Chemistry 221

Master Class: Reprise of the Pocket NMR

noreply@blogger.com (Michelle M Francl) — Sat, 1 Apr 2006 08:32:00 -0500

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

noreply@blogger.com (Michelle M Francl) — Fri, 2 Dec 2005 09:56:00 -0500

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

noreply@blogger.com (Michelle M Francl) — Mon, 28 Nov 2005 14:40:00 -0500

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

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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

noreply@blogger.com (Michelle M Francl) — Mon, 21 Nov 2005 14:34:00 -0500

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).

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Lumos! The Quantum Mechanics of Harry Potter

noreply@blogger.com (Michelle M Francl) — Fri, 18 Nov 2005 14:29:00 -0500

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.

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A Pocket NMR?

noreply@blogger.com (Michelle M Francl) — Wed, 16 Nov 2005 21:20:00 -0500

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.

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

Magnetic Personalities: NMR

noreply@blogger.com (Michelle M Francl) — Mon, 14 Nov 2005 20:46:00 -0500

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

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What's a cow magnet?
Accidents with MRIs

Degrees of Freedom

noreply@blogger.com (Michelle M Francl) — Fri, 11 Nov 2005 15:59:00 -0500

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!

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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

noreply@blogger.com (Michelle M Francl) — Wed, 9 Nov 2005 09:56:00 -0500

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".

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

noreply@blogger.com (Michelle M Francl) — Mon, 7 Nov 2005 22:10:00 -0500

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.

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

noreply@blogger.com (Michelle M Francl) — Fri, 4 Nov 2005 22:17:00 -0500

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.

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Pure Vibrational Spectroscopy

noreply@blogger.com (Michelle M Francl) — Wed, 2 Nov 2005 14:57:00 -0500

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.]

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An Exciting Lecture: An Introduction to Molecular Spectroscopy

noreply@blogger.com (Michelle M Francl) — Mon, 31 Oct 2005 14:47:00 -0500

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".

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Spinning Around: The Pauli Principle and Slater Determinants

noreply@blogger.com (Michelle M Francl) — Fri, 28 Oct 2005 14:03:00 -0400

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

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Linear Variation Theory: Building a Better Wavefunction Piece by Piece

noreply@blogger.com (Michelle M Francl) — Tue, 25 Oct 2005 20:38:00 -0400

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.

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

Using Variational Theory

noreply@blogger.com (Michelle M Francl) — Sun, 23 Oct 2005 22:03:00 -0400

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

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Test Drive of the Variational Theorem

noreply@blogger.com (Michelle M Francl) — Thu, 20 Oct 2005 21:19:00 -0400

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?

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Variational Theory Exercise Worksheet (PDF)

Variations on a Theme

noreply@blogger.com (Michelle M Francl) — Tue, 18 Oct 2005 21:26:00 -0400

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

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t-shirts

noreply@blogger.com (Michelle M Francl) — Mon, 17 Oct 2005 15:47:00 -0400

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

noreply@blogger.com (Michelle M Francl) — Mon, 17 Oct 2005 15:31:00 -0400

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.

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Chemical Urban Legends: Hydrogen Atomic Orbitals

noreply@blogger.com (Michelle M Francl) — Fri, 7 Oct 2005 09:47:00 -0400

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

noreply@blogger.com (Michelle M Francl) — Wed, 5 Oct 2005 13:12:00 -0400

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.

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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

noreply@blogger.com (Michelle M Francl) — Mon, 3 Oct 2005 11:48:00 -0400

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.

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Problem Set 5

noreply@blogger.com (Michelle M Francl) — Sun, 2 Oct 2005 21:21:00 -0400

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

noreply@blogger.com (Michelle M Francl) — Fri, 30 Sep 2005 14:32:00 -0400

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.

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