1 INTRODUCTION THE VIBRATIONAL SPECTRUM OF A POLYATOMIC MOLECULE (Revised 4/7/2004) 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 analysis, or interpretation, of an infrared spectrum to obtain structural information is the objective of this lab. In this experiment you will obtain an infrared spectrum of a polyatomic molecule, predict its 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 60 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. 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) . This number represents the maximum number of vibrational modes for the molecule. However, often the observed number is less 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. 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. A fundamental vibrational mode will involve a transition from the υ = 0 level to the υ = 1 level. 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
2 can result in a seemingly complex IR 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 of each (allowed or forbidden) 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 species will absorb can be obtained from group frequency charts, which summarize empirical information. Vibrational transitions may also be observed using Raman spectroscopy. 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. To be Raman active (i.e. allowed), there must be a change in polarizability (α ij ) during the vibration. This can be better understood as a changing induced dipole. 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, 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 the Raman spectrum, assume 2-3 cm -1 for a sharp peak. For broad bands in either spectrum, the precision will be less; assume ± 3-
3 4 cm -1 for this lab experiment. The criterion for coincidence is then the sum of the two uncertainties. Group Theoretical Analysis The first point group that you use 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. 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. 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 IR and Raman activity from your group theoretical calculations, you can identify the Raman activity of each mode yourself. You can predict the approximate Raman intensities from the group theoretical predictions (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 if you use the 6-13G* basis set . This correction will allow you to compare your experimental values with your calculated values. Be sure to place this information in a table and discuss it. This correction factor is an empirically determined value which can vary from 0.89 to 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 harmonic, calculated values, you can calculate the precise coefficient needed for each mode. (These could be placed in a second 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)?
4 If a negative frequency (an imaginary energy) is obtained, this means that you are on 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. ASSIGNED COMPOUNDS GROUP 1a: methyl iodide, CH 3 I GROUP 1b: formamide, HCONH 2 nitromethane, CH 3 NO 2 GROUP 2a: acetonitrile, CH 3 CN GROUP 2b: nitromethane, CH 3 NO 2 formamide, HCONH 2 GROUP 3a: chloroform, CHCl 3 GROUP 3b: methylene chloride, CH 2 Cl 2 EXPERIMENTAL PROCEDURE 1. Obtain an IR spectrum of your group s assigned compound using the Mattson FTIR located in room 601B. (Instructions for it s operation are also available in 601B.) You will need to schedule the IR with the lab supervisor with at least 1 day s notice. Prepare your sample by placing it on an adsorbent polymer sample card (obtained from the lab supervisor). 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. You may want 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. Use HyperChem to draw your molecule, calculate the minimum energy configuration, and generate the vibrational energies for your assigned compound. A negative frequency means that you have not obtained the minimum energy configuration. Ideally, the symmetry of this configuration and that used for your group theoretical analysis should be the same; however, the former symmetry may be of lower symmetry. 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). Raman spectral data will be provided for each molecule. 4. 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 absorptions generated by HyperChem. Evaluate how well the
5 three compare. LABORATORY REPORT FORMAT Although you are experimentally obtaining the IR spectrum yourself, your report will interpret both the IR spectrum and the Raman spectrum. The reason for this is that the two, together, provide more detailed structural information than either by itself. The report should include the following: a) a results section with Figure(s) of your IR spectrum (with the IR spectral band positions written in near each major band). Your Raman spectral data should be included as a Figure (or in a table) as well. b) a Table containing the IR and Raman band positions for your compound and your assignments for each band (these should include ν, δ, or τ, the atoms involved, and the symmetry species). Indicate the intensity of each band in your table by using the following symbols: vs=very strong, s=strong, m=medium, w=weak, vw=very weak 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 conclusion as to what the actual structure of your molecule is, based on your data. NOTE: No preliminary report is required for this experiment. Reports submitted by Monday, April 26 th will be proof-read (for style, formatting, grammar, punctuation, etc.) and returned to you. This report must then be re-submitted to the lab s faculty advisor, Professor Cornilsen, for grading. The final report is due no later than Thursday, April 29 th. REFERENCES 1. Ira N. Levine, Physical Chemistry, 5 th edn., McGraw-Hill Co. Inc., N.Y., 2002, pp J. B. Foresman and A. Frisch, Exploring Chemistry with Electronic Structure Methods, 2 nd Edn., Gaussian, Inc., Pittsburgh, PA, 1995, p. 64.
6 Applying Group Theory: Representations of Vibrational Motion April 12, BCC The following is an introduction to the use of group theory. To apply the 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 vibrational freedom in this case. 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) for the basis set. Cotton* also obtains reproducible representations (Γ) for the CH stretches and HCH bends. How these reducible representations are obtained and how the reducible representations are reduced will be discussed in class (and in Levine s text, Chap. 21). * F. A. Cotton, Chemical Applications of Group Theory, 3 rd edn., Wiley, NY, 1990.
7 HOW IS GROUP THEORY USED? April 12, 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, 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.)