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Overview  Module 20  Quantum Mechanics and Orbitals
Dr. MJ Patterson
The first 5 sections of this module provide background material to help the rest of quantum mechanics make more sense. You will not be tested directly on this material, so do not be concerned about the equations and calculations that pop up in these sections. It is more important that you get a feel for some of the names and ideas associated with quantum.
Starting with section 6, the rest of this module describes orbitals. We have already seen that atoms contain electrons. We are now going one step farther and investigating how those electrons are arranged within an atom. The locations where electrons can be found within an atom are called orbitals. The next module will investigate how the electrons are distributed among the orbitals.
Quantization of Energy
The main idea of quantum mechanics is that the energy of an electron in an atom can only take on certain allowed values. A marble on a staircase is a good analogy. The marble can be in a stable position on the flat part of a stair, but if you try to place it anywhere else, it will simply fall to the first available step. Each step has a different gravitational energy associated with it. The highest step has the highest gravitational energy, and the lowest step has the lowest gravitational energy. If you add enough energy to the marble by lifting, you can raise the marble to a higher step. If you take out energy, you can move the marble to a lower step.
In an atom, the electron can have only certain certain energy values, and each energy value is associated with a three dimensional space known as an orbital. In the stair analogy, the energy levels were due to gravity. In an atom, the energy levels are due to the electrical attraction of a positive nucleus for a negative electron. In the staircase, each energy level was associated with a step. In an atom, each energy level is associated with an orbital. Just as a marble can only be in a stable position on a step, an electron can only be found in an orbital.
This module and the next are mostly concerned with describing the orbitals available in an atom, and how the electrons will arrange themselves in these orbitals.
Quantization of energy simply means that only certain energy levels are allowed. If this idea seems strange, think about another case of quantization  matter. The smallest quantum of an element is an atom. The smallest quantum of a compound is a molecule.
Orbitals
Orbitals are three dimensional regions near the nucleus where an electron is likely to be found. Each orbital also has a specific energy associated with it.
s orbitals
s orbitals are spherical in shape (refer to the multimedia lesson for good drawings of all of the orbitals). Each s orbital also has a number associated with it, called the principal quantum number, and abbreviated n. For the elements in the periodic table, n = 1, 2, 3, 4, 5, 6, 7. Theoretically, n can go up to infinity, but physicists have not yet created any atoms that require a principal quantum number bigger than 7. The principal quantum number is also associated with the row of the periodic table, and note that there are 7 rows, if you slide the lanthanides and actinides back in where they belong. So, the most frequently used names for the s orbitals are 1s, 2s, 3s, 4s, 5s, 6s and 7s.
Each orbital is spherical, with the nucleus at the center of the sphere. The main difference between s orbitals is in the size. The 1s orbital is the smallest, and the 7s orbital is the largest. Also, the s orbitals occur singly. In other words, when we talk about the 3s orbital, it is just a single orbital.
p orbitals
p orbitals look like a cartoon dumbell, with two lobes, one on each side of the nucleus. p orbitals occur in sets of three. In other words, when you talk about the 2p orbitals, you are really talking about 3 different orbitals  2p_{x}, 2p_{y} and 2p_{z}. The set of 2p orbitals is called the 2p subshell.
d orbitals
d orbitals look like a 3 dimensional cloverleaf, or two p orbitals stuck together. Each d subshell contains 5 orbitals.
f orbitals
f orbitals are even more complicated than the d orbitals, starting with double cloverleafs. Each f subshell contains 7 orbitals.
Example 1:
How many orbitals are in an s, p, d and f subshell? What general shape does each type of orbital have?
Solution 1:
Subshell Type  Number of Orbitals  Shape of Orbitals 
s  1  sphere 
p  3  dumbbell 
d  5  cloverleaf 
f  7  double cloverleaf 
Aufbau Principle Diagram
Trying to remember which orbitals exist and which orbitals do not can be tricky. The simplest way I have found is to refer to the Aufbau Principle Diagram shown below. If the orbital is listed on the diagram, it exists. Otherwise, it doesn''t exist. We will expand on this diagram in the next module.
You start this diagram by writing down the 1s subshell at the bottom. Then, you move up a row and write down the 2s and 2p subshells. You continue this pattern of moving up a row and adding one more column until you hit the 4f subshell. After than, you can stop each row at the f subshell column. Theoretically, additional subshells such as the g, h and so on, can exist, but they are not required for any real atoms.
6s  6p  6d  6f 
5s  5p  5d  5f 
4s  4p  4d  4f 
3s  3p  3d 

2s  2p 


1s 



Example 2:
Which of the following orbitals exist? Which do not?
1p, 2d, 4d, 6s, 3f
Solution 2:
If an orbital exists, it will be in the Aufbau Principle Diagram. Otherwise, it does not exist.
exist: 4d, 6s
do not exist: 1p, 2d, 3f
Quantum numbers
We have already seen the principal quantum number which is associated with the row of the periodic table as we will see in the next module. The next quantum number is called the angular momentum quantum number and is abbreviated with a script L (Since web pages do not easily support a script L, I will use a capital L instead). L distinguishes between types of orbitals. The smallest value L can take is 0, and the largest needed for real elements is 3, although theoretically L can go on up to infinity just like n.
Historically, quantum mechanics developed later than another branch of science known as spectroscopy. In the early 1900''s, scientists realized that spectroscopy and quantum mechanics were largely dealing with the same concepts, but using different terminology. In the case of the angular momentum quantum number L, scientists realized that the different numerical values for L could be associated with the spectroscopy terms which we have already seen  s, p, d and f. The following table shows the relationship:
L value (quantum)  0  1  2  3 
Spectroscopy notation  s  p  d  f 
Over time, a hybrid terminology incorporating some spectroscopic terms developed for describing orbitals. The principal quantum number, such as the 2 in 2p, comes from quantum, while the value of the azimuthal quantum number is specified with the letter p. This kind of notation 2p combines both families of symbols, and is the most commonly used.
Shells and Subshells
Now that we have defined the first two quantum numbers n and L, we can easily define shell and subshell. A shell consists of all of the orbitals which have the same value for the principle quantum number n. A subshell consists of all of the orbitals with the same value of n and L.
To identify all of the orbitals in a subshell, look at the Aufbau Principle Diagram. Each entry in the diagram is a subshell. Just remember that an s subshell has 1 orbital, a p subshell has 3 orbitals, a d subshell has 5 orbitals and an f subshell has 7 orbitals.
To identify all of the orbitals in a shell, look at an entire row of the Aufbau Principle Diagram. Each row shows all of the subshells with the same value of n.
Example 3:
How many orbitals are in the n=3 shell?
How many orbitals are in each subshell in the n=3 shell?
Solution 3:
Let''s start by looking at the diagram for the row with n=3. The following subshells are listed: 3s, 3p and 3d. The 3s subshell has 1 orbital, the 3p subshell has 3 orbitals and the 3d subshell has 5 orbitals. The total number of orbitals in the n=3 shell is 1 + 3 + 5 = 9 orbitals.