# Visualization of Atomic Orbitals

## Orbitals

Quantum mechanics employs a wave function, ψ, to describe the physical state of an atom or molecule. The value of the wave function (which may be complex) depends upon the positions of the electrons and the nuclei in the system. This tutorial looks at wave functions for a hydrogen atom, and the wave function ψ depends upon the position of the one electron in the atom.

Cartesian coordinates ( *x, y, z* ) may be used, but it is frequently more convenient to use spherical coordinates (*r*, θ, φ ). Imagine a line segment connecting the origin (which is the position of the nucleus in the atom) and the position of the electron. The variable *r* is the length of the line segment, θ is the angle between the *z* axis and the line segment, and φ is the angle between the *x* axis and the projection of the line segment onto the *xy* plane.

The term "orbital" refers to a wave function for an electron. The quantity ψ^{2} (or ψ*ψ for complex wave functions) describes the probability of interacting with the electron at a particular point in space, ( *x, y, z* ) or ( *r*, θ, φ ). For this reason the wave function can be used to predict regions of high and low electron density.

## Quantum Numbers

Each orbital is identified and characterized by three quantum numbers: *n*, *l*, and *m*

**Principal Quantum Number**, *n*

The principal quantum number describes the size of the orbital and may have the value of any positive (nonzero) integer. Each orbital has an energy,E, and for a hydrogen-like atom (an atom with only one electron) the energy is dictated solely by the principal quantum number and the charge,_{n}Z, on the nucleus. For the hydrogen atom,Z= 1.

*E*= - 13.61 eV (

_{n}*Z / n*)

^{ 2}

**Angular Momentum Quantum Number**, *l*

The electron possesses angular momentum by virtual of its motion around the atom (sometimes calledorbital angular momentum). It is tempting to envision this angular momentum in the same terms as that arising from the motion of a planet around the sun, but this view is incorrect. One example of the invalidity of this view is that forl= 0, there is no orbital angular momentum, a behavior that does not exist for a planet in orbit.lmay have the value of zero and of any positive integer that is less than the value ofn.

**Magnetic Quantum Number**, *m*

The magnetic quantum number is any integer (positive, negative, or zero) whose absolute value does not exceed the value ofl. The magnetic quantum number characterizes the component of the orbital angular moment that lies along thezaxis. In practice, the value ofmdetermines theshapeof the orbital.

## Orbital Notation

Orbitals are designated by the notation: * nS_{g}*. As indicated above,

*n*is the principal quantum number. The symbol

*indicates the orbital angular momentum. That is, the value of*

**S***l*determines

*. The subscript*

**S***provides information on the angular geometry of the orbital. An s orbital is spherically symmetric, thus no extra description is required. For p, d, and f orbitals, however, the geometry of the orbital (which depends upon the value of*

**g***m*) is described by the subscript

*.*

**g**l | S | g |
---|---|---|

0 | s | |

1 | p | x, y, z |

2 | d | z^{2}, xy, xz, yz, x^{2}-y^{2} |

3 | f | 5z^{3}-3zr^{2}, 5xz^{2}-xr^{2}, 5yz^{2}-yr^{2},
zx^{2}-zy^{2}, xyz, x^{3}-3xy^{2}, y^{3}-3yx^{2} |

The lowest energy orbital in the hydrogen atom is the 1s orbital, which corresponds with *n* = 1, *l* = 0, and *m* = 0.

## Visualization of Orbitals

Geometrically, orbitals are three dimensional constructs with complicated features that make visualization challenging. Chemists employ a variety of graphical representations to depict the shape and structure of an orbital. Each representation provides a different perspective on the orbital.

### Wave Function Plots

Wave functions, and thus orbitals, are functions of three coordinates. One option for visualization is to hold two variables constants and plot the variation of ψ with the third variable. The wave function plot shown at the bottom of this page sets *y* = z = 0 and plots how ψ varies with *x*.

### Radial Distribution Plots

The wave functions for an atom (but not a molecule) can be separated into two functions: *R _{nl}(r)* and

*Y*.

_{lm}(θ,φ)*R*depends only upon the distance from the nucleus and is called the radial function.

_{nl}(r)*Y*depends only upon the angular orientation about the nucleus and is called the angular function.

_{lm}(θ,φ)Sometimes one is only interested in the likelihood of interacting with an electron a certain distance from the nucleus. In this case, one can integrate the electron density (ψ*ψ) over all values for θ and φ, which is equivalent to integrating the electron density over the surface of a sphere. The remaining, unintegrated portion of the electron density is 4 π *r*^{2} *R _{nl}*

^{2}d

*r*, which is a

*radial distribution function*(rdf). The rdf shows the likelihood of interacting with an electron a distance

*r*from the nucleus.

### Electron Density Plots

An electron density plot depicts the electron density (ψ*ψ) on a particular plane. In this plot, the electron density is represented by the intensity of the color. If the color at a particular point is very bright, there is a high probability of interacting with the electron at that point. If the color is dim, the probability is low. As is clear from a plot ψ vs *x*, the wave function can have either a positive or negative sign. (The sign is sometimes called the phase of the wave function.) The sign can have important implications when orbitals from separate atoms interact. In the electron density plot, the two different colors represent different values for the sign of ψ .

### Isosurfaces

An orbital isosurface is a surface on which all points have the same ψ*ψ value (called the isovalue). The isosurface encloses a region of high electron density. An isosurface plot may use an isovalue that characterizes the how close other molecules can approach. Or an isosurface plot might show the surface enclosing a certain fraction (*e.g.*, 90%) of the electron density.

Isosurfaces are three-dimensional, so a 3D virtual reality representation is necessary to visualize isosurfaces. As with electron density plots, the isosurface is usually shaded to indicate the sign of ψ in a particular region.

## Visualization of Hydrogen Atomic Orbitals

*AtomicOrbitals-Visualization.html version 3.0*

*© Copyright 2020, 2023 David N. Blauch*