Electronic Structure of Atoms

Orbitals in many-electron atoms

The hydrogen atom is a simple system having only one electron.

The quantum mechanical description of the hydrogen atoms places all subshells (i.e. l quantum number, or the s, p, d and f subshells) with the same principle quantum number (n) on the same energetic level.

An atom with more than 1 electron is called a many-electron atom.

Although the shape of electronic orbitals for many-electron atoms are the same as those for the hydrogen atom, the presence of more than 1 electron influences the energy levels of the orbitals (due to electron-electron replusion).

For example, the 2s orbital is a lower energy state than the 2p orbital in a many-electron atom: (note: this is a qualitative representation for an "average" many-electron atom)

In a many-electron atom, each electron is simultaneously:

• attracted to the protons in the nucleus
• repelled by other electrons (like-charge repulsion)

What is the average environment, created by the nucleus and all the other electrons in the atom, which is "felt" by a particular electron in the atom?

The net positive charge attracting the electron is called the effective nuclear charge

• Any electron density between the nucleus and the electron of interest will reduce the nuclear charge acting on that electron
• The effective nuclear charge (Z_eff) equals the number of protons in the nucleus (Z), minus the average number of electrons (S) that are between the electron in question and the nucleus

• The positive charge "felt" by the outer electrons is always less than the full nuclear charge (inner electrons "screen" the nuclear charge).

The extent to which an electron will be screened by the other electrons depends on the shape of the electron distribution as we move out from the nucleus

• Probability of being closer to the nucleus (based on orbital shapes) is as follows:

• For the 3^rd principle quantum number, for example, the 3s electrons experience the least shielding and the 3d electrons the most
• Conversely, the 3s electrons experience a greater Z_eff and the 3d electrons the least

The energy of an electron depends on Z_eff

• Because Z_eff is larger for 3s electrons (in the above n=3 example) they have a lower energy than 3p electrons (which in turn have lower energy than 3d electrons)

Note: all the orbitals of a give sub-shell still have the same energy level (e.g. all the 3d orbitals (with different m_l quantum values)

The sodium atom has 11 electrons, two in a 1s orbital, two in a 2s orbital, six in 2p orbitals and one in a 3s orbital. As far as the electrons in s type subshells, which experiences the smallest effective nuclear charge (Z_eff)?

Answer: the outermost electron, or the one in the 3s orbital

• What determines the orbitals in which the electrons reside?
• How do the electrons populate the available orbitals?

• Lines which were thought to be single lines actually were composed of two very closely spaced lines
• Thus, there were twice as many energy levels as there were "supposed" to be

It was proposed (Uhlenbeck and Goudsmit, 1925) that electrons have yet another quantum property called electron spin:

• A new quantum number for the electron called the electron spin quantum number, or m_s
• m_s has a value of +1/2 or -1/2
• The electron spin quantum number characterized the "direction of spin" of the electron
• A spinning charge produces a magnetic field
• The opposite spins produce opposite magnetic fields which results in the splitting of the line spectrum into two closely spaced lines

Electron spin is crucial for understanding the electron structures of atoms:

• The Pauli exclusion principle (Wolfgang Pauli, 1925) states that no two electrons in an atom can have the same set of four quantum numbers (n, l, m_l and m_s)
• For a given orbital (e.g. 2p_z) the values of n, l and m_l are fixed. Thus, if we want to put more than one electron into an orbital we must assign unique values to the magnetic spin (m_s quantum number)
• the m_s quantum number can only have two values (+1/2, and -1/2) therefore, only two electrons at most can occupy the same orbital, and they have opposite values for magnetic spin

What are the consequences of magnetic spin quantum number and the Pauli exclusion principle?

• If we know the number of electrons in an atom we can assign probable quantum numbers and know something about their orbital shapes
• Provides an understanding for the periodic nature of the elements