ATOMIC ORBITALS: WHAT THEY CONSIST OF AND TYPES - CHEMISTRY - 2025

The atomic orbitals are those regions of the atom defined by a wave function for electrons. Wave functions are mathematical expressions obtained from solving the Schrödinger equation. These describe the energy state of one or more electrons in space, as well as the probability of finding it.

This physical concept, applied by chemists to understand the bond and the periodic table, considers the electron as a wave and a particle at the same time. Therefore, the image of the solar system is discarded, where the electrons are planets rotating in orbits around the nucleus or the sun.

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This outdated visualization comes in handy when illustrating the energy levels of the atom. For example: a circle surrounded by concentric rings representing the orbits, and their static electrons. In fact, this is the image with which the atom is introduced to children and young people.

However, the true atomic structure is too complex to even have a rough picture of it.

Considering then the electron as a wave-particle, and solving the Schrödinger differential equation for the hydrogen atom (the simplest system of all), the famous quantum numbers were obtained.

These numbers indicate that electrons cannot occupy any place in the atom, but only those that obey a discrete and quantized energy level. The mathematical expression of the above is known as a wave function.

Thus, from the hydrogen atom, a series of energy states governed by quantum numbers was estimated. These energy states were called atomic orbitals.

But, these only described the whereabouts of an electron in a hydrogen atom. For other atoms, polyelectronics, from helium onwards, an orbital approximation was made. Why? Because solving the Schrödinger equation for atoms with two or more electrons is very complicated (even with current technology).

What are atomic orbitals?

Atomic orbitals are wave functions that consist of two components: one radial, and one angular. This mathematical expression is written as:

Ψ _nlml = R _nl (r) Y _lml (θϕ)

Although it may seem complicated at first, note that the quantum numbers n, l, and ml are indicated in small letters. This means that these three numbers describe the orbital. R _nl (r), better known as the radial function, depends on nyl; while Y _lml (θϕ), angular function, depends on l and ml.

In the mathematical equation there are also the variables r, distance to the nucleus, and θ and ϕ. The result of all this set of equations is a physical representation of the orbitals. Which one? The one seen in the image above. There a series of orbitals are shown which will be explained in the following sections.

Their shapes and designs (not the colors) come from graphing the wave functions and their radial and angular components in space.

Radial wave function

As seen in the equation, R _nl (r) depends on both n and l. So, the radial wave function is described by the main energy level and its sublevels.

If the electron could be photographed regardless of its direction, an infinitely small point could be observed. Then, taking millions of photographs, it could be detailed how the point cloud changes as a function of distance to the core.

In this way, the density of the cloud in the distance and near the core can be compared. If the same operation were repeated but with another energy level or sublevel, another cloud would form that encloses the previous one. Between the two there is a small space where the electron is never located; this is what is known as a radial node.

Also, in the clouds there are regions with higher and lower electron density. As they get larger and farther from the nucleus, they have more radial nodes; and furthermore, a distance r where the electron rounds more frequently and is more likely to be found.

Angle wave function

Again, it is known from the equation that Y _lml (θϕ) is mainly described by the quantum numbers l and ml. This time it participates in the magnetic quantum number, therefore, the direction of the electron in space is defined; and this direction can be graphed from the mathematical equations involving the variables θ and ϕ.

Now, we do not proceed to take photographs, but to record a video of the trajectory of the electron in the atom. Contrary to the previous experiment, it is not known exactly where the electron is, but where it is going.

As the electron moves, it describes a more defined cloud; in fact, a spherical figure, or one with lobes, like those seen in the image. The type of figures and their direction in space are described by l and ml.

There are regions, close to the nucleus, where the electron does not transit and the figure disappears. Such regions are known as corner nodes.

For example, if you look at the first spherical orbital, you quickly come to the conclusion that it is symmetric in all directions; however, this is not the case with the other orbitals, whose shapes reveal empty spaces. These can be observed at the origin of the Cartesian plane, and in the imaginary planes between the lobes.

Probability of finding the electron and chemical bond

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To determine the true probability of finding an electron in an orbital, the two functions must be considered: radial and angular. Therefore, it is not enough to assume the angular component, that is, the illustrated shape of the orbitals, but also how their electron density changes with respect to the distance from the nucleus.

However, because the directions (ml) distinguish one orbital from another, it is practical (though perhaps not entirely correct) to consider only the shape of the orbital. In this way, the description of the chemical bond is explained by the overlapping of these figures.

For example, above is a comparative image of three orbitals: 1s, 2s, and 3s. Note its radial nodes inside. The 1s orbital has no node, while the other two have one and two nodes.

When considering a chemical bond, it is easier to keep in mind only the spherical shape of these orbitals. In this way, the ns orbital approaches another, and at a distance r, the electron will form a bond with the electron of the neighboring atom. From here several theorists arise (TEV and TOM) that explain this link.

How are they symbolized?

Atomic orbitals are explicitly symbolized as: nl _ml.

Quantum numbers take integer values ​​0, 1, 2, etc., but to symbolize the orbitals only a numerical value is left n. While for l, the whole number is replaced by its corresponding letter (s, p, d, f); and for ml, a variable or mathematical formula (except for ml = 0).

For example, for the 1s orbital: n = 1, s = 0, and ml = 0. The same applies to all ns orbitals (2s, 3s, 4s, etc.).

To symbolize the rest of the orbitals, it is necessary to address their types, each with its own energy levels and characteristics.

Types

Orbitals s

The quantum numbers l = 0, and ml = 0 (in addition to their radial and angular components) describe an orbital with a spherical shape. This is the one that heads the pyramid of orbitals of the initial image. Also, as can be seen in the image of the radial nodes, it can be expected that the 4s, 5s and 6s orbitals have three, four and five nodes.

They are characterized by being symmetrical and their electrons experience a greater effective nuclear charge. This is because its electrons can penetrate inner shells and hover very close to the nucleus, which exerts a positive attraction on them.

Therefore, there is a probability that a 3s electron can penetrate the 2s and 1s orbital, approaching the nucleus. This fact explains why an atom with sp hybrid orbitals is more electronegative (with a greater tendency to attract electronic density from its neighboring atoms) than one with sp ³ hybridization.

Thus, the electrons in the s orbitals are the ones that experience the nucleus charge the most and are more energetically stable. Together, they exert a shielding effect on electrons in other sublevels or orbitals; that is, they decrease the actual nuclear charge Z experienced by the outermost electrons.

Orbitals p

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The p orbitals have the quantum numbers l = 1, and with values ​​of ml = -1, 0, +1. That is, an electron in these orbitals can take three directions, which are represented as yellow dumbbells (according to the image above).

Note that each dumbbell is located along a Cartesian x, y, and z axis. Therefore, that p orbital located on the x axis is denoted as p _x; the one on the y, p _y axis; and if it points perpendicular to the xy plane, that is, on the z axis, then it is p _z.

All the orbitals are perpendicular to each other, that is, they form an angle of 90º. Likewise, the angular function disappears in the nucleus (the origin of the Cartesian axis), and there is only the probability of finding the electron within the lobes (whose electron density depends on the radial function).

Poor shielding effect

Electrons in these orbitals cannot penetrate inner shells as easily as s orbitals. Comparing their shapes, the p orbitals appear to be closer to the nucleus; however, the ns electrons are found more frequently around the nucleus.

What is the consequence of the above? That an np electron experiences a lower effective nuclear charge. Furthermore, the latter is further reduced by the shielding effect of the s orbitals. This explains, for example, why an atom with sp ³ hybrid orbitals is less electronegative than one with sp ² or sp orbitals.

It is also important to note that each dumbbell has an angular nodal plane, but no radial nodes (the 2p orbitals only). That is, if it were to be sliced, there would be no layers inside as with the 2s orbital; but from the 3p orbital onwards, radial nodes would begin to be observed.

These angular nodes are responsible for the outermost electrons experiencing a poor shielding effect. For example, the 2s electrons shield those in the 2p orbitals better than the 2p electrons shield those in the 3s orbital.

Px, Py and Pz

Since the values ​​of ml are -1, 0, and +1, each represents a Px, Py, or Pz orbital. In total, they can accommodate six electrons (two for each orbital). This fact is crucial for understanding the electronic configuration, the periodic table, and the elements that make up the so-called p-block.

Orbitals d

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The d orbitals have values ​​of l = 2, and ml = -2, -1, 0, +1, +2. There are therefore five orbitals capable of holding ten electrons in total. The five angular functions of the d orbitals are represented in the image above.

The first ones, the 3d orbitals, lack radial nodes, but all the others, except the d _z2 orbital, have two nodal planes; not the planes of the image, as these only show in which axes the orange lobes with shapes of clover leaves are located. The two nodal planes are those that bisect perpendicular to the gray plane.

Their shapes make them even less effective in shielding the effective nuclear charge. Why? Because they have more nodes, through which the nucleus can attract external electrons.

Therefore, all the d orbitals contribute to a less pronounced increase in atomic radii from one energy level to another.

Orbitals f

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Finally, the f orbitals have quantum numbers with values ​​of l = 3, and ml = -3, -2, -1, 0, +1, +2, +3. There are seven f orbitals, for a total of fourteen electrons. These orbitals become available from period 6, superficially symbolized as 4f.

Each of the angular functions represents lobes with intricate shapes and several nodal planes. Therefore, they shield the external electrons even less and this phenomenon explains what is known as lanthanide contraction.

For that reason, for heavy atoms there is no pronounced variation in their atomic radii from one level n to another n + 1 (6n to 7n, for example). To date, the 5f orbitals are the last found in natural or artificial atoms.

With all this in mind, a gulf opens up between what is known as orbit and the orbitals. Although textually they are similar, in reality they are very different.

The concept of the atomic orbital and the orbital approximation have made it possible to explain the chemical bond, and how it can, in one way or another, affect the molecular structure.

References

1. Shiver & Atkins. (2008). Inorganic chemistry. (Fourth edition., Pp. 13-8). Mc Graw Hill.
2. Harry B. Gray. (1965). Electrons and Chemical Bonding. WA Benjamin, Inc. New York.
3. Quimitube. (sf). Atomic orbitals and quantum numbers. Recovered from: quimitube.com
4. Nave CR (2016). Visualizing Electron Orbitals. Recovered from: hyperphysics.phy-astr.gsu.edu
5. Clark J. (2012). Atomic Orbitals. Recovered from: chemguide.co.uk
6. Quantum tales. (August 26, 2011). Atomic orbitals, a high school lie. Recovered from: cuentos-cuanticos.com