In this appendix we list some of the commonly needed physical constants in spectroscopy. We also calculate the emitted field intensity directly from the third-order response functions for standard experimental conditions and molecular properties. The purpose of this calculation is not to obtain a highly accurate result, but to serve as a (somewhat interesting) exercise of units. In the following, we use units of meters-kilograms-seconds (SI units) unless otherwise noted.

One of the strengths of 2D IR spectroscopy is the ability to quantitatively link experimental results to computer simulations, be it molecular dynamics simulations, quantum chemistry calculations, or ideally a combination of both on the level of mixed quantum mechanics/molecular mechanics (QM/MM) calculations. In the present chapter, we outline how such simulations are performed and present some examples with computer code that can be reproduced on a personal computer. We also describe more sophisticated models that have been developed. The motivation of the chapter is not to get the most accurate agreement with experiment, but to outline the essential concepts with working examples.

In this chapter we use the molecular dynamics simulation package Gromacs 3.3 [183] (which can be downloaded for free from http://www.gromacs. org), the quantum chemistry program Gaussian09 for electronic structure calculations [58], and simple Mathematica or C codes (together with Numerical Recipes routines [152]). All the relevant computer programs in this chapter can be downloaded from the book webpage (http://www.2d-ir-spectroscopy.com), so the reader has operational programs to start with which can then be modified at will. For each of the Mathematica programs, Matlab versions are available on the book webpage as well.

2D lineshapes: Spectral diffusion of water

Perhaps the most accurate quantities that can currently be modeled are 2D IR lineshapes.

The concepts outlined in the previous chapters lay a foundation from which new pulse sequences can be designed. We have covered two types of 2D IR spectra that can be generated with third-order pulse sequences, the so-called rephasing and non-rephasing pulse sequences. We start this chapter by presenting the third type of third-order 2D IR pulse sequence, which we term the two-quantum (2Q) pulse sequence, for reasons that will become apparent. We then improve upon this two-quantum pulse sequence by adding two more laser pulses to generate a fifth-order two-quantum coherence, which also enables 3D IR experiments. Purely absorptive 3D IR experiments are described as are transient 2D IR spectroscopies which are also fifth-order nonlinear experiments. Currently, 3D IR pulse sequences are largely unexplored, and the ones that have been implemented have only been applied to a few molecules. This chapter is written to illustrate some of the basic concepts to serve as a platform for more sophisticated experiments in the future.

Two-quantum pulse sequence

Third-order 2D IR spectra are generated from pulse sequences that interact three times with the sample. The electric field for one of the pulse interactions must be the complex conjugate of the other two, which gives rise to the third basic types of third-order 2D IR spectra: E*1E2E3, E1E*2E3, E1E2E.

In this appendix we list some textbooks that we found helpful in learning the subjects contained in this book.

Quantum mechanics and/or density matrices

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Volume II, John Wiley, New York, 1977.

G. C. Schatz andM. A. Ratner, Quantum Mechanics in Chemistry, Prentice Hall, New Jersey, 1993.

I. N. Levine, Quantum Chemistry, Pearson Prentice Hall, New Jersey, 2009.

M. D. Fayer, Elements of Quantum Mechanics,Oxford University Press, Oxford, 2001.

Electrodynamics

D. J. Griffiths, Introduction to Electrodynamics,Addison Wesley, New York, 1999.

J. D. Jackson, Classical Electrodynamics,John Wiley, New York, 1999.

Nonlinear optical spectroscopy

S. Mukamel, Principles of Nonlinear Optical Spectroscopy,Oxford University Press, Oxford, 1995.

M. Cho, Two-Dimensional Optical Spectroscopy, CRC Press, Boca Raton, 2009.

R. W. Boyd, Nonlinear Optics, Academic Press, Amsterdam, 2008.

General spectroscopy

J. L. McHale, Molecular Spectroscopy, Pearson Education, New Jersey, 1999.

W. S. Struve, Fundamentals of Molecular Spectroscopy, Wiley Inter-Science, New York, 1989.

D. A. McQuarrie, Quantum Chemistry, University Science Books, California, 1983.

The most common picture of vibrational spectroscopy is that of normal modes. However, the normal mode picture is not sufficient to describe 2D IR spectroscopy without modification, because normal modes are harmonic. Anharmonicity is necessary to create 2D IR spectra. This fact can be seen from Fig. 1.5, where the negative and positive peaks of the peak pairs would overlap and cancel if the anharmonic shifts Δij were all zero. However, describing anharmonicity with normal modes is somewhat complicated.

The purpose of this chapter is to introduce a local mode description of molecular vibrations. It is a useful description for simulating 2D IR spectra and providing a conceptual framework for visualizing the vibrations of molecules, especially of molecules built from repeating units, like proteins. In the local mode description, we treat each repeat unit as a local coordinate. In a 3D structure, these local modes will be coupled, so that they vibrate in unison, forming delocalized states which are called vibrational excitons. The coupling between local modes depends on their relative distances and orientations, and thus are probes of the 3D structure.

Vibrational excitons

The term vibrational exciton is borrowed from molecular excitons, which come from studies of closely packed aggregates of optical chromophores that create delocalized electronic excitations [36]. Vibrational excitons, which are sometimes also called vibrons, deal with vibrational rather than electronic excitations. In fact, their Hamiltonians (Eq. 6.2 below) look formally the same.

Polarization plays a central role in the measurement and interpretation of 2D IR spectra. In standard pump–probe spectroscopies, polarization has been used for many years to measure the rotational times of molecules or eliminate rotational motion from dynamics measurements. The polarization dependence of the diagonal peaks provide the same capabilities, but polarization can do much more in 2D and 3D spectroscopies. Recall Fig. 4.11 from Chapter 4, which is a schematic of a rephasing 2D IR spectrum with each peak labeled by its respective Feynman pathway. The objective of using polarization in 2D IR spectroscopy is to enhance or suppress particular Feynman pathways based on the relative angles of the transition dipoles. Selection is possible because each Feynman pathway has a different ordering of quantum states (e.g. j →j →i →i versus j →i → j →i). Thus, the ordering of polarized pulses in a pulse sequence will scale one pathway differently from another, thereby altering the intensities and phases of the diagonal and cross-peaks. By measuring these effects, the relative angles between transition dipoles can be measured [69, 190, 202]. Angles are an extremely insightful tool for monitoring the structures of molecules, perhaps more so than actual couplings. In fact, properly polarized pulses can actually eliminate the diagonal peaks from the 2D IR spectra [201], thereby better resolving the cross-peaks, which is illustrated in Fig. 5.1.

In the preceding chapters we have discussed many of the mathematical and logical underpinnings of 2D IR spectroscopy. In this chapter we focus on some of the practical aspects of implementing 2D IR spectroscopy in the laboratory and methods of processing the data. We have written this chapter assuming that our audience is already familiar with basic laser and ultra fast spectroscopic techniques and is interested in transitioning into the 2D IR field. The field of 2D IR spectroscopy is still growing and evolving. Improvements are continuously being made. Thus, some parts of this chapter may go out of date quickly, but at this time there is no commercial 2D IR spectrometer available nor is there a consensus on the best way of collecting multidimensional spectra, and so we think it prudent to discuss the design aspects of current 2D IR spectrometers, especially since the spectrometer design is tied closely with the theory.

Shown in Fig. 9.1 are some of the designs that have been invented so far for measuring 2D IR spectra. We have separated the designs into three categories, based on whether they operate in the frequency or time domains, or either. We have also divided this chapter into these same three categories. And, embedded throughout this chapter, are discussions pertaining to phase stability, data workup, and other issues pertinent to deciding how to implement 2D IR spectroscopy.

In Chapter 7 we found that a time dependent Hamiltonian (Eq. 7.2) is the source of infrared lineshapes. It also causes energy transfer between eigenstates that will alter the intensity of cross- and diagonal peaks or create completely new peaks. As for lineshapes, the time dependence of the Hamiltonian results from the fluctuations of the solvent and molecular structure. Three types of relaxation occur, all of which have been observed experimentally: population relaxation, population transfer, and coherence transfer. Figure 8.1 shows an example of a Feynman diagram for each of these cases. In contrast to the Feynman diagrams that we have seen so far, these diagrams time-propagate with more than just the usual oscillatory terms. They flip quantum states (indicated by the dotted lines and the arrows in Fig. 8.1). Population relaxation, population transfer, and coherence transfer correspond to flipping an excited population state |1〈〉1| back into the ground state |0〈〉0| (Fig. 8.1a), flipping an excited population state |1〈〉1| into another excited state |1'〈〉1'| (Fig. 8.1b), and flipping a coherence state |0〈〉1| into another coherence state |0〈〉1'| (Fig. 8.1c), respectively. A rigorous treatment of relaxation processes has become quite demanding, and we outline here only the simplest cases. For a more comprehensive discussion, we refer readers to Ref. [106].

Population transfer

As the simplest case, consider a dimer of coupled oscillators that is sitting in a fluctuating bath.

In this chapter we apply the mathematical methodology that we have developed in the preceding chapters to predict what the 1D and 2D IR spectra will look like for some generic systems. It turns out that 2D IR line shape and cross-peak patterns depend upon the experimental setup chosen to measure the 2D IR spectra, and some are better than others. Thus, this chapter is organized according to the common ways of collecting 2D IR spectra.

Linear spectroscopy

Before discussing 2D IR spectra, we illustrate the concepts of the preceding chapters by applying the methodology to linear infrared spectroscopy. For linear spectra measured using weak infrared light, and assuming that all the molecules are in their ground vibrational state before the laser pulse interacts with the sample, we only need to consider two vibrational levels and one Feynman diagram (Fig. 4.1a, b). Using this Feynman diagram, we develop the response function step by step:

• At negative times, the system is in the ground state, described by the density matrix ρ =|0〈 〉0|.

• At time t = 0, we generate a ρ10 off-diagonal matrix element of the density matrix (we also generate ρ01 element from the corresponding complex conjugate Feynman diagram, which is not necessary to consider because it is redundant). The probability that this happens is proportional to the transition dipole moment μ10.

• […]

Scientific questions encompassing both the structure and dynamics of molecular systems are difficult to address. Take the case of a folding protein, a fluctuating solvent environment or a transferring electron. In each case, one wants to know the reaction pathway, which requires time-resolving the structure. But the range of time-scales can easily span from femtoseconds to hours, depending on the system. If time-scales are slow, then exquisite structural information can be obtained with nuclear magnetic resonance (NMR) spectroscopy. If time-scales are fast, then fluorescence or absorption spectroscopy can be used to probe the dynamics with a corresponding tradeoff in structural resolution. In between, there is an experimental gap in time- and structure-resolution. The gap is even broader when the dynamics takes place in a confined environment like a membrane, which makes it especially difficult to apply many standard structural techniques.

2D IR spectroscopy is being used to fill this gap because it provides bond specific structural resolution and can be applied to all relevant time-scales (see these Special Issues [96, 143, 144] and review articles [19, 26, 27, 56, 63, 67, 80, 87, 103, 108, 142, 165, 191, 200, 208]). It has the fast time-resolution to follow electron transfer and solvent dynamics, for instance, or can be applied in a “snapshot ” mode to study kinetics to arbitrarily long time-scales. Moreover, it can be applied to any type of sample, including dilute solutions, solid-state systems, or membranes.

So far, we have implicitly assumed that the transition frequency ω01 of a vibrational mode is infinitesimally sharply defined and does not vary as a function of time. In an actual sample, this will not be the case because the solvent molecules will push and pull at the molecule, thereby deforming the molecular potential energy surface of the vibrational transition under study and hence modulating its transition frequency ω01 (Fig. 7.1). The time dependence of the transition frequency leads to pure and inhomogeneous dephasing. So far, we have considered pure dephasing by just including a phenomenological T2 damping term whenever the system is in a coherent state. In what follows, we will develop a microscopic theory that explains dephasing and relates it to the microscopic motion of the solvation shell or the molecule itself. Measuring dephasing processes turns out to be a powerful tool to study the dynamics of molecular systems in the solution phase.

Microscopic theory of dephasing

The theory we outline was originally formulated by Kubo to describe the dephasing of NMR transitions [116], but has also proved adequate for describing dephasing of vibrational transitions. Kubo's stochastic theory of lineshapes leads to a microscopic theory of dephasing. It treats the vibrational transitions quantum mechanically, and the solvent classically (it is therefore sometimes called a semiclassical theory of dephasing).

Researchers that are new to the field of 2D IR spectroscopy will find an enormous literature on the mathematical formalism behind the technique. To fully understand the capabilities of 2D IR spectroscopy, one needs to know nonlinear optics, lineshape theory, quantum mechanics and density matrices, to name a few topics (see Appendix E for recommended reading material). It can take years to learn all of these topics, but for many applications such a detailed understanding is not necessary. On a day-to-day basis, researchers in the field do not dwell on these topics, but instead rely on a few methods to design and interpret experiments. In later chapters, we focus on many aspects of the detailed mathematical formalism. In this chapter, we outline a view of 2D IR spectroscopy that we think provides intuition for the interpretation and design of 2D IR experiments based on physical phenomena. We will end up with double sided Feynman diagrams that are a useful tool for designing multiple pulse experiments.

Eigenstates, coherences and the emitted field

We begin in the same way that one would do an experiment; we shine light on a molecule. Consider a molecule like the one shown in Fig. 2.1(a). It has many vibrational modes and can be oriented in any direction in the laboratory frame. Describing the vibrational modes of this molecule is the subject of Chapter 6 and calculating the signal strength for an isotropically distributed sample is the subject of Chapter 5.