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Unformatted text preview: SPECTROSCOPY C.H.Townes 3 A.L.Schawlow 114 MICROW’AVE SPECTROSCOPY TABLE 48. STRUCTURES OF ASYMMHTRIC ROTORS WHICH IIAVE BEEN
OBTAINED FROM MICROWAVE SPECTRA (Concluded) Molecule Structure ltcfm'lmcc l H\ 3:“39' . 11—.
us}. was“) my 7,, lid]
1.203 "a a
i ‘h' . 3
ms 5?: 178 l . , .,
ozs (son 5 "W. [em [5m] .‘e O
[875] p
m/ 03 O n °a9' advent of highresolution microwave spectroscopy. A large number have
now been worked out by microwave techniques, and it seems feasible to
solve the rotational spectrum of any asymmetric rotor which does not
have serious complications due to internal motions (of. Chap. 12) or
an exceptionally complex hyperfine structure (of. Chap. 6). The Stark
effect has been extremely valuable in identifying and working out this type
0f SDGClI‘Um (of. Chap. 10). The structures of asymnmtric rotors which
have been obtained from microwave results are given in Table 4—8. CHAPTER 5 ATOMIC SPECTRA While most microwave spectra have their origin in absorption by
molecules, certain types of atomic spectra may fall in the microwave
region. Atomic theory is important even for molecular spectroscopy
because it is often convenitnit to consider a molecule as being composed
of atoms “’llUSB properties are not too greatly different from their proper
ties in the free state. Moreover, many molecular phenomena. are
sufficiently analogous to phenomena in atoms so that it is worthwhile to
study ﬁrst the simpler atomic case. This chapter will present a summary of those parts of the theory of
atomic spectra which are needed for microwave spectroscopy. More
extensive treatments are given in the several books devoted to the subject
(c.g., A. C. Candler 79], G. llerzberg [[24], L. Pauling and S. Goudsmit
[23], H. E. “iliite [53], and for a more advanced, quantum—mechanical
treatment, R. U. (I‘ondon and G. .H. Shortley [56]). 5—1. The Hydrogen Atom. The. simplest atom is that of hydrogen,
consisting of a. single proton and an electron. It is described by the wave
equation 8 Q , r
We + L,“ (or e W = 0
or. in spherical coordinates, 10 ,at 1 are lift/.614!»
735 (r 01') + sin! 56¢? + 7'2 sin 6 <36 \sm 9 06 + 8—}? (W ~ w = 0 (51) where the nucleus, or more exactly the center of mass of the electron and
nucleus, is taken as the origin of coordinates, and ,u = mill/(M + m.) is
the reduced mass of the atom. W is the total energy of the atoma
V = ~Z63/r is the potential energy, Z is the nuclear charge in units of
the proton charge. and c is the proton charge. By EL process of separating the variables similar to that used for the
dintomit}. molecule (Chap. 1), the wave equation may be solved [62.
giving 4’ : Ri")®(6l¢m(¢l (5'2)
115 116 MICROWAVE SPECTROSCOPY where
1 r .
CIJ_ = —— 6”“ (5—3)
H ‘\/er
(21+ no — m!)1 * , ,
= —. m , ~ ,4
GM: [ 2(£+ lm‘)! _ Pi (00:. 0) () )
_ [IL—.13 27’"? t Aer'nug 2H1 EZT r i“
R”: — l [(n + l)!]3n"a‘,‘ (no.0) (’ L”! man (1)“)
and where n = l, 2, . . . is the principal quantum number
I = U, 1, 2, . . . , (n — 1) is the orbital quantum number
m : ——l, ml + 1, . . . , Z? '1, l is the magnetic quantum number and should not be confused with the same sym
bol used for the mass of the electron Pi“' = associated Legendre polynomial
L35} = associated Laguerre polynomial
on = h2/4r2pez is the radius of the tiret orbit of the hydrogen atom in the Bohr theory It may be observed that Eq. (51) for the hydrogen atom is exactly
the same as the wave equation (111) for a diatomic molecule with the
potential V(r) replacing the molecular potential (Tr). The hydrogen
atom may, in fact, be regarded as a diatomic molecule with the proton
and electron as the two atoms. The parts of the wave function (53) and
(54) which depend on angle are identical with those for a diatomic or
linear molecule (15) and (16). They are the some for any spherically
symmetric potential, since these functions represent conservation of total
angular momentum (Z or J) and of the projection of the angular momen
tum (m or ill) on a chosen axis. Unlike the diatomic molecule, the
dependence of the potential on r in this atomic case is very simple and
the radial wave function (55) can he determined. In more complex
atoms, the potcnticl for a. single electron may often be considered spher
ically symmetric, so that the angular parts of the wave function are
unchanged. However, the dependence of the potential on r is usually
very difﬁcult to determine, so that the radial wave function and energy
cannot be exactly obtained. Figures 51, 52, and 53 show the radial and angular distribution of the
electrons. The S electron wave function is seen to he the only one which
does not vanish at the center of the nucleus. The s electron is also the
only one which has a spherical charge distribution. The allowed energy levels for the hydrogen atom are given by 2 2
— L2“? = —Rhc a (56) nigh? n? W: where R : 21r3pc4/ch3 is the Rydberg constant in cm“. W is in ergs;
t0 conx'ert to cm", divide by he. On this model of the hydrogen atom, ATOMIC srscrna 117 0 5 1000 0 5 10 i5'a —r—>
FIG. :51. The radial part of the hydroe‘
function of the distance between the on electronic wave function Rm: plotted as a
electron and the nucleus. 15 25 m
L_ J—
O 1 ‘6‘" r ‘4 _l_ l 900 O 6 12 18 a0 2p L41: 4p 70 2 4 6 800 O 6 12 1890 O 8 16 24 3200 1500 O 9 18 2300 O 12 24 36 4800 _I_J ‘ F0“ 7l0 20 3300 O 15 30 4500 O 12 24 36 48 60 7200
I'm. .‘i—J. the electronic density distribution HINHJF  1H I’lﬂllf‘d :15 ll. fimc .c .
(flé‘Cfrournucleal' distance for Sllx'o Um 0f the rui status of the hvdrom‘n atom Tl ' ‘ 7‘ a ‘ I _ ‘ r _ h . . ie ordinate
shin: thL piohnlnhty of ﬁnding the electron between spherical shells of radii r and
r , (T. 118 MICROWAVE SPECTROSCOPY which neglects electron spin and relativistic effects, the energy is inde
pendent of l and m, and depends only on the principal quantum number it. Since the angular dependence of the wave functions is the same as for
the diatomic molecule, the selection rules and intensity relations which
depend on angular momentum are identical. As in Chap. 1, it can be
shown that transitions occur between energy levels such that l’=li1 and m’:m,mi1 A state with l = 0 is called an S state; states for which I a 1, 2, 3, . . . are, respectively, P, D, F, G, H, . . . states.
a?
3 ll?" :7
"efecirens E 3);, 9 _ '69— electrons
I=O m=0 1:1 m=+1 m:0 m=—I
99 ' l d
l 7 I —‘ k _ electrons
l l
I=2 m=+2 m=+i m=0 m=1 m:—2
l
$0 I f
‘ I "' electrons
[=3 m=+3 m=+2 1713+: m=0 m=—l m=—2 =3 FIG. 5—3. The probability density distribution factor (4)2 plotted as a function of 9
for s, p, 05, f electrons. For states with m = 0 the scale is approximately 1(3 + 1)
times that. of the other states having the same value of 1. (After White [53].) 52. Atoms with More Than One Electron. While a wave equation can
be written for manyelectron atoms, it is not possible to solve it exactly.
To a good approximation, however, each electron may be considered to
move in a spherically symmetrical ﬁeld produced by the nucleus and all
the other electrons. Then the angular part of the solution will be just
the same as for the hydrogen atom, and the electron can be characterized
by quantum numbers I and m as before. The radial part of the wave function can only be approximated. Some
useful methods of making the approximation and obtaining numerical
or sometimes analytical wave functions have been discussed by Hartree,
Fock, Fermi, Thomas, Slater, and others {305}. Particularly simple are the alkali—type atoms which have one electron
outside a spherically symmetrical core of “closed electron shells.” As ATOMIC SPECTRA 119 long as the valence electron stays out beyond all other electrons in the
atom, it moves in a Coulomb ﬁeld with Zen = 1. To this extent the wave
functions and energy levels resemble those of the hydrogen atom, and
each state is characterized by quantum numbers n, l, m. The energy
is still independent of m and for large values of 1 does not depend greatly
on I. But for small values of I, especially for 8 states where l = 0, the
wave functions (Figs. 51 and 5—2) show a relatively large probability
of the electron’s being near the nucleus. In these states, the electron —55—5p——_50’—5f——59 0
~45 —4p— qd—M
10,000 10000
—35 —3p— 30' '
20,000 zopoo
30.000 —2s ——2; 30,000 _2"
7 40.000 40,000
E E —23
U U
a; 50,000 — 50,000
a 3
C} _
’ so 000 3
g » E 60,000
.2 r,
70,000 '— r0000
30.000 80,000
90,000 90,000
100,000 100,000
110,000 13 110,000
1" LF FIG. 54. Energy levels of hydrogen and an alkali atom. The term value, or energy,
of the atom is referred to a zero which represents the energy after the valence electron
has been completely removed or the atom ionized. spends considerable time inside the closed electron shells, in a region of
larger 2,“, so that its binding energy is increased. Figure 54 shows the
energy levels of a hydrogen atom and those of an alkali atom. The energies for levels in a complex atom are often given by a formula
analogous to Eq. (56) for the hydrogenic case. However, since Eq. (56)
no longer applies exactly, it can be made to agree with the observed
energies only by modifying the value of n or the value of Z, or both.
Thus for the first case, _ Hthﬁ _ i RthEL W = n*2 _ (n — a}? (57) 120 MICROWAVE. SPECTROSCOPY where R = Rydberg constant _
Z5, = effective nuclear charge in the outer region of the atom. Zo = 1 for alkali atoms, but may be 2, 3, 4, . . . for ionized
alkalilike atoms such as Be+, B”, Cttt, etc.
it effective principal quantum number; not an integer
a principal quantum number, and is an integer
a' : quantum defect
From Eq. 5—7) it is seen that :k I. *=n—a' (0—8) ﬂ If n. is retained and Z is modiﬁed, the term value or energy depends on an
effective Z, Zn”, such that '
__ li’nchZE,f (54)) W = n, Equations (57) and (5—9) are written down by purely formal aniilligy
with the expression which applies to a hydrogenic atom, Since n in
Eq. (57) or Z,” in Eq. (59) are not integers but are empirical constants.
these equations are useful only when the same constant can be used for
several purposes. Equation (5?) is quite useful for describing the ten:
values of alkalilike atoms because the quantum defect, or : n i n ,
varies only slowly with n. and 1. Thus if 71* is known for one level,
its value, and hence the term value for another level, can be found at least
approximately. ' I An effective value of Z is often used in connection \vith atomic ﬁne
structure, or for the energy levels of dilferent atoms in an isoelectronic
sequence. it is then sometimes convenient to express Z.“ in terms of
ZN,” and anr, which are effective nuclear charges in the inner and outer
region of the atom. 7 53. Fine Structure, Electron Spin, and the Vector Model. When
spectral lines are examined with instruments of moderate resolving power,
thev are found to have a structure. 219., to consist of several components.
This “ﬁne structure" is explained by attributing to the electron a spin
angular momentum and a magnetic moment. The angular momentum
is related to a spin quantum number s, and its magnitude is given by in units h.,/'27r. For a single electron .9 always has the value 57‘
This 5' should not be confused with the same letter used to denote state
with t : 0. The corresponding spin magnetic moment is l + (cg/210,
in units of the. Bohr magneton* lieflarmr, where m is the electron mass.
The ﬁne structure constant a 27H“?th and approximately equals T37.
A further quantum number no : i7} is rerouted to describe the projec MTl‘ﬂ' “alll' niagncton is usually taken us a posilive quantity even though the nicetron (barge is negative. ir'gurdlt'ss of this convention, it should be remembered
“mi “1“ "lf‘t'lron magnetic morncnt is opposite to the direction 0! its spin. ATOMIC SPECTRA 121 tion of s on some ﬁxed direction, just as 7m gives the projection of the
orbital angular momentum on a ﬁxed direction. The electron’s orbital angular momentum and spin angular momentum
are vectors and will therefore add in some way similar to the ordinary
rules of vector addition to form a resultant denoted by j. The magnitude
of j is also quantized. It is speciﬁed by the total—angularmomcntum
quantum numborj which has the valuesj = l + s andj : ll — 8' (when
.9 = é). The corresponding vector equation is l=1+s ' (5.10) where the two possible values of j correspond to l and 5 being nearly
parallel or nearly antiparallel. The magnitudes of j, 1, and s are, respectively, m, +71),
and Vets Howcver, sometimes equations are written as if the
magnitudes were 3', l, and s. This convention is frequently employed
because of its conciseness (cf. [23]). Using it, one may speak of 1 and 5
being parallel or antiparallel for then the magnitude of their resultant
would just be I + s or ll ﬂ The method leads to equations which
must finally be corrected by replacing 3'2 by j(j + 1), l2 by I“ + l), and
32 by 3(3 + 1). The two orientations of 8 relative to I produce levels with slightly
different energies. They are distinguished by appending the value of j
as a subscript to the term symbol, for example, Si', Pi, Pg, 1),, I);. Since
there are in general two possible orientations of 5 relative to l, for any
given 1 there are two possible j values. The multiplicity of the term is
therefore said to be 2, and this is denoted by a superscript 2 to the left
of the term symbol. The doublet symbol is used even for 8 states where
only one value of j is possible. Some typical states in a oneelectron
(hydrogenic or alkalilike) spectrum are 98;, 210;, 31);, QDé, EDS, . . . .
The two levels 2P; and 2P; form a ﬁne—structure doublet term. The splitting between the two levels with different values of j which
occurs when l is not zero is caused primarily by the magnetic interaction
between the electron spin and electron orbital magnetic moments. This
interaction can be derived from the magnitude of the magnetic ﬁeld
at the electron caused by the relative motion of the nucleus with respect
to the electron. The splitting, however, is reduced by another contribu—
tion to the energy exactly half as large and in the opposite direction which
is connected with a precession of the electron axis due to relativistic
effects [7]. When both are taken into aceount, the energy level for a
hydrogenic atom with nuclear charge Z is displaced by an amount 22
W‘_l ehZ (1 _ 4r2m3c2 _ SI COS (5,1) (511) GHAP'J‘ER 12 THE AMRIONIA SPECTRUIVI AND IIINDERED h'IOTIONS 12—1. Introduction. Because of the intensity and richness of its
spectrum. :uninonin has played 3, major role in the development. of micro—
wave speetroneopy. It has provided :1. lurge number of msin observable
lilittri on whieh to try both experimental tei..l1nirgue>; and theory. NHa
also provides the Simplest and most thoroughly worksd out. (txnmple of .‘L
clans of Spt‘t'll‘fl which will continue to omupy and puzzle inirirmrave
spawhti'oscopists for many years The moat. important hindered motions all involve the quantum
Himhanitul tunneling eil'eet. That is. the“! are motions Wllit'll (:unnot
oreur in chin: cal mechanics because of energy («insidomtiona but are
allowed by the wave nature of quantum meehnnim. For example. in
the ground vilimtionnl state of Nlla, the molei'ule does not. have enough
energy to allow the. nitri'ig‘en to be found in the plune oi the hydrogens
because of the large potoiuialeuerg}: hump in this pmiiion. as indicated
in Fig: lBl. its nl‘tunl penetration of the. plnnn ol' the liyclroueus and
rapid vihrntion from one bide of it. to the other is (ﬁlled the tunnel ell'eet
heunwe, if the nitrogen cannot (limb lll(:}')ill'(‘11i'l£1llllll. it. mi.th "tunnel "
in Ol‘th‘I‘ to get through. In principle, the .\'ll;‘ inversion is n vilirutionnl
motion. Although vibrations normnlly give frequencies in the iufl‘z'u‘m'i
region. the .\f ll. invernion so much slowed down by the hindering: poten
tinl thnt its frequenty lies in themicrowaveregion. 'l‘he(lunlitntivei‘liniige
from ordinary viln‘ntionnl levels where there is no hump in the potential to spectra involving; hindered n'iotions. vilu‘utionul levels owurring,r in pairs when the. hindering liunip orctll‘s‘ iF<
diwiiussed in (hip. When the. Vibrntionnl enemy in so grout that the
NH; molestile has Hilflltf'lt‘lll'. energy to invert ('lfl:~1>lt‘:!ll_\’1 the vilarntionnl
levels no longer ()(‘tilll‘ in Close palm but. tll't.‘ similar to more normal
equally Hpaeed \‘il.i1‘e.i.tit.innl levels. lloii'ever. tlngv are Mill markedly
inlluent'cd by the preHenee of the potential hump. and heme in this (use
of no “tunneling.” one may still spent; of some hindering; of the ITlOilOl'l. .Xnut'llm' type of liind<_i'e(l motion is the. rotntion of one part of a. mole
cule with rrspm‘d. to the remnimler. ’l‘hun the hydrogen nltnvhed to
oxym'iu in (."ll,,i Jli (Fig. 122) has three pwwihle ponitions of equal energy
lint must tunnel through n. 1)::t':‘1‘.ti;il hump lvtwwu‘: them :‘is innlirntod in
Fig. 123. llL‘IlLiC the rotation (if this hydrogen LLJ‘UUIlti Ll'n.‘ C l) liond Ron THE AMMONIA SPECTRITM AND HENDERJ‘JD MOTIONS 301 is hindered. Similarly, the methyl group in CHI/T‘s (Fig. 124) has
three positions of equal energy separated by potential humps. Usually
hindered motions do involve two or more positions of equal energy. If
these positions are only Ininimn, which are not equal in energy. many of O
«0.000
T
E
5 —20,000
3
a;
E,
.73
E 450000
*5
D.
440,000
i
l
'50‘0003 a t 0 1 2 3
S ——3'
i (a)
__. 40,000
E
§
2 745,000
.,
E
I?
7500001
. s_>
(bl Fro. 121. Potential curve of NHL The variable 3 is a measure of the distance
between the nitrogen and the plane of the hydrogens. (b) shows the. lower part of the.
potential curve in more detail, and the. energy levels. the characteristic properties and interesting features of the hindered
motions Considered here do not occur. When hindered motions have frequencies in or near the microwave
region. the symctrn are often quite diliieult to interpret l.)(‘f‘2111Fi(2 the num—
ber of lines falling in the, rnizrou'ave region is greatly inereahed. and
he muse there in no exact solution for the energy levels in these cases. In 304 MICROWAVE SPECTROSCOPY barrier in terms of the frequency of inversion. This rate of penetration
may also be calculated from the “tunnel effect.” Consider now the nitrogen striking the barrier at s = —so from the
left. Its wave function will partly penetrate the barrier and extend
over into the righthand potential minimum. The amount of penetration
can be roughly evaluated by examining Schrodinger’s equation in the
classically forbidden region. dhp_2u E h, (V — WM» = 0 (125) An approximate solution to (125) is a = exp [a % f more) — News: (12—6)
if V does not vary too rapidly with 5'. Since V — W is positive in the
classically forbidden region, this corresponds to an exponentially decreas
ing function. which has unity value at the boundary where the nitrogen
strikes (the increasing exponential which is also a solution is omitted,
since it would give the N the largest probability of being found on the
right). The amplitude of the wave function which penetrates the
boundary is then absn. = exp {— [2ttv — W)]ids} —80 u = exp [— if)" [2M s words} = 313 (127) Now in a time t which is large enough for the nitrogen to oscillate on the
left side many times, but small compared with the time required for
inversion, the nitrogen will strike the left side of the potential barrier
vat times, where :20 is the vibrational frequency. Each time it may be
considered to partially penetrate and transmit an amplitude 1/ A? to the
right side. The transmitted amplitudes add up to give a total amplitude
after time t of pot/A2, so that the probability of penetration in time t is
(yet/A2)? This is similar to, but differs slightly from, the usual expres
sion for radioactive decay or other types of barrier penetration by
tunneling ([305], p. 22), because the transmitted wave on the righthand
side of the barrier is trapped in an identical potential minimum and is
added to by successive transmitted waves. Equating the amplitude
pot/A2 to the amplitude for in; in (12—4), we have rot V
1rvt 2 s or V : —U
«A2 A2 (128) which is the result obtained by Dennison and Uhlenbeck.
In the ground state Of N113, vﬁvo 9: 1200, so that. :19 ’m' 400 or c". THE AMMONIA SPECTRUM AND HINDERED MOTIONS 305 Because A.2 is large and varies exponentially with air or (V — WV, changes
in either of these quantities can very drastically affect the inversion
freQLIency v. For example, if the reduced mass a is increased by a factor
of 2, such as would be roughly done by changing from NH3 to IlD3, v
decreases by eﬁ‘iﬁ—l) or a factor of 1]. The inversion frequency is cor
respondingly sensitive to the potential, and most molecules have such
high potentials and heavy masses that the inversion frequencies are less
than 1 cycle/sec. hIany molecules invert so slowly that they have not
succeeded in inverting during the few billion years’ life of our planet. N113 in the ﬁrst excited level of this vibrational mode has a much higher
inversion frequency because of the increased value of W. Dennison and
Uhlenbeck L33] were able to assume a simple shape for the potential
barrier and obtain values for the inversion frequencies in the ground and
excited states which agreed roughly with experimental observations. The exact form of the potential barrier assumed is not critical, since A
depends only on an integral of this energy, and not on details of shape.
However, hianning [61] found a potential function which has the general
shape to be expected for NHa, and for which the wave equation could be
more easily solved and accurate values for the energies obtained. Blan
ning’s potential function is V . _ _ —— = 6(3,ool seeh4
he i_
29 Ct: 109,619 sech2
2;: (12—9)
where V/hc = potential in units of cm—1 3: = a coordinate which is dependent on the distance of the
nitrogen from the plane of the hydrogens
6.98 X libs/pi Where ,u is the reduced mass in atomic mass
units
This potential is zero for large 5 (or r), is symmetric about 3 = 0, and
has a peak at s : 0, where the potential is 743,068 cm—l. It has two
minima where the potential is — 45,140 cm‘l. rl‘he constants in (129)
are also chosen to give close to the correct equilibrium conﬁguration for
the molecule, the vibration frequency, and the correct inversion frequency
for one level. If the distance between hydrogens is assumed to stay constant during
the motion so that they move together as a rigid triangle, then .r is taken
as equal to s, the distance between the nitrogen and the plane of this
triangle. If m is the mass of the hydrogens and ﬂI that of nitrogen, then
the reduced mass for this case is simply ,u = BrmM/(Sm —}— M). However,
probably a better approximation to the motion is to assume the N7H
distance remains constant, and only the HiNiH angles change with
vibration [110]. In this case a: is taken as the distance along an are
moved by the hydrogens from the median plane, i.e., a plane through the
nitrogen and perpendicular to the molecular axis. If the angle between ,0 306 MICROWAVE SPECTROSCOPY this plane and the NﬁH bond is taken as a, the proper reduced mass can
be shown to be ,u = 377101! + 3m sin2 a)/’(3m + .11). This reduced
mass varies only a small amount with 0:, and its value may be assumed
to be that at the equilibrium angle of a0 = 21°49. Different reduced
masses in hianning’s potential will give different equilibrium heights for
the NH;; pyramid. I'loweverJ these heights do not deviate signiﬁcantly
from the observed value. The values of inversion frequency for various isotopic species of
ammonia calculated from Manning’s potential are shown in Table 121 TABLE i21. vaEnsioN Momennems or AMMONIA IN MEGACYCLEs
The “ﬁrst excited vibrational state" (a; = 1) corresponds to approximately 050
cm—1 excitation of the vibration of the nitrogen against the plane of the. liydrogens.
The calculated values are given by Manning [61] and Newton and Thomas [306]
except for the calculation of NUD; with the Manning potential, which was done by
A. Javan and J. Lutspeich. ————._*__—_—Jﬁ* ll“ 9 ext‘t d
(iI‘UllIld state . mi \ I P
Vlbralmnnl state. Nun. Nuna mun Nun. Nuin Observed value, Mc. . . , _ . . . . . . , , 23,780 1000 22,705 1.095.000
Calculated from Manning poten— tial . . . . . . . . . . . . . . . . . . . . . . . . . 25.000 1250
Calculated from NewtonThomas
potential..................... 1 17 , 000 . . . . . . 780 . 000 83 _, 000 23 , 800 690, 000 and compared with experimental results. The potential constants have
been chosen to ﬁt some of the data, including the energy of the ﬁrst
vibrational level. The excited vil.)rational levels are fitted rather satis
factorily by this potential [61]. Although the inversion frequencies are
given approximately correctly, it does not seem possible to obtain a
really satisfactory ﬁt of all the data with a potential of D'Iaiining’s type.
This will be seen again when ﬁne structure of the inversion spectrum is
discussed.
Newton and Thomas [306] used a potential of the form V i [[(0.377)2 — 82 2
i i  — — ” 4 . —1 _
he i 0.536 + i 1i X 3'1“ X 10 L111 (12 10) Where 3 is measured in angstrom units. Here the hydrogens are con—
Sidured to move as a rigid triangle. Values for some of the inversion
ff‘3‘ltleiicies calculated by an approximate method [3013] from this poten—
tial are given in Table 131. Form (1210) for the potential appears to SW0 results which are comparable in accuracy with those from the, pomp
tial (129}, THE AMMONIA SPECTRUM AND HINDERED MOTIONS 307 12—3. Inversion of Other Symmetric Hydrides. A very simple form
of potential has been found by Costain and Sutherland [678] to give fairly
accurate values for the inversion frequency of N113, and has been used
to estimate the inversion frequencies of RH; and AsH3. This is a poten tial of the form
V = Skim)? + SicAM)? (12—11) where Ar = change in the N—II bond length A6 : change in the H N H bond angle
The force constants k, and 15.5, as well as the ratio of Ar to A8, can be
obtained by a normal coordinate treatment of the observed vibrational
frequencies of N113. Their evaluation gives V = 3.89 X 10“(A6)2 cm—1 (12—12) where 5 is measured in radians. This gives a potential hill of 2077 cm—I
between the two minima in good agreement with Manning’s value of 2072
cm”. In fact, the entire potential curve given by (1212) is very close to
that given by (129). A similar evaluation of the potential hill from vibrational constants
and molecular geometry for PH3 and Asli3 gives the following results [678].
For PHa: V = 5.3 X 104035)” cnr1 (12—13)
Height of potential hill abdve minimum = 6085 end—1
Inversion frequencies: Ground state = 0.14 Me First excited state 2 7.2 hie For Asllaz V = 41.55 X 1.040382 cm—1 (12—14)
50 = 0.585 radian
Height of potential hill above minimum
Inversion frequencies: Ground state : 1/2 cycle/year First excited state = 1 cycle/day 11,220 em—1 Thns the inversion splitting of 13113 is probably large enough to give a
splitting in microwave measurements of the rotational spectrum, whereas
Asl—Iz would take two years to go through a cycle of inversion and
should hence have no observable splitting. These numbers illustrate
the very rapid variation in frequency as the potential barrier height 13
changed. .
124. Fine Structure of the Ammonia Inversion Spectrum Rotation
Vibration Interactions. Discussion of the inversion spectrum has so far ...
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 Fall '11

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