Multi-Electron Atoms

While Quantum Theory gives exact equations describing the H-atom, which has only one electron, it runs into problems trying to give exact equations of atoms with many electrons. This is because, in addition to the electrostatic attraction between the electron and the positively charged nucleus, there are electrostatic repulsions between electrons. The problem starts to get complicated quickly. In spite of this problem, approximate solutions can be obtained, which can, in fact, be quite accurate. For a multi-electron atom, the energy of a particular electron in the atom is given by

This looks the same as for a single electron atom except that now we use an effective charge for the positively charged nucleus. The effective charge is reduced from the full charge due to the shielding of the nuclear charge by other electrons in the atom.

The effective nuclear charge equates to the number of protons in the nucleus, Z, minus the average number of electrons, S, between the nucleus and the electron of interest.

In a multi-electron atom, it turns out that the effective charge, Z_eff, decreases with increasing value of ℓ, the azmuthial quantum number. This is because electrons in the s-orbital have a greater probability of being near the nucleus than a p-orbital, so the s-orbital is less shielded than a p-orbital. Likewise, a p-orbital is less shielded than a d-orbital.

In a multi-electron atom, the energy of an orbital increases with increasing value of ℓ for a given value of n.

• Electron shell: All the orbitals with the same value of n
• Electron subshell: All the orbitals with the same value of n and ℓ. Electrons in the same subshell are degenerate (i.e., have the same energies).

Aufbau Principle

The energy structure of a many-electron atom is obtained by filling the orbitals one-electron at a time, in order of increasing energy starting with the lowest energy. This is called the Aufbau principle.

The ordering of orbital energy levels is

These are two ways to indicate the electronic structure of an atom: the Electronic Configuration and the Orbital Diagram.

1. Electronic Configuration

e.g., for the H-atom, the electronic configuration is 1s¹. This notation is compact, and describes how the electrons are distributed with principal, n, and azimuthal, ℓ, quantum numbers, but does indicate the magnetic, m_ℓ and spin, m_s quantum numbers for the electrons.

2. Orbital Diagram

Orbital Diagrams give a more complete indication of the electron quantum numbers. Each orbital represented by a box and each electron by a half-arrow.

For example Hydrogen has 1 electron. That electron goes into the lowest energy orbital, that is the 1s orbital. Thus, we write...

For Boron we start with 5 electrons. Again, we start by filling the lowest energy orbital 1s, then the 2s orbital, and finally putting one electron in the 2p orbital...

Now let's look at Carbon, which has 6 electrons.

• Hund's Rules: Electrons occupy different orbital or a given subshell with spins in the same direction before spin pairing occurs.

Therefore for carbon we have

Another example, Neon has 10 electrons.

For Neon, both the n=1 and n=2 shells are completely full.

Let's look at Sodium, which has 11 electrons.

Na has one electron in its outermost shell. For convenience, we often represent the electron configuration of the closed shells with the corresponding noble gas symbol. So for sodium, we would write:

In fact, we distinguish between electrons depending on whether they are part of the closed shell or in the outermost shell.

• Valence Electrons: Electrons in the outermost shell (principal quantum level) of an atom.
• Core Electrons: Electrons in the inner closed shells.

If you look at the periodic table, you will notice that elements in the same group have the same number of valence electrons. That is the main reason why elements in the same group have such similar chemical and physical properties.

Let's look at Argon, which has 18 electrons. It has the configuration

Now you might be tempted for Potassium (the next element) to put the extra electron in the 3d orbital, however, it turns out that the 4s orbital is slightly lower in energy than the 3d, so the electron configuration of Potassium is:

After the 4s orbital is filled, then you can start to fill the 3d orbital.

For example, Titanium has 22 electrons. Its configuration is

The triangular diagram below can be used to simplify memorizing the order in which the orbitals are filled.