Energy States of Molecules

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Resource Papers VIIIPrepared under the sponsorship of

The Advisory Council on College Chemistry

W o r k in a modern chemical laboratoryincreasingly depends upon instruments that allow themeasurement of the absorption or emission of radiationby a sample. All such work depends on an energyseparation between states of the atoms or molecules ofthe sample. The variety of states that can he studiedor used in this way leads to the many areas of spec-troscopy that are now of importance: nmr eprinfrared ultraviolet and so forth. A treatmenthere of all these areas of spectroscopy and the energylevel patterns on which they are based would encompassfar too much territory.Here the discussion will he restricted to those energiesthat are partially populated and therefore need to beconsidered in any deduction of the thermodynamicproperties of the material or of the reactions in whichit is involved. The energy levels that must he treatedare in this way restricted to rotational vibrationaland electronic and the corresponding spectroscopicstudies are reduced to microwave infrared and Ramanand visible and ultraviolet. The scope so defined is

J Leland HollenbergUniversity of RedlandsRedlands, California 92373

outlined in Table 1.nneem will he focused on the determination of the

Energy States of Molecules

spectral patterns corresponding to these three types of

energy levels and to the relationships that can be estah-lished between these observed patterns and the param-eters of the molecular model. The analytical uses ofthese types of spectroscopy will not he considered norwill any serious treatment of the experimental tech-niques be attempted.Rotational Energies

If a model of a molecule that is a simple rigid rotor isconsidered and if the appropriate restriction on theallowed angular momentum is imposed or the rotationalmotion is treated by use of the Schroedinger equation

This series of Resource Papers is being preparedunder t,he sponsorship of the Advisory Council on CollegeChemistry. The Advisory Council is supporbed by theNational Science Foundation. Professor L. Carroll King,of Northwestern University, is the chairman.Single copy reprint ^ of this paper are being sent tochemistry depwtment chairmen of every U. S. Institu tionoffering college chemistry courses and to ot,hers on themailing lis t far the ACa Newsletter.This is Serial Publication No. 46 of t he Advisory Council.

Table 1 Types of Spectroscopy CoveredType of Range of energiesspectros- Frequency mo?cularf Information

C O P Y (see-1) em- kcal mole-'* energy obtainedMicrowave 10'-10 3 X 10-'-3 10-e10-2 Rotation of Interatomic distances,heavy mole- dipole moments,cules nuclear interactionsFar 10'L1O1a 3-300 10--1 Rotation of Interatomic distances,infrared light mole- bond force con-cules, vibra- stantstions of

he aw mole-Infrared 10'8-lox4 300-3000 1-10 culeiVibrations of Inte ratomic distances,light mole- hond force con-eules. vibrsr stants, moleculartion-iotation charge distributionsRaman ~O'LIO'~ 3-3000 10-P-10 Rotation, Interatomic distances,vibratiotions bond force con-stants. molecularchar& distributions(for energy changesnot ohsemable withinfrared)Visible- 1OlL10 3 x lo3- l lo5 Electronic All above propertiesultra- 3 X 106 transitions pl w bond d~ssociarviolet tlon energiesCompare to the average thermal kmetic energy per degree of freedom, 12RT g0.3 kcal mole-', or -100 om-'.

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we deduce that only the rotational energies E thatconform to

are allowed. Here I is the moment of inertia of themolecule, B h2/8a21 is known as the rotationalconstant, and J is the rotational quantum number.For a diatomic molecule I pr2 where p is the re-duced mass mlm2/(ml ma) and r is the distance be-tween the two atoms. The pattern of energy levelsbased on this model and treatment is shown in Fiaure 1.

Figure 1. Energy of rotation sfunction of J quantum numberfor the rigid rotor.

-For each energy level in-dicated by avalue of J thereare2J 1rotationalstates.These can be looked on ascorresponding to the pos-sible angular momentumcomponents along a partic-ular direction in space.Alternatively one can recog-nize at this stage that therotating-molecule problemis formally the same as therotational aspects of thehydrogen-atom problem.Thus J is identified s the

counterpart of the orbital angular momentum quantumnumber 1, and the several states included in a particularJ designation correspond to the atomic states indicatedby the possible values of m for a given value of 1.Analysis of the quantum mechanical rigid rotor, asoutlined above, is carried out in greater detail in avariety of physical chemistry, spectroscopy, and quan-tum mechanics books. In addition to the bookslisted in the bibliography, reference can be madeto several others for this introductory material 1-3).The t)xoretical treatment can be extended to allowthis calculated pattern to be confronted with experi-mental spectra. The selection rules governing allowedtransitions between rotational states require thatJ I, or, for absorption spectra, J +1, andthat the permanent dipole y of the molecule be non-zero. (Thus homonuclear diatomic molecules likeH, and O2 are expected to exhibit no pure rotational

spectra.)For an absorption spectrum, the above treatmentpredictsAE, E, J 1) E, J) = 2 B J I),

J = 0 , 1 , 2 , 3 . . . 2 )Thus the rigid-rotor model predicts that the spectrumshould consist of a series of absorption lines at 2B, 4B,6B, 8B, etc., i.e., a series of equally spaced lines 2Bapart. That this is approximately borne out by ob-served spectra is shown by the pure rotational spectrumof HF, a portion of which is presented in Figure 2.If known internuclear distances are used to computeI substitution into eqn. (2) soon indicates that for allbut the lowest moment of inertia molecules, like HF,the rotational transitions, except for those with ex-tremely high J values, have such small AE hat thespectra occur in the microwave region. Microwavestudies of many relatively small molecules have beenmade since World War 11 and this area of investigation

Figure 2. Pure rototion spectrum of HF(g) run on Beckmen IR-12. About2 6 0 mm pressure in 10-cm brass cell with polyethylene windows. Ex-traneous line ne r 3 8 5 c m - I s due primarily to SiF* impurity.

Figure 3. Microwave rotmtion spectrum of CHsCHKHnl obtoined withHewleft-Pockard 84 00 C spectrometer. Sample pressure war 50 irRelative proportions of gauche and trans mta men re related to re sunder each line. Note the regular spacing of lines. (Courtesy of DrHoward Harrington, Hewlett-Packard Corporation, Polo Alto, Colifornim.)

is well described by Sugden and Kenney 4). Thetype of spectrum that can be obtained by this tech-nique is illustrated in Figure 3 which shows a smallpar t of the microwave spectrum of CH,CH,CH,I.Because of the very great frequency accuracy attain-able with microwave spectrometers, internuclear dis-tances of six significant figures can, at least formally, be

lMicrawave spectroscopy remains a much less frequently en-countered tool than infrared and visible-uv, which will be dealtwith later. Two principal obstacles have prevented the wide-spread adoption of this technique in chemical laboratories.In the first place, commercial equipment has, until recently,not been available although now a unit is advertised by Hewlett-Packard.The second obstacle ttrisks from the difficulty of making quan-titative deductions of concentration from line intensities and,therefore, t he difficulty of doing quantit ative analysis.A very significant development due t Harrington 6 ) nowgives s method of determining separately the molecular eoncen-tration N, and the relaxation-broadening time r. N is number ofmolecules per unit volume, and is the mean time between col-lisions which broaden the lines. N and r vary widely with gaspressure, molecular properties, and J . The product of N andgoverns absorption. Experimentally, Harrington has showntha t measurement of the Star k modulated imposing an electricfield on the sample) microwave absorption signal amplitude andthe incident microwave power level allows calculation of Nand r. Thus a wide range of new uses for microwave spectros-copy such as quant itat ive analysis, isotope effecta, kinetic studies;and chemical process monitoring is now becoming possible, andmicrowave spectroscopy may become a mare frequently usedtool in the chemical laboratory.Volume 47 Number 1 January 1970 3


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calculated if the observed absorptions can be related tothe molecular model.Careful analysis of rotational spectra of a linearmolecule reveals that the line spacings are not quiteconstant and therefore the rigid-rotor model is notentirely adequate. Inclusion of a term, D, for centrif-ugal distortion, which represents the increasing mo-ment of inertia as the bond stretches with greater energyof rotation, leads to the modified energy pattern

Details of the development of this relation are given byHerzberg 6). This modified energy pattern can betested against the observed spectral transitions if theexpression for the transitions allowed by the selectionrule AJ +1 is again obtained. We have nowAE, 2 B J 1 ) 4 D J 1): 0 1 2 , . . (4)

The improvement in fit with the observed spectra is,as shown by early studies of KC1 reported in Herz-berg, 7) such that no additional refinement of themodel for simple, linear molecules is necessary. Wethus have, for certain molecules, a useful expressionfor the allowed rotational energies.A physical chemistry infrared experiment analyzingthe rotational spectra of simple molecules and utilizingeqns. 2 ) and 4) bas been described by Hollenberg8). The increasing availability of far infrared in-struments, which extend the infrared range to the orderof 10 cm- , will allow a variety of rotational spectraof small molecules to be obtained and studied.So far no mention has been made of an experimentalconfirmation of the 2J degeneracy of the rotationalenergy states. Qualitative confirmation stems from theintensity distribution found in the pure rotationalspectrum. The intensity of a given component de-pends primarily on the population of the initial energy

level. For increasing J values the population increasesin response to the 2J 1 multiplicity term hut ulti-mately the exponential Boltzmann factor which worksagainst the population of higher energy states over-comes this trend. Detailed considerations of this aregiven in Herzberg 9). The two factors that operateto give a maximum in the population-energy curve are,it can be noticed, comparable with those that apply totranslational energies.Finally, in this consideration of linear molecules wereturn to those nonpolar examples, like H,, 02 , COf,that defy direct study of their rotational spectra byabsorption spectroscopy. Fortunately a different spec-troscopic techn~que hat depends on a scattering, orsimultaneous absorption-emission, process is available.This technique, Raman spectroscopy, provides theinformation not available through absorption spec-trosopy.Historically, Raman spectroscopy was developedlong before infrared became popular. However, thelong exposure times needed to detect Raman lines on aphotographic plate contributed to the rapid rise inpopularity of recording infrared spectrophotometers.Recently continuous gas lasers have been adapted asRaman sources l o ) , and it seems likely that interestin this technique will grow among chemists.The Raman spectrum is obtained by irradiating thesample with intense monochromatic light of frequency

uo. Examination of the scattered light with a mono-chromator shows that in addition to the incident fre-quency vo, there are much weaker lines displaced fromvo by various amounts Au,. It is found that each Avinvolves transitions between molecular energy levels.Both rotational and vibrational energy transitionscan be observed by this technique. In Raman spectrathe interaction of the radiation with the molecule de-pends on a dipole tha t is induced by the electromagneticfield of the light energy

pl d aEwhere r is the polarizability of the molecule, and Eis the electric field vector. As shown by Herzbergl l ) ,only when there is a periodic change in a is thequantum mechanical transition moment y .d/ non-zero, permitting energy transfer between states andn. The necessary, periodic changes in a can be pro-duced by rotation of the molecule, except for those ofnear spherical shape like CH,, resulting in a rotationalRaman spectrum.The pattern of rotational energy levels expected formolecules that can only be studied by Raman spectros-copy can be approached as before on the basis of therigid or non-rigid rotor. To this must be added theselection rules governing transitions between theseenergy levels. Now, since the transition is not re-stricted to an increase in energy as in absorption ex-periments the selection rules are different in kind anddegree. Calculations show that transitions with J0, + 2 are allowed.The comparison of the rotational Raman spectra ofN2 0 and Cop, as shown in Figure 4, which are bothlinear and have similar moments of inertia, and theappearance of the rotational Raman spectrum ofacetylene, Figure 5, suggest that some additional feature

Figure 4. Representations of pure mtationol Roman spectra of CO,(vpper) ond N 2 0 lowerl. Note that all of the odd J lines for COz areabsent. Reprodurtiom of octvol spectra ore given in G. W. King, Spec-troscopy and Molecu lar Structure. Holt, Rinehort and Win don, Inc., Ne wYork, 1964, p. 294 and p. 195.

I , ' . ' 's d 3

1Figure 5. Represontation of a porBon of the pure rotational Ramonspectrum of C2Hzlgb Note the strong-weak-stmng.weak intenrilypattern. Bored on work of J. H. Collomon and 8. P. Saicheff, Can J.Phyr., 35. 373 (1 9571.

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must be added to the molecular model. This featurestems from considerations of nuclear spins and thePauli Exclusion principle. Although it is most familiarto chemists in connection with electron pairs and withortho and para hydrogen, the restriction on allowedstates in terms of the properties of identical nucleiand the symmetries of the states shows up most clearlyin rotation and vibration-rotation spectra. The sub-ject is very clearly discussed in Herzberg (12).The consequences of this restriction for statisticalcalculations of thermodynamic properties is that, forsystems with relatively small energy separations, thenumber of states that exists must be reduced by a fac-tor, known as the symmetry number, equal to thenumber of equivalent nuclei that can be interchanged bya molecular rotation. Thus, for homonuclear diatomicmolecules and linear molecules such as CO, and H-C=C-H, there are only half as many rotational statesin the rotational pattern as there would have been ifthe nuclei were not identical. (For HZ,where the ro-tational states are widely separated, more detailedconsideration of the consequences of the exclusionprinciple must be applied.) The inclusion of suchsymmetry considerations in thermodynamic calcula-tions is presented in most statistical mechanics texts, as,for example in that by Mayer and Mayer (IS).For polyatomic molecules all the above considera-tions can, with appropriate extensions and additionalcomplexity, be carried over. For symmetric topmolecules, which have two of their three moments ofinertia equal, the analogy with the linear examples isquite close and the rotational energy patterns, selectionrules, and rotational spectra are often little differentfrom that for a linear molecule. For a general asym-metric top molecule the allowed rotational energypattern is complex and cannot be expressed by a simpleequation. The spectrum is likewise complex and diffi-cult to analyze. Nevertheless such spectra can beinterpreted in terms of a molecular model, and themoments of inertia and the rotational energy patternhave, in many cases, been deduced. Generally thisrequires recourse to a variety of isotopic substitutionsand to the imposition of an electric field to producea Stark splitting to help identify the states involvedin a transition. Treatment of these asymmetric mole-cules is discussed in considerable detail by Sugden andKenney (4).Other complications can occur that interfere withthe deduction of an energy pattern and molecularparameters. For example, a significant fraction of themolecules may occupy the first excited vibrationdstate if the vibrational energy spacing is quite small.Such molecules have a slightly different moment ofinertia, resulting in new microwave lines. The natureof this excited vibrational state is revealed by the micro-wave spectra. For example, in linear OCS, a stretch-ing vibration, which causes a slightly larger I, producesweak satellite lines on the low frequency side. In theOCS bending vibration, where the excited st ate resultsin a slightly smaller I, extra microwave lines are foundat somewhat higher frequencies.Although such situations lead to interesting spectralstudies, they are not a serious impediment to the collec-tion of data needed for most statistical studies.Thus, directly from spectroscopic data, or by calcu-lations using the rotational parameters, the moments of

inertia, the rotational states, and energies of gas phasemolecules can be readily deduced.By contrast, for the liquid state, even the idea ofmolecular rotation seems strange and only recentlyhas -much attention been paid to the possibility ofspectral studies of the rotation-like motion of moleculesin condensed states. The special cases of HZ and Pin inert solvents have received attention by Ewingand coworkers (14), and for these it is clear that rota-tional energy and perhaps even translational energychanges are responsible for the observed absorptions.(Neither species has a permanent dipole or a chargeand such spectra must owe their existence to inducedelectrical effects resulting from interactions with thesolvent.) Less direct evidence for rotation-like motionof other molecules in relatively inert solvents has beengiven in other spectral studies. However, our in-formation on the rotational, and translational, energiesof molecules in liquids remains less substantial thandoes tha t for gas phase molecules.Vibrational nergy States

Let us now turn our attention to the vibrationalaspects of a non-rotating, ball-and-spring molecularmodel. The simplest assumption for the nature of thesprings is that they obey Hooke's law, i.e., that theyexert a restoring force proportional to the displacementfrom the equilibrium position. The classical behaviorof such a system, that is the counterpart of a diatomicmolecule, is described by a characteristic vibrationfrequency of

where k is the force constant of the spring and, again,the reduced mass is r = m,mz/(m~+ mn).For such a system, subject to quantum restrictions,as can be imposed by using the Hooke's Law potentialfunction = 1/2kxz in the Schroedinger equation,only certain allowed vibrational energies are foundand these are given by the expressionThus, vibrational energies of

.. etc. are deduced. Theseenergies may be simplified.. to 1/2hv, 3/2hv, etc. The. I pattern is shown in Figure6A For each energy level

z only one state exists.E . E The selection rules that

. , are deduced for transitionsbetween states, according tothis model, give v = slv o . which, for absorption, im-

.-......------..lies Aw = l . The rules, a > ,a) require also that b y b

Figure 6. A, Vibrational one,- 0, that is, the electric dipolegier of a harmonic o~cillotor.8 Vibrational energies on the changeanharmonic oxi~ ~a to r. as the molecule vibrates.Volume 47 Number 1 January 1970 / 5


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Figure 7 Fundamental vibmtion.mtatbn spectrum of HClfg) run anBeckman IR-12. rerrurswa r .bout 1 0 0 mm Hg in o 10-c m cell. The lessintense line shifted just to the right of each of tho main liner ir due to thenatura l isotopic obvndonce of about 25 HCI .

Figure 8 Overtone vibration-rototion spectrum of HCllgl run on a Cory1 4 using the M . 1 obsorbonce slidewire. Sample pressure was obovt 10 0mm g in 10-cm cell. The less intense line shifte d just to the right of eachmain line is due to the nat urd isotopic abundanceof about 25 HCIJ'.

Thus, for heteronuclear diatomic molecules the modelpredicts a single absorption due to the change invibrational energy.Although such diatomic molecules do show a singlestrong absorption band, usually in the region 200-4000cm-' in the infrared, closer inspection shows the need

of modifying the Hooke's Law model. If, for the mo-ment the fine structure of the HC1 bands in Figures 7and 8 is ignored, we see that in addition to the funda-mental band attributable to a = 0 to u = 1 transi-tion, centered at 2886 cm-', there is another band atnearly twice this frequency, 5667 cm-', that suggests au = 0 to u = 2 transition. Thus, the selection ruleAu = l is not strictly followed.To allow for the existence, and frequency, of suchan overtone band the model must be modified, andthis is done by replacing the parabolic Hooke's Lawpotential function by a more general one. One caneither adopt a new, hopefully more satisfactory func-tion, the Morse function

= D . l -alr- l]Pbeing particularly convenient, or one can simply write aTaylor's series expansion to represent a general potentialfunction with a minimum at the equilibrium bondlength.With a Taylor's series

we can arbitrarily set V o= 0 , and the slope at the mini-mum ( b V / b r ) = 0 . Thus, the first nonzero term is of

the form ' / x 2 , and when only this value for V isused in the Schroedinger equation, the energies arethose of eqn. ( 6 ) for the harmonic oscillator. Higherderivatives of than the quadratic k are called cubic,quartic, etc., force constants.If the cubic term in is retained, that is, an anhar-monic potential is used, and only the first additionalterm that is introduced into the energy level expressionis retained, the predicted energy pattern is given by

where o is the harmonic vibration frequency andog eads to an addition term and is known as an-.harmonicity (15 ) . With such an expression the fre-quencies of the fundamental and the first few over-tones can be well fitted. The anharmonicity is gener-ally small but far from negligible. Thus, for HCI oneobtains from the two observed frequencies mentionedabove, the parameters w, = 2990 and o.x, = 52 cm-I.Such parameters are tabulated for most diatomicspecies by Herzberg I@ , both for ground and excitedelectronic states. One is usually most concerned withground state molecular properties.This simple anharmonicity treatment can be applied,in a student experiment, written by Boobyer and Cox( 1 7 ) , to a particular vibration of the polyatomic mole-cules CHCL and CDCb. The C-H and C-D stretch-ing vibrations are sufficiently ndependent of the vibra-tions of the rest of the molecule tha t one can proceed asthough an X-H or X-D molecule were being dealtwith. A set of overtone bands can be observed, ifnear infrared equipment is available, and the use ofthe anharmonicity term can be illustrated.The values of the force constants, which can becalculated from the o values, or approximately fromthe observed fundamental frequencies, lie mughlyaround 5 X l o 5 dyne cm-' for single bonds, 10X 10for double bonds, and 15 X l o 5 for triple bonds. Atable in the book by Davies ( 1 8 ) shows that anapproximate relation between bond order and forceconstant also applies to polyatomic molecules.Empirical correlations of Such bond parameters areuseful both in considerations of chemical bonding and inextending the vibrational energy pattern analysis tonew molecules. An early and often referred to rule,given by Badger ( 1 9 ) , gave the relation k(r , d)3 =1.86 where d is a parameter governed by the positionsof the atoms in the periodic table.More recently Herschbach and Laurie ( 0) aveshown a valuable correlation between w,, w x andquadratic, cubic, and quartic force constants on the onehand and bond lengths on the other. Their predictionsbased on diatomic data agree with available measure-ments on polyatomic species, regardless of the type ofchemical bonding involved.For polyatomic molecules we must first considerthe ball and spring counterpart on which our molecularmodel will be based. f such a model is given the free-dom of a gas phase molecule, as by being thrown in theair, we might try to analyze the complicated motionthat could result in terms of three translational andthree rot,ntional modes of the ent,ire system and theremaining modes of vibrational energy. Since thetotal degrees of freedom of an n body system is 3n,this implies n 6 vibrational modes must be de-

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scribed. (For linear systems there would be n 5such modes.)For a ball and spring model we might soon find certaincharacteristic, or natural vibrations in which each ballmoved in a straight line with simple harmonic motion.Analysis would show that any vibrational motion couldbe analyzed in terms of these natural or normalmodes (21).The separation of the kinetic and potential energiesof the vibrating system along normal coordinatestha t t he above implies allows us to carry over the treat-ment given for diatomic molecules to each normal mode.We expect, for a molecule with n atoms, n 6 vi-brational en6rgy patterns each of which is the counter-part of the pattern deduced for diatomic molecules.For some molecules the above deduction is im-mediately verified. For Hz0, for example, threestrong absorption regions are found in the infraredand these can be attributed to = to = vibrationaltransitions of each of the n 6 = 3 normal modes.For such molecules, particularly if overtone and com-bination bands are analyzed, one obtains a completepicture of the vibrational energy pattern of the mole-cule.For other molecules, the n 6 fundamentals cannotbe immediately recognized and this difficulty may stemeither from the complexity of the spectrum which con-sists of unidentified fundamentals, overtones and com-binations, or from the symmetry of the moleculewhich,swe will see, prevents some fundamentals from appear-ing. For such situations a calculation of the fre-quency and the amplitude of motion of each atom in aparticular normal mode can be determined by per-forming a complete normal coordinate analysis (22)which requires formulating the kinetic and potentialenergies of the molecule in matrix form. To adequatelyexpress the potential energy, a system of force constantsmust be chosen which allows for rather complete inter-actions between parts of the molecule. One of themost successful in terms of wide applicability is theUrey-Bradley potential function (23). Another typeof potential commonly used, known as the valence-force function, is illustrated in the calculations byBegun, et al (24) on some XAFl type molecules. Astill more general force field can be used, as illustratedby Krynauw and Schutte 35) in their study of C103-.Accurate determination of the vibrational energies of anumber of isotopically substituted species is usuallynecessary in such normal coordinate analyses. Inaddition the assignment of observed absorption fre-quencies to particular modes of vibration is aidedgreatly by studying the effect of isotopic substitutionon the spectra and hence on the molecular energies.

Interpretation of the spectra of HNO, by Hisatsune(26) is a good example of this spectroscopic use ofisotopes.Some guides to the identification of fundamentals inspectra of more complicated molecules come from theconcept of functional group frequencies, which areprincipally of value in qualitative analysis. Thecarbonyl group absorption near 1700 em-' is the stan-dard example.The physical basis for such group frequencies hasbeen discussed in an understandable way by Dows(27). The frequency-structure correlations are widelyused by chemists, and a text such s that by Silverstein

and Bassler (28) may be used for introducing this topicin the organic chemistry course. A book by Nakanishi(29) gives more depth and provides many examples ofinterpreted infrared spectra. An even more completecoverage is provided by Rao (SO), who considers with apractical emphasis the usual organic subdivisions, suchas heterocyclic compounds and hydrocarbons. Alsoincluded are quantitative analysis, high polymers, andbiochemical materials such as nucleic acids and proteins.In addition there is a chapter on inorganic compounds,but this topic is covered much more thoroughly byNakamoto (31), who also provides considerable materialon coordination compounds. The book by Colthup,Daly, and Wiberly (32) gives a much more theoreticalcoverage of infrared absorptions, but still gives con-siderable attention to the frequency-structure correla-tions. A brief discussion of group theory and thecalculation of thermodynamic functions is included.Each type of molecular vibration may be pictured ashaving its own set of vibrational energy levels with acharacteristic spacing. If a particular vibration in-volves a changing dipole, it may readily absorb photonsand give an infrared spectrum; the absorption fre-quency is governed by the spacing of vibrationallevels, E2 El = hv However, for many symmetricmolecules there are considerable numbers of the n 6vibrations which do not involve a changing dipole andtherefore do not appear in the infrared spectrum.Fortunately, many of these missing lines can beobserved in the Raman spectrum.A periodically changing polarizability is the re-quirement for a Raman band. The possibility of sucha change can be visualized for some vibrations, as iseasily seen by considering the stretching modes of COzshown in Figure 9. We assume that we can think of

Figure 9. The lhree vibration01modes o CO . The n 5 r u l e

Y~ i667cm? for linear molecules indicatesour vibration= should be presentThis apparent dbcrepmcy is ex-ploined by t h e foct that ur isV oubly degenerate. Thus, theres another vibration un in a planeu, 12349cm 1 perpendicular to t ha t shown.

bond polarizabilities, and we assume that the bondpolarizability gets greater (or smaller) when a bond isstretched and smaller (or greater) when a bond iscompressed. For the asymmetric stretching mode thechange for one bond cancels that for the other bond andthis mode, which is infrared-active, is expected to beRaman-inactive. For the syrnrnetric stretching, thepolarizability changes in each bond add, and we expecta net periodic polarizability change. This mode, whichis infrared-inactive, is expected to be Raman-active.A similar conclusion is reached by recognizing thatthe polarizability has units of cm3 and can be inter-preted as proportional to the volume of the niolecule.Any vibrational energy states that involve a mode ofmotion of the molecule resulting in a changing volume

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are Raman-active. The totally symmetric stretchobviously changes the volume and hence is Raman-active, whereas the bending mode vz and the asym-metric stretch v do not change the molecular volume.This example suggests the way in which Ramanspectroscopy augments infrared spectroscopy to pro-vide a complete picture of the energy spacings of the3 n normal modes. It should suggest also theprime importance of molecular symmetry in determin-ing which modes will be infrared- and which Raman-active. For example, since COz has a center of sym-metry, the rule of mutual exclusion applies, which statesthat for molecules with a center of symmetry, anyRaman-active vibrations will not be infrared-active,and vice versa. This rule is confirmed by experiment,which in fact shows vz and us to he infrared-active,and to he Raman-active.Raman spectra can in an additional way he a powerfultool for characterization of vibrational energy states.By 1111a1winghe ,x,l:iriz>ltio~~roperrier of t h e scnrtercdRnmnn lighr, totoll? synirnctric vibrations (tlm*r whichpreserve the symmetry of the undistorted molecule)can be distinguished from less symmetric vibrations.Polarization studies are thus of great value in assigning,even within the class of Raman-active modes, spectrallines to transitions between particular types of energystates. Some recent articles and reviews of Ramanspectroscopy have been given by Hawes, et al. (SS),Beattie SG), nd Tobias (35). A review of the use oflasers in Raman spectroscopy has been included in thehook by Szymanski (10).An exciting new technique for determining vihra-tional energies, called molecular photoelectron spec-troscopy, is being developed by Al-Jobory and Turner(36) and coworkers in England. The effect s based onthe measurement of energies of photoelectrons in the5-15 eV range (40,000-120,000 cm-') formed in theprocess of photoionization. This method is providingsome previously unavailable data on the vibrational(and electronic) energy states of the resultant molecularions. An excellent illustration of the spectra and re-sults obtainable with the method is given by Turnerand May (37), who report on Hz+,CO+,02+,nd NO+.The instrumental and theoretical limitations of thistype of spectroscopy are discussed elsewhere by Turner38).

Vibration Rotation EnergiesIt is not immediat.ely clear that our success in estab-lishing energy patterns on the basis of rotating-but-not-vibrating and vibrating-but-not-rotating models im-

plies that we can handle the situation presented by gasphase molecules which must be assumed to be free toundergo either, or both, or some other, type of motion.Here we investigate the spectral evidence that indi-cates the extent to which we are justified in simplycombining the analyses of the two types of motion thathave been given, and we consider the interpretation ofthe spectra that involve simultaneous rotational andvibrational energy changes.The key to the analysis is the recognition that thecalculated periods of vibration and rotation would betypically of the order of 10-l4 and 10-l2 sec, respectively.A rotation-vibration system with such relative periodswould be described in terms of a clearly distinguished

rotation accompanied by a rapid vibration which os-cillated through a hundred cycles for each rotationalcycle. The analysis of such a system would beginwith a simple combination of descriptions of therotational-vibrational motions.If we proceed in this manner we write

E,. I / z W . Z L/~ 2B J ( J 1 ) D J a (J 8)

The combination of the rotation and vibrationselection rules,2 aJ = l and Av = + l would leadus to expect a fundamental band with, from AJ =1, an R branch, very much like a pure rotationalspectrum extending out to higher frequencies fromthe band center and, from J = -1, a correspondingP branch extending to lower frequencies. SeeFigure 10 for the energy level diagram and anticipatedtransitions.Inspection of actual vibration-rotation spectra,Figure 7, shows something of this expectation. Themarked closing up of the spacings at higher frequenciesand opening up at lower frequencies suggest, however,the need for a refined model.We must now insert into our model the expectationthat the effective moment of inertia, or rotationalconstant, which governs the rotational energies, canbe expected to depend on the degree of vibrationalexcitation. We can express this most simply by writingthe dependence of the rotational constant on the vibra-tional quantum number as

Be = Be '/d (9)where 0 is an additional parameter that represents thechange in effective internuclear distance due to vibra-tional excitation. With this addition, we write for theenergy pattern expected for the rotating-vibrating mole-culeE m , ,= W.(U msxs(v /?

[ B . a .( v l / n ) l J (J 1 ) DJ P (J I =W.(V /*I W ~ Z 1/z 2 B J J 1 )

D J V l I a - ' / t ) J ( J 1 ) ( 1 0 )

These selection rules are derived by expressing the quantummechanical transition moment connecting the energy statesIdnR +n~Jlmdr

Here = + J , and = +o P, that is, a product of one-dimensional harmonic oscillator vibrational wavefunctions andrigid-rotor wavefunctions. These wavefunctions are governedby t,he quantum numbers v and J , and the single and doubleprimes designate t,he upper and lower energy states, respectively.To show the effect of vibration, the dipole is expanded in aTaylor s series using the displacement f rom equilibrium z,giving a = ( b r r / b z ) ~ z neglected terms. These substi-tutions give

The rotational selection rules come from the first and thirdterms. The first term is zero unless ro 0 and o' = v U , dueto orthogonality. The third term is zero unless J' = J 1,due to the properties of J . The vibration-rotation selectionrules come from the second and th ird terms. The second termis zero unless a r / b z 0 and v' = v 1, due to properties of JlAgain the t,hird term requires for vibration-rotation spectra J '= J' il

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Figure 10. Vibration-rotation energy stater and dlo we d transitions ina diwo mic molecule. The fundamental absorption vibration-ro:ationspectrum ~ e eig. 7 s esult of th selection ruler 4 v = +I, J = I.The P-bronch gmup of lines arises f ro m 4 1 = -1 ond the R-bronchmmAJ = l.

I t is on the success of this expression in interpreting theobserved spectra th at our model of nearly independentrotational and vibrational motions and energy patternsis based.The deduction of the five parameters in the aboveequation requires the manipulation of considerabledata. Since such calculations can be part of a labora-tory experiment tha t seems particularly effective ininvolving students with spectroscopic calcuIations, wepresent an outline of one way this can be done in anorganized manner.First it is convenient to label the spectral lines asR or P to indicate the spectral branch and to indexwith an integer in parenthesis which is the lower of thetwo J values. Thus the third line of the P branchfrom the band center is labelled P ( 3 ) . In addition weindicate that this line is a component of the fundamentalband by writing P(3),+., or of the first overtone band

by writing P(3)2,0. In reference to the energy levelpattern the symbol J designates a rotational level ofthe v = 0 vibrational state.With these stipulations we write using eqn. (lo),for the components of the fundamental band, whichrequire Av = +1, J = 1

R(J) , -o = we mz. 2B A J 1 )4 J 1 ) (J 3 ) W J 1)' ( 1 1 )

P ( J = w w s 2B 7 1 )a . (J 1 ) ( J 1 ) 4 D ( J ( 1 2 )

P ( J 1) is written, rather than P ( J ) , o achieve great-est similarity between eqns. ( 1 1 ) and ( 1 2 ) .

For the overtone vibration-rotation absorption band,as in Figure 8 , the selection rules are Av = +2 , AJ =* I . Substitution of these quantum conditions into eqn.(10) gives

Suitable treatment of these four equations whichrelate the model parameters to the frequencies of therotational components of the fundamental and over-tone bands provides values for the parameters.From subtractions of eqns. ( 1 1 ) and 12) and ofeqns. ( 1 3 ) and ( 1 4 ) we obtain

and

From corresponding additions of these pairs of equa-tions we obtain

andR(J)sco P (J l )aco = 3,,, . =, J 1)' ( 1 8 )4For any chosen value, the left sides of these equa-tions can be evaluated and one can see that subtractionof the first pair of equations will lead to a value for

a whereas subtraction of the second pair will give theanharmonicity constant w x .These calculations3may be repeated if desired for allavailable lines and average values of ry, and w gdetermined. Equations ( 1 5 ) and ( 1 6 ) may now berearranged into a form that will allow graphical evalua-tion of additional parameters. After transposing thevalue of -a or -3/2a to the left side, the value of theleft side is plotted against 2 ( J I ) , and the slope ofthe line determines -D and the intercept givesBe. Aleast squares treatment can be used to calculate thebest D and B values.Finally, substitution into any of the above equationsleads to values of o . (The equilibrium internucleardistance r may easily be calculated from the relationsBe = h 2 / 8 r 2 1 and I = fir, , where these quantitieswere defined a t the beginning of the section on Rota-tional Energies.)Commercial spectrophotometers usually do not pro-vide sufficient precision to justify application of suchan elaborate model as that based on eqn. ( 1 0 ) . Bymeans of an interferometric technique, Rank, et al.59) have obtained overtone and fundamental data forHC1 to the nearest 0.0001 cm- and for DC1 to thenearest 0.001 cm- . I have sometimes assigned juniorstudents to use the spectral data from Rank, et al.for the above calculations. Typical student results

for HClS6 are w e = 2989.96 om- , w,x, = 51.977

I am indebted to ProfessorDavid A. Daws of the ChemistryDepartment of ,he Universi ty of Southern California fnr des-scribing the above treatment.Volume 47, Number 1 January 1970 / 9


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Figure 12. Elechonic trmrit ionr betw een various vibrat ional energy stoles of a diatomic molecule. Vertico l transitions ore explained by the Franck-Condon principle. A1 indicoter the common origin from the ground vibration level of the most intense gmup of lines in Figure 1 1 0) accounts for the nexlsttongert gmup of lines in Figure 11 marked with orrowr Cl accounh for th third stmngert gmup of lines in Figure 11 marked with o.

achieved by dealing with the relatively easily observedand analyzed absorption spectrum of I1 The suit-ability of this example has already been recognizedby Staffoid 65).Electronic transitions are most often observed in theultraviolet, and occasionally the visible, spectral re-gion. The violet color of iodine indicates a visibleabsorption band, and this is found, as shown in Figure11, to have considerable structure, even if a relativelylow resolution instrument is used.The absorption of the large amount of energy thatis implied by even a visible spectrum can produce amolecule with a quite differentelectronic binding energyfrom that of the ground state. The expectation isthat the excited state will have an equilibrium inter-nuclear distance and a force constant, i.e., it will berepresented by a potential energy function that is quitedifferent from that of the ground state. The situationthat is expected is illustrated in Figure 12.Careful scrutiny of the I, spectrum gives experi-mental support to the conclusion that there are nosimple selection rules governing allowed changes in thevibrational quantum number v when electronic energychanges are involved. Rather, the relative intensitiesof observed transitions may be understood through theoperation of the Franclc-Condon principle 56). Thisprinciple recognizes that electronic transitions occurmuch more rapidly than vibrational motions and there-fore only vertical transitions, as drawn in Figure 12,which maintain the same internuclear distance r ,are likely to occur.Those vertical transitions for which the transitionmoment can be appreciable are those for which thewavefunctions of the two different states have, a t thesame internuclear distance, appreciable magnitudes.I t follows that no simple selection rule exists but tha t,as suggested by Figure 12, some relation can be estab-lished between the potential functions of the twostates and the absorption or emission components th atare expected to be observed.

To proceed we need to consider again the observedspectrum. The three quite easily distinguished sets oflines or progressions, actually band heads of

Figure 11 can, since this is an absorption spectrum,be attributed to transitions beginning with v 0,v v = 1, and v = 2, with the set a t highest energybeing identified with v = 0 Additional progressionscan be recognized particularly at higher sample tempera-tures. The analysis begins by making columns of thefrequencies of the components in the progressions thatare observed. These columns can now be related toone another by sliding them vertically with referenceto each other until the differences in frequency betweenhorizontal pairs show a gradual decrease as one movesacross the table to the right. The arrangement tha t isobtained from student results at the University ofRedlands, for a variety of sample temperatures, ispresented in Table 2.The arrangement of Table 2 is called a Deslandrestable. It remains, to show the development of sucha table, to indicate how the v' numbers are deduced.For I2 his is a matter of some considerable difficultysince, for all the sets of lines, there is no abrupt ter-mination at long wavelength, which would indicatethat the v' = 0 level had been reached. Rather there isa fading out of the lines. The assignment must de-pend on fluorescent or isotope effect studies, and thecorrect 1 ssignment has only recently been arrivedat by these techniques by Steiufeld, et al. 57) and byBrown and James 58). This assignment has beenused to assign the quantum numbers in Table 2.With the organization provided by the Deslandrestable one can proceed to the deduction of the propertiesof the ground and excited electronic states involvedin the transition.To represent adequately the vibrational energyspacing in either the excited or the ground electronicstate we can write eqn. 7) again

E = u (V >/ ) W L Z ~V / d a ( 1 7 )The successive horizontal entries in the Deslandrestable give vibrational energy differences for the groundstate whereas the vertical differences give the valuesfor the excited state. Thus, allowing for the introduc-tion of either single or double primes we write

AE, = E(u 1 ) E ( v ) 18)Volume 47 Number I January 1970 1


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Table 2. Deslandres Table of 12 Bands (om-1)0 0 1 2 3 4 5 6 7

111.88 16241.7108.28 16330.3103.110 18463.4102.611 16566.0106.912 16672.9106.613 16991.1 211.6 16779.5101.4 104.014 17306.4 213.9 17092.5 209. 0 16883.5101.6 101.4 99.815 17408.0 214.1 17193.9 210.6 16983.398 .7 97 .0 96 .616 17506.7 215.3 17290.9 211.0 17079.997.1 100.4 98.917 17603.8 212 .5 17391.3 212.5 17178.89 4 . 8 9 5 . 2 9 5 . 318 17698.6 212.1 17486.5 212.4 17274.194.5 92.3 94.619 17793.1 214.3 17578.8 210.1 17368.792.6 YO4 94.920 17885.7 216.5 17669.2 205.6 17463.6a Btudent data obtained from the Cary 14using a 10 cm cell containing 11 ryatala at 35 to 85-C. Wavelengthswere read from the apaotra withoutcalibration. Conversion to vacuum wavenumberwasma ss from C. D. Coleman. WR. Boasnun sod W F. Mesgera Conversion Tablcsfor Wauelewlhsto Vacuum Wouenuntber~National Bureau of Standards, 1960.

which simplifiestoAE. = 0 2vw.z. %A* (2 0 )

This expression suggests that plotting AE, versus vshould give a straight line with slope -2wje and inter-cept at v = of we Note that no primes or doubleprimes have been used since the plotting procedure is,at least in principle, applicable to either vertical orhorizontal differences in the Deslandres table. Sucha plot is termed a Birge-Sponer plot 59) and is shownin Figure 13 for the excited electronic state of I,.From Figures 12 and 13 it is also apparent that sum-ming the AE, over all v i.e., obtaining the area from

Figure 13. Birge-Sponer plot for the upper alestmnis state of I .

Table 3. Molecular Constants of lz g)Student results Accepted values(cm-LI l rm- l>

O, Z. 0 . 9 4 0 o . 7 o i6 *D 4260 4391 .OaweX 214 214.5ZbD. 12420d 125RlY

*STEINPELD, J. I., ZARE, R . N. JONES,L., LESK,M.,AND KLEMPERER, 'W., J Chem. Phys., 42, 25 (1965).ERMA, R. D.,J Chem. Phys., 3 2 , 7 3 8 (1 9 6 0 ) .0.75 from slope in u = 0 t = 16 region.Reanires the additiand informstion t,hat,t.ha diswain-~ ~ ~~~~~ ~tion products of the excited st& are atoms in the Pa/,gmund state md PI/, excited state. Subtraction of theatomic excitation energy of 4280 cm- leads to the de-duced dissociation energy for I. in the gmund state.

v = o the point a t which AE, = 0 ives the depthof the potential well, that is, the dissociation energy of2 in the excited electronic state to whatever atomicstates are formed.Similar treatments can be given the ground statedata but, as is typical of absorption spectra, so fewvibrational levels are involved that the results are veryunreliable. Here, in fact, it must be emphasized thatthe near linear relation of Figure 12 cannot be countedon as it depends on the assumption that a simple an-harmonic term in the vibrational energy pattern isadequate.Some University of Redlands student results areshown in Table 3 along with results from the literature.More extensive use of the data can be made, for ex-ample calculation of potential energy curves, as illus-trated in the work of Weissman, Vanderslice, andBattino 60) on I2 and HI and by Clyne and Coxon61) on Brz.

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(16) Ref. 181.no. 501-81.~ . . . .i l j j B O O ~ T E R ,. J AN COX. A. P. J CXEM. OITC.. 45,18 (1968).(18) D ~ v r ~ s ,., (Edttor). Infrared Spectroscopy and Molecular Struo-tore:' Elsevier Publi sbine Co., Amsterdam, 1963, D. 192: JncoaE. J.. T e o M ~ r 0 B .H. B.. and B A ~ S L G .. S.. J C h e m . P h y a . 47,R7RR llQR71~(19) B ma en, R. M., J C h e m . P h y r . 2, 128 (1934): 3, 710 (1936); P h y s .Rcu. 48, 284 (1935).(20) Henscnnhce, D. R., AND Lnunm, V. W., J. Chem. Phus . 35, 458

-.(221 WIZBON B.. DECIVB. . C., A N D CEO +. . C.. molecular Vihra-tions. MoGrav.-Hill nook Co., New York. 1955.(23) D*vms.M., (Ed ilo rl. Inf r~r ed peetroseopyand Molecular Structure;'Elsevier Puhlishine Co., Amsterdam. 1963, Chap. Y.(24) ~ ~ o u n. M., FLETCIIER.. H., AND SMITX,n. F.,J cham . ~ h y a42,2236 (1965).

(251 K n u n ~ u w . . N.. AND SCXOTTE,. J. H.. S p ~ o t m c h i m . c t. 21. 1947(19651.(261 McGnnv. G. E.. B m n m ~ , . L . . AND HIBATSDNE,. C., J . Cham.P l iys . 42,237 (1965).(27) DO WB . . A., J. CHIM. E DUC. , 5,6 29 (1958).(28) S z L v m a ~ m N .R. M.. AND Bnasmn, G. C.. Speotrometrio Identifioa-cation of Orzanie Com ~ou nda , ohn Wiley iu Sons New York 1967.(291 N x n ma s r . I

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Energy States of Molecules