1.4: Representations of Atomic Orbitals

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Atomic Orbitals

Three things happen to atomic orbitals as n increases:

1. They become larger, extending farther from the nucleus.
2. They become higher in energy because of their increased distance from the nucleus.
3. They contain more radial nodes.

s Orbitals (l=0)

s orbitals are spherically symmetrical. They have no angular nodes but may have radial nodes.

Orbitals are generally drawn as three-dimensional surfaces that enclose 90% of the electron density, as was shown for the hydrogen 1s, 2s, and 3s orbitals in Figure 1.3.2b. Although such drawings show the relative sizes of the orbitals, they do not normally show the spherical nodes in the 2s and 3s orbitals because the spherical nodes lie inside the 90% surface.

p Orbitals (l=1)

As the value of l increases, the number of orbitals in a given subshell increases, and the shapes of the orbitals become more complex. p orbitals have a two-lobed shape split by an angular nodal plane. p orbitals may have radial nodes, depending on the value of n.

For the 2p subshell l = 1, so there are three values of m_l (−1, 0, and +1), and there are three 2p orbitals. Figure \(\PageIndex{1}\) depicts a 2p orbital.Because this orbital has two lobes of electron density arranged along the z axis, with an electron density of zero in the xy plane (i.e., the xy plane is a nodal plane), it is a \(2p_z\) orbital.

Figure \(\PageIndex{1}\): Electron probability distribution for a 2p Orbital. The nodal plane of zero electron density separates the two lobes of the 2p orbital. The colors correspond to regions of space where the phase of the wave function is positive (orange) and negative (blue).

Figure \(\PageIndex{2}\) shows the contour surfaces, which enclose 90% of the total electron probability, for all three 2p orbitals. All orbitals have identical shapes just one nodal plane. In each case, the phase of the wave function for each of the 2p 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 the nodal plane that is perpendicular to that axis bisects each 2p orbital. The shapes of the 90% probability surfaces of the 3p, 4p, and higher-energy p orbitals are essentially the same as those shown in Figure \(\PageIndex{2}\).

Figure \(\PageIndex{2}\): Three equivalent 2p orbitals of the hydrogen atom. The surfaces shown enclose 90% of the total electron probability for the three hydrogen 2p orbitals. These three orbitals have the same shape but different orientations. Each orbital consist of two lobes arranged along an axis that is bisected by a perpendicular nodal plane. The colors correspond to regions of space where the phase of the wave function is positive (orange) and negative (blue).

d Orbitals (l=2)

Subshells with l = 2 have five d orbitals; the first principal shell to have a d subshell corresponds to n = 3. The five d orbitals have m_l values of −2, −1, 0, +1, and +2.

The hydrogen 3d orbitals, shown in Figure \(\PageIndex{3}\), have more complex shapes than the 2p orbitals. All five 3d orbitals contain two nodal surfaces, as compared to one for each p orbital and zero for each s orbital. For three of the d orbitals, the lobes of electron density are oriented between the xy, xz, and yz planes; these orbitals are referred to as the \(3d_{xy}\), \)3d_{xz}\), and \(3d_{yz}\) orbitals, respectively. A fourth d orbital has lobes lying along the x and y axes; this is the \(3d_{x^2−y^2}\) orbital. The fifth 3d orbital, called the \(3d_{z^2}\) orbital, has a unique shape: it looks like a \(2p_z\) orbital combined with an additional doughnut of electron probability lying in the xy plane. Despite its peculiar shape, the \(3d_{z^2}\) orbital is mathematically equivalent to the other four and has the same energy. In contrast to p orbitals, the phase of the wave function for d orbitals is the same for opposite pairs of lobes. For the (dz^2\) orbital, the phase of the wave function is positive for the two lobes that lie along the z axis, and the phase of the wave function is negative for the doughnut of electron density in the xy plane. Like the s and p orbitals, as n increases, the size of the d orbitals increases, but the overall shapes remain similar to those depicted in Figure \(\PageIndex{3}\).

Figure \(\PageIndex{3}\): Five equivalent 3d orbitals of the hydrogen atom. The surfaces shown enclose 90% of the total electron probability for the five hydrogen 3d orbitals. Four of the five 3d orbitals consist of four lobes arranged in a plane that is intersected by two perpendicular nodal planes. These four orbitals have the same shape but different orientations. The fifth 3d orbital, \(3d_{z^2}\), has a distinct shape even though it is mathematically equivalent to the others. The colors correspond to regions of space where the phase of the wave function is positive (orange) and negative (blue).

f Orbitals (l=3)

Principal shells with n = 4 can have subshells with l = 3 and m_l values of −3, −2, −1, 0, +1, +2, and +3. These subshells consist of seven f orbitals. Each f orbital has three nodal surfaces, so their shapes are complex. Because f orbitals are not particularly important for our purposes, we do not discuss them further, and orbitals with higher values of l are not discussed at all.

Contributors and Attributions

• Modified by Joshua Halpern (Howard University)