1.3: Probability Density and Nodes

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Atomic orbitals are mathematically derived regions of space with different probabilities of containing an electron. An orbital can be viewed as a probability map of finding an electron at a specific point in space.

Probability Density

One way of representing electron probability distributions was illustrated previously for the 1s orbital of hydrogen. Ψ² gives the probability of finding an electron in a given volume of space, and a plot of Ψ² versus distance from the nucleus (r) is a plot of the probability density. The 1s orbital is spherically symmetrical so the probability of finding a 1s electron at any given point depends only on its distance from the nucleus. The probability density is greatest at \(r = 0\) (at the nucleus) and decreases steadily with increasing distance. At very large values of r, the electron probability density is very small but is not equal to zero.

Radial Probability Density

The radial probability density is the probability of finding an electron at a distance r from the nucleus. It is calculated by adding together the probabilities of an electron being at all points on a series of spherical shells of radius r₁, r₂, r₃,…, r_{x − 1}, r_x. In effect, the atom is divided into very thin concentric shells, much like the layers of an onion (Figure \(\PageIndex{1a}\)), and the radial probability density calculates the probability of finding an electron on each spherical Layer. Although the electron probability density is greatest at r = 0 (Figure \(\PageIndex{1b}\)), is lowest at r = 0(Figure \(\PageIndex{1c}\)). The surface area of the spherical shells increases more rapidly with r than the electron probability density decreases; therefore, the plot of radial probability has a maximum at a particular distance (Figure \(\PageIndex{1d}\)).

Overall, when r is very small, the surface area of a spherical shell is so small that the total probability of finding an electron close to the nucleus is very low; at the nucleus, the electron probability vanishes (Figure \(\PageIndex{1d}\)).

Figure \(\PageIndex{1}\): Most Probable Radius for the Electron in the Ground State of the Hydrogen Atom. (a) Imagine dividing the atom’s total volume into very thin concentric shells as shown in the onion drawing. (b) A plot of electron probability density Ψ² versus r shows that the electron probability density is greatest at r = 0 and falls off smoothly with increasing r. The density of the dots is therefore greatest in the innermost shells of the onion. (c) The surface area of each shell, given by 4πr², increases rapidly with increasing r. (d) If we count the number of dots in each spherical shell, we obtain the total probability of finding the electron at a given value of r. Because the surface area of each shell increases more rapidly with increasing r than the electron probability density decreases, a plot of electron probability versus r (the radial probability density plot) shows a peak. This peak corresponds to the most probable radius for the electron, 52.9 pm, which is exactly the radius predicted by Bohr’s model of the hydrogen atom.

For the hydrogen atom, the peak in the radial probability plot occurs at r = 0.529 Å (52.9 pm), which is exactly the radius calculated by Bohr for the n = 1 orbit. Thus the most probable radius obtained from quantum mechanics is identical to the radius calculated by classical mechanics. In Bohr’s model, however, the electron was assumed to be at this distance 100% of the time, whereas in the Schrödinger model, it is at this distance only some of the time. The difference between the two models is attributable to the wavelike behavior of the electron and the Heisenberg uncertainty principle.

Nodes

Radial Nodes

Figure \(\PageIndex{2}\) compares the electron probability densities for the hydrogen 1s, 2s, and 3s orbitals. Note that all three are spherically symmetrical. Figure \(\PageIndex{2c}\) shows that the 2s and 3s orbitals, the radial probability density does not fall off smoothly with increasing r. Instead, a series of minima and maxima are observed in the radial probability plots. The minima correspond to spherical nodes, which are regions with zero electron probability that alternate with spherical regions of nonzero electron probability. The existence of these nodes is a consequence of changes of wave phase in the wavefunction Ψ. These spherical nodes, which separate electron shells, are known as radial nodes. The number of radial nodes depends on the energy level of the orbital (n) and on the shape of the orbital (l). The number of radial nodes is equal to n − l − 1 .

Figure \(\PageIndex{2}\): Probability Densities for the 1s, 2s, and 3s Orbitals of the Hydrogen Atom. (a) The electron probability density in any plane that contains the nucleus is shown. Note the presence of circular regions, or nodes, where the probability density is zero. (b) Contour surfaces enclose 90% of the electron probability, which illustrates the different sizes of the 1s, 2s, and 3s orbitals. The cutaway drawings give partial views of the internal spherical nodes. The orange color corresponds to regions of space where the phase of the wave function is positive, and the blue color corresponds to regions of space where the phase of the wave function is negative. (c) In these plots of electron probability as a function of distance from the nucleus (r) in all directions (radial probability), the most probable radius increases as n increases, but the 2s and 3s orbitals have regions of significant electron probability at small values of r.

Angular Nodes

Angular nodes are planar nodes that are a result of the shape of an orbital. The number of angular nodes in any orbital is equal to l. For s-orbitals (\(l=0\)) so they have zero angular nodes. For p-orbitals (\(l=1\)) so they have one angular node, and for d-orbitals (\(l=2\)) so they have two angular nodes. Planar nodes can be flat planes (like the nodes in all p orbitals) or they can have a conical shape, like the two angular nodes in the \(d_{z^2}\) orbital. Angular nodes in some p and d orbitals are shown in Figure \(\PageIndex{3}\).

Figure \(\PageIndex{3}\): The angular nodes in p-orbitals and d-orbitals. Every p orbital has one angular node, and every d orbital has two angular nodes. The number of angular nodes in any orbital is equal to the value of the quantum number, \(l\). (CC BY-NC-SA; anonymous by request)

Exercise \PageIndex{1}\)

Exercise \(\PageIndex{1}\)

How many radial nodes are in the s, p, d, and f orbitals in the first four shells? \(n=1,2,3,4\)? Do you notice any patterns in your answers?

Hint

You can use the mathematical relationship between the number of radial nodes and the values of \(n\) and \(l\): number of radial nodes is \(n-1-l\).

Answer: s orbitals

1s: \(n-1-l = 1-1-0 = 0\), zero radial nodes

2s: \(n-1-l = 2-1-0 = 1\), one radial node

3s: \(n-1-l = 3-1-0 = 2\), two radial nodes

4s: \(n-1-l = 4-1-0 = 3\), three radial nodes

Answer: p orbitals

2p: \(n-1-l = 2-1-1 = 0\), zero radial nodes

3p: \(n-1-l = 3-1-1 = 1\), one radial node

4p: \(n-1-l = 4-1-1 = 2\), two radial nodes

Answer: d orbitals

3d: \(n-1-l = 3-1-2 = 0\), zero radial nodes

4d: \(n-1-l = 4-1-2 = 1\), one radial nodes

Answer: f orbitals

4f: \(n-1-l = 4-1-3 = 0\), zero radial nodes

Answer: patterns

You may notice the pattern that the first orbital of any type (1s, 2p, 3d...) has zero radial nodes, the second orbital of a type (2s, 3p, 4d...) has one radial node, the third orbital of a type has two radial nodes (3s, 4p, 5d...)...etc.

Contributors and Attributions

• Modified by Joshua Halpern (Howard University)