Quantum Mechanics of Infrared Spectroscopy

Miah Turke, University of Chicago

Copyright (c) Miah Turke 2020

OverviewInfrared 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!).

IntroductionWhen 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 informationon 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.

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(2.1.2)(2.1.2)

(2.1.3)(2.1.3)

(2.1.1)(2.1.1)> >

(2.1.4)(2.1.4)

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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 structureWe first need to set the digits of our computations to 10, and then load our quantum chemistry package.

We can now obtain the molecular geometry of ethanol using our Molecular Data command:

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(2.1.4)(2.1.4)

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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 ( . Becausethis is a wave equation, we should be able to take the second derivative with respect to x, and still get thewavefunction back.

(2)

This can be further simplified as follows.

(2.1.4)(2.1.4)

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(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 bracketSometimes 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 ofour 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)(2.1.4)

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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 ValuesThe 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?

momentumdipole momentpositionand 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)(2.1.4)

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TheoryThe 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 OscillatorIntroduction

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 andenergies.

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.

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(2.1.4)(2.1.4)

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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 vibrationalmode. Not all transitions are allowed, however.

Selection RulesIn 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. The dipole transition moment can be represented as follows, where we are evaluating the possible transition from state 1 to 2, and the molecule is being irradiated in the z direction. is z component of the dipole moment.

(19)

When a vibrational mode changes the dipole moment of a molecule, that mode will be IR active. The

of the molecule, x is the displacement from the equilibrium distance.

(20)

Assuming we are working with an orthonormal basis set, we have:

(21)

This term is only nonzero when the dipole moment changes upon a change in bond distance, ,

Normal ModesRecall our structure of ethanol:

(3.1.2.1)(3.1.2.1)

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(2.1.4)(2.1.4)

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Ethanol is a nonlinear molecule consisting of 9 atoms. This means it will have 3(9)-6=21 vibrational modes. Lets calculate what these would be using Maple commands.

*Please note that this calculation may take a long time (it varies based on your computer). Feel free to take a little break while maple works its magic. Be patient, it will be worth it!

Okay, so we now have a bunch of numbers. Is this what we waited so long for? The answer is yes. BUT these numbers are actually really cool. Lets try to visualize these vibrations on an actual ethanol molecule. Using the VibrationalModeAnimation command, we can look at how the molecule is moving at each normal mode. Those who are more familiar with chemistry may know that ethanol molecules arenotorious for closely resembling dogs. I challenge you to find the vibrational mode that is most dog-like!(Perhaps one that looks like a wagging tail?).

Change the number at the end of the following command line to look at different modes.

(2.1.4)(2.1.4)

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Anharmonicity

Second QuantizationWhile the harmonic oscillator is a good approximation for the lower vibrational energy levels, as we increase in energy, the model deviates from experiment. This is called anharmonicity. As the energy of the vibrations increase, the potential curve becomes less symmetrical, and the energy levels get closer together. There are several ways to address anharmonicity, but for the sake of this worksheet, we will focus on one. We can use perturbation theory to introduce a small changes into the system (the harmonicoscillator) which can account for the anharmonicity. To do this, we will need to discuss something calledSecond Quantization which involves defining two new operators: the creation and annihilation operators.

We begin by defining the creation operator, , as a linear combination of the position ( ) and momentum ( operators. This operator creates an electron in a certain orbital or energy state. This electron can be thought of as a wavefunction.

= (22)

Similarly, the creation operator, can be defined as follows:

= (23)

This leads to the expression:

= =

(24)

Where is the commutator of and :

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= (25)

We know from a previous section (equation 1) that our Hamiltonian for the harmonic oscillator is:

(26)

This can be rewritten as follows knowing that = , and :

= (27)

When the harmonic oscillator Hamiltonian is rewritten in this way, we can substitute it into our expression for to get:

= (28)

Conversely, if we solved for , we get:

= (29)

We can then express our Hamiltonian in terms of creation and annihilation operators:

= (30)

This equation can be rewritten using the commutation relation, [ ] as follows:

= (31)

In the above equation, the harmonic oscillator quantum state (or eigenstate) is determined by . Let'stake a moment to reflect on what this operation means by defining it as This operator is "destroying" what it operates on (whatever is to the right), and then "creating" again in the same state. This can be defined as the number operator.

= (32)

Does this look familiar? This is very similar to our energy expression in equation 2! When we operate this Hamiltonian on a wavefunction, we will obtain the energy for the harmonic oscillator. This is an excellent way to derive the energy of the harmonic oscillator without dealing with calculus at all! This isshown below.

(2.1.4)(2.1.4)

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= E (33)

= (34)

Perturbation TheoryPerturbation theory is a method used to approximate the energy of a new system by perturbing the energy slightly of a known one. A common example of perturbation theory is used to better approximatethe energies of molecular vibrations. As mentioned previously, the harmonic oscillator does not match experimental results well at higher quantum states. This can be remedied by changing the Hamiltonian toaccount for this anharmonicity. To address the anharmonicity of the potential, we can take the well-known harmonic oscillator Hamiltonian, and gently perturb it by adding terms that adjust for the anharmonicity. While this may seem abstract, we will now show this mathematically.

The basic principle of perturbation theory is that we can write a Hamiltonian for an unknown system, is the unperturbed

Hamiltonian, and is the perturbation operator. (35)

Let's impose a quartic perturbation on the oscillator: = (36)

Using second quantization, we can rewrite as follows:

= (37)

We now use this Hamiltonian to solve for the energy of the perturbation. We will denote our wavefunction, as

(38)

We can save ourselves from tedious amount of math by recognizing that we need to preserve the numberof particles, and that the original states of our harmonic oscillator Hamiltonian are orthogonal. This means we only need to write the terms from the expansion of the Hamiltonian that have 2 of each operator.

= (39)

We can simplify this expression further by rewriting the creation and annihilation operators as follows:

(40)

(41)

This then becomes:

(2.1.4)(2.1.4)

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= (42)

(43)

Now going back to our original energy expression, we write our first order in perturbative correction to the harmonic oscillator energy!

= (44)

+ (45)

Applications of IR SpectroscopyThere are many applications of IR spectroscopy. Arguably the most common application is molecular characterization. Many students perform IR in organic chemistry labs as a supplement to NMR or as a means to identify a sample. The signature vibrational modes of organic functional groups make it easy toidentify if your sample has those functional groups.

While it may seem like IR is only used in organic chemistry labs for pedagogical /molecular identification purposes, IR has many useful applications in the "real world" i.e. in industry, government, and academic labs.

Pulse Oximeters

IR has a number of medical applications, one of them being near-IR spectroscopy (NIRS). This technique is used to measure the oxygenation, or amount of oxygen through tissue . You have probably had a pulse oximeter on your finger at the doctors office at some point to measure your blood oxygenation- this is a form of NIR spectroscopy! But how does it work?

These oximeters shine near infrared light through your skin, and are absorbed by the hemoglobin in yourblood. This hemoglobin can adopt two different conformations: one when it is bound to oxygen, and another when it is not. Hemoglobin absorbs different wavelengths of light in each of its conformations, so the relative concentrations of each conformation can be compared using the principles of Beer's law, giving you the percent of oxygen dissolved in your blood

NIR light penetrates the layers of the skin more effectively than IR light because of its shorter wavelength (higher energy). NIR spectroscopy uses light of a slightly higher energy than IR to probe a sample's vibrational overtones. These overtones correspond to "forbidden transitions" according to the quantum mechanical harmonic oscillator selection rules shown above.

Because these transitions are "forbidden", they happen less frequently, making this a less sensitive technique.

Blood Alcohol Testing

(2.1.4)(2.1.4)

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We have all probably heard of breathalyzer tests used to measure the blood alcohol content of individuals, particularly those suspected of drunk driving. The breathalyzer was invented in 1953 and sits alongside a blood and urine test. All of these methods are effective, but with the increase in alcoholism, present a greater biohazard risk .

A less invasive technique was developed in the last 20 years that utilizes diffuse reflectance near-infrared spectroscopy through a person's skin.

While this is a more grim side to our beloved ethanol molecule, it shows how infrared spectroscopy can be applied creatively to fix a problem facing the general public.

Discussion and ConclusionsThis worksheet doesn't begin to describe all of the applications of IR or ways of understanding its theory. While its basis is in the harmonic oscillator model in quantum mechanics, it can also be derived in second quantization. While the harmonic oscillator model is excellent for a lot of scenarios, it is not perfect and begins to fail at higher energy levels. This can be remedied by addressing the anharmonicity of the potential. There are many different ways to address this, but we discussed using perturbation theory here as a means to connect some higher-level quantum mechanics. Infrared Spectroscopy is vastlyapplicable and is still commonly used for characterization of samples and understanding the energetic landscape of the vibrations of a system.

Selected References1) Ethanol. https://webbook.nist.gov/cgi/cbook.cgi?ID=C64175&Type=IR-SPEC&Index=2 (accessed Mar 3, 2020).2) McQuarrie, D. A.; Simon, J. D. Physical chemistry a molecular approach; Viva Books: New Delhi, 2015.3) Townsend, J. S. A modern approach to quantum mechanics; Viva Books: New Delhi, 2017.4) Tom Lancaster and Stephen J. Blundell,Quantum Field The-ory for the Gifted Amateur, (Oxford University Press, 2014) 5) Moerman, A.; Wouters, P. Near-infrared spectroscopy (NIRS) monitoring in contemporary anesthesiaand critical care. https://www.ncbi.nlm.nih.gov/pubmed/21388077 (accessed Mar 3, 2020).6) Grabska, J.; Czarnecki, M. A.; Be, K. B.; Ozaki, Y. Spectroscopic and Quantum Mechanical Calculation Study of the Effect of Isotopic Substitution on NIR Spectra of Methanol. The Journal of

7) Iowa Head and Neck Protocols. https://medicine.uiowa.edu/iowaprotocols/pulse-oximetry-basic-principles-and-interpretation (accessed Mar 3, 2020).8) Today's DUI and Its Evolution. https://www.idrivesafely.com/defensive-driving/trending/todays-dui-and-its-evolution (accessed Mar 3, 2020).