Quantum Numbers and Atomic Orbitals

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follow the rules:

 for the principal q.n.  n
         azimuthal q.n.  l = 0, 1, 2, ..., n-1
          magnetic q.n.  m = -l, -(l-1), ..., (l-1), l

Rules are algorithms, by which we generate possible quantum numbers. The lowest value of n is 1 (NOT zero). For n = 1, the only possible value for quantum number l is 0, and m = 0. Each set of quantum numbers is called a state. Thus, for n = 1, there is only one state (1,0,0). The states are represented by symbols, and special symbols have been used to represent the quantum number l as follows:

           l = 0, 1, 2, 3, 4, ...
      symbol = s, p, d, f, g, ...

Using symbols, the valid quantum states can be listed in the following manner:

1s
2s 2p
3s 3p 3d
4s 4p 4d 4f
5s 5p 5d 5f 5g
6s 6p 6d 6f 6g 7h
7s 7p 7d 7f 7g 7h 8i

For hydrogen-like atoms, that is atoms or ions with one electron, the energy level is solely determined by the principle quantum number n, and the energy levels of the subshells np and nd etc. are the same as the ns. For these species, the energy levels have the reverse order as the list given earlier:

 = 7s 7p--- 7d----- ....  ]  These are very close together!
 = 6s 6p--- 6d----- ....  ]
 = 5s 5p--- 5d----- 5f------- 5g---------
 - 4s 4p--- 4d----- 4f-------

 - 3s 3p--- 3d-----




 - 2s 2p---







 - 1s

Confidence Building Questions

How many possible orbitals are there if n = 3?

Hint: There are nine (9) possible orbitals.

Discussion:
3s, 3 3p, 5 3d; total = 1 + 3 + 5 = ?
How many possible orbitals are there in the subshell [n=5, l=4]?

Hint: There are 9 such orbitals.

Discussion:
For 5g, l = 4; ml = -4, -3, -2, -1, 0, 1, 2, 3, 4. A total of 9.
How many electrons can be accommodated in the subshell 4f?

Hint: A total of 14 electrons.

Discussion:
Each of the 7 4f orbitals accommodates a pair of electrons. There are 14 elements in the lanthanides, the 4f block elements.
How many atomic orbitals are there for the subshell with [n = 3, l = 2]?

Hint: There are 5 3d orbitals.

Discussion:
The magnetic q.n. = -2, 1, 0, 1, 2. The number of orbitals associated with a given value of l is (2l + 1).
What is the symbol representing the set of orbitals in the subshell with [n = 3, l = 2]?

Hint: The symbol is 3d.

Discussion:
For symbols, consider the following:
```
Symbol = s  p  d  f  g  correspond to
     l = 0  1  2  3  4
```
The d-block in each period of the periodic table has 10 elements.

Contributors and Attributions

Chung (Peter) Chieh (Professor Emeritus, Chemistry @ University of Waterloo)