Figure 7.3. Three vibrational modes of H2O

Figure 7.3. Three vibrational modes of H2O

0-©-© va electronic Schrôdinger equation which provides the equilibrium configuration and energy Ue and also departures from that in the form

Often the equilibrium configuration and coordinates qi can be defined in terms of the bond lengths and bond angles. Thus, for the pyramidal NH3 molecule with N = 4, the six coordinates may be chosen as the three N-H band lengths {Ri, R2, R3} and the three HNH bond angles so that

+ \ka [(<*! - a\f + («2 - ae2f + (a3 - a^)2] , (7.9)

the two force constants and obtained from a quantum chemistry calculation of the electronic motion or often by fitting to vibrational or Raman spectra. It is useful to utilize symmetries as in the above example where the same constant k is expected for each of the NH bonds and the same ka for the three identical angles. A whole elaborate technology of point group symmetries is available for this purpose built on the symmetries of rotations and reflections for molecules having such geometrical symmetries.

Because nuclear masses are large, a classical-mechanical analysis of normal modes often suffices. This involves a Lagrangian constructed from T and the U in (7.9) and a matrix diagonalization to obtain the coordinates of the normal modes in terms of which the Hamiltonian is of one-dimensional harmonic oscillators. The vibrational energies are, therefore, with wave functions as in (6.17). Thus, in the water molecule, we have u\_ = 3832 cm-1, i/2 = 1648 cm'1, and = 3942 cm-1, and the zero-point energy is 4636 cm"1. Fig. 7.5 is a sketch of the ground and low-lying excited states labeled as As with quantum chemistry calculations for the electronic motion, tables and computer programs are available for carrying out the normal mode analysis of even complex polyatomic molecules. Anharmonic corrections to (7.10) arising from further terms in the expansion (7.9) are also available.

Turning to vibrational spectra, the analysis of the matrix element of the dipole moment d between an initial and final vibrational state is as in other cases considered earlier. Thus, just as in (6.23) for diatomic molecules, the dipole moment can be expanded around its equilibrium value along any generalized coordinate

Once again, the linear term in the expansion gives the dominant contribution although higher order terms in (7.11) as well as anharmonicities in vibrations also prove important at times. But, within the harmonic approximation of (7.9), with oscillator wave functions independently in each coordinate qk, the relevant matrix elements of the linear operator

(9k ~ it) are non-zero only for Vk —Vk ± 1. An important consideration, however, is that the multiplicative (dd/dqk)e be non-zero for such a vibrational transition to be exhibited. Here again, symmetries of the molecule and the associated group theory prove helpful in determining which transitions take place. As an example, in a linear triatomic molecule like CO2 (Fig. 7.4), the k = 1 symmetric stretch clearly has no dipole moment for any value of and the corresponding vibrational mode is "inactive". Since vibrational energy spacings fall in the infrared, the usual terminology is that such a mode is "infrared inactive". The other two modes of bending and asymmetric stretch do have a non-vanishing (dd/dqk)e and are "infrared active".

Infrared radiation was itself first discovered in an astronomical context, when Sir William Herschel in 1800 saw the response of a thermometer when placed just outside the red region of the solar spectrum. It took over a century before infrared absorption in molecules began to be studied. It is now an active branch of both laboratory and astronom ical molecular studies. Intense infrared bands of CH4 are seen in the atmospheres of the major planets, bands of C02 in Venus and II20 in Mars. One talks of near infrared (A = 0.8 - 2.5//m), mid infrared (2.5 - 50 /jm) and far infrared (50 - 1000 fim) regions. In the C02 molecule with states \v1v2vz), the k = 1 is inactive but transitions 010-^000 and 001-+000 have frequencies 667 cm-1 and 2349 cm-1, respectively, that is, wavelengths 15 and 4.3 ¡.im. Generally, with the ground state having the highest occupancy, these Av = 1 transitions involving the ground vibrational state and one unit of excitation are the strongest, and are called fundamental frequencies. The 001 (2349 cm"1) and 100 (1388 cm"1) levels are the basis of the C02 laser. When mixed with N2 whose molecules are excited by an electric discharge, collisions between N2 and C02 populate the 001 level, thus creating a population inversion from which stimulated emission at 10.59 pm is possible (Fig. 7.6).

Superposed on vibrational transitions is rotational fine structure, which is analyzed as in Section 7.2.2. The resultant close grouping of lines gives rise to "bands" as observed. Thus, for symmetric tops, any allowed vibrational transition changes the dipole moment along the symmetry axis or in one of the directions perpendicular to it. For the "parallel" vibration-rotation transition, the selection rules on J and K are as in (7.8) for pure rotation spectra:

A J = 0, ±1, A K = 0, with A J = 0 forbidden for K = 0. (7.12) For the perpendicular transition, on the other hand,

For asymmetric tops, the changes in rotational quantum numbers are as before in Section 7.2.2, with K changing in steps of 2 and A J = 0, ±1. Vibration-rotation absorption between 100 and 750 cm-1 in CO2, 1000 and 1080 cm"1 in 03, and 1300 to 1600 cm"1 and below 600 cm-1 in H20, all contribute to the greenhouse effect of the Earth's atmosphere.

A linear polyatomic molecule has a ground electronic state and is a symmetric top with K = 0. As a result, A J = ±1 for the parallel vibration-rotation band and A J — 0,±1 for the perpendicular band. In CO2, V's changes the dipole moment component along the symmetry axis and is hence a parallel band whereas changes perpendicular to the symmetry axis, forming a perpendicular band. The latter shows all P, Q, and R branches whereas the Q branch (A J = 0) is absent in the v3 parallel band. Rotational substructure on the 001-+100 C02 transition mentioned above gives rise to a group of transitions around thus providing partial tunability of the CO2 laser.

Again, as for diatomic molecules, Raman spectroscopy is an important and useful adjunct to pure vibration-rotation spectra. Since Raman transitions are due to the induced dipole moment by the electric field of the radiation, it is the polarizability a rather than the permanent dipole moment, and changes in a that govern the transitions. Complementary information is provided so that, for instance, the symmetric stretch in CO2 which has d = 0 and is infrared inactive is Raman active. Similarly, in the tetrahedral molecule, which has one nondegenerate vibration one doubly degenerate and two triply degenerate vibrations, only and are infrared active but all four are Raman active. Molecules with a center of symmetry obey a "rule of mutual exclusion", infrared active modes being Raman inactive, and vice versa. This is because such molecules have the inversion through the center as a good symmetry so that states are either or u under it. The electric dipole operator is u (a vector) whereas the polarizability is g so that no pair of states can have simultaneously non-vanishing values for both.

Even for the simplest polyatomic molecule, our discussion so far in terms of a product wave function of electronic, vibrational, and rotational motions is only approximate, although it provides the basic understanding of the electronic and nuclear motions. Couplings between these, arising from anharmonicities in the vibrations of the nuclei or centrifugal distortions, as well as the entanglement of electronic with nuclear degrees of freedom as, for instance, when vibrations even change the geometrical arrangement of the molecule, complicate the discussion. In particular, the presence of degeneracies which have already been noted has the implication that the corresponding states may be mixed by any additional coupling no matter how weak. The full wave function is a "rovibronic" function, its factorization as a product of rotational, vibrational and electronic functions only an approximation.

A first example is in linear molecules such as C02 where the two bending modes z/2a and v2h are degenerate (Fig. 7.4). With z the molecular axis, these vibrations in the xz- and yz- planes are like two orthogonal simple harmonic motions of a pendulum in x and in y. Equivalently, such a two-dimensional pendulum can be viewed in circular coordinates (the familiar conical pendulum or Lissajous figures) with an angular momentum I in an angular coordinate <p = arctan(y/a;). Although these are the conventionally used symbols, they should not be confused with the electronic angular momentum or the azimuthal coordinate of nuclear rotation in Section 7.2.2. The terms I and <p here refer purely to the vibrational motions orthogonal to the molecular axis. In a state with such a vibrational angular momentum t, interactions of the bending with the rotation of the molecule gives a rotational energy as in (7.7) but with I replacing ¡A'|:

with A the inverse of the moment of inertia of the nuclei about the molecular axis for the degenerate vibration.

Each rotational level undergoes "¿-doubling", the two states | + I) and | - I) forming the superpositions [| +1) ± | - ¿)]/V2 under the interaction between rotation and vibration, and having slightly different energies. Thus, in HCN, with B - 1.477 cm-1 and vibration frequency v2 = 711.7 cm-1, the splitting is 112.2 (v2 + I) J (J + 1) MHz. Transitions between the levels of any doublet are dipole allowed, leading to microwave absorption, the levels themselves indicated by \vi v2v3).

Exact degeneracy is not necessary for such mixings by weak coupling terms, and near-degeneracies can suffice. Thus, in C02, 2v2 is very close to v\ with the consequence that states with the same value of 2vi + v2 are nearly degenerate and mixed by anharmonicities or other couplings. This is called a Fermi resonance. A similar example obtains in benzene, a stretching C-H vibrational frequency lying close to the sum of two other vibrational frequencies. This gives rise to two strong bands at 3099 and 3045 cm-1. In larger molecules, with more vibrational frequencies, such accidental near degeneracies are more common and this phenomenon, therefore, quite ubiquitous.

Another spectacular example of degenerate configurations, leading to a so-called "inversion doubling", is provided by molecules such as NH3 with two equivalent pyramidal configurations, the N atom on one or the other side of the plane formed by the three H atoms. The two configurations are separated by a potential barrier of kcal/mole), the maximum energy corresponding to the N lying in the plane of the H atoms. As in other such examples, starting from the initial one (Section 6.2.1) of Hj" binding (with two protons separated to infinity and an electron bound to one or the other providing two degenerate configurations), the same basic quantum-mechanical theme of physical eigenstates being the plus/minus superpositions obtains, any vi-brational state of being split into two states

The two levels are split by an energy representing the tunneling of the system between the two configurations I and II. In it is an electron that tunnels over atomic-scale distances so that the splitting lies correspondingly in the eV range. But in NH3, with the entire massive N atom having to tunnel through the intervening barrier, the splittings are considerably smaller, approximately 24 GHz (wavelength = 1.25 cm) for the ground vibrational state. Indeed, in most molecules with two equivalent geometrical configurations of this type, the barrier is so large relative to the vibrational energy that the tunneling probability and, therefore, doublet spacing is quite negligible and the molecule can be considered effectively frozen in one of the configurations, but in there is an appreciable effect.

In the JWKB approximation, the splitting of a vibrational level of frequency is given by

AE = (ïz/tt ) exp j- y j [2/i (V - Evih)]1'2 ds j, (7.15)

where ¿¿is the reduced mass (of N relative to the three H) and the integral runs between the turning points defining the classically forbidden region

V > Uyib. Fig. 7.7 gives the doublet structure of the rotational spectrum of NH3. With the rotational constant B approximately 298 GHz, the doublet splitting of 24 GHz (0.67 cm-1) is an order of magnitude smaller than the spacing between successive rotational levels. The energy levels are well represented by v = 23787 - 151.3 J{J + 1) + 211 K2 MHz.

One important application of the doublet spectrum is that it serves as the basis for a maser (microwave amplification by stimulated emission of radiation). Population inversion between any two levels, such that the upper energy state has a higher population, can give rise to amplification of radiation, stimulated emission from the upper level dominating over absorption from the lower. The first maser was demonstrated by Townes and by Basov and Prokhorov in the ammonia molecule, followed soon after by hydrogen masers operating at 1420 MHz between the F = 1 and F = 0 hyperfine levels.

Besides the various diatomic species considered in Section 6.4, a large number of polyatomic molecules have been seen through their microwave spectral lines in interstellar space. These include H20, C02, S02, OCS,

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