### Table of Contents

# Quantum mechanics and atomic structure (a simple introduction)

## A very quick introduction

The shape of individual atoms is in general only representative. To describe the shapes of atoms we use their characteristic principal quantum numbers along with three other quantum numbers that are representative for the so called atomic shells or energy levels on where the electrons are located, the atomic orbitals that describe the probability (of a certain %, not discussed here) of locating an electron within a specific range and finally these basic concepts of quantum mechanics are used to describe the shape of atoms. However, remember that their shape and it‘s description is only representative, but not a exact description. We can only (with a very high degree of precision) describe the shapes of molecules that consist of two or more atoms that are connected to each other with chemical bonds.

## The hydrogen atom – the simplest atom of all

The hydrogen atom is the simplest atom regarding to atomic structure, consisting of only one proton and one electron (e-). The hydrogen atom is also the simplest atom of all atoms that contain only one electron. However, the elements He (helium) and Li (lithium) can form ions (ions are charged atoms or molecules) that contain only one electron: He forms the +1 (positively charged) ion He+ and Li forms the ion Li2+. These atoms differ only in the positive charge on the nucleus. Note that a single He atom has two electrons but has lost one of them, thus the He+ ion has a positive charge on it‘s nucleus that equals the charge of an electron, only with a positive sign (as one electron with it‘s negative charge has been lost). In the same way the Li2+ ion has a positive charge on its nucleus that equals two times the charge of an electron.

### Energy levels

Atoms only contain electrons on specific energy levels and each energy level is described by a principal quantum number, denoted n. The first energy level of a atom is denoted with n = 1, the second with n = 2 and so on. For the simplest one-electron atom, the hydrogen (H) atom, we can calculate the energy of each level with the following equation:

Where n = 1, 2, 3, …

The letter n denotes the energy level of which we want to calculate the energy of. In this equation we calculate the energy of a specific level in the unit of electron volts (eV). The unit of electron volts equals 1 eV = 1.602*10E-19 J (joule).

**Example of calculations:**
Let‘s calculate the energy of the first energy level in a H (hydrogen) atom:

We can convert this value of energy into joules (J) using the given value of joules per. each eV:

Note however that the electrons on a particular energy level „possess“ a certain amount of energy so the final answer would have a positive sign.

### The quantum numbers

Each energy level (on which the electrons „array“) is described by four different quantum numbers: n, l, ml and ms. The first three describe the particular orbital which we are considering, but the fourth quantum number, ms, specifies how many electrons can occupy that orbital.

### The descriptive characteristics of the four quantum numbers

**1.** The principal quantum number, n: n = 1, 2, 3, …
It specifies the energy of an elecron and the size of the orbital (the distance from the nucleus). All orbitals that have the same value of the principal quantum number n are in the same shell (energy level).
An electron that is on the n = 1 energy level is said to be in its ground state, but electrons on n = 2 and higher levels are said to be in an excited state.

**2.** The quantum number l describes so called angular momentum. It describes the shape of a energy level on which the electron is situated. This second quantum number, l, divides the energy level or the shells of the electrons into smaller groups of orbitals called subshells.
The allowed values of l are: l = [0, …, n-1], so for the second energy level (n = 2), the allowed values of l are: [0, 1]. That is the lowest allowed value of l is 0, and the highest allowed value of it is n-1 (one lower than the value of n). All whole numbers in between these lowest and highest allowed values are also allowed.

The angular momentum quantum number, l, is usually also described by a letter. The most common letters are s, p, d and f (though g, h and so on also exist). These letters describe so called orbitals, that is s-orbitals, p-orbitals, d-orbitals and so on. Each of these ofbitals has its specific shapes and with these information about the allowed values of l, and which orbital eachs of them stands for we can determine the shape of the orbital in which a specific electron is located. To apply the letters s, p, d, f, g and h to the value of the quantum number l, we use the following rules:

**Value of l →** 0 1 2 3 4 5
**Appropriate letter for each value of l →** s p d f g h

Thus, for the value of l = 1, we denote the orbital on which a specific electron is located on with p, that is: this electron is on a p orbital. In the same way, l = 4 denotes a electron on a f orbital.

For a better description of the location of a specific electron, we also include the principal quantum number n. If we have n = 2 and l = 1 for a specific electron, then we denote that it is located on the np orbital, and as n = 2 then it‘s the 2p orbital. The shapes of each of the first four types of orbitals are shown below.

**3.** The magnetic quantum number ml has the allowed values: ml = [-l, l]. That is, the allowed values are from the lowest value of l to the highest value of l, and also contain all whole number integers in between.
For the value of l = 4, we have the allowed values of ml = [-4, 4] or [-4, -3, -2, -1, 0, 1, 2, 3, 4]
The magnetic quantum number describes the orientation in space of an orbital with a given energy (n) and shape (l). There can be a total of 2l+1 orbitals in each energy level (subshell). Thus the s subshell has only one orbital, the p subshell has three orbitals and so on.

**4.** The spin quantum number ms has the allowed values:

The spin quantum number describes the spin of the electrons. An electron can only spin in two directions. It‘s usually said that an electron either spins up or down.

According to a principle called The Pauli exclusion principle no two electrons in the same atom can have identical values for all of their four quantum numbers.

### The shapes of the s, p and d atomic orbitals

#### s-orbitals

All s-orbitals are spherical, and if the midpoint of the sphere is put at the origin of a 3-axis (x, y and z axis) cortesian plane, then it‘s symmetric about the origin.

**An s-orbital is always spherical and symmetric about the origin of a 3-axis corteisan plane.**

#### p-orbitals

The shape of p-orbitals is often described as the shape of a dumbbell. The number of p-orbitals on each shell is always three. Each orbital lay on one of the three axis: x, y or z. Thus, as the number of them is a total of three, we always have one p-orbital laying on each of the axis. To get a better idea of the shape of the p-orbitals, see the picture below.

**p-orbitals are said to be „dumbbell-shaped“**

On each shell, the number of p-orbitals is three, so the picture of all of the three p-orbitals with each dumbbell shaped p-orbital aligned along one of the axis is as shown on the picture of them combined in the three axis cortesian plane here below:

**The shape of all of the three p-orbitals combined on a 3-axis cortesian plane**

#### d-orbitals

The d-orbitals have much more complex shapes than the s- and p-orbitals. The total number of d-orbitals on each shell (or energy level) is five. It‘s hard to describe their shape with only words, and to make things even more complicated they don‘t all have the same shape. The shapes of the d-orbitals are shown on the following picture:

**The shapes of the five d-orbitals**

If all the five d-orbitals are combined in a 3-axis cortesian plane like we have discussed for the s- and p-orbitals, then we get a rather complicated picture which can hardly be described only in words:

**On this picture we can see the shape of all the five d-orbitals and how each of them contributes to a complicated shape in a cortesian plane when they‘re combined in a picture in the same way as they appear in atoms**

NOTE: the total number of orbitals on each shell (or energy level) is equal to the square of the quantum number n, or simply equal to n^2.

**Example/problem:**
From what we have learned so far, let‘s give the names of all the orbitals with the values: n = 4, and state how many values of the magnetic quantum number ml correspond to each type of orbital.

**Answer:**
As the principal quantum number n = 4, we have the allowed values of l = [0, 3] or simply 0, 1, 2 and 3. As l = 0 represents an s-orbital, l = 1 denotes an p-orbital, l = 2 gives us a d-orbital and l = 3 gives us a f-orbital.
Thus the name of the orbitals are:
l = 0 and we have a 4s orbital (n = 4, and l = 0 gives us the s-orbital: so this s-orbital is a 4s orbital).
l = 1 and we have a 4p orbital (as for l = 1 we have a p-orbital according to the rules we have discussed).
l = 2 and we have a 4d orbital.
l = 3 and we have a 4f orbital.

Finally, the magnetic quantum number, ml, has the total number of values 2l+1, so: - For l = 0, we have one 4s orbital. - For l = 1, we have three 4p orbitals. - For l = 2, we have five 4d orbitals. - For l = 3, we have seven 4f orbitals. And if we use the rule for a total number of orbitals on each shell, we get that we have a total of n2 orbitals on shell number four: and as n = 4, we have a total of 42 = 16 orbitals on the fourth shell. To test our result we can count the number of orbitals we got from using the equation for the magnetic quantum numbers: 1x 4s orbital + 3x 4p orbitals + 5x d-orbitals + 7x f-orbitals = a total of 1+3+5+7 = 16 orbitals.

# Summary

Atoms can only contain electrons on specific energy levels or shells. These energy levels are described with a principal quantum number n. Electrons on the first such level, n = 1, are said to be in their ground state while electrons on higher energy levels are said to be in an excited state. Electrons are arranged on these energy levels, which each has its characteristic amount of energy. The energy levels are then further divided into orbitals. The atomic orbitals are the place in „space“ within the atom where there‘s a certain (and the highest) probability of finding a specific electron that has been arranged on that orbital of the shell (or energy level). Furthermore, each type of a orbital has its characteristic shape(s). We can use a simple equation to calculate the energy of the energy levels in the hydrogen atom. To calculate the levels of more complicated atoms or molecules we need to use a much more complicated equation that we do not discuss here. To sum up our discussion of atomic orbitals in a few words: the shapes of atoms are only representative as mentioned earlier and we use these basic ideas of quantum mechanics to describe these shapes.