### Electron Orbitals, Quantum Numbers, and Orbital Filling

Let's start our discussion of Schrödinger's wave equation by examining the rules regarding the quantum numbers. Note that Pauli's exclusion principle dictates that no two electrons can have the same set of four quantum numbers.

n is called the principle quantum number; it describes how far away from the nucleus the electron shell under consideration is. The larger the value of n, the farther away the shell is; since shells are concentric, farther shells are also larger. n can have any whole-number value greater than zero, but any number bigger than about 4 is uncommon (all the electrons in the largest atoms can fit into about this many shells).

l is the angular momentum quantum number, and it describes the shape of the subshell (each electron shell except n=1 is composed of several subshells). The maximum number of subshells in any electron shell is equal to n - 1; for instance, the 3rd electron shell has permissible l values of 0, 1, and 2. Before the quantum number system was instituted, subshells were referred to with letters: 0 was s, 1 was p, 2 was d, 3 was f, 4 was g, and so on. Many texts will still use the "s-p-d-f..." system, so be sure you understand how the letters and l values correspond.

ml, the magnetic quantum number, describes which orbital an electron belongs to. Orbitals in a subshell but differ in their orientation--one might point "up," another to the "left," etc. Possible orbital values range from the negative value of l through zero to the positive value of l. For instance, possible ml values for an l value of 3 are -3, -2, -1, 0, 1, 2, and 3.

ms is the spin quantum number. Two electrons are possible for each orbital, so the ms value is necessary to fulfill the exclusion principle. Possible values for this number at +1/2 or -1/2. If only one electron occupies an orbital, the electron may have either value (the assignment is arbitrary, since this number serves only to separate the electrons).

Using these four quantum numbers, you can precisely describe any electron in any element. The chart below gives examples of all the orbitals within the first four principle quantum numbers:

 Principle Quantum Number, n Angular Momentum Quantum Number, l Magnetic Quantum Number, ml Electron Summary 1 0 (s) 0 1 1s orbital, 2 electrons 2 0 (s) 0 1 2s orbital, 2 electrons 1 (p) 1, 0, -1 3 2p orbitals, 6 electrons 3 0 (s) 0 1 3s orbitals, 2 electrons 1 (p) 1, 0, -1 3 3p orbitals, 6 electrons 2 (d) 2, 1, 0, -1, -2 5 3d orbitals, 10 electrons 4 0 (s) 0 1 4s orbitals, 2 electrons 1 (p) 1, 0, -1 3 4p orbitals, 6 electrons 2 (d) 2, 1, 0, -1, -2 5 4d orbitals, 10 electrons 3 (f) -3, -2, -1, 0, 1, 2, 3 7 4f orbitals, 14 electrons

By adding up the electrons in each orbital, we can find how many fit in each shell: 2 for the first shell, 8 for the second, 18 for the third, and 32 for the fourth, for a total of 70 in just the first four shells! All of the known elements fit in the first five shells, which can accommodate 120 electrons in total. To find out how many electrons are possible for any shell, use the formula 2n2, where n is the principle quantum number.

Next, we'll examine the shape of each orbital, starting with the "s" subshell. As some simple rules, the principle quantum number determines the size of the subshell and orbitals. The l quantum number determines the number of zero-electron-density planes that pass through each subshell. We'll explain what this means to each orbital below, and a chart of orbital shapes is also below.

Each "s" subshell has only one orbital; it is a simple sphere, which increases in size with each quantum number. Since l is zero for these orbitals, there is no plane in the orbital.

"P" subshells are shaped like barbells, whose orientation changes with each orbital. For example, in a hypothetical 3D space, one orbital in the "p" subshell may have its nodes facing along the left-right axis, the second along the front-back axis, and the third from top to bottom. The l value requires one plane.

"D" subshells have more complex shapes; two zero-density planes are required for each orbital. One "d" orbital is particularly strange, appearing as a ring with two spherical zones lying above and below. The rest of the orbitals are more normal, with four balloon-shaped regions of high electron density extending along various planes.

"F" orbitals require three planes, so they have very complex shapes. Visual knowledge of this orbital is reserved to higher-level chemistry courses, so we'll just include one as an example.

To write out the electron configuration of an atom, we use the principal quantum number and the letter term for each subshell; a superscript number indicates how many electrons are present in each subshell. For instance, hydrogen has one electron in its 1s subshell, making its electron configuration 1s1. Lithium has two electrons in its 1s subshell and one in its 2s subshell, making its configuration 1s22s1. This method is called spectroscopic notation.

However, electrons do not simply fill all the orbitals of one shell and then move onto the next; life isn't that simple! They obey a very specific filling order, minimizing the value of the "n + l" quantum numbers. In the event of a "tie" between subshells, the subshell with the lower n value will be filled first. This fill order is called Hund's rule. The diagram below is a convenient reminder of how subshells fill; notice the backward-diagonal order. Keep filling using this order.

As an example, the spectroscopic diagram of neon is 1s22s22p6. When even more detail about electron configurations is needed, orbital box diagrams are used. In this type of notation, every electron in the atom is represented; atoms with positive ms values (+1/2) are represented by an upward arrow; those with a -1/2 value are shown with a down arrow. Since arrows are difficult to represent in a web-page format, we'll use "plus" and "minus" symbols instead. For example, the orbital box diagram for neon is:

1s [+-] 2s [+-] 2p [+-][+-][+-]

There is one further tangle to the rules of electron-filling: in an atom like nitrogen, with three electrons in the 2p subshell, you might think the box structure would be [+-][+ ][  ]. However, slight electron-electron repulsion exists between electrons in the same orbital, so if it can be avoided, all slots in the current subshell must be filled with one electron before they begin doubling-up. Therefore, the spectroscopic and electron-box diagrams for nitrogen are, respectively:

1s22s22p3 and
1s [+-] 2s [+-] 2p [+ ][+ ][+ ]

Note that subshells where all the orbitals are filled with one electron are more stable than would be expected. For instance, chromium does not follow the "n+l" rule, electing instead to take one electron from its 4s subshell to half-fill the 3d subshell, as shown below:

Cr Expected: [Ar] 3d [+ ][+ ][+ ][+ ][  ] 4s [+-]

Cr Actual: [Ar] 3d [+ ][+ ][+ ][+ ][+ ] 4s [+ ]

Several atoms, including chromium, copper (which fills the 3d subshell by taking an electron from the 4s subshell), and silver, violate the usual filling order to increase stability in filled or half-filled subshells.

If you look at the Periodic Table, you will notice that the table is set up around the orbital-filling guidelines: the alkali metals and alkaline earth metals fill the "s" subshells; because we don't reach the "p" subshells until the 2nd main shell, there is no "p" group in the first row (hydrogen is followed directly by helium). Once the "p" subshells become available, they are filled by the boron, carbon, nitrogen, oxygen, and fluorine columns. The "d" subshells appear later still, and are filled by the transition metal elements. Finally, the "f" subshells are occupied by the lanthanide-actinide series when necessary. If we discover enough new elements, we will have to add another group onto the table to represent the "g" subshells as well. So, simply using the Periodic Table, one can find what the electron configuration of an atom is.

Consider the noble gases, the final column of the table. Each noble gas (except helium) has completely filled "s" and "p" orbitals. This arrangement is extremely stable, and is responsible for the almost totally inert behavior of the noble gases. As we shall see, every atom "wants" to have these subshells filled with electrons--for example, fluorine needs one more electron to reach the noble gas configuration of neon, which is why it always forms the F- ion. Conversely, sodium must lost one electron to reach a stable configuration, the reason why it becomes the Na+ ion so frequently.

When one considers the structure of subshells and orbitals, it becomes apparent that only the outermost electrons of an atom can be affected by outside influences, because shells are concentric. The electrons in the other subshells and orbitals are inaccessible "beneath" the outermost subshells; they are called the core electrons. For example, in the example of nitrogen above, the 1s subshell electrons are "submerged" beneath the 2s subshell. Therefore, to reduce the amount of writing we have to do, it is customary to represent the "buried" electrons by using a noble gas configuration, in which the core electrons are replaced by the symbol of a noble gas in brackets. Again using nitrogen as an example, the 1s2 subshell is included in the helium atom, so the noble gas configuration for nitrogen is [He]2s22p3. The electrons outside the noble gas core are the valence electrons, and they determine the chemical properties of an element.

The next page will use the concept of orbitals to describe how atoms come together to form chemical bonds.

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