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By C. H. Townes, A. L. Schawlow
Dover Publications, Inc.Copyright © 1975 Dover Publications, Inc.
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ROTATIONAL SPECTRA OF DIATOMIC MOLECULES
1-1. The Rigid Rotor. If the distance between nuclei in a diatomic molecule is considered fixed, the possible frequencies of the end-over-end rotation of this "rigid rotor" can be rather simply obtained. Using assumptions of the "old" quantum mechanics, the angular momentum must be some integral multiple of h/2π, so that
2πvI = Jh/2π
where h is Planck's constant, I is the molecular moment of inertia about axes perpendicular to the internuclear axis, v is the frequency of rotation, and J is a positive integer giving the angular momentum in units of h/2π. Hence the frequencies expected from such a system are
v = Jh/4π2I (1-1)
The moment of inertia I comes largely from the nuclei, where most of the molecular mass is concentrated, and for diatomic molecules of ordinary masses is of such size that for small integral values of J, the frequency v is of the order 10,000 to 100,000 Mc, or the wavelength in the region 3 cm to 3 mm.
On this simple basis one might expect a rotation about the molecular axis to occur also and to have characteristic frequencies a few thousand times greater because the moment of inertia about this axis is produced by electrons, which are very much lighter than the nuclei. These frequencies lie then near the optical region, and in a very rough way the electronic frequencies may be regarded as due to this type of rotation about the molecular axis. Since these frequencies are very high, they lie far beyond the microwave range and are not ordinarily excited at room temperature. They will therefore be neglected in most of the following treatment. A somewhat more sophisticated and rigorous determination of the frequencies produced by a rigid diatomic molecule can be obtained by finding the permitted energy levels from wave mechanics (see , p. 271, or , p. 60). As the molecule rotates about its center of gravity, its orientation in space may be specified by the spherical polar coordinates θ and φ. The wave equation may then be written
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-2)
where ψ is the wave function and W the rotational energy of the molecule. The variables θ and φ may be separated by substituting
ψ = Θ(θ)Φ(φ)
d2Φ/dφ2 = -M2Φ (1-3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-4)
where M2 is an arbitrary constant.
Solutions of these equations which are single-valued and normalized can be obtained only when
W = h2/8π2I J(J + 1)
where J is a positive integer and M is an integer such that |M| ≤ J. Such solutions are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-6)
where P|M|J (cos θ) is an associated Legendre function. [J (J + 1)](h2/4π2) is the square of the total angular momentum, so that the angular momentum may for convenience be designated by J. Similarly the projection of the angular momentum on the polar axis is given by M (h/2π), or simply by the integer M.
The frequency observed when the molecule makes a transition between a lower state of energy W1 and an upper state of energy W2 is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-7)
From the correspondence principle, these frequencies may be expected to approximately equal the frequencies given by expression (1-1); hence J2 should equal J1 + 1, and
v = 2B(J + 1) (1-8)
where J is the angular-momentum quantum number for the lower state (J1), and B = (h/8π2I) is called the rotational constant. The quantity B is often expressed in units of cm-1 for infrared spectroscopy. In that case B = (h/8π2Ic). For microwave spectroscopy, B will generally be given in cycles per second, or B = h/8π2I. However, numerical values will usually be quoted in megacycles, or 106 cycles/sec. The selection rule that J2 = J1 + 1 or ΔJ = ±1 for dipole radiation of a diatomic molecule will be more rigorously demonstrated in the discussion of intensities later in this chapter.
1-2. Energy Levels of the Diatomic Molecule. From Eq. (1-8) it is seen that the spectrum of a rigid rotor consists of absorption lines equally spaced in frequency with an interval 2B. Although the rigid rotor is an idealization to which actual molecules conform to a good approximation, accurate spectroscopic measurement reveals many deviations from this approximation. As J increases and the molecule rotates faster, it stretches so that the moment of inertia increases. Moreover, the nuclei vibrate back and forth along the line joining them even in the lowest vibrational state. A much greater difficulty from the point of view of obtaining a complete theoretical treatment is that the entire molecular system, composed of interacting electrons as well as nuclei, is so complicated that an exact quantum-mechanical solution is impossible.
However, since the electrons are very much lighter than the nuclei and move in electric fields of approximately the same intensity, the electron motion is very much faster than that of the nuclei; i.e., many cycles of the electronic motion occur during a small portion of a cycle of the nuclear motion. It is therefore reasonable to treat first the electronic motion, considering the nuclei as fixed. Then the internuclear distance r appears as a parameter. In this way the electrons are found to be capable of occupying several states, each giving the molecule a particular value of the energy U, for each internuclear distance. Generally in microwave spectroscopy only the lowest of these electronic states is important.
As the internuclear distance is slowly varied, the electronic energy varies. Because the electronic motion is so fast in comparison with the nuclear motion, at each instant the electronic energy may be considered to have reached its equilibrium value corresponding to that distance. Thus we are justified in treating the vibration and rotation of the nuclei separately from the electronic motion. In this treatment U(r), which is the sum of the electron energy plus energy of electrostatic interaction between the two nuclei, appears as the potential energy. The validity of this approximation was discussed by Born and Oppenheimer (; see also , pp. 259–274, and , Chap. I). They showed that the entire molecular energy, including that due to electronic motion, can be expanded in powers of (m /M)½, where m is the electronic mass and M an average nuclear mass. Separation of nuclear and electronic motions hence corresponds to selecting the larger terms of the series expansion and neglecting those which are smaller by (m/M)½ or more. In some cases the neglected terms lead to observable effects, but they can only with difficulty be taken into account.
Using the approximation that the variation in electron energy with nuclear motion may be included in the potential U(r), the wave equation for vibration and rotation of a diatomic molecule becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-9)
in which ψ is the wave function for the nuclear motion, M1 and M2 are the nuclear masses, and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-10)
xi, yi, and zi being Cartesian coordinates of the ith nucleus relative to axes fixed in space.
Transforming to spherical polar coordinates r, θ, φ of the second nucleus relative to the first as origin (cf. , p. 264),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-11)
where µ is the reduced mass, M1M2 /(M1 + M2). The variables may be separated by the substitution
ψ = R(r)Θ(θ)Φ(φ) (1-12)
Θ(θ) and Φ(φ) turn out to be the same as the wave functions found above for the rigid rotor.
The radial wave function R (r ) obtained by the separation process is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-13)
The term J(J + 1)/r2 may be regarded as a potential energy associated with the centrifugal force due to the rotational angular momentum J. Substituting the expression
R(r) = 1/r S(r) (1-14)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-15)
The solutions of Eq. (1-15) will obviously depend on the form of U(r). Since it is seldom possible actually to solve the electronic wave equation, it is customary to use an empirical expression for U(r).
From experimental studies of molecular spectra and from calculations on simple molecules, the general form of U(r) is known to be that of Fig. 1-1 (see ). At large distances the atoms are independent, and the force between them is negligible. Their energy is then just the sum of the energies of the individual atoms. At very small distances, when the atoms are "in contact," they must repel each other. At some intermediate distance there must be a potential minimum, corresponding to the equilibrium distance of the atoms.
Solution for Morse Potential. A potential which fulfills these requirements is the Morse function 
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-16)
where D = dissociation energy of the molecule
re = equilibrium distance between nuclei
a = a constant
The Morse function differs from the true potential at r = 0, where the actual potential would be extremely large. However, the Morse potential is also quite large at r = 0 and this is a region where the wave function of the vibrating rotor is expected to be small so that the discrepancy is not serious.
Using the Morse potential function, the radial equation (1-15) becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-17)
The solution of this equation for J = 0 has been given by Morse  and for any J by Pekeris . Substituting
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-18)
in Eq. (1-17), we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-19)
For A ≠ 0, it is necessary to expand [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in terms of y:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-20)
If the first three terms of this Taylor expansion are retained, Eq. (1-19) becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-21)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-22)
Eq. (1-21) can be further simplified by the substitutions
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-23)
so that it becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-24)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-25)
As in the usual quantum-mechanical treatment of the simple harmonic oscillator or of the hydrogen atom (cf. ), for the solution of Eq. (1-24) to be finite and vanish at the ends of its range, it must be given by a terminating series, i.e., a polynomial. In fact, Eq. (1-24) is identical in form with the equation for Laguerre polynomials found in the solution of the hydrogen atom. This requirement can be shown to restrict v to the values 0, 1, 2,.... Strictly speaking, the solutions thus obtained satisfy the boundary condition S -> 0 as r -> - ∞ rather than the proper condition S -> 0 as r -> 0. Ter Haar  has examined this approximation and shown that it is usually a good one.
It is possible to solve for W using Eqs. (1-25), (1-23), (1-22), and (1-18), which give
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-26)
Expanding Eq. (1-26) in powers of c1/D and c2/D, it takes the form:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-27)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-28)
ωe, αe, Be in (1-27) and (1-28) are expressed in cycles per second. The terms in Eq. (1-27) can be identified with the solutions of more specialized problems, so that each can be given a physical significance. Thus the first term involving (v + 1/2) has the form of the solution of the wave equation of a pure vibrator with a harmonic potential. The second term is obtained when the vibrator potential is made anharmonic by the addition of a cubic term in the potential energy. A term of the form BJ(J + 1) is just that obtained in Eq. (1-4), the solution of the rigid rotor problem, while the next to last term comes from centrifugal stretching of the rotating molecule. The last term allows for the change in average moment of inertia due to vibration and the consequent change in rotational energy.
Dunham's Solution for Energy Levels. Some other more refined potentials have been used for problems in optical spectra involving excited rotational or vibrational states (, pp. 102, 108). Dunham  has calculated the energy levels of a vibrating rotor, by a Wentzel-Kramers-Brillouin method, for any potential which can be expanded as a series of powers of (r - re) in the neighborhood of the potential minimum. This treatment shows that the energy levels can be written in the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-29)
where l and j are summation indices, v and J are, respectively, vibrational and rotational quantum numbers, and Ylj are coefficients which depend on molecular constants. The effective potential function of the vibrating rotor may be written in the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-30)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The term involving BeJ(J + 1) allows for the influence of the rotation on the effective potential.
Excerpted from Microwave Spectroscopy by C. H. Townes, A. L. Schawlow. Copyright © 1975 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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