### An Exciting Lecture: An Introduction to Molecular Spectroscopy

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Physical Chemistry I: Quantum Chemistry

Bryn Mawr College

This semester we will be covering quantum chemistry, including an introduction to quantum mechanics, molecular quantum mechanics, and spectroscopy.

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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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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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

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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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Mathematica notebook

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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)

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Variational Theory Exercise Worksheet (PDF)

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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Mathematica notebook

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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?

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:

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

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- 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 **L**^{2}. Knowing that linear operators that commute share at least one set of eigenfunctions, we tested to see if the Hamiltonian and **L**^{2} 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.

We talked about the origins of the familiar orbital designations,

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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, L^{2}. 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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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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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.