Electron Configuration of Many-Electron Atoms

1. Many-Electron Atoms
1.1. Aufbau Principle
1.2. Madelung's rule
1.3. Hund's Rules

In the previous article we have seen that the electron in a hydrogen atom always occupies one of various different orbitals depending on its energy. These orbitals were received as solutions of the Schrödinger equation.

Many-Electron Atoms

Aufbau Principle

It is not possible to proceed in an analogous manner for many-electron atoms because the Schrödinger equation cannot be solved analytically for such systems. However, one can think of the atom as a composition of the nucleus and all its electrons, constructed in the following way:^[1]

The spatial arrangement of the atom is characterised by the same set of orbitals as for the one-electron hydrogen atom.
The orbitals are filled successively with the atom's electrons. In the ground state of the atom each electron prefers to fill the orbital which is lowest in energy. There are well-working rules (Madelung's rule, Hund's rules) of thumb for the ordering of the orbitals and will be presented more detailed below.
The number of electrons within a specific orbital is limited to a maximum of two by the Pauli exclusion principle: A quantum state specified by the four quantum numbers $n$, $l$, $m$ and $s$ can be possessed by at most one particle at the same time.^[2] Due to the two spin quantum numbers $s$ there are two quantum states associated with each orbital. Therefore it can be filled with up to two electrons.

Madelung's rule

The energy of a specific orbital is in principle determined by the mean distance to the positively charged nucleus. But in many-electron atoms it is also dependent on the mutual repulsion of the electrons and the screening of the nuclear charges by inner electrons.

For the atoms with low atomic numbers the following rule of thumb works quite good:

The energy ordering is from lowest value of $n + l$ to the largest.
When two orbitals have the same value of $n + l$ the one with smaller $n$ is filled first.

This rule is called Madelung's rule.^[3] A mnemonic is depicted below in fig. 1.

Mnemonic for Madelung's rule — Mnemonic for the ordering of the orbitals due to Madelung's rule. The diagonal lines correspond to constant values for $n + l$ and are ordered as labelled. Within a certain diagonal the orbitals are ordered by ascending value of $n$.

Hund's Rules

The scheme is not yet complete since until now there is no rule regarding the $m$-degeneracy. For nonzero $l$-values the orbitals (e.g. p, d, f) exhibit a $2l+1$-fold degeneracy due to the magnetic quantum number $m$. For those a filling order is required as well. The corresponding rules known as Hund's rules:^[4]

In filling a set of degenerate orbitals

the number of unpaired electrons is maximized
and such unpaired electrons possess parallel spins, thus maximising the total spin of the outermost orbital.

How can this be understood? First of all, occupying different orbitals is energetically favourable since it reduces the mutual repulsion and the screening of the nucleus. The second point is less intuitive. However, when having parallel spin these electrons have a zero probability of possessing the same orbital which would be unfavourable.

This rule is visualised in fig. 2 with a box-arrow diagram:^[5]

Element	1s	2s	2p_x	2p_y	2p_z
H	$\uparrow$
He	$\uparrow\downarrow$
Li	$\uparrow\downarrow$	$\uparrow$
Be	$\uparrow\downarrow$	$\uparrow\downarrow$
B	$\uparrow\downarrow$	$\uparrow\downarrow$	$\uparrow$
C	$\uparrow\downarrow$	$\uparrow\downarrow$	$\uparrow$	$\uparrow$
N	$\uparrow\downarrow$	$\uparrow\downarrow$	$\uparrow$	$\uparrow$	$\uparrow$
O	$\uparrow\downarrow$	$\uparrow\downarrow$	$\uparrow\downarrow$	$\uparrow$	$\uparrow$
F	$\uparrow\downarrow$	$\uparrow\downarrow$	$\uparrow\downarrow$	$\uparrow\downarrow$	$\uparrow$
Ne	$\uparrow\downarrow$	$\uparrow\downarrow$	$\uparrow\downarrow$	$\uparrow\downarrow$	$\uparrow\downarrow$

Box-arrow diagram for the first 10 elements in the periodic table. The three p-orbitals are equivalent and can be exchanged arbitrarily.

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References

[1]	S. H. Simon Oxford Solid State Basics Oxford 2013 (ch. 5.2)
[2]	J. Barrett, A. G. Davies, D. Phillips, E. W. Abel, J. Woollins Atomic Structure and Periodicity The Royal Society of Chemistry 2007 (p. 39)
[3]	S. H. Simon Oxford Solid State Basics Oxford 2013 (ch. 5.2)
[4]	J. Barrett, A. G. Davies, D. Phillips, E. W. Abel, J. Woollins Atomic Structure and Periodicity The Royal Society of Chemistry 2007 (p. 48)
[5]	J. Barrett, A. G. Davies, D. Phillips, E. W. Abel, J. Woollins Atomic Structure and Periodicity The Royal Society of Chemistry 2007 (p. 49)