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Quantum Mechanics of Infrared Spectroscopy

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Quantum Mechanics of Infrared Spectroscopy Quantum Mechanics of Infrared Spectroscopy Miah Turke, University of Chicago Copyright (c) Miah Turke 2020 Overview Infrared Spectroscopy is an incredibly versatile experimental technique that allows us to gather structural information about gaseous, liquid, or solid samples. While interpreting IR spectra is a basic practice in any undergraduate chemistry curriculum, understanding the theory requires a deeper grasp of the principles of quantum mechanics. Here we will explore the fundamental quantum mechanics that allows us to understand how IR spectroscopy works, and we will then take a deeper dive into more advanced topics that connect the theory and practice of IR spectroscopy. *This worksheet is intended for college and graduate level chemistry students (and beyond!). Introduction When a sample is irradiated by infrared light, it can either be absorbed or scattered. Both processes can be used to probe a molecule's vibrational states. We know from quantum mechanics, that the energy of matter is quantized, meaning it can only take on discrete values. Therefore, if energy is quantized, then there are set frequencies at which a molecule can vibrate. If we were to shine light on a molecule, and that light is of the correct frequency, it may be absorbed by the molecule, and can thus excite the molecule to a higher vibrational energy level. Infrared absorption spectroscopy is the process by which we irradiate a sample with infrared light, and look at the peaks in transmission. This gives us information on what energies can be absorbed by a molecule, and can then give us insight into how a given molecule moves and interacts in a system. While infrared spectroscopy can be used to understand the vibrations of a molecule, it can also be used in a more cursory manner to identify unknown substances. Certain types of bonds have characteristic frequencies at which they oscillate, so IR spectroscopy can be used to identify functional groups (certain types of bonds), and can be an invaluable aid in molecular identification. An example IR spectrum for ethanol (a popular molecule among college and graduate students alike!) is shown below. Each "dip" in transmittance corresponds to an absorbance peak. Figure 1. Infrared Spectrum of ethanol from NIST database In the following sections, we will learn how to understand the information enclosed in this spectrum, and use it to to obtain structural information about ethanol which we will generate below using the Maple Quantum Chemistry package. Warm-up: ethanol structure We first need to set the digits of our computations to 10, and then load our quantum chemistry package. > (2.1.1) > (2.1.2) We can now obtain the molecular geometry of ethanol using our Molecular Data command: > (2.1.3) > (2.1.4) > (2.1.4) Now that we have determined the geometry of ethanol, we can plot it as follows. > We will return to our investigation of ethanol later on by looking at its vibrations! Fundamental Quantum Mechanics Crash Course If you are brand new to quantum mechanics- that's okay! Vibrational spectroscopy is for everyone and we are happy to have you here. I recommend checking out the Quantum Mechanics Crash Course on youtube for an introduction into quantum mechanics, in order to get the most out of this worksheet. From there, you can explore the basics of the math below, then dive right into IR! Why does that H have a hat? In brief, quantum mechanics was developed to explain a number of phenomena that could not be explained with classical physics. This involved treating electrons as waves using wavefunctions to (1) De Broglie Wavelength ( . Because this is a wave equation, we should be able to take the second derivative with respect to x, and still get the wavefunction back. (2) This can be further simplified as follows. > (2.1.4) (3) (4) We can then divide each side by (2m) where m is mass. (5) We can now recognize that is kinetic energy, T. This means that the expression to the left of this equality gives us the kinetic energy! This doesn't account for all energy, however...we must also include potential energy, V. Lets rewrite the equation to give an expression that will yield our total energy of the system (potential+kinetic). We can also simplify this using =h/2 (6) This is the Schr dinger equation! This equation is arguably the most important equation of quantum mechanics. It is often written in a simplified fashion as follows: (7) Where (H-hat) is our Hamiltonian operator. We operate the Hamiltonian onto our wavefunction in order to obtain the wavefunction's energy. You can think of it in a way as a function, you put in your operator and wavefunction, get out your energy and the same wavefunction you put in. Generally our Hamiltonian is written like this, = (8) and when you operate it on a wavefunction, you obtain its energy as shown in the Schr dinger equation. Its bra-ket not bracket Sometimes it is easier for us to think about quantum mechanics through the lense of linear algebra, where we are not working necessarily with wavefunctions, but instead with wavevectors. We can think of our Hamiltonian as a matrix containing all of our possible energies for each energy state, and our wavefunction is merely a vector that can "select" which energy states to include in the total energy expression. Don't worry too much of grasping this right now- a deep understanding of linear algebra is not required to explore this worksheet. Because people sometimes think about wavefunctions as wavevectors, two different notations are commonly used. The "linear algebra" notation is called bra-ket notation (AKA Dirac notation). We can rewrite our Hamiltonian in bra-ket notation as follows: = E (9) where is our ket wavevector. > (2.1.4) If we wanted to solve this expression for energy, we would need to take the transpose of the wavevector, and multiply it to the other side. This means that we will now be multiplying our Hamiltonian from the left with the bra wavevector, and from the right with the ket wavevector. This gives us our energy. E= (10) This is also equal to taking the integral of the complex conjugate of the wavefunction times the wavefunction (square modulus E= (11) Expectation Values The last topic that will be useful for you to proceed with this worksheet is an understanding of expectation values. While the quantum mechanics up to this point is all fine and good, the only piece that is experimentally measurable is the energy. We can't see an operator or a wavefunction, so we require tools to obtain information from these principles that can be measured, thus proving these principles to be true. These "observables" are called expectation values in quantum mechanics. Believe it or not, you have already found an expectation value... energy! For practice, what other quantities can we observe? momentum dipole moment position and more! For example, let us find the expression for the momentum. We must use the momentum operator, which is defined as (12) We then operate this on our wavefunction, = p (13) = p (14) where p is our momentum! This can also be written in bra-ket notation: = p (15) p= (16) > (2.1.4) Theory The most logical place to begin our discussion of infrared spectroscopy is with the simplest model for diatomic vibrations: the harmonic oscillator. This model can be solved exactly, and is a great approximations for many systems. From there, we will discuss how the harmonic oscillator isn't perfect, and introduce one of many ways to deal with this using second quantization and perturbation theory. The Harmonic Oscillator Introduction We begin with the general Hamiltonian of a particle by summing its kinetic and potential energy operators, where the potential energy is modeled by Hooke's Law. Putting this Hamiltonian in the time- independent Schr dinger equation gives us equation 17. The energies that we get out of this equation are quantized as seen in equation 18 and figure 2. The harmonic potential is drawn in black, and the wavefunctions for each vibrational levels are given in purple and red. The different colors correspond to different phases of the wavefunctions. Each of the vibrational states have a corresponding quantum number, . Figure 2. Depiction of the harmonic oscillator wavefunctions with their respective quantum numbers and energies. These vibrational modes are referred to as normal modes because the excitation of any given mode does not affect the others, or in mathematical terms, these vibrational modes are orthogonal. > (2.1.4) All molecules have 3N total degrees of freedom, where N is the number of atoms in the molecule. They additionally have 3 translational degrees of freedom, and 3 rotational degrees of freedom (2 rotational degrees of freedom if the molecule is linear). The number of normal modes corresponds to the number of vibrational degrees of freedom, thus, linear molecules have 3N-5 vibrational modes, and nonlinear molecules have 3N-6. Spectroscopy can be used to look at these vibrational energy levels of a molecule, and because infrared light spans frequencies in the range that bonds oscillate, IR spectroscopy can be used to determine the frequencies of the vibrational modes of molecules. If the frequency of incident IR radiation matches a molecule's vibrational mode, it can be absorbed, allowing the molecule to excite to a different vibrational mode. Not all transitions are allowed, however. Selection Rules In order for a transition to be "IR active" it must have a transition probability that is nonzero. When a molecule absorbs light, it is subjected to an oscillating electric field. This electric field interacts with the transition dipole of the molecule.
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