Chapter 8
Using Computational Chemistry to Extend the Acetylene Rovibrational Spectrum to C2T2
Using Computational Methods To Teach Chemical Principles Downloaded from pubs.acs.org by UNIV OF ROCHESTER on 05/15/19. For personal use only.
William R. Martin and David W. Ball* Department of Chemistry, Cleveland State University, 2121 Euclid Avenue, Cleveland, Ohio 44115, United States *E-mail:
[email protected]. Phone: 216-687-2456.
Although an analysis of the rovibrational spectrum of acetylene (C2H2) and deuterated acetylene (C2D2) are known physical chemistry laboratory experiments, an extension of the exercise to tritiated acetylene, C2T2, has never been proposed. While an experimental measurement of the rovibrational spectrum of C2T2 may be out of reach, computational methods do exist that allow us to predict the spectrum using data from the C2H2 and C2D2 spectra. Here, we detail the method of performing just such an exercise and show that the resulting spectrum agrees very well with an actual experimental spectrum available in the literature.
Introduction The spectroscopic measurement of the rovibrational spectra of acetylene (C2H2) and deuterated acetylene (C2D2) is an established physical chemistry laboratory experiment (1). Although similar experiments exist for HCl and HBr and their isotopomers (D for H, naturally-occurring 35Cl and 37Cl for chlorine, and 79Br and 81Br for bromine) (2), because the acetylene molecule is centrosymmetric, its spectrum shows intensity variations due to impact of nuclear spin on rotational degeneracies (3), making it one of the few direct observations of the influence of the nuclear wavefunction. Starting with calcium carbide, students can use H2O and D2O to generate acetylene isotopomers in situ and measure the spectra easily. Students can then fit the frequencies of the absorptions to mathematical equations to determine the rotational constants B′ and B″ as well as the centrifugal distortion constant De (1), much like they would do for the diatomic molecules HCl and DCl. One of the reasons this type of analysis works for acetylene, a polyatomic molecule, is because it is a linear molecule that has some vibrations whose oscillating dipole moments are parallel to the molecular axis. Such vibrations show rotational structures that follow the same spacing patterns as do © 2019 American Chemical Society
heteronuclear diatomic molecules; i.e. equally-spaced absorption lines separated by 2B but getting slightly farther apart (for the P branch) or slightly closer together (for the R branch) that can be modeled as centrifugal distortion. In the case of acetylene, the three parallel vibrations are labeled as ν1, ν2 and ν3 (Figure 1), which for C2H2 occur at 3374, 3289, and 1974 cm-1, respectively (4). Some combination bands also share these characteristics.
Figure 1. The three parallel normal vibrational modes of acetylene. The other two normal modes are the symmetric and asymmetric bending modes, both of which are doubly degenerate. This experiment is one of several instances where a (relatively rarer) isotope is prominent in an undergraduate lesson, in this case deuterium. Other examples include the mention of isotopes as tracers in reaction mechanisms, as radioactive signatures because of their specific decay properties, and in their use in atomic power. In organic chemistry, 13C becomes important in the discussion of NMR spectroscopy (as are deuterated solvents, if experimental NMR is available in the undergraduate curriculum) and in interpreting mass spectral patterns. Mössbauer spectroscopy is another isotope-specific experimental technique (most commonly, 57Fe) that is occasionally mentioned in upper-level inorganic curricula, and discussions of nuclear partition functions can arise in statistical thermodynamics, although experimental demonstration of its impact is rare – hence the value of the acetylene rovibrational spectrum experiment. Seldom, however, does tritium come up in the undergraduate curriculum, and even less frequently in an experimental way. There is the occasional mention of tritium as a radioactive tracer in water supplies, and possibly its application in nuclear sciences, especially ordnance and radioluminescence. Beyond this, however, tritium’s mention in the curriculum is negligible. We recently considered the reasons why students don’t do this experiment using C2T2 as well. There are several obvious reasons: • Tritium is exceedingly rare. Although it can be formed from 6Li in special breeder reactors, only kilogram quantities are produced each year, much of that going to nuclear ordnance. Tritium occurs naturally as cosmic rays produce fast neutrons that interact with 14N in the upper atmosphere, producing 12C and 3H; however, its natural occurrence is estimated at 1 in 1018 hydrogen atoms. • Tritium is radioactive, with a half-life of 12.32 y (5). As a low-energy beta emitter, it is not especially hazardous, but as a hydrogen isotope it has the chemical ability to replace 1H anywhere, and experimenters are appropriately cautious. Having pointed this out, it is also worth mentioning that the experimental rovibrational spectrum of C2T2 has been reported, by Jones et al. in 1967 (6). 94
But why not calculate the rovibrational spectrum of C2T2, and compare it to the available literature? Consider the data that the students have at their disposal from the C2H2/C2D2 lab. They calculate, by a fitting of the rovibrational frequencies: • • • •
ν0, the pure vibrational frequency assuming no rotation; B″, the rotational constant of the lower vibrational state; B′, the rotational constant of the upper vibrational state; and De, the centrifugal distortion constant.
Students also have access to modern computational chemistry software (in our case, the GAUSSIAN suite of programs (7)) that can be used to calculate data, data that can vary by isotope. The question then becomes: Can students who have determined the four molecular constants from C2H2 and C2D2, along with using computational chemistry software, reasonably calculate the rovibrational spectrum of C2T2 as an add-on to the existing experiment? Actually, the answer is “yes”. With a few reasonable assumptions or approximations, and thanks to some mathematical luck, it can be done rather easily. In fact, our analysis results in an absolute error less than 6 cm-1 from the experimental rovibrational spectral lines of C2T2.
Some Computational Background In addition to determining the appropriate molecular constants from the experimental data, students can use computational chemistry software to determine, either directly or indirectly, the appropriate molecular constants for C2T2 to predict its spectrum. In the course of determining the geometry of molecule, the program can (and in the case of GAUSSIAN, does) output a value of the rotational constant B. Because it is a linear molecule, acetylene has a single non-zero value for its rotational constant. This value depends on certain molecular parameters, namely the distances of the nuclei from the molecular center and the masses of those nuclei. In going from H to D to T, the relevant distances do not vary; the only things that vary are the masses of the two hydrogen atoms in the acetylene molecule. As such, these numbers are easy to calculate for the program and are part of the standard output file, in the case of GAUSSIAN having units of GHz. Students merely copy the value of B for the energy-minimized structure. The issue of using computational chemistry programs to predict vibrational frequencies is a bit trickier. This is because, unlike for a simple diatomic molecule (e.g. HCl or HBr), a threedimensional vibrational mode of a polyatomic molecule does not necessarily follow a simple mathematical rule that allows one to just re-substitute new isotopic masses and calculate a new vibrational frequency. However, the word “necessarily” is important here, because it turns out that in some case this is possible to make an accurate prediction. To understand why, it is useful to review how the computational chemistry program calculates vibrational frequencies. Here we follow the lead of a “white paper” discussion of vibrational modes available at the GAUSSIAN website (8), although our variables may be different. The program begins by numerically calculating what is known as the Hessian matrix, which is a 3N × 3N (N = number of atoms) matrix of force constants, evaluated at the minimum-energy geometry:
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where V is the potential energy of the molecule in 3N space, and qi and qj represent the 3N possible changes in Cartesian coordinates of the N atoms in the molecule. (The terms of this matrix are determined numerically, which is one reason vibrational calculations sometimes require a lot of computational time.) Then, the terms in this matrix are divided by the square roots of the masses of the atoms involved, generating a “mass-weighted coordinate” force constant matrix fMWC. As a matter of choice, the GAUSSIAN program then determines the principal axes of the moments of inertia of the molecule and transforms the coordinates to internal coordinates to remove translations and rotations. This matrix transformation (using a transformation matrix D; this will be used again later) generates a new version of the Hessian matrix labeled fint. This new matrix is diagonalized by determining a transformation matrix L such that
where L is the transformation matrix of eigenvectors and Λ is the diagonal matrix of mass-weighted frequency-equivalents. Each entry on the diagonal of Λ needs only one index, and so is designated λi. The program then calculates the frequency of the (now normal mode of) vibration as
These are the frequencies reported by the program. This is where the concept of “imaginary” frequencies arises, because if λi is negative (it is, after all, ultimately a second derivative of a potential energy curve) then the square root of a negative yields an imaginary number. What about the other characteristics of the vibration, like the reduced mass? The program now reverses to make a Cartesian coordinate-based version of L, labeled lcart:
where D is the original Cartesian-to-internal-coordinate matrix and M is a diagonal matrix of the square root of the masses of the atoms. The matrix lcart is important to understand how the mathematics of this analysis works, but it is important to clear up a misconception about reduced mass. What GAUSSIAN (and perhaps other programs) report as a “reduced mass” in vibrational calculations is more appropriately termed an “effective mass,” and this is what we get from lcart: the effective mass, still labeled μi, for each vibrational mode is the reciprocal of the sum of the squares of each column matrix entry, lcart,k,i in lcart:
The “effective” force constant for this normal mode, ki, is then calculated from the standard equation that relates the force constant of a vibration to the (effective) mass of the vibrator:
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From this analysis, we have vibrational frequencies, effective (or “reduced”) masses, and force constants. In performing this analysis for the three isotopomers of acetylene, the only input data that changes is the masses (or more directly, the square root of the masses) of the atoms in the molecule. Now we have to ask a crucial question: How do the values of ki, as determined by the computational chemistry program, relate to the force constants of the actual motions – stretching and bending – of the molecule? From a physical point of view, the treatment of the acetylene molecule can be found in Herzberg (9). Figure 2 shows the acetylene molecule and how its stretches and bends, and the resulting force constants, are defined. Based on this, we can define three force constants, which are listed and described in Table 1. Analyzing the normal modes, the following equations can be defined (9):
Note that only for ν3 (equation 9), ν4 (equation 10), and ν5 (equation 11) do the vibrational frequencies depend directly on the force constants. On the other hand, for ν1 (equation 7) and ν2 (equation 8), the dependence on the force constants is more complicated. From this, we conclude that only for these three normal modes (ν3, ν4, and ν5) is there a simple and direct relationship between a force constant and the frequency.
Figure 2. The definition of the internal coordinates for the vibrational motions of acetylene. See equations 7–11. Table 1. Definitions of the three force constants based on the internal coordinates of acetylene (see Figure 2)
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We can verify this by looking at some of the characteristics that the program provides about the frequencies. Because of the way GAUSSIAN determines the vibrations, rather than comparing the force constants it calculates with the frequencies, we take advantage of the classical relationship
and look at the relationship between the frequencies and the reciprocal square of the effective mass as reported by the calculation. Figure 3 shows graphs of the frequencies of the normal modes (in cm-1) for each isotopomer versus the effective mass (in g/mol), along with the linear best fit and its correlation factor. For ν1, the trend is almost perfectly linear, which one might expect for the symmetric C-H stretching mode of acetylene. However, ν2, the C-C stretch, is obviously not even close to linear. The modes ν3, ν4, and ν5 show a perfect linear relationship between the reciprocal square of the reduced mass and the vibrational frequency. The normal mode ν3 is the only parallel stretching mode that demonstrates this correlation, as ν4 and ν5 are the C-C-H bending modes.
Figure 3. Graphs of the calculated frequency of the normal mode (in cm-1, y axis) versus the reciprocal of the square root of the effective mass of the mode (in g/mol, x axis). The axes have been left unlabeled for clarity. What this demonstrates is that we can use values of ν0 for the ν3 normal mode of C2H2 and C2D2 and linearly extrapolate a value of ν0 for C2T2. Then by using the rotational constants for C2T2 (extracted from the output file) and the B values for acetylene, we can calculate a predicted rovibrational spectrum for C2T2. 98
We make one additional approximation in graphing the positions of the rotational structure of C2T2. According to the published procedure (1), students measure the frequencies of the rotational structure and fit them according to the cubic equation
(Here, the index m equals -J for the P branch and J + 1 for the R branch.) In predicting the spectrum of C2T2, students are asked to use the rotational constant from the computational output, assume B′ = B″, and finally assume De = 0. This introduces negligible error in the predicted results (see below). The final step is to use the Boltzmann distribution and the degeneracies of the rotational states to predict the intensity alterations of each rotational line. The intensities are proportional to the populations of the initial states, which are populated thermally according to the expression
where gJ is the degeneracies of the Jth rotational energy level, h is Planck’s constant, c is the speed of light, B is the rotational constant, k is Boltzmann’s constant, and T is the absolute temperature (3). It is the degeneracies that concern us here, because for this particular system, the nuclear spin has an impact on the rotational degeneracies in order that the Pauli principle be satisfied, i.e. that the total wavefunction of the molecule be antisymmetric. Tritium has a nuclear spin quantum number I of ½, making it a fermion. As such, when the rotational quantum number J is odd, the rotational wavefunction is antisymmetric with a rotational degeneracy of 2J + 1 and the nuclear spin wavefunction must be symmetric with a degeneracy of (I + 1)(2I + 1):
However, when J is even, the rotational wavefunction is symmetric, requiring the nuclear wavefunction to be antisymmetric and having a degeneracy of 2I2 + I:
Thus, we should see a rough 3:1 ratio of adjacent absorptions in the rovibrational spectrum of C2T2. Students can either use the intensity values as calculated from equation 14, or alternately divide all intensity values by the lowest value, essentially normalizing it to 1 and scaling all other intensities relative to that one.
Procedure The following Procedure is available verbatim as a handout to the students performing this experiment. Note that different fonts are used to distinguish between the instructions and the input/ output text for the computational chemistry program.
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Addendum to Acetylene Rovibrational Spectrum Experiment There are many places in the chemistry curriculum that invoke deuterium, 2H (or D), as an example of an isotope. Generating C2D2 and measuring its rovibrational spectrum in this experiment is one of them. However, rarely does the undergraduate curriculum invoke tritium, 3H (or T), in discussions of isotopic substitution. Here is an additional exercise that uses tritium-substituted acetylene, C2T2. 1. Using the GAUSSIAN09 program, set up and perform an optimization and frequency analysis of C2H2. Use “B3LYP/6-31G**” as the method and basis set. For the Molecular Specification, use:
2. After the job is finished, perform two additional calculations by substituting D and T atoms for H in the optimized structure. The easiest way to do this is to open your original job file and add “(Iso=x)” next to each hydrogen atom:
Of course, Iso=2 is for deuterium, so use Iso=3 for tritium. 3. In the list of vibrational frequencies, the mode you are studying (labeled ν3 in the literature) is the final vibrational frequency, as GAUSSIAN09 lists vibrational frequencies in increasing order of wavenumber. As part of the vibrational frequency output, GAUSSIAN09 also prints the reduced mass of the normal mode. Record this reduced mass for this vibration for all three isotopomers of acetylene. For example, here is part of a GAUSSIAN output file for another molecule:
The frequencies (in units of cm-1, or wavenumbers) and force constants (in units of millidynes/ Angstrom) are highlighted in bold.
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Also, record the rotational constant that GAUSSIAN09 calculates for the optimized geometry. Note that the units are gigahertz. Here is where you find the rotational constants – MAKE SURE YOU USE THE VALUES FROM THE OPTIMIZED GEOMETRIES:
Not far after this in the output file, there is a section showing the optimized molecular geometry (“Standard orientation:”), followed by the rotational constants:
Note that, unlike the molecule above, a linear molecule will have only two non-zero rotational constants, and they should be exactly the same (otherwise the molecule isn’t perfectly linear). 4. Perform your analysis of your C2H2 and C2D2 experimental data, following the original lab instructions from the text. Part of your results should be the ν0 values for the ν3 vibration of acetylene. Graph these values for C2H2 and C2D2 versus 1/√μ, where μ is the reduced mass of the vibration from the GAUSSIAN09 output. You should have two points in your graph. 5. Extrapolate the line you get to the value of 1/√μ for C2T2 from the appropriate GAUSSIAN output file, and read off the corresponding value of ν0 for C2T2. 6. Assuming that the two rotational constants are equal (that is, B′ = B″) and that the centrifugal distortion constant De = 0, determine the wavenumbers of the rovibrational spectra for ν3 of C2T2 using equation (11) in your handout – don’t forget that m is defined differently for the P and R branches! Using a spreadsheet and entering the equation would be easiest. Calculate the rovibrational values up to J = 30. 7. Now comes the fun part. The intensities of the C2T2 rovibrational lines are determined by two factors: thermal energy distributions and rotational degeneracies (see your lab handout). Because the molecule is centrosymmetric, the nuclear wavefunction impacts the degeneracies of the 101
rotational states. For 3H, the nuclear spin I = ½. This makes it a fermion (as opposed to a boson), and the degeneracies for fermions depend on whether J(lower) is odd or even. If J(lower) is odd, the degeneracies are
If J(lower) is even, the degeneracies are
Because I does not change, the “I” contribution to the degeneracies are the same for all odd J values and all even J values, so the two expressions above become
These degeneracies need to be combined with the thermal intensity contribution. The proper, complete form of equation (12) in your handout is then
where h is Planck’s constant (normal units), c is the speed of light (in cm/s!!!), B is the rotational constant (in cm-1!!), k is Boltzmann’s constant (normal units) and T is the absolute temperature. (Note that you will need to convert the B values from the GAUSSIAN output; that number has units of gigahertz, GHz). In two separate calculations, calculate the value of the intensity expression for the odd values of J and the even values of J (again, using Excel’s equation editor can be useful. Separating your J values into separate sections might be helpful as well.). For example, a sample calculation using the rotational constant of acetylene (B = 29.72524 MHz, converted to cm-1), the speed of light in the proper units (2.9979×1010 cm/s), J = 1, and the standard values and units of h and k, and at 298 K (approximate room temperature, maybe a bit warm), the first equation gives
(Verify this.) Perform a similar calculation for all 30 values of J, using a spreadsheet. (Question: Does the intensity pattern resemble C2H2 or C2D2 better? Can you explain your choice?) 8. Take all numbers and divide them by the value you get for Int (J = 1). This will make Int (J = 1) = 1.000…. exactly and will scale all other intensities to this one. 9. Graph your predicted rovibrational spectrum – both the P branch and the R branch – of the ν3 mode of C2T2; that is, graph lines with wavenumbers along the x axis with the height of the lines equaling Int (J = whatever). This should give you a stick spectrum that shows the proper wavenumbers and relative intensities for the rovibrational spectrum of C2T2. 102
To graph a stick spectrum in Excel: Sort your data by descending wavenumber. Select the column for wavenumber as well as your column for intensity, select Insert in Excel, then select Recommended Charts. On the All Charts tab, select the Column option on the left, and select Clustered Column at the top of the window. Ensure you have selected the chart with wavenumber as the x-axis (you should see a preview and it should look like a rovibrational spectrum), and click ok to add your chart. It is important that the data be sorted by descending wavenumber, or your chart will likely not graph correctly. For comparison, the experimental rovibrational spectrum can be found at: L.H. Jones, M. Goldblatt, R. S. McDowell, D.E. Armstrong. J. Mol. Spectrosc. 1967, 23, 9 – 14. 10. Turn in your calculations and your predicted spectrum as part of your report. (End of addendum.)
Results Figure 4 shows a representative summary of the experimental measurement of the rovibrational spectra of C2H2 and C2D2, along with the values of B′, B″ and De as determined by the best fit to a cubic equation. Figure 5 shows the linear fit of the calculated frequency parameters that can be used to predict ν0 for C2T2: ν̃0 = 2066.32 cm-1 (compared to an experimental value of 2072.4 cm1 (6)). Using the value of B from the computational output and using the simplified expression for ν̃
(simplified equation 13), and following the instructions for generating a stick spectrum, we can get the predicted spectrum shown in Figure 6. How does this compare to the experimental rovibrational spectrum of C2T2? Figure 7 shows the value of the absolute error versus J value, using the assignments from the previously published spectrum of C2T2 (6). The ~6 cm-1 error occurs at the band head and decreases as J deviates from 0. Sources of error include:
• error in the calculated values of the vibrational frequencies, including errors inherent in the computational model (including scale factors); • assumption that B′ = B″; • use of a calculated value for B for C2T2 rather than an extrapolated value; • neglect of De; and • possible inaccuracies in the experimentally-measured spectrum. That said, an error of ≤ 6 cm-1 should be considered extraordinarily low and an additional verification that this sort of exercise is justified. We requested feedback from the students who performed this experiment in two semesters of our physical chemistry lab. Specific comments on this lab were minimal, but one student did write “It’s kinda cool how a few quick calculations using Gaussian allows you to pretty accurately predict the IR spectrum for isotopes using the spectra for the other isotopes!” Other students gave oral feedback on the first draft of the Addendum, which allowed us to revise it to the form presented here.
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Figure 4. A representative graph of the frequencies of the rovibrational lines versus the index m (see equation 13) and the molecular parameters derived from the coefficients of the cubic curve fitting.
Figure 5. Fitting used to extrapolate the value of ν0 of C2T2. Substitute the value of 1/sqrt(μ) from the C2T2 vibrational calculation to obtain the predicted value of ν0. 104
Figure 6. A graph of the predicted rovibrational spectrum of C2T2 without the impact of nuclear degeneracy (top) and with nuclear degeneracy (bottom). The relative intensities are in arbitrary units, but they have been scaled as described in the Addendum.
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Figure 7. Error between the calculated frequencies for the rovibrational spectrum and the experimental frequencies for the rovibrational spectrum. See text for the definition of the variable m.
Conclusion We have demonstrated that it is possible to predict the rovibrational spectrum of C2T2, extending a known physical chemistry lab experiment to an isotope that is seldom considered in the undergraduate curriculum. This extension requires computation of the optimized geometry and vibrational frequencies of C2H2 and its isotopomers; for such a small molecule, even a reasonablysized basis set and basic computer resources allow for these calculations to be done on the order of minutes (if not faster). Most of the additional work is numerical work that can be done after analysis of the experimental data and using a spreadsheet. The accuracy of the prediction is excellent. If there is one lesson to be learned in this exercise, it is to understand as best as possible the mathematical manipulations being performed by a computational chemistry program. Without a better understanding of how the GAUSSIAN program performed its vibrational frequency analysis, we would never have understood the relationship between the effective masses and the vibrational frequencies; only a step-by-step analysis of the linear algebra involved gave us the clue to being able to confidently extrapolate to a spectrum of C2T2. The prevalence of inexpensive and easy-to-use computational chemistry programs, combined with the almost ubiquitous presence of computers in the modern workplace, have conspired to allow many users of these resources to treat them, effectively, as a black box. Put some numbers in, get some numbers out, and assume those numbers make sense. However, it should be better than that: someday we may be required to defend those numbers, and if we don’t know how they were generated as well as we should, we cannot defend them properly. Caveat utilitor. (If different computational chemistry packages are used—and there is no reason not to use them—users should understand how those packages calculate the normal modes of vibration.) Finally, the word “luck” was used in the Introduction, and although as scientists we don’t believe in luck per se, it is worth pointing out two facts, both of which had to be true for this procedure to work. First is the fact that the vibration involved (ν3, the asymmetric C-H stretching motion) is a parallel absorption. This allows the rovibrational spectrum to adopt a simple pattern similar to those found in heteronuclear diatomic molecules, i.e. the regular spacing of rovibrational absorptions that can be used to determine molecular parameters. Second is the fact that this is one of the five normal 106
modes, and the only parallel absorption, whose vibrational frequency is related to its effective mass in a straightforward way. It is probable that similar exercises can be performed on the other vibrations of acetylene, but likely the mathematics is much more complicated. Were either of these issues not satisfied, this exercise would not have been developed as easily as it was.
Acknowledgements The authors would like to thank the following students of our physical chemistry lab courses who performed the original experiment and the extension described here, pointing out issues and making valuable suggestions that improved the exercise: Hamoud Asi, Paul Benton, Nathan Canterbury, Stacey Cargill, Brian Davis, Teya Eshelman, Maranda Florjancic, Megan Frey, Rachel Grabowski, Kathryn Kiesel, Drew Kingery, William Knight, Marek Kowalik, Chris Lattanzio, Keith LeHotan, Carrie Lewis, James Maher, Connor Meek, Cody Orahoske, Abboud Sabbagh, Sydney Simpson, Daniel Terrano, Jocelyn Thompson, Ilona Tsuper, and Morgan Worthley.
References 1. 2. 3. 4. 5. 6. 7.
8. 9.
Garland, C. W.; Nibler, J. W.; Shoemaker, D. P. Experiments in Physical Chemistry, 8 ed. McGraw-Hill: Boston, MA, 2009; pp 424−436. Garland, C. W.; Nibler, J. W.; Shoemaker, D. P. Experiments in Physical Chemistry, 8 ed.; McGraw-Hill: Boston, MA, 2009; pp 416−424. Ball, D. W. Physical Chemistry, 2 ed.; Cengage Publishing: Stamford, CT, 2015; pp 644−648. Shimanouchi, T. Tables of Molecular Vibrational Frequencies; NBS Circular NSRDS-NBS 39, 1972. Tritium. https://en.wikipedia.org/wiki/Tritium (accessed 14 February 2018). Jones, L. H.; Goldblatt, M.; McDowell, R. S.; Armstrong, D. E.; Scott, J. F.; Williamson, J. G.; Rao, K. N. Vibration Rotation Bands of C2T2: Part I. ν3. J. Mol. Spectrosc. 1967, 23, 9–14. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, Ö.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian, Inc. Gaussian 09, Revision D.02; Wallingford, CT, 2009. Vibrational Analysis in Gaussian; White paper available at www.gaussian.com/vib/ (accessed 15 February 2018). Herzberg, G. Molecular Spectra and Molecular Structure: Infrared and Raman of Polyatomic Molecules; D. Van Nostrand Company, Inc.: Princeton, NJ, 1966.
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