Chemistry 221

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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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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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.

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

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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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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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 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

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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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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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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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

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 Yl,m , we tried a solution to the one-electron atom Schrodinger equation of the form R(r)Yl,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.

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 Yl,m . We noted that the solutions depended on two quantum numbers, l and ml .

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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