Introduction
The l-C3H data
were summarized in the publications:
(1) M. Caris, T. F. Giesen, C. Duan, H. S. P. Müller,
S. Schlemmer, and K. M. T. Yamada,
Sub-Millimeter Wave Spectroscopy of the C3H radical:
Ro-Vibrational Transitions from Ground to the Lowest Bending State,
J. Mol. Spectrosc. 253 (2009) 99–105.
View the abstract.
Additional data were taken from the following laboratory
spectroscopic publications:
(2) S. Yamamoto, S. Saito, H. Suzuki, S. Deguchi, N. Kaifu, S.-I.
Ishikawa, and M. Ohishi,
Laboratory Microwave Spectroscopy of the Linear C3H and
C3D Radicals and Related Astronomical Observation,
Astrophys. J. 348, 363–369 (1990).
View the abstract.
(3) C. A. Gottlieb, J. M. Vrtilek, E. W. Gottlieb, P. Thaddeus,
and Å. Hjalmarson,
Laboratory Detection of the C3H Radical,
Astrophys. J. 294, L55–L58 (1985).
View the abstract.
Further transition frequencies were taken from astronomical observations:
(4) N. Kaifu, M. Ohishi, K. Kawaguchi, S. Saito, S. Yamamoto, T.
Miyaji, K. Miyazawa, S.-I. Ishikawa, C. Noumara, S. Harasawa, M.
Okuda, and H. Suzuki,
A 8.8 – 50 GHz Complete Spectral Line Survey toward TMC-1
I. Survey Data,
Publ. Astron. Soc. Japan 56, 69–173 (2004).
View the abstract.
Recent theoretical calculations indicate that the Σ components
of the CCH bending vibration (ω4 ≈
589 cm–1) are shifted apart by the Renner-Teller effect
by such a large amount that the lower component
2Σμ is only 27.17 cm–1
above the Π1/2 component of the vibrational ground state.
(The experimental value from (1) is 27.19107 (27) cm–1.)
(5) M. Peric, M. Mladenovic, K. Tomic, and C. M. Marian,
Ab Initio Study of the Vibronic and Spin-Orbit Structure
in the X 2Π Electronic State of CCCH,
J. Chem. Phys. 118, 4444–4451 (2003).
View the abstract.
In (2), transitions for v4 = 1, 2Σμ have been recorded for the first time besides additional data of v = 0. In addition, the Coriolis interaction between these two states has been analyzed. This analysis has been extended in (1) where, in addition, rovibrational transitions of l-C3H have been recorded for the first time, namely between the 2Π3/2 spin component of the ground state and the loqw-lying Renner-Teller component of v4 = 1. Further down, simulations of the C3H spectrum are shown for various temperatures.
It is rather straightforward to fit a 2Π radical in
Hund's case (b) employing the SPFIT/SPCAT programs
even if Hund's case (a) were more appropriate. Examples for a
molecule close to Hund's case (a) and (b) on this page are SH and CH,
respectively. Of course, things get a bit trickier if data (or hyperfine
split data) are available only for the 2Π1/2 or
the 2Π3/2 spin component; see the examples
C7H and C9H or C8H and C10H,
respectively !
Problems with the assignment of quantum numbers will be dealt with
below !
The C7H to C9H data were taken from
(6) M. C. McCarthy, M. J. Travers, A. Kovács, C. A. Gottlieb,
and P. Thaddeus,
Astrophys. J. Suppl. Ser. 113, 105–120 (1997).
The Parameter File
C3H Fri Jun 04 12:06:23 2010
30 216 5 0 0.0000E+000 3.0000E+003 1.0000E+000 1.0000000000
t -22 2 1 1 0 1 1 1 -1 -1 0
-22 1 0 0 0 1 1 1 0 -1 0
Having two rather very different states to deal with, it is a question of taste which one is defined as the default and which as the "exeption". In the present case, the 2Πr ground vibrational state (with the state number 0) was defined as default and the vibrationally excited 2Σ state as the "exeption" (state number 1).
1 1 0 1 0 -1 0. .00001
This line from the line file . . .
0 2.051945468181433E+005 1.00000000E+036 /E
-11 2.051945468181433E+005 1.00000000E-037 /E
. . . and this parameter afford the lowest rotational state to be set to zero energy and all other states accordingly higher. The –1 as lower quantum number signals that the line describes a state and not a transition ! In the present case, the value of the energy is largely determined by the value of A/2.
It is quite common to expand the angular momentum of a linear molecule in N(N+1) – l2 (or – K2 or . . .) rather than in N(N+1). Therefore, so-called l-corrections have been introduced for B, D, and H:
100 1.118910330525249E+004 1.00000000E+036 /B0
-1000 -1.118910330525249E+004 1.00000000E-037 /l-Korr
200 -5.233971497534610E-003 1.00000000E+036 /-D0
-1100 1.046794299506925E-002 1.00000000E-037 /l-Korr
-2000 -5.233971497534610E-003 1.00000000E-037 /l-Korr
300 2.276006064723007E-008 1.00000000E+036 /H0
-1200 -6.828018194169020E-008 1.00000000E-037 /l-Korr
-2100 6.828018194169020E-008 1.00000000E-037 /l-Korr
-3000 -2.276006064723007E-008 1.00000000E-037 /l-Korr
The – in front of a parameter code signals, as usual, that the ratio of this parameter to the one before in the parameter list should be kept fixed in the fit.
B and D of the upper, Σ, state are defined in a more common way:
111 1.121267027473720E+004 1.00000000E+036 /B1
211 -4.866466407213966E-003 1.00000000E+036 /-D1
as is the energy:
11 6.099742272101132E+005 1.00000000E+036 /E1
This energy (20.34655 (14) cm–1) corresponds to the energy difference between the Σ state and the average Π state taking into consideration the l correction. The energy difference to the Π1/2 state is 27.20 cm–1, as mentioned above.
The Λ splitting is described by p and q along with possible distortion terms. While q is defined positively, p is defined negatively. The parity is governed by Π ↔ Σ transitions.
10040000 3.534067862932181E+000 1.00000000E+036 /-p/2
10040100 -2.521667384761119E-004 1.00000000E+036 /-pD/2
40000 -6.498882311834095E+000 1.00000000E+036 /q/2
40100 -7.164615990107675E-005 1.00000000E+036 /qD/2
The Coriolis interaction is described by:
400001 2.468396380271915E+003 1.00000000E+036 /Gb
400101 -3.833283824580967E-002 1.00000000E+036 /GbD
It should be mentioned that this value is twice that of β
in (1). This is similar to the situation in a symmetric top molecule
were Ga = 2Aζ holds. (For an asymmetric
top the factor is usually slightly different from 2 depending on the
energies of the vibrations involved.)
The sign of Gb is not relevant for the fit;
however, it does matter, in combination with the signs and magnitudes
of the transition and permanent dipole moments, for the intensities of
some strongly perturbed lines.
GbD expresses the N dependence
of Gb, it is thus its distortion correction.
20010000 1.236109105343095E+001 1.00000000E+036 /a(0)
120000000 -1.366855236204242E+001 1.00000000E+036 /bF(0)
120010000 2.818115489638794E+001 1.00000000E+036 /c(0)
120040000 8.128932779249276E+000 1.00000000E+036 /d(0)/2
120000011 1.397275114853093E+000 1.00000000E+036 /bF(1)
120010011 2.850120353305769E+001 1.00000000E+036 /c(1)
There is little to say about the hyperfine constants except that one
should consider that bF = b + c/3
holds and that the sign of d should be consistent with those of
p und q.
It gets a bit trickier if hyperfine splitting has been resolved only for
one spin component, e.g. for C7H or C8H. In these
cases, only three and one parameter, respectively, of a,
bF, c, and d are determinable.
In the case of C7H, a + bF/2 +
c/3 = 0 has been constrained, for C8H it was
a + bF/2 = 0. Comparison with C3H
indicates that these assumptions are only somewhat reasonable. Of course,
other constraints may be imposed.
C7H:
120000000 -1.603371107130468E+001 1.00000000E+036 /bF
-20010000 8.016855535652327E+000 1.00000000E-037 /a
120010000 8.638471820414173E+000 1.00000000E+036 /c
-20010000 -2.879490606804726E+000 1.00000000E-037 /a
120040000 3.055924176719215E+000 1.00000000E+036 /-d/2
C8H:
120000000 -1.832071477027064E+001 1.00000000E+035 /bF
-20010000 9.160357385135310E+000 1.00000000E-037 /a
On the Quantum Number Assignment
In Hund's case (a), the quantum number N is redundant, in Hund's
case (b) it is not. Using J and N, one can evaluate wether
a certain state belongs to the spin component Π1/2 or
Π3/2:
For 2B (J – 1/2)(J + 1/2) < A, states
with J + 1/2 = N belong to Π1/2; for
2B (J – 1/2)(J + 1/2) > A they belong
to Π3/2.
For 2B (J – 1/2)(J + 1/2) ≈ A
some transitions occur that seem to have strange selection rules:
97995.1917 0.0033 -3.2190 2 5.4486 11 375011325 4 1 0 5 5 4-1 0 4 4 97995.9423 0.0029 -3.3184 2 5.4489 9 375011325 4 1 0 5 4 4-1 0 4 3 98005.1714 0.0044 -4.8624 2 5.4486 9 375011325 4 1 0 5 4 4-1 0 4 4 98005.4653 0.0046 -4.8622 2 5.4454 9 375011325 4-1 0 5 4 4 1 0 4 4 98011.6485 0.0034 -3.2188 2 5.4454 11 375011325 4-1 0 5 5 4 1 0 4 4 98012.5862 0.0030 -3.3182 2 5.4452 9 375011325 4-1 0 5 4 4 1 0 4 3 103311.8121 0.1051 -4.8903 2 19.4368 9 375011325 5-1 0 5 4 3 1 0 4 4 103319.3201 0.0144 -3.2469 2 19.4368 11 375011325 5-1 0 5 5 3 1 0 4 4 103319.8404 0.0165 -3.3462 2 19.4365 9 375011325 5-1 0 5 4 3 1 0 4 3 103365.2961 0.1053 -4.8899 2 19.4386 9 375011325 5 1 0 5 4 3-1 0 4 4 103372.5486 0.0149 -3.2464 2 19.4386 11 375011325 5 1 0 5 5 3-1 0 4 4 103373.1610 0.0167 -3.3458 2 19.4383 9 375011325 5 1 0 5 4 3-1 0 4 3
The 1H Hyperfine Structure
is well determined for v = 0 because of the astronomical observation of six J = 1.5 – 0.5 components. In the case of v4 = 1, μ2Σ the lowest N transition has N = 3 – 2; extrapolation to lower quantum numbers should be viewed with some caution.
Extrapolation to Higher Quantum Numbers
should always be viewed with caution ! Because of the Coriolis interaction particular caution had been advised prior to (1) since trasitions with the highest quantum numbers had N ≈ 15, while the strongest interaction occurs at J = 24.5 and 25.5, as can be seen in the energy file:
50 6 251.006716 0.000274 0.576925 49: 25 1 0 25 24
52 4 251.006852 0.000274 0.576753 51: 25 1 0 25 25
50 3 251.958606 0.000272 0.624306 49: 24 0 1 25 24
52 2 251.958629 0.000272 0.624111 51: 24 0 1 25 25
51 3 269.993733 0.000274 0.695283 51: 25 0 1 26 25
53 2 269.993739 0.000273 0.695452 53: 25 0 1 26 26
51 6 271.005804 0.000272 0.612013 51: 26 -1 0 26 25
53 4 271.005958 0.000272 0.612170 53: 26 -1 0 26 26
Since the spin-orbit coupling parameter ASO has an uncertainty of almost 10 MHz, transitions having Σ ← Π1/2 can be predicted less well.
The C3H Spectrum up to About 2 THz
is dominated by the pure rotational spectrum
at room temperature. The highlighted lines of v4 = 1,
μ2Σ are only slightly weaker than those of
the ground vibrational state because of the relatively small energy difference.
Since the transition dipole moment is with 0.5 D (estimated)
considerably smaller than the permanent dipol moment
with calculated 3.1 D (3.55 D according to a more sophisticated calculation),
the vibration-rotation spectrum is significantly
weaker – with the exception of two substantially perturbed lines which
borrow intensity from v = 0. However, the intensity
does not pose a severe problem for finding the lines. The main
problem is its rather large uncertainty of the band origin which was
determined from the fit to be 500 MHz. Because of possible correlation
effects and because of the effects of missing spectroscopic parameters
which can not be determined at the present stage, the true uncertainty of
the band center is likely larger by a factor of 3 to 10 !
Accurate transition frequncies with higher quantum numbers will reduce
these uncertainties.
At 75 K, the pure rotational spectrum and the
vibration-rotation spectrum occur in regios
that do not overlap very much. The lines of v4 = 1,
μ2Σ are noticeably weaker than those of v = 0.
At 18.75 K, the pure rotational spectrum and
the vibration-rotation spectrum appear in
almost distinct regions. The region around 300 GHz being an exception.
The rotational spectrum in the submillimeter range now consists mainly of
transitions between Π1/2 and Π3/2; the
transitions with N' = N" = 1 are near 444.9 GHz and have
a predicted uncertainty of 10 MHz which originates from the uncertainty
in A. The strong rovibrational transitions are described
by the selection rules ΔJ = 0, ±1;
ΔN = ΔJ ±0, 1. 6 out of the 9 strong
branches can be discerned easily.
