2. If the attractive forces become stronger than the repulsive forces, a molecule is created. Based on the quantum mechanical model, there’s a higher probability you’ll detect electrons between two atomic nuclei, due to their nuclear attraction.
In many diagrams and plots of molecules, the nuclei are assumed to be fixed in space, and we study how the electrons move and change positions. These huge nuclei move much slower compared to the electrons. There are multitudes of ways to visualize the electron distribution in a molecule. One useful way is with contour plots. You may have seen contour plots in a geography textbook describing elevation or heat maps. They basically indicate increasing probability with more rings and decreasing probability with fewer rings.
In another way to express electron density, a Cartesian plane slices through the nuclei. The z-axis is for the electron probability. The plot is reminiscent of a castle with many low and high towers.
There are 3-D computer molecular models, which lets us see the distribution inside a mesh cage, where there is less electron distribution in the outer layers than in the inner layers nearer the nuclei.
This image was first published in Nature Structural & Molecular Biology and showcases the mesh-like electron density map for a structure that helps unwind DNA.
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