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The Aufbau Principle

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The Aufbau Principle

To determine the electron configuration for any particular atom, we can “build” the structures in the order of atomic numbers. Beginning with hydrogen, and continuing across the periods of the periodic table, we add one proton at a time to the nucleus and one electron to the proper subshell until we have described the electron configurations of all the elements. This procedure is called the Aufbau principle, from the German word Aufbau (“to build up”).

Each added electron occupies the subshell of lowest energy available (in the order shown in the figure above), subject to the limitations imposed by the allowed quantum numbers according to the Pauli exclusion principle. Electrons enter higher-energy subshells only after lower-energy subshells have been filled to capacity.

The figure below illustrates the traditional way to remember the filling order for atomic orbitals. Since the arrangement of the periodic table is based on the electron configurations, the electron configuration table below provides an alternative method for determining the electron configuration. The filling order simply begins at hydrogen and includes each subshell as you proceed in increasing Z order. For example, after filling the 3p block up to Ar, we see the orbital will be 4s (K, Ca), followed by the 3d orbitals.

This figure includes a chart used to order the filling of electrons into atoms. At the top is a blue circle labeled “1 s.” In a row beneath this circle are 6 additional blue circles labeled “2 s” through “7 s.” A column to the right begins just right of 2 s and contains pink circles labeled 2 p through 7 p. A column to the right begins just right of 3 p and contains yellow circles labeled 3 d through 6 d. No circles are placed to the right of the 7 s and 7 p circles. A final column on the right begins right of 4 d. It includes grey circles labeled, “4 f” and, “5 f.” No circles are placed right of 6 d. Through these circles, arrows are included in the figure pointing down and to the left. The first arrow begins in the upper right and passes through 1 s. The second arrow begins just below and passes through 2 s. The third arrow passes through 2 p and 3 s. The fourth arrow passes through 3 p and 4 s. This pattern of parallel arrows pointing downward to the left continues through all circles completing the pattern 1 s 2 s 2 p 3 s 3 p 4 s 3 d 4 p 5 s 4 d 5 p 6 s 4 f 5 d 6 p 7 s 5 f 6 d 7 p.

The arrow leads through each subshell in the appropriate filling order for electron configurations. This chart is straightforward to construct. Simply make a column for all the s orbitals with each n shell on a separate row. Repeat for p, d, and f. Be sure to only include orbitals allowed by the quantum numbers (no 1p or 2d, and so forth). Finally, draw diagonal lines from top to bottom as shown.

This periodic table shows the electron configuration for each subshell. By “building up” from hydrogen, this table can be used to determine the electron configuration for any atom on the periodic table.

We will now construct the ground-state electron configuration and orbital diagram for a selection of atoms in the first and second periods of the periodic table. Orbital diagrams are pictorial representations of the electron configuration, showing the individual orbitals and the pairing arrangement of electrons.

We start with a single hydrogen atom (atomic number 1), which consists of one proton and one electron. Referring to any of the two figures above, we would expect to find the electron in the 1s orbital. By convention, the \({m}_{s}=+\frac{1}{2}\) value is usually filled first. The electron configuration and the orbital diagram are:

In this figure, the element symbol H is followed by the electron configuration is 1 s superscript 1. An orbital diagram is provided that consists of a single square. The square is labeled below as, “1 s.” It contains a single upward pointing half arrow.

Following hydrogen is the noble gas helium, which has an atomic number of 2. The helium atom contains two protons and two electrons. The first electron has the same four quantum numbers as the hydrogen atom electron (n = 1, l = 0, ml = 0, \({m}_{s}=+\frac{1}{2}\)). The second electron also goes into the 1s orbital and fills that orbital. The second electron has the same n, l, and ml quantum numbers, but must have the opposite spin quantum number, .\({m}_{s}=-\frac{1}{2}.\)

This is in accord with the Pauli exclusion principle: No two electrons in the same atom can have the same set of four quantum numbers. For orbital diagrams, this means two arrows go in each box (representing two electrons in each orbital) and the arrows must point in opposite directions (representing paired spins). The electron configuration and orbital diagram of helium are:

In this figure, the element symbol H e is followed by the electron configuration, “1 s superscript 2.” An orbital diagram is provided that consists of a single square. The square is labeled below as “1 s.” It contains a pair of half arrows: one pointing up and the other down.

The n = 1 shell is completely filled in a helium atom.

The next atom is the alkali metal lithium with an atomic number of 3. The first two electrons in lithium fill the 1s orbital and have the same sets of four quantum numbers as the two electrons in helium. The remaining electron must occupy the orbital of next lowest energy, the 2s orbital. Thus, the electron configuration and orbital diagram of lithium are:

In this figure, the element symbol L i is followed by the electron configuration, “1 s superscript 2 2 s superscript 1.” An orbital diagram is provided that consists of two individual squares. The first square is labeled below as, “1 s.” The second square is similarly labeled, “2 s.” The first square contains a pair of half arrows: one pointing up and the other down. The second square contains a single upward pointing arrow.

An atom of the alkaline earth metal beryllium, with an atomic number of 4, contains four protons in the nucleus and four electrons surrounding the nucleus. The fourth electron fills the remaining space in the 2s orbital.

In this figure, the element symbol B e is followed by the electron configuration, “1 s superscript 2 2 s superscript 2.” An orbital diagram is provided that consists of two individual squares. The first square is labeled below as, “1 s.” The second square is similarly labeled, “2 s.” Both squares contain a pair of half arrows: one pointing up and the other down.

An atom of boron (atomic number 5) contains five electrons. The n = 1 shell is filled with two electrons and three electrons will occupy the n = 2 shell. Because any s subshell can contain only two electrons, the fifth electron must occupy the next energy level, which will be a 2p orbital. There are three degenerate 2p orbitals (ml = −1, 0, +1) and the electron can occupy any one of these p orbitals. When drawing orbital diagrams, we include empty boxes to depict any empty orbitals in the same subshell that we are filling.

In this figure, the element symbol B is followed by the electron configuration, “1 s superscript 2 2 s superscript 2 2 p superscript 1.” The orbital diagram consists of two individual squares followed by 3 connected squares in a single row. The first square is labeled below as, “1 s.” The second is similarly labeled, “2 s.” The connected squares are labeled below as, “2 p.” All squares not connected contain a pair of half arrows: one pointing up and the other down. The first square in the group of 3 contains a single upward pointing arrow.

Carbon (atomic number 6) has six electrons. Four of them fill the 1s and 2s orbitals. The remaining two electrons occupy the 2p subshell. We now have a choice of filling one of the 2p orbitals and pairing the electrons or of leaving the electrons unpaired in two different, but degenerate, p orbitals. The orbitals are filled as described by Hund’s rule: the lowest-energy configuration for an atom with electrons within a set of degenerate orbitals is that having the maximum number of unpaired electrons. Thus, the two electrons in the carbon 2p orbitals have identical n, l, and ms quantum numbers and differ in their ml quantum number (in accord with the Pauli exclusion principle). The electron configuration and orbital diagram for carbon are:

In this figure, the element symbol C is followed by the electron configuration, “1 s superscript 2 2 s superscript 2 2 p superscript 2.” The orbital diagram consists of two individual squares followed by 3 connected squares in a single row. The first blue square is labeled below as, “1 s.” The second is similarly labeled, “2 s.” The connected squares are labeled below as, “2 p.” All squares not connected to each other contain a pair of half arrows: one pointing up and the other down. The first two squares in the group of 3 each contain a single upward pointing arrow.

Nitrogen (atomic number 7) fills the 1s and 2s subshells and has one electron in each of the three 2p orbitals, in accordance with Hund’s rule. These three electrons have unpaired spins. Oxygen (atomic number 8) has a pair of electrons in any one of the 2p orbitals (the electrons have opposite spins) and a single electron in each of the other two. Fluorine (atomic number 9) has only one 2p orbital containing an unpaired electron. All of the electrons in the noble gas neon (atomic number 10) are paired, and all of the orbitals in the n = 1 and the n = 2 shells are filled. The electron configurations and orbital diagrams of these four elements are:

This figure includes electron configurations and orbital diagrams for four elements, N, O, F, and N e. Each diagram consists of two individual squares followed by 3 connected squares in a single row. The first square is labeled below as, “1 s.” The second is similarly labeled, “2 s.” The connected squares are labeled below as, “2 p.” All squares not connected to each other contain a pair of half arrows: one pointing up and the other down. For the element N, the electron configuration is 1 s superscript 2 2 s superscript 2 2 p superscript 3. Each of the squares in the group of 3 contains a single upward pointing arrow for this element. For the element O, the electron configuration is 1 s superscript 2 2 s superscript 2 2 p superscript 4. The first square in the group of 3 contains a pair of arrows and the last two squares contain single upward pointing arrows. For the element F, the electron configuration is 1 s superscript 2 2 s superscript 2 2 p superscript 5. The first two squares in the group of 3 each contain a pair of arrows and the last square contains a single upward pointing arrow. For the element N e, the electron configuration is 1 s superscript 2 2 s superscript 2 2 p superscript 6. The squares in the group of 3 each contains a pair of arrows.

 

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