## Understanding Quantum Theory of Electrons in Atoms

The goal of this section is to understand the electron orbitals (location of electrons in atoms), their different energies, and other properties. The use of quantum theory provides the best understanding to these topics. This knowledge is a precursor to chemical bonding.

As was described previously, electrons in atoms can exist only on discrete energy levels but not between them. It is said that the energy of an electron in an atom is quantized, that is, it can be equal only to certain specific values and can jump from one energy level to another but not transition smoothly or stay between these levels.

The energy levels are labeled with an *n* value, where *n* = 1, 2, 3, …. Generally speaking, the energy of an electron in an atom is greater for greater values of *n*. This number, *n*, is referred to as the principal quantum number. The **principal quantum number** defines the location of the energy level. It is essentially the same concept as the *n* in the Bohr atom description. Another name for the principal quantum number is the shell number.

The **shells** of an atom can be thought of concentric circles radiating out from the nucleus. The electrons that belong to a specific shell are most likely to be found within the corresponding circular area. The further we proceed from the nucleus, the higher the shell number, and so the higher the energy level (see the figure below). The positively charged protons in the nucleus stabilize the electronic orbitals by electrostatic attraction between the positive charges of the protons and the negative charges of the electrons. So the further away the electron is from the nucleus, the greater the energy it has.

This quantum mechanical model for where electrons reside in an atom can be used to look at electronic transitions, the events when an electron moves from one energy level to another. If the transition is to a higher energy level, energy is absorbed, and the energy change has a positive value. To obtain the amount of energy necessary for the transition to a higher energy level, a photon is absorbed by the atom. A transition to a lower energy level involves a release of energy, and the energy change is negative. This process is accompanied by emission of a photon by the atom. The following equation summarizes these relationships and is based on the hydrogen atom:

\(\begin{array}{l}\text{Δ}E={E}_{\text{final}}-{E}_{\text{initial}} \\ =-2.18\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-18}\left(\phantom{\rule{0.2em}{0ex}}\frac{1}{{n}_{\text{f}}^{2}}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{1}{{n}_{\text{i}}^{2}}\phantom{\rule{0.2em}{0ex}}\right)\phantom{\rule{0.2em}{0ex}}\text{J}\end{array}\)

The values *n*_{f} and *n*_{i} are the final and initial energy states of the electron. A previous lesson demonstrates calculations of such energy changes.

The principal quantum number is one of three quantum numbers used to characterize an orbital. An **atomic orbital**, which is distinct from an *orbit*, is a general region in an atom within which an electron is most probable to reside. The quantum mechanical model specifies the probability of finding an electron in the three-dimensional space around the nucleus and is based on solutions of the Schrödinger equation. In addition, the principal quantum number defines the energy of an electron in a hydrogen or hydrogen-like atom or an ion (an atom or an ion with only one electron) and the general region in which discrete energy levels of electrons in a multi-electron atoms and ions are located.

Another quantum number is *l*, the **angular momentum quantum number**. It is an integer that defines the shape of the orbital, and takes on the values, *l =* 0, 1, 2, …, *n* – 1. This means that an orbital with *n* = 1 can have only one value of *l*, *l* = 0, whereas *n* = 2 permits *l* = 0 and *l* = 1, and so on. The principal quantum number defines the general size and energy of the orbital. The *l* value specifies the shape of the orbital. Orbitals with the same value of *l* form a **subshell**. In addition, the greater the angular momentum quantum number, the greater is the angular momentum of an electron at this orbital.

Orbitals with *l* = 0 are called ** s orbitals** (or the

*s*subshells). The value

*l*= 1 corresponds to the

**. For a given**

*p*orbitals*n*,

*p*orbitals constitute a

*p*subshell (e.g., 3

*p*if

*n*= 3). The orbitals with

*l*= 2 are called the

**, followed by the**

*d*orbitals*f-,*

*g-, and h-*orbitals for

*l*= 3, 4, 5, and there are higher values we will not consider.

There are certain distances from the nucleus at which the probability density of finding an electron located at a particular orbital is zero. In other words, the value of the wavefunction *ψ* is zero at this distance for this orbital. Such a value of radius *r* is called a radial node. The number of radial nodes in an orbital is *n* – *l* – 1.

Consider the examples in the figure above. The orbitals depicted are of the *s* type, thus *l* = 0 for all of them. It can be seen from the graphs of the probability densities that there are 1 – 0 – 1 = 0 places where the density is zero (nodes) for 1*s* (*n* = 1), 2 – 0 – 1 = 1 node for 2*s*, and 3 – 0 – 1 = 2 nodes for the 3*s* orbitals.

The *s* subshell electron density distribution is spherical and the *p* subshell has a dumbbell shape. The *d* and ** f orbitals** are more complex. These shapes represent the three-dimensional regions within which the electron is likely to be found.

If an electron has an angular momentum (*l ≠* 0), then this vector can point in different directions. In addition, the *z* component of the angular momentum can have more than one value. This means that if a magnetic field is applied in the *z* direction, orbitals with different values of the *z* component of the angular momentum will have different energies resulting from interacting with the field. The **magnetic quantum number**, called *m _{l,}* specifies the

*z*component of the angular momentum for a particular orbital. For example, for an

*s*orbital,

*l*= 0, and the only value of

*m*is zero. For

_{l}*p*orbitals,

*l*= 1, and

*m*can be equal to –1, 0, or +1.

_{l}Generally speaking, *m _{l}* can be equal to –

*l*, –(

*l –*1), …, –1, 0, +1, …, (

*l*– 1),

*l*. The total number of possible orbitals with the same value of

*l*(a subshell) is 2

*l*+ 1. Thus, there is one

*s*-orbital for

*ml = 0*, there are three

*p*-orbitals for

*ml = 1*, five

*d*-orbitals for

*ml = 2*, seven

*f*-orbitals for

*ml = 3*, and so forth. The principal quantum number defines the general value of the electronic energy. The angular momentum quantum number determines the shape of the orbital. And the magnetic quantum number specifies orientation of the orbital in space, as can be seen in the figure above.