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Lecture 3
Many - electron atoms
The orbital approximation
Putting electrons into orbitals similar to those in
the hydrogen atom gives a useful way of
approximating the wavefunction of a manyelectron atom.
Electron spin
Electrons have an intrinsic rotation called spin,
which may point in only two possible
directions, specified by a quantum number
ms. Two electrons in the same orbital with the
opposite spin are paired. Unpaired electrons
give rise to paramagnetism.
ms can take only one of two values: +1/2 or -1/2
Pauli exclusion principle
When the spin quantum number ms is included,
no two electrons in an atom may have the
same set of quantum numbers. Thus a
maximum of two electrons can occupy any
orbital .
Effective nuclear charge
The electrostatic repulsion between electrons
weakens their binding in an atom; this is
known as screening or shielding. The combined
effect of attraction to the nucleus and
repulsion from other electrons is incorporated
into an effective nuclear charge.
The electrostatic repulsion between negatively
charged electrons has a large influence on the
energies of orbitals. Thus the ionization energy of a
neutral helium atom (two electrons) is 24.58 eV
compared with 54.40 eV for that of He+ (one
electron). The effect of repulsion is described as
screening or shielding. The combined effect of
attraction to the nucleus and repulsion from
other electrons gives an effective nuclear charge
Zeff, which is less than that (Z) of the ‘bare’
nucleus.
ε(ionization energy) = - Z2effR/n2
σ = Z –Zeff . Where σ is the screening constant
Zeff = Zactual - (electron/electron repulsion effect)
Orbital Penetration and Shielding Effects
The radial distribution function of s orbitals is said to penetrate the
core electron density. The probability of finding s orbital electrons
from the valence shell is significant in the core. Because of this, the
effective nuclear charge felt is higher, and thus the energy level of
the orbital is lower.
This makes the filling order: 3s, then 3p, then 3d.
This affects the order of filling between the 4s and 3d orbitals as well.
Since the 4s orbital allows more penetration of the electron close to
the nucleus, it is favoured over the 3d orbital for filling.
The filling order becomes:
1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p, 8s
The Aufbau Principle
The Aufbau principle states (quite simply) that electrons will
preferential occupy the lowest energy orbital possible, and
therefore when assigning the electrons of a given atom to
subshells, one starts at the lowest empty orbital (1s) and
continues to fill in order of increasing energy until all of the
electrons for that atom are accommodated.
Because not all electrons feel the same “pull” from the
nucleus (Zeff), the order in which the subshells are filled rely
on both the principal quantum number n, and the angular
quantum number l.
Hund’s Rule
The Aufbau principle describes how subshells must be
filled with respect to energy, but it doesn’t address the
order of filling of the individual orbitals.
Hund’s Rule of Maximum Multiplicity states that orbitals
should be filled to give the maximum number of
unpaired electrons:
E.g. What is the orbital filling pattern for
carbon?
Given the Aufbau Principle, carbon has: 1s2 2s2
2p2. How are the p orbitals filled?
Two possible solutions arise: