Aufbau Principle | Hund’s Rule

Aufbau Principle

The Aufbau principle (also known as the building up principle) is a procedure that leads to plausible ground-state configurations. It is not infallible, but it is an excellent starting point for the discussion. Moreover, as we shall see, it provides a theoretical framework for understanding the structure and implications of the periodic table.

According to the Aufbau principle, orbitals of neutral atoms are treated as being occupied in the order determined in part by the principal quantum number and in part by penetration and shielding:

Order of occupation: 1s    2s    2p    3s    3p    4s     3d     4p . . .

Aufbau Principle

Each orbital can accommodate up to two electrons. Thus, the three orbitals in a p subshell can accommodate a total of six electrons and the five orbitals in a d subshell can accommodate up to 10 electrons. The ground-state configurations of the first five elements are therefore expected to be

Aufbau Principle

This order agrees with the experiment.

Hund’s Rule

When more than one orbital of the same energy is available for occupation, such as when the 2p orbitals begin to be filled in boron and carbon, we adopt Hund’s rule :

When more than one orbital has the same energy, electrons occupy separate orbitals and do so with parallel spins (↑↑).

The occupation of separate orbitals of the same value of l (such as a px orbital and a py orbital) can be understood in terms of the repulsive interactions that exist between electrons occupying different regions of space (electrons in different orbitals) being weaker than between those occupying the same region of space (electrons in the same orbital).

The requirement of parallel spins for electrons that do occupy different orbitals is a consequence of a quantum mechanical effect called spin correlation, the tendency for two electrons with parallel spins to stay apart from one another and hence to repel each other less.

An additional factor that stabilizes arrangements of electrons with parallel spins is exchange energy. The exchange energy is the extra stability that a parallel spin configuration (↑↑) gains because the electrons are indistinguishable and interchangeable. If one of the electrons of a pair with parallel spins is removed then the exchange energy is lost, so that arrangements of electrons in degenerate orbitals with large numbers of parallel spins are stabilized relative to those without. The highest exchange energy occurs with a half-filled shell when the largest number of electrons with parallel spins exists.

Consequence

A consequence of this is that half-filled shells, such as d5 and f7, are particularly stable arrangements as removing an electron from these configurations requires the input of energy to overcome the greatest amount of exchange energy. Removing one electron from the d5 configuration (↑↑↑↑↑) to give (↑↑↑↑) reduces the number of pairs of electrons with parallel spins from 10 to 6. One result of this preference for arrangements with half-filled shells is that the ground state of the chromium atom is 4s1 3d5 rather than 4s2 3d4 as the former maximizes the exchange energy.

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