H atom excited states (2s, 2p, 3d), Shielding, Aufbau
Sanja Pudar
Study Guide Outline
1. Hydrogen Atomic Orbitals
2. Shielding
3. The Aufbau Principle
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Hydrogen Atomic Orbitals
Atomic orbitals of hydrogen atom can be obtained by solving the Schrdinger equation,which happens to
be separable into a radial and an angular components. Unfortunately, the Schrdinger equation can be
solved exactly only for simple systems, such as hydrogenic atoms. Such orbitals are classified according
to their energy levels (which depend on the total number of nodes -radial andangular) and also according
to the number of angular nodes and symmetry. The suggestions for further reading at the end of this guide
will help you visualize these orbitals better (you can also find them in almost every quantum mechanics
textbook).
ζ2
The energy of hydrogen orbitals is calculated to be E = − 2n
2 . Knowing that 2s and 2p orbitals both
have the same total number of nodes, their energy is the same, where this is regarded to as ”accidental
degeneracy.“ The same happens in case of 3s, 3p, and 3d orbitals.
In order to classify orbitals according to symmetry, we consider how they change sign when rotated about
the z axis. If there is no sign change (pz , dz2 ) , then the symmetry of orbital is σ. If there are two sign
changes in order to get to the original function (px , py , dxz , dyz ), then the symmetry is π. Finally, if there
are four sign changes in order to get to the original function (dxy , dx2 −y2 ), then the symmetry is δ. It is
very important that you familiarize yourself with the different orbitals and their properties (geometry and
symmetry) since this course depends on understanding what orbitals do to form bonds.
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Shielding
We looked at lithium as an example of a system with more than two electrons and said that the energy
of each level doesn’t depend only on the total number ofnodes anymore, but that we must evaluate each
orbital in the presence of the others to make energy comparisons between orbitals that would be degenerate
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in a hydrogenic atom. It is particularly important to evaluate the degree of electron-electron repulsion (i.e.
to which degree both electrons occupy the same space) and the degree of shielding of nuclear charge.
In the case of lithium, the 1s orbital is doubly occupied, so the third electron can go to either the 2s or a 2p
orbital. The 2p orbital has a node through the nucleus, while 1s electron density is highest at the nucleus.
Because of this, the 1s electrons completely shield the nucleus so that 2p electron experiences effective
nuclear charge (Zef f ) of +1. In addition, the 2p orbital has long lobes that would overlap less with the
already filled 1s orbital, which means that there is less electron-electron repulsion.
On the other hand, 2s orbital is nonzero at the nucleus, and therefore 1s electrons partially shield the
nucleus, so that 2s electron experiences larger effective nuclear charge. However, due to the significant
overlap of 1s and 2s electron density, the electron repulsion is greater than in a case of 2p orbital.
The calculated effective nuclear charges for 2s and 2p electrons are 1.28 and 1.00 respectively. Overall,
for this system, the 2s has a lower energy, which tells us that the larger effective charge dominates the
higher electron-electron repulsion effect. The ground state of Li atom is (1s)2 (2s)1 , followed by the first
excited state (1s)2 (2p)1 .
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The Aufbau Principle
Similar shielding effects cause splitting of states with the same n value resulting in the following ordering
of orbital energies:
2s < 2p
3s < 3p < 3d
4s < 4p < 4d < 4f
etc.
Higher number of angular nodes leads to less penetration of the core orbitals, which leads to greater
shielding by core orbitals. Ground states of most atoms can be qualitatively predicted by the Aufbau
Principle which arises from shielding effects. Below is given the order in which atomic orbitals are filled
according to the Aufbau Principle:
1s,
2s, 2p,
3s, 3p,
4s, 3d, 4p,
5s, 4d, 5p,
6s, 4f, 5d, 6p,
etc.
Notice that n-splitting is in some cases smaller than the l-splitting.
Suggested further reading:
• Excited states of H atom 5.1
2
• Hydrogenlike wave functions 5.1
• Shielding and the aufbau principle 5.2.1a to 5.2.2
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