Chemistry 221

Master Class: Reprise of the Pocket NMR

2006-04-01T08:32:00.000-05:00

Why does MRI require high magnetic fields? Why is it such a low energy technique compared to X-ray?

Mathematica notebook to explore the question.

Wrapping Up: Reviewing normalization, orthogonality; A closer look at basis functions

2005-12-02T09:56:00.000-05:00

A Mathematica exercise to review the finer points of orthonormality. We explore these concepts by comparing the behaviors of Slater type orbital basis functions and Gaussian basis functions (the latter are widely used in quantum calculations of molecular wavefunctions).

Mathematica notebook with answers

The last lecture

2005-11-28T14:40:00.000-05:00

In which we say good-bye...and consider how a laser "amplifies" light.

MP3 podcast
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The truly dedicated student can build a laser by following the directions at Sam's Laser site. Lasers can be built from a number of different materials, including Jello.

Fiat Lux! Population inversion is the key to successful lasing

2005-11-21T14:34:00.000-05:00

Population inversion is a key feature of a system which be used to construct a laser. A system in thermal equilibrium follows Boltzmann's statistics, in which the number of molecules in higher energy states is smaller than the number in the lowest energy state. Lasers require that you have a non-equilibrium situation established, in which more molecules are "stuck" in an excited state than are currently in a lower energy state. This phenomenon is called population inversion. A second feature of lasers is that the emission process(the release of a photon when a molecule or atom relaxes from an excited state to a lower energy state) can be stimulated, or enhanced by the emissions from other molecules. This is where the "se" in the name comes from! (LASER = Light Amplification by Stimulated Emission of Radiation).

MP3 podcast
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Lumos! The Quantum Mechanics of Harry Potter

2005-11-18T14:29:00.000-05:00

We wrap up NMR and begin to consider the quantum mechanics behind lasers. Lasers are magic wands for chemists, making it possible to explore what happens in chemical processes on very short time scales. Lasers are ubiquitous tools in everyday life, too. Grocery store scanners and CD players use lasers to read information, an when you "burn" a CD, a laser is used to literally score the material.

MP3 podcast
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A Pocket NMR?

2005-11-16T21:20:00.000-05:00

Could you build an NMR that could fit in your pocket? The effect of magnetic field on the splitting between nuclear spin states. What would happen if you walked through a very strong magnetic field? Say a million Tesla field? Are there such fields? We propose building a pocket-sized NMR from a cow magnet. It could be done, if you're not interested in very high resolution.

MP3 podcast
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A list of very strong magnetic fields, the strongest are found in rare stars.

Magnetic Personalities: NMR

2005-11-14T20:46:00.000-05:00

The quantum mechanics of nuclear spins. How a magnetic field splits degenerate spin states of at nuclei, setting the stage for NMR.

MP3 podcast
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What's a cow magnet?
Accidents with MRIs

Degrees of Freedom

2005-11-11T15:59:00.000-05:00

The vibrational spectra of most molecules is very complex. We considered how additional lines arise in diatomic spectra including isotopic substitution and "hot bands". There are many more vibrational modes available to polyatomic molecules. How many? 3N-6!

MP3 podcast
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Check out the vibrational modes of CO₂ at Purdue's site. You need CHIME for this.
Animations of infrared vibrational modes.
and more animations

A Matter of Moment

2005-11-09T09:56:00.000-05:00

The rotational spectra of polyatomic molecules depend on the moments on inertia about the principal axes. We considered 4 cases: linear molecules, spherical tops, oblate symmetric tops and prolate symmetric tops.

We backtracked to vibration spectroscopy to discuss the Franck-Condon principle, or the principle of vertical excitation. It adds a third rule to our list: What goes up must come down; You can't always get there from here; When you go up, go straight up!

Answers to the exercise to determine the type of molecular "top".

MP3 podcast
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Out of Tune: The Effects of Anharmonicity and Centrifigual Distortion on Rotational/Vibrational Spectra

2005-11-07T22:10:00.000-05:00

We noted in our demonstration on Friday that rotation affected vibration. We quantified this, including a term in the energy to account for centrifugal distortion. The effect is small, but noticeable, as we saw with HCl. We consider the appearance of overtones in the vibrational spectrum, and the shifts in equilibrium bond length that occur as a result of the anharmonicity of the vibrational potential.

MP3 podcast
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Shake, Rattle and Roll: Simultaneous Excitation of Vibrational and Rotational States

2005-11-04T22:17:00.000-05:00

Why are there all those lines in the HCl spectrum? Why is there no line at the fundamental frequency? We consider the interplay of rotation and vibration and their respective selection rules to see why.

MP3 podcast
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Pure Vibrational Spectroscopy

2005-11-02T14:57:00.000-05:00

Using the harmonic oscillator to model vibrational energy transitions can be done, but has its limits. Consider the observed high resolution spectrum of gaseous HCl.

[Figure from hyperphysics.phy-astr.gsu.edu.]

MP3 podcast
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An Exciting Lecture: An Introduction to Molecular Spectroscopy

2005-10-31T14:47:00.000-05:00

Why does your white shirt glow under a blacklight? What makes the glow in the dark stars glow? How does a glow stick work? We look at the absorbtion and emission of light by molecules. This is an appropriate lecture for Halloween since the first meaning of "spectrum" is "ghost".

MP3 podcast
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Spinning Around: The Pauli Principle and Slater Determinants

2005-10-28T14:03:00.000-04:00

Electron spin is generally viewed as an ad hoc development in wave mechanics (though it arises naturally in other forumations, such as Dirac's). Using a general statement of the Pauli Exclusion Principle, we showed that Slater's suggestion of using wavefunctions constructed from determinants would insure that the Pauli's principle was satisfied.

Some biographical information on J.C. Slater

MP3 podcast
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Linear Variation Theory: Building a Better Wavefunction Piece by Piece

2005-10-25T20:38:00.000-04:00

Linear combinations of functions are a good way to build wavefunctions. The goal is to have "off the shelf" sets of functions that we can use to build wavefunctions for molecules. We show how linear variation theory can be used to find the variational energy of the ground AND excited states and how to find the coefficients for the linear expansion of the wavefunction.

MP3 podcast
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Mathematica notebook for lecture demonstration
worksheet
Mathematica notebook with solution to worksheet

Using Variational Theory

2005-10-23T22:03:00.000-04:00

Different trial functions yield different energies, the quality of the energy doesn't necessarily predict the quality of other properites predicted from the wavefunction (such as average position). We looked at the framework for linear variation theory, since this is the backbone of one of the standard methods for computational molecular quantum chemistry.

Mathematica notebook

MP3 podcast
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Test Drive of the Variational Theorem

2005-10-20T21:19:00.000-04:00

A Mathematica exercise based on the one-dimensional particle in the box explores the variational principle. Does a function with a lower energy necessarily do better at predicting other quantities, such as the average position of the particle within the box?

MP3 podcast
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Variational Theory Exercise Worksheet (PDF)

Variations on a Theme

2005-10-18T21:26:00.000-04:00

Though the Schrodinger equation cannot be solved exactly, robust approximate techniques exist for finding solutions to problems of interest to chemist. The variational theorem is the foundation for much of computational chemistry. Using a Mathematica notebook we explore how a simple gaussian function can be used to find an approximation to the wavefunction and the energy.

Mathematica notebook

MP3 podcast
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t-shirts

2005-10-17T15:47:00.000-04:00

Time to decide on a class t-shirt! Send your ideas to mfrancl@brynmawr.edu and I'll post them up here. Comments and suggestions?

2s Orbitals Really are Bigger than 2p and other Urban Legends of Atomic Orbitals Debunked

2005-10-17T15:31:00.000-04:00

How big is an orbital? What measures do chemists use for orbital size and how are they computed using quantum mechanics? Why would an orbital on carbon be smaller than one on lithium? Is is purely an electrostatic effect, or would the presences of other electrons change the sizes? How? We introduced the first multi-electron atomic system we will study: He. The problem? It can't be solved! Why? Electron-electron repulsion makes the Hamiltonian inseparable.

MP3 podcast
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Chemical Urban Legends: Hydrogen Atomic Orbitals

2005-10-07T09:47:00.000-04:00

We look at several common ideas about atomic orbitals:

The principal quantum number, n, controls the size of the orbital.
s orbitals have no nodes
Which is larger (extends further from the nucleus), a 2s or a 2p orbital?
Which is larger, the 2s orbital in C5+ or the 2s orbital in Li2+?

Are they all true? Use the Mathematica notebook posted below to uncover the mysteries....

Mathematica notebook
worksheet

The Mystery of s,p,d and f Revealed

2005-10-05T13:12:00.000-04:00

We rewrote the Hamiltonian for a one-electron atom in terms of the operator L². Knowing that linear operators that commute share at least one set of eigenfunctions, we tested to see if the Hamiltonian and L² did commute. They do, and so there must exist a common set of eigenfunctions. Since we already know one set of eigenfunctions for angular momentum operator, the the spherical harmonics or Y_l,m, we tried a solution to the one-electron atom Schrodinger equation of the form R(r)Y_l,m(θ,φ). Such solutions do work and allow us to derive a differential equation in a single variable, r, to solve for the radial part of the wavefunction.

We talked about the origins of the familiar orbital designations, s, p, d nd f.

MP3 podcast
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See rotatable images of s, p, d, f and g orbitals.

Morphing Quantum Spheres into Atoms: The Spherical Harmonics and Associated Legendre Functions

2005-10-03T11:48:00.000-04:00

As a step along the path to creating a quantum mechanical model of an atom, we considered the solution to the problem of a single particle moving on the surface of a sphere. We saw that the Hamiltonian for the motion could be written simply in terms of the angular momentum operator, L². The eigenfunctions of this operator are well known and called the spherical harmonics or Y_l,m. We noted that the solutions depended on two quantum numbers, l and m_l.

We wrote down the Hamiltonian for a one-electron atom (the archetype would be the hydrogen atom) and discussed the form of the potential energy (Coloumbic or electrostatic attraction). We noted that we could simplify matters by assuming that nuclear motion was very slow compared to the motion of the electrons and therefore could be (to a first approximation) ignored.

MP3 podcast
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Problem Set 5

2005-10-02T21:21:00.000-04:00

NOTE: The answer in the back of the book is computed assuming that the fundamental line is at 2559 cm-1, not the 2630 cm-1 the authors give. With thanks to Jennifer Gerfen who noticed this! The actual value is 2648.97 cm-1, as reported in the NIST database, so the authors' value is closer than the 2559 value Jennifer found in other texts.

Around and around in circles: the rigid rotor

2005-09-30T14:32:00.000-04:00

We consider one more model problem, this one concerning the rigid rotation of a diatomic molecule. Though the problem is simple compared to most molecular systems chemists are interested in, it yielded our first example of a wavefunction that was complex and not real valued. It will also provide a basis for an atomic model problem - the hydrogen atom.

MP3 podcast
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MicroTest #5

2005-09-28T21:15:00.000-04:00

You can pick any "particle" you like for the problem. Pertinent data are in the notebook we used in class and in your text.

Problem 5-22

2005-09-26T20:14:00.000-04:00

Problem 5-22 refers you to Problem 5-20 for the rms displacement. Note that the formula in 5-20 is for v=2 and question 5-22 asks about v=0! You can derive the formula for v=0 by following 5-22, but I hadn't intended that. The formula for v=0 is the same as for v=2 except it's 1/2 instead of 5/2 as the factor.

Quantum Mechanical Escapes: Tunneling Through Walls

2005-09-26T09:55:00.000-04:00

What happens when you shoot an electron at a wall? We consider the case of a particle impinging on a rectangular potential barrier to discover that under many conditions - the particle goes right through! An electron with an energy of 4 eV has about a 70% chance of going "through" a 5eV wall and coming out the other side.

MP3 podcast
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Mathematica notebook
Worksheet

Overstepping the boundaries: Harmonic Oscillators Defy the Classical Limits

2005-09-23T11:44:00.000-04:00

How can parity work for you in quantum mechanics? We see that the parity of the harmonic oscillator solutions can make some problems trivial (the average value of the position, for example). We computed the probability of a harmonic oscillator being stretched or compressed further than classical mechanics would permit, to discover that in the ground state of HCl the probability was 15% that the classical limits would be exceeded.

MP3 podcast
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Connecting the Dots: Quantum Dots and the Particle in the Box

2005-09-23T08:45:00.000-04:00

Read about how the particle in the box quantum mechanical model and quantum dots are connected in a short article from the Strouse group at UC Santa Barbara. The use of quantum dots as biosensors is being explored at Livermore National Labs. Such dyes are now commercially available!

Photo of the dyes from the gallery at theONR

Error on the MicroTest #5

2005-09-21T18:22:00.000-04:00

The boundary conditions on z in the MicroTest should be 20 to 200 - not 200 to 200 as stated.

With apologies!

mmf

Shaking Things Up: The Harmonic Oscillator

2005-09-21T09:54:00.000-04:00

Moving beyond the particle in the box, a model for simple molecular vibrations is constructed. The solutions depend on the Hermite polynomials and exhibit parity. Wavefunctions for states with an even quantum number have even parity (are symmetric about the y axis) and those with odd quantum numbers are odd. Parity considerations can simplify quantum chemical calculations.

MP3 podcast
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Mathematica notebook

The Real Product: Particle in a 3-Dimensional Box

2005-09-19T12:26:00.000-04:00

We show that the product of one-dimensional functions is indeed a solution to the 3-dimensional problem. We find the energy and show that the energy and wavefunction now depend on 3 independent quantum numbers. When the box is symmetric, for example, a cube, some energy levels are degenerate. We noted that symmetry generally leads to degeneracy, though not all degeneracy is a result of symmetry (accidental symmetry). We extended the concept of the product wavefunction to systems of more than one particle. We drew a quick concept map of where we have been in the course so far.

MP3 podcast
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Mathematica notebook - same as previous lecture

Trout Fishing in America - the band
Six

What do you get when you add three plus three?
I believe the answer is six.
And how ‘bout seven when take away one now?
I believe the answer is six.

Well, how do you do that in your head?
I would need a pencil all filled with lead,
A huge piece of paper ‘bout the size of my bed.
Now…You must be a mathematician.....

and on...to

What is the dimension of the field of complex numbers over
the real numbers, times the order of the alternating group on
three elements divided by the definite integral from zero to pi
over two of sine of X D X?

Class t-shirts

2005-09-19T09:48:00.000-04:00

If you have an idea for a class shirt - let me know in the next couple of weeks and I'll get things organized over fall break.

Mathematica Boot Camp

2005-09-18T22:05:00.000-04:00

Monday night, 7-9 pm in Park 354. Improve your Mathematica skills, whatever level you are!

Feedback

2005-09-18T21:58:00.000-04:00

Thanks for all your feedback on the course so far! Overall, you'd like me to continue using the tablet instead of the white board (even if you don't listen to the podcasts or watch the screencasts). About 1/4 of you are using the recorded materials, and are happy to have them.

The pace seems to be alright for some to slightly fast for others. If you lose the thread in lecture, be sure to stop me and ask. It's very likely that if you have lost the thread or missed a point, two other people have as well.

Interesting questions you raise:

Can I connect all the different equations?
Why if there is a node in a 2s, is it still classifed as "s"? Does the node have any significance?
Why an eigenvalue?
What does ψ measure?
Mathematica is confusing!
Why are Cheetos orange?
Where does Schrodinger's equation come from and why should I just accept it?
How to find uncertainty and probablity?
How to deal with electrons in rings in the free particle model?
What's the point of quantum mechanics?
What math should we review?
How can you predict the color of something using PIB?
What does orthonormality have to do with anything?

Watch this space for the answers!

Particles in Real Boxes

2005-09-18T21:09:00.000-04:00

A "real" quantum mechanical box - that is, one that is 3-dimensional. We consider the case of a particle confined to a rectangular parallelepiped. The Schrodinger equation for this system separates neatly into 3 one dimensional cases and we propose that the solutions to these problems are the 1-D particle in a box wavefunctions. We will verify this in the next lecture.

MP3 podcast
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Mathematica notebook - same as previous lecture

The wavefunction is a complete description of the system.

2005-09-14T21:41:00.000-04:00

Our first postulate of quantum mechanics is that the wavefunction is a complete description of the system. This is great in principle, but what are the practical details? How do we use the wavefunction to produce information that is useful to chemists? The difference between average (or expectation) values and the probability density are explored. We consider the radial distribution of electron density in 1s and 3s orbitals using Mathematica.

MP3 podcast
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Mathematica notebook

Why color are flamingos?

2005-09-12T09:28:00.000-04:00

An in-class Mathematica exercise using the particle in the box wavefunctions. Expectation values, the free-electron model and electronic transitions.

PIBWavefns.nb: Mathematica notebook to accompany the lesson.
PIBWavefnsWorksheet.pdf: Worksheet

What to "expect" from quantum mechanics?

2005-09-10T19:52:00.000-04:00

How can we use the framework of quantum mechanics to tell us something useful to chemists? The expectation value and the probability density are the keys. The eigenfunctions of the Hamiltonian are an orthonormal set. We graphed the probability density and wavefunctions for the particle in the box. We noticed that as n increased, the probabilty profile appeared more classical - a manifestation of the Bohr Correspondance Principle.

MP3 podcast
webcast
Mathematica notebook to plot the wavefunctions and probality densities

Open Hours in Park 354

2005-09-09T11:09:00.000-04:00

Open hours in Park 354 are Tuesdays 7-9pm and Wednesdays 6-8pm, beginning next Tuesday, September 13th.

Practice Problems

2005-09-09T11:06:00.000-04:00

If you are looking for practice problems, a selection is posted at the course blackboard site. Practice problems for credit are due one week after they are posted. I will post solutions one week after the problem is posted. If you would like problems of a particular sort, please let me know and I'd be happy to create some to order!

Making Up the Rules: Three Postulates of Quantum Mechanics

2005-09-07T12:54:00.000-04:00

If Hψ=eψ, why can't you just cancel the ψs? Operators, rules that change one function into another, play a key role in quantum chemistry. Each measurable quantity has a corresponding operator, and the operator, used in tandem with the wavefunction, can be used to calculate expected values of these quantities. We introduced the notion of the wavefunction as a vector in a function space and used Dirac's bra and ket notation to express the wavefunction and the expectation value.

MP3 podcast
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What happens when you confine a very small particle?

2005-09-05T14:28:00.000-04:00

Schrodinger's equation provides a way to describe the wave nature of matter, most important for the small bits of matter that concern chemists. We solve Schrodinger's equation for a model problem of a particle in a 1-D universe, confined to a small line segment, to get the wave functions and the energy. The solution requires using the boundary conditions for the problem, including the condition that the total probability of finding the particle somewhere in the universe is 1. We notice that not every wavefunction of the proper form is allowed, nor is every energy. The solutions are characterized by a quantum number "n".

MP3 podcast
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Office Hours, real and virtual/Mathematica Boot Camp

2005-09-05T12:02:00.000-04:00

Office hours are W 9-10, 12-1 and Th 9-10.

Virtual hours are S-Th 8:30 to 9:30 by IM (quantophrenic)

Anyone interested in a Mathematica boot camp later this week, please send me an e-mail with some good times.

The Rise of Quantum Mechanics: Schrodinger's Wave Equation

2005-09-02T18:54:00.000-04:00

In late 1925 Erwin Schrodinger, prompted by a question asked by Sommerfeld in a seminar Schrodinger had given, developed the wave equation. This lecture discusses the 1-dimensional, time independent Schrodinger equation for a single particle. We set up a sample problem, for a single particle trapped in an infinitely deep potential energy well and found solutions (by inspection) for the wave equation in 3 different regions (outside the box, where it is zero, and inside the box). We discussed the basic form of the Hamiltonian operator.

MP3 podcast
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Error on MicroTest 1

2005-08-31T20:26:00.000-04:00

My apologies, there is a micro problem with the micro test, as two alert students have pointed out:

The electron velocity should be 1.5 x 10^+8 m/sec.

Reminder: No Lecture

2005-08-30T19:15:00.000-04:00

Just a reminder, there is no lecture scheduled for Wednesday, August 31. See you all on Friday!

The Downfall of Classical Physics

2005-08-29T21:05:00.000-04:00

In the late 19th century the reputation of Newtonian physics with its ability to describe the macroscopic behavior of matter was beyond reproach. As the end of the century approached, scientists began to be able to make accurate observations of very small pieces of matter, such as atoms. Suddenly, the fabric of physics began to fray. What is the UV catastrophe? How much damage did it do and to who? Listen to the podcast.

MP3 podcast
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Room Change for Integrated Lab

2005-08-26T17:03:00.000-04:00

If you are enrolled in integrated lab, there has been a room change. The class is now scheduled to meet in Park 337, Tuesday 1-2.

iTunes

2005-08-26T16:53:00.000-04:00

You can subscribe to the podcast for this course through Apple iTunes. The software is free and can be downloaded from Apple for either PC or Mac. Subscribing is also free. To find the course, click on Podcasts, then Education, Higher Education. The course is titled "Introduction to Quantum Chemistry".

Sample Podcast of Lecture

2005-08-23T10:21:00.000-04:00

To hear a sample podcast of a lecture, click on the title. You can also subscribe to this podcast through its feed or through the iTunes store (software and lectures are free and run on both Mac and PC platforms). You can listen on any MP3 player, or on your computer.

How to Access Audio and Video of the Lectures

2005-08-22T14:18:00.000-04:00

This course is broadcast from Bryn Mawr College's Park Science Center as audio and video. You can also access the instructor's notes via BlackBoard.

To see a recording of the lecture, click on the link given in the lecture you would like to review. A screen will appear. You can fast forward, rewind and pause (all the control you wish you had in the live lecture, but don't!).

To hear the lecture, without video, click on the link to the audio post. Individual lectures are also available at iTunes, under Education->Higher Education->Introduction to Quantum Chemistry, or you can subscribe to the 'cast using your favorite aggregator at http://feeds.feedburner.com/Chemistry221.