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Forbidden rotational spectra of axially symmetric polar molecules
JOURNAL OF MOLECULAR
Forbidden
SPECTROSCOPY
Rotational
49, 15-26 (1974)
Spectra
of Axially
Symmetric
Polar
Molecules1
M. R. ALIEV AND V. M. MIKHAYLOV~ Institute of Spectroscopy, Academy of Sciences of the USSR, A kademgorndok, Moscow region 112092, l[SSR In this paper we consider the rotational transitions induced by centrifugal distortion in polar or quasipolar symmetric top molecules belonging to the point groups C, and C,, (n 2 3). It is shown that in this series only the molecules of point groups CZ, Ca,,, Cd, and Can may possess rotational spectra induced by first-order centrifugal distortion. A general expression is given for the effective dipole moment operator and for its matrix elements. The peak absorption coefficients for some of the strongest AK = 3 transitions of the CHID molecule have been calculated and compared with the peak absorption coefficients of allowed transitions. 1. INTRODUCTION
The study for forbidden transitions has long been an interesting and important task for molecular spectroscopists (1, 2). An important example has recently been found in rotational spectra of molecules in the vibronic ground state (3-9). The detailed theory for these transitions, based on a centrifugal distortion mechanism, has been worked out by Watson et al. (3, 4, 7) and Fox (5, 8), and by us (6, 9) independently. From the estimates in (3-9) it is clear that the purely rotational spectra of nonpolar molecules, introduced by centrifugal distortion, are very much weaker than ordinary rotational spectra of polar molecules. In spite of this, the rotational spectrum of methane has already been observed in the far infrared (10). The derivation of the effective dipole moment operator and its matrix elements for the nonpolar molecules and polar CaU molecules has been given in detail in (3, 9). In this paper, detailed calculations and results will be given, for the entire class of polar C, and C,, (z > 3) molecules. We focus special attention on quasipolar molecules, i.e., molecules with a small permanent dipole moment. A main point about considering quasipolar molecules is that the similar magnitudes of the permanent and centrifugally induced dipole moments make it likely that the forbidden and allowed transitions may have comparable intensities. The case of CHSD is briefly treated. 2. CENTRIFUGALLY INDUCED ROTATIONAL TRANSITIONS C, AND C,, MOLECULES (n b 3)
The intensity quantity
of the transition
i-f
between
states
IN
i and j is proportional
to the
1Presented at the 2nd International Seminar on High Resolution Infrared Spectroscopy in Prague, September 4-8, 1972, and at the Symposium on High Resolution Molecular Spectroscopy in Novosibirsk, September 11-14, 1972. a Permanent address: Institute of Atmospheric Optics, Academy of Sciences of the USSR, Siberian Branch, Tomsk, USSR.
Copyright A11 rights
@ 1974 by Academic of reproduction
Press.
in anv form
Inc. reserved.
FORBIDDEN
ROTATIONAL
I(GW)P+
SPECTRA
19
I(+dif)l2,
IN.4f)/*+
(1)
where 11x”, pr”, pz” are the space-tixed components of the dipole moment related to the molecule-fixed components pz”, py”, pz” by PA” = c IIm”XAn, d = X, Y, z; a
operator
a = x, y, 2.
(2)
XAu is the direction cosine between the space-fixed A asis and the molecule-nixed o axis. Espressed as a function of the normal coordinates Ok, IL,,” is given to first order by
where the bLolo are the components of the permanent dipole moment. In order to calculate the matrix elements (i : PA” 1f) we use the contact transformation method (II, 12). In this method the matrix elements (~/PA” j f) are calculated on the basis of rigid-rotor wavefunctions, from an effective dipole moment operator which is obtained from the true dipole moment operator p” in Eq. (3) by the same unitary transformation as is applied to the rotational Hamiltonian. To first order in the normal coordinates the rotational Hamiltonian may be written as (II) H” = C B,Pa2 f LI
C C Bk”@qkP,PB a,0 x-
(4)
where B, is the rotational constant for the a-axis, Y,, Pb are molecule-fixed components of the total angular momentum operator (dimensionless), the (Ik are dimensionless normal coordinates, and BkUBare given by i
_&a@ B,
AkaB Bkd = - _---~--_-2I,I$dk’
(21,)+ ( wk >
BP
Here Ak”’ = (N,,/aQk), are the partial derivatives of the instantaneous inertia tensor with respect to the normal coordinates evaluated at the equilibrium configuration of the molecule, 1, are the principal moments of inertia and Wk are the vibrational frequencies (or wave numbers) : B,, Bkaa and Wkare in frequency or wave number units and .IkaLBand 1,; are in a.m.u. IA units. The Hamiltonian H” is subjected to a contract transformation H’ = eiSuH”eciSv = H” + i[S,, H”]_
-
[S,, H”]_]_
(1/2)[S,,
+ ...;
[S, H]_ = SH - HS,
(5)
where the Hermitian operator S, is chosen to remove the second term from H’, to first order. It can be shown that (II)
BkaP --pkPrnpp wk Substitution
of (4) and (6) into (5) gives
;
pk
= -
i&.
(6)
20
ALIEV
where the well-known
centrifugal
AND MIKHAYLOV
distortion
constants
BlsUflBky6
7a@y6= - 2 c
(8) k
PAf = c LL?~~A~ LX where, using Watson’s
wk
PA” is subjected
The dipole moment operator in Eq. (S), thereby yielding
notation
are given by
+
c
to the same
transformation
as H”
(9)
~7asP&?hA~
USB!“f
(3)
e,4
=
-
C
(10)
k
The units of (+-,/a&), are lO’*D/g’ cm and the or@ are in Debye units. The quartic terms in (7) are not diagonal in the symmetric top representation in the case of some symmetric-top molecules. In the calculation of the effective dipole moment operator hA’ we must take account of the nondiagonal contribution H,,d’ from II’. The terms in H’ are of two kinds (13). One kind represents the corrections to the rotational energy of Cs and CI,, molecules and may be written as (see also (3, 4))3 ~zmz
cpz, Pzl+l+- ccpz, Pt/l+,CPU, PJ+l+)
Hnd)(Cd= __ 4 (CVZ”- PI/?,
(11)
where [A, B], = A B + BA. The other kind of terms occurs for C-1 and Cd, molecules and is given by
H’(G) = Y[PZ’
P,]+2 - Y(P2
- P,2)2.
(12)
In the case of C5, Cs, Cju, and Cga molecules Hnd’ = 0. It is apparent that no terms in (11) and no nondiagonal elements in (12), except the (-2 1 i- 2) element, contribute to the energies in the first order of approximation. These terms can be eliminated from H’ to the first order of approximation by the contact transformation e+isRH’e-iSR = H (12,14). The dipole moment operator PA’ in (9) is subjected to the same transformation, thereby yielding an effective dipole moment operator PA to first order: 1uA
=
1
pmo{hAm
+
i[sR,
hAa]-}
+
(I The transformation form as (3, 12) SR
=
function
(~mm
c
(13)
e7”%pB~Ay.
a,P,r
SB which removes H,~‘(CG,) can be chosen in operator
/4(&
The terms in Hnd’(C4”) contain e1ement, except the (-21+2)
- Bz)){CPz2,
P,l+ -
(2/3)P,3
-
(1/3)P,}
(14)
only the matrix elements (K 1K&4), and all of these, can also be eliminated by a contact transformation.
8 Extra terms of type s,,~ for CS and Cg groups are excluded in the canonical form of the T tensor (13) used here.
FORBIDDEN
However, the SR functions matris form
ROTATIOSAL
corresponding
SPECTRA
to these elements
21 must
(I< 1Hd 1K * 4) (K1S,<~Kf 4) = i--&*4
&O
0 -
be chosen in the
(15)
where EKO is the rigid-rotor energy. Thus the required transition moments are obtained by taking matrix elements of the effective dipole moment operator pC(Ain (13) between symmetric-top wavefunctions. Transition probabilities for the particular Stark components are proportional to the squares of the matrix elements (JKM j PA / J’K’M’). In the fieldfree molecular rotor, the Stark components are degenerate, and the intensities of the pure rotational lines are proportional to the squares of these matrix elements summed over all M’ values of the final J’ levels and, if the radiation is isotropic, over the three coordinates, _Y, Y, Z. The result is $5
I(J,k’,MIll.~jJ’,K’,M’)12=
where, using the symmetry
[M(G)]2.F(J-.7’,K-K’)
(16)
rules of Henry and Amat (15)4
M(C,,)
= OP(E)
M(C4J
=
+ PzO?zzzz(E)/2(B,
llzo J(B,
-
T.LrUU(BL)
+
- B,)
Tzll,Il(B29
(17)
(18)
&)(
and the expressions for F(_7 - J’, K - K’) are listed in Table I (we have not summed over two-fold K degeneracy in (16) or in Table I). It may be noted that in the case of plane polarized radiation commonly employed in microwave and laser spectroscopy we can choose the coupling vector of the radiation along the Z axis and can drop the X and 1’ components from (16) or we can divide (16) by 3. 3. ROTATIONAL
SPECTRUM
OF CHID
Due to Td symmetry, the CH, molecule has no permanent electric dipole moment. The centrifugally induced dipole moment and corresponding rotational spectrum of CH, was recently considered by Watson (3), Fox (5, 8), and Dorney and Watson (7). When one of the protons is replaced with a deuteron, the symmetry is reduced to Cs,,, and the associated changes in the zero-point vibration and molecular charge distribution lead to the generation of a small permanent electric dipole moment pi’- 5X10P3 D. The allowed rotational spectrum of CHID in the ground vibrational state associated with pzo has been observed in the frequency range from 40 to 120 cm-’ b\. Ozier cl 01. (IQ, at low resolution and with a signal-to-noise ratio of about 10. In order to estimate the absorption coefficient of the centrifugally induced transitions, the parameter M in (17) was calculated from the dipole moment derivatives and force field of Sverdlov (17,18). Depending upon the relative signs of +/aQk, the four possible values of Oz.rzare found and listed in Table II, where the values of all parameters used 4The 6 term for Ce and Cdv groups is zero due to symmetry (15).
22
ALIEV
AND MIKHAYLOV
FORBIDDEN
ROTATIONAL
SPECTRA
23
TABLE II KUNXIAR
species Ql
PARMETER
poll CH3D’
Al
species
Qz
b
E
Q4
Q5
Qb
-
0.169
0.107
0.062
-
0.215
Q3 b
B$
-
0.330
-
0.187
B;=
-
0.026
-
0.565
4.28
5.48
G
-
0.064
BXX k
0.219
Bf
8.48
PI:
exx x
z
-
0.103
-
8.17
3.29
0.278
7.23
exx
exxz
Z’
7
t
f
7
1.02
+
+
+
7
0.28
f
0.27
+ 1.88
?- 0.13
t
0.67
0.47
‘F 0.95
*
7
0.52
t
F
*
i
+
r
7
t
%nits of Debye b
BF
and unita
In the
‘Thin
of
calculation
wl = 2306, and
cm-l,
are
colrmm
of
parameters of
w2 = 3146,
normal-coordinate
for ep
unita
corresponds
these
u)3
7
0.92
+ 2.08
2.42
i
0.64
+ 2.20
p;
=
ag ‘ay
are
(~&j1’2
10
parameters
= 1352,
u4 =
transformation
to
+ 1.48
pi;
(I& -
-5
the 1512, matrix
1,2,3)
(g?Je
1.44
are
lO-2
Debye. ‘harmonic” u+, L are
for
8;
frequenciee
3154,
w6 = 1197
taken
and 8:”
from
or
cm
-1
Sverdlov
to
(17 -,-
18)
cl;: (k = 4,5,6)
and eEZ .
are also given.5 The largest value M = 1.02 X 10p5D corresponds to the preferred sign choice of Sverdlov (17, 18) and will be used here. The formula for the peak absorption coefficient a I,,axf of the centrifugally induced transitions can be obtained from Eq. (3.2i) of Gordy and Cook (19) if (rnl,.~] n)” is replaced by the squared matrix elements M”F(J - J’, K - K’) from (16). Then the formula for cy,*,’ specialized to the CHsD molecule becomes al,laXf(cm-l) = 1.804~10-17v2(g~ug~)g~(1 - 0.00008~) X exp(-0.00016E~~“)F(J
- J’, K - K f
3)
(19)
where c is the symmetry number, gJ = 2J + 1, gk = 1 for K = 0 and 2 for K # 0, gr is the nuclear spin weight factor (19), Yis the line frequency in GHz, and the temperature T = 300°K and line-width parameter Av/P = 1 MHz/torr (cf. (3, 9)) are used; (gkUg1) = 3 for k’ = 3, 6, 9, . . . , and (gkagr) = 3/2 for all other values of K. The 6Secondterm M=e,=s.
in
(17) is neglected for the CHaD molecule, since it is of the order lo-’
Debye, and
9 - 10
13 - 14
2600-2790
2830-3020
2.0 x 10-6
1826.113
7.6 x lO-(j 1.2 x 1o-5
2184.505 2415.421 2686.562 2876.075 3191.887 3420.779 3649.099 3876.810
3-o 3-o o-3 3-o 3-6 3-6 3-6 3-6
2.6 x 1O-5
2.3 x 1O-5
1.8 x 1O-5
5.9 x 10-6
6.2 x 1O-6
4.9 x 10-6
1953.247
3-o 3.4 x 1o-6
1584.169
5-8 6-9
1.1 x 10-6
1338.627
4-7
7.7 x 1o-7 -6 2.0 x 10 -6 1.1 x 10
4.7 x 1o-7
1.9 x 10-7
0. max cm-l
16 -17
15 -16
14 -15
13 -14
12 -13
11 -12
10 -11
9 -10
8-9
7-8
6-7
5-6
4-5
3-4
2-3
l-2
J - 3'
3-3
3-3
3-3
3-3
3-3
3-3
3-3
3-3
3-3
3-3
3-3
3-3
3-3
o-o
o-o
o-o
DK = -23.02 x 10-4 (all in GHz) from Olson (2).
3923.335
3695.827
3467.722
3239.060
3009.876
2780.208
2550.094
2319.570
2088.674
1857.443
1625.914
1394.125
1161.852
930.180
697.765
465.239
GHz
a Y
Allowed Transition K - K'
+h e calculated frequencies were obtained using ground state constants Bx = 116.322, BZ = 157.412, DJ = 15.51 x 10-4 , DJK = 37.09 x 10-4 and
11 - 12
11 - 12
2330-2560
3750-4000
10 - 11
2130-2325
10 - 11
9 - 10
3520-3700
14 - 14
1680-1860
1910-2090
9 - 10
13 - 13
1449-1640
8-9
13 - 13
1200-1400
3290-3470
12 - 12
1050-1110
3090-3240
852.138
2-5
12 - 12 1096.601
609.591
l-4
11 - 11
588-700
830-930 3-6
365.589
o-3
11 - 11
v
355-465
K - K' GHz
J - J'
a
Forbidden Transition
GHz
Range
Frequency
SOME ROTATIONAL TRANSITIONS OF CH3D
TABLE III
2.5 x 10-4
3.6 x 10-4
4.9 x 10-4
6.2 x 10-4
7.9 x 10-4
9.8 x 10-4
1.0 x 1o-3 1.0 x 10-3
1.0 x 10-3
9.0 x 10-4
7.2 x 10-4
5.1 x 10-4
30 x 10-5
15 x 10-5
6.9 x 10-5
2.2 x 10-5
max cm-1
a
FORBIDDEN
ROTATIONAL
23
SPECTRA
in several calculated values of (Y,,& for some of the strongest K-K f 3 transitions frequency ranges are presented in Table III. The peak absorption coefficient (Y,,~ for the allowed transition J, K-J + 1, K of CH,D molecule may be written as (19) alnaxO = 6.844.1OV(l
- 0.00008~)~~ exp(-0.00016E~~“) (J + 1)” - R2 X (&‘gr)
where the dipole matrix element factor M, for an arbitrary the parameters 0 and p” as follows
Jfl
Ma2,
Csu molecule is related
iM, = /.4zo- ez=* + (e,= - 0,= - 20,TZ)K2 + e,=(J + 1)2.
(20) to
(21)
The calculated values of Br”fl corresponding to all sign combination of r+/aQk are given in Table II. Using pzo = 5.64 X 1P3 (ZO), ezzz = -1.48 X 10-5, &** = -2.08 X lo-“, and 0,“” = 0.92 X lop5 (all in Debye units) we obtain the values of (~,,p listed in Table III. Such a choice of e-values provide the correct rotational dependence of pz0 established by Wofsy et al. (20) and discussed previously (21). Moreover, this choice gives a slight K dependence of M, in agreement with the statement of Ozier et al. (16). The results show that a,,,,’ for the strongest lines in the low frequency region (-400 GHz) are of the order lo+ X LY,,~~ but in the high frequency region (-4000 GHz) c&J - 10-l x %z3xQ.One can show that in the region -6000 GHz some of the calculated forbidden transitions of CHSD are stronger than the allowed transitions in the same region. However, vibration-rotation “hot-band” transitions may be expected to occur in this region, although these two types of transitions should be readily distinguishable by their selection rules, by their different temperature dependence and by the different dependence of their intensities on J and K. A rough estimate shows that the absorption coefficients for other deuterated methanes are of the same order of magnitude as for CHID. There seems a possibility that the forbidden rotational transitions of quasipolar molecules like deuterated methanes may be observed even by existing methods (22). ACKNOWLEDGMENTS The authors are grateful to Dr. W. B. Olson for providing the ground state constants of CHID prior to publication and to Dr. J. K. G. Watson for valuable comments on the first draft of the manuscript. RECENED:
February
28, 1973 KEFERENCES
1. G. HERZBERG,“Electronic Spectra of Polyatomic Molecules.” Van Nostrand, Princeton, N. J., 1966.
2. R. H. GARSTANG,“Atomic and Molecular Processes” (D. R. Bates, Ed.) Academic Press, New York and London, 1962. J. K. G. WATSON,J. Mol. Spectrosc. 39, 364 (1971). T. OI(A, F. 0. S~IZU, T. SHI~ZU, AND J. K. G. WATSON,Astrophys. J. 195, L15 (1971). K. Fox, Phys. Rev. Let,!. 27, 233 (1971). M. R. ALIEV, ZhETF Ph. Red. (JETP Lett.) 14, 600 (1971). 7. A. J. DORNEY~AND J. K. G. WATSON, J. Mol. Spectrosc. 42, 135 (1972).
3. 4. 5. 6.
26
ALIEV
AND MIKHAYLOV
8. K. Fox, Pkys. Rev. A, 6, 907 (1972). 9. M. R. ALIEV AND V. M. MIKHAYLOV,Optika i Spektroskopiya (submitted for publication, February 1972). 10. A. ROSENBERG,I. OZIER, AND A. K. KUDIAN,J. Chem. Phys. 57, 568 (1972). 11. H. H. NIELSEN, Rev. Mod. Phys. 23, 90 (1951). 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
M. R. ALIEV ANDV. T. ALEKSANYAN,Optika i Spektroskopiya 24, 520, 695, and 388 (1968). A. P. ALEKSANDROV ANDM. R. ALIEV, J. Mol. Spectrosc., 47, 1 (1973). J. K. G. WATSON,J. Chem. Pkys., 46, 1935 (1967). L. HENRY ANDG. AYAT, Cahiers Phys., 14, 230 (1960). I. OZIER, W. Ho, AND G. BIRNBAUM,J. Ckem. Phys., 51,4873 (1969). L. M. SVERDLOV,Optika i Spektroskopiya, 10, 33 (1961). L. M. SVERDLOV,M. A. KOVNER,AND E. P. KRAINOV, “Vibrational Spectra of Polyatomic Molecules.” Science Press, Moscow, 1970. W. GORDY AND R. L. COOK, “Microwave Molecular Spectra.” pp. 41, 57, 154, Interscience, New York, 1970. S. C. WOFSY, J. S. MUENTER,ANDW. KLEMPERER,J. Chem. Phys. 53, 4005 (1970). V. M. MIKHAYLOVANDM. R. ALIEV, Optika i Spektroskopiya (submitted for publication, July 1972). A. F. KRUPNOV,Submillimeter Spectroscopy of Gases, Preprint, NIRFI, Gorky, USSR, 1972. W. B. OLSON,J. Mol. Spectrosc. 43, 190 (1972).
Forbidden
SPECTROSCOPY
Rotational
49, 15-26 (1974)
Spectra
of Axially
Symmetric
Polar
Molecules1
M. R. ALIEV AND V. M. MIKHAYLOV~ Institute of Spectroscopy, Academy of Sciences of the USSR, A kademgorndok, Moscow region 112092, l[SSR In this paper we consider the rotational transitions induced by centrifugal distortion in polar or quasipolar symmetric top molecules belonging to the point groups C, and C,, (n 2 3). It is shown that in this series only the molecules of point groups CZ, Ca,,, Cd, and Can may possess rotational spectra induced by first-order centrifugal distortion. A general expression is given for the effective dipole moment operator and for its matrix elements. The peak absorption coefficients for some of the strongest AK = 3 transitions of the CHID molecule have been calculated and compared with the peak absorption coefficients of allowed transitions. 1. INTRODUCTION
The study for forbidden transitions has long been an interesting and important task for molecular spectroscopists (1, 2). An important example has recently been found in rotational spectra of molecules in the vibronic ground state (3-9). The detailed theory for these transitions, based on a centrifugal distortion mechanism, has been worked out by Watson et al. (3, 4, 7) and Fox (5, 8), and by us (6, 9) independently. From the estimates in (3-9) it is clear that the purely rotational spectra of nonpolar molecules, introduced by centrifugal distortion, are very much weaker than ordinary rotational spectra of polar molecules. In spite of this, the rotational spectrum of methane has already been observed in the far infrared (10). The derivation of the effective dipole moment operator and its matrix elements for the nonpolar molecules and polar CaU molecules has been given in detail in (3, 9). In this paper, detailed calculations and results will be given, for the entire class of polar C, and C,, (z > 3) molecules. We focus special attention on quasipolar molecules, i.e., molecules with a small permanent dipole moment. A main point about considering quasipolar molecules is that the similar magnitudes of the permanent and centrifugally induced dipole moments make it likely that the forbidden and allowed transitions may have comparable intensities. The case of CHSD is briefly treated. 2. CENTRIFUGALLY INDUCED ROTATIONAL TRANSITIONS C, AND C,, MOLECULES (n b 3)
The intensity quantity
of the transition
i-f
between
states
IN
i and j is proportional
to the
1Presented at the 2nd International Seminar on High Resolution Infrared Spectroscopy in Prague, September 4-8, 1972, and at the Symposium on High Resolution Molecular Spectroscopy in Novosibirsk, September 11-14, 1972. a Permanent address: Institute of Atmospheric Optics, Academy of Sciences of the USSR, Siberian Branch, Tomsk, USSR.
Copyright A11 rights
@ 1974 by Academic of reproduction
Press.
in anv form
Inc. reserved.
FORBIDDEN
ROTATIONAL
I(GW)P+
SPECTRA
19
I(+dif)l2,
IN.4f)/*+
(1)
where 11x”, pr”, pz” are the space-tixed components of the dipole moment related to the molecule-fixed components pz”, py”, pz” by PA” = c IIm”XAn, d = X, Y, z; a
operator
a = x, y, 2.
(2)
XAu is the direction cosine between the space-fixed A asis and the molecule-nixed o axis. Espressed as a function of the normal coordinates Ok, IL,,” is given to first order by
where the bLolo are the components of the permanent dipole moment. In order to calculate the matrix elements (i : PA” 1f) we use the contact transformation method (II, 12). In this method the matrix elements (~/PA” j f) are calculated on the basis of rigid-rotor wavefunctions, from an effective dipole moment operator which is obtained from the true dipole moment operator p” in Eq. (3) by the same unitary transformation as is applied to the rotational Hamiltonian. To first order in the normal coordinates the rotational Hamiltonian may be written as (II) H” = C B,Pa2 f LI
C C Bk”@qkP,PB a,0 x-
(4)
where B, is the rotational constant for the a-axis, Y,, Pb are molecule-fixed components of the total angular momentum operator (dimensionless), the (Ik are dimensionless normal coordinates, and BkUBare given by i
_&a@ B,
AkaB Bkd = - _---~--_-2I,I$dk’
(21,)+ ( wk >
BP
Here Ak”’ = (N,,/aQk), are the partial derivatives of the instantaneous inertia tensor with respect to the normal coordinates evaluated at the equilibrium configuration of the molecule, 1, are the principal moments of inertia and Wk are the vibrational frequencies (or wave numbers) : B,, Bkaa and Wkare in frequency or wave number units and .IkaLBand 1,; are in a.m.u. IA units. The Hamiltonian H” is subjected to a contract transformation H’ = eiSuH”eciSv = H” + i[S,, H”]_
-
[S,, H”]_]_
(1/2)[S,,
+ ...;
[S, H]_ = SH - HS,
(5)
where the Hermitian operator S, is chosen to remove the second term from H’, to first order. It can be shown that (II)
BkaP --pkPrnpp wk Substitution
of (4) and (6) into (5) gives
;
pk
= -
i&.
(6)
20
ALIEV
where the well-known
centrifugal
AND MIKHAYLOV
distortion
constants
BlsUflBky6
7a@y6= - 2 c
(8) k
PAf = c LL?~~A~ LX where, using Watson’s
wk
PA” is subjected
The dipole moment operator in Eq. (S), thereby yielding
notation
are given by
+
c
to the same
transformation
as H”
(9)
~7asP&?hA~
USB!“f
(3)
e,4
=
-
C
(10)
k
The units of (+-,/a&), are lO’*D/g’ cm and the or@ are in Debye units. The quartic terms in (7) are not diagonal in the symmetric top representation in the case of some symmetric-top molecules. In the calculation of the effective dipole moment operator hA’ we must take account of the nondiagonal contribution H,,d’ from II’. The terms in H’ are of two kinds (13). One kind represents the corrections to the rotational energy of Cs and CI,, molecules and may be written as (see also (3, 4))3 ~zmz
cpz, Pzl+l+- ccpz, Pt/l+,CPU, PJ+l+)
Hnd)(Cd= __ 4 (CVZ”- PI/?,
(11)
where [A, B], = A B + BA. The other kind of terms occurs for C-1 and Cd, molecules and is given by
H’(G) = Y[PZ’
P,]+2 - Y(P2
- P,2)2.
(12)
In the case of C5, Cs, Cju, and Cga molecules Hnd’ = 0. It is apparent that no terms in (11) and no nondiagonal elements in (12), except the (-2 1 i- 2) element, contribute to the energies in the first order of approximation. These terms can be eliminated from H’ to the first order of approximation by the contact transformation e+isRH’e-iSR = H (12,14). The dipole moment operator PA’ in (9) is subjected to the same transformation, thereby yielding an effective dipole moment operator PA to first order: 1uA
=
1
pmo{hAm
+
i[sR,
hAa]-}
+
(I The transformation form as (3, 12) SR
=
function
(~mm
c
(13)
e7”%pB~Ay.
a,P,r
SB which removes H,~‘(CG,) can be chosen in operator
/4(&
The terms in Hnd’(C4”) contain e1ement, except the (-21+2)
- Bz)){CPz2,
P,l+ -
(2/3)P,3
-
(1/3)P,}
(14)
only the matrix elements (K 1K&4), and all of these, can also be eliminated by a contact transformation.
8 Extra terms of type s,,~ for CS and Cg groups are excluded in the canonical form of the T tensor (13) used here.
FORBIDDEN
However, the SR functions matris form
ROTATIOSAL
corresponding
SPECTRA
to these elements
21 must
(I< 1Hd 1K * 4) (K1S,<~Kf 4) = i--&*4
&O
0 -
be chosen in the
(15)
where EKO is the rigid-rotor energy. Thus the required transition moments are obtained by taking matrix elements of the effective dipole moment operator pC(Ain (13) between symmetric-top wavefunctions. Transition probabilities for the particular Stark components are proportional to the squares of the matrix elements (JKM j PA / J’K’M’). In the fieldfree molecular rotor, the Stark components are degenerate, and the intensities of the pure rotational lines are proportional to the squares of these matrix elements summed over all M’ values of the final J’ levels and, if the radiation is isotropic, over the three coordinates, _Y, Y, Z. The result is $5
I(J,k’,MIll.~jJ’,K’,M’)12=
where, using the symmetry
[M(G)]2.F(J-.7’,K-K’)
(16)
rules of Henry and Amat (15)4
M(C,,)
= OP(E)
M(C4J
=
+ PzO?zzzz(E)/2(B,
llzo J(B,
-
T.LrUU(BL)
+
- B,)
Tzll,Il(B29
(17)
(18)
&)(
and the expressions for F(_7 - J’, K - K’) are listed in Table I (we have not summed over two-fold K degeneracy in (16) or in Table I). It may be noted that in the case of plane polarized radiation commonly employed in microwave and laser spectroscopy we can choose the coupling vector of the radiation along the Z axis and can drop the X and 1’ components from (16) or we can divide (16) by 3. 3. ROTATIONAL
SPECTRUM
OF CHID
Due to Td symmetry, the CH, molecule has no permanent electric dipole moment. The centrifugally induced dipole moment and corresponding rotational spectrum of CH, was recently considered by Watson (3), Fox (5, 8), and Dorney and Watson (7). When one of the protons is replaced with a deuteron, the symmetry is reduced to Cs,,, and the associated changes in the zero-point vibration and molecular charge distribution lead to the generation of a small permanent electric dipole moment pi’- 5X10P3 D. The allowed rotational spectrum of CHID in the ground vibrational state associated with pzo has been observed in the frequency range from 40 to 120 cm-’ b\. Ozier cl 01. (IQ, at low resolution and with a signal-to-noise ratio of about 10. In order to estimate the absorption coefficient of the centrifugally induced transitions, the parameter M in (17) was calculated from the dipole moment derivatives and force field of Sverdlov (17,18). Depending upon the relative signs of +/aQk, the four possible values of Oz.rzare found and listed in Table II, where the values of all parameters used 4The 6 term for Ce and Cdv groups is zero due to symmetry (15).
22
ALIEV
AND MIKHAYLOV
FORBIDDEN
ROTATIONAL
SPECTRA
23
TABLE II KUNXIAR
species Ql
PARMETER
poll CH3D’
Al
species
Qz
b
E
Q4
Q5
Qb
-
0.169
0.107
0.062
-
0.215
Q3 b
B$
-
0.330
-
0.187
B;=
-
0.026
-
0.565
4.28
5.48
G
-
0.064
BXX k
0.219
Bf
8.48
PI:
exx x
z
-
0.103
-
8.17
3.29
0.278
7.23
exx
exxz
Z’
7
t
f
7
1.02
+
+
+
7
0.28
f
0.27
+ 1.88
?- 0.13
t
0.67
0.47
‘F 0.95
*
7
0.52
t
F
*
i
+
r
7
t
%nits of Debye b
BF
and unita
In the
‘Thin
of
calculation
wl = 2306, and
cm-l,
are
colrmm
of
parameters of
w2 = 3146,
normal-coordinate
for ep
unita
corresponds
these
u)3
7
0.92
+ 2.08
2.42
i
0.64
+ 2.20
p;
=
ag ‘ay
are
(~&j1’2
10
parameters
= 1352,
u4 =
transformation
to
+ 1.48
pi;
(I& -
-5
the 1512, matrix
1,2,3)
(g?Je
1.44
are
lO-2
Debye. ‘harmonic” u+, L are
for
8;
frequenciee
3154,
w6 = 1197
taken
and 8:”
from
or
cm
-1
Sverdlov
to
(17 -,-
18)
cl;: (k = 4,5,6)
and eEZ .
are also given.5 The largest value M = 1.02 X 10p5D corresponds to the preferred sign choice of Sverdlov (17, 18) and will be used here. The formula for the peak absorption coefficient a I,,axf of the centrifugally induced transitions can be obtained from Eq. (3.2i) of Gordy and Cook (19) if (rnl,.~] n)” is replaced by the squared matrix elements M”F(J - J’, K - K’) from (16). Then the formula for cy,*,’ specialized to the CHsD molecule becomes al,laXf(cm-l) = 1.804~10-17v2(g~ug~)g~(1 - 0.00008~) X exp(-0.00016E~~“)F(J
- J’, K - K f
3)
(19)
where c is the symmetry number, gJ = 2J + 1, gk = 1 for K = 0 and 2 for K # 0, gr is the nuclear spin weight factor (19), Yis the line frequency in GHz, and the temperature T = 300°K and line-width parameter Av/P = 1 MHz/torr (cf. (3, 9)) are used; (gkUg1) = 3 for k’ = 3, 6, 9, . . . , and (gkagr) = 3/2 for all other values of K. The 6Secondterm M=e,=s.
in
(17) is neglected for the CHaD molecule, since it is of the order lo-’
Debye, and
9 - 10
13 - 14
2600-2790
2830-3020
2.0 x 10-6
1826.113
7.6 x lO-(j 1.2 x 1o-5
2184.505 2415.421 2686.562 2876.075 3191.887 3420.779 3649.099 3876.810
3-o 3-o o-3 3-o 3-6 3-6 3-6 3-6
2.6 x 1O-5
2.3 x 1O-5
1.8 x 1O-5
5.9 x 10-6
6.2 x 1O-6
4.9 x 10-6
1953.247
3-o 3.4 x 1o-6
1584.169
5-8 6-9
1.1 x 10-6
1338.627
4-7
7.7 x 1o-7 -6 2.0 x 10 -6 1.1 x 10
4.7 x 1o-7
1.9 x 10-7
0. max cm-l
16 -17
15 -16
14 -15
13 -14
12 -13
11 -12
10 -11
9 -10
8-9
7-8
6-7
5-6
4-5
3-4
2-3
l-2
J - 3'
3-3
3-3
3-3
3-3
3-3
3-3
3-3
3-3
3-3
3-3
3-3
3-3
3-3
o-o
o-o
o-o
DK = -23.02 x 10-4 (all in GHz) from Olson (2).
3923.335
3695.827
3467.722
3239.060
3009.876
2780.208
2550.094
2319.570
2088.674
1857.443
1625.914
1394.125
1161.852
930.180
697.765
465.239
GHz
a Y
Allowed Transition K - K'
+h e calculated frequencies were obtained using ground state constants Bx = 116.322, BZ = 157.412, DJ = 15.51 x 10-4 , DJK = 37.09 x 10-4 and
11 - 12
11 - 12
2330-2560
3750-4000
10 - 11
2130-2325
10 - 11
9 - 10
3520-3700
14 - 14
1680-1860
1910-2090
9 - 10
13 - 13
1449-1640
8-9
13 - 13
1200-1400
3290-3470
12 - 12
1050-1110
3090-3240
852.138
2-5
12 - 12 1096.601
609.591
l-4
11 - 11
588-700
830-930 3-6
365.589
o-3
11 - 11
v
355-465
K - K' GHz
J - J'
a
Forbidden Transition
GHz
Range
Frequency
SOME ROTATIONAL TRANSITIONS OF CH3D
TABLE III
2.5 x 10-4
3.6 x 10-4
4.9 x 10-4
6.2 x 10-4
7.9 x 10-4
9.8 x 10-4
1.0 x 1o-3 1.0 x 10-3
1.0 x 10-3
9.0 x 10-4
7.2 x 10-4
5.1 x 10-4
30 x 10-5
15 x 10-5
6.9 x 10-5
2.2 x 10-5
max cm-1
a
FORBIDDEN
ROTATIONAL
23
SPECTRA
in several calculated values of (Y,,& for some of the strongest K-K f 3 transitions frequency ranges are presented in Table III. The peak absorption coefficient (Y,,~ for the allowed transition J, K-J + 1, K of CH,D molecule may be written as (19) alnaxO = 6.844.1OV(l
- 0.00008~)~~ exp(-0.00016E~~“) (J + 1)” - R2 X (&‘gr)
where the dipole matrix element factor M, for an arbitrary the parameters 0 and p” as follows
Jfl
Ma2,
Csu molecule is related
iM, = /.4zo- ez=* + (e,= - 0,= - 20,TZ)K2 + e,=(J + 1)2.
(20) to
(21)
The calculated values of Br”fl corresponding to all sign combination of r+/aQk are given in Table II. Using pzo = 5.64 X 1P3 (ZO), ezzz = -1.48 X 10-5, &** = -2.08 X lo-“, and 0,“” = 0.92 X lop5 (all in Debye units) we obtain the values of (~,,p listed in Table III. Such a choice of e-values provide the correct rotational dependence of pz0 established by Wofsy et al. (20) and discussed previously (21). Moreover, this choice gives a slight K dependence of M, in agreement with the statement of Ozier et al. (16). The results show that a,,,,’ for the strongest lines in the low frequency region (-400 GHz) are of the order lo+ X LY,,~~ but in the high frequency region (-4000 GHz) c&J - 10-l x %z3xQ.One can show that in the region -6000 GHz some of the calculated forbidden transitions of CHSD are stronger than the allowed transitions in the same region. However, vibration-rotation “hot-band” transitions may be expected to occur in this region, although these two types of transitions should be readily distinguishable by their selection rules, by their different temperature dependence and by the different dependence of their intensities on J and K. A rough estimate shows that the absorption coefficients for other deuterated methanes are of the same order of magnitude as for CHID. There seems a possibility that the forbidden rotational transitions of quasipolar molecules like deuterated methanes may be observed even by existing methods (22). ACKNOWLEDGMENTS The authors are grateful to Dr. W. B. Olson for providing the ground state constants of CHID prior to publication and to Dr. J. K. G. Watson for valuable comments on the first draft of the manuscript. RECENED:
February
28, 1973 KEFERENCES
1. G. HERZBERG,“Electronic Spectra of Polyatomic Molecules.” Van Nostrand, Princeton, N. J., 1966.
2. R. H. GARSTANG,“Atomic and Molecular Processes” (D. R. Bates, Ed.) Academic Press, New York and London, 1962. J. K. G. WATSON,J. Mol. Spectrosc. 39, 364 (1971). T. OI(A, F. 0. S~IZU, T. SHI~ZU, AND J. K. G. WATSON,Astrophys. J. 195, L15 (1971). K. Fox, Phys. Rev. Let,!. 27, 233 (1971). M. R. ALIEV, ZhETF Ph. Red. (JETP Lett.) 14, 600 (1971). 7. A. J. DORNEY~AND J. K. G. WATSON, J. Mol. Spectrosc. 42, 135 (1972).
3. 4. 5. 6.
26
ALIEV
AND MIKHAYLOV
8. K. Fox, Pkys. Rev. A, 6, 907 (1972). 9. M. R. ALIEV AND V. M. MIKHAYLOV,Optika i Spektroskopiya (submitted for publication, February 1972). 10. A. ROSENBERG,I. OZIER, AND A. K. KUDIAN,J. Chem. Phys. 57, 568 (1972). 11. H. H. NIELSEN, Rev. Mod. Phys. 23, 90 (1951). 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
M. R. ALIEV ANDV. T. ALEKSANYAN,Optika i Spektroskopiya 24, 520, 695, and 388 (1968). A. P. ALEKSANDROV ANDM. R. ALIEV, J. Mol. Spectrosc., 47, 1 (1973). J. K. G. WATSON,J. Chem. Pkys., 46, 1935 (1967). L. HENRY ANDG. AYAT, Cahiers Phys., 14, 230 (1960). I. OZIER, W. Ho, AND G. BIRNBAUM,J. Ckem. Phys., 51,4873 (1969). L. M. SVERDLOV,Optika i Spektroskopiya, 10, 33 (1961). L. M. SVERDLOV,M. A. KOVNER,AND E. P. KRAINOV, “Vibrational Spectra of Polyatomic Molecules.” Science Press, Moscow, 1970. W. GORDY AND R. L. COOK, “Microwave Molecular Spectra.” pp. 41, 57, 154, Interscience, New York, 1970. S. C. WOFSY, J. S. MUENTER,ANDW. KLEMPERER,J. Chem. Phys. 53, 4005 (1970). V. M. MIKHAYLOVANDM. R. ALIEV, Optika i Spektroskopiya (submitted for publication, July 1972). A. F. KRUPNOV,Submillimeter Spectroscopy of Gases, Preprint, NIRFI, Gorky, USSR, 1972. W. B. OLSON,J. Mol. Spectrosc. 43, 190 (1972).