An Aufbau Methodology for the Modeling of Rotational Fine Structure of Infrared Spectral Bands Boyd A. Waite United States Naval Academy, Annapolis, MD 21402
The mapping of molecular energy levels onto a spectrum is mediated bv the rules that govern transitions between those levels: molecular transition X energy spectrum (1) rules levels
The "Aufbau" Methodology
The rules that govern transitions between levels include the selection rules (which indicate the levels that can be connected through optical interaction),the rules that govern probabilities of selection-rule-allowedtransitions, as well as the rules that govern line shapes of transitions. Selection rules determinethe locations 02 spectral lines, while probability rules determine the intensities of spectral transitions. or the numoses of this work.line-shane rules are concerned with line-broadening effects involved in transitions (natural line width..Dooolereffect..etc.). as well as resolution parameters of spectrometers. The objective of this work is to provide a methodology for ascertaining the effects of these various rules on the resultant spectrum. The methodolow involves a sequential "build& up" or "aufhau" of the &ection, probability, and Line-shape rules that eovern spectral transitions. The insight provided by this methodology should assist students of spectroscopy in unraveling the apparently complex connection between quantum mechanical energy levels and observed molecular spectroscopy. A simulation of a given high-resolution infrared spectrum can be built by sequentially invoking the various rules that eovern transitions. For example. for the high-resolution gasphase infrared spectrum of methyl i o d i d e ' i ~ ~ s ~whicb ) ' , is the spectrum studied in detail in this work, a possible choice for this aufhau simulation methodology in&des, in order of operation on the molecular energy levels, (1) line-shape rules, (2) selection rules, (3) Boltzmann population rules, (4) transition moment rules, (5)nuclear symmetry rules, (6) rotational/vibrational coupling effects, and (7) Coriolis effects. Of course, selection rules are the major governing rules f o r spectral line positions, with rotational/vibrational coupling and Coriolis effects being of minor (but significant) imoortance. Intensities of soectral lines are governed prim a h y by Boltzmann popula'tion rules, with interesting and verv sienificant effects also arising from transition moment rulls and nuclear symmetry rules.%he purpose of this study is todemonstrate the effects (bothmajor and minor) of these physically relevant rules. Previous work in infrared band simulation has focused primarily on fitting parametrized curves describing positions, widths, and intensities of infrared spectra, thereby extracting the physically relevant rotational. couoline. -. Coriolis.. etc... constants.2 The discussion that follows demonstrates bow the simulation technique of this study can, giuen the relevant physical parameters, yield highly accurate spectra as well as serve as an instructive tool for understanding the complexities of molecular spectroscopy. Some of the computational details (e.g., how line width and resolution are incor~orated)are described in the appendix.
where o and u are the vibrational frequency and quantum number, respectively, for the particular vibrational mode involved, J and K are the two rotational quantum numbers, and B, and A, are the rotational constants. B, and A, depend parametrically on u, the vibrational quantum number, due to rovibrational coupling in excited vibrational states, a typical u dependence being given by
-
. .
&.
.
Molecular Energy Levels For a symmetric top molecule like CHJ, the energy levels for rovibrational states are given by3 E(v,J.K) = (u
+ l/Z)w + BJ(J + 1)+ (A, - B,)P+ ...
(2)
where Be and A, are the rotational constants corresponding to the equilibrium geometry of the appropriate molecular electronic state. For the purpose of building up, it is convenient to restrict the u dependence of B, and A, until after some of the selection rule and probabilitv rule effects have been investigated, so as not to bbscure these effects among slightly shifted transition lines. Thus, until that point in the simulation, the coupling constants d. andg, are set to zero. Line-Shape Rules In this model, the rules governing line shapes are invoked from the outset and are held constant during the entire building process. The lifetimes of excited vibrational states in small gas-phase polyatomic molecules like CH3I (in the range of 10-8s) combine with the collisional broadening and Doppler broadening to yield line widths for rovibrational spectra on the order of 0.001 cm-l.4 The modeled resolution of the spectrometer is initially chosen to simulate a very high-resolution setting (0.1 cm-') in order that a detailed accounting of line shifts due to rovibrational counline. etc.. will not be obscured. The anpendix describes the &ple interpolative approach utilized in this study for incorporating the line-shape information in the finite-resolution graphical representation of the spectrum. A~odizationsimulation (e.e.. boxcar or triangular aDodizatio$ or slit width effects forsimulating dispe&ive measurements are not incor~oratedin this studv and are not expected to be significant for the methyl iodide system with typical rotational fine-structure spacings of 0.5 cm-' comG e d to the resolution setting of0.l cm-'. Such technical
' McNaught, I. J. J. Chem. Educ 1982,59,879-882.
See, for example. Lin. C. L.; Shaw. J. H.; Calvert. J. G. J. Quant. Spectrosc. Radiat Transfer 1980, 23.387-398. Herzberg, G. H. Molecular Spectra and MoIecular Structure 11. hfraredand Raman Spectra of Polyatomic Molecules; Van Nostrand: New York, 1945. Steinfeld. J. I. Molecu1esandRadiation:An introduction to Modern Molecular Spectroscopy; MIT: Cambridge. MA. 1979. Gritfiths,P. R.; DeHaseth. J. A. Fourier Transform h f r a d S p e e tmrnetry; Wiley: New York, 1986:pp 15-25. Volume 66 Number 10 October 1989
805
Parameters for "Aufbau" Simulation ot the Rotational Flne Structure Band Spectrume
l band
0.10 0.10
1 2
salection rules Boltzmann factw wansition prob. nuclear symmetry rovib coupling rovlb coupling Coriolis coupling magnified Fig. 7 experimental magnified Fig. 8 ,esoIUtion temperawe line widm
1 band
0.10 0.001 298.15 0.10 298.15 bsndataazom-'and 1 I bandat 1251om-'. /I = 5.W1 cm-'. B. = 0.255 om-'. 12
-0.002
0.0145
0.0
parallel band experimental
13
a
Figwe 1. --band spectrum of CHj. showing lhs mappimJot the anergl levels re~Jtlnghom selection ruler. All bansltlonr are asslgned an t lntensm
Figue 2. Specrmm Including the enects of Bollzmann population Nles lor inlial slates. Compare to Figwe 1
aspects of infrared spectroscopy, while significant from an analytical point of view, are not the primary focus of this study. Variations in Line shape and spectrometer resolution will be demonstrated after the basic model is developed in order to demonstrate the effects of condensed phase (i.e., short excited state lifetimes for rovibrational states) and lower spectrometer resolution (see Figs. 9 and 11).
pendent rotational constants as tabulated in the table (which includes parameters for all subsequent figures). All of the possible values of J and K (of the initial state) and J' and K' (of the final state) are sampled, those transitions that satisfy the selection rules being assigned unit intensity (arhitrarv units). Thus, there is a slight variation in the "intensities%f the$-branch transitions due to the fact that up to an arbitrarily high set of rotational quantum numbers J and K, there are more A J = 0 transitions for low IKI than for high IKI. The weakly intense transitions near the base Line (Rand P-branch transitions) correspond to the relatively few connections between energy levels via the A J = f1selection mle.
Selection Rules The selection rules for rovibrational transitions in symmetric too molecules are distinguished hy whether thevibrational trksition is of the paraliel or perpendicular type. For CH31,there are three parallel vibrational transitions (nondegenerate) and three perpendicular vibrational transitions (doubly degenerate). The selection rules governing these transitions are given by3
1 transitions Itransitions
AK = 0 AK=O AK = i l
LW = 0, f1 AJ=H AJ = 0, i 1
ifK f 0 ifK=O
(4) (5)
Throughout most of the remainder of this simulation, we will focus on one of the perpendicular vibrational transitions of CHJI,centered at 882 em-'. However, the results of applying the simulation methodology to one of the parallel hands (1251 ern-') are also resented (see Fics. 12 and 13). ' Figure 1 shows themapping that re<s from invoking the oemendicular hand selection rules for CHd, using the specirai line half-widths, spectrometer resolution, and u-inde-
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Journal of Chemical Education
Boltzmann Pooulation Rules The simplest extension of the selection rule mapping of Figure 1is inclusion of initial state population distributions as governed by the Boltzmann population distribution func-
where P J K is proportional to the population of the rotational K the degeneracy of the (JJO level (J,K) in the sample, ~ J is rotational level given by g,=W+1 =2(W+1)
ifK=O
(7)
ifKZO
and EJKis the rotational part of the energy expression given
Figure 3. Spectrum including me e n e m of bansltionmoment rules fw selec. tion-rul%allowed bansnians. Compare to Figure 2.
Figure 5. Specbwn including Um effecl of mtationallvibratlonaImupling, with 4 # 0. See table. Compare to Figure 4.
Figure 4. Spectrum including Um etfects of nuclear symmeby rules. Fw me case of CH.1. a G,molecule, Um ene* is wch Umt every third (1 branch is twice as intense as predicled using only Boltzmann and transition mment rules. Compare to Figure 3.
F W s 8. Same as Figure 5, now including.g f 0.See table. Rlmary e f f m is a sllghl shilling of posnion of 1ransRions. giving rise to an apprent ' broadeh Ing" of me s p e m m bansitions.
in eq 2. Intensities of spectral transitions are assumed to he orooortional (at this staee of the simulation) to the oopulaLions of the initial states.-Figure 2 shows the expected r<s of including this Boltzmann population rule in the mapping of the energy levels into the spectrum for CH31. Since the Boltzmann factor exponentially falls to zero for large (J,K) values. this ereatlv simnlifiesthe calculational asnects of the simul&ion,&eioopkg over all possible (J,K)'values can now he limited so as to include onlv those states that contribute greater than some small factor 6 to the intensity of a given transition.
Nuclear Symmetry Rules
It is well known3 that for C?., molecules where the off-axis nuclei have spin 1/2, there is &I intensity modulation in the oooulations of states that manifests itaelf in the rotational it& populations. This effect is independent of the Boltzmann population effects descrihed ahove, and for the specific casi of CH31the resulting modification to the rotational state populations is that states withK = 0,3,6, etc., are twice as states with K = 1, 2, 4, 5, etc. Figure 4 as demonstrates the effect of these nuclear symmetry rules on the maooine of the soectrum. Note the aonearance of the typical intensity pattern for Q branches, &ch are characteristic of the I hands of Ca molecules. &
Transition Moment Rules
For the case of symmetric top molecules, the transition moment p (i.e., the probability of transition between selection-rule allowed states) is given by3 n-tupe
f o r u = +I
= [(J+ I)? - K~I/[(J+ I)(ZJ+
01 18)
where the upper sign in the perpendicular hand case refers to AK = +1, and the lower sign refers to AK = -1, and (J,K) refer to the lower state of the transition. In addition, the transition prohahility is proportional to the frequency v. These effects are depicted in Figure 3, which indicates that the dependence of transition moment on rotational state is very insignificant, the greatest effects being seen in states of low (J,K) values.
&
The next step in the "aufbau" simulation of high-resolution infrared soectra is to allow the centrifueal couoline . to be "turned on". +he effect of the relaxation 2 the approximation d. = e. = 0.0 is exoected to manifest itself as a slieht shifting o F n ~ a nof~the spectral lines (depending on the magnitude of d. and g.), which up to this point have been isoenergetic. Figure 5 shows the effect of relaxing the coupling constant d, to its experimental value1 for the 882 cm-I hand of C H J (see table). Fieure 6 continues the relaxation by setting g i t d its experimeital value' (see table). Converselv. it mav he oossihle to adiust these oarmeters the experimentally by fitting thk similatei spectrum determined soectrum. In either case. note how the couoline . constant d , Gfects the change in the spacing between adjaand the cent lines m o n n the R and P branches ( A J = flJ coupling const&t g. affeds the change in the spacing hetween adjacent lines among the Q hranches (AK = f1, A J = 0).
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Figure 7. Campleteiy "built-up" specbum, including the effect of the Corioiis coupling. Forlhis 882cm-' band, the Corioiisconstanl(is positive. giving rise 10 an effective reduction in spacing betweenenergy levels. Compareto Figure 6. a. (below) Magnified inset of Figure 7, showing lhe detailed hansitions in the range 650-900 cm-'.
Figure 8. Experlmentai 0.1 Em-' resolution specmrm otCH.i, showing me 882 band Campars to Figure 7. a. (below1 Llagnilied "set of the sxpsricm-' , mental apecbvm ot Figure 8. Compare to Fagwe 7a
Coriolis Effects
The perpendicular hands in symmetric top molecular spectroscopy arise from doubly degenerate vibrational states. For molecules with more than one set of degenerate perpendicular hands, there arises a Coriolis coupling that affects the rotational energy levels, yielding a corrected energy level expression E,,(JJO given by3 E,(Jrn
= E,(Jrn
-
2A"W
(10)
where is the Coriolis coupling constant for the ith perpendicular hand and E,(J,K) is the rotational part of the uncoupled energy level expression of eq 2. These constants are governed by summation rules, e.g., in the molecule CH3I (where there are three perpendicular hands) the summation rule takes the fonn3 (11) i-1
Figure 7 demonstrates how this Coriolis coupling modulates the previous spectrum of Figure 6, the major effect in the case of the 882 cm-' band (f = 0.2103)' heing a compression of the uncoupled spectrum, indicative that the Coriolis forces for this mode reduce the spacing between rotational energy levels. Comparison to Experiment
The spectra of Figures 7 and l a (which is a magnified inset of Fig. 7) can now be compared to the experimentally ohtained high-resolution spectrum of the 882 em-' hand of CH31, depicted in Figures 8 and 8a (magnified inset of Fig. 8). All experimental spectra were ohtained on the Digilah FTS-65 Fourier transform infrared spectrometer, the high resolution (0.1 cm-') being ohtained by using a MCT lowtemperature detector system. Both the simulated spectra 808
Journal of Chemical Education
Figure 9. Simulated spectrum showing Uw effen of lower resolution (1.0 cm-I). Compare to Figure 7.
and the experimentally ohtained spectrum are for resolutions of 0.1 cm-1 and room temDerature. Notice the accuraw of the simulation, including some of the finest details df Figure la, such as thegeneral variance in intensity of the low intensity R and P hranch peaks between the high intensity Q branches. The annarent broadenine of the s~ectrallines (comnare of Fig. 7 adi Figs. 1-4) arises-prim=& from the in&& the correct vibrational state denendence of the rotational energy levels, giving rise to small' shifts in locations of many of the transitions. Thus, rather than there heing many coincident transitions (giving rise to the S-function-like peaks of Fias. 1-4). there are many nearly coincident transitions at ~ locations (giving rise to both the Q and the R , hranch apparently broadened features of the actual spectrum). Rememher, however, that the actual line-width broadening of individual spectral transitions due to natural excited-state lifetimes.. Doonler broadenine. -. etc.. is on the order of 0.001 em-' for this system.4 &.
Figrre 10. Simulated spemum showing effeot ol increased temperawe (1000 K). Note increased lntensntesdbanshionscorrespondlngto higher (J,Q initial stetes. Compare to Figure 9.
Figwe 11. Speceum shawing lhs effect of bah low-nraolution and greatly increased line width, simulating a condensed-phaweinfraredspecbum. S m I a l Effects Figure 9 demonstrates how the simulated spectrum of Figure 7 is modified by settine the resolution t o 1.0 cm-1. ~ i & e 10 shows the effect of increased temperature, e.g., T = 1000 K, where the primary modification is in the Boltzmann population rulesuch that transitions involving initial states of higher (Jmare more intense. Finally, Figure 11 shows the effect of not only poorer spectral resolution (3 cm-I), but also of broadening t h e line-width (2 cm-'), such as might be observed in a condensed phase spectrum. Parallel Band Exam~le The same methodology of building u p an infrared specto other bands trum as demonstrated above is easilv . applied .. of CHJ or to other molecular systems. Figure 12 shows the final result of a high-resolution simulation for one of the parallel bands (1251 cm-l) of CH31, including all of the effects described above. Comparison with the experimentally obtained spectrum in Figure 13 indicates an excellent mapping, not only in spectral transition location, but in intensities as well.
Figure 12.Mmplete simulated high-resolution spectrum of lion of CH.1 a1 1251cm-'.
11-band bansl-
Figue 13. Experimemal hlgh-resolution specbum of Ib 11-band transhim ol CHd. Canpare to Figure 12. of the mapping of energy levels onto spectra, especially for spectroscopies involving more than one type of molecular motion. Appendix There are several computational methodologies for treating line width and resolution in spectroscopy. The following gives a simple approachutilized in thecomputer simulationsdepicted in this work. Assuming the range of the spectrum (q, to "higb) and the speetrometer resolution (n) to be specified, an array r of length (ukph v~,)lo + 1is estahfished in which will be stored the intensity information generated as described in the text. Consider a given transition (connectinginitial and final states) of frequency u and intensity 7 (as cnlculakd based on the rulps de~ h to scribed in text). The I'-mray indices II,. and ~ b , corresponding the bins surrounding frequency 1 are grven by
ihigb=it,
+1
where int(x) retains the integer part of x. The intensity of the transition y is assigned to these surrounding bins of the r array according to
Summary I t has been demonstrated how a simple computational simulation of relatively complex rovibrational infrared spectra can be both highly accurate and instructive. The "aufbau" methodology of stepwise building of infrared spectra, from line-shape&les, t o selection ruie mapping rdes, to transition probability (intensity) rules, etc., is an excellent tool in understanding the influence of each on the overall spectrum. Such "augau" simulations are easily extended to other molecular svstems. including vibronic simulations (e.g., 1%absorptionapectrkn). I t is ioped that such efforts would assist students in overcoming the conceptual barrier
where account has been taken of the fact that the r-array elements may already have intensity assigned to them from previously treated transitions. The half-width {(as determined by line-shaperules) is accounted for simply hy repeatingthe above process for transitionsof frequency u f t each with intensity 712. The generation of the graphical representation of the spectra is accomplished by a simple XY plot (available on mast commercial graphics applications systems) of the array as a function of v.
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