Resource Pa pers-VIII Prepared under the sponsorship of The Advisory Council on College Chemistry
J. Leland Hollenberg
Energy States of Molecules
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University of Redlands Redlands, California 92373
W o r k in a modern chemical laboratory increasingly depends upon instruments that allow the measurement of the absorption or emission of radiation by a sample. All such work depends on an energy separation between states of the atoms or molecules of the sample. The variety of states that can he studied, or used, in this way leads to the many areas of spectroscopy that are now of importance: nmr, epr, infrared, ultraviolet and so forth. A treatment here of all these areas of spectroscopy and the energy level patterns on which they are based would encompass far too much territory. Here the discussion will he restricted to those energies that are partially populated and therefore need to be considered in any deduction of the thermodynamic properties of the material or of the reactions in which it is involved. The energy levels that must he treated are, in this way, restricted to rotational, vibrational, and electronic, and the corresponding spectroscopic studies are reduced to microwave, infrared and Raman, and visible and ultraviolet. The scope so defined is outlined in Table 1. Cnneem .. - - .. - - - will he focused on the determination of the spectral patterns corresponding to these three types of Table 1.
Type of spectrosCOPY
Microwave
energy levels and to the relationships that can be estahlished between these observed patterns and the parameters of the molecular model. The analytical uses of these types of spectroscopy will not he considered, nor will any serious treatment of the experimental techniques be attempted. Rotational Energies
If a model of a molecule that is a simple rigid rotor is considered and if the appropriate restriction on the allowed angular momentum is imposed, or the rotational motion is treated by use of the Schroedinger equation, This series of "Resource Papers" is being prepared under t,he sponsorship of the Advisory Council on College Chemistry. The Advisory Council is supporbed by the National Science Foundation. Professor L. Carroll King, of Northwestern University, is the chairman. Single copy reprint,^ of this paper are being sent to chemistry depwtment chairmen of every U. S. Institution offering college chemistry courses and to ot,hers on the mailing list far the ACa Newsletter. This is Serial Publication No. 46 of the Advisory Council.
Types of Spectroscopy Covered
?
Range of energies Frequency (see-1) 10'-10"
em-' 3 X 10-'-3
kcal mole-'* 10-e10-2
Far infrared
10'L1O1a
3-300
10--1
Infrared
10'8-lox4
300-3000
1-10
Raman
~O'LIO'~
3-3000
10-P-10
Visibleultraviolet
1OlL10"
3
x
lo33 X 106
l&lo5
Of mo ecular energy
Information obtained
Rotation of heavy molecules Rotation of light molecules, vibrations of h e a w moleculei Vibrations of light moleeules. vibrsr tion-iotation Rotation, , vibratiotions
Interatomic distances, dipole moments, nuclear interactions Interatomic distances, bond force constants
Electronic transitions
Interatomic distances, hond force constants, molecular charge distributions Interatomic distances, bond force constants. molecular char& distributions (for energy changes not ohsemable with infrared) All above properties plw bond d~ssociar tlon energies
* Compare to the average thermal kmetic energy per degree of freedom, '12RT g0.3 kcal mole-', or -100 om-'.
2 / Journal of Chemical Education
we deduce that only the rotational energies E, that conform to
are allowed. Here I is the moment of inertia of the molecule, B = h2/8a21 is known as the rotational constant, and J is the rotational quantum number. For a diatomic molecule I = pr2, where p is the reduced mass mlm2/(ml ma) and r is the distance between the two atoms. The pattern of energy levels based on this model and treatment is shown in Fiaure 1. For each energy level indicated by avalue of J there are2J 1rotationalstates. These can be looked on as corresponding to the possible angular momentum components along a particular direction in space. Alternatively one can recognize a t this stage that the rotating-molecule problem is formally the same as the Figure 1. Energy of rotation as rotational aspects of the a function of J quantum number hydrogen-atom problem. for the rigid rotor. Thus J is identified as the counterpart of the orbital angular momentum quantum number 1, and the several states included in a particular J designation correspond to the atomic states indicated by the possible values of m for a given value of 1. Analysis of the quantum mechanical rigid rotor, as outlined above, is carried out in greater detail in a variety of physical chemistry, spectroscopy, and quantum mechanics books. I n addition to the books listed in the bibliography, reference can be made to several others for this introductory material (1-3). The t)xoretical treatment can be extended to allow this calculated pattern to be confronted with experimental spectra. The selection rules governing allowed transitions between rotational states require that AJ = *I, or, for absorption spectra, AJ = +1, and that the permanent dipole y of the molecule be nonzero. (Thus homonuclear diatomic molecules like H, and O2 are expected to exhibit no pure rotational spectra.) For an absorption spectrum, the above treatment predicts
+
-
+
AE, = E,(J
+ 1) - E,(J)
=
2B(J
+ I),
J = 0 , 1 , 2 , 3, . . . ( 2 )
Thus the rigid-rotor model predicts that the spectrum should consist of a series of absorption lines at 2B, 4B, 6B, 8B, etc., i.e., a series of equally spaced lines 2B apart. That this is approximately borne out by observed spectra is shown by the pure rotational spectrum of HF, a portion of which is presented in Figure 2. If known internuclear distances are used to compute I, substitution into eqn. (2) soon indicates that for all but the lowest moment of inertia molecules, like HF, the rotational transitions, except for those with extremely high J values, have such small AE, that the spectra occur in the microwave region. Microwave studies of many relatively small molecules have been made since World War 11, and this area of investigation
Figure 2. Pure rototion spectrum of HF(g) run on Beckmen IR-12. About 2 6 0 mm pressure in 10-cm brass cell with polyethylene windows. Extraneous line near 3 8 5 cm-I is due primarily to SiF* impurity.
Figure 3. Microwave rotmtion spectrum of CHsCHKHnl obtoined with Hewleft-Pockard 8 4 0 0 C spectrometer. Sample pressure war 50 ir. Relative proportions of gauche and trans "mtamen" are related to areas under each line. Note the regular spacing of lines. (Courtesy of Dr. Howard Harrington, Hewlett-Packard Corporation, Polo Alto, Colifornim.)
is well described by Sugden and Kenney (4). The type of spectrum that can be obtained by this technique is illustrated in Figure 3, which shows a small part of the microwave spectrum of CH,CH,CH,I. Because of the very great frequency accuracy attainable with microwave spectrometers,' internuclear distances of six significant figures can, at least formally, be
lMicrawave spectroscopy remains a. much less frequently encountered tool than infrared and visible-uv, which will be dealt with later. Two principal obstacles have prevented the widespread 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 HewlettPackard. The second obstacle ttrisks from the difficulty of making quantitative deductions of concentration from line intensities and, therefore, the difficulty of doing quantitative analysis. A very significant development due ta Harrington ( 6 ) now gives s method of determining separately the molecular eoncentration N, and the relaxation-broadening time r. N is number of molecules per unit volume, and 7 is the mean time between collisions which broaden the lines. N and r vary widely with gas pressure, molecular properties, and J . The product of N and r governs absorption. Experimentally, Harrington has shown that measurement of the Stark modulated (imposing an electric field on the sample) microwave absorption signal amplitude and the incident microwave power level allows calculation of N and r. Thus a wide range of new uses for microwave spectroscopy such as quantitative analysis, isotope effecta, kinetic studies; and chemical process monitoring is now becoming possible, and microwave spectroscopy may become a mare frequently used tool in the chemical laboratory.
Volume 47, Number 1, January 1970
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calculated if the observed absorptions can be related to the molecular model. Careful analysis of rotational spectra of a linear molecule reveals that the line spacings are not quite constant and therefore the rigid-rotor model is not entirely adequate. Inclusion of a term, D, for centrifugal distortion, which represents the increasing moment of inertia as the bond stretches with greater energy of rotation, leads to the modified energy pattern
uo. Examination of the scattered light with a monochromator shows that in addition to the incident frequency vo, there are much weaker lines displaced from vo by various amounts Au,. It is found that each Av involves transitions between molecular energy levels. Both rotational and vibrational energy transitions can be observed by this technique. I n Raman spectra the interaction of the radiation with the molecule depends on a dipole that is induced by the electromagnetic field of the light energy pl.d = aE
Details of the development of this relation are given by Herzberg (6). This modified energy pattern can be tested against the observed spectral transitions if the expression for the transitions allowed by the selection rule AJ = +1 is again obtained. We have now AE, = 2B(J
+ 1 ) - 4D(J + 1):
J
=
0, 1, 2,
...
(4)
The improvement in fit with the observed spectra is, as shown by early studies of KC1 reported in Herzberg, (7) such that no additional refinement of the model for simple, linear molecules is necessary. We thus have, for certain molecules, a useful expression for the allowed rotational energies. A physical chemistry infrared experiment analyzing the rotational spectra of simple molecules and utilizing eqns. ( 2 ) and (4) bas been described by Hollenberg (8). The increasing availability of far infrared instruments, which extend the infrared range to the order of 10 cm-', will allow a variety of rotational spectra of small molecules to be obtained and studied. So far no mention has been made of an experimental confirmation of the 2J 1 degeneracy of the rotational energy states. Qualitative confirmation stems from the intensity distribution found in the pure rotational spectrum. The intensity of a given component depends primarily on the population of the initial energy level. For increasing J values the population increases 1 multiplicity term hut ultiin response to the 2J mately the exponential Boltzmann factor which works against the population of higher energy states overcomes this trend. Detailed considerations of this are given in Herzberg (9). The two factors that operate to give a maximum in the population-energy curve are, it can be noticed, comparable with those that apply to translational energies. Finally, in this consideration of linear molecules we return to those nonpolar examples, like H,, 02, COf, that defy direct study of their rotational spectra by absorption spectroscopy. Fortunately a different spectroscopic techn~quethat depends on a scattering, or simultaneous absorption-emission, process is available. This technique, Raman spectroscopy, provides the information not available through absorption spectrosopy. Historically, Raman spectroscopy was developed long before infrared became popular. However, the long exposure times needed to detect Raman lines on a photographic plate contributed to the rapid rise in popularity of recording infrared spectrophotometers. Recently continuous gas lasers have been adapted as Raman sources ( l o ) , and it seems likely that interest in this technique will grow among chemists. The Raman spectrum is obtained by irradiating the sample with intense monochromatic light of frequency
+
where or is the polarizability of the molecule, and E is the electric field vector. As shown by Herzberg ( l l ) ,only when there is a periodic change in a is the quantum mechanical transition moment 1 y,.d/, nonzero, permitting energy transfer between states m and n. The necessary, periodic changes in a can be produced by rotation of the molecule, except for those of near spherical shape like CH,, resulting in a rotational Raman spectrum. The pattern of rotational energy levels expected for molecules that can only be studied by Raman spectroscopy can be approached as before on the basis of the rigid or non-rigid rotor. To this must be added the selection rules governing transitions between these energy levels. Now, since the transition is not restricted to an increase in energy as in absorption experiments the selection rules are different in kind and degree. Calculations show that transitions with AJ = 0, + 2 are allowed. The comparison of the rotational Raman spectra of N20 and Cop, as shown in Figure 4, which are both linear and have similar moments of inertia, and the appearance of the rotational Raman spectrum of acetylene, Figure 5, suggest that some additional feature
+
4
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Journol o f Chemical Educofion
Figure 4. Representations of pure mtationol Roman spectra of CO, (vpper) ond N 2 0 llowerl. Note that all of the odd J lines for COz are absent. Reprodurtiom of octvol spectra ore given in G. W. King, "Spectroscopy and Molecular Structure." Holt, Rinehort and Windon, Inc., New York, 1964, p. 294 and p. 195.
+
4sd
I , ' . " ' " " ' 3
I
1 -
Figure 5. Represontation of a porBon of the pure rotational Ramon spectrum of C2Hzlgb Note the "strong-weak-stmng.weak" intenrily pattern. Bored on work of J. H. Collomon and 8. P. Saicheff, Can. J. Phyr., 35. 373 (1 9571.
must be added to the molecular model. This feature stems from considerations of nuclear spins and the Pauli Exclusion principle. Although it is most familiar to chemists in connection with electron pairs and with ortho and para hydrogen, the restriction on allowed states in terms of the properties of identical nuclei and the symmetries of the states shows up most clearly in rotation and vibration-rotation spectra. The subject is very clearly discussed in Herzberg (12). The consequences of this restriction for statistical calculations of thermodynamic properties is that, for systems with relatively small energy separations, the number of states that exists must be reduced by a factor, known as the symmetry number, equal to the number of equivalent nuclei that can be interchanged by a molecular rotation. Thus, for homonuclear diatomic molecules and linear molecules such as CO, and H-C= C-H, there are only half as many rotational states in the rotational pattern as there would have been if the nuclei were not identical. (For HZ,where the rotational states are widely separated, more detailed consideration of the consequences of the exclusion principle must be applied.) The inclusion of such symmetry considerations in thermodynamic calculations is presented in most statistical mechanics texts, as, for example in that by Mayer and Mayer (IS). For polyatomic molecules all the above considerations can, with appropriate extensions and additional complexity, be carried over. For symmetric top molecules, which have two of their three moments of inertia equal, the analogy with the linear examples is quite close and the rotational energy patterns, selection rules, and rotational spectra are often little different from that for a linear molecule. For a general "asymmetric top" molecule the allowed rotational energy pattern is complex and cannot be expressed by a simple equation. The spectrum is likewise complex and difficult to analyze. Nevertheless such spectra can be interpreted in terms of a molecular model, and the moments of inertia and the rotational energy pattern have, in many cases, been deduced. Generally this requires recourse to a variety of isotopic substitutions and to the imposition of an electric field to produce a Stark splitting to help identify the states involved in a transition. Treatment of these asymmetric molecules is discussed in considerable detail by Sugden and Kenney (4). Other complications can occur that interfere with the deduction of an energy pattern and molecular parameters. For example, a significant fraction of the molecules may occupy the first excited vibrationd state if the vibrational energy spacing is quite small. Such molecules have a slightly different moment of inertia, resulting in new microwave lines. The nature of this excited vibrational state is revealed by the microwave spectra. For example, in linear OCS, a stretching vibration, which causes a slightly larger I, produces weak satellite lines on the low frequency side. I n the OCS bending vibration, where the excited state results in a slightly smaller I, extra microwave lines are found a t somewhat higher frequencies. Although such situations lead to interesting spectral studies, they are not a serious impediment to the collection of data needed for most statistical studies. Thus, directly from spectroscopic data, or by calculations using the rotational parameters, the moments of
inertia, the rotational states, and energies of gas phase molecules can be readily deduced. By contrast, for the liquid state, even the idea of molecular rotation seems strange and only recently has -much attention been paid to the possibility of spectral studies of the rotation-like motion of molecules in condensed states. The special cases of HZ and DP in inert solvents have received attention by Ewing and coworkers (14), and for these it is clear that rotational energy and perhaps even translational energy changes are responsible for the observed absorptions. (Neither species has a permanent dipole or a charge and such spectra must owe their existence to induced electrical effects resulting from interactions with the solvent.) Less direct evidence for rotation-like motion of other molecules in relatively inert solvents has been given in other spectral studies. However, our information on the rotational, and translational, energies of molecules in liquids remains less substantial than does that for gas phase molecules. Vibrational Energy States
Let us now turn our attention to the vibrational aspects of a non-rotating, ball-and-spring molecular model. The simplest assumption for the nature of the springs is that they obey Hooke's law, i.e., that they exert a restoring force proportional to the displacement from the equilibrium position. The classical behavior of such a system, that is the counterpart of a diatomic molecule, is described by a characteristic vibration frequency 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 potential function V = 1/2kxz in the Schroedinger equation, only certain allowed vibrational energies are found and these are given by the expression
+
Thus, vibrational energies of
, "..-
etc. are deduced. These energies may be simplified to 1/2hv, 3/2hv, etc. The , . I pattern is shown in Figure - . ' 6A. For each energy level only one state exists. ..z E" The selection rules that are deduced for transitions " . ,-.. between states, according to this model, give Av = sl, v.o .o which, for absorption, im.-......------.. plies Aw = +l. The rules ,a> ,a) require also that b y / b Z Figure 6. A, Vibrational one,0, that is, the electric dipole gier of a harmonic o~cillotor. 8, Vibrational energies d on the change anharmonic o x i ~ ~ a t o r . as the molecule vibrates. '' '
... . . . E
8
.
Volume 47, Number 1 , January 1970
/
5
the form '/&x2, and when only this value for V is used in the Schroedinger equation, the energies are those of eqn. ( 6 ) for the harmonic oscillator. Higher derivatives of V than the quadratic k are called cubic, quartic, etc., force constants. If the cubic term in V is retained, that is, an anharmonic potential is used, and only the first additional term that is introduced into the energy level expression is retained, the predicted energy pattern is given by Figure 7. Fundamental vibmtion.mtatbn spectrum of HClfg) run an Beckman IR-12. Prerrurs war .bout 1 0 0 mm Hg in o 10-cm cell. The less intense line shifted just to the right of each of tho main liner ir due to the natural isotopic obvndonce of about 25% HCI".
where o, is the "harmonic vibration frequency" and to an addition term and is known as anharmonicity (15). With such an expression the frequencies of the fundamental and the first few overtones can be well fitted. The anharmonicity is generally small but far from negligible. Thus, for HCI one obtains from the two observed frequencies mentioned above, the parameters w, = 2990 and o.x, = 52 cm-I. Such parameters are tabulated for most diatomic species by Herzberg ( I @ , both for ground and excited electronic states. One is usually most concerned with ground state molecular properties. This simple anharmonicity treatment can be applied, in a student experiment, written by Boobyer and Cox (17), to a particular vibration of the polyatomic molecules CHCL and CDCb. The C-H and C-D stretching vibrations are sufficientlyindependent of the vibrations of the rest of the molecule that one can proceed as though an X-H or X-D molecule were being dealt with. A set of overtone bands can be observed, if near infrared equipment is available, and the use of the anharmonicity term can be illustrated. The values of the force constants, which can be calculated from the o, values, or approximately from the observed fundamental frequencies, lie mughly around 5 X lo5 dyne cm-' for single bonds, 10X 10" for double bonds, and 15 X lo5 for triple bonds. A table in the book by Davies (18) shows that an approximate relation between bond order and force constant also applies to polyatomic molecules. Empirical correlations of Such bond parameters are useful both in considerations of chemical bonding and in extending the vibrational energy pattern analysis to new molecules. An early and often referred to rule, given by Badger (19), gave the relation k(r, - d)3 = 1.86 where d is a parameter governed by the positions of the atoms in the periodic table. More recently Herschbach and Laurie ($0) have shown a valuable correlation between w,, w,x,, and quadratic, cubic, and quartic force constants on the one hand and bond lengths on the other. Their predictions based on diatomic data agree with available measurements on polyatomic species, regardless of the type of chemical bonding involved. For polyatomic molecules we must first consider the ball and spring counterpart on which our molecular model will be based. If such a model is given the freedom of a gas phase molecule, as by being thrown in the air, we might try to analyze the complicated motion that could result in terms of three translational and three rot,ntional modes of the ent,ire system and the remaining modes of vibrational energy. Since the total degrees of freedom of an n body system is 3n, this implies 3n - 6 vibrational modes must be deleads . og, .
Figure 8. Overtone vibration-rototion spectrum of HCllgl run on a Cory 1 4 using the M . 1 obsorbonce slidewire. Sample pressure was obovt 1 0 0 mm Hg in 10-cm cell. The less intense line shifted just to the right of each main line is due to the naturd isotopic abundance of about 25% HCIJ'.
Thus, for heteronuclear diatomic molecules the model predicts a single absorption due to the change in vibrational energy. Although such diatomic molecules do show a single strong absorption band, usually in the region 200-4000 cm-' in the infrared, closer inspection shows the need of modifying the Hooke's Law model. If, for the moment the fine structure of the HC1 bands in Figures 7 and 8 is ignored, we see that in addition to the "fundamental" band attributable to a v = 0 to u = 1 transition, centered at 2886 cm-', there is another band at nearly twice this frequency, 5667 cm-', that suggests a u = 0 to u = 2 transition. Thus, the selection rule Au = * l is not strictly followed. To allow for the existence, and frequency, of such an "overtone" band the model must be modified, and this is done by replacing the parabolic Hooke's Law potential function by a more general one. One can either adopt a new, hopefully more satisfactory function, the Morse function V
=
D . [l -
e-alr-%l]P
being particularly convenient, or one can simply write a Taylor's series expansion to represent a general potential function with a minimum at the equilibrium bond length. With a Taylor's series
we can arbitrarily set Vo = 0 , and the slope a t the minimum ( b V / b r ) = 0 . Thus, the first nonzero term is of 6
/
Journol of Chemicol Education
scribed. (For linear systems there would be 3n - 5 such modes.) For a ball and spring model we might soon find certain characteristic, or natural vibrations in which each ball moved in a straight line with simple harmonic motion. Analysis would show that any vibrational motion could be analyzed in terms of these natural or "normal" modes (21). The separation of the kinetic and potential energies of the vibrating system along "normal coordinates" that the above implies allows us to carry over the treatment given for diatomic molecules to each normal mode. We expect, for a molecule with n atoms, 3n - 6 vibrational en6rgy patterns each of which is the counterpart of the pattern deduced for diatomic molecules. For some molecules the above deduction is immediately verified. For Hz0, for example, three strong absorption regions are found in the infrared and these can be attributed to v = 0 to v = 1 vibrational transitions of each of the 3n - 6 = 3 normal modes. For such molecules, particularly if overtone and combination bands are analyzed, one obtains a complete picture of the vibrational energy pattern of the molecule. For other molecules, the 3n - 6 fundamentals cannot be immediately recognized and this difficulty may stem either from the complexity of the spectrum which consists of unidentified fundamentals, overtones and combinations, or from the symmetry of the moleculewhich, as we will see, prevents some fundamentals from appearing. For such situations a calculation of the frequency and the amplitude of motion of each atom in a particular normal mode can be determined by performing a complete "normal coordinate analysis" (22) which requires formulating the kinetic and potential energies of the molecule in matrix form. To adequately express the potential energy, a system of force constants must be chosen which allows for rather complete interactions between parts of the molecule. One of the most successful in terms of wide applicability is the Urey-Bradley potential function (23). Another type of potential commonly used, known as the valenceforce function, is illustrated in the calculations by Begun, et al. (24) on some XAFl type molecules. A still more general force field can be used, as illustrated by Krynauw and Schutte (35) in their study of C103-. Accurate determination of the vibrational energies of a number of isotopically substituted species is usually necessary in such normal coordinate analyses. I n addition the assignment of observed absorption frequencies to particular modes of vibration is aided greatly by studying the effect of isotopic substitution on 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 of isotopes. Some guides to the identification of fundamentals in spectra of more complicated molecules come from the concept of functional group frequencies, which are principally of value in qualitative analysis. The carbonyl group absorption near 1700 em-' is the standard example. The physical basis for such group frequencies has been discussed in an understandable way by Dows (27). The frequency-structure correlations are widely used by chemists, and a text such as that by Silverstein
and Bassler (28) may be used for introducing this topic in the organic chemistry course. A book by Nakanishi (29) gives more depth and provides many examples of interpreted infrared spectra. An even more complete coverage is provided by Rao (SO), who considers with a practical emphasis the usual organic subdivisions, such as heterocyclic compounds and hydrocarbons. Also included are quantitative analysis, high polymers, and biochemical materials such as nucleic acids and proteins. In addition there is a chapter on inorganic compounds, but this topic is covered much more thoroughly by Nakamoto (31), who also provides considerable material on coordination compounds. The book by Colthup, Daly, and Wiberly (32) gives a much more theoretical coverage of infrared absorptions, but still gives considerable attention to the frequency-structure correlations. A brief discussion of group theory and the calculation of thermodynamic functions is included. Each type of molecular vibration may be pictured as having its own set of vibrational energy levels with a characteristic spacing. If a particular vibration involves a changing dipole, it may readily absorb photons and give an infrared spectrum; the absorption frequency is governed by the spacing of vibrational levels, E2 - El = hv. However, for many symmetric molecules there are considerable numbers of the 3n 6 vibrations which do not involve a changing dipole and therefore do not appear in the infrared spectrum. Fortunately, many of these "missing" lines can be observed in the Raman spectrum. A periodically changing polarizability is the requirement for a Raman band. The possibility of such a change can be visualized for some vibrations, as is easily seen by considering the stretching modes of COz shown in Figure 9. We assume that we can think of
-
t Y~
-
t
F
0
i667cm?
V u, 12349cm'1
Figure 9. The lhree vibration01 modes of CO*. The 3n - 5 rule for linear molecules indicates four vibration= should be present. This apparent dbcrepmcy is exploined by the foct that ur is doubly degenerate. Thus, there is another vibration un in a plane perpendicular to that shown.
bond polarizabilities, and we assume that the bond polarizability gets greater (or smaller) when a bond is stretched and smaller (or greater) when a bond is compressed. For the asymmetric stretching mode the change for one bond cancels that for the other bond and this mode, which is infrared-active, is expected to be Raman-inactive. For the syrnrnetric stretching, the polarizability changes in each bond add, and we expect a net periodic polarizability change. This mode, which is infrared-inactive, is expected to be Raman-active. A similar conclusion is reached by recognizing that the polarizability a has units of cm3 and can be interpreted as proportional to the volume of the niolecule. Any vibrational energy states that involve a mode of motion of the molecule resulting in a changing volume Volume 47, Number
I,
Jonuory 1970
/ 7
are Raman-active. The totally symmetric stretch n obviously changes the volume and hence is Ramanactive, whereas the bending mode vz and the asymmetric stretch v8 do not change the molecular volume. This example suggests the way in which Raman spectroscopy augments infrared spectroscopy to provide a complete picture of the energy spacings of the 3n - 6 normal modes. It should suggest also the prime importance of molecular symmetry in determining which modes will be infrared- and which Ramanactive. For example, since COz has a center of symmetry, the rule of mutual exclusion applies, which states that for molecules with a center of symmetry, any Raman-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 n to he Raman-active. Raman spectra can in an additional way he a powerful tool for characterization of vibrational energy states. By 1111a1wing rhe ,x,l:iriz>ltio~~ properrier of the scnrtercd Rnmnn lighr, totoll? synirnctric vibrations (tlm*r which preserve 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, spectral lines to transitions between particular types of energy states. Some recent articles and reviews of Raman spectroscopy have been given by Hawes, et al. (SS), Beattie (SG),and Tobias (35). A review of the use of lasers in Raman spectroscopy has been included in the hook by Szymanski (10). An exciting new technique for determining vihrational energies, called molecular photoelectron spectroscopy, is being developed by Al-Jobory and Turner (36) and coworkers in England. The effectis based on the measurement of energies of photoelectrons in the 5-15 eV range (40,000-120,000 cm-') formed in the process of photoionization. This method is providing some previously unavailable data on the vibrational (and electronic) energy states of the resultant molecular ions. An excellent illustration of the spectra and results obtainable with the method is given by Turner and NO+. and May (37), who report on Hz+,CO+, 02+, The instrumental and theoretical limitations of this type of spectroscopy are discussed elsewhere by Turner (38). Vibration-Rotation Energies
It is not immediat.ely clear that our success in establishing energy patterns on the basis of rotating-but-notvibrating and vibrating-but-not-rotating models implies that we can handle the situation presented by gas phase molecules which must be assumed to be free to undergo either, or both, or some other, type of motion. Here we investigate the spectral evidence that indicates the extent to which we are justified in simply combining the analyses of the two types of motion that have been given, and we consider the interpretation of the spectra that involve simultaneous rotational and vibrational energy changes. The key to the analysis is the recognition that the calculated periods of vibration and rotation would be typically of the order of 10-l4 and 10-l2 sec, respectively. A rotation-vibration system with such relative periods would be described in terms of a clearly distinguished 8
/
Journal o f Chemical Education
rotation accompanied by a rapid vibration which oscillated through a hundred cycles for each rotational cycle. The analysis of such a system would begin with a simple combination of descriptions of the rotational-vibrational motions. If we proceed in this manner we write E,.
.
=
4
+
I/z)
- W.Z&
+
L/~)2
BJ(J
+
+ 1 ) - DJa(J +
(8)
The combination of the rotation and vibration selection rules,2 aJ = *l and Av = +l would lead us to expect a fundamental band with, from AJ = +1, an "R" branch, very much like a pure rotational spectrum extending out to higher frequencies from the band center and, from AJ = -1, a corresponding "P" branch extending to lower frequencies. See Figure 10 for the energy level diagram and anticipated transitions. Inspection of actual vibration-rotation spectra, Figure 7, shows something of this expectation. The marked closing up of the spacings at higher frequencies and opening up at lower frequencies suggest, however, the need for a refined model. We must now insert into our model the expectation that the effective moment of inertia, or rotational constant, which governs the rotational energies, can be expected to depend on the degree of vibrational excitation. We can express this most simply by writing the dependence of the rotational constant on the vibrational quantum number as Be = Be - 4 '/d (9) where 0, is an additional parameter that represents the change in effective internuclear distance due to vibrational excitation. With this addition, we write for the energy pattern expected for the rotating-vibrating molecule - msxs(v '/? E m , ,= W.(U
+
+
[B. - a.(v W.(V
+ '/*I
+ + + + BJ(J + 1) D J V + l I a - 4+ '/t)J(J + 1)
+ l/n)lJ(J + 1 ) - D J P ( J+ I ) ¶
=
1/z)2
W ~ Z &
(10)
These selection rules are derived by expressing the quantum mechanical transition moment connecting the energy states
IdnR=
+,
+,
S
+n~Jlmdr
Here = +, + J , and = +o" +P, that is, a product of onedimensional harmonic oscillator vibrational wavefunctions and rigid-rotor wavefunctions. These wavefunctions are governed by t,he quantum numbers v and J , and the single and double primes designate t,he upper and lower energy states, respectively. To show the effect of vibration, the dipole is expanded in a Taylor's series using the displacement from equilibrium z, giving a = w ( b r r / b z ) ~ z neglected terms. These substitutions give
+
+
The rotational selection rules come from the first and third terms. The first term is zero unless ro # 0 and o' = v U , due to orthogonality. The third term is zero unless J' = J" + 1, due to the properties of + J . The vibration-rotation selection rules come from the second and third terms. The second term is zero unless a r / b z # 0 and v' = v" &1, due to properties of Jl.. Again the t,hird term requires for vibration-rotation spectra J' = J' il.
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 which relate the model parameters to the frequencies of the rotational components of the fundamental and overtone bands provides values for the parameters. From subtractions of eqns. (11) and (12) and of eqns. (13) and (14) we obtain
and
From corresponding additions of these pairs of equations we obtain
and R(J)sco Figure 10. Vibration-rotation energy stater and dlowed transitions in a diwomic molecule. The fundamental absorption vibration-ro:ation Fig. 7 ) is o result of the selection ruler 4 v = +I, 4 J = *I. spectrum ( ~ e e The P-bronch gmup of lines arises from 4 1 = -1, ond the R-bronch fmm AJ = +l.
I t is on the success of this expression in interpreting the observed spectra that our model of nearly independent rotational and vibrational motions and energy patterns is based. The deduction of the five parameters in the above equation requires the manipulation of considerable data. Since such calculations can be part of a laboratory experiment that seems particularly effective in involving students with spectroscopic calcuIations, we present an outline of one way this can be done in an organized manner. First it is convenient to label the spectral lines as R or P to indicate the spectral branch and to index with an integer in parenthesis which is the lower of the two J values. Thus the third line of the P branch from the band center is labelled P(3). In addition we indicate that this line is a component of the fundamental band by writing P(3),+., or of the first overtone band by writing P(3)2,0. In reference to the energy level pattern the symbol J designates a rotational level of the v = 0 vibrational state. With these stipulations we write using eqn. (lo), for the components of the fundamental band, which require Av = +1, AJ = + 1 R(J),-o = P(J
+
+
we
- 2mz. + 2BAJ
= w.
+
1) 4 J 1)(J 3) - W J - 2w.s. - 2B&7 1 ) a.(J 1)(J - 1 ) 4D(J
+
+
+ +
+
+ 1)'
(11)
+
(12)
P(J 1) is written, rather than P ( J ) , to achieve greatest similarity between eqns. (11) and (12).
f
P(J f l)aco 4
=
- 3,,,&.
-
=,(J
+ 1)'
(18)
For any chosen J value, the left sides of these equations can be evaluated and one can see that subtraction of the first pair of equations will lead to a value for a, whereas subtraction of the second pair will give the anharmonicity constant w,x,. These calculations3may be repeated if desired for all available lines and average values of ry, and w g , determined. Equations (15) and (16) may now be rearranged into a form that will allow graphical evaluation of additional parameters. After transposing the value of -a, or -3/2a, to the left side, the value of the left side is plotted against 2(J I ) % , and the slope of the line determines -D and the intercept gives Be. A least squares treatment can be used to calculate the best D and B, values. Finally, substitution into any of the above equations leads to values of o,. (The equilibrium internuclear distance r, may easily be calculated from the relations Be = h2/8r21 and I = fir,%, where these quantities were defined a t the beginning of the section on Rotational Energies.) Commercial spectrophotometers usually do not provide sufficient precision to justify application of such an elaborate model as that based on eqn. (10). By means of an interferometric technique, Rank, et al. (59) have obtained overtone and fundamental data for HC1 to the nearest 0.0001 cm-' and for DC1 to the nearest 0.001 cm-'. I have sometimes assigned junior students to use the spectral data from Rank, et al. for the above calculations. Typical student results for HClS6 are we = 2989.96 om-', w,x, = 51.977
+
a I am indebted to ProfessorDavid A. Daws of the Chemistry Department of +,he University of Southern California fnr desscribing the above treatment.
Volume 47, Number 1, January 1970
/
9
cm- 1, a, = 0.3017, D = 5) X 10- 4 cm- 1, B, = 10.590 cm-1, r, = 1.2747 A. A similar treatment using a computer program has been described by Richards (40). We have found that this pro~ram gives excellent results using fundamental data obtained with the Beckman IR-12 and overtone data obtained with the Cary 14. Another computer approach is given by Bader (41). Similar experiments have been described by Stafford et al. (42), Roberts (43), and Shoemaker and Garland (44). A different method of vibration-rotation band analysis ba sed on "first differences" and "second differences" can be used for experiments and is clearly described by Dunford (45). The above material suggests that the thermodynamic properties of molecules can be calculated by combining the contributions made by the rotational motions with those from the vibrational motions. In arriving at this conclusion attention has been paid to the energy pattern that is needed to explain the observed spectra. Of interest also is a consideration of the statistical weights that must be ascribed to each energy level of the vibrating-rotating molecule. As in pure rotational spectra, the consequences of identical nuclei and the Exclusion Principle are easily recognized. A particularly good example is provided by acetylene and deutero acetylene where the effect of the different statistics obeyed by the sets of indistinguishable nuclei has been incorporated into a student experiment by Richards (46). Another interesting case is the "strong, weak, weak, strong" intensity variation in the perpendicular bands of the methyl halides, as illustrated by the high resolution study of CH 3F and CD 3F by Jones, et al. (47). An interesting example of the use of vibrational frequencies to calculate thermodynamic functions by statistical methods is provided by Berkowitz, et al. (48). These authors used the Raman and infrared spectra of S6, which is a puckered ring, to assign the eight low energy fundamentals of vibration. Such low energy transitions, through Boltzmann factors in the statistical expressions, contribute rather heavily to the thermodynamic functions. The agreement of their calculated values with experimenta l entropies confirms the assignments and shows that no lower energy fundamentals were overlooked. Thus spectral studies can lead to detailed patterns of vibrational energy levels and can show the extent to which the energies of gas phase molecules can be interpreted with separate terms for rotational and vibrational energies. One finds, it should be pointed out, that unless some quite specific solute -solute or solute-solvent interaction is occurring, the vibrational spectrum, and therefore vibrational energy pattern, is little different in the gas and the condensed states. · Most thermodynamic uses of information on energy states assume an equilibrium distribution. Interesting variations from this can occur and examples are given by chemical lasers, such as those studied by Pimentel and coworkers (49, 50) for HCl and for HF. For example, in HF the population inversion 4 was achieved
4 Population inversion is a metastable condition in which, contrary to the Boltzmann factor, higher energy states have greater density of occupation than lower states.
10 / Journal of Chemical Education
by photolyzing mixtures of UF 6 and H 2 with a xenon flash tube. This reaction produces HF molecules whose laser action is emission of vibration-rotation energies from v = 2 to v = 1 in the P-branch, with J having values of 4-8. As a final note on vibration-rotation spectra, it is somet imes surprising to chemists that species which are not ordinarily thought of as molecules may yet be analyzed in great detail by spectroscopic methods. For example, Gloersen and Dieke (51) have recent ly studied the diatomic molecules 3He2 and 4 He2 which are formed in a helium discharge tube. These molecules are in existence in sufficient number and length of time to permit their vibration-rotation spectra to be measured. Electronic Energies
For the vast majority of chemical materials dealt with at ordinary temperatures the single ground state of the species involved is all that needs to be considered in thermodynamic matters. There are, however, a few species where complexities due to the electronic states do enter and at higher temperature consideration of excited electronic states can be of major concern. In such cases, which include flames, plasmas, and solar and stellar conditions, one usually deals with atoms and sma ll molecules, or their ions, and it is enough here to indicate some aspects of the study of the electronic states of diatomic molecules. Emission, and to some extent absorption, spectra that depend on changes in the electronic energy of a molecule generally present a wealth of experimental information that can lead to a detailed knowledge of the ground and the variety of excited electronic states that usually exist. The detail that is observed, due to the simultaneous rotational and vibrational energy changes that occur, demands an intricate and complex analysis for the unraveling of the various patterns. The classic presentation of this material is given by Herzberg (52) with less complete but very useful treatments also being given by Gaydon (53) and by King (54). An introduction to some of the results that are useful for studies of the energy states of molecules can be
Figure 11. Electronic spectrum of h(g) run on the Cary 14 with the 0-1 absorbance slidewire. A 1 0-cm cell containing 12 crystals was heated to about 50°C. The lines are actually band heads in which the many closely spaced rotational transitions are not resolved. The strongest group of lines originates from the ground electronic state vibrational level v" = 0 (see Fig. 12A). The group marked with arrows originates from v" = 1 (see Fig. l 2B). The group marked "o" originates from v" = 2 (see Fig. 1 2C).
Figure 12. Elechonic trmritionr between various vibrational energy stoles of a diatomic molecule. Verticol transitions ore explained b y the FranckCondon principle. (A1 indicoter the common origin from the ground vibration level of the most intense gmup of lines in Figure 11. (0) accounts for the nexl sttongert gmup of lines in Figure 11, marked with orrowr (Cl accounh for the third stmngert gmup of lines in Figure 11, marked with "o."
achieved by dealing with the relatively easily observed and analyzed absorption spectrum of I1. The suitability of this example has already been recognized by Staffoid (65). Electronic transitions are most often observed in the ultraviolet, and occasionally the visible, spectral region. The violet color of iodine indicates a visible absorption band, and this is found, as shown in Figure 11, to have considerable structure, even if a relatively low resolution instrument is used. The absorption of the large amount of energy that is implied by even a visible spectrum can produce a molecule with a quite differentelectronic binding energy from that of the ground state. The expectation is that the excited state will have an equilibrium internuclear distance and a force constant, i.e., it will be represented by a potential energy function that is quite different from that of the ground state. The situation that is expected is illustrated in Figure 12. Careful scrutiny of the I, spectrum gives experimental support to the conclusion that there are no simple selection rules governing allowed changes in the vibrational quantum number v when electronic energy changes are involved. Rather, the relative intensities of observed transitions may be understood through the operation of the Franclc-Condon principle (56). This principle recognizes that electronic transitions occur much more rapidly than vibrational motions and therefore 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 transition moment can be appreciable are those for which the wavefunctions of the two different states have, at the same internuclear distance, appreciable magnitudes. I t follows that no simple selection rule exists but that, as suggested by Figure 12, some relation can be established between the potential functions of the two states and the absorption or emission components that are expected to be observed. To proceed we need to consider again the observed spectrum. The three quite easily distinguished sets of lines 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 energy being identified with v" = 0. Additional progressions can be recognized particularly at higher sample temperatures. The analysis begins by making columns of the frequencies of the components in the progressions that are observed. These columns can now be related to one another by sliding them vertically with reference to each other until the differences in frequency between horizontal pairs show a gradual decrease as one moves across the table to the right. The arrangement that is obtained from student results at the University of Redlands, for a variety of sample temperatures, is presented in Table 2. The arrangement of Table 2 is called a Deslandres table. It remains, to show the development of such a table, to indicate how the v' numbers are deduced. For I2 this is a matter of some considerable difficulty since, for all the sets of lines, there is no abrupt termination at long wavelength, which would indicate that the v' = 0 level had been reached. Rather there is a fading out of the lines. The assignment must depend on fluorescent or isotope effect studies, and the correct 1%assignment has only recently been arrived a t by these techniques by Steiufeld, et al. (57) and by Brown and James (58). This assignment has been used to assign the quantum numbers in Table 2. With the organization provided by the Deslandres table one can proceed to the deduction of the properties of the ground and excited electronic states involved in the transition. To represent adequately the vibrational energy spacing in either the excited or the ground electronic state we can write eqn. (7) again E" = ua (V >/%)- W L Z ~(V '/da (17) The successive horizontal entries in the Deslandres table give vibrational energy differences for the ground state whereas the vertical differences give the values for the excited state. Thus, allowing for the introduction of either single or double primes we write
+
AE, = E(u Volume
+
+ 1 ) - E(v)
47, Number I,
January
(18)
1970 / 1 1
Table 2. 0'
0
1
2
16991.1 101.4 17092.5 101.4 17193.9 97.0 17290.9 100.4 17391.3 95.2 17486.5 92.3 17578.8 YO4 17669.2
16241.7 108.2 16330.3 103.1 18463.4 102.6 16566.0 106.9 16672.9 106.6 16779.5 104.0 16883.5 99.8 16983.3 96.6 17079.9 98.9 17178.8 95.3 17274.1 94.6 17368.7 94.9 17463.6
8 10 11 12 13
15 16 17 18 19 20
"" 3
12
Bands*
(om-1)
4
5
6
7
111.8
8
14
Deslandres Table of
17306.4 101.6 17408.0 98.7 17506.7 97.1 17603.8 94.8 17698.6 94.5 17793.1 92.6 17885.7
213.9 214.1 215.3 212.5 212.1 214.3 216.5
211.6 209.0 210.6 211.0 212.5 212.4 210.1 205.6
a Btudent data obtained from the Cary 14 using a 10 cm aa cell containing 11oryatala at 35 to 85-C. Wavelengths were read from the apaotra without Conversion to vacuum wavenumber was mass from C. D. Coleman. W. R. Boasnun sod W. F. Mesgera, Conversion Tablcsfor Wauelewlhs to Vacuum Wouenuntber~,National Bureau of Standards, 1960.
calibration.
Table 3.
O,'Z.'
D . '
which simplifiesto AE. =
0.
- 2vw.z.
- %A*
weX D.' (20)
This expression suggests that plotting AE, versus v should give a straight line with slope -2wje and intercept a t v = - 1, of we. Note that no primes or double primes have been used since the plotting procedure is, at least in principle, applicable to either vertical or horizontal differences in the Deslandres table. Such a plot is termed a Birge-Sponer plot (59) and is shown in Figure 13 for the excited electronic state of I,. From Figures 12 and 13 it is also apparent that summing the AE, over all v , i.e., obtaining the area from
Figure 13.
12
/
Birge-Sponer plot for the upper alestmnis state of I$.
Journol of Chemical Education
. Molecular Constants of lz (g) Student results (cm-LI
Accepted values
0.940 4260 214 12420d
o.7oi6* 4391 .Oa 214.5Zb 125RlY
lrm-l>
*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., 32,738 (1960).
0.75 from slope in u' = 0 to v' = 16 region. Reanires the additiand informstion t,hat, t.ha diswain~-~~~~ ~~-~ tion products o f the excited st& are atoms in the 'Pa/, gmund state m d PI/, excited state. Subtraction of the atomic excitation energy of 4280 cm-' leads to the deduced dissociation energy for I. in the gmund state. ~~~
~
v = 0 to the point at which AE, = 0,gives the depth of the potential well, that is, the dissociation energy of 12 in the excited electronic state to whatever atomic states are formed. Similar treatments can be given the ground state data but, as is typical of absorption spectra, so few vibrational levels are involved that the results are very unreliable. Here, in fact, it must be emphasized that the near linear relation of Figure 12 cannot be counted on as it depends on the assumption that a simple anharmonic term in the vibrational energy pattern is adequate. Some University of Redlands student results are shown in Table 3 along with results from the literature. More extensive use of the data can be made, for example calculation of potential energy curves, as illustrated in the work of Weissman, Vanderslice, and Battino (60) on I2 and HI and by Clyne and Coxon (61) on Brz.
(16) Ref. 1 1..no. .8~ . . .501-81. iljj B O O ~ T E R G., J.. AN; COX. A. P., J. CXEM.EOITC.. 45,18 (1968). (18) D ~ v r ~ M., s , (Edttor). "Infrared Spectroscopy and Molecular Struotore:' Elsevier Publisbine Co., Amsterdam, 1963, D. 192: Jncoa, E. J.. T e o M ~ r 0 B .H. B.. and B A ~ S L GL.. S.. J. Chem. Phya., 47, R7RR ~llQR71~ (19) B m a e n , R. M., J . Chem. Phyr., 2, 128 (1934): 3, 710 (1936); Phys. Rcu. 48, 284 (1935). (20) Henscnnhce, D. R., A N D Lnunm, V. W., J. Chem. Phus., 35, 458
Tnmzn. D. W.. A N D A k l o ~ a n r ,M. I., J . Chem. Phva. 37, 3007 (19621: J . Chem. Soc., 5141 (19631. 4434 (1964). tonne^, D. W., AND MAY.D. P.. J . Chem. P h u ~ .45,471 , (19661. T o n N E n , D , W , """YT*. I,.' ",*a (IU",,. ""-~ 1.3, R m n , D. H.. I~ B T M A N . D. P.. RAO,B. 6.. A R O W I D O I N ~T. . A,. J . opt. Soc. Am.. 5 2 , l (19621. RICXARDS, L. W., J. C x e ~ Eouo., . 43, 552 (1960). n."..nt 7 , - EDnc.. 46, 206 (1969). &nu. TAFFORD, F. E.. IIOLT,C. W.. AND PAULBON, G. L.. J. CHEM.EDUC.,
-.. ,..,.
....,..
-. (221 W I Z B O N E., B.. DECIVB.J. C., A N D CEO%+.P. C.. molecular Vihrations." MoGrav.-Hill nook Co., New York. 1955. (23) D*vms.M., (Edilorl."Infr~redSpeetroseopyand Molecular Structure;' Elsevier Puhlishine Co., Amsterdam. 1963, Chap. Y. (24) ~ ~ o uG.n M., , FLETCIIER. W. H., A N D SMITX,n. F., J . cham. ~ h y a42, 2236 (1965). (251 K n u n ~ u w G. . N.. A N D SCXOTTE, C. J. H.. Sp~otmchim.Act., 21. 1947 (19651. (261 McGnnv. G. E.. B m n m ~ D. , L.. AND HIBATSDNE, I. C., J . Cham. Pliys. 42,237 (1965). (27) D O W BD. . A., J. CHIM. EDUC.,35,629 (1958). (28) S z L v m a ~ m N .R. M.. A N D Bnasmn, G. C.. "Speotrometrio Identifioacation of Orzanie Com~ounda,"John Wiley iu Sons,New York, 1967. (291 N*xnmasr. I
