Lecture #6

 CURMUDGEON GENERAL'S WARNING. These "slides" represent highlights from lecture and are neither complete nor meant to replace lecture. It is advised not to use these as a reliable means to replace missed lecture material. Do so at risk to healthy academic performance in 09-105.
Lecture Outline Many-electron systems

Pauli Exclusion Principle

Photoelectron spectra, ionization energies and Zeff

Hund's Rule

Electron configurations

Another mathematical consequence of wave behavior can be expressed in a simple statement. This is crucial as soon as we deal with more than one electron.

Illustrating the Pauli Principle for two electrons.
Continuing with our illustratiion of two electrons.
More discussion on two electrons.
A reminder that an electron in a 2s orbital has a different distribution with respect to the nucleus (origin) than when it is in the 2p orbital.
A 2s electron will be attracted by a greater net positive charge than a 2p electron because of the different effectiveness of the screening by inner core electrons.
Unlike one electron systems where the 2s and 2p orbitals have exactly the same energy, in many-electron systems, the 2s is lower in energy than the 2p.
Recalling that the photoelectric effect can be exploited to obtain "binding energies" of an electron to some anchor (metal surface, gaseous molecule or atom), we can look at binding energies of an electron.
When we go from H to He, despite the fact that the nuclear charge has doubled and that Bohr's energy formula suggests the binding energy might then have quadrupled, it increases by only 80%. You should be able to show that the effective nuclear charge holding a 1s electron in He is about 1.3.
For element number 3, we note two photoelectron peaks, one from each of the orbitals containing electrons. Depending on which single electron is ejected by the photon, the electron has one of two possible final kinetic energies.
Boron is element 5. There are 5 electrons distributed as: two in the inner core, 1s orbital; 2 in the valence 2s orbital; and 1 in the valence 2p orbital. The latter is easiest to remove and, when kicked out by the photon, shows up with the greatest kinetic energy therefore.
The enery levels of the one electron in the hydrogen atom.
The energy levels of the various s-states in a one-electron system. The energy needed to remove an electron from hydrogen in its "ground state" is shown. This can be determined through photoelectron spectroscopy (an application of the photoelectric effect), for example.
From the values of the electrons's energies, you can also get the photon energies (and wavelengths) involved in transitions between initial and final electron states.
Consider lithium's ground state (Z=3). The "inner core" of electrons comrpises the filled 1s shell. The "outer electrons" are the valence electrons. In this case, the 2s electron is the valence electron for lithium.
We can indicate how electrons are placed in orbitals by configuration diagrams. The one shown here corresponds to the ground state for lithium (Z=3).
Because the effective nuclear charge is determined by how much the nucleus is screened by other electrons, the 2p orbitals are not as tightly bound as the 2s orbital. This is the ground state for 5-electron systems.
The arrangement of orbitals in increasing energy can be shown horizontally too, as in the text. For the sixth electron, there are several locations to chose from. This is the lowest energy choice, the most stable.
This arrangement is another possible choice for 6 electrons, but is an excited state for a 6-electron system.
The arrangement of orbitals in increasing energy can be shown horizontally too, as in the text. For the sixth electron, there are several locations to chose from. This is the lowest energy choice, the most stable.
This arrangement is another possible choice for 6 electrons, but is an excited state for a 6-electron system.
Here are seven and eight electron systems (in addition to six) corresponding to ground state arrangements of electrons in nitrogen and oxygen. On the last line is written another form of the electron configuration for atomic oxygen in its ground state.
Other electron configuration examples: fluorine and sodium.
Excited states for the valence electron in lithium
The idea that an effective nuclear charge can embody all the various "penetration" and "screening" effects due to the many electrons around an atomic nucleus. Penetration refers to an electron having some of its density inside the region where inner core electrons are most likely to be found. Screening refers to the reduction of positive charge felt by an outer electron due to the presence of inner, negatively charged electrons.
The ionization energy of the highly excited valence electron, starting in the 5s state of lithium, corresponds very closely to the ionization energy of a hydrogen atom which has Z = 1. This is because the lithium 5s electron is far away from the nucleus with charge +3 and the inner core of electrons which contribute a charge of -2 and through which the far-away electron penetrates only very slightly.
That excited valence electron, if in a p-orbital, penetrates the inner core even less than if an s-orbital. The effective nuclear charge is much closer to unity. The effectiveness of penetration decreases as the electron in question moves to higher and higher orbitals. The transition shown corresponds to 2p --> 2s and involves red light.