THE VIBRATIONAL SPECTRA OF A POLYATOMIC MOLECULE (Revised 3/27/2006)

THE VIBRATIONAL SPECTRA OF A POLYATOMIC MOLECULE (Revised 3/27/2006) 1) INTRODUCTION The vibrational motion of a molecule is quantized and the resulting energy level spacings give rise to transitions in the mid-ir portion of the electromagnetic spectrum (4000 to ca. 400 cm -1 ). As you know from study of the diatomic harmonic oscillator, the energies (or wavenumber positions, cm -1 ) of these transitions are related to the bond strength (force constant), bond length, and atomic masses (reduced mass). In polyatomic spectra, the positions and relative intensities of the vibrational modes depend on the symmetry (i.e. shape or structure) of the molecule, as well as the bond strengths and masses. For this reason, vibrational spectra (IR and Raman) can provide detailed structural information. This structural information is the objective of this lab, and it is obtained by this analysis, or interpretation, of the infrared and Raman spectra. In this experiment you will obtain infrared and Raman spectra of a polyatomic molecule, predict the selection rules, assign vibrational modes, and then compare these with the vibrational mode positions and intensities predicted for that molecule using HyperChem. Using group theory, we shall predict the spectral selection rules, i.e. predict the spectra for a particular structural model. Assignment of vibrational modes in a spectrum involves relating the experimental spectrum and the predicted spectrum so that each observed vibrational band is identified as to its theoretical origin. A series of empirical rules is provided below to aid in this assignment. Also, a chart defining the well-known positions of group frequencies will be available. (These charts summarize the vast knowledge obtained from the extensive, experimental spectral database that has been collected, literally, over the past 65 years.) Finally, the results of the HyperChem calculation will be compared with the above. Because these calculated normal mode positions will be harmonic frequencies, they must be multiplied by a constant to relate them to the empirical, anharmonic band positions. This constant depends upon the orbital basis set that you use. 2) BACKGROUND For a non-linear polyatomic molecule containing n atoms, there will be 3n-6 vibrational degrees of freedom (3n-5 for a linear molecule) [1]. This number represents the maximum number of vibrational modes for the molecule. However, often the observed number is smaller because of degeneracies and selection rules. Because of the symmetry (i.e. structure), some transitions may be degenerate, and some transitions may be forbidden and not observed; others will be allowed and observed. Allowed or forbidden transitions are also referred to as active or inactive vibrational modes, respectively. Vibrational transitions may also be observed using IR or Raman spectroscopy. For an IR absorption to be allowed between two vibrational levels, a change in dipole moment (µ) must occur as the atoms move, and υ must equal + 1. To be Raman active (i.e. allowed), there must be a change in polarizability (α ij ) during the vibration and υ must equal ± 1. This polarizability can be better understood as an induced dipole. A fundamental vibrational mode will involve a transition from the υ = 0 level to the υ = 1 level. The selection rules for IR and

for Raman spectra differ so that the two techniques provide complementary information, not redundant information. Therefore, the Raman spectrum provides significant new structural information, in addition to that provided by the IR spectrum. Group theory is used to predict the characteristic or normal modes of vibration for a molecule. (Normal refers to the fact that these modes of vibration are orthogonal to each other, i.e. independent of each other.) For molecules with many atoms, 3n-6 becomes very large, and this can result in a seemingly complex spectral pattern. However, the presence of symmetry in a molecule often simplifies the vibrational spectrum. Recognition of this symmetry simplifies and allows the interpretation of even complex vibrational spectra, both IR and Raman. The predicted symmetry species define the activity (allowed or forbidden) of each vibrational mode in the IR and in the Raman spectra, and these are referred to as selection rules. Also, the stretching modes can be predicted to distinguish them from the bending and torsional vibrations. The atom masses and force constants determine the precise region of the spectrum at which each normal mode vibrates, i.e., the energy (or frequency or wavenumber) for each. The general region in which various molecular groups will occur can be obtained from group frequency charts, which summarize extensive collections of empirical data. Article II. Empirical Rules Used to Interpret Spectra When we interpret a spectrum, we shall relate (or assign) each band in the spectrum to its origin as predicted by the group theoretical selection rules. The following empirical rules are used in this assignment process for molecular species. The band position (cm -1 ) provides information about the type of vibration. Vibrational modes can be of three basic types, stretching modes (ν), bending modes (δ) or torsional modes (τ). Stretching modes of vibration occur at higher energy, i.e. higher wavenumber, than bending modes. Torsional modes appear at even lower energy. Both the type of vibration and the symmetry species of each vibrational mode influence the relative band intensities. The intensities of asymmetric stretching modes will be greater than asymmetric bending (or torsional) modes in the IR. If one compares asymmetric species with symmetric species of each type (stretch or bend), the asymmetric species will, in general, be stronger in the IR spectrum. For a Raman spectrum, the opposite is true; the symmetric species are stronger. The spectra, both the IR and Raman, will contain some weak bands that are fundamental vibrations and some weak bands that are overtone or combination bands. (Overtone and combination bands are not allowed in the harmonic approximation, and therefore exhibit reduced intensities.) To distinguish the two, compare the IR and Raman spectra. If a band is weak in both the IR and the Raman, it is more likely to be an overtone. A fundamental is expected to be strong in one and weak in the other. However, it is possible, on occasion, to have a mode be weak in both. When comparing IR and Raman band positions, how do you decide if the bands may be ascribed to the "same" vibrational mode, i.e. that these are coincident in the IR and the Raman? This is defined by the experimental error in each spectrum. The maximum precision (for a sharp band) is defined by the spectral resolution used for the scan. For example, if you used 2 cm -1

resolution, then your precision (or uncertainty) is ±2 cm -1. For broad bands, the precision will be less. The criterion for coincidence is then the sum of the two uncertainties. Article III. Group Theoretical Analysis The first choice of point group used to predict the IR and Raman spectral selection rules should be the ideal structure, the highest symmetry point group possible. Sometimes, more than one structure (and point group) is possible. Do the group theoretical calculation for both and determine which applies to the real spectrum. For example, when calculating the minimum energy geometry, you may find that this geometry has lower symmetry than the ideal and belongs to a lower symmetry point group. You will then need to predict the IR and Raman selection rules for this point group, as well as the ideal, high symmetry point group. When comparing the two sets of predictions with the experimental data, you may then decide which agrees better with the experimental results. There are two "background" pages that will be given to you as handouts in class. These are a) How is Group Theory Used? and b) Applying Group Theory: Representations of Vibrational Motion. Copies of these follow this experiment for your convenient reference. Article IV. HyperChem Output, Anharmonicity Correction, Negative Frequencies, and Degeneracy HyperChem calculates all of the fundamental modes, both IR and Raman, as well as the IR intensities. Thus it outputs two spectra, one with all of the bands plotted, and the second with just the IR active modes. Since it does not calculate Raman intensities, it cannot plot a Raman spectrum. HyperChem also identifies the symmetry species of each mode. Since you know the assignments from your group theoretical results, you can predict the approximate Raman intensity expected for each calculated mode. (The totally symmetric species will be stronger in the Raman.) The frequencies calculated using HyperChem (or any other ab initio calculation) are harmonic frequencies. To correct for anharmonicity the calculated, harmonic frequencies are multiplied by a constant. This constant depends upon the basis set used. Professor Brown recommends use of the HF/6-13G* basis set (select the 6-13G* basis in the setup) to minimize the energy and to calculate the geometry. Multiply your calculated values by 0.8929, if you use the 6-13G* basis set [2]. This correction will allow you to compare your experimental values with these corrected values. These should agree quite will, but will not give a perfect agreement. Any drastic disagreement here may indicate an error in your group theoretical analysis and empirical assignments. Be sure to describe and discuss this comparison. This correction factor is an empirically determined value which can vary from 0.89 to 0.99. The precise value needed for each molecule or even for each fundamental vibrational mode may differ slightly. This is especially true for the lower frequency vibrational modes such as bending and torsional modes. After you "assign" each fundamental vibration and compare the experimental and calculated harmonic values, you can calculate the precise coefficient needed for each mode. (These could be placed in another table.) The overall average coefficient for

your molecule, as well as the average values for just the stretching modes and for just the bending plus torsional modes will be interesting to calculate and discuss in your report. How much do these differ from the literature value reported above (0.8929)? A negative frequency means that you have not obtained a minimum. A negative frequency indicates an imaginary energy, which occurs if you are at a saddle point. You are not at the minimum energy structure. Go back and recalculate the minimum energy structure. If your molecule exhibits degenerate vibrational modes (E or T species), HyperChem will calculate 2 or 3 essentially identical (i.e. within experimental error) frequencies for doubly or triply degenerate vibrational modes, respectively. Article V. ASSIGNED COMPOUNDS SEE LAB SUPERVISOR Article VI. EXPERIMENTAL PROCEDURE 1. Obtain an IR spectrum of your group s assigned compound using the Mattson FTIR located in room 601B. (Instructions for its operation are also available in 601B.) You will need to schedule the IR with the lab supervisor with at least 1 day s notice. The supervisor will explain how to prepare your sample and operate the instrument, if needed. The strongest absorption bands in your IR spectrum should absorb no more than 1 absorbance unit (A 1); at least 10% transmittance (10-15% is better) is required to insure resolution of multiple peaks within very strong peaks. It s a good idea to record and/or report two IR spectra, one that clearly resolves all bands, even the strongest, and a second (more concentrated or thicker), which provides good definition of the weaker bands in the spectrum. 2. Obtain a Raman spectrum of your group s assigned compound using the Raman spectrometer in the Chemical Engineering Department. Please contact Professor Bahne Cornilsen to set up an appointment for this. He will explain how to prepare a sample and operate the instrument. 3. Predict the infrared and Raman spectra for your molecule (using group theory) and assign the observed vibrational modes (i.e. relate the experimental spectra to the group theoretical predictions). 4. Use HyperChem to draw your molecule, calculate the minimum energy configuration, and generate the vibrational energies for your assigned compound. Often, the point group symmetry of this structure and that used for your group theoretical analysis will be the same; however, the former may be of lower symmetry. You will then need to predict the selection rules using both point groups. Alternatively, but not necessarily, this may indicate you have not reached the global minimum in your calculation. 5. Submit a group report (see below) that presents the above data and results in tables and figures, and includes a comparison of the group theoretical analysis of the experimental

spectra and the vibrational modes generated by HyperChem. Evaluate how well the three compare. Article VII. LABORATORY REPORT FORMAT The report should include the following: a) a results section with Figures of your IR and Raman spectra (with the band positions writtenin near each major band). b) the following results from your calculations and experimental IR and Raman spectra i) one or two tables comparing the experimental IR band positions, the experimental Raman band positions, and the calculated band positions (in decreasing order, highest energy on top) for your compound. It is often convenient to include symmetry (species) and band assignment information for each band in your table(s). Use the following symbols: ν, δ, or τ, and the atoms involved; e.g. ν CH or δ CHC. Indicate the intensities of each band in your table(s) by using the following symbols for the IR: vs=very strong, s=strong, m=medium, w=weak, vw=very weak, and 1-100 for Raman. It is often convenient to compare the calculated results in a second table. ii) a table of calculated bond distances and angles. If there is experimental data in the literature, these can be compared. iii) the calculated total energy (in Kcal/mole and in atomic units) for the optimized geometry. c) a concise discussion to explain how you made your band assignments. d) a discussion relating part c (your empirical band positions and assignments) to your calculated frequencies. e) a table of calculated bond distances and angles. If there is experimental data in the literature, these can be compared. f) a conclusion as to what the actual structure of your molecule is, based on your data. g) Finally, a copy of your HyperChem output should be sent to Professor Cornilsen as an e-mail attachment (change first to a *.txt file so it can be e-mailed). NOTE: No preliminary report is required for this experiment. The report must be-submitted (by email) to the lab s faculty advisor, Professor Cornilsen, for grading. Article VIII. REFERENCES 1. Ira N. Levine, Physical Chemistry, 5 th edn., McGraw-Hill Co. Inc., N.Y., 2002, pp. 738-816. 2. J. B. Foresman and A. Frisch, Exploring Chemistry with Electronic Structure Methods, 2 nd edn., Gaussian, Inc., Pittsburgh, PA, 1995, p. 64.

Applying Group Theory: Representations of Vibrational Motion March 24, 2006 - BCC The following is an introduction to the use of group theory. To apply group theory, we must represent some physical entity that defines the property of interest. For vibrational spectroscopy, we represent the vibrational motion of the molecule in terms of 3 unit Cartesian displacement vectors on each atom. Thus 3n degrees of motion (or degrees-of-freedom) are defined, where n is the number of atoms in the molecule. This is referred to as the basis set. The 3n-6 normal modes of vibration will be combinations of these Cartesian displacement vectors. Six of the 3n degrees of freedom will be rotations (3) and translations (3) of the molecule as a whole. The remaining 3n-6 degrees of freedom represent the normal (i.e. mathematically unique and independent) modes of vibration. After the point group is defined for a molecule, a reducible representation (Γ) is produced by operating on the basis set with each symmetry operation, R. For each atom that is not shifted during the operation (i.e. left invariant) the vectors are summed, with a +1 if the vector is not inverted, and a -1 if the vector is inverted. These sums produce the character for that operation, χ(r). This reducible representation actually represents the chosen basis set, which is the 3n degrees of freedom in the analysis of vibrational motion. For example, see how this representation and the vibrational selection rules are obtained for a tetrahedral molecule (or tetrahedral ion) in Cotton s text, in a section entitled Tetrahedral Molecules, Such as Methane. * A second method of representing molecular motion is to use internal coordinates (bond lengths, bond angles, or torsional angles) as the basis set. Cotton* also obtains reproducible representations (Γ) for the CH stretches and HCH bends of methane, and shows how the reducible representations are reduced. * F. A. Cotton, Chemical Applications of Group Theory, 3 rd edn., Wiley, NY, 1990.

HOW IS GROUP THEORY USED? March 24, 2006 - BCC After we learn to classify molecules in terms of their symmetry, i.e. assign a molecule to the appropriate point group, we wish to apply it to specific examples. This general application process is outlined in the following outline, using vibrational spectroscopy as an example. Other examples are included in parentheses. 1. First, the point group is assigned. 2. Secondly, a basis set is chosen to represent the physical entities that we wish to study, e.g. vibrational modes of a molecule. (Other examples include atomic orbital basis sets to represent and determine hybrid orbitals or molecular orbitals.). For vibrational modes two different basis sets are useful. The first is a Cartesian coordinate basis set, which represents all motions (translation, as well as rotation & vibration about the center-of-mass) of a molecule. The second type of basis set uses internal coordinates, which represent specific inter-atomic motions. In each case our final result will be a group theoretical representation of this basis set which will allow determination of the symmetry of the excited state, vibrational energy levels (and of their corresponding wavefunctions). 3. The symmetry operations of the molecule are applied to the basis set to obtain a reducible representation which represents all of the properties in question. 4. The reducible representation is reduced to form irreducible representations (i.e. it is reduced into "symmetry species") which represent the results of our application. These species represent the excited vibrational states of the molecule, the normal modes of vibration. 5. The irreducible representations (or symmetry species) obtained are then used to determine selection rules for transitions between the energy levels in question, i.e. to predict spectral selection rules for infrared and Raman spectra. The ground state vibrational level is always represented by the totally symmetric symmetry species. (If we represent electronic energy levels or molecular orbitals, selection rules for electronic transitions can be predicted.)