Orbitals
Unit:  Atomic Theory (Quantum/Electronic Structure)
Knowledge/Understanding:
•	what orbitals are
•	the hierarchy of energy levels, sub-levels, and orbitals
•	how the hierarchy corresponds with the periodic table of the elements
orbital:  a region in an atom (outside the nucleus) where there is a high probability of finding an electron.
These regions are called orbitals because these regions correspond with the spherical orbits in the Bohr model.
Note that orbitals are not physical objects with boundaries.  They are simply the space that an electron can move around in because of the amount of energy it has.  When an electron approaches the “boundary” of its orbital, the forces pulling it back are strong enough to overcome the energy that the electron has, and it cannot get any farther away.
The locations and geometric shapes of these orbitals correspond with the solutions to the Schrödinger wave equation, a complex mathematical formula that would require mathematics far too advanced for a high school class.

Energy Level Hierarchy
Electrons have energy levels, which roughly correspond to the solutions (wavefunctions) of the wave equation.
Each particular wavefunction (which corresponds with the energy of a particular electron) is specified by a set of four quantum numbers.  It’s easiest to think of these quantum numbers in a hierarchy, sort of like an outline:
1.	n = principal energy level (1, 2, 3, ...)
a.	ℓ = sub-level, which governs the shape of the orbital (s, p, d, f, ...)
i.	mℓ = inclination (magnetic moment) orbitals
a.	ms = spin of individual electron
Energy Level
The main or principal energy level is a measure of total distance from the nucleus.  The levels are assigned the principal quantum numbers 1-7.  The periodic table of the elements is arranged so that the Period (row number) that an element is in equals the energy level of the electron that has the most energy.
For example, helium is in period (row) #1.  That means its highest energy level is 1, which means both of its electrons have to be in level 1.
Sulfur is in period (row) #3, which means it has electrons in levels 1, 2, and 3.

Sub-Levels & Orbitals
There are four types of sub-levels.  Each type has a specific number of orbitals with specific shapes.  Each of those orbitals can hold up to 2 electrons.
There are four kinds of sub-levels:  s, p, d, and f, corresponding with the ℓ-values from 0, 1, 2, and 3, respectively.  Each sub-level contains orbitals whose mℓ-values are integers ranging from −ℓ to +ℓ.
The shapes, numbers, ℓ-values and mℓ-values for each type of sub-level are shown in the table below.:
sub-level	shape(s) of orbital(s)	value of ℓ	# orbitals
values of mℓ	total # electrons	
s		0	1 orbital
mℓ = 0	2	
p		1	3 orbitals
mℓ = −1, 0, +1	6	
d	  	2	5 orbitals
mℓ = −2, −1, 0, +1, +2	10	
f	    	3	7 orbitals
mℓ = −3, −2, −1, 0, +1, +2, +3	14	
Notice that each sub-level contains an odd number of orbitals.
Arrangement of Levels and Sublevels
Each energy level in an atom has at least one of these kinds of sub-levels—each level contains an s sub-level, each level starting with level 2 contains s and p sub-levels, each level starting with level 3 contains s, p, and d sub-levels, etc.
Level	1	2	3	4	5	6	7	
sub-levels	s	s, p	s, p, d	s, p,
d, f	s, p, d, f, (g)	s, p,
d, f, (g), (h)	s, p, d, f, (g), (h), (i)	
values
of ℓ	0	0, 1	0, 1, 2	0, 1,
2, 3	0, 1, 2, 3, (4)	0, 1,
2, 3, (4), (5)	0, 1, 2, 3, (4), (5), (6)	

Note that g, h, and i are in parentheses because no atom is large enough that it actually has any electrons in g, h, or i sublevels, but mathematically we know that those sub-levels will exist if we “discover” (create) a large enough atom.
Note also that these sub-levels overlap.  For example, the levels and sublevels in a sodium (Na) atom might look like the picture to the right:
Electron Spin and the Pouli Exclusion Principle
In 1925, Wolfgang Pauli formulated the Pauli Exclusion Principle, which states that no two electrons within an atom can occupy the same quantum state simultaneously.  This means that every electron must have a unique set of quantum numbers.
Pauli theorized that in order to occupy the same orbital, two electrons must have rotations that balance each other.  To describe the rotation or “spin”, a fourth quantum number, ms, was defined.  The first electron in an orbital is assigned an ms value of −½.  The second electron to occupy the same orbital is assigned an ms value of +½.
Quantum State of an Electron
The quantum state of every electron in an atom can be fully described by the value of the four quantum numbers n, ℓ, mℓ, and ms:
Quantum Number	Description	Allowed Values	
n	principal energy level	1, 2, 3, …	
ℓ	type of sub-level	0 = s; 1 = p; 2 = d; 3 = f	
mℓ	which orbital?	−ℓ, …, −1, 0, +1, …, +ℓ	
ms	electron spin	−½, +½	

Sub-levels and the Periodic Table
Note that the sub-level of the electron that has the highest energy  corresponds with the location of the element on the periodic table:
Notice, for example, that the “s section” of the periodic table is two columns wide.  This is because each s sub-level has one orbital that can hold two electrons.
type of sub-level	

× 2 =

# orbitals	# electrons 	# columns on P.T.	
s	1	2	2	
p	3	6	6	
d	5	10	10	
f	7	14	14