- Email: [email protected]
An investigation of centrifugal distortion in the microwave spectrum of formamide
An
Investigation the Microwave
of Centrifugal Distortion Spectrum of Formamide
in
Measurements of the microwave spectrum of formamide have been extended in order to account accurately for the effects of centrifugal distortion. A total of 22 new transitions involving J 5 29 have been measured for “NHPCH160 in the ground vibrational state. Combined with previous observations, these transitions have been fit to a model containing five quartic distortion terms and seven sextic terms with a rms deviation of 64 kHz. A large number of resolved quadrupole shifts were fit with an rms deviation of 42 kHz. The remainder of the spectrum for J _< 30 has been calculated with standard deviations less then 3 MHz. Correct weighting of the observed transitions has been found to be important.
The detection of emission signals from interstellar formamide (&H&HO) by Rubin d al. (I) in March 1971 has created a renewed interest in the laboratory microwave spectrum of this molecule. Previous laboratory investigations of formamide by l&-land and Wilson (2) in 1957 and by Costain and Dowling (3) in 1960 have produced accurate structural parameters, quadrupole coupling constants and dipole moment components Aong the molecular principal axes. Recently, Kukolich and Nelson (4) have remeasured the loI+- 000 and the 5,, + 515 transitions using a molecular beam maser spectrometer and have obtained refined values for the quadrupolc coupling constants. The purpose of the work reported in this paper is to extend previous measurements to transitions of possible astrophysical interest. The effects of centrifugal distortion have been accounted for with sufficient accuracy that all transitions of astrophysical interest which could not be measured can now be calculated with low and reliable uncertainti. limits. The microwave spectrum was fit to a model which included both centrifugal distortion and quadrupole coupling in the following w;tJ’. Csin g the previously determined values of the quadrupole coupling constants obtained by Kurland and Wilson, the hypothcticall~, unsplit frequencies of all measured transition were calculated. These h>-pathetic.ally unsplit frequencies were then fit to IVatson’s centrifugal distortion model (5) using I)rocedures described by Kirchhoff (6). The calculated frequencies obtained from thr distortion fit were then subtracted from the observed quadrupole components and these differences (for completely resolved components only) were fit to an expression of tht form Av = x,,Q”
J
., ‘PF’
+
XhhC)b_r.r’FF”
+
%~~~.lJ’b’r.‘,
(1)
-
-
-
-
-
5
40,4
41,4
4 2,3
-
-
4
3
41,3
I,1
-
1
21,2
-
3
-
-
2
1,l I-
o,2
2
2
-
32,2
31,3
30,3
5
4
41,4 3
20,2
Il.1
lo.1
3
2
21.2 1
Do,0
1
-
2
00.0 1
1
-
2
l-
1,o 0
%,l
1
-
2
2
l-
-1
1
2
0
l-
- 11,l -1
l-
11,O 0
18956.091
84542.375
81693.502
0.1
0.13
0.08
0.21
15392.71
84542.40
81693.54
84807.94
04807.686
15392.503
15390.553
0.1
15393.004
CI5391.963)
15390.80
(15392.093)
0.1
0.04
(0.25)
18956.06
15392.88
0.2
40874.91
40875.461
42386.072
0.03
42366.07
4610.966
0.02
4618.970
4617.126
4619.988
0.02
0.02
4619.988
( 4618.557
82549.600
21207.338
21207.932
4617.118
(0.02)
0.17
( 4618.555)
0.02
0.02
21207.922
21207.334
0.1
21206.560
82549.37
(21207.437)
(0.02)
(21207.432)
21206.447
1539.830
0.02
1539.269
1538.675
0.02
0.02
1538.113
0.02
1539.517
1541.002
( 1539.544)
1539.851
1539.295
1538.693
1538.135
0.05
0.02
1541.018
1539.570
(0.02)
( 1539.571)
Table I.
0.055
0.038
0.026
0.207
0.247
-0.124
(0.110)
-0.031
-0.555
-0.002
0.004
-O.OOE
O.OOO
(-0.002)
-0.230
-0.004
-0.010
0.113
(-0.005)
0.021
0.026
0.018
0.022
0.053
0.016
14 N"2%%
1.5
1.9
1.3
1.7
1.5
1.5
d,e
d.e
d,s
d,e
d,e
d,e
(2.0) (1
af
d,f
b
d,S
d.g
J
h.j
J
d,i
d.i
d.i
0.4
0.8
0.3
2.9
3.4
1.7
f
f
f
(0.7) .
-1.3
-3.9
-0.1
0.3
-0.6
0.0
(-0.1) .
-2.0
-0.3
-0.8
1.6
(-0.4) a
Spectrum
(0.026)
nicrovsve
- 112,10
- 10
213,1a - 213.19
19 - 20
192,17 - 201.20 la - 19 20 - 21 >
19 - la
192,18 - 1a3,15 18 - 17 20 - 191
18 - 19
"2.16 - 191.19 17 - la 19 - 20
174.13 - '*3,16
l54,12 - 163.13
103.7
10
a
7 9I
'2,7
9
91.9
- lO1.10 - 11 - 9)
-
9 101.9 11 9
-
8 10
-
-
91.8
:I?
a
-
;
'2.6
Fn the Ground
Sfe.te.*
(37189.37)
(0.04)
0.05
0.05 54058.54
(0.05) 54o61.09
0.08
0.06
(0.06)
0.15
0.15
(0.15)
0.03
0.21
0.15
0.06
0.06
(0.06)
0.04 0.04
(0.020)
(37189.368)
(0.002)
-0.060
-0.044 0.077 54061.134 { 54061.013 54058.6oa
(-0.007) (54o60.247)
0.001
-0.003 -0.o49 36827.493 { 38827.539 38828.419
(-0.018) (38827.8la)
0.067
0.037 0.163
40940.203 ( 40940.077 40937.693
(0.088)
-0.010
0.136
0.073
(40939.322)
51208.510
84384.614
82578.367
-0.011
0.057 -0.145
84073.703 t 84073.905 84071.691
(-0.053)
0.039
0.070 -0.033
(0.019)
0.008
-0.066 0.110
(84073.093)
37566.901
37567.740 37567.843
(37567.491)
37259.732
0.04
(0.01)
37261.506 t 37261.330
(37260.850)
0.04
(0.04)
(54O60.24)
38828.42
38827.49
(38827.80)
40937.76
40940.24
(40939.41)
51208.50
84384.75
82578.44
84071.68
84073.76
(84073.04)
37566.94
37567.81
(37567.51)
37259.74
37261.44
(37260.87)
Vibrational
d
d
d
d
d (1.2) .
-1.7
-1.2 2.1
(-1.5) *
0.0
-0.1 1.1
(-0.7) .
0.6
0.3 1.5
(1.6) .
-2.3
2.1
1.0
-0.2
(-2.1) . 1.3 -3.3
1.4
2.4 -1.1
(1.2)
0.3
d
(1.2) . -2.1 3.0
5
r '% z
LI
c'
i;
-
$ c 7
z c
z
-
-
61.5 5 7
6
7
‘1.6 6\ a
6
6 1,6 51 78
6
5
51,5 4
4
0.02
(0.25)
0.1
0.1
(0.1)
23081.762
(32297.451)
3229.9.19
32297.u
(16961.25)
Ihie
see
Reported
Reported
Included
0.
ia
the
b,
3.
the
16961.766
by C. C. Co)staln
and J.
al.,
C.
Nclm,,,
B. Wilson,
1.
M. Dwling,
ref.
distortion
end A.
and E.
Padford,
fit
ref.
for
ref.
Jr.,
ref.
0.102 -0.025
(-0.036)
2.
See
text far
(voba-vcale
3.
ta
the
) ld
1.4 -0.3
(0.6)
18.5
0.2 4.8
(1.1)
-0.6
-0.5
0.5
(-0.3)
0.5
-4.4 3.4
(-0.4)
16.9
0.4
0.3
f f
.
f
f f
.
d.8
d.6
d,g
1
d
a
f.h
im not
text.
diacu.aio,,
of
286,22
“5,25
24
25 23 1
243,21
244,21
20
21 191
203.17
21
the of
numbers.
(0.05)
(52890.38)
it
estimated deviations
ie
quantum
51087.42
54471.51
52891.38
by which
frm
~oodoesl, voba-vcalc
the
fit. differe
of
unsplit
51087.420
54471.513
frw
ita
0.2
value
uncertainty.
expected
frequency,
0.000
-0.003
0.007
0.6 -0.8 0.025 -0.033
52889.855 r 52889.913 51891.373
(-0.4)
1.8
1.0
(-0.001)
0.004
0.7 -0.7
(0.7)
0.1
-0.3 0.7
(52890.381)
40665.806
0.037
0.025 0.026
53131.235 53131.286 53132.363
(0.011)
0.004
-0.008 0.019
(53131.629)
37188.976
37189.578 437189.551
The hypothetically
0.04
0.05
0.06
0.06
0.03
52809.80
0.05
40655.81
0.05
53131.26 53132.40
0.04
(0.05)
0.04
37188.90
37189.57
(53131.64)
ltandrrd
authora,
traaitiom.
psmber
by
asymmetric rotor qu.ntum lu&ers.
- 295,24
-
25 -
286,23
theee
-
26 24 --
2s2,24
23 5.18
21 -
22 20 -
-
21 21 2.20
the
provided
tbum repre.eat.
quantity
further
Uhen this
rezal~on, described
4.
ref.
8.
of
frequency.
deviatim
me.sured
mtand.rd
the
and H. E.
fit.
cantrifugal
Kukolich
Kvrlend
Cottlieb
by R. H. Rubin .-_et
in
S. C.
by R.
in
) divided by the rther discussion.
u.cert.inty
quadzupole
the
by C. A.
included
Reported
llot
J Reported
i
h
%parCed
f
e
d
(v tesP&h
represents
Ct(A”) -
b
(16961.212)
16960.858 r 16960.985
1.338
0.013 0.350
32295.642
(0.182)
32298.177 r 32297.040
-0.009
-0.007
0.007
(-0.003)
(32297.269)
23081.771
23079.792
23082.174
(23081.221)
0.013
-0.129 0.100
50693.157
(-0.009)
50694.099 r 50693.870
1.151
0.021
0.014
(50693.669)
26922.699
84891.130
84889.136
resolved quadrupole etruclmr8, the P quantum numbers are given belm the etc., are given in pmra~~tlmmm 011 the mame lime (LB the anymetric rotor
0.1
0.02
23079.785
16960.96
0.02
23082.181
0.04
50693.97
0.04
(0.02)
(50693.68)
(0.02)
0.10
26924.05
50693.17
0.10
84891.15
(23081.218)
0.10
84889.15
‘For transiticms with calculated frequency,
-
-
6
6
-
5
-
-
51.4 4
5 7
-
5
%,5
-
50.5 4 6
41.4 3; 5,
31,3
-
33,l
40,4
-
33,o
3,2
(13.1
4
d
.
d
l
d
of
c
” =
z :,
z
/, T
L h
2
KIKCHHOI;I“
162
AND JOHNSON
where
= [2(Y,2).Ifl/J’(J’ Q"JJtFL't
+ l)]j((I,
J’, f;‘) - [2(P,x)J/J(J
+ l)]f(r,
J, f;).
(2j
In Eq. (2), (Y,“).P and (Y,“)J are the expectation values of the square of the angular momentum operator component along the g principal axis (g = a, b, c) for the two rotational states involved in the transition J’K,,,K~~ +- JK+, and f(1, J, I;) is Casimir’s function as defined in Appendis T of Townes and Schawlow (7), viz., f(1, J,F)
= [(3/4)C(C
+ 1) - 1(1+
l)J(J
+ 1)]/21(21
- 1)(2J - 1)(2J + 31,
(3)
where c’ = Z+‘(i;+ 1) - I(1 + 1) - J(J + 1).
(1)
In performing the least-squares fit of the quadrupole shifts, the constraint of Laplace’s equation, C, X,, = 0, was imposed so that the quadrupole coupling constants, X,, and X,,, were obtained by fitting Av = &~(Q=.IJ~FP~ - Q*JPFF,) + x<.c(Qc~.,,~,~”- Q*,,PFP>.
(5)
Once retined values for the quadrupole coupling constants were obtained, improved values for the hypothetically unsplit frequencies were calculated and the entire procedure was repeated. Because the observed transitions were obtained from a variety of sources, the assumption of equally probable measurement errors for each transition could not be made. The results of a fit in which each transition was weighted by the inverse square of its measurement uncertainty were compared with the results of a fit in which all transitions were given equal weighting. The differences between the calculated transition frequencies obtained from the two fits were greater than two standard deviations (of the calculated frequencies) in many cases. Newly measured transitions were found to agree better with the frequencies calculated from the nonuniformly weighted fit. In a few instances, transition frequencies were reported in the literature without corresponding estimates of the measurement uncertainty. For these transitions, uncertainties were estimated from the values of AV = v,,bsd - v,,led obtained from a fit using uniform weighting. There were two exceptions to this procedure. The measurements of Kukolich and Nelson (4) using beam maser techniques gave measured frequencies of sufficient accuracy to account for the effects of spin-rotation interactions. The spin of the nitrogen nucleus can interact with the overall rotation of the molecule through its magnetic moment as well as its electric quadrupole moment. This magnetic interaction is typically two orders of magnitude smaller than the electric quadrupole interaction and manifests itself as a shift of the hyperfine pattern from that expected for an electric quadrupole interaction alone. In addition, Kukolich and Nelson have tenatively assigned an observed spectral line as a component of a doublet arising from an interaction of the spin of one of the hydrogen atoms with the overall rotation of the molecule. Unfortunately, the number of transitions measured was insufficient to allow for the effects of spinrotation to be calculated for the remainder of the spectrum. Thus, in weighting the Kukolich and Nelson data, uncertainties of 20 kHz, representative of the effects of the spin-rotation interaction, were used rather than the measurement uncertainties which were on the order nf 0.5 kHz.
A second exception to the weighting scheme described above was the exclusion from the fit of the 11,1c O,,“, 2,.! +- 11,1 and 4,,., c- 31,a transitions. The hyperfine splitting was not resolved, but the observed line shape did nol of the 1,,1 +- O0,0 transition duplicate the calculated shape indicating that this transition was overlapped by another line. Hence, the frequency measurement, made at the peak intensity of the transition. was an unreliable representation of the true frequcnc!, of t hc 11,1+- Ocl,otransition evc’n though the observed frequent)- differed from the calculated frequency b!- only 0.2.3 MHz. (The uncertainty in the measurement of the peak frequency was 0.17 MHz. 1 +11,1 transition reported by Costain and I)owSimilar comments also apply to the 21.2 t 31,3 transition, reported b!, Kurland and MYson, differed from the calling. The &,,r culated value I,!, more than 1 MHz and it was assumed that the frequency for this t txnsition was reported incorrectly. The measured microwave spectrum of formamide is presented in Table I. The entries in this table :tre self-explanatory with the exception of /(Au,) which is the value of &I,( = v.,~,~~, ~,:,~~.tf for the i-th transition) divided by ai&;), the standard deviation of L,. I;or those transitions included in the tit, ~(AY,) = {S?/ll’, - CT+, ,,il,edj}:, m:here II’, is the weight given to the i-th transition in the L&-squares tit, 9 is the standard deviation of the fit itself and G(Y~ cnlcil) is the standard deviation of the calculated frequency of the i-th transition. For transitions not included in the tit B(AY,) = (.Y?,,%~,+ u”(Y, r:,,<<,))I. The quantity i(&;) follows a Student’s t distribution and c’an be interpreted as the nunber of standard deviations b!- which Av, differs from its expected value of 0. In the situation of a weighted fit, the standard deviation is the espected error of a measurement with unit weight. If the standard deviation is less than unity, then the probable error for each transition as estimated by the standard deviation is, on the average, less than the uncertainty assigned to that transition. For the formamide data the standard deviation of the weighted fit was 0.672 MHz indicating that the uncertainties used in the weightings and reported in Table I were somewhat overestimated. The rms deviation for those transitions included in the fit was 61 kHz for the centrifugal distortion calculation and 34 kHz for the quadrupole calculation. The derived rotational and centrifugal distortion constants arc presented in Table [I. ‘l‘hc, number of significant figures quoted for each constant is such that round-off errors in each cnlcuIated transition frequency will be less than the calculated standard devialion for that frequency. It should be noted that the number of significant figures quotetl for each parameter is two or three greater than required by the standard deviation of that {x~rumeter, i.e., the necessary precision appears to be two or three orders of magnitude greater than the accuracy. This is required by the fact that the errors in the reported parameters are highly correlated so that certain linear combinations of the parameters are better determined than the parameters themseIves. In particular, this “exaggerated” precision is absolutely necessary for the calculation of the predicted transitions. Using the parameters of Table II it is possible to calculate all of the transition frequencies of I hc normal isc)(ol)ic sptv.itLs nf form;~rnid~ in t ht. ~TPII~CI \~il)r;\tinn;ll st:ltts hctween 1 OOC)
164
KIKCHHOJ~I: AND JOHNSOK Table II. The RotationalConstantsof Fonnamide
Watson's DeterminableParameters=
Kivelson-WilsonParametersC
A"
=
72716.9496to.022 MHZ
A'
= 72716.93520.022 MHZ
B"
=
11373.4541to.0039 MHZ
B'
= 11373.509to.004 MHZ
C"
=
C'
=
T1
=
T2
=
r3
=
9833.90416fO.0036 MHZ
188.353 ? 3.1 kHz
Tibcc = 7' = ccaa
2.8776 + 0.44 kHz 1701 ? 12 kHzb
'Aabb =
9833.957 to.003 MHZ
-27.85 kO.17 kHz 110.1
t 1.1 kHz
106.1
? 2.0 kHz
7aaaa = -5373.00f 15 kHz 'bbbb = = Tcccc HK HJK %I HK hJ hJK hK
-44.0702+ 0.20 km -18.9373+ 0.17 kHz
= (-0.9815+2) x 1o-7 MHZ = (+0.5787?6) x lO'6 MHZ = (-0.3260+O.S) ~10'~ MHZ = (+0.2454f 0.4) ~10-~ MHZ = (+0.899t4) x1o-8 MHZ = (-0.1972+0.3)~10-'MHZ = (+1.01221.0) x10-4 MHZ
a7=7
CCCC
-T2
-
z&
(~~-7~)
=
-0.12 +0.07 kHz
Axnming planarity,the followingsets of constantswere calculated: 'aabb = 279 + 2 kHz (from 71 and TV); = 274 f 4 kHz (from 71 or 72 and 7cccc) 'abab - -86.3 +_0.6 kHz (from 71 and 72); = -82.3 -I2.4 kHz (from 71 and 7cccc) = -81.8 ? 2.7 kH.z(from ~2 and 7cccc) A = 72716.892k0.022 MHz
Ia =
B = 11373.466to.004 MHz
Ib = 44.43597+0.00002 ux2
c =
Xc = 51.39210?0.00002 ug2
9834.02220.003 MHZ
A
=
6.95018 ?0.000002ui2e
0.00601 +O.OOOOl ug2
* The number of significant figures quoted are necessary to reproduce all the calculated frequencies for J = 1 -+ 30 and 1 GHz < P < 90 GHz, within their standard deviations. h The value of 7~ is srt using the planarity conditions and is not, strictly speaking, a deternlinable parameter. c These parameters arc claculate~l from .\“, IS”, C“‘, 71, Q and ra and thus oljey the planarit>conditions used to calculate 73. d LC’atson (5) uses 2hJ for this ~~arumctcr. * The conversion factor = 505391 .O.
and !JO 000 MHz
for .I
XSyo of these transitions
2 31 with an accuracy
of better than 2 ,\I Hz and 11)~~~lculat~~ to better than 0.5 MHz. Copies of the raIclIIated spectrum nl:+!
be obtained from the authors on request. ICinally, the values of the quadrupole couplin, (r UJnstatltS X IL,, =
l.c)iOi & 0.03 MHz,
X ,,* =
1.8i22 k 0.02 .\IHz,
x,, = -3.8.510 ;IS ~omp;tr~l
with the values obtained
f
by Kukolich
;~nd Nelson:
0.002 .IIHz,
1.888 f
0.003 MHz,
x ,.,. = -3.848
from thr tit werx
0.02 .\IHz.
Y,,,, = 1.060 f X I,,, =
ot)tained
-c 0.004 .\lHz
(It should be noted that the values of the X’s given b!, Kukolich and Nelson are in agreement with the values of the frequencies which the!- report in the texit, hut that the intermediate parameters the\- report in ‘Table I I ;tre not. 1
The authors measurements
2. .?. 4. i. 0. 7.
would like to express their gratituclr to C. .i. Gr)ttliel) ant1 H. I-. Katlford of the hvperfine pattern of the I 11o - I ,., transition.
I~nion Circ. So. 2319, 1971. I<. J. RUKLAKD AND E. H. WILSON, JR., J. C‘l~enl. Pirys. 27, .%S il9.iil. (‘. (‘. COSTMK AND J. M. DOWLING, J. C’lretn. Ploys. 32, 290 (1960~. S. G. Kukolich and 4. C. NELSON, Clxm. Plrys. /Al. 11. 383 ( 1971 I. J. R. G. WATSON, J. Chew Ploys. 48, 4517 (19681. LV. H. KIRCHHOFF, J. Vol. Spectrosc. 41, 333 (19i2). (‘. H. TOWNES ,~h‘n ;\. I,. %CHAWLOR., “IRIicrowave SJW~ ~0s~ OJ)J.,“ 1). -409. 10.55.
Md;raw
for
I)rovitlin::
Hill. Sew \~ork.
Investigation the Microwave
of Centrifugal Distortion Spectrum of Formamide
in
Measurements of the microwave spectrum of formamide have been extended in order to account accurately for the effects of centrifugal distortion. A total of 22 new transitions involving J 5 29 have been measured for “NHPCH160 in the ground vibrational state. Combined with previous observations, these transitions have been fit to a model containing five quartic distortion terms and seven sextic terms with a rms deviation of 64 kHz. A large number of resolved quadrupole shifts were fit with an rms deviation of 42 kHz. The remainder of the spectrum for J _< 30 has been calculated with standard deviations less then 3 MHz. Correct weighting of the observed transitions has been found to be important.
The detection of emission signals from interstellar formamide (&H&HO) by Rubin d al. (I) in March 1971 has created a renewed interest in the laboratory microwave spectrum of this molecule. Previous laboratory investigations of formamide by l&-land and Wilson (2) in 1957 and by Costain and Dowling (3) in 1960 have produced accurate structural parameters, quadrupole coupling constants and dipole moment components Aong the molecular principal axes. Recently, Kukolich and Nelson (4) have remeasured the loI+- 000 and the 5,, + 515 transitions using a molecular beam maser spectrometer and have obtained refined values for the quadrupolc coupling constants. The purpose of the work reported in this paper is to extend previous measurements to transitions of possible astrophysical interest. The effects of centrifugal distortion have been accounted for with sufficient accuracy that all transitions of astrophysical interest which could not be measured can now be calculated with low and reliable uncertainti. limits. The microwave spectrum was fit to a model which included both centrifugal distortion and quadrupole coupling in the following w;tJ’. Csin g the previously determined values of the quadrupole coupling constants obtained by Kurland and Wilson, the hypothcticall~, unsplit frequencies of all measured transition were calculated. These h>-pathetic.ally unsplit frequencies were then fit to IVatson’s centrifugal distortion model (5) using I)rocedures described by Kirchhoff (6). The calculated frequencies obtained from thr distortion fit were then subtracted from the observed quadrupole components and these differences (for completely resolved components only) were fit to an expression of tht form Av = x,,Q”
J
., ‘PF’
+
XhhC)b_r.r’FF”
+
%~~~.lJ’b’r.‘,
(1)
-
-
-
-
-
5
40,4
41,4
4 2,3
-
-
4
3
41,3
I,1
-
1
21,2
-
3
-
-
2
1,l I-
o,2
2
2
-
32,2
31,3
30,3
5
4
41,4 3
20,2
Il.1
lo.1
3
2
21.2 1
Do,0
1
-
2
00.0 1
1
-
2
l-
1,o 0
%,l
1
-
2
2
l-
-1
1
2
0
l-
- 11,l -1
l-
11,O 0
18956.091
84542.375
81693.502
0.1
0.13
0.08
0.21
15392.71
84542.40
81693.54
84807.94
04807.686
15392.503
15390.553
0.1
15393.004
CI5391.963)
15390.80
(15392.093)
0.1
0.04
(0.25)
18956.06
15392.88
0.2
40874.91
40875.461
42386.072
0.03
42366.07
4610.966
0.02
4618.970
4617.126
4619.988
0.02
0.02
4619.988
( 4618.557
82549.600
21207.338
21207.932
4617.118
(0.02)
0.17
( 4618.555)
0.02
0.02
21207.922
21207.334
0.1
21206.560
82549.37
(21207.437)
(0.02)
(21207.432)
21206.447
1539.830
0.02
1539.269
1538.675
0.02
0.02
1538.113
0.02
1539.517
1541.002
( 1539.544)
1539.851
1539.295
1538.693
1538.135
0.05
0.02
1541.018
1539.570
(0.02)
( 1539.571)
Table I.
0.055
0.038
0.026
0.207
0.247
-0.124
(0.110)
-0.031
-0.555
-0.002
0.004
-O.OOE
O.OOO
(-0.002)
-0.230
-0.004
-0.010
0.113
(-0.005)
0.021
0.026
0.018
0.022
0.053
0.016
14 N"2%%
1.5
1.9
1.3
1.7
1.5
1.5
d,e
d.e
d,s
d,e
d,e
d,e
(2.0) (1
af
d,f
b
d,S
d.g
J
h.j
J
d,i
d.i
d.i
0.4
0.8
0.3
2.9
3.4
1.7
f
f
f
(0.7) .
-1.3
-3.9
-0.1
0.3
-0.6
0.0
(-0.1) .
-2.0
-0.3
-0.8
1.6
(-0.4) a
Spectrum
(0.026)
nicrovsve
- 112,10
- 10
213,1a - 213.19
19 - 20
192,17 - 201.20 la - 19 20 - 21 >
19 - la
192,18 - 1a3,15 18 - 17 20 - 191
18 - 19
"2.16 - 191.19 17 - la 19 - 20
174.13 - '*3,16
l54,12 - 163.13
103.7
10
a
7 9I
'2,7
9
91.9
- lO1.10 - 11 - 9)
-
9 101.9 11 9
-
8 10
-
-
91.8
:I?
a
-
;
'2.6
Fn the Ground
Sfe.te.*
(37189.37)
(0.04)
0.05
0.05 54058.54
(0.05) 54o61.09
0.08
0.06
(0.06)
0.15
0.15
(0.15)
0.03
0.21
0.15
0.06
0.06
(0.06)
0.04 0.04
(0.020)
(37189.368)
(0.002)
-0.060
-0.044 0.077 54061.134 { 54061.013 54058.6oa
(-0.007) (54o60.247)
0.001
-0.003 -0.o49 36827.493 { 38827.539 38828.419
(-0.018) (38827.8la)
0.067
0.037 0.163
40940.203 ( 40940.077 40937.693
(0.088)
-0.010
0.136
0.073
(40939.322)
51208.510
84384.614
82578.367
-0.011
0.057 -0.145
84073.703 t 84073.905 84071.691
(-0.053)
0.039
0.070 -0.033
(0.019)
0.008
-0.066 0.110
(84073.093)
37566.901
37567.740 37567.843
(37567.491)
37259.732
0.04
(0.01)
37261.506 t 37261.330
(37260.850)
0.04
(0.04)
(54O60.24)
38828.42
38827.49
(38827.80)
40937.76
40940.24
(40939.41)
51208.50
84384.75
82578.44
84071.68
84073.76
(84073.04)
37566.94
37567.81
(37567.51)
37259.74
37261.44
(37260.87)
Vibrational
d
d
d
d
d (1.2) .
-1.7
-1.2 2.1
(-1.5) *
0.0
-0.1 1.1
(-0.7) .
0.6
0.3 1.5
(1.6) .
-2.3
2.1
1.0
-0.2
(-2.1) . 1.3 -3.3
1.4
2.4 -1.1
(1.2)
0.3
d
(1.2) . -2.1 3.0
5
r '% z
LI
c'
i;
-
$ c 7
z c
z
-
-
61.5 5 7
6
7
‘1.6 6\ a
6
6 1,6 51 78
6
5
51,5 4
4
0.02
(0.25)
0.1
0.1
(0.1)
23081.762
(32297.451)
3229.9.19
32297.u
(16961.25)
Ihie
see
Reported
Reported
Included
0.
ia
the
b,
3.
the
16961.766
by C. C. Co)staln
and J.
al.,
C.
Nclm,,,
B. Wilson,
1.
M. Dwling,
ref.
distortion
end A.
and E.
Padford,
fit
ref.
for
ref.
Jr.,
ref.
0.102 -0.025
(-0.036)
2.
See
text far
(voba-vcale
3.
ta
the
) ld
1.4 -0.3
(0.6)
18.5
0.2 4.8
(1.1)
-0.6
-0.5
0.5
(-0.3)
0.5
-4.4 3.4
(-0.4)
16.9
0.4
0.3
f f
.
f
f f
.
d.8
d.6
d,g
1
d
a
f.h
im not
text.
diacu.aio,,
of
286,22
“5,25
24
25 23 1
243,21
244,21
20
21 191
203.17
21
the of
numbers.
(0.05)
(52890.38)
it
estimated deviations
ie
quantum
51087.42
54471.51
52891.38
by which
frm
~oodoesl, voba-vcalc
the
fit. differe
of
unsplit
51087.420
54471.513
frw
ita
0.2
value
uncertainty.
expected
frequency,
0.000
-0.003
0.007
0.6 -0.8 0.025 -0.033
52889.855 r 52889.913 51891.373
(-0.4)
1.8
1.0
(-0.001)
0.004
0.7 -0.7
(0.7)
0.1
-0.3 0.7
(52890.381)
40665.806
0.037
0.025 0.026
53131.235 53131.286 53132.363
(0.011)
0.004
-0.008 0.019
(53131.629)
37188.976
37189.578 437189.551
The hypothetically
0.04
0.05
0.06
0.06
0.03
52809.80
0.05
40655.81
0.05
53131.26 53132.40
0.04
(0.05)
0.04
37188.90
37189.57
(53131.64)
ltandrrd
authora,
traaitiom.
psmber
by
asymmetric rotor qu.ntum lu&ers.
- 295,24
-
25 -
286,23
theee
-
26 24 --
2s2,24
23 5.18
21 -
22 20 -
-
21 21 2.20
the
provided
tbum repre.eat.
quantity
further
Uhen this
rezal~on, described
4.
ref.
8.
of
frequency.
deviatim
me.sured
mtand.rd
the
and H. E.
fit.
cantrifugal
Kukolich
Kvrlend
Cottlieb
by R. H. Rubin .-_et
in
S. C.
by R.
in
) divided by the rther discussion.
u.cert.inty
quadzupole
the
by C. A.
included
Reported
llot
J Reported
i
h
%parCed
f
e
d
(v tesP&h
represents
Ct(A”) -
b
(16961.212)
16960.858 r 16960.985
1.338
0.013 0.350
32295.642
(0.182)
32298.177 r 32297.040
-0.009
-0.007
0.007
(-0.003)
(32297.269)
23081.771
23079.792
23082.174
(23081.221)
0.013
-0.129 0.100
50693.157
(-0.009)
50694.099 r 50693.870
1.151
0.021
0.014
(50693.669)
26922.699
84891.130
84889.136
resolved quadrupole etruclmr8, the P quantum numbers are given belm the etc., are given in pmra~~tlmmm 011 the mame lime (LB the anymetric rotor
0.1
0.02
23079.785
16960.96
0.02
23082.181
0.04
50693.97
0.04
(0.02)
(50693.68)
(0.02)
0.10
26924.05
50693.17
0.10
84891.15
(23081.218)
0.10
84889.15
‘For transiticms with calculated frequency,
-
-
6
6
-
5
-
-
51.4 4
5 7
-
5
%,5
-
50.5 4 6
41.4 3; 5,
31,3
-
33,l
40,4
-
33,o
3,2
(13.1
4
d
.
d
l
d
of
c
” =
z :,
z
/, T
L h
2
KIKCHHOI;I“
162
AND JOHNSON
where
= [2(Y,2).Ifl/J’(J’ Q"JJtFL't
+ l)]j((I,
J’, f;‘) - [2(P,x)J/J(J
+ l)]f(r,
J, f;).
(2j
In Eq. (2), (Y,“).P and (Y,“)J are the expectation values of the square of the angular momentum operator component along the g principal axis (g = a, b, c) for the two rotational states involved in the transition J’K,,,K~~ +- JK+, and f(1, J, I;) is Casimir’s function as defined in Appendis T of Townes and Schawlow (7), viz., f(1, J,F)
= [(3/4)C(C
+ 1) - 1(1+
l)J(J
+ 1)]/21(21
- 1)(2J - 1)(2J + 31,
(3)
where c’ = Z+‘(i;+ 1) - I(1 + 1) - J(J + 1).
(1)
In performing the least-squares fit of the quadrupole shifts, the constraint of Laplace’s equation, C, X,, = 0, was imposed so that the quadrupole coupling constants, X,, and X,,, were obtained by fitting Av = &~(Q=.IJ~FP~ - Q*JPFF,) + x<.c(Qc~.,,~,~”- Q*,,PFP>.
(5)
Once retined values for the quadrupole coupling constants were obtained, improved values for the hypothetically unsplit frequencies were calculated and the entire procedure was repeated. Because the observed transitions were obtained from a variety of sources, the assumption of equally probable measurement errors for each transition could not be made. The results of a fit in which each transition was weighted by the inverse square of its measurement uncertainty were compared with the results of a fit in which all transitions were given equal weighting. The differences between the calculated transition frequencies obtained from the two fits were greater than two standard deviations (of the calculated frequencies) in many cases. Newly measured transitions were found to agree better with the frequencies calculated from the nonuniformly weighted fit. In a few instances, transition frequencies were reported in the literature without corresponding estimates of the measurement uncertainty. For these transitions, uncertainties were estimated from the values of AV = v,,bsd - v,,led obtained from a fit using uniform weighting. There were two exceptions to this procedure. The measurements of Kukolich and Nelson (4) using beam maser techniques gave measured frequencies of sufficient accuracy to account for the effects of spin-rotation interactions. The spin of the nitrogen nucleus can interact with the overall rotation of the molecule through its magnetic moment as well as its electric quadrupole moment. This magnetic interaction is typically two orders of magnitude smaller than the electric quadrupole interaction and manifests itself as a shift of the hyperfine pattern from that expected for an electric quadrupole interaction alone. In addition, Kukolich and Nelson have tenatively assigned an observed spectral line as a component of a doublet arising from an interaction of the spin of one of the hydrogen atoms with the overall rotation of the molecule. Unfortunately, the number of transitions measured was insufficient to allow for the effects of spinrotation to be calculated for the remainder of the spectrum. Thus, in weighting the Kukolich and Nelson data, uncertainties of 20 kHz, representative of the effects of the spin-rotation interaction, were used rather than the measurement uncertainties which were on the order nf 0.5 kHz.
A second exception to the weighting scheme described above was the exclusion from the fit of the 11,1c O,,“, 2,.! +- 11,1 and 4,,., c- 31,a transitions. The hyperfine splitting was not resolved, but the observed line shape did nol of the 1,,1 +- O0,0 transition duplicate the calculated shape indicating that this transition was overlapped by another line. Hence, the frequency measurement, made at the peak intensity of the transition. was an unreliable representation of the true frequcnc!, of t hc 11,1+- Ocl,otransition evc’n though the observed frequent)- differed from the calculated frequency b!- only 0.2.3 MHz. (The uncertainty in the measurement of the peak frequency was 0.17 MHz. 1 +11,1 transition reported by Costain and I)owSimilar comments also apply to the 21.2 t 31,3 transition, reported b!, Kurland and MYson, differed from the calling. The &,,r culated value I,!, more than 1 MHz and it was assumed that the frequency for this t txnsition was reported incorrectly. The measured microwave spectrum of formamide is presented in Table I. The entries in this table :tre self-explanatory with the exception of /(Au,) which is the value of &I,( = v.,~,~~, ~,:,~~.tf for the i-th transition) divided by ai&;), the standard deviation of L,. I;or those transitions included in the tit, ~(AY,) = {S?/ll’, - CT+, ,,il,edj}:, m:here II’, is the weight given to the i-th transition in the L&-squares tit, 9 is the standard deviation of the fit itself and G(Y~ cnlcil) is the standard deviation of the calculated frequency of the i-th transition. For transitions not included in the tit B(AY,) = (.Y?,,%~,+ u”(Y, r:,,<<,))I. The quantity i(&;) follows a Student’s t distribution and c’an be interpreted as the nunber of standard deviations b!- which Av, differs from its expected value of 0. In the situation of a weighted fit, the standard deviation is the espected error of a measurement with unit weight. If the standard deviation is less than unity, then the probable error for each transition as estimated by the standard deviation is, on the average, less than the uncertainty assigned to that transition. For the formamide data the standard deviation of the weighted fit was 0.672 MHz indicating that the uncertainties used in the weightings and reported in Table I were somewhat overestimated. The rms deviation for those transitions included in the fit was 61 kHz for the centrifugal distortion calculation and 34 kHz for the quadrupole calculation. The derived rotational and centrifugal distortion constants arc presented in Table [I. ‘l‘hc, number of significant figures quoted for each constant is such that round-off errors in each cnlcuIated transition frequency will be less than the calculated standard devialion for that frequency. It should be noted that the number of significant figures quotetl for each parameter is two or three greater than required by the standard deviation of that {x~rumeter, i.e., the necessary precision appears to be two or three orders of magnitude greater than the accuracy. This is required by the fact that the errors in the reported parameters are highly correlated so that certain linear combinations of the parameters are better determined than the parameters themseIves. In particular, this “exaggerated” precision is absolutely necessary for the calculation of the predicted transitions. Using the parameters of Table II it is possible to calculate all of the transition frequencies of I hc normal isc)(ol)ic sptv.itLs nf form;~rnid~ in t ht. ~TPII~CI \~il)r;\tinn;ll st:ltts hctween 1 OOC)
164
KIKCHHOJ~I: AND JOHNSOK Table II. The RotationalConstantsof Fonnamide
Watson's DeterminableParameters=
Kivelson-WilsonParametersC
A"
=
72716.9496to.022 MHZ
A'
= 72716.93520.022 MHZ
B"
=
11373.4541to.0039 MHZ
B'
= 11373.509to.004 MHZ
C"
=
C'
=
T1
=
T2
=
r3
=
9833.90416fO.0036 MHZ
188.353 ? 3.1 kHz
Tibcc = 7' = ccaa
2.8776 + 0.44 kHz 1701 ? 12 kHzb
'Aabb =
9833.957 to.003 MHZ
-27.85 kO.17 kHz 110.1
t 1.1 kHz
106.1
? 2.0 kHz
7aaaa = -5373.00f 15 kHz 'bbbb = = Tcccc HK HJK %I HK hJ hJK hK
-44.0702+ 0.20 km -18.9373+ 0.17 kHz
= (-0.9815+2) x 1o-7 MHZ = (+0.5787?6) x lO'6 MHZ = (-0.3260+O.S) ~10'~ MHZ = (+0.2454f 0.4) ~10-~ MHZ = (+0.899t4) x1o-8 MHZ = (-0.1972+0.3)~10-'MHZ = (+1.01221.0) x10-4 MHZ
a7=7
CCCC
-T2
-
z&
(~~-7~)
=
-0.12 +0.07 kHz
Axnming planarity,the followingsets of constantswere calculated: 'aabb = 279 + 2 kHz (from 71 and TV); = 274 f 4 kHz (from 71 or 72 and 7cccc) 'abab - -86.3 +_0.6 kHz (from 71 and 72); = -82.3 -I2.4 kHz (from 71 and 7cccc) = -81.8 ? 2.7 kH.z(from ~2 and 7cccc) A = 72716.892k0.022 MHz
Ia =
B = 11373.466to.004 MHz
Ib = 44.43597+0.00002 ux2
c =
Xc = 51.39210?0.00002 ug2
9834.02220.003 MHZ
A
=
6.95018 ?0.000002ui2e
0.00601 +O.OOOOl ug2
* The number of significant figures quoted are necessary to reproduce all the calculated frequencies for J = 1 -+ 30 and 1 GHz < P < 90 GHz, within their standard deviations. h The value of 7~ is srt using the planarity conditions and is not, strictly speaking, a deternlinable parameter. c These parameters arc claculate~l from .\“, IS”, C“‘, 71, Q and ra and thus oljey the planarit>conditions used to calculate 73. d LC’atson (5) uses 2hJ for this ~~arumctcr. * The conversion factor = 505391 .O.
and !JO 000 MHz
for .I
XSyo of these transitions
2 31 with an accuracy
of better than 2 ,\I Hz and 11)~~~lculat~~ to better than 0.5 MHz. Copies of the raIclIIated spectrum nl:+!
be obtained from the authors on request. ICinally, the values of the quadrupole couplin, (r UJnstatltS X IL,, =
l.c)iOi & 0.03 MHz,
X ,,* =
1.8i22 k 0.02 .\IHz,
x,, = -3.8.510 ;IS ~omp;tr~l
with the values obtained
f
by Kukolich
;~nd Nelson:
0.002 .IIHz,
1.888 f
0.003 MHz,
x ,.,. = -3.848
from thr tit werx
0.02 .\IHz.
Y,,,, = 1.060 f X I,,, =
ot)tained
-c 0.004 .\lHz
(It should be noted that the values of the X’s given b!, Kukolich and Nelson are in agreement with the values of the frequencies which the!- report in the texit, hut that the intermediate parameters the\- report in ‘Table I I ;tre not. 1
The authors measurements
2. .?. 4. i. 0. 7.
would like to express their gratituclr to C. .i. Gr)ttliel) ant1 H. I-. Katlford of the hvperfine pattern of the I 11o - I ,., transition.
I~nion Circ. So. 2319, 1971. I<. J. RUKLAKD AND E. H. WILSON, JR., J. C‘l~enl. Pirys. 27, .%S il9.iil. (‘. (‘. COSTMK AND J. M. DOWLING, J. C’lretn. Ploys. 32, 290 (1960~. S. G. Kukolich and 4. C. NELSON, Clxm. Plrys. /Al. 11. 383 ( 1971 I. J. R. G. WATSON, J. Chew Ploys. 48, 4517 (19681. LV. H. KIRCHHOFF, J. Vol. Spectrosc. 41, 333 (19i2). (‘. H. TOWNES ,~h‘n ;\. I,. %CHAWLOR., “IRIicrowave SJW~ ~0s~ OJ)J.,“ 1). -409. 10.55.
Md;raw
for
I)rovitlin::
Hill. Sew \~ork.