H Atom Wave Functions

H Atom Wave Functions

• Last day we mentioned that H atom wave functions can be factored into radial and angular parts. We’ll directly use the radial part most in this course. (The “sigma” introduced for H wave functions in the text is for printing convenience only.)

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Probability Plots for the H Atom

• We often describe the probability of finding the electron in the H atom as a function of its position in three dimensional space. This requires an evaluation of Ψ2 and three dimensional plots. Due to the wave like properties of electrons the maximum value of r that should be used in such plots is not obvious (there is a small likelihood that the electron will be found far from the nucleus).

Final Note on Coordinates• For the motion of a particle described using

spherical polar coordinates the r, θ and φ can vary over the ranges:

• 0 ˂ θ ˂ ; 0 ˂ φ ˂ 2 and 0 ˂ r ˂ • For electron motion in an atom or molecule we

expect the most likely r values to be very small. Why? (Aside: Using Cartesian coordinates we’d have - ˂ x ˂ + , etc.)

Orbitals and Electron Density

• In practice it is customary to draw a boundary surface enclosing the smallest volume which has, say, a 95% probability of containing the electron. Chemists also speak in using these plots of electron density. The s orbitals are again a special case. The wave functions for s orbitals, the Ψ(r,ϴ,φ), have in this case no angle dependence – the probability of finding the electron somewhere in space depends “only” on the r value.

Orbitals have Different Shapes

• It follows from the previous slide that, for s orbitals, the smallest volume that will have a 95% probability (say) of containing the electron will necessarily always be a sphere. Other orbitals have associated wave functions which show dependence on all of r, ϴ and φ. As a result, these orbitals have more complex shapes. Nodal planes are seen for p orbitals as seen on the next slides (d orbitals later).

s orbitals

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FIGURE 8-24

•Three representations of the electron probability density for the 1s orbital

2s orbitals

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FIGURE 8-24

•Three-dimensional representations of the 95% electron probability density for the 1s, 2s and 3s orbitals

Three representations of electron probability for a 2p orbital FIGURE 8-27

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The three 2p orbitalsFIGURE 8-28

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Electron Spin – Another Quantum Number

• Charged particles in motion can generate magnetic fields or act as magnets. Electrons in an atom can move rapidly around the nucleus (orbital angular momentum) and can also have spin angular momentum. Experiments show that spin angular momentum is also quantized. For electrons we introduce a fourth and final quantum number, ms, which can have values of +½ or – ½.

8-9 Electron Spin: A Fourth Quantum Number Number

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FIGURE 8-32

•Electron spin visualized

Stern Gerlach Experiment - Spin

• In light of the previous slide one might be tempted to pass a beam of H atoms through a magnetic filed and see if the beam splits into two parts corresponding to the two possible ms values. In fact, an experiment analogous to this was first done with a beam of silver (Ag) atoms which also have a single unpaired electron. The result is shown on the next slide.

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The Stern-Gerlach ExperimentFIGURE 8-33

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8-10 Multi-electron Atoms

• Schrödinger equation was for only one e-.• Electron-electron repulsion in multi-

electron atoms.• Hydrogen-like orbitals (by approximation).

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Radial Probability Distributions

• A final plot, the radial probability distribution, is used to gain insight into an electrons likely location in space. Here we are describing not the likelihood that an electron will be found at a particular point in space but rather at a particular distance from the nucleus. One can imagine constructing spheres of differing sizes inside the H atom. Imagine that each of these spheres is covered with very small “boxes” (volume elements).

Radial Probability Distributions (cont’d)

• It follows that, if the electron distribution within the atom is uniform, then the “boxes” on a large sphere should be more likely to contain the electron than the (necessarily fewer) boxes on a small sphere. The number of boxes on a sphere should be proportional to the surface area. Asphere = 4πr2. The radial distribution function, P(r) then takes the form

• P(r) = 4πr2R(r)2 [We should write Rn,l(r) !]

Radial probability distributions

• FIGURE 8-35

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Zeff is the effective nuclear charge.

Paradise (and Degeneracy!) Lost ?

• Experiments show that for H, He+, Li2+ and other one electron species there are many degenerate energy levels. For example, the 2s and 2p subshells have the same energy (3s, 3p and 3d subshells also have the same energy). In many electron atoms the new electron-electron interactions cause the degeneracy of subshells to disappear. The result is portrayed (qualitatively) on the next slide.

Orbital energy-level diagram for the first three electronic shells

FIGURE 8-36

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Electron Configurations

• Electrons can be distributed amongst the subshells/orbitals of an atom in different ways –producing different electron configurations. The most stable configuration has the lowest energy – corresponding to the situation where electrons get as close to the nucleus as possible while staying as far away from each other as possible.

Electron Configurations (cont’d)

• We will deal with familiar (?) material for the rest of this lecture. The relevant concepts – Aufbau Principle, Pauli Exclusion Principle and Hund’s Rule are summarized on the next slide. Hund’s rule tells us that electrons occupy equivalent orbitals singly when possible and with their spins parallel. Do the (initially at least) singly occupied orbitals make sense in terms of the coulombic interactions between electrons? (Think, for example, of the 3 distinct p orbitals).

Electron Configurations• Aufbau process– Electrons occupy orbitals in a way that

minimizes the energy of the atom.• Pauli exclusion principle– No two electrons can have all four quantum

numbers alike.• Hund’s rule• When orbitals of identical energy (degenerate

orbitals) are available, electrons initially occupy these orbitals singly.

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Allowed Quantum Numbers• Class example: Write (a) two possible sets of

the four quantum numbers (n, l, ml and ms) for the H atom (b) the four quantum numbers for each of the two electrons in He and (c) two possible sets of the four quantum numbers for the Li atom.

Populating Orbitals

• If we continued with the previous exercise we’d see that for each value of n there is one s orbital (unique ml value), three p orbitals (three ml values) and five d orbitals (five ml values: -2, -1, 0, +1, +2). This means that s, p, d subshells can contain (at most) 2, 6 and 10 electrons respectively. Relative subshell energies comes from experiment – next slide.

The order of filling of electronic subshellsFIGURE -37

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Representing Electron Configurations

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spdf notation (condensed)

spdf notation (expanded)

spdf notation – Orbital Diagram

1s22s22p2

1s22s22px12py

1

Writing Electron Configurations

• Class examples: Use the Aufbau principle to write condensed configurations and orbital diagrams for F, F-, P, Na, Na+ and Bill Gates favourite atom. Which atoms have unpaired electrons? Can an atom with an even number of electrons have unpaired electrons?

The Aufbau process

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The Aufbau Process – Sc through Zn

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Electron Configurations and the Periodic Table

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FIGURE 8-38

Valence Shell Configurations

• The occupied shell with the highest value of n is called the valence shell. When atoms undergo chemical change electrons in the valence shell can be lost or shared with other atoms. The valence shell can also pick up electrons. Atoms with similar chemical properties often have the “same” valence shell electron configuration. For example, Li, Na, K, Rb, Cs and Fr have an ns1 valence shell configuration.

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