# Assessing the Molecular Specificity and Orientation Sensitivity of Infrared, Raman, and Vibrational Sum-Frequency Spectra

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Molecular and Ensemble Response Functions

#### 2.2. Accessing Elements of the Response Functions Using Polarized Light

**IR absorption.**In an infrared absorption experiment, one measures diagonal elements of the rank-2 tensor Ⅱm$\left\{{\chi}_{II}^{\left(1\right)}\right\}$ that originates from a sum and orientational average over Ⅱm$\left\{{\alpha}_{II}^{\left(1\right)}\right\}$. As shown in Figure 3, these quantities have nine elements, but can almost always be diagonalized into three principle components. This is the origin of the typically single-subscripted refractive index and absorption coefficient. If we consider $\mathbf{D}(\theta ,\varphi ,\psi )$ to be the $3\times 3$ direction cosine matrix (DCM) incorporating the Euler angles $\theta $, $\varphi $, and $\psi $ defined in Figure 1, we can then carry out the projection [26,46]

**Sum frequency generation.**In a vibrational SFG experiment, we can independently control the polarization of the incoming visible and infrared beams, and select a component of the emitted SFG field polarization. This enables measurement of all non-zero components of the 27-element rank-3 tensor ${\chi}_{IJK}^{\left(2\right)}$. From the molecular response, we can again project into the laboratory frame to obtain

**Raman scattering.**The Raman scattering process is the most complex and interesting among the techniques we compare, owing to the four-dimension response function. In a spontaneous Raman experiment, although we probe components of the 81-element rank-3 tensor ${\chi}_{IJKL}^{\left(3\right)}$, only the elements ${\chi}_{IJIJ}^{\left(3\right)}$ are accessible. This readily lends itself to the contracted notation as shown in Figure 3, where the transition polarizability $\overline{\alpha}$ (a rank 2 tensor with dimensions $3\times 3$ are used. In analogy to our description of the IR absorption experiment, we note that the transformation from molecular to laboratory coordinates can then be carried out on ${\alpha}^{\left(3\right)}$ directly [48]:

## 3. Methods

#### 3.1. Generation of the Candidate Spectra

#### 3.2. Linear Programming

#### 3.3. Construction of Test Cases

## 4. Results and Discussion

#### 4.1. Known Scaling Factors

#### 4.2. Arbitrary Scaling Factors

#### 4.3. Exploring the Origins of Orientation Sensitivity

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DCM | direction cosine matrix |

IR | infrared |

LP | linear programming |

SFG | sum-frequency generation |

## References

- Ward, I.M. Determination of Molecular Orientation by Spectroscopic Techniques. Adv. Poly. Sci.
**1985**, 66, 81–115. [Google Scholar] - McHale, J.L. Molecular Spectroscopy; Pearson Education: New York, NY, USA, 2008. [Google Scholar]
- Kliger, D.S.; Lewis, J.W.; Randal, C.E. Polarized Light in Optics and Spectroscopy; Academic Press, Inc.: San Diego, CA, USA, 1990. [Google Scholar]
- Pelletier, I.; Laurin, I.; Buffeteau, T.; Pézolet, M. Determination of Molecular Orientation in Biaxially Oriented Ultrathin Films. J. Phys. Chem. B
**2004**, 108, 7162–7169. [Google Scholar] [CrossRef] - Umemura, J.; Kamata, T.; Kawai, T.; Takenaka, T. Quantitative Evaluation of Molecular Orientation in Thin Langmuir–Blodgett Films by FT-IR Transmission and Reflection-Absorption Spectroscopy. J. Phys. Chem.
**1990**, 94, 62–67. [Google Scholar] [CrossRef] - Brunner, H.; Mayer, U.; Hoffmann, M. External Reflection Infrared Spectroscopy of Aniostropic Adsorbate Layers on Dielectric Substrates. Appl. Spectrosc.
**1997**, 51, 209–217. [Google Scholar] [CrossRef] - Chollet, P.A.; Messier, J.; Rosilio, C. Infrared Determination of the Orientation of Molecules in Stearamide Monolayers. J. Chem. Phys.
**1976**, 64, 1042–1050. [Google Scholar] [CrossRef] - Debe, M.K. Extracting Physical Structure Information from Thin Film Organic Films with Reflection Absorption Infrared Spectroscopy. J. Appl. Phys.
**1984**, 55, 3354–3366. [Google Scholar] [CrossRef] - Dluhy, R.A. Quantitative External Reflection Infrared Spectroscopic Analysis of Insoluble Monolayers Spread at the Air–Water Interface. J. Phys. Chem.
**1986**, 90, 1373–1379. [Google Scholar] [CrossRef] - Greenler, R.G. Infrared study of adsorbed molecules on metal surfaces by reflection techniques. J. Chem. Phys.
**1966**, 44, 310–315. [Google Scholar] [CrossRef] - Hasegawa, T.; Takeda, S.; Kawaguchi, A.; Umemura, J. Quantitative Analysis of Uniaxial Molecular Orientation in Langmuir-Blodgett Films by Infrared Reflection Spectroscopy. Langmuir
**1995**, 11, 1236–1243. [Google Scholar] [CrossRef] [Green Version] - Mendelson, R.; Brauner, J.W.; Gericke, A. External Infrared Absorption Spectrometry of Monolayer Films at the Air-Water Interface. Annu. Rev. Phys. Chem.
**1995**, 46, 305–333. [Google Scholar] [CrossRef] - Sourisseau, C. Polarization Measurements in Macro- and Micro-Raman Spectoscopies: Molecular Orientations in Thin Films and Azo-Dye Polymer Systems. Chem. Rev.
**2004**, 104, 3851–3892. [Google Scholar] [CrossRef] [PubMed] - Tanaka, M.; Young, R.J. Polarised Raman Spectroscopy for the Study of Molecular Orientation Distributions in Polymers. J. Mater. Sci.
**2006**, 41, 963–991. [Google Scholar] [CrossRef] - Bower, D.I. Investigation of Molecular Orientation Distributions by Polarized Raman Scattering and Polarized Fluorescence. J. Poly. Sci.
**1972**, 10, 2135–2153. [Google Scholar] [CrossRef] - Lagugné Labarthet, F. Polarized Measurements in Raman Microscopy. Annu. Rep. Prog. Chem.
**2007**, 103, 326–350. [Google Scholar] [CrossRef] - Richard-Lacroix, M.; Pellerin, C. Novel Method for Quantifying Molecular Orientation by Polarized Raman Spectroscopy: A Comparative Simulations Study. Appl. Spectrosc.
**2013**, 67, 409–419. [Google Scholar] [CrossRef] [PubMed] - Richard-Lacroix, M.; Pellerin, C. Accurate New Method for Molecular Orientation Quantification Using Polarized Raman Spectroscopy. Macromolecules
**2013**, 46, 5561–5569. [Google Scholar] [CrossRef] - Tsuboi, M.; Benevides, J.M.; Tomas, G.J., Jr. Raman Tensors and their Application in Structural Studies of Biological Systems. Proc. Jpn. Acad. Ser. B
**2009**, 85, 83–97. [Google Scholar] [CrossRef] - Yang, S.; Michielsen, S. Orientation Distribution Functions Obtained via Polarized Raman Spectroscopy of Poly(ethylene terephthalate) Fibers. Macromolecules
**2003**, 36, 6484–6492. [Google Scholar] [CrossRef] - Takenaka, T.; Fukuzaki, H. Resonance Raman Spectra of Insoluble Monolayers Spread on a Water Surface. J. Raman Spectrosc.
**1979**, 8, 151–154. [Google Scholar] [CrossRef] - Takenaka, T.; Nakanaga, T. Resonance Raman Spectra of Monolayers Adsorbed at the Interface between Carbon Tetrachloride and an Aqueous Solution of a Surfactant and a Dye. J. Phys. Chem.
**1976**, 80, 475–480. [Google Scholar] [CrossRef] - Takenaka, T. Effect of Electrolyte on the Molecular Orientation in Monolayers Adsorbed at the Liquid–Liquid Interface: Studies by Resonance Raman Spectra. Chem. Phys. Lett.
**1978**, 55, 515–518. [Google Scholar] [CrossRef] - Nakanaga, T.; Takenaka, T. Resonance Raman Spectra of Monolayers of a Surface–Active Dye Adsorbed at the Oil–Water Interface. J. Phys. Chem.
**1977**, 81, 645–649. [Google Scholar] [CrossRef] - Morita, A. Theory of Sum Frequency Generation Spectroscopy; Springer: Singapore, 2018. [Google Scholar]
- Hall, S.A.; Jena, K.C.; Covert, P.A.; Roy, S.; Trudeau, T.G.; Hore, D.K. Molecular-Level Surface Structure from Nonlinear Vibrational Spectroscopy Combined with Simulations. J. Phys. Chem. B
**2014**, 118, 5617–5636. [Google Scholar] [CrossRef] [PubMed] - Richmond, G.L. Molecular Bonding and Interactions at Aqueous Surfaces as Probed by Vibrational Sum Frequency Spectroscopy. Chem. Rev.
**2002**, 102, 2693–2724. [Google Scholar] [CrossRef] [PubMed] - Bain, C.D. Sum-Frequency Vibrational Spectroscopy of the Solid/Liquid Interface. J. Chem. Soc. Faraday Trans.
**1995**, 91, 1281–1296. [Google Scholar] [CrossRef] - Lambert, A.G.; Davies, P.B.; Neivandt, D.J. Implementing the Theory of Sum Frequency Generation Vibrational Spectroscopy: A Tutorial Review. Appl. Spectrosc. Rev.
**2005**, 40, 103–145. [Google Scholar] [CrossRef] - Vidal, F.; Tadjeddine, A. Sum-Frequency Generation Spectroscopy of Interfaces. Rep. Prog. Phys.
**2005**, 68, 1095–1127. [Google Scholar] [CrossRef] - Buck, M.; Himmelhaus, M. Vibrational Spectroscopy of Interfaces by Infrared-Visible Sum Frequency Generation. J. Vac. Sci. Technol. A
**2001**, 19, 2717–2736. [Google Scholar] [CrossRef] - Shen, Y.R. Basic Theory of Surface Sum-Frequency Generation. J. Phys. Chem. C
**2012**, 116, 15505–15509. [Google Scholar] [CrossRef] - Pezzotti, S.; Serva, A.; Gaigeot, M.P. 2D-HB Network at the Air–Water Interface: A Structural and Dynamical Characterization by Means of ab initio and Classical Molecular Dynamics Simulations. J. Chem. Phys.
**2018**, 148, 174701. [Google Scholar] [CrossRef] - Ojha, D.; Kühne, T.D. “On-The-Fly” Calculation of the Vibrational Sum-Frequency Generation Spectrum at the Air-Water Interface. Molecules
**2020**, 25, 3939. [Google Scholar] [CrossRef] [PubMed] - Shen, Y.R. Surfaces Probed by Nonlinear Optics. Surf. Sci.
**1994**, 299–300, 551–562. [Google Scholar] [CrossRef] - Boyd, R.W. Nonlinear Optics, 2nd ed.; Academic Press: San Diego, CA, USA, 2003. [Google Scholar]
- Superfine, R.; Huang, J.Y.; Shen, Y.R. Phase Measurement For Surface Infrared-Visible Sum-Frequency Generation. Opt. Lett.
**1990**, 15, 1276–1278. [Google Scholar] [CrossRef] [PubMed] - Shen, Y.R.; Ostroverkhov, V. Sum-frequency Vibrational Spectroscopy on Water Interfaces: Polar Orientation of Water Molecules at Interfaces. Chem. Rev.
**2006**, 106, 1140–1154. [Google Scholar] [CrossRef] [PubMed] - Mondal, J.; Nihonyanagi, S.; Yamaguchi, S.; Tahara, T. Structure and Orientation of Water at Charged Lipid Monolayer/Water Interfaces Probed by Heterodyne-Detected Vibrational Sum Frequency Generation Spectroscopy. J. Am. Chem. Soc.
**2010**, 132, 10656–10657. [Google Scholar] [CrossRef] [PubMed] - Nihonyanagi, S.; Yamaguchi, S.; Tahara, T. Direct Evidence for Orientational Flip–Flop of Water Molecules at Charged Interfaces: A Heterodyne-Detected Vibrational Sum Frequency Generation Study. J. Chem. Phys.
**2009**, 130, 204704. [Google Scholar] - Jena, K.; Hung, K.K.; Schwantje, T.; Hore, D.K. Methyl groups at dielectric and metal surfaces studied by sum-frequency generation in co- and counter-propagating configurations. J. Chem. Phys.
**2011**, 135, 044704. [Google Scholar] [CrossRef] - Jena, K.C.; Covert, P.A.; Hore, D.K. Phase Measurement in Non-Degenerate Three-Wave Mixing Spectroscopy. J. Chem. Phys.
**2011**, 134, 044712. [Google Scholar] [CrossRef] - Long, D.A. The Raman Effect: A Unified Treatment of The Theory of Raman Scattering by Molecules; John Wiley & Sons: Hoboken, NJ, USA, 2002. [Google Scholar]
- Potma, E.O.; Mukamel, S. Coherent Raman Scattering Microscopy; CRC Press: Boca Raton, FL, USA, 2013; Chapter 1; pp. 3–42. [Google Scholar]
- Mukamel, S. Principles of Nonlinear Optical Spectroscopy; Oxford University Press: New York, NY, USA, 1995. [Google Scholar]
- Hung, K.K.; Stege, U.; Hore, D.K. IR Absorption, Raman Scattering, and IR-Vis Sum-Frequency Generation Spectroscopy as Quantitative Probes of Surface Structure. Appl. Spectrosc. Rev.
**2015**, 50, 351–376. [Google Scholar] [CrossRef] - Lagugné Labarthet, F.; Buffeteau, T.; Sourisseau, C. Orientation Distribution Functions in Uniaxial Systems Centered Perpendicularly to a Constraint Direction. Appl. Spectrosc.
**2000**, 54, 699–705. [Google Scholar] [CrossRef] - Wang, Y.; Cui, Z.F.; Wang, H.F. Experimental Observables and Macroscopic Susceptibility/Microscopic Polarizability Tensors for Third and Fourth-Order Nonlinear Spectroscopy of Ordered Molecular System. Chin. J. Chem. Phys.
**2007**, 20, 449–460. [Google Scholar] [CrossRef] - Hall, S.A.; Hickey, A.D.; Hore, D.K. Structure of Phenylalanine Adsorbed on Polystyrene From Nonlinear Vibrational Spectroscopy Measurements and Electronic Structure Calculations. J. Phys. Chem. C
**2010**, 114, 9748–9757. [Google Scholar] [CrossRef] - Schmidt, M.W.; Baldridge, K.K.; Boatz, J.A.; Elbert, S.T.; Gordon, M.S.; Jensen, J.H.; Koseki, S.; Matsunaga, N.; Nguyen, K.A.; Su, S.; et al. General atomic and molecular electronic structure system. J. Comput. Chem.
**1993**, 14, 1347–1363. [Google Scholar] [CrossRef] - Linder, R.; Nispeal, M.; Häber, T.; Kleinermanns, K. Gas-phase FT-IR spectra of natural amino acids. Chem. Phys. Lett.
**2005**, 409, 260–264. [Google Scholar] [CrossRef] - Merrick, J.R.; Moran, D.; Radom, L. An Evaluation of Harmonic Vibrational Frequency Scale Factors. J. Phys. Chem. A
**2007**, 111, 11683–11700. [Google Scholar] [CrossRef] - Matousek, J.; Gärtner, B. Understanding and Using Linear Programming; Springer: Heidelberg/Berlin, Germany, 2007. [Google Scholar]
- Dantzig, G.B. Reminiscences about the origins of linear programming. Oper. Res. Lett.
**1982**, 1, 43–48. [Google Scholar] [CrossRef] - Karmarkar, N. A new polynomial-time algorithm for linear programming. Combinatorica
**1984**, 4, 373–395. [Google Scholar] [CrossRef] - Chvatal, V. Linear Programming; W. H. Freeman and Company: New York, NY, USA, 1983. [Google Scholar]
- Roy, R.; Sevick-Muraca, E. Truncated Newton’s Optimization Scheme for Absorption and Fluorescence Optical Tomography: Part I Theory and Formulation. Opt. Exp.
**1999**, 4, 353–371. [Google Scholar] [CrossRef] - Partovi-Azar, P.; Kühne, T.D.; Kaghazchi, P. Evidence for the Existence of Li
_{2}S_{2}clusters in Lithium-Sulfur Batteries: ab initio Raman spectroscopy simulation. Phys. Chem. Chem. Phys.**2015**, 17, 22009–22014. [Google Scholar] [CrossRef] [Green Version] - Li, D.H.; Fukushima, M. A Modified BFGS Method and its Global Convergence in Nonconvex Minimization. J. Comput. Appl. Math.
**2001**, 129, 15–35. [Google Scholar] [CrossRef] [Green Version] - Cormen, T.H.; Leiserson, C.E.; Livest, R.L.; Stein, C. Introduction to Algorithms; MIT Press: Cambridge, MA, USA; McGraw-Hill: New York, NY, USA, 2001; Chapter The Simplex Algorithm; pp. 790–804. [Google Scholar]
- GNU Linear Programming Kit, Version 4.64. 2017. Available online: http://www.gnu.org/software/glpk/glpk.html (accessed on 19 February 2015).
- Chen, F.; Hung, K.K.; Stege, U.; Hore, D.K. Linear Programming Applied to Polarized Raman Data for Elucidating Molecular Structure at Surfaces. Chemometr. Intell. Lab.
**2020**, 196, 103898. [Google Scholar] [CrossRef] - Buffeteau, T.; Lagugné Labarthet, F.; Sourisseau, C.; Kostromine, S.; Bieringer, T. Biaxial orientation induced in a photoaddressable azopolymer thin film as evidenced by polarized UV-visible, infrared, and Raman spectra. Macromolecules
**2004**, 37, 2880–2889. [Google Scholar] [CrossRef] - Lagugné-Labarthet, F.; Sourisseau, C.; Schaller, R.D.; Saykally, R.J.; Rochon, P. Chromophore orientations in a nonlinear optical azopolymer diffraction grating: Even and odd order parameters from far-field Raman and near-field second harmonic generation microscopies. J. Phys. Chem. B
**2004**, 108, 17059–17068. [Google Scholar] [CrossRef] - Rodriguez, V.; Lagugné Labarthet, F.; Sourisseau, C. Orientation Distribution Functions based upon both 〈P
_{1}〉, 〈P_{3}〉 Order Parameters and upon the Four 〈P_{1}〉 up to 〈P_{4}〉 Values: Application to an Electrically Poled Nonlinear Optical Azopolymer Film. Appl. Spectrosc.**2005**, 59, 322–328. [Google Scholar] [CrossRef]

**Figure 1.**Illustration of the molecule-fixed $(x,y,z)$ and laboratory frame $(X,Y,Z)$ coordinates, related through three Euler angles. Here $\theta $ is the tilt angle that projects the molecular long axis z onto the surface normal Z, $\varphi $ is the azimuthal angle that describes rotation about Z and $\psi $ is the twist angle that describes rotation about z. The surface is represented by the $(X,Y)$-plane.

**Figure 2.**Energy level and double-sided Feynman diagrams of the IR absorption, visible–infrared sum-frequency generation, and spontaneous Stokes Raman scattering processes.

**Figure 3.**The tensoral nature of the IR absorption, visible–infrared sum-frequency generation, and Raman scattering processes illustrated. In the top row, the material response is arranged according to the rank of the process. In the bottom row, the most-often used contracted notation for an absorption process and Raman scattering are illustrated.

**Figure 4.**A two-dimensional example of a minimization problem subject to constraints whose linear programming solution can be visualized graphically.

**Figure 5.**An illustration of the semi-discrete parameter space that comprises a single mixture. In the case of molecules with known polarity, we consider the range $\theta ={0}^{\xb0}$ to $\theta ={90}^{\xb0}$ in discrete steps of ${10}^{\xb0}$. In the case of unknown polarity, the tilt angles in the range $\theta ={0}^{\xb0}$ to $\theta ={180}^{\xb0}$ may be selected for each molecule. A fraction of each molecule (yellow squares) is then chosen as a weighting factor to generate the mixture. The mixture in the example shown contains 55% methionine tilted at ${30}^{\xb0}$ as its largest component, and 2% isoleucine tilted at ${30}^{\xb0}$ as its smallest component.

**Figure 6.**Average spectra (blue) computed from all tilt angles in the range ${0}^{\xb0}$–${80}^{\xb0}$ and the standard deviation (grey) about the mean. The extent of the vertical axis is the same in all plots, as the spectra are vector normalized for this comparison only. The standard deviation averaged over all frequencies is indicated in red in the inset of each panel.

**Table 1.**A summary of the evaluated test cases indicating the ability of the spectral data to reveal the target composition in terms of the identity and orientation of the constituent molecules. An ✗ indicates that the data set was insufficient to determine the target composition. A ✓ indicates that sufficient data was available for spectral unmixing, while a parenthetical (✓) indicates success, but a data set that is unnecessary since a subset of the data has been shown to be sufficient.

Spectral Data | Known Scaling Factors | Arbitrary Scaling Factors | ||
---|---|---|---|---|

(a) Known Polarity | (b) Unknown Polarity | (c) Known Polarity | (d) Unknown Polarity | |

IR only | ✗ | ✗ | ✗ | ✗ |

SFG only | ✓ | ✗ | ✗ | ✗ |

Raman only | ✓ | ✗ | ✓ | ✗ |

IR + Raman | (✓) | ✗ | (✓) | ✗ |

IR + SFG | (✓) | ✓ | ✗ | ✗ |

SFG + Raman | (✓) | ✓ | (✓) | ✓ |

IR + SFG + Raman | (✓) | (✓) | (✓) | (✓) |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, F.; Gozdzialski, L.; Hung, K.-K.; Stege, U.; Hore, D.K.
Assessing the Molecular Specificity and Orientation Sensitivity of Infrared, Raman, and Vibrational Sum-Frequency Spectra. *Symmetry* **2021**, *13*, 42.
https://doi.org/10.3390/sym13010042

**AMA Style**

Chen F, Gozdzialski L, Hung K-K, Stege U, Hore DK.
Assessing the Molecular Specificity and Orientation Sensitivity of Infrared, Raman, and Vibrational Sum-Frequency Spectra. *Symmetry*. 2021; 13(1):42.
https://doi.org/10.3390/sym13010042

**Chicago/Turabian Style**

Chen, Fei, Lea Gozdzialski, Kuo-Kai Hung, Ulrike Stege, and Dennis K. Hore.
2021. "Assessing the Molecular Specificity and Orientation Sensitivity of Infrared, Raman, and Vibrational Sum-Frequency Spectra" *Symmetry* 13, no. 1: 42.
https://doi.org/10.3390/sym13010042