Atomic Orbital

The atomic orbital is a complex mathematical function called a wave function, which decides the energy, angular momentum, and location of an electron. A better way to define the atomic orbital is the space around the nucleus which has a high probability of finding the electron.

The simplest orbital of all is 1s orbital, which is spherical in shape (see figure below). The nucleus is at the center of the orbital, and the electron revolves around the nucleus.

The region surrounding the nucleus is the probabilistic region, not discrete. The probability of finding the electron in this region is high (say 90 %), and the electron might exist beyond it. Thus, an orbital is an electronic probabilistic cloud around the nucleus.

Wave function

In quantum mechanics, a wave function is a complex mathematical description of a quantum state. It provides the probability distribution of an electron and is denoted by Ψ. Every orbital has its own Ψ. |Ψ|² represents the probability density function. The probability of an electron at the position x between a and b is given by the equation below.

We can normalize the equation.

Thus, the size of the orbital, when the probability of finding an electron is 100 %, is equal to the size of the universe, i.e. −∞ to ∞. Practically, we define the orbital based on the probability of 90 %, not 100 %.

Hydrogen atom

The hydrogen atom has only one electron and only atom for which the Schrodinger equation can be solved exactly. The wave function for the hydrogen atom in the spherical coordinate system is as follows:

R(r) is the radial function, which decides the nuclear distance while Y^m
_l(θ, ϕ) is spherical harmonics, which determine the direction of the orbital. Here, n, l, and m are the principal, azimuthal, and magnetic quantum number. R(r) is expressed below.

Here, a₀ is the Bohr radius, , and L^2l + 1
_{n − l − 1} is a generalized Laguerre polynomial.

As you can see from the above, the quantum numbers n, l, and m decide the equation of the wave function Ψ. For each set of these three quantum numbers, we get a new wave function. Hence, we can conclude these three quantum numbers represent an orbital.

The principal quantum number n is a set of positive integers; n = 1, 2, 3… The azimuthal quantum number l is a set of non-negative integers ranging from 0 to (n − 1); l = 0, 1, 2 … (n − 2), (n − 1). Finally, the magnetic quantum number m is a set of integers from −l to l; m = −l, −(l − 1) … −1, 0, 1 … (l − 1), l.

Quantum numbers

An atom can have a large number of orbitals. The three quantum numbers mentioned in the section above are used to identify an orbital in an atom.

In an atom, the electronic space is divided into shells. Each shell is represented by a value of the principal quantum number n. For example, n = 1 is the lowest energy shell, called K shell, n = 2 is the second shell, L shell, and so on.

Within every shell, there are subshells. The number of subshells in a shell is decided by the azimuthal quantum number l. For every value of n, the possible values of l are 0, 1, 2 … n − 2, n − 1. Thus for K shell (n = 1), l = 0, for L shell (n = 2), l = 0, 1; for M shell (n = 3), l = 0, 1, 2, and so forth. Each value of l corresponds to a subshell. l = 0 is an s subshell, l = 1 is a p subshell, l = 2 is a d subshell, l = 3 is an f subshell etc. The table below summarizes the same.

From the above table, the n-th shell has n's subshells. Each of these subshells consists of a number of orbitals, which is determined by the magnetic quantum number m. For every l, the possible m's are 0, ±1, ±2 … ±(l − 1), ±l. Each value of m corresponds to an orbital.

For an s subshell (l = 0), m = 0. Thus, it has only one orbital, called s orbital. For a p subshell (l = 1), m = 0, ±1. Therefore, a p subshell has three orbitals: p_z (m = 0), and p_x and p_y (m = ±1).

In a d subshell (l = 2), there are five orbitals: d_z² (m = 0), d_xz and d_yz (m = ±1), and d_xy and d_{x² − y²} (m = ±2).

In an f subshell, (l = 3), there are seven orbitals: f_z³ (m = 0), f_xz² and f_yz² (m = ±1), f_xyz and f_{z(x² − y²)} (m = ±2), and f_{x(x² − 3y²)} and f_{y(3x² − y²)} (m = ±3).

Other higher orbitals, viz. g, h, i, and so on, are very complicated and hardly encountered.

If notice from the previous table, the number of orbitals increases oddly with the azimuthal quantum number. The number of orbitals of a given l is 2l + 1.

For a given value of the principal quantum number n, the number of orbitals is given by n².

Proof of n²

The possible values of l for given n are 0, 1, 2 … n − 2, n − 1, and for every l there are 2l + 1 orbitals. The number of orbitals for a given n:

An orbital can hold the upmost two electrons, so the number of electrons for a given n is 2n².

Naming of orbitals

Commonly, the orbital is named with a combination of numbers and alphabets. Consider 4d_z² orbital; the first number represents the principal quantum number (or shell). Here, it is 4 (N shell). The number is followed by an alphabet, here d, it stands for a subshell. The third is the subscript to the alphabet. It tells the orientation of an orbital. Thus, 4d_z² orbital is in the d subshell of the 4th shell oriented along z-axis.

Orbitals can also be represented simply by the three quantum numbers, for example, 4d_z² is 420 (n = 4, l = 2, m = 0).

Let take one more example: n = 5, l = 3, m = ±1 corresponds 5f_xz² and 5f_yz² orbitals. Both lie in the f subshell of the 5th shell. 5f_xz² rests in the xz plane while 5f_yz² rests in the yz plane.

Shapes of orbitals

Every orbital has a unique shape, and the shape becomes more complex and difficult to follow as we move toward higher orbitals.

The s orbital is spherical in shape; the nucleus resides at the center of the sphere. It does not orient itself in any direction. In other words, it is non-directional.

There are three dumbbell-shaped p orbitals. Each orbital has two lobes aligned in one of the three axes. The p_z orbital is aligned with the z-axis, p_x with the x-axis and finally p_y with the y-axis.

The d_z² orbital has two lobes in the z-axis and a doughnut in the xy plane. The remaining four d orbitals have two dumbbells to each. d_xz and d_yz lie in the xz and yz planes while the other two, d_xy and d_x² −y², are in the xy plane. d_xz and d_yz are identical to each other. They have the same radial component of the wave function and only differ by an angle of 90°. Similarly, d_xy and d_x² −y² are identical to each other and differ by an angle of 45°.

The f_z² orbital has two lobes along the z-axis and two doughnuts in-between. f_xz² and f_yz² are similar to the d_xz and d_yz except they have two extra bean-shaped lobes, which are aligned in the x-axis in f_xz² and in the y-axis in f_yz². f_xyz and f_{z(x² − y²)} have eight lobes; four of them are below the xy plane and the rest four above. f_{x(x² − 3y²)} and f_{y(3x² − y²)} have six lobes each; all the lobes are placed in the xy plane. All the above three pairs are identical to each other and have the same radial component but a different angular component.

Except for the s orbital, all orbitals are directional—they orient themselves in a specific direction.

s orbital

For s orbitals, l and m are always zero, the only n varies. When n = 1, it is 1s orbital. The wave function of 1s orbital for the hydrogen atom can be obtained by substituting n, l, and m as 1, 0, 0 in the generalized wave function mentioned earlier.

Ψ² is the probability density function. The probability density function is the probability of finding an electron per unit volume. It is maximum at r = 0 and tends to zero with increasing r (see the first graph below). We can obtain the radial probability if we multiply the area of the sphere, 4πr², to the probability density. The third graph shows the radial probability with the radius. At r = 0 and r = ∞, the probability is zero. From the graph, the probability reaches a peak at r = 52.9 pm, which is also the Bohr's radius (a₀).

When n = 2 and n = 3, we get 2s and 3s orbitals. Their wave functions are as follows:

The graph below is the plot of the radial probability (Ψ²r²) versus the radius for 1s, 2s, and 3s orbitals.

As seen from the above graph, 1s orbital has only one peak while 2s and 3s have two and three peaks respectively. As a consequence, there is point of the minimum probability between every two peaks. Besides the extrema, the probability becomes zero at these points. These points, where the probability becomes zero, are called nodes. 1s orbital has zero nodes, 2s orbital has one node, and 3s has two nodes; in general, we can say ns orbital has (n − 1) nodes.

The nodes in orbitals are similar to the nodes of the standing wave, where they are defined as the points of zero amplitudes. One can imagine the nodes as the gap between spheres that placed one within another.

One can also observe from the above diagram the size of the orbital increasing with the principal quantum number—i.e., 3s > 2s > 1s.

p orbital

The p orbital corresponds to l = 1. The p orbital, unlike the s orbital, is not spherical in shape; it is dumbbell-shaped. As discussed earlier, there are three p orbitals: p_z (l = 1; m = 0), and p_x and p_y (l = 1; m = ±1).

For n = 2, the wave functions are as follows:

Ψ₂₁₀ is the wave function for 2p_z orbital and Ψ_21±1 is for 2p_x and 2p_y. The θ and ϕ part of Ψ decides the orientation of the orbital. In the graphs below, only the radial part of Ψ is considered, i.e., θ and ϕ is ignored.

Unlike the s orbital, the probability density of the p orbital is not maximum at r = 0. We can see from the above figure it is zero at r = 0. The value attained the peak in-between. The radial probability graph also shows a similar behavior. Since the probability density is zero at the nucleus, the electron will spend more time away from the nucleus in the p orbital in comparison to the s orbital, where the probability density is maximum at the nucleus.

Also, the probability is zero in the xy plane for 2p_z orbital. Similarly, the corresponding planes for 2p_x and 2p_y are yz and xz. Such planes, where the probability becomes zero, are called nodal planes. In p orbitals, there are three nodal planes as shown below.

These nodal planes are known as angular nodes. Thus, p orbitals have three angular nodes. The s orbital does not have angular nodes; they only have radial nodes. p orbitals have both radial and as angular nodes.

The wave functions of 3p_z, 3p_x and 3p_y are as follows:

The plot below shows the radial node in 3p_z orbital. 2p_z does not have any radial nodes.

In 4p_z orbital, not shown in the above figure, has two radial nodes. In short, the np orbital has (n − 2) radial nodes.

d orbital

There are five d orbitals: d_z², d_xz, d_yz, d_xy, and d_{x² − y²}. Of these orbitals, d_z² is unique; it has two lobes in the z-axis and a doughnut-shaped lobe in the xy plane. d orbitals start from the 3rd shell and their wave functions are mentioned below.

In all the above wave functions, the angular parts are different while the radial parts are a multiple of one another.

Like p orbitals, d orbitals have angular nodes. Each d orbital has 2 angular nodes.

The radial nodes for the nd orbital are (n − 3) nodes. So 3d has no radial nodes. The radial nodes in the d orbital start from the 4d.

f orbital

The f orbital is far more complex than the d orbital and observed in heavy elements. There are seven f orbitals: f_z³, f_xz², f_yz², f_xyz, f_{z(x² − y²)}, _{x(x² − 3y²)}, and f_{y(3x² − y²)}.

Each of these orbitals has 3 angular nodes. The radial nodes start from 4f; nf orbital has (n − 4) radial nodes.

Nodes

Nodes, as discussed above, are the region with zero probabilities of finding electrons. They can be divided into two types: angular and radial. The radial nodes are obtained from the radial component of the wave function while the angular nodes from the angular component, i.e., spherical harmonics.

From the above table, we generalized the formula for nodes. The angular nodes only depend on the value of l and is equal to l. On the other hand, the radial nodes depend on both n and l and is given by n − l − 1. Therefore, the total number of nodes is the sum of both and is n − 1.

Orbit vs orbital

Orbits and orbitals are often confused with each other, particularly by beginners. They sound the same but are completely different concepts and should not be interchangeably used. The table below explained the difference between both.

Associated articles

• Principal quantum number
• Azimuthal quantum number
• Bohr's atomic model
• Rutherford's atomic model