Rotation of Molecules

Chapter 2 Rotation of Molecules C. C. COSTAIN

I. Introduction II. Diatomic Molecules A. The Rigid Rotor B. The Nonrigid Rotor III. Linear Polyatomic Molecules A. Rotational Energy B. Vibrational Angular Momentum C. Fermi Resonance IV. Symmetric Tops A. Rotational Energies of the Prolate Top; IA
20 22 22 27 29 29 30 30 31 31 32 4

3

34 6

36 38 42 42 44 44 49 51 52 55 56 56 58 59 59 62 °5

3

20

C. C. Costain

I. Introduction Much of our knowledge of the structure of molecules has been obtained from the analysis of their rotation spectra. T h e information has been obtained from studies over a broad region of the electromagnetic spectrum : the pure rotation spectra in the microwave and far-infrared regions ; the vibration-rotation bands in the infrared; the Raman rotational spectrum in the visible region; and the vibration-rotation fine structure of electronic transitions in the visible and ultraviolet regions. Each region has its particular advantages and limitations. T h e resolving power that can be achieved is approximately the same in all regions of the spectrum. T h e microwave region, with its low frequencies or low energies, therefore has a very great advantage in the resolution of small energy differences. T h e pure rotation spectra of a great many molecules fall in the microwave region, and the spectra of different isotopic species are easily resolved. Several books have been published which emphasize those aspects of rotation spectra which are of importance in microwave spectroscopy. T h e advantages of microwave spectroscopy are such that it is also important to remember that there are severe limitations in the microwave techniques. Lighter molecules must be studied in the far-infrared region. Also, studies in both the microwave and far-infrared are limited to relatively stable polar molecules in the ground or low-lying vibrational states. T h e study of rotation in the higher vibrational states is dependent on the analysis of the vibration-rotation bands in the infrared region. The accuracy of the constants obtained is not as high as in the microwave region, but nonpolar molecules can also be investigated. The lifetime of molecules in excited electronic states is normally so short that information on rotation can only be obtained from the vibrationrotation fine structure of electronic spectra in the visible and ultraviolet regions. These spectra become very complex, particularly when one or both of the electronic states have electronic orbital and spin angular momentum to combine with the angular momentum of the molecular frame. No attempt has been made to summarize these complex investigations: discussions in this chapter have been limited to molecules in the λ Σ electronic state, for which the electronic orbital and spin angular momentum are zero. T h e Raman rotational spectrum has also been neglected, although it is of primary importance in the study of nonpolar molecules, and its scope has been greatly extended by laser techniques. In many ways a rotating molecule in the gas phase provides a nearly

2. Rotation of Molecules

21

ideal system for quantum-mechanical studies. At low gas pressure the collisions with other molecules have little effect on the energy levels, and the molecule approaches the idealized isolated system. Exact solutions of the Schrödinger wave equation are possible for the simpler forms of the rigid molecule, and the solutions provide a very good approximation to the observed spectra. T h e deviations from the rigid-rotor theory—for example, those which arise from centrifugal distortion or rotation-vibration interaction—are usually small enough that they can be adequately treated by perturbation theory. Further, these deviations can be measured experimentally with high accuracy, and thus this second stage of the theoretical development can be precisely checked. Many higher-order interactions, initially of only academic interest, are now essential for the interpretation of observed spectra. This interplay stimulates continual extension of both theory and experiment, and the study of rotation spectra continues to be an active field of research. In the first approximation it is assumed that the atoms in the molecule are mass points fixed in position. T h e spectrum is interpreted in terms of the moments of inertia of this rigid molecule, and from the moments of inertia the molecular structure is deduced. In fact, no molecule is rigid, and this fact limits the accuracy of the structural determination. However, the effects of nonrigidity can be analyzed to give information on the vibrational potential function. A pure-rotation microwave spectrum exists only for molecules which have a permanent dipole moment to interact with the electromagnetic radiation field. An approximate value of the dipole moment can be derived from the intensities of the absorption lines. Much more accurate values of the dipole moment are determined from the measurements of the perturbations or displacements of the energy levels in an applied electric field. These studies of the "Stark effect'' are also very useful for the identification of rotational transitions. Information on the nature of the chemical bond can be obtained from the quadrupole hyperfine structure of rotational lines. T h e nucleus of an atom with spin greater than \ has a quadrupole moment which interacts with the gradient of the electric field at the nucleus. T h e magnitude and asymmetry of the components of the gradient give information on the bond order, ionicity, and hybridization. Of necessity, the discussions of these and other topics can only be very brief. References will be made to specialized books on rotational spectra. These contain detailed studies of most topics, and many of them list complete bibliographies of original papers.

22

C. C. Costain

II. D i a t o m i c Molecules A. T H E RIGID ROTOR

1. Wave

Functions

In classical wave motion, if the wavelength is properly chosen to fit the boundary conditions, the constructive interference of the waves produces a standing wave pattern. T h e most familiar examples of this "stationary state" are the loops and nodes of the vibrating string and of the resonant organ pipe. In quantum mechanics a system in which there is repetitive motion has stationary states which are produced in a similar manner. There is the de Broglie wavelength, λ = hjmv, associated with a particle of mass m and velocity v. This wave nature will lead to constructive interference if the frequency of the repetitive motion is such that a standing wave pattern, or wave function, is produced. At other frequencies it will lead to destructive interference, i.e., the wave, and therefore the particle, cannot exist. T h e wave function ψ and the stationary states of quantum mechanics are obtained by solving the Schrödinger wave equation for the particle, dhp

dhp

2

8π τη

dhp

For pure rotation the potential energy V is zero. T h e two masses m1 and m2 of the rigid diatomic molecule can be replaced by one equivalent particle by introducing the moment of inertia of the molecule /

=

-

»

^

= ^ ,

(2.2)

where r is the bond length between m1 and m2. T h e Schrödinger wave equation in terms of the spherical polar coordinates 0, φ, and r becomes 1

d ( .

^ïnT ~w (

s m θ

β

3 ψ \ -π)

Λ

1

+ ifonr

θ*ψ

Ί ψ

8π*Ετ +

- ^

Ι

ψ

=

-

0

·

(2 3}

The variables θ and φ are easily separable, r (or / ) is a constant, and the solution for ψ can be expressed as a product of the solutions of the two equations : ψ = Θ(θ)Φ(φ) = NjMPj»(B)

e
(2.4)

2. Rotation of Molecules

23

M

where NJM is a normalization factor and Pj {Q) is the associated L e gendre polynomial of cos θ (see Pauling and Wilson, 1935). For these solutions to exist the kinetic energy Ε is restricted to the energy levels or eigenvalues E=

1),

(h*l8n*I)J{J+

7=0,1,2,3,...,

(2.5)

The angular momentum corresponding to the energy levels in Eq. (2.5) is J , which has a magnitude

υΐ = (Α/2π)[7(7+1)]«·;

(2.6)

J represents the vector of the angular momentum with a fixed direction in space for the free rotor. If a preferred direction in space is defined by an electric field ε, then M(| M I = Mhßn) is the projection of J in the direction of the field, i

FIG. 1. The vector relation between the total angular momentum vector J and the projection M in the direction of the electric field ε.

24

C. C. Costain

M = J, J — 1, . . . , — J. T h e vector J precesses about ε at a rate determined by the interaction between the field and the molecule (Fig. 1). T h e wave functions for the low J values have simple forms: / = 0 ,

Μ = 0,

J = 1, M = 0, y = 1, I M I = 1,

= \\l\Tk

;

Ψ

y> = | ( 3 / π ) =

v

1 /2

cos 0 ;

1/2

i(3/jr) sin(9cos^;

ψ = i(^M) sin θ sin 0; 1/2

Figure 2 shows pictorial representations of the J = 0 and J = I wave functions.

J=0,M=0 J = l,M = 0

Z

2

•Y

FIG. 2. The wave functions of the rigid diatomic rotor in three dimensions. These have the same dependence on θ and φ as the s, px, p v , and pz orbitals of the Η atom.

T h e magnitude of the wave functions in a particular direction 0, φ is given by the length of the vector from the origin to the surface. T h e probability distribution function is obtained by squaring this quantity. T h e χ and y axes in this representation are not fixed in space but precess about the ζ axis as / precesses about M . T h e precession may be made as slow as desired by reducing the interacting field.

25

2. Rotation of Molecules

2. Selection

Rules and Intensities of

Transitions

The intensity of a transition between two rotational levels i and j is proportional to the square of the dipole-moment matrix element μ^. T h e matrix element, like any observable in quantum mechanics, is evaluated by the expression

μα = !ν>ί*μψι >

i')

ατ

2 7

where μ is the electric dipole moment, which must be along the internuclear axis of a diatomic molecule. T h e integration is over all threedimensional space. T h e matrix element is written in several equivalent forms :

μα = <μ>α =

<*Μ.7>·

The dipole-moment matrix elements are zero except for the transitions AJ= ± 1 , ΔΜ= 0 =fc 1:

Σ 0\μΜ+

1> = μ\]

+ l)/(2 / + 1 ) ,

(2.8)

Σ < / + 1 I μ I J Y = μ\]

+ l)/(2 / + 3 ) .

(2.9)

2

In these equations the matrix element is summed over the M values of 2 the final state. When the | μϋ | of (2.8) and (2.9) are multiplied by the M degeneracy of the i state, respectively 2J + 1 and 2J + 3, it is seen that the probability of induced absorption is identical with the probability of induced emission. In pure-rotation spectra there is a net absorption only because of the larger Boltzmann factor of the lower state. Rotational energies are small compared to kT> and the difference in the Boltzmann factor is -E(J)/kT

e

_

— hvjkT.

-E(J+V/kT e

In this equation hv = ΔΕ = E(J + 1) — E(J). Transitions to particular M levels may be observed if the levels are sufficiently separated by the applied field ε. Usually, the applied field is in the same direction as the polarized microwave radiation field, and the selection rule AM = 0 applies. T h e intensity of the transitions is then proportional to ( Τ _L i\2

< / , M | H / + l . M > ^ ^

(

^ +

1

)

>

_ 2

^ 2 /

+

3

.

)

26

C. C. Costain

3. Transition

Frequencies

It is convenient to divide both sides of the expression for the energy - 1 levels, Eq. (2.5), by he to convert the energy to units of c m , or by h to express the energy in cycles/sec (Hz). I f the latter unit is used, = E\h = - g ^ - / ( / + 1) = BJU + 1 ) ,

F(J)

(2.10)

2

where Β = Α/8π / is the rotational constant. If we now apply the selection rule / + 1 « - / , we obtain for the frequency ν of the absorbed photon F(J + 1) - F(J) = 2B(J +l) 4. The Heisenberg

Uncertainty

= v.

(2.11)

Principle

The Heisenberg uncertainty principle states that the uncertainty in the simultaneous measurement of the coordinate χ and the conjugate angular momentum p x is at least Α/2π, that is, AxApx>hßn.

(2.12)

Several features of the quantum-mechanical results given above are a consequence, or an illustration, of the Heisenberg uncertainty principle: 1 / 2

a. T h e angular momentum is given by | J | = (hßn)[J(J + 1)] . The simple standing wave picture of classical mechanics gives the result (Α/2π) / for the magnitude of J . b. T h e vector J cannot be aligned with the external electric field because this would imply an exact knowledge of both the magnitude and direction. In Figs. 1 and 2 we can represent the quantum-mechanical results by a classical diagram only if the precession is included or implied in the diagram. c. Classically, the diatomic molecule would be entirely in a plane perpendicular to the axis of rotation. In Fig. 2 the wave functions are not confined to a plane. T h e differences in the two pictures are greatest for J = 0 and J = 1, because the total angular momentum is zero or not much greater than the uncertainty. d. T h e absorption or emission frequency is given by ν = 2B(J + 1 ) . A rotating dipole in classical mechanics would radiate at its rotation

2. Rotation of Molecules

27

frequency / . We can define / only in terms of the angular momentum 1

( Α / 2 π ) [ / ( / + I)] '»

Ιω = Ι2π/= or

/ = (hl4n*I)[JU

= 2B[J(J

+

+ 1)] . 1/2

Once again, the difference between the classical and the quantum-mechanical result is a consequence of the difference in the angular momentum of part (a) above. In all these examples, in agreement with the correspondence principle, differences between the classical and quantum-mechanical results become negligible for very large values of the quantum number / . T h e essential uncertainty of h\2n is then negligible compared to the total angular momentum.

B . T H E NONRIGID ROTOR

1. Centrifugal

Distortion

In the solution given in the preceding subsection it was assumed that the interatomic distance r was fixed, but in any real molecule the centrifugal forces on the atoms will stretch the bond between them. T h e increase in r (and the moment of inertia) will decrease the effective value 2 of Β = η/8π Ι. T h e centrifugal distortion constant Dj is introduced to include the effects of stretching, while Β retains the value in Eq. (2.10). T h e rotational energy is then given by

F(J)=[B-DjJ(J+l)]JU+l)

= BJU+l)-DJ*(J+\)\

(2.13)

T h e magnitude of the centrifugal distortion is dependent on the force constant, or restoring force, between the atoms, or, more precisely, on the potential function V. For most molecules it is sufficient to assume 2 a harmonic potential function V = ik(re — r ) , because higher terms make negligible contributions for the very small displacements involved. When this form of V is included in the Schrödinger wave equation, Dj is found to be 3

2

Dj = 4£ /o> .

(2.14)

C. C. Costain

28

The ω in Eq. (2.14) is the harmonic oscillator frequency. T h e same value of Dj can be obtained from a classical treatment of the centrifugal distortion. 2. Vibration-Rotation

Interaction

For the nonrigid molecule suitable steps must be taken to separate vibration and rotation. It is obvious that the changes in r in the vibrating molecule will change the moment of inertia, and therefore the Β values. I f the vibrational frequency is very much higher than the rotational frequency, so that many vibrational cycles occur during one period of 2 rotation, then an effective Β value is obtained in which 1/r is averaged over the vibration: £ β„ = ( Α / 8 π ν * ) < 1 / Γ « > .

(2.15)

The vibrational energy levels are given by (Chapter 3 ) 2

G(v) = œe(v + i ) + ω Λ ( * + è ) + - - - . The effective Β value for the given vibrational state is also expressed by an expansion in ν + è: Bv = Be — a{v + i ) - γ(ν + \f

+ . ·. ,

(2.16)

where Be is the value of the rotational constant which is determined by the value of r at the lowest or equilibrium point of the potential energy function. These equilibrium values of the constants are thus precisely defined. However, they are hypothetical values in that they can be determined only by extrapolation from the higher vibrational levels, because the zero-point vibration remains even for the lowest vibrational level. All that can be determined experimentally are the Β values B0, B1, B2, etc., in the corresponding vibrational states ν = 0, ν = \ y ν = 2y etc. The ground state, ν = 0, is of primary interest in pure rotation studies, B0 = B

e

- i a

.

(2.17)

The value of a can be calculated from the potential function V, or, conversely, an experimental value of a can be helpful in the determination of V. The harmonic and anharmonic contributions to a are usually considered separately. T h e displacements for harmonic oscillations are sym-

2. Rotation of Molecules

29

metrical about r e , and we have 2

= r e ,

> r e ,

and

2

2

> l / r e .

These inequalities can be verified readily by evaluating the functions for r = r e ( l ±<5). T h e harmonic part of a is therefore negative. The anharmonic contribution is of the opposite sign. As the bond length is increased, the force constant becomes smaller and the displacements for r > r e are larger than for r < re. T h e average bond length is increased, and therefore the effective Β value decreases. T h e anharmonic terms are usually large enough to overcome the harmonic contribution and the resultant a is positive. The rotational energy of the vibrating molecule is finally F(J)

=

B JU y

+

1) - DjJ\J

+ 1)«,

(2.18)

and, for Δ] = 1, the frequency of the rotational transition is v = 2Bv(7+l)-4Z)y(y+l)».

(2.19)

A more exact treatment is necessary for light molecules with high rotational frequencies, such as the hydrides. In Dunham's treatment the potential V is expressed as a power series in r — r e , and the solution of the Schrödinger equation introduces higher-order corrections into the expressions for Bv and Dj. A list of many of the terms is given by Townes and Schawlow (1955).

III. L i n e a r P o l y a t o m i c M o l e c u l e s A. ROTATIONAL ENERGY

The expression for the rotational energy and the transition frequencies of a linear molecule in the ground state are identical with those of the diatomic molecule. However, the rotational spectrum in excited vibrational states becomes much more complex even for triatomic linear molecules. For such a molecule there are four vibrational modes, two of which are degenerate. I f these are labeled vx, v2 ( 2 ) , and i>3, the effective Β value is written By = Be — ax{vx = B

e

+ i)

— a 2(*>2 + 1) —

- Σ «.(«.+ Κ ) ,

«3(^3 +

i) (3-1)

30

C. C. Costain

where d{ is the degeneracy of the vibration. T h e a's can be determined experimentally from the rotational spectrum in the ground state and the excited vibrational states, from the analysis of infrared vibration-rotation bands, or from the vibration-rotation structure of electronic transitions. Theoretical expressions for the a's have been obtained, but seldom are the potential functions well enough known for precise calculations. Usually, the experimental values of the a's are used together with the information on the vibrational frequencies to determine the potential functions. B . VIBRATIONAL ANGULAR MOMENTUM

The degenerate vibration v2 of a triatomic linear molecule is a bending vibration in which the displacements of the atoms are perpendicular to the linear axis of the molecule. T h e displacements for the two identical frequencies are at right angles (orthogonal) for v2 = 1, and the relative phases are such that atoms move in circles about the axis at the vibrational frequency. Therefore the vibration is, in effect, a rotation of the molecule about the linear axis. The angular momentum is quantized, and equal to Ihßn. T h e quantum number / can take the values / = ν, ν — 2, . . . , — v. The total angular momentum J is now the vector sum of the rotational and vibrational angular momenta, and / need no longer be perpendicular to the linear axis. T h e rotational energy is dependent on the perpendicular component, and becomes F(J)

=

B [ju y

+1) -

η-

DJUU

+ i) -

/']*·

The selection rules AJ = ± 1 , ZlZ = 0 leave the transition frequency unchanged by the vibrational angular momentum. There is an additional factor introduced by the Coriolis interaction between vibration and rotation. This interaction splits the / = 1 level by Av=

kqi{v, +

1)·

(3-2)

The constant ql is a function of the harmonic vibrational frequencies and the atomic masses and is normally of the same order of magnitude as the a's. C.

FERMI RESONANCE

T h e anharmonic terms in the potential function lead to a degree of mixing or perturbation between vibrational levels of the same symmetry

31

2. Rotation of Molecules

(see Chapter 3 ) . This Fermi resonance may be a purely vibrational perturbation, but since the a's or the Bv values are dependent on the vibrations, a mixing of the vibrational levels will alter the values of By. T h e sum of the two Β values is unchanged by the perturbation, but if one of the levels is not a fundamental vibration, the unperturbed values of By must be obtained before Eq. (3.1) is applied.

IV. S y m m e t r i c Tops A nonlinear molecule has three principal moments of inertia about the three orthogonal axes a, b> and c. By convention, the axes are chosen with the moments of inertia IA < IB < Ic. A symmetric top has two of the three moments equal. A.

ROTATIONAL ENERGIES OF THE PROLATE T O P ; IA


B

=

Ic

The molecule C H 3 F is a prolate top in which the C—F bond defines the 1 /2 symmetry axis. T h e total angular momentum is J , | J | = [ / ( / + 1 ) ] Χ Α/2π, but now there is a component Κ of magnitude Khßn along the symmetry axis (Fig. 3 ) . Κ can take the integral values K = J , J - \ , . . . ,

-J.

The rotational energy is F(J,

K) = B[J(J

+1)-K*]

(4.1)

+ AK\

The first term represents the energy of rotation about the b axis, perpendicular to the symmetry axis (B = hßnHB), and the second term the 2 energy of rotation about the a axis {A = Α/8π /^). Equation (4.1) is usually written F(JK)

= BJU

+ 1) + (A -

B)K\

The selection rules for pure rotation are Δ J = 0, ± 1 ; ΔΚ = 0. For 4 / = + l , ΔΚ = 0 the transition frequency is ν = 2B(J + 1). T h e transitions for the various Κ values are coincident except for the effects of centrifugal distortion. Three terms must be added to the energy to include centrifugal distortion: F(JK)

= BJU -

+1)

DJKJU

+ (A-

2

B)K

- Djß(J

+ \)K* - DKK\

+ 1)» (4.2)

32

C. C. Costain

::VT_ _

F I G . 3.

. _ _. - - - M J + I )

T h e vector relations connecting J , K , a n d M

Κ Φ 0 there is a c o m p o n e n t μΚΜ/JiJ

for a s y m m e t r i c top. F o r

+ 1 ) of the dipole m o m e n t in the direction of

the field. T h e Κ and M m a y have the values J, J — 1, . . .,

—J.

T h e frequency for / + 1 <— Jy Δ Κ = 0 is r = 2B(J

+ 1) -

4Z>,(7 + l ) - 2 Z ) . ( / + l)K\ 3

i / A

(4.3)

T h e dipole-moment matrix elements for the symmetric top are

Ο Κ \ μ \ ^

B.

S Y M M E T R Y AND S P I N

= μ*[

],

FUNCTIONS

T h e wave function of the symmetric top may be written

Ψ = OJKAW*****

(·) 4 4

where χ is the angle of rotation about the top axis. Since Κ can be positive or negative, all levels for K^£0 are doubly degenerate. If the molecule

2. Rotation of Molecules

33

has a threefold symmetry about the top axis, a rotation of 120°, or 2π/3, exchanges the identical particles. For Κ a multiple of 3, this rotation X —• χ + (2π/3) leaves the wave-function unchanged, i.e., it is of species A. There are therefore two A levels which might be split by vibrationrotation interaction. For Κ not a multiple of 3, the wave function is changed by a 2π/3 rotation, and is of the degenerate species E. T h e rotation, in exchanging the three equivalent protons in C H 3 F , also exchanges their spins. T h e three spins of ± \ can be combined in eight ways to give four totally symmetric spin functions (4^4) and two doubly degenerate spin functions (22?). T h e product of the symmetry species of the rotation and spin functions must be of species A. T h e multiplication rules are A χ A = Α, Ε X A = £, Ε χ Ε = 2A + Ε. T h e multiplication of the species A(4A + 2E)

2,000

,8

l,500k

L5 1,000

500

FIG. 4. The rotational energy-level diagram of a C 3 V prolate symmetric top in the ground vibrational state or a totally symmetric vibrational state. The scale is approximately correct for the C H 3 F molecule.

C. C. Costain

34

= jA + 2E and species E(\A + 2E) = \E + 2(2A + E) shows that both A and Ε rotational levels have four allowed functions of species A> and have the same statistical weight. If, as frequently happens, the two levels for Κ = 3, 6, 9, . . . are not resolved, these coincident levels will have twice the weight of the remaining Κ levels. The above features of the prolate symmetric top are summarized in a Herzberg energy level diagram (Fig. 4 ) .

C . T H E OBLATE SYMMETRIC T O P ; IA

= IB

<

Ic

T h e rotational energy of the oblate top, with A = Β > C, is F(J,K)

= BJU+1)

+ (C-

(4.5)

B)K\ 2

The energy-level diagram differs because the K term is negative for the oblate top. T h e pure rotation transition frequency with ΔJ = ± 1 , ΔΚ = 0 is given by Eq. (4.3), and the previous discussions all apply equally to the oblate top.

D . VIBRATION-ROTATION INTERACTION: / - T Y P E DOUBLING

The degenerate vibrations of a symmetric top, of species Ε for a C 3 v molecule, will each contribute vibrational angular momentum with quantum number l{ = ν{, ν ι — 2, . . . , — vit However, the displacement vectors for the vibrations will not in general be at right angles to the symmetry axis, and only the projection of the corresponding angular momentum on the symmetry axis contributes to the resultant angular momentum. This projection is given by ζϊΙϊ, where — 1 <; ζϊ < 1. T h e total angular momentum about the axis must still be Kh/2n, and therefore the rotational contribution is (Κ — ζ^^Ηβπ. T h e rotational energy becomes F(J,

Ky I) = B[J(J = BJ(J+l)

2

+ 1) - K ] + C(K 2

+ (C - B)K

ζΐ)

2

- 2CKCI + Οζψ.

(4.6)

T h e last term is unchanged by rotational quantum numbers and can be ignored. Since Κ and / have been defined as signed ( + or — ) quantum numbers, the product Kl can be positive or negative. Figure 5 illustrates a rotational energy-level diagram for a degenerate vibration of an oblate top.

2. Rotation of Molecules K= 0 I

K= 0

I

+1 -I

I

2

+t -I

2

35

3

4

-/

3

+Z

-I

4

FIG. 5. The rotational energy-level diagram of a C 3 v oblate symmetric top molecule in a doubly degenerate vibrational state E. The scale is approximately correct for the CHF 3 molecule, with Β = 10,349 MHz, C = 5670 MHz, and ζ = - 0 . 8 1 .

T h e species of the levels in Fig. 5 are derived from those of Fig. 4 by the multiplication rules Ε X A = Ε and Ε χ Ε = Al + A2 + Ε. T h e A levels in the ground state (or A vibrational states) occur for Κ a multiple of 3. For the degenerate vibration of Fig. 5, A levels exist only for | Κ — /1 a multiple of 3. Thus, for Κ = 2, the A species exist for I 2 - ( - 1 ) I = I - 2 - ( + 1 ) | = 3, or for the - / levels. For Κ = 4 the A species are the + / levels. It must be noted that + / or — / does not designate the sign of /, but the sign of the product Kl. Ä^-type doubling, or the splitting of the Ax, A2 levels, is usually detected only for the Κ = 1 levels. T h e splitting is given by an equation analogous to that for the linear molecule [Eq. (3.2)]. T h e / + 1 <- J, Κ = 1 transitions form a symmetrical triplet with the three lines of equal intensity. T h e transitions between the members of the Αλ, A2 doublets for Κ = 1 are also allowed (AJ = 0, ΔΚ = 0 ) , and for ν = 1 the frequency is ν = qJ(J + 1).

C. C. Costain

36

V. T h e A s y m m e t r i c Top; IA

A . T H E SYMMETRY OF ROTATIONAL ENERGY LEVELS

The energy levels of the asymmetric top are defined in terms of the prolate and oblate symmetric-top limits. On the left-hand side of Fig. 6 the energy levels of the prolate top A > Β = C are given. Here Β increases linearly from the left until at the right-hand side the oblate limit is reached, with A = Β > C. T h e energy levels are labeled by the J value, and by two subscripts, i^(prolate) and ^(oblate). In the asymmetric top the Κ is no longer a good quantum number, i.e., it is not a constant of the motion. T h e two values i£(prolate) and ^(oblate) serve primarily as labels for the level. Several different conventions are used for the designation of K. ^(prolate) can be written Ka for identification with the a axis, which is the Κ quantization axis in the prolate top limit. It is also written K_x, where the —1 is the value of Ray's asymmetry parameter [see Eq. (5.4)] Prolate

Asymmetric

Oblate

J Κ

2

2

2

I

2

0

I

I

I

0

0

0

A>B = C

Β

0

0

A = B>C

FIG. 6. The designation of the asymmetric-top energy levels. The K__x subscript of J is the value of Κ at the prolate symmetric-top limit, and the Kx subscript is the value of Κ at the oblate symmetric-top limit.

37

2. Rotation of Molecules

for the prolate top. ^(oblate ) is designated as Kc for identification with the c axis of the limiting oblate top, or as K1, where + 1 is the limiting value of Ray's asymmetry parameter for the oblate top. A third method a n of labeling the energy levels is / T , where r = K_x — Kx. T h e ]κακ0 d JK κχ conventions are entirely equivalent, and while the JKAKC is more a s explicit in its designation of the a and c axes, the JK_1,K1 ^ been used in recent books and tables and will be used in this chapter. T h e energy levels for the low / values can be expressed as simple functions of the rotational constants. Figure 7 is a schematic diagram of the energy levels for / = 0, 1, and 2. It includes the JK_1,K1 label, the symmetry of the levels, which is simply the even (e) or odd (o) character of the K_x and Kx values, and the expressions for the energy. T h e wave functions for the asymmetric top are formed from linear combinations of the appropriate symmetric-top functions Ä

K M

-i> \

κ

T h e behavior of the rotational wave function for rotation about a principal axis can be determined from the portion of the symmetric-top K function, ë *. For a rotation of π about the a axis the wave function K_, K,

t.

Energy levels 2

ee

2A + 2B+2C + 2[(B-C) +(A-C)(A-B)]'

2„

e ο

4A + B + C

2„

0 0

A+4B+C

2,2

οe

A+B+4C

2()2

e e

2A+2B+2C-2[(B-Cf+(A-C)(A-B)]'

ο e

A+B

ο ο

A+C

eο

B+C

^20

, 1 1

k,k,

/2

FIG. 7. A schematic diagram of the J = 0, 1, and 2 rotational energy levels of an a n asymmetric top. The levels are not equally spaced as shown. The labeling JK_1A\ d JT are both given. The symmetry of the levels is given by the even, ey or odd, o, character of K_v and K,.

C. C. Costain

38

changes sign if Ky i.e., K_ly is odd, and is unchanged if is even. Similarly, the symmetry with respect to the c axis depends on whether Kx is even or odd. Because the rotation about the b axis is equivalent to successive rotations about the a and c axes, the symmetry is completely described by the ey ο character in Fig. 7, and can be written down for any level from inspection of the K_ly Kx subscripts. There are four possible combinations, eey eoy oey and ooy and therefore the wave functions of an asymmetric top are separable into these four distinct symmetry species. In Fig. 7 it can be seen that if a given species occurs once for a given / , the energy levels are a simple function of the rotational constants. I f a species occurs twice, as for the ee species of 2 2 0 and 2 0 2 , the energies are determined from the solution of a quadratic. For J = 4 the nine energy levels are determined from three quadratic and one cubic equation. T h e derivation of the equations will be discussed briefly in the next section. The general asymmetric top can have a dipole-moment component along each of the principal axes. T h e allowed transitions for each of μα, μh, and μ0 for / = 0 and / = 1 levels are included in Fig. 7. T h e allowed transitions are derived from the symmetric-top selection rules Δ] = 0, ± 1 , Δ Κ = 0 where the * Quantization' ' axis for Κ is successively the ay by and c axes. B.

DETERMINATION OF THE ENERGY LEVELS

The rotational energy in the classical form can be written Ρ 1

χ 2L

ττ 2

2

Ρ

2

\1

Ρ

2

2Iyy

\1

2 τ

2IZ

2

where \Ιω = Pœ /2I = Ρ /2Ι. A preferred axis, zy is chosen as the quantization axis for Κ of symmetric-top basis set (tpJKM)' T h e terms of the Hamiltonian may then be rewritten as

={-k +
H

p2 p

2

2

2

+

[-k-4^) * - ' (Ρ

2 ρ β)

·

2

2( )5

where Ρ = Px + Ρ + Ρ . T h e diagonal matrix elements of Hy which become the eigenvalues (or energy levels) in the symmetric-top limit, Ix = Iy9 have already been given in a slightly different form for the symmetric-top,
jKy

= (JL + ±-)uu

+ l) - κψ*

+

-g-*:

39

2. Rotation of Molecules

T h e only other nonzero elements, which are off-diagonal in K> are Κ ± 2> = ( - L -

iJK\H\J,

+ 1) - K(K ± 1)]

^ - {[7(7

χ

UU+i)-(K±\)(K±2)]y>*.

With the option of a left- or right-handed axis system, and with three choices for identification of the ζ axis, there are six possible ways to correlate the x> yy ζ axes with the a, b> c axes in a molecule. T h e axis chosen to be the ζ axis should be that most nearly approximating the symmetric-top axis, in order to make the off-diagonal matrix element as small as possible. For a near-prolate symmetric top a proper choice would be ζ —• a> χ -> by and y -> c. T h e matrix elements then become

(JKI HI JK> = \{B + C)[7(7 + i) = i ( B + C)7(7 ijK\H\j,K±iy

=

+ AK*

+ 1) + [A -

\{B +

-ï{B-C)[7(7

C))K>,

(5.3)

+i)-K(K±

χ [7(/+i)-(^±i)(^±2)]^. T h e matrix elements are arranged with the elements Κ = — J to J on the diagonal. For 7 = 1

M =

Κ

—Ι

0

—1

A + i(B + C)

0

0

S + C

0 Ι

- HB

0

- C)

T h e Wang transformation, X'MX,

A+t(B

where

1

-1 -1 X = X'

V2

1 1

vT

1 1 1

i(B 0

C)

+ C)

C. C. Costain

40

gives for J = 1 A + Β B+C A+C In general, the Wang transformation gives a matrix which can be factored into four submatrices, = E+ + E- + 0+ + 0 - .

X'MX

These four submatrices correspond to the four symmetry species discussed in reference to the K_1K1 nomenclature of the asymmetric-top levels. The Ε and Ο refer, respectively, to the even or odd value of the Κ on the ζ quantization axis. T h e calculation of the matrix elements is simplified by the relations Ε

Ε

Ε

Ε

κκ = -κ-κ> κ,κ+2 = κ+2,κ = E-K-K-2 = E_K_2_K. submatrices may therefore be written down directly: E

λ/2

oo

V2E02 +

E

E

E

E

2\

2\ 0

•

0

#02

E

22

=

\ \

E~ is obtained by omitting the first row and column of E

U

± E

E

-ll

3(B + C)

and

0

^13

1S

#33

£35

0

#35

£55

Again, for the trivial example of J = 1, E Ο- = A + B, and / = 2

- V T ( ß - C )

T h e four

+

- v T ( f i - C ) + ß +

C

0

Β + C,0+ = A + C,

2 0 2 and 2 2 0 levels,

£~ = \A + Β + C == 2 „

level,

0 + = A + Β + 4C = 2 12

level,

O- = A + 4 £ + C = 2

level.

n

2. Rotation of Molecules

41

T h e calculation of the energy levels is a matter of diagonalization of the four tridiagonal submatrices of the approximate order J/2. This once formidable task is now a routine procedure where high-speed computers are available. It is still convenient to use tables of reduced energies for many purposes. T o construct the tables, it is necessary to make a change of variables to obtain a reduced energy matrix dependent only on a dimensionless asymmetry parameter. T h e most commonly used expansion is ^(Λ-.,.α-,) =

Ï(A

+ C)JU

+ 1) +

\{A

-

(5.4)

C)E{x),

where κ is Ray's asymmetry parameter, which varies from —1 for the prolate top to + 1 for the oblate top, κ = (2B — A — C)/(A — C ) . T h e matrix elements for the reduced energy Ε(κ) are

(JKI (JK

E(x) I J K } = F [ J ( J +l)-K*]

I E(x) \J,K±2>

= H[J(J

+ GK\ \)nJU

+\)-K(K±

-(K±\)(K±

+ 1)

2)Y'\

Again, six representations are possible. With ζ —> α, χ —• b, y —• c as used above, F = (κ — l ) / 2 , G = 1, H = -(κ + l ) / 2 . T h e reduced energies may be computed in the range of κ appropriate for the repre—Ε_τ(—κ) sentation, and are well tabulated. T h e symmetry Ετ(κ) = makes it unnecessary to tabulate for more than the positive values of κ. T h e Wang expansion is an alternative expression which is very useful for slightly asymmetric tops. T h e energy is written in the symmetric top form. For the prolate case : F(J)

= i(B + C)J(J

+ 1) + [A - i(B + C)]w,

= \(A + B)J{J

+ 1) + [C -

for the oblate case F(J)

\{A + B)]w.

T h e reduced energy w can be computed from the matrix elements F = 0, G = 1, H = bf in the representation of Eq. (5.3) for specific values of by where 6 p r o l a et = (C - B)I(2A - Β - C) and 6 0 b l a te = (A - B)j{2C — Β — A). T h e reduced energy w can also be expressed as a power series, 3

4

w = K* + Cxb + C2b* + C 3 6 + C4Z> + - · -.

(5.5)

42

C. C. Costain

T h e coefficients C1, C 2 , C 3 , . . . are available from tables. For sufficiently small values of b the series is convergent, and w can be determined from Eq. (5.5) with no interpolation. C.

INTENSITIES

T h e intensities of asymmetric top transitions can be obtained from tables of line strengths. T h e large number of possible transitions limits the number of intervals for κ that it is practical to tabulate, and the errors in the interpolation are rather larger than for the energy levels. High accuracy is not very important for the intensities themselves, but accurate dipole-moment matrix elements are extremely important in the determination of the dipole moments from the Stark effect. This problem is also readily solved with high-speed computers. Once the energy matrix is diagonalized to produce the eigenvalues a matrix can be determined which transforms the tridiagonal energy matrix to diagonal form. T h e columns of this transformation matrix are the eigenvectors, and each vector expresses the asymmetric-rotor wave function in terms of the symmetric-rotor basis set. In other words, the relevant expansion of Eq. (5.1) is known. T h e dipole-moment matrix element for the asymmetric top can therefore be determined from the linear combination of the symmetric-top elements.

VI. Internal Rotation Molecules in which internal rotation occurs have been extensively studied in recent years, and much interest has centered on the internal rotation of the methyl group, — C H 3 . When the internal rotor is a symmetric top of threefold symmetry a potential function of the form V = iF8(l -

cos 3a) + * F e ( l -

cos 6a) + · · ·

(6.1)

is assumed. T h e V3 term is dominant unless it vanishes for reasons of symmetry. In the classical limit of a very high barrier the methyl top will execute torsional vibrations about its symmetry axis in the Vs potential well, and since the contributions to the moments of inertia are independent of the angle of orientation of the methyl group, no influence on the rotational spectrum would be observed. However, with a lower barrier

2. Rotation of Molecules

43

the possibility of tunneling through the barrier exists, and the wave functions of the three equivalent wells are mixed by the tunneling. T h e mixing separates the triple degeneracy of each torsional level into a nondegenerate level of species A and a degenerate level of species E. T h e tunneling constitutes a rotation of the internal top with its associated angular momentum p . T h e effective rotational Hamiltonians for the A and Ε species are slightly different, and two rotational lines, A and Z?, appear as a doublet. T h e separation of the two lines, as with all tunneling phenomena, is strongly dependent on the height of the barrier. There have been many treatments of the internal rotation problem for particular ranges of barrier heights. For the example of a symmetric rotor rotating against an asymmetric molecular frame perhaps the most elegant formalism has been given by Herschbach [see Wollrab (1967), p. 168]. The Hamiltonian is H=Ht

+

(6.2)

F(p-P)*+V{a),

where HT is the rigid rotor Hamiltonian of Eq. (5.2); p — Ρ is the relative 2 2 angular momentum between the top and the frame; and F = Α /8π Γ/ Α is the reduced rotational constant of the internal top, with Ix the moment of inertia of the top, r = 1 — 2* ^iQi (* = b, c)y Κ the direction cosine between the top axis and the ith principal axis, and Qt = kJJIi; Ρ = 2» X PiQt is a reduced angular momentum. T h e expansion of the p — Ρ term gives H=HT

2

+ FP -

2

2FpP + Fp + \V*{\ - 3 cos a) + . . ·. (6.3)

2

P is quadratic in PA etc., and modifies the rotational constants, A' — A 2 2 + FQA ' T h e Hamiltonian for the internal rotation is the portion Fp + \V9{\ — 3 cos a ) + . . . , and the eigenfunctions are obtainable from the Mathieu equation in terms of a reduced barrier, s = (4/9) V3jF. The rotation and the internal rotation are not separable, because of the cross terms 2FpP. However, if the spacings between the torsional energy levels are large compared to those of the rotational levels, it is possible to transform the Hamiltonian so that it can be factored into a rotational submatrix for each torsional level v, HyA

(

= Hr + Σ W%P *\ η

The matrix elements of Eq. , (K\H\K±2>,

HvE

M

= Hr + Σ W%P -

(6-4)

η

(6.4) now include (Κ | Η | K}, , etc., and have

44

C. C. Costain

been tabulated by Herschbach. Diagonalization by computer is now practical, but for many internal rotation problems the tables for pern turbation coefficients Wy 2 and are adequate for the solution. The rotational levels of the A species follow a pseudorigid-rotor form, and if the odd order terms in the expansion, Eq. (6.4), are negligible, the Ε levels are also pseudorigid rotor. This occurs for moderately high barriers, and in this limit the barrier can be determined from the differences in the rotational constants. For example, ΔΑ

= Α

Λ

- Α

Β

= FQA*[W?1

-

W$].

The observed value of Δ A and the geometric factors F and ρΑ fix the value of the difference of the perturbation coefficients. This in turn gives the value of s = (4/9) VJF from the tables, and therefore the barrier Vz. T h e splitting of the A and Ε components of the individual lines can also be expressed in terms of the coefficients to give the barrier. As the effective barrier to internal rotation is lowered, it is necessary to use higher and higher orders of perturbation to obtain satisfactory solutions. For the very low barrier the problem again simplifies when the internal rotation is considered from the perturbed free-rotor limit. This limit normally applies when twofold (or higher) symmetry of the framework of the molecule eliminates the Vz term of the potential function.

VII. Spectroscopic D e t e r m i n a t i o n o f M o l e c u l a r Structure A. T H E RIGID MOLECULE

1. The Calculation

of the Moments of Inertia

The calculation of the moment of inertia of a model of a rigid molecule is a simple procedure if the molecule is assumed to be made up of a number of mass points. T h e axis system chosen is one in which it is most convenient to determine the coordinates of the atoms. T h e coordinates of the z'th atom of mass m{ are x/, y{y and z/9 where the prime denotes the arbitrary axis system. T h e moments of inertia,

IL = Σ »«(/,* + *?), I'yy = Σ «,(*? + ζ\% Ι' = Σ «K/i* + *:)> 2

α

*

i

i

and the products of inertia, I%y

Σ ^'v^iy% ?

lyz > etC.,

2. Rotation of Molecules

45

are calculated for the arbitrary coordinate system. T h e free molecule rotates about its center of mass, and in the center-of-mass system the m x m m z first moments are zero, i.e., Σ i i — Σ %y% — Σ % % — 0. I f we make the conversion to the center-of-mass system X{9 yi, ziy we find 4 * = Σ miy'?

+ *?) -

(1/M)( Σ m&ï)*

i; = Σ ηι,χ{γ{ - (1/M)( Σ y

(1/M)( Σ ™ ^ ) \

-

Σ («tf/),

i m

where M = Yii i- T h e moments of inertia in the principal axis system, in which the products of inertia are zero, are obtained by solving the determinantal equation Ixx — λ

—Lxy

- 4 =

(7.1)

o.

For the planar molecule, with all z{ = 0, IA and IB are determined from Ixx

0,

or IA,IB=

WXX + lyy) ± K 4 r - hvY +

^y]

1/2

(7.2)

and IC = IA + I B . It is worth noting that in Eq. (7.2) both terms under the square root are positive, and the sum can be zero only if the two terms go simultaneously to zero. T h e nonlinear triatomic molecule X Y X with < Χ Υ Χ = Θ (such as H 2 0 ) will become a symmetric top with IA = IB only if cos θ = mxj{mx + my). For the molecule X Y Z , < X Y Z = Θ, however, two conditions must be satisfied if the molecule is to be a symmetric top: mx r(XY) _ mz r(YZ) (7.3) cos θ = mx + my r{YZ) m y + mz ^ ( X Y ) It is obvious that Eq. (7.3) cannot be satisfied for the H D O molecule. T h e bond lengths r ( O H ) = r ( O D ) , but mu\{m0 + mH) φ mB\(mQ + mD) In general, Eq. (7.3) can never be satisfied for the X Y Z molecule. A plot of IA and IB against θ has the appearance of the "avoided crossing" of the two-level perturbation system, as might be expected from Eq. (7.2). A cubic equation must be solved for the general nonplanar asymmetric top, and a computer solution is worthwhile if many models are to be tested.

46

C. C. Costain

2. Equilibrium

Moments of Inertia

In a polyatomic molecule the equilibrium coordinates of the atoms are those which correspond to the lowest point of the potential energy function. In the concept of the equilibrium structure it is assumed that the potential function is dependent only on the electronic wave function, and that it is therefore invariant for isotopic substitution. T h e equilibrium structures of different isotopic species are identical. T h e principal axes are changed by isotopic substitution, but the changes in the coordinates are easily calculated. In the next section it is assumed that the equilibrium moments of inertia have been obtained from the spectrum. 3. Kraitchman*s

Equations

a. The Linear Molecule. The One-Dimensional Problem. Obviously, if a given Ie is dependent on only one structural parameter, this parameter can be obtained directly. I f more than one unknown structural parameter is included in the expression for Ie, then Ie must be known for other isotopic species, and the unknown parameters are determined by solving the simultaneous equations. This procedure is quite straightforward, but it is greatly simplified by using Kraitchman's equations. In this method the 7 e are expressed in terms of the coordinates of the atoms in the principal axis system. For a linear molecule e

IB

2

2

2

= m,r, + m2r2

where r{ is the distance of the this definition

+ m3r

+ · · ·,

(7.4)

atom from the center of mass. From

m1r1 + m2r2 + m3r3 + . - - = 0 . If an isotope of mass m, become

(7.5)

Am, is substituted for m,, the two equations

1% = 0 » i + Am,)(r, + Ar,)

2

2

+ ...

(7.6)

(m, + Am,)(r, + Ar,) + m2(r2 + Ar,) + . . . = 0.

(7.7)

+ m2(r2 + Ar,)

and

Subtracting

(7.5)

from

(7.7),

we obtain the value of Ar,

where M = m1 4 - m2 4 - m3 • • • is the total mass of the original molecule.

2. Rotation of Molecules

47

Here Ar, is the shift in the center of mass caused by the isotopic substitution. When this expression for Ar, is substituted in (7.6), and (7.4) is subtracted from (7.6), the simple relation V

= [ ( M + Am^MAm,]^

-

//)

=

(7.8)

is obtained, where μ is the "reduced mass" for the substitution. Equation (7.8) is equivalent to Eq. (7.6) and (7.7), and clearly shows that the isotopic substitution gives the square of the distance of the substituted atom from the center of mass of the original molecule, and nothing else. T h e value of the coordinate r,, and the r{ obtained from single substitutions on other atoms, are then used in the equations for the original molecule to solve for the remaining coordinates. For a linear molecule of Ν atoms, substitutions on Ν — 2 atoms must be made for a complete determination of the structure, but any particular bond length, for example, r,2 = r, — r2, can be determined from two substitutions. T h e probable error in the value of r, is readily obtained from Eq. (7.8). e An error of δ, in the value of AIB results in an error in r, of or, = (\^)ô,l2r,.

(7.9)

T h e error is inversely proportional to r,, and consequently is large when the substitution is near the center of mass. b. The Planar Asymmetric Top. The Two-Dimensional Problem. With t ne the convention IA moments of inertia about the two principal axes a and b in the plane of the molecule are IA and IB, respece tively, and for the equilibrium state IA + IB = T h e position of an atom is defined by two coordinates a{ (measured along the a axis) and bi (measured along the b axis). I f an isotopic substitution is made on an e e atom which does not lie on a principal axis, both IA and IB will be changed. T h e two coordinates of the atom in the principal axis system of the original molecule are given by Kraitchman's equations for the two-dimensional problem :

? =j

a

4

AI

1

+

I^j?]>

> =7 * I Ä 1 · ·

b

1+

Δ1

e

e

(7

10)

e

e

As before, μ = M Am/(M + Am\ AIB = 1% — IB , and ΔΙΑ* = I i—IA . T h e terms in brackets in Eq. (7.10) arise from the rotation of the prin-

C. C. Costain

48

cipal axes with respect to the molecular frame in going from the original species to the isotopic species. T h e angle of rotation θ is given by £

—2a Jbi

9 t a n

imvA-h)-«<*

=

+ *>?'

There are five equations for the original molecule which can be used in the determination of the 2N coordinates of the Ν atoms : 9

IA =

Σ*ηΑ , 2

h* =

i

Σ i

Σ ™ Α

2

>

i

= Σ ™Α = Σ fn^ibi = 0 . i

i

Two coordinates are obtained from each substitution, and therefore (2N — 5)/2 substitutions must be made to obtain a complete structure. This means Ν — 2 substitutions are required, and one relation is redundant. Frequently, an atom lies too close to a principal axis for an accurate coordinate to be determined by substitution and the redundant equation is required. c. The General Asymmetric Top. The Three-Dimensional Problem. T o extend Kraitchman's equations to three dimensions, it is convenient to change from the principal moments of inertia to the planar moments of inertia. These are defined as PA = WB + IC

-h)

= Σ mi<*i , 2

i

PB = WA + Io -h)

= i

Pc = WA + IB -Io)

= Σ WiC?. i

An isotopic substitution on an atom will give

d. The Symmetric Top. A substitution on the symmetry axis of a symmetric top does not alter the symmetry properties or the off-axis

49

2. Rotation of Molecules

contributions to the moment of inertia. T h e coordinate of the substituted atom can therefore be determined from Eq. (7.8) for the linear molecule. When a single substitution is made on an atom off the symmetry axis the molecule is no longer a symmetric top. For example, in substituting D for H in CH 3C1, the D atom defines a plane of symmetry in the asymmetric CH 2 DC1 species, and Eq. (7.10) for the planar asymmetric top may be used to determine the coordinates of the substituted atom. In solving the simultaneous equations for the moments of inertia, the coordinates of the atoms were obtained in two different ways: (1) T h e coordinates of atoms on which isotopic substitutions have been made are obtained using the AI in Kraitchman's equations; and (2) the remaining coordinates are obtained by inserting the values of these coordinates in the equations for the original molecule. This difference is of no consequence for the exact equations of the rigid molecule, but it is of importance to the following discussion of the vibrating molecule. It should also be emphasized that the use of Kraitchman's equations is not optional, but constitutes an intermediate step in the solution of the equations regardless of the algebraic procedure used.

B . T H E SUBSTITUTION OR r s STRUCTURES DETERMINED FROM GROUNDSTATE CONSTANTS

T h e determination of the equilibrium rotational constants requires a great amount of experimental work for even a triatomic molecule. When the constants are needed for several isotopic species, it requires years to accumulate sufficient data. For most molecules it is practical to obtain only the ground-state constants. These are the constants in the lowest vibrational state, the ground state for which the zero-point vibrations always exist. In Eq. (2.15) the effective Β value of a molecule is defined. In general, the r 0 coordinates are defined only by summations of the form το l B

_

-

h

_

y

8𻣠0 -

i

2 m

r

1 1 0

1

_

"

(1/ Σ

'

i

where the average is over the zero-point vibrations. T o a first approximation, the r{ is the instantaneous position of the mi atom, but the a's which contribute to B0 also contain terms arising from the Coriolis interactions between the vibrations. When an isotopic substitution is made the

C. C. Costain

50

vibrational displacement vectors are altered by the changes in the reduced mass of the vibrations, with the largest changes—approximately 1/2 [niiKnii - f ZLM)] —occurring for the substituted atom. T h e averages over y the changed zeropoint vibrations result in r0 s that are slightly different for each isotopic species. Since a unique r0 does not exist, it obviously cannot be determined from the improper set of simultaneous equations. T h e two different ways of determining the coordinates does become important. T h e coordinates determined using the ground-state 1° in Kraitchman's equations cannot be r 0 , and have been called r s u b s t i t u t ni o or r s . For a linear molecule r s is defined as τ*=(\Ιμ)ΔΙΒ\

(7.11)

If rs is determined for each atom of a linear molecule by successive s m r c a n e isotopic substitutions, t h e n / B = 2 i % \% b determined. For several S 2 molecules it has been found experimentally that IB° — IB « 0.1 amu  e and IB° > IB* > IB . On the average, therefore, r 0 > r s > r e for a linear molecule. From this inequality it is easy to see the nature of the errors which arise from the normal solution of the simultaneous equations for IB°. I f the rs coordinates of Ν — 1 atoms have been obtained from the AI0i all less than the r 0 , then the one coordinate determined from the equation for IB° will be considerably greater than its r0 value because it must make up the accumulated difference IB° — IB*. T h e error will be àu=

( ν - / / ) / 2 ^ .

(7.12)

T h e variation in the solutions of different sets of simultaneous equations depend on the particular ôr{ of the unsubstituted atom of the set. T h e error will be large if r{ or ηι{ is small. It is particularly important, therefore, to determine the position of Η atoms by D substitution, and not from the equation for IB°. T h e error in the r s coordinate ôra = (Ι/μ) ô(AIB°)l2ra

(7.13)

is, in contrast, independent of m, but can be large for substitution near a principal plane. In several molecules substitutions have been made on atoms known from symmetry to be on a principal axis. T h e coordinate, and AI°9 should be zero for such substitutions, but the value of AI° 2 found is about—0.0024 amu  . From this measure of the effects of zero-

51

2. Rotation of Molecules

point vibrations the approximate error in the r s coordinate is expected to be Ά

1

0.0012 Α

Λ η

where rx is the true position (in  ) . In some molecules it is possible to r use the first moment equation Σ ™i %i = 0 to locate an atom near the center of mass. T h e first moment equation is never exactly satisfied by the r s coordinates, but experiments have shown that the error should not be more than 0.002  for a carbon atom. T h e large errors in the AI° structural determination are avoided if it is possible to determine a complete ra structure from the AI° by isotopic substitution on each atom in the molecule. T h e r s structure is, to a high degree, independent of the isotopic species used in the determination. This invariance is an important feature, but more important is the fact that an accurately determined ra structure provides a close approximation to the equilibrium structure. T h e deviations from the equilibrium structure appear from a few examples to be up to 0.002  for the heavier atoms and perhaps 0.005  for H atoms, but these limits are subject to some controversy.

C . AVERAGE STRUCTURES

T h e r 0 are defined by summations (1/ Σ " W * ) a v

9

and represent a rather involved average over the zero-point vibrations of the ground state. They cannot be determined, as was shown in the preceding section, if isotopic substitutions are required. T h e r 8 are de0, fined by Kraitchman's equations using the ground-state Z l / s , and are readily obtained by isotopic substitutions, but their physical interpretation is rather vague. A third parameter used to describe the ground state structures is r a v e r a g .e T h e r a v e r a eg (also called r 2 , , and rg) structure represents the average atomic distance in the ground state. T h e a's in E q . (3.1) can be written as ax = ^(harmonic) + a^anharmonic). T h e a^anharmonic) is dependent on the cubic and higher anharmonic terms in the potential function, and these are known for only a few molecules. T h e (^(harmonic) can be calculated from the quadratic

Λ

Α

C. C. Costain

52

force constants, which can be obtained with reasonable accuracy from the fundamental vibrational frequencies. When this partial (harmonic) correction is applied to the B0, the r a v e r a eg distances can be determined. Figure 8 is a qualitative graph of the various r. T h e r a v is further displaced from the ideal r e than the r0 or r s , but it has the advantage of being a self-consistent parameter (Σ w^av = 0, etc.) like re. harmonic

H average

Ν

anharmonic

1

FIG. 8. The approximate relations between r e , r s , r0 and ^average- The r can be taken as coordinates of the atoms or as bond lengths. The equilibrium value r e is invariant. The r s is determined by isotopic substitution. The r0 and r a v e r ae g are slightly different for each isotopic species.

The r a v is also close to the rg used in the electron-diffraction determination of structures. T h e rg is the time average of the bond length determined at a particular temperature, and is therefore observed for a mixture of ground and vibrational states determined by the Boltzmann distribution. T h e r a v is normally computed for a particular vibrational state, and usually only for the ground state.

VIII. T h e Stark Effect When an electric field is applied to a rotating molecule the components of the total angular momentum in the direction of the field will have the integral values M = J,J— 1, / — 2, . . . , — / , with | M | = MH = Mhßn. The rotational energy is the same for all M values for a sufficiently small field, but as the field increases, the M degeneracy is partially or fully removed. This perturbation of the rotational energy by the electric field is the Stark effect. In this section the only Stark effects considered will be those interactions resulting from a permanent dipole moment in the molecule. T h e higher-order effects of polarizability will not be discussed. In this limit the Stark effect perturbations can be described in a very simple statement: I f any two rotational levels / and j are connected by a dipolemoment matrix element μ^, they are mixed by an applied field ε, and the interaction energy μ^ε causes the levels to repel one another.

2. Rotation of Molecules

53

If we consider only the two levels i and / , the Stark effect is obtained from the roots of the secular determinant for the usual two-level perturbation equation: Ei-Λ

-Pip

=

0.

(8.1)

T h e solutions of Eq. (8.1) can be divided into three ranges which depend on the relative values of E{ — Ej and μ^ε. These ranges correspond to three commonly observed Stark effects. 1. Ei — Εj = 0. The Linear

Stark

Effect

T h e degenerate levels are completely mixed by the field, and the Stark displacements of the mixed levels are given by

(8.2)

AW = ±μ ε. υ

This linear Stark effect (the displacements are linearly proportional to β) is observed for symmetric-top levels with Κ φ 0. T h e dipole matrix element connecting the degenerate Κ levels is μJKM = μ^/ΙΚ\]{ J + 1 ), and therefore AW = ±μ{>ΜΚευ(] + 1). T h e Stark effect observed for a transition between the levels / , Κ, M and / + 1, -Κ", M is the difference in the displacements for the two levels:

A

MK

. _ "

(1

1

\

2μοΜΚε ±

0

/ ( / + ! ) ( / + 2 ) *

2. μ^ε <^ Ei — Ej. The Quadratic

Stark

Ρ· *

Effect

When the separation of the levels is large compared to the interaction energy only the first term in the expansion of the square root of Eq. (8.5) need be retained, and the displacements are given by the second-order perturbation equation: A

W

=

fr-G'

·

·

4)

The displacement of the levels is proportional to the square of the applied electric field, and is normally much less than for the linear Stark effect.

( 8

54

C. C. Costain

3. μ^ε ^ Ei — Ej In this range the exact solution must be used Ε/, Ε/ = m

+ Ef) ± i[(Ei - EjY + 4\μν\*

(8.5)

εψ\

If ε is varied from zero to a high value, it is apparent that the limiting case (2) will apply at low fields, but a large or "fast" second-order Stark effect will be observed because of the small value of E{ — Ej. At high fields, when μ^ε > E% — Ejy a near linear Stark effect is observed, and the two levels are nearly completely mixed by the field. T h e mixing of the levels as they are displaced can drastically alter the observed spectrum. For example, at zero field a transition may be allowed to only one of two closely spaced levels. As the two levels are mixed by a steadily increasing field, the intensity of the transition may gradually shift to the originally "forbidden" level. T h e apparent change in the selection rules is the consequence of the new states formed by a linear combination of the two original levels. This changing or "borrowing" of intensity is characteristic of strong perturbations in all fields of spectroscopy. For the degenerate or near-degenerate examples (1) and (3) the interaction between the two levels is so great that the contribution from the interactions with other levels is usually negligible. For example (2), however, there will be more than one level/for which μ^ will be important. If the displacements are small compared to the values of Ei — Ejy the total Stark effect for each level is determined from the summation over all Δ] = 0, ± 1 dipole-matrix elements: Αΐν{=Σ

j

ψ

ί

Λ

Χ .

(8.6)

&i — &j

In an asymmetric top μα, and μ0 may exist, and a particular μ^ will belong to only one of these components. For simplicity in tabulating line strengths, the M dependency is factored from the dipole matrix element. For example,

I iJxM

I μ I J - 1, τ, My |» = χ

J

{

£ j Y \ ~ ™ 2 1)

S

Jr.J-l.r.

Tables of S for specific values of the asymmetry parameter κ are available. T h e complete expression for the Stark energy in terms of the line strength is given in Townes and Schawlow (1955).

2. R o t a t i o n of M o l e c u l e s

55

If a near-degeneracy exists such that perturbation of the energy levels changes a particular E{ — Ej appreciably, a simple s u m cannot be used, and a secular determinant of the form of E q . (8.1) must be solved for the affected levels. As was previously mentioned, when computer calculations of the eigenvalues and eigenfunctions are possible these can be stored for subsequent calculations of the line intensities, or, using E q . (8.6), the Stark energies. T h i s method avoids interpolation errors, and can save a great deal of time in a complex problem.

IX. Quadrupole Hyperfine Structure T h e techniques of microwave spectroscopy are limited to stable molecules or to molecules with long lifetimes. With rare exceptions the moleΧ cules studied have been in a Σ state, for which the resultant electronic angular momentum and spin are zero. For most of these molecules only the electric quadrupole hyperfine structure can be resolved. 1 /2 A nucleus with spin J has a spin angular momentum [/(/ + 1 ) ] Χ Λ/2π, and if / is greater than | , has an electrical quadrupole moment Q. T h e moment Q is positive if the nucleus is prolate with respect to the spin axis, and negative if it is oblate. T h e quadrupole is not influenced by a uniform electric field, but does interact with the gradient of the electric field of the molecule. T h e gradient is normally expressed by the second derivative of the electric potential with respect to a convenient set of orthogonal axes. T h e spin angular momentum is combined with the rotational angular momentum to give a total angular momentum F with magnitude

| F | = [F(F+ \)Õ'%Ι2π;

(9.1)

F is the vector s u m of J and I, and F = J + I,J

+ I - i , . . . , \ J - I \ .

/ and / remain good quantum numbers, but the projections Mj Mf are replaced by MF. T h e interaction energy between Q and the field gradient qj is W

(9.2) and

< = Τ / / ( 2 / - W - D [ T < + õ - /(/ + 1)7(7 + i)] c

c

(9.3)

C. C. Costain

56

where C = F ( F + 1) - / ( / + 1) - / ( / + 1). T h e field gradient qd is evaluated from an integration over all charge density outside the nucleus, 2 3 qj = j ρ [(3 cos θ — l ) / r ] dr. This integral is the average value of the second derivative of the potential at the nucleus in the direction of / , 2 This direction is designated by Zj. T o be meaningqj = [dWjdzj ]^. ful, the gradient must be related to an axis fixed in the molecule. T h e qj must therefore be evaluated for the various classes of molecules. A . T H E LINEAR MOLECULE

The charge distribution in a linear molecule is cylindrically symmetric with respect to the linear axis zm. Therefore 2

dV 2 dx

2

_

dV

_

_

J _

2

2

dV 2 dz

9

(9.4)

since the sum in the three orthogonal directions is zero (Laplace equation). If 6mJ is the angle between the molecular axis and / , dW

1J = dz*

2

ι

3cos 0,

This average is evaluated from the wave functions to give - J 2J+ 3

1J

=

-

In

J 2/+3

The quadrupole energy for the linear molecule is therefore

SC(C+!)-/(/+ i)/(/+i)

WQ=-eqmQ =

2 I I{ 2

eqmQf(I,J,F).

l)(2/-l)(2/+3) (9.5)

The product eqmQ is the quadrupole coupling constant, and / ( / , / , F) is known as Casimire function. T h e selection rules for transition are Δ] = 0, ± 1 ; A F = 0, ± 1 ; ΔΙ = 0. T h e hyperfine components with Δ F = Δ] have the greatest relative intensity. Tables of / ( / , / , F) and the relative intensities are available in Townes and Schawlow (1955). B . T H E SYMMETRIC T O P

T h e averaging of qj over the symmetric-top wave functions intro2 duces an additional 1 — 3 cos θ factor. Equation (9.5) and relevant

57

2. Rotation of Molecules

tables may be applied to the symmetric top if multiplied by the term

l-[3^//(/+l)]. The hyperfine structure becomes increasingly complex if more than one nucleus contributes, and this problem will not be discussed here. 35 The molecule C D 3 C C C 1 has the D atoms with I = 1 in addition to 35 C1 with 7 = f , but the value of Q for the D nucleus is so small that its contributions to the hyperfine structure are much less than the line35 width. Figure 9 is a tracing of the / = 6 «— 5 transition of C D 3 C C C 1

CD 3 CC

5 5

CI

^ = 23,747.47 Mc/sec

D JK = I5.0 Kc/sec

egQ=-79.6 Mc/sec

H J KK = 0.0I3 Kc/sec

κ

op

35

FIG. 9. The J = 6<-5 transition of C D 3 C C C 1 . Only the AF = AJ = + 1 transitions have sufficient intensity to be identified. The lines with Κ were recorded with a low Stark voltage. The Κ = 0 lines were recorded with high Stark voltage specially modified to remove the Κ Φ 0 lines.

35

and shows only the C1 hyperfine structure for the Κ values of 0 - 5 . Of the many allowed transitions, only the AF = AJ = + 1 transitions originating in the F = ψ, ^ , f , and \ levels of / = 5 have sufficient intensity to be seen. The hyperfine structure of Eq. (9.5) is a first-order splitting of the F levels degenerate in a field of zero gradient. I f eqQ is large enough to be

58

C. C. Costain

a significant fraction of the rotational energy level intervals, secondorder perturbations must be included. For quadrupole interactions the matrix elements connect levels with ΔJ = ± 2 in addition to ΔJ = ± 1 .

C . ASYMMETRIC T O P S

T h e determination of qj for asymmetric tops involves much the same procedure as that for dipole moments, and, in fact, the tables of line strengths can be used. T h e hyperfine structure must now be fitted 2 2 2 2 2 2 and d Vldxm — à V\dyr y where xmy ymy by two parameters d V/dzm and zm are the principal axes of the molecule. T o interpret the gradients in terms of chemical bonding, it is necessary to make a rotational transformation to the principal axes of the quadrupole tensor; the major axis will usually coincide with the chemical bond to the atom of interest. I f the structure of the molecule is known, and therefore the required angles for rotation of the axis system, the transformation to bond direction will show whether or not the charge distribution is symmetrical about the bond, i.e., if Eq. (9.4) is satisfied. Alternatively, if only single bonds are present in the molecule and cylindrical symmetry is assumed, the rotation required to satisfy Eq. (9.4) determines the bond angle with respect to the principal axes. Results have generally confirmed cylindrical symmetry within experimental error. T h e determination of the quadrupole tensor is of equal interest when cylindrical symmetry is not found or expected. In a molecule such as monochloroethylene, the planar structure resulting from the carboncarbon double bond makes it unlikely that the in-plane and out-of-plane 2 2 2 2 components d V\dx and d V\dy would be equal. It is usually assumed + that a structure with a C = C 1 bond is one of the resonating structures of C H 2 = C H C 1 , and the measurement of the quadrupole hyperfine structure makes a quantitative assessment of the theory possible. T h e magnitude of eqQ along or nearly along the bond is of course of primary inportance in studies of chemical bonding. T h e experimental 35 values of eqQ for C1, for example, vary by more than a factor of 100 in different molecules; the "normal" covalent value is about —80 MHz, but the value in the ionic NaCl is less than 1 MHz. T h e quadrupole coupling constant is obviously a sensitive probe for the study of bond character, and might be expected to be increasingly useful as a monitor in complex molecular-orbital calculations.

2. Rotation of Molecules

59

X . Instrumentation A.

MICROWAVE SPECTROMETERS

T h e basic microwave spectrometer consists of a source, or oscillator, an absorption cell, a crystal detector, and a display system. It differs from the infrared spectrometer or optical spectrograph in that the source is monochromatic. T h e scanning of the frequency is accomplished by tuning the source, rather than by using a dispersing prism or grating with a slit system to resolve the different frequencies. 1. Reflex Klystron

Oscillators

T h e reflex klystron oscillator is still the most widely used source in 9 microwave spectroscopy. T h e frequency range from 8 GHz (8 Χ 1 0 cycles/sec, wavelength λ = 3.75 cm) to 80 GHz (λ = 0.375 cm) can be covered with 15 commercially available tubes. T h e frequency of the reflex klystron is determined primarily by the resonant frequency of the cavity in the tube, and the frequency is changed by mechanically changing the size of the cavity. T h e cavity is of relatively high Q, which is of importance in the production of a monochromatic or spectrally pure frequency, but the cavity size, and therefore the frequency, is sensitive to temperature. T h e thermal drift can be largely prevented if the klystron can be immersed in a constant-temperature oil bath. T h e klystron can also be tuned or swept electronically over a range of 30-100 MHz. A sawtooth waveform is usually applied to the reflector electrode to produce an approximately linear sweep. All of the voltages from the klystron power supply must be well regulated. 2. Backward-Wave

Oscillators

T h e backward-wave oscillator, or B W O , is a more recent development than the reflex klystron and has several advantages. It is voltage-tuned over its entire range of oscillation, and therefore the frequency can be rapidly and accurately set, and readily swept. T h e frequency range of a single tube is two or three times that of a reflex klystron. T h e sensitivity to temperature is considerably less than that of a klystron. T h e voltage-tuning characteristic of the backward-wave oscillator, however, has the disadvantage that the frequency is much more sensitive to voltage instability and ripple in the power supply. It is virtually im-

C. C. Costain

60

possible to regulate the voltage well enough to produce an oscillation with the same spectral purity as a reflex klystron. T h e incidental frequency modulation on the backward-wave oscillator makes it unsuitable for use in a sensitive spectrometer unless some form of frequency stabilization is applied. 3. Frequency

Stabilization

of

Oscillators

T h e frequency variation of microwave oscillators arising from temperature and voltage instabilities can be overcome by locking the microwave oscillator to a more stable frequency source, a low-frequency crystal oscillator. T h e crystal frequencies are multiplied until its harmonics lie near the frequency of the microwave oscillator. T h e difference frequency between the crystal harmonic and the oscillator is held constant by a feedback system which effectively locks the microwave oscillator frequency to crystal frequency. T h e sweeping of the oscillator frequency is accomplished by altering the crystal frequency, or by adding a stable interpolation oscillator frequency at some point in the crystal multiplier chain. Because no mechanical tuning is involved, the backward-wave oscillator is most easily stabilized and swept over wide regions of the spectrum. However, the involved electronics makes a stabilizing system an expensive addition to a microwave spectrometer. 4. Microwave

Harmonic

Generation

T h e construction of any microwave oscillator requires very precise fabrication of parts which are a small fraction of a wavelength, and when the wavelength itself becomes less than 0.5 cm, it becomes increasingly difficult and expensive to make the oscillator tubes. It is more practical for microwave spectroscopy purposes to produce frequencies beyond 80 GHz by harmonic generation. When a pure sine wave—for example, 30 GHz—is rectified by a crystal detector, harmonics at 60, 90, 120, 150, . . . GHz are produced with powers sufficient for microwave spectroscopy. T h e techniques of producing and detecting the high frequencies are exacting. Most successful in this have been Gordy and his group at Duke University, who have made measurements above 600 GHz at wavelengths of 0.043 cm. 5. The Hughes-Wilson

Stark

Spectrometer

T h e intensity of absorption lines in the microwave region is small, _8 with γ = 3 X 1 0 / c m a typical value for the peak absorption of an

2. Rotation of Molecules

61

asymmetric top in the 30-GHz region. I f the cell length is 300 cm, the total absorption is only about 10 ppm of the power incident from the source. T h e presence of source and detector noise and the frequencydependent transmission characteristic of the microwave components makes the detection of such small signals difficult. Much of the difficulty is overcome by using the Stark modulation spectrometer of Hughes and Wilson. The principles of operation are shown in Figs. 10 and 11. A squarewave voltage from the Stark modulator is applied to an electrode or septum in the center of the waveguide absorption cell. T h e square wave Waveguide

Stark electrode

I

1 1

I Frequency sweep + I stabilizer

Stark Modulator

-Ι 1

Phase reference

r-i Detector

Amplifier and display

FIG. 1 0 . A block diagram of a microwave Stark spectrometer.

is zero-biased, so that during one half the cycle the absorption line is unperturbed, but during the opposite half-cycle a nearly uniform field is applied to the absorbing molecule, which results in a splitting and displacement of the absorption line into its various M components. I f the Stark modulation is at a frequency / , and the amplifier of the detector output is tuned to amplify only this frequency, a signal will appear only when there is a molecule present which has a Stark effect. I f a phasesensitive detector is used with a reference signal at the Stark frequency / , a true subtraction of the on-off absorption signal occurs and the Stark components or lobes are displaced with the opposite polarity to the unperturbed line. Power variations in the microwave source, etc., have little effect unless they occur at the modulation frequency / . This * Molecular-modulation* ' technique of the Stark spectrometer not only separates the signal from the noise, but for low J values it displays the M components of J for positive identification of the / value of the transition (Fig. 12). Further, the accurate measurements of the Stark field in the waveguide and the measurement of the Stark displacements

62

C. C. Costain

—0

600

FIG. 1 1 . The "molecular-modulation" principle of the Stark spectrometer. The J = l<-0 absorption line in (a) is displaced by 6 MHz by the 600-V Stark field in (b). The frequency of the klystron is swept very slowly (compared to the Stark modulation frequency) through the line (a) and its Stark lobe (b), and only the difference signal is detected by a phase-sensitive detector.

of the line makes possible an accurate determination of the dipole moment of the molecule. B . FAR-INFRARED SPECTROMETERS

The generation of microwave frequencies above 300 GHz by harmonic generation becomes very difficult, and while measurements at more than twice this frequency have been made, these measurements have been of the nature of special projects rather than as a matter of routine. Un- 1 fortunately, at 300 GHz, λ = 0.1 cm, or ν = 10 c m , and the techniques of optical spectroscopy are also very difficult. There are two fundamental reasons for these difficulties. First, the energy radiated by a black body at long wavelengths is proportional to

2. Rotation of Molecules

63

i

1 Me/s H—H

FIG. 1 2 . The linear Stark effect of a J = 5 < - 4 , Κ = 4 , zlM = 0 transition. The absorption line (zero field) is upward, and the displacements of the Stark lobes M = 1, 2, 3 , and 4 are proportional to M [Eq. ( 8 . 3 ) ] . 4

l/λ , and a great deal of power in the source is required to produce a - 1 useful amount of radiation at 10 c m . Second, a diffraction instrument must be scaled in proportion to the wavelength if a given resolution is to be maintained. T h e rotation spectrum of water makes it essential that a far-infrared spectrometer be entirely evacuated, and vacuum tanks of diameters greater than 2 m are not very practical for spectroscopic studies. - 1 T h e resolution obtainable approaches 0.05 c m , which gives a resolving - 1 power of 200 at 10 c m . This must be compared with a resolving power greater than 200,000 for a microwave spectrometer, so that the overlap between the microwave and far-infrared regions is not very satisfactory. In spite of these difficulties, a number of far-infrared spectrometers -1 have been built for the region from 10 to 300 c m and have produced very useful results that cannot be obtained from studies in other spectral regions. A recent development, the Fourier transform spectrometer, has been shown to be superior in performance to the conventional spectrometer employing diffraction gratings. T h e Fourier transform spectrometer does not employ a dispersing element, such as a grating, and a slit system to scan the spectral region

64

C. C. Costain

and detect the absorption lines. Instead, the infrared radiation, with its absorption lines at various frequencies, is split into two equal beams in an interferometer. A path difference χ is introduced between the two beams, and the spectrometer is scanned by changing the path difference χ from zero to a maximum, L. T h e two beams are combined at the detector, and the interference between the two beams produces a Fourier cosine transform of the absorption spectrum. This Fourier cosine transform, which is a function of the path difference χ between the two beams, is F(x) =

I(y) cos 2πχν dv,

Jο

( 10.1 )

where I(v) is the absorption spectrum, and ν = 1/λ. T h e transform or interferogram is then itself Fourier-cosine transformed by a computer to recover the original spectrum, I(v) =

F(x) cos 2πχν dx.

jο

If the absorption spectrum consists of a single line, 1

{

)v

=

o/

("o -

Δν vf + {Δνγ '

where Δ ν is the half-width and v0 the frequency of the line, the interferogram has the form F(x) = e -

2 n x Av

2nxAv

cos 2πχν0 = e~

c o s ( 2 ^ / A 0) ,

where λ0 is the wavelength of the absorption line. T h e signal at the detector is a cosine function with a period A0 in x. T h e amplitude decays exponentially, and the narrower the line (the smaller Δν), the more extended is the interferogram. A series of equally spaced lines from a rotation spectrum with fre-1 quencies ν = 2B(J + 1 ) c m produces an interferogram with equally spaced features of decreasing amplitude, and the spacing in χ is (1/22?) cm. These simple spectra can be recognized from the interferogram, but for accurate measurements and the recognition of finer details, like centrifugal distortion, the computer transform of Eq. (10.1) back to the original spectrum is required. In an actual spectrometer the path difference can only be extended to some distance L, and the limiting resolution and the limiting linewidth that can be attained is approximately \\L. Also, only a finite number of

2. Rotation of Molecules

65

points can be used in the input and output of the computer, and the integrals in the Fourier transforms become summations. The Fourier transform spectrometer would appear to be, and is, a complex machine, but it has two very important advantages. In the diffraction spectrometer the input (and output) slits must be narrower than the line to be observed, and this requirement can be a severe limitation on the energy which reaches the detector. In the Fourier transform spectrometer the limitations on the apertures are those dictated by the aberrations of the optical system and the usable area of the detector. A second and most important feature is the "Fellgett advantage" of the Fourier transform spectrometer. In the diffraction spectrometer the noise can be averaged over the time required to scan over the half-width, Avy of a line. In the Fourier transform spectrometer a line is reconstituted by the transform of the total interferogram, and the signal is "averaged" over the Ν observation points spaced at intervals of Av. I f the noise is ll2 Imperfectruly random, this gives a gain in signal to noise of N \2. tions in the instrumentation, such as in the scanning and measurement of the path difference x, introduce noise which is not random, and reduce the advantage. T o date, advantages of 10 and more have been achieved, and further improvements are possible. The resolution in the far-infrared region is still much lower than that in the microwave region, but this is partially overcome by the possibility of extending the pure-rotation studies to much higher / values. I t is to be expected that the importance of far-infrared studies of pure rotation will steadily increase.

REFERENCES GORDY, W., SMITH, W. V., and TRAMBARULO, R. (1953). "Microwave Spectroscopy."

Wiley, New York. HERZBERG, G . (1945). "Infrared and Raman Spectra." Van Nostrand, Princeton, New Jersey. HERZBERG, G . (1950). "Spectra of Diatomic Molecules." Van Nostrand, Princeton, New Jersey. HERZBERG, G . (1966). "Electronic Spectra and Electronic Structure of Polyatomic Molecules". Van Nostrand, Princeton, New Jersey. "Microwave Spectral Tables." Natl. Bur. Std. Monograph 70, Vol. I. Diatomic Molecules (1964) II. Line Strengths of Asymmetric Rotors (1964) III. Polyatomic Molecules with Internal Rotation (1969) IV. Polyatomic Molecules without Internal Rotation (1968) V. Spectral Line Listing. (1968)

66

C. C. Costain

PAULING, L., and WILSON, Ε . Β . , Jr. ( 1 9 3 5 ) . "Introduction to Quantum Mechanics".

McGraw-Hill, New York. STRANDBERG, M. W . P. ( 1 9 5 4 ) . "Microwave Spectroscopy." Methuen, London. SUGDEN, T . M., and KENNEY, C . N. ( 1 9 6 5 ) . "Microwave Spectroscopy of Gases." Van

Nostrand, Princeton, New Jersey. TOWNES, C . H., and SCHAWLOW, A. L . ( 1 9 5 5 ) . "Microwave Spectroscopy." McGraw-

Hill, New York. WOLLRAB, J. E. ( 1 9 6 7 ) . "Rotational Spectra and Molecular Structure." Academic Press, New York. Far-infrared : CONNES, J . , and CONNES, P. ( 1 9 6 6 ) . J. Opt. Soc. Am. 56, 8 9 6 . FELLGETT,

P.

(1958).

LOEWENSTEIN, Ε .

V.

J. Phys. Radium 19, Appl. Opt. 5,

(1966).

187. 845.