# 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.

## Presentation on theme: "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."— Presentation transcript:

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.)

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 Ψ 2 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

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,,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, 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 Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8Slide 7 of 50 FIGURE 8-24 Three representations of the electron probability density for the 1s orbital

2s orbitals Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8Slide 8 of 50 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 Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8Slide 9 of 50

The three 2p orbitals FIGURE 8-28 Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8Slide 10 of 50

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, m s, which can have values of +½ or – ½.

8-9 Electron Spin: A Fourth Quantum Number Number Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8Slide 12 of 50 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 m s 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. Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8Slide 13 of 50

The Stern-Gerlach Experiment FIGURE 8-33 Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8Slide 14 of 50

8-10 Multi-electron Atoms Schrödinger equation was for only one e -. Electron-electron repulsion in multi- electron atoms. Hydrogen-like orbitals (by approximation). Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8Slide 15 of 50

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. A sphere = 4πr 2. The radial distribution function, P(r) then takes the form P(r) = 4πr 2 R(r) 2 [We should write R n,l (r) !]

Radial probability distributions FIGURE 8-35 Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8Slide 18 of 50 Z eff is the effective nuclear charge.

Paradise (and Degeneracy!) Lost ? Experiments show that for H, He +, Li 2+ 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 Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8Slide 20 of 50

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. Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8Slide 23 of 50

Allowed Quantum Numbers Class example: Write (a) two possible sets of the four quantum numbers (n, l, m l and m s ) 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 m l value), three p orbitals (three m l values) and five d orbitals (five m l 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 subshells FIGURE -37 Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8Slide 26 of 50

Representing Electron Configurations Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8Slide 27 of 50 spdf notation (condensed) spdf notation (expanded) spdf notation – Orbital Diagram 1s22s22p21s22s22p2 1s22s22px12py11s22s22px12py1

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 – Sc through Zn Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8Slide 31 of 50

Electron Configurations and the Periodic Table Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8Slide 32 of 50 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 ns 1 valence shell configuration.