Fine-structure spectrum of the FO radical, observed by far-infrared laser magnetic resonance

Article abstract of DOI:10.1063/1.478630

The fine-structure transition ²∏_1/2→ ²∏_3/2 of the free radical FO has been detected by far-infrared laser magnetic resonance. All the observed transitions are magnetic dipole in character. The spin-orbit constant A₀ has been determined experimentally; its value of -196.108686(50) cm^-1 is consistent with previous estimates. The analysis of a set of 290 transitions leads to the determination of a number of molecular parameters including rotational, centrifugal distortion, spin-orbit, lambda-doubling, magnetic hyperfine, and Zeeman terms. All four magnetic hyperfine structure constants a, b_F, c, d for the ¹⁹F nucleus have been determined and are discussed in terms of the expectation values of the appropriate operators over the electronic wave function.

Full text of DOI:10.1063/1.478630

Fine-structure spectrum of the FO radical, observed by far-infrared laser magnetic
resonance
Filippo Tamassia, John M. Brown, and Kenneth M. Evenson
Citation: The Journal of Chemical Physics 110, 7273 (1999); doi: 10.1063/1.478630
View online: http://dx.doi.org/10.1063/1.478630
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 110, NUMBER 15
15 APRIL 1999
Fine-structure spectrum of the FO radical, observed by far-infrared laser
magnetic resonance
Filippo Tamassia and John M. Browna)
Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford OX1 3QZ, United Kingdom
Kenneth M. Evenson
National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80303
͑
Received 1 December 1998; accepted 20 January 1999͒
2
2
The fine-structure transition ⌸1 ← ⌸ of the free radical FO has been detected by far-infrared
/2
3/2
laser magnetic resonance. All the observed transitions are magnetic dipole in character. The spin–
orbit constant A0 has been determined experimentally; its value of Ϫ196.108 686͑50͒ cm is
Ϫ1
consistent with previous estimates. The analysis of a set of 290 transitions leads to the determination
of a number of molecular parameters including rotational, centrifugal distortion, spin–orbit,
lambda-doubling, magnetic hyperfine, and Zeeman terms. All four magnetic hyperfine structure
19
constants a, b , c, d for the F nucleus have been determined and are discussed in terms of the
F
expectation values of the appropriate operators over the electronic wave function. © 1999
American Institute of Physics. ͓S0021-9606͑99͒01315-X͔
2
2
INTRODUCTION
direct detection of the fine-structure transition ⌸1 ← ⌸
/2
3/2
in the far-infrared region. This is the aim of the present work.
The LMR technique was chosen because of its high sensitiv-
ity and its ability to discriminate between open-shell and
closed-shell molecules, which are usually present in much
higher amounts; also, the only intense sources in the far-
infrared ͑FIR͒ region are fixed-frequency lasers. In a Hund’s
case ͑a͒ limit, the fine-structure transition is electric dipole
forbidden but magnetic dipole allowed and hence it is ex-
pected to be three or four orders of magnitude weaker than a
normal, electric dipole allowed transition. On the other hand,
The first experimental detection of the FO radical was
made by McKellar in 1979 through the observation of its
infrared spectrum. He recorded the fundamental vibration–
rotation band associated with the lower spin component
1
2
⌸3/2
by CO -laser magnetic resonance ͑LMR͒ spectroscopy
2
and determined the band origin ␯ , the rotational constant
0
B , the centrifugal distortion correction D , the hyperfine
0
0
1
2
parameter h3/2ϭaϩ (bϩc), the vibrational differences B
1
ϪB , h Ϫh , and the spin–orbit constant A . Since the
0
1
0
0
5
observations were confined to the ⍀ϭ3/2 spin component,
Brown, Cole, and Honey, in their work on the fine-structure
A could not be determined directly, and in fact the reported
value of Ϫ177.3͑57͒ cm was not very accurate. Subse-
spectrum of NO, pointed out that magnetic dipole allowed
transitions were about 18 times stronger than electric dipole
0
Ϫ1
6
quently, the same vibrational band was reinvestigated by di-
forbidden transitions. The same ratio is 20 for SeH and 140
2
7
ode laser spectroscopy, leading to a better determination of
for BrO; it depends on the magnitude of the electric dipole
B , B , D , D because it was now possible to study FO in
moment and the spin–orbit splitting.
0
1
0
1
3
both spin components.
Burkholder et al. have also considered the production
3
Burkholder et al. recorded the 1–0 and 2–0 bands in
of FO by the reaction between F atoms and ozone and em-
phasized that, in the presence of excess ozone, there is no net
loss of FO radicals. FO is produced by the reaction
2
2
both ⌸ and ⌸ spin components by Fourier transform
3
/2
1/2
IR spectroscopy. In 1988 the same infrared bands and a num-
ber of ⌬vϭ1,2,3 bands were recorded in a Fourier transform
emission experiment by Hammer et al. The set of reported
4
FϩO3→FOϩO2
͑1͒
data was fitted to an effective Hamiltonian which included
rotational, centrifugal distortion, spin–orbit, and lambda-
and destroyed by the reaction
FOϩFO→2FϩO2.
͑2͒
doubling terms. Nuclear hyperfine structure was not re-
Ϫ1
solved. A was estimated to be Ϫ198.3͑67͒ cm in Ref. 3
0
Thus two F atoms are produced in the latter process and
can react with further ozone molecules, thereby regenerating
the FO radical. This was confirmed in our experiments where
we observed a surprisingly long lifetime of the signal ͑see
the experimental section͒. These considerations and the high
intrinsic sensitivity of LMR gave us confidence in our ap-
proach to the experiment, despite the expected low intensity
of the magnetic dipole transitions.
Ϫ1
and Ϫ193.28͑97͒ cm in Ref. 4, still indirectly and subject
to quite a large uncertainty. As pointed out by the authors of
Ref. 4, the comparison of all the available experimental de-
terminations and theoretical calculations suggested that A0
Ϫ1
could lie between Ϫ180 and Ϫ200 cm ; that is to say, a
major parameter of FO was still very poorly determined.
The best way to measure the spin–orbit splitting is the
No pure rotational or electron paramagnetic resonance
a͒Electronic mail: jmb@physchem.ox.ac.uk
spectra of FO have been reported in the literature. McKellar8
0021-9606/99/110(15)/7273/14/$15.00
7273
© 1999 American Institute of Physics
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7274
J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Tamassia, Brown, and Evenson
combined LMR spectroscopy and Stark spectroscopy to
demonstrate that the electric dipole moment of FO is
0.0043͑4͒ D for the vϭ0 level and 0.0267͑9͒ D for vϭ1.
The exceptionally small value of ␮ has therefore precluded
any detection of rotational spectra so far but the strong varia-
tion of ␮ with the vibrational excitation makes the infrared
spectra relatively easy to observe. This is supported by the-
oretical calculations by Langhoff, Bauschlicher, and
9
Partridge. Their predictions for ␮vϭ0 ͑Ϫ0.0089 D with the
negative end on the F atom͒ and ␮vϭ1 ͑Ϫ0.0318 D͒ show
quite good agreement with McKellar’s experimental values.
The computed curve of the dipole moment as a function of
the internuclear distance shows a steep slope for bond
lengths less than 0.16 nm, confirming the large change of ␮
passing from vϭ0 to vϭ1.
In the present paper, we report the successful detection
of magnetic dipole, fine-structure transitions of the FO radi-
2
cal in the vϭ0 level of the X ⌸ state. The measurements
are more accurate than any which precede it and allow,
among other things, the first direct determination of the
spin–orbit coupling constant and of all four magnetic nuclear
hyperfine parameters.
EXPERIMENTAL DETAILS
The far-infrared spectrometer used for the experiment
10
11
has been described elsewhere. Very recently some
changes have been made: the Zeeman modulation frequency
has been raised to 40 kHz to improve the sensitivity and the
diameter of the pump tube has been reduced in order to pro-
duce better performance of the laser at wavelengths shorter
than 100 ␮m. This has been proved to be particularly impor-
tant for our experiment since the fine structure transitions of
FO are around 50 ␮m. The detector used was a liquid-
helium-cooled gallium–germanium photoconductor.
FIG. 1. Experimental ͑the upper trace͒ and simulated ͑the lower trace͒ far-
1
infrared laser magnetic resonance spectrum of the R(4 ) fine-structure tran-
2
Ϫ1
sition of FO observed at 209.819 231 cm in perpendicular polarization
(⌬M ϭϮ1).
J
O , even in the absence of further F atoms. The interesting
3
result is that, if the microwave discharge is switched off and
the flow of ozone is maintained, the signal is maintained for
a few minutes. This is not due to a very long lifetime of FO
but to the regeneration of fluorine atoms by the reaction of
two FO molecules. Signals of lower intensity were also ob-
served reacting F atoms ͑42.7 Pa 10% F /He͒ with O ͑100
FO was produced by reacting F atoms with O . The F
3
atoms were generated by flowing a 10% F in He mixture
2
through a microwave discharge. Ozone was introduced into
the cell by flowing He over silica gel, on which ozone was
2
2
adsorbed. The O /silica gel trap was placed in an ethanol–
Pa͒.
3
dry ice bath to regulate the amount of O evaporated. The
The spectra were recorded on an xy plotter as a function
3
best signals were observed with 24 Pa ͑180 mTorr͒ of F /He
of the magnetic flux density. Since 1Ϫf detection with mag-
netic modulation was employed, the first derivative of the
absorption profile was observed. Figure 1 shows a long sur-
2
and 9.3 Pa ͑70 mTorr͒ of O /He. As described in the Intro-
3
duction, the production of FO was sustained by an excess of
TABLE I. FIR laser lines used to record LMR spectra for FO.
a
CO laser line
Laser gas
Wavelength/␮m
Frequency/MHz
Assigned FO transitions
2
1
2
1
1
1
1
1
1
0R(46)
0R(56)
0R(20)
0R(24)
0R(52)
CH OH
52.004
51.478
50.629
49.973
49.694
49.455
48.721
47.660
46.165
5 764 826.7
5 823 660.9*
5 921 370.4*
5 999 028.8*
6 032 811.3*
6 061 914.8*
6 153 279.0*
6 290 222.3*
6 493 911.5
3
P(2 )
CD OH
3
P(1 )
2
1
1
2
CD OH
3
Q(1 ), Q(2 )
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
CD OH
3
Q(11 ), Q(12 ), Q(13 ), Q(14 )
CH OH
3
Q(15 ), Q(16 ), Q(17 )
1
3
1
2
9
9
9
9
P(12)
R(6)
CH OH
3
Q(17 ), Q(18 ), Q(19 ), Q(20 )
1
CD OH
3
R(2 )
2
1
R(8)
CH OD
3
R(4 )
2
1
P(10)
CH OH
3
R(7 )
2
aReference 28. The lines marked with an asterisk have been frequency measured for this work.
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J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Tamassia, Brown, and Evenson
7275
typical spectra are shown in Figs. 1–3. We have recorded
and assigned 290 lines corresponding to 17 rotational transi-
1
1
2
tions with J ranging from 2 to 20 . All the observed reso-
nances are magnetic dipole in origin. The spectrum of the
1
2
1
2
1
2
(
Jϭ18 , Fϭ19, M ϭϪ19,Ϫ)←(Jϭ18 , M ϭϪ , M
F
I
J
3
2
9
ϭϪ ,Ϫ) transition showed a Lamb dip, at low pressure
and modulation amplitude, which indicates that the signals
are strong enough to reach saturation and also that there is no
underlying structure in the Doppler profile ͑see Fig. 4͒. No
evidence for FO in vibrationally excited states has been
found and all of the recorded lines have been assigned to
fine-structure transitions in vϭ0.
By comparison of our spectra with the LMR spectra re-
ported by Mizushima, Evenson, and Wells13 in their work on
the corresponding fine-structure transition of NO, we identi-
fied the different Q branches of our spectrum and obtained a
rough idea of the value of J. Using the parameters of FO
already known from literature3 and hyperfine constants
,4
1
4
scaled from the corresponding parameters of CF, the high-J
Q branch was tentatively assigned. A was changed step by
0
step in order to match the calculated pattern with the experi-
mental pattern. Once the first transition was identified, the
spectroscopic constants were refined and new predictions
made. The analysis of the spectra was lengthy and elaborate
because of our ignorance of some basic parameters and the
7
unusual Zeeman patterns, which arise from the different
nuclear spin coupling schemes in the upper and lower levels.
2
For a molecule in a ⌸ state there are two possible val-
ues, 3/2 and 1/2, for magnitude of the projection ⍀ of the
angular momentum J on the internuclear axis. The two spin
FIG. 2. Experimental ͑the upper trace͒ and simulated ͑the lower trace͒ far-
2
2
components, ⌸ and ⌸ , show very different behavior
1
3/2
1/2
infrared laser magnetic resonance spectrum of the P(1 ) fine-structure tran-
2
in the presence of an external magnetic field. For a molecule
Ϫ1
sition of FO observed at 194.256 417 cm in perpendicular polarization
which conforms closely to Hund’s case ͑a͒ coupling, such as
(
⌬M ϭϮ1). Note the complicated hyperfine structure because of the dif-
J
Ϫ1
Ϫ1
ference in coupling schemes in the lower and upper states.
FO ͑A ӍϪ196 cm , B Ӎ1 cm ͒ the first-order Zeeman
0 0
contribution to the total energy is given by
1
2
EZeeϭgJ␮BBMJ ,
͑3͒
vey scan of the R(4 ) transition. The spectra have been re-
corded in both parallel (B␻
0 ␻ 0
B ) and perpendicular (B ЌB )
ʈ
where g is the g factor for the rotational level J, ␮ is the
J B
polarizations. Typical measurement scans covered 40 mT
and were performed back and forth to correct shifts due to
the time constant. The magnet was periodically calibrated
with a NMR gaussmeter: the overall fractional uncertainty is
Bohr magneton, B is the magnetic flux density, and M is the
projection of J on the laboratory-fixed Z axis. The g factor is
given by
J
Ϫ4
Ϫ5
Ϯ1ϫ10 above 0.1 T and Ϯ1ϫ10 below 0.1 T. We
͑⌳ϩ⌺͒͑⌳ϩgS⌺͒
Ϫ4
gJϭ
,
͑4͒
estimate an experimental uncertainty of Ϯ2ϫ10 T for an
͑
J Jϩ1
͒
individual resonance.
where ⌳ and ⌺ are, respectively, the projections of L and S
on the internuclear axis. According to Eq. ͑4͒ g is nearly
We used nine FIR laser lines to make our observations
͑see Table I͒ and seven of their frequencies have been mea-
J
2
2
zero for the ⌸ state, while it is nonvanishing for the ⌸
sured for the first time in this experiment. Their frequency
was determined by measuring the beat frequency when the
FIR radiation was mixed with the radiation from two
1/2
3/2
2
state; in other words, the ⌸ state is essentially diamag-
1/2
netic. As a result, the energy level patterns for the two types
of state in a magnetic field are considerably different. In Fig.
frequency-stabilized CO lasers in a metal–insulator–metal
2
1
2
1
2
͑MIM͒ diode.
5 the energy level diagram for the P(1 ) transition at a flux
density of 1.5 T is given.
For the lower state (Jϭ3/2) the interaction with a mag-
RESULTS AND ANALYSIS
netic field produces four M components, 3/2, 1/2, Ϫ1/2,
J
Ϫ1
Nine laser lines, listed in Table I, have been used to
record all of the spectra reported in this paper. The total
range explored is about 24 cmϪ1 and covers part of the P, Q,
and R branches of the fine-structure transition. Examples of
Ϫ3/2, spread over 1.7 cm . Each M level is in turn split
J
into two levels by magnetic hyperfine interaction (I ϭ1/2)
F
and a further two by lambda-doubling effects. The finer split-
tings between these four sublevels cannot be appreciated on
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7276
J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Tamassia, Brown, and Evenson
1
2
1
2
Ϫ1
FIG. 4. A spectrum of the (Jϭ18 , Fϭ19, M ϭϪ19,Ϫ)←(Jϭ18 , M
F
I
1
39
ϭϪ , M ϭϪ ,Ϫ) transition of FO observed at 202.203 713 cm in
2
J
2
parallel polarization (⌬M ϭ0) showing a single Lamb dip. Total pressure
J
22 Pa ͑165 mTorr͒, scan time 5 min, time constant 1 s, sensitivity 100 ␮V,
modulation 0.1 mT.
FIG. 3. Experimental far-infrared laser magnetic resonance spectrum of the
1
2
1
2
1
2
Ϫ1
Q(15 ), Q(16 ), Q(17 ) transitions of FO observed at 201.232 924 cm
in parallel polarization (⌬M ϭ0). The measured lines are identified as
J
1
2
1
2
1
2
1
2
follows: ᮀ—Q(15 ) , ᭝—Q(15 ) , ᭺—Q(16 ) , ᭹—Q(16 )
,
ϩ
Ϫ
ϩ
Ϫ
1
*
—Q(172)Ϫ .
the scale of the diagram, being much smaller than the sepa-
ration between the MJ components. In this situation the
nuclear spin I is decoupled from J so that J,M ,I,M and
J
I
the parity are all good quantum numbers. For the upper J
ϭ1/2 state in the same field, the Zeeman levels are spread
Ϫ1
over a much narrower range of energy ͑0.036 cm ͒ and
there is no clear magnetic structure. In this case the nuclear
spin remains coupled and only F (FϭJϩI), its projection on
the field axis M and the parity are good quantum numbers.
F
The Jϭ1/2 eigenfunctions in the coupled representation can
be expressed as a linear combination of the decoupled basis
1
2
set
͉
JM IMI
͘
and, in the specific case of the P(1 ) transi-
J
tion in FO they are, neglecting J and I:
1
2
1
2
͉
͉
Fϭ1, M ϭϪ1,Ϫ
͘
ϭ
͉
M ϭϪ ,M ϭϪ ,Ϫ
͘
,
F
J
I
1
1
2
Fϭ1, M ϭ0,Ϫ
͘
ϭ0.735
͉
M ϭϪ ,M ϭ ,Ϫ
͘
F
J
2
I
1
2
1
2
ϩ0.677
͉
M ϭ ,M ϭϪ ,Ϫ
͘
͘
,
,
J
I
1
2
1
2
͉
͉
Fϭ1, M ϭ1,Ϫ
͘
͘
ϭ
͉
M ϭ ,M ϭ ,Ϫ
͘
,
F
J
I
1
2
1
2
Fϭ1, M ϭ0,ϩ
ϭ0.609
͉
M ϭϪ ,M ϭ ,ϩ
͘
F
J
I
1
2
1
2
ϩ0.792
͉
M ϭ ,M ϭϪ ,ϩ
J
I
1
2
1
2
͉
͉
͉
Fϭ1, M ϭ1,ϩ
͘
ϭ
͉
M ϭ ,M ϭ ,ϩ
͘
,
F
J
I
1
2
1
2
1
Fϭ1, M ϭϪ1,ϩ
͘
ϭ
͉
M ϭϪ ,M ϭϪ ,ϩ
͘
,
F
J
I
FIG. 5. Energy level diagram for the P(1 ) fine-structure transition of FO
2
2
calculated in a flux density of 1.5 T. For the ⌸3 component, the fluorine
/2
1
2
1
2
Fϭ0, M ϭ0,Ϫ
͘
ϭϪ0.677
͉
͉
͉
͉
M ϭϪ ,M ϭ ,Ϫ
͘
͘
͘
͘
F
J
I
nuclear spin is decoupled and every MJ level is split into two levels by the
nuclear hyperfine interaction, each of which is further split by the effect of
lambda doubling. The four energy levels associated with each single MJ
1
2
1
2
ϩ0.735
M ϭ ,M ϭϪ ,Ϫ
,
,
J
I
component are not visible on the present scale; an enlargement is shown for
1
2
1
2
2
͉
Fϭ0, M ϭ0,ϩ
͘
ϭϪ0.792
ϩ0.609
M ϭϪ ,M ϭ ,ϩ
M ϭϪ3/2. For the ⌸1/2 component, the nuclear spin remains coupled to
F
J
I
J
the overall rotational angular momentum J; the quantum numbers F and M
F
1
2
1
2
M ϭ ,M ϭϪ ,ϩ
are therefore appropriate. The energy levels in the 2⌸1/2 component have
been expanded, since their separation is not visible on the present scale.
Note that the levels ͑Fϭ1, M ϭ0, ϩ͒ and ͑Fϭ1, M ϭ1, ϩ͒ are over-
J
I
where the calculations have been made for a flux density of
.5 T and use the molecular parameters in Table III. The
F
F
1
lapped.
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J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Tamassia, Brown, and Evenson
7277
1
2
P(1 ) transition is shown in Fig. 2, where the simulated
lambda-doubling term and HcdLD its centrifugal distortion
correction. Hhfs represents the magnetic hyperfine interaction
Hamiltonian and HZ is the Zeeman Hamiltonian.
spectrum is also given. The six peaks correspond to the tran-
sitions:
The selection rules for LMR transitions are ⌬Jϭ0, Ϯ1,
3
1
͑
͑
͑
͑
͑
͑
i͒
͉
Fϭ0, M ϭ0,Ϫ
͘
←
J I
2 2
͘
͉M ϭ ,M ϭϪ ,Ϫ ,
F
⌬
M ϭ0, ⌬M ϭ0, Ϯ1. If the laser magnetic field ͑or electric
I J
field in case of electric dipole transitions͒ is parallel to the
B ϭ1489.17 mT,
0
external magnetic field the selection rule ⌬M ϭ0 applies. If
J
3
1
ii͒
iii͒
iv͒
v͒
͉
Fϭ1, M ϭ1,ϩ
͘
←
J I
2 2
͘
͉M ϭ ,M ϭ ,ϩ ,
F
they are perpendicular ⌬M ϭϮ1 holds. As pointed out by
J
5
Brown, Carrington, and Sears, the optimum polarization is
B ϭ1501.60 mT,
0
parallel for Q transitions and perpendicular for R and P tran-
3
2
1
2
͉
Fϭ1, M ϭ0,ϩ
͘
͘
←
←
͉
M ϭ ,M ϭϪ ,ϩ
͘
͘
,
,
sitions. This rule has been experimentally confirmed by our
F
J
I
1
2
1
2
1
2
spectra. Only for the R(4 ), Q(1 ), and Q(2 ) transitions
not measured,
have equally intense lines been detected in both parallel and
perpendicular polarizations.
3
2
1
2
͉
Fϭ0, M ϭ0,ϩ
͉
M ϭ ,M ϭϪ ,ϩ
F
J
I
The data have been analyzed using a linearized least-
squares procedure. Since the Zeeman effect mixes energy
levels with different J according to the selection rule ⌬Jϭ
Ϯ1, a suitably truncated basis set for the matrix of the
Hamiltonian must be chosen ͑⌬JϭϮ2 in our case͒ in order
B ϭ1528.97 mT,
0
3
1
͉
Fϭ1, M ϭ1,Ϫ
͘
←
͉
J I
2 2
͘
M ϭ ,M ϭ ,Ϫ ,
F
B ϭ1545.84 mT,
0
3
2
1
2
to ensure that the calculations are accurate enough. H was
0
vi͒
͉
Fϭ1, M ϭ0,Ϫ
͘
←
͉
M ϭ ,M ϭϪ ,Ϫ
͘
,
F
J
I
fixed to a calculated value ͑see Ref. 17͒ and p ϩ2q was
D
D
B ϭ1571.73 mT.
constrained to the previous value determined in Ref. 3. The
observed lines are given in Table II. For each rotational tran-
sition, identified by the usual notation P(J), Q(J) or R(J),
0
Transitions ͑i͒, ͑iii͒, ͑iv͒ and ͑vi͒ involve heavily mixed basis
functions and are in part forbidden because of the decoupled
the lower level is labeled by the quantum numbers M and
J
selection rule ⌬M ϭ0.
I
M , and the upper level by F and M . The parity is defined
I
F
As mentioned before, we cannot analyze the data in
terms of a single coupling scheme. We chose to calculate the
eigenvalues for the two spin components in the most appro-
for both levels, according to the selection rule for magnetic
dipole transitions Ϯ←Ϯ. For each transition the laser fre-
quency, the differences observed–calculated from the fit, the
tuning rates, and the weights are also given. The tuning rate,
defined as the variation of the transition frequency with re-
priate basis sets, that is I-decoupled
͉
͉
␩⌳
͘
JIFMF for ⌸ . The
1/2
͘ ͘ ͘ ͘
͉S⌺ ͉JM ⍀ ͉IMI
J
2
2
for ⌸ and I-coupled
͉
␩⌳
͘
͉
S⌺
͘
3
/2
15,16
set of data has been fitted with an effective Hamiltonian
of the form
spect to the magnetic flux density B , is in MHz/mT. We
0
chose to give a weight of 0.1 to the transitions with o–c
between 6 and 10 MHz and a weight of 0 to the transitions
with o–c greater than 10 MHz; virtually all these data points
corresponded to resonances at high magnetic fields where the
measurements are less accurate. The quality of the fit ͑Table
III͒ is satisfactory and only a few lines, mostly partially over-
lapped or at high fields, have been given a lower weight. The
standard deviation of the fit is 1.9 MHz, consistent with the
expected experimental uncertainty.
We fitted AD and constrained ␥ to zero, but in principle
an equivalent fit can be made with the opposite choice. When
we tried such a calculation, we obtained a significant wors-
ening of the quality of the fit with a standard deviation of 3.0
MHz. Since we are confident that our set of data is not af-
fected by severe systematic errors, this suggests that there are
may be other contributions from parameters not accounted
for in the effective Hamiltonian ͑see Discussion and Conclu-
sions͒.
H ϭH ϩH ϩH ϩHLDϩHcdLDϩHhfsϩHZ ,
͑5͒
eff
so
rot
cd
where
H ϭA L S ϩ A ͓N ,L S ͔ ,
1
2
2
͑6͒
͑7͒
͑8͒
so
0
z
z
D
z z ϩ
2
H ϭB N ,
rot
0
4
6
H ϭϪD N ϩH N ,
cd
0
0
1
2
2i␾
Ϫ2i␾
HLDϭ ͑pϩ2q͒͑e S J ϩe
S J ͒
Ϫ
Ϫ
ϩ
ϩ
1
2
2i␾
2
Ϫ2i␾
2
Ϫ q͑e J ϩe
J ͒,
͑9͒
Ϫ
ϩ
1
4
2
2i␾
Ϫ2i␾
HcdLDϭ ͑p ϩ2q ͓͒N ,e S J ϩe
S J ͔ ,
ϩ ϩ ϩ
D
D
Ϫ
Ϫ
͑
10͒
HhfsϭaI L ϩb͑I S ϩI S ͒ϩ͑bϩc͒I S
z
z
x
x
y
y
z
z
1
2
Ϫ2i␾
2i␾
ϩ d͑e
I S ϩe I S ͒,
͑11͒
ϩ
ϩ
Ϫ Ϫ
H ϭg ␮ B L ϩg ␮ B S Ϫg ␮ B N
Z
Z
L
B
0
Z
S
B
0
Z
r
B
0
ϩg ␮ ͑S B ϩS B ͒Ϫg ␮ B I
l
B
x
x
y
y
N
N
0
Z
DISCUSSION AND CONCLUSIONS
Ϫ2i␾
2i␾
ϪgЈ␮ ͑e
N B ϩe N B ͒
ϩ ϩ Ϫ Ϫ
r
B
The main goal of the present work was the determination
of the spin–orbit coupling constant A for FO in its ground
Ϫ2i␾
2i␾
0
ϩgЈ␮ ͑e
S B ϩe S B ͒.
ϩ ϩ Ϫ Ϫ
͑12͒
l
B
2
⌸
state. This parameter has been determined with great
Hso is the spin–orbit operator including the centrifugal
distortion correction. Hrot is the rotational operator and Hcd is
the rotational centrifugal distortion contribution. HLD is the
accuracy and the fit as a whole shows a marked improvement
in the precision of most of the other parameters. The two
exceptions are the rotational and centrifugal distortion con-
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7278
J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Tamassia, Brown, and Evenson
2
2
TABLE II. The ⌸ ← ⌸ transitions of FO observed by far-infrared LMR.
1
/2
3/2
␯L
B0
o–c
ץ␯/ץB0
F
MF
←
MI
1/2
Ϫ1/2
Ϫ1/2
Ϫ1/2
1/2
MJ
5/2
5/2
5/2
5/2
5/2
p
͑MHz͒
͑mT͒
͑MHz͒ ͑MHz/mT͒
wt
1
2
2
Ϫ
Ϫ
Ϫ
ϩ
ϩ
5 764 826.7 1778.74
1807.11
Ϫ0.9
Ϫ0.0
Ϫ0.0
Ϫ8.7
Ϫ0.6
Ϫ10.9
Ϫ10.9
Ϫ10.9
Ϫ10.9
Ϫ10.9
1.0
1.0
1.0
0.1
1.0
P(2 )
2
2
1
1
2
1
1
1
2
1831.51
1839.71
1880.46
1
2
0
1
0
1
1
0
1
0
1
0
Ϫ1/2
1/2
Ϫ1/2
1/2
3/2
3/2
3/2
3/2
3/2
Ϫ
ϩ
ϩ
Ϫ
Ϫ
5 823 660.9 1489.17
1501.60
Ϫ2.0
Ϫ2.3
1.5
Ϫ0.2
Ϫ0.3
Ϫ15.1
Ϫ15.0
Ϫ15.0
Ϫ15.0
Ϫ14.9
1.0
1.0
1.0
1.1
1.0
P(1 )
1528.97
1545.84
1571.73
Ϫ1/2
1
2
2
2
1
1
2
2
2
1
1
2
1
1
2
2
2
1
1
1
2
1
1
1
1
0
0
0
1
0
1/2
Ϫ1/2
Ϫ1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
1/2
1/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
3/2
Ϫ
Ϫ
Ϫ
ϩ
ϩ
ϩ
Ϫ
Ϫ
ϩ
Ϫ
Ϫ
ϩ
ϩ
ϩ
5 921 366.7 1295.46
1321.56
3.7
Ϫ2.1
3.2
Ϫ0.1
Ϫ0.9
2.2
Ϫ15.4
Ϫ15.4
Ϫ15.4
Ϫ15.4
Ϫ15.4
Ϫ15.3
Ϫ15.4
Ϫ15.4
Ϫ15.4
Ϫ15.3
Ϫ15.3
Ϫ15.3
Ϫ15.3
Ϫ15.2
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0.0
Q(1 )
1339.43
1346.74
1369.56
1400.17
1301.42
1318.74
1326.25
1328.14
1348.31
1360.90
1379.70
1408.18 Ϫ41.9
3.6
Ϫ0.6
Ϫ2.4
Ϫ2.1
3.6
Ϫ0.5
Ϫ1.1
1/2
Ϫ1/2
Ϫ1/2
Ϫ1/2
1/2
Ϫ1/2
1
2
3
3
2
2
3
3
3
2
3
2
2
3
3
2
2
2
3
2
2
2
1
1
1
2
1/2
Ϫ1/2
Ϫ1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
1/2
Ϫ1/2
Ϫ1/2
Ϫ1/2
1/2
5/2
5/2
5/2
5/2
5/2
5/2
5/2
5/2
5/2
5/2
5/2
5/2
ϩ
ϩ
ϩ
Ϫ
Ϫ
Ϫ
ϩ
ϩ
ϩ
ϩ
Ϫ
Ϫ
5 921 366.7 1908.66
Ϫ1.6
Ϫ0.2
7.6
Ϫ10.9
Ϫ10.9
Ϫ10.9
Ϫ10.9
Ϫ10.9
Ϫ10.8
Ϫ10.9
Ϫ10.9
Ϫ10.8
Ϫ10.7
Ϫ10.8
Ϫ10.8
1.0
1.0
0.1
0.1
0.0
0.0
1.0
1.0
1.0
0.1
0.0
0.0
Q(2 )
1940.85
1968.26
2017.10
2051.37
2091.93
1924.56
1951.24
1958.09
1990.43
2046.38
2075.70
9.8
18.9
30.5
1.7
Ϫ0.0
Ϫ0.2
8.0
17.3
25.6
1
2
12
11
12
11
12
12
11
12
12
Ϫ12
Ϫ11
Ϫ11
Ϫ10
Ϫ10
Ϫ9
Ϫ9
Ϫ9
Ϫ8
Ϫ1/2 Ϫ23/2
1/2 Ϫ23/2
Ϫ1/2 Ϫ21/2
1/2 Ϫ21/2
Ϫ1/2 Ϫ19/2
1/2 Ϫ19/2
1/2 Ϫ19/2
Ϫ1/2 Ϫ17/2
Ϫ1/2 Ϫ15/2
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
5 999 028.8 1202.87
1259.35
1.8
Ϫ0.4
Ϫ0.3
10.2
1.3
Ϫ4.8
Ϫ0.4
1.1
7.0
7.0
6.4
6.4
5.8
5.7
5.8
5.2
4.6
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0.0
Q(11 )
1315.06
1379.02
1449.60
1460.84
1527.57
1615.16
1819.88
16.4
1
2
12
12
12
12
12
12
12
12
12
12
12
13
12
13
13
12
13
Ϫ12
Ϫ11
Ϫ10
Ϫ9
Ϫ8
Ϫ7
Ϫ6
Ϫ5
Ϫ4
Ϫ12
Ϫ11
Ϫ13
Ϫ10
Ϫ12
Ϫ11
Ϫ9
1/2 Ϫ25/2
1/2 Ϫ23/2
1/2 Ϫ21/2
1/2 Ϫ19/2
1/2 Ϫ17/2
1/2 Ϫ15/2
1/2 Ϫ13/2
1/2 Ϫ11/2
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
5 999 028.8
76.05
82.72
91.19
0.3
1.4
Ϫ0.2
0.0
0.5
1.8
Ϫ1.8
Ϫ0.2
0.6
3.9
2.4
2.5
0.3
Ϫ0.2
1.1
1.8
6.8
6.2
5.7
5.1
4.6
4.0
3.5
2.9
2.4
6.9
6.3
6.9
5.8
6.3
5.8
5.2
5.3
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Q(12 )
101.14
113.47
129.21
151.01
179.55
221.75
801.68
872.30
886.49
956.49
962.55
1052.16
1057.70
1160.24
1/2
Ϫ9/2
1/2 Ϫ25/2
1/2 Ϫ23/2
Ϫ1/2 Ϫ25/2
1/2 Ϫ21/2
Ϫ1/2 Ϫ23/2
Ϫ1/2 Ϫ21/2
1/2 Ϫ19/2
Ϫ10
Ϫ1/2 Ϫ19/2
2.5
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J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Tamassia, Brown, and Evenson
7279
TABLE II. ͑Continued.͒
␯L
B0
o–c
ץ␯/ץB0
F
MF
←
MI
MJ
p
͑MHz͒
͑mT͒
͑MHz͒ ͑MHz/mT͒
wt
1
2
3
2
3
Ϫ8
Ϫ9
Ϫ7
Ϫ8
1/2 Ϫ17/2
Ϫ1/2 Ϫ17/2
1/2 Ϫ15/2
ϩ
ϩ
ϩ
ϩ
1182.77
1294.27
1341.81
1460.84
2.6
Ϫ1.2
0.0
4.7
4.7
4.1
4.2
1.0
1.0
1.0
1.0
1
1
1
Ϫ1/2 Ϫ15/2
4.5
1
2
14
14
14
14
14
14
13
13
14
13
14
13
14
13
14
13
14
13
14
13
13
13
14
13
14
13
14
13
14
13
14
13
13
12
12
11
13
13
11
12
10
12
10
11
9
11
9
10
8
10
9
13
14
12
13
11
12
10
11
9
1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
Ϫ1/2
Ϫ1/2
1/2
27/2
27/2
25/2
25/2
23/2
23/2
27/2
25/2
21/2
25/2
21/2
23/2
19/2
23/2
19/2
21/2
17/2
21/2
17/2
19/2
19/2
27/2
27/2
25/2
25/2
23/2
23/2
21/2
21/2
19/2
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
5 999 028.8
344.58
363.21
373.29
391.33
406.33
425.02
430.28
445.45
445.45
464.82
464.82
483.47
493.09
504.70
514.34
530.20
552.67
552.67
575.27
586.77
610.99
1217.35
1275.29
1315.06
1378.55
1431.23
1499.58
1569.16
1643.86
1737.59
Ϫ0.8
Ϫ0.2
Ϫ1.5
Ϫ1.7
Ϫ2.8
Ϫ1.1
Ϫ2.5
5.8
Ϫ6.7
Ϫ6.6
Ϫ6.3
Ϫ6.1
Ϫ5.8
Ϫ5.6
Ϫ6.7
Ϫ6.4
Ϫ5.3
Ϫ6.2
Ϫ5.1
Ϫ5.9
Ϫ4.8
Ϫ5.7
Ϫ4.6
Ϫ5.4
Ϫ4.3
Ϫ5.2
Ϫ4.1
Ϫ4.8
Ϫ4.7
Ϫ6.7
Ϫ6.7
Ϫ6.2
Ϫ6.2
Ϫ5.6
Ϫ5.7
Ϫ5.1
Ϫ5.2
Ϫ4.6
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Q(13 )
1/2
Ϫ3.7
1.1
Ϫ1/2
Ϫ1/2
1/2
Ϫ3.2
Ϫ2.8
Ϫ2.4
Ϫ0.5
Ϫ0.2
Ϫ2.5
2.8
Ϫ0.9
0.6
Ϫ1.5
Ϫ1.6
1.1
Ϫ1.6
Ϫ2.1
Ϫ0.2
0.7
Ϫ0.6
Ϫ1.4
Ϫ1.4
Ϫ0.8
1/2
Ϫ1/2
Ϫ1/2
1/2
1/2
Ϫ1/2
Ϫ1/2
1/2
Ϫ1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
Ϫ1/2
1
2
15
15
14
15
15
14
14
14
1/2
Ϫ1/2
Ϫ1/2
1/2
29/2
29/2
29/2
27/2
ϩ
ϩ
ϩ
ϩ
5 999 028.8 1674.97
Ϫ0.1
Ϫ6.7
Ϫ6.5
Ϫ6.6
Ϫ6.2
1.0
0.0
1.0
1.0
Q(14 )
1709.27 Ϫ23.2
1763.83
1799.81
0.3
Ϫ0.1
1
2
16
15
16
15
16
15
16
15
16
15
16
15
16
15
16
15
16
Ϫ16
Ϫ15
Ϫ14
Ϫ14
Ϫ14
Ϫ13
Ϫ13
Ϫ12
Ϫ12
Ϫ11
Ϫ11
Ϫ15
Ϫ16
Ϫ14
Ϫ15
Ϫ13
Ϫ14
Ϫ1/2 Ϫ31/2
1/2 Ϫ31/2
1/2 Ϫ29/2
1/2 Ϫ29/2
Ϫ1/2 Ϫ27/2
1/2 Ϫ27/2
Ϫ1/2 Ϫ25/2
1/2 Ϫ25/2
Ϫ1/2 Ϫ23/2
1/2 Ϫ23/2
Ϫ1/2 Ϫ21/2
1/2 Ϫ31/2
Ϫ1/2 Ϫ31/2
1/2 Ϫ29/2
Ϫ1/2 Ϫ29/2
1/2 Ϫ27/2
Ϫ1/2 Ϫ27/2
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
6 032 811.3
810.90
863.66
865.51
924.37
928.42
2.9
3.1
Ϫ1.8
Ϫ0.7
1.1
0.6
0.5
Ϫ1.0
Ϫ1.0
1.1
7.2
7.2
6.6
6.8
6.3
6.3
5.9
5.8
5.4
5.4
5.0
7.3
7.3
6.8
6.9
6.4
6.4
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0.0
Q(15 )
993.28
1000.87
1073.72
1085.70
1167.48
1185.35
1718.98
1798.69
1836.54
1920.37
1972.59
1.4
Ϫ3.2
Ϫ1.4
Ϫ0.1
Ϫ4.3
Ϫ5.7
2060.53 Ϫ13.8
1
2
16
16
16
17
16
16
17
16
16
16
15
17
15
14
16
13
Ϫ1/2
1/2
Ϫ1/2
1/2
1/2
Ϫ1/2
1/2
33/2
31/2
31/2
33/2
29/2
29/2
31/2
27/2
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
6 032 811.3
609.55
626.29
648.58
662.77
669.42
693.86
706.80
745.23
Ϫ0.8
0.3
Ϫ1.5
Ϫ2.5
Ϫ3.4
2.0
Ϫ7.1
Ϫ6.9
Ϫ6.7
Ϫ7.1
Ϫ6.4
Ϫ6.3
Ϫ6.7
Ϫ5.8
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Q(16 )
1.5
0.9
Ϫ1/2
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7280
J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Tamassia, Brown, and Evenson
TABLE II. ͑Continued.͒
␯L
B0
o–c
ץ␯/ץB0
F
MF
15
12
11
13
12
10
11
9
10
Ϫ16
Ϫ15
Ϫ14
Ϫ13
Ϫ17
Ϫ12
Ϫ16
Ϫ15
Ϫ14
Ϫ10
Ϫ13
Ϫ9
←
MI
1/2
Ϫ1/2
Ϫ1/2
1/2
1/2
Ϫ1/2
1/2
MJ
p
͑MHz͒
͑mT͒
͑MHz͒ ͑MHz/mT͒
wt
1
7
6
6
7
7
6
7
6
7
6
6
6
6
7
6
7
7
7
6
7
6
7
6
7
7
29/2
25/2
23/2
25/2
23/2
21/2
21/2
19/2
19/2
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
756.03
805.15
876.15
878.95
955.30
960.10
1047.05
1062.83
1158.74
293.31
312.55
334.15
359.29
374.34
388.48
397.93
424.52
455.51
463.75
491.54
513.53
533.08
575.01
582.56
641.78
0.5
0.4
1.8
Ϫ6.3
Ϫ5.4
Ϫ4.9
Ϫ5.4
Ϫ5.0
Ϫ4.5
Ϫ4.5
Ϫ4.0
Ϫ4.1
7.2
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
3.7
Ϫ0.5
Ϫ1.1
Ϫ1.2
Ϫ1.8
Ϫ0.2
Ϫ0.5
Ϫ0.5
1.5
Ϫ1/2
1/2
1/2 Ϫ33/2
1/2 Ϫ31/2
1/2 Ϫ29/2
1/2 Ϫ27/2
Ϫ1/2 Ϫ33/2
1/2 Ϫ25/2
Ϫ1/2 Ϫ31/2
Ϫ1/2 Ϫ29/2
Ϫ1/2 Ϫ27/2
1/2 Ϫ11/2
Ϫ1/2 Ϫ25/2
1/2 Ϫ19/2
Ϫ1/2 Ϫ23/2
1/2 Ϫ17/2
Ϫ1/2 Ϫ21/2
Ϫ1/2 Ϫ19/2
6.8
6.3
1.7
5.9
1.4
7.2
2.0
5.4
2.0
6.8
3.8
6.4
2.5
5.9
2.3
4.6
0.2
5.5
1.7
4.1
Ϫ12
Ϫ8
Ϫ11
Ϫ10
1.0
5.1
1.4
3.7
0.9
4.6
2.6
4.2
1
2
18
18
17
18
17
18
17
18
17
18
17
18
18
17
17
16
16
1/2
1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
Ϫ1/2
Ϫ1/2 Ϫ35/2
1/2 Ϫ35/2
Ϫ1/2 Ϫ33/2
1/2 Ϫ33/2
Ϫ1/2 Ϫ33/2
35/2
33/2
35/2
31/2
33/2
29/2
31/2
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
ϩ
ϩ
ϩ
ϩ
ϩ
6 032 811.3 1122.30
1190.75
1.9
1.4
0.0
Ϫ1.3
2.7
0.3
Ϫ7.1
Ϫ6.7
Ϫ7.1
Ϫ6.3
Ϫ6.7
Ϫ5.9
Ϫ6.3
7.7
7.7
7.2
7.2
6.8
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Q(17 )
1204.42
1267.72
1278.89
1356.06
15
15
1362.49
0.5
Ϫ18
Ϫ17
Ϫ17
Ϫ16
Ϫ16
6 061 914.8 1752.14
1801.61
Ϫ0.5
Ϫ0.6
Ϫ1.7
Ϫ1.6
Ϫ4.4
1855.28
1911.05
1971.43
1
2
19
19
19
19
18
19
18
19
18
19
18
19
18
18
18
18
18
19
18
19
19
18
19
18
19
18
19
Ϫ19
Ϫ18
Ϫ17
Ϫ16
Ϫ18
Ϫ15
Ϫ17
Ϫ14
Ϫ16
Ϫ13
Ϫ15
Ϫ12
Ϫ14
Ϫ13
Ϫ12
Ϫ18
Ϫ17
Ϫ19
Ϫ16
Ϫ18
Ϫ17
Ϫ15
Ϫ16
Ϫ14
Ϫ15
Ϫ13
Ϫ14
Ϫ1/2 Ϫ39/2
Ϫ1/2 Ϫ35/2
Ϫ1/2 Ϫ33/2
Ϫ1/2 Ϫ31/2
1/2 Ϫ39/2
Ϫ1/2 Ϫ29/2
1/2 Ϫ35/2
Ϫ1/2 Ϫ27/2
1/2 Ϫ33/2
Ϫ1/2 Ϫ25/2
1/2 Ϫ31/2
Ϫ1/2 Ϫ23/2
1/2 Ϫ29/2
1/2 Ϫ27/2
1/2 Ϫ25/2
1/2 Ϫ37/2
1/2 Ϫ35/2
Ϫ1/2 Ϫ39/2
1/2 Ϫ33/2
Ϫ1/2 Ϫ37/2
Ϫ1/2 Ϫ33/2
1/2 Ϫ31/2
Ϫ1/2 Ϫ33/2
1/2 Ϫ29/2
Ϫ1/2 Ϫ29/2
1/2 Ϫ27/2
Ϫ1/2 Ϫ27/2
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
6 061 914.8
187.80
198.06
209.73
222.81
236.68
236.68
250.22
255.23
265.56
275.34
283.25
298.39
303.08
325.72
352.30
1193.59
1261.52
1270.12
1338.14
1341.41
1421.10
1424.26
1511.31
1521.91
1612.11
1634.15
1728.14
Ϫ0.9
0.7
1.1
1.9
Ϫ0.9
8.5
0.8
0.2
1.5
Ϫ1.2
0.1
Ϫ0.4
0.6
2.1
2.2
0.1
7.6
7.2
6.8
6.4
7.6
6.0
7.2
5.6
6.7
5.2
6.3
4.8
5.9
5.5
5.1
7.7
7.2
7.7
6.8
7.3
6.9
6.4
6.5
6.0
6.1
5.6
5.7
1.0
1.0
1.0
1.0
1.0
0.1
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Q(18 )
1.7
2.1
Ϫ0.3
0.8
Ϫ0.1
Ϫ0.6
Ϫ4.4
0.1
0.4
Ϫ1.4
0.9
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On: Thu, 04 Dec 2014 10:52:57

J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Tamassia, Brown, and Evenson
7281
TABLE II. ͑Continued.͒
␯L
B0
o–c
ץ␯/ץB0
F
MF
←
MI
MJ
p
͑MHz͒
͑mT͒
͑MHz͒ ͑MHz/mT͒
wt
1
1
1
8
9
8
Ϫ12
Ϫ13
Ϫ11
1/2 Ϫ25/2
Ϫ1/2 Ϫ25/2
1/2 Ϫ23/2
ϩ
ϩ
ϩ
1763.25
1862.28
1914.85
0.7
0.2
Ϫ0.2
5.2
5.3
4.8
1.0
1.0
1.0
1
2
20
20
20
20
20
20
20
20
19
20
19
20
19
19
20
19
20
19
20
19
20
19
19
20
19
20
19
20
20
19
19
18
18
17
17
16
19
16
19
15
18
18
15
17
14
19
20
18
19
17
16
17
15
16
14
15
1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
Ϫ1/2
Ϫ1/2
1/2
39/2
39/2
37/2
37/2
35/2
35/2
33/2
33/2
39/2
31/2
37/2
31/2
37/2
35/2
29/2
35/2
29/2
39/2
39/2
37/2
37/2
35/2
33/2
33/2
31/2
31/2
29/2
29/2
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
6 061 914.8
309.61
325.72 Ϫ18.0
Ϫ3.0
Ϫ7.6
Ϫ7.3
Ϫ7.4
Ϫ7.0
Ϫ6.9
Ϫ6.6
Ϫ6.5
Ϫ6.3
Ϫ7.6
Ϫ6.1
Ϫ7.4
Ϫ5.9
Ϫ7.2
Ϫ7.0
Ϫ5.7
Ϫ6.8
Ϫ5.5
Ϫ7.5
Ϫ7.5
Ϫ7.1
Ϫ7.1
Ϫ6.7
Ϫ6.3
Ϫ6.3
Ϫ5.9
Ϫ5.9
Ϫ5.5
Ϫ5.5
1.0
0.0
0.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Q(19 )
328.28
341.22
347.45
360.55
368.91
382.74
386.61
392.31
392.31
407.15
407.15
414.90
420.80
430.09
434.72
1384.91
1434.02
1460.83
1512.73
1545.44
1640.69
1698.87
1748.60
1810.30
1872.13
1937.50
30.7
0.1
Ϫ0.2
0.8
Ϫ0.1
4.0
Ϫ2.5
Ϫ4.4
Ϫ0.8
2.8
Ϫ2.3
Ϫ3.5
2.3
Ϫ3.0
Ϫ0.1
0.2
Ϫ1.2
0.9
Ϫ0.0
0.1
1/2
Ϫ1/2
Ϫ1/2
1/2
1/2
Ϫ1/2
Ϫ1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
Ϫ1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
Ϫ0.3
Ϫ0.1
Ϫ0.9
Ϫ0.2
Ϫ0.4
Ϫ0.4
Ϫ1/2
1/2
1
2
21
21
1/2
41/2
ϩ
6 061 914.8 1941.29
Ϫ0.8
Ϫ7.5
1.0
Q(20 )
1
2
4
4
3
3
4
4
4
3
3
3
4
4
3
3
3
4
3
2
3
2
2
3
1/2
Ϫ1/2
Ϫ1/2
Ϫ1/2
1/2
1/2
Ϫ1/2
1/2
5/2
5/2
5/2
5/2
5/2
3/2
3/2
3/2
3/2
3/2
3/2
Ϫ
Ϫ
Ϫ
ϩ
ϩ
Ϫ
Ϫ
Ϫ
Ϫ
ϩ
ϩ
6 153 279.0
992.23
1021.04
1049.00
1137.92
1164.58
1619.03
1656.23
1659.37
1702.18
1838.23
1886.63
Ϫ0.1
Ϫ1.3
Ϫ3.7
Ϫ7.5
Ϫ5.3
3.5
0.6
0.2
Ϫ1.1
Ϫ7.7
Ϫ8.0
Ϫ11.9
Ϫ11.8
Ϫ11.9
Ϫ11.9
Ϫ11.9
Ϫ7.6
Ϫ7.5
Ϫ7.6
Ϫ7.5
Ϫ7.6
1.0
1.0
1.0
0.1
1.0
1.0
1.0
1.0
1.0
0.1
0.1
R(2 )
Ϫ1/2
Ϫ1/2
1/2
Ϫ7.6
1
2
6
6
5
6
6
5
5
5
6
5
6
6
6
5
6
5
5
6
6
5
4
4
4
3
4
4
3
5
3
3
4
2
3
3
2
2
3
6
1/2
Ϫ1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
Ϫ1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
1/2
Ϫ1/2
1/2
Ϫ1/2
Ϫ1/2
Ϫ1/2
1/2
9/2
9/2
9/2
7/2
7/2
7/2
9/2
7/2
9/2
7/2
5/2
7/2
5/2
5/2
7/2
5/2
5/2
5/2
9/2
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
ϩ
Ϫ
ϩ
ϩ
Ϫ
ϩ
Ϫ
Ϫ
ϩ
Ϫ
ϩ
ϩ
Ϫ
6 290 222.3
779.39
808.68
854.26
Ϫ1.1
Ϫ3.2
Ϫ1.5
Ϫ0.6
Ϫ2.7
Ϫ1.2
Ϫ1.2
Ϫ1.8
1.6
Ϫ1.0
0.0
0.2
Ϫ0.6
Ϫ1.3
Ϫ0.2
Ϫ0.0
Ϫ0.5
2.8
Ϫ8.6
Ϫ8.4
Ϫ8.5
Ϫ6.7
Ϫ6.6
Ϫ6.8
Ϫ8.5
Ϫ6.6
Ϫ8.5
Ϫ6.6
Ϫ4.9
Ϫ6.7
Ϫ4.8
Ϫ4.9
Ϫ6.5
Ϫ4.8
Ϫ4.9
Ϫ4.9
Ϫ8.7
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
R(4 )
1000.02
1033.72
1055.61
1085.58
1092.89
1126.59
1387.42
1387.42
1441.53
1432.74
1465.07
1485.11
1517.31
1918.91
1991.95
768.28
1/2
Ϫ1.2
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7282
J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Tamassia, Brown, and Evenson
TABLE II. ͑Continued.͒
␯L
B0
o–c
ץ␯/ץB0
F
MF
←
MI
MJ
p
͑MHz͒
͑mT͒
͑MHz͒ ͑MHz/mT͒ wt
6
5
6
6
5
5
5
6
6
5
5
6
6
6
5
6
5
5
5
6
6
5
5
5
4
5
5
4
6
5
5
4
4
3
5
4
4
3
4
3
4
3
Ϫ1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
Ϫ1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
9/2
9/2
7/2
7/2
7/2
9/2
7/2
9/2
9/2
7/2
7/2
5/2
5/2
7/2
5/2
7/2
5/2
5/2
5/2
5/2
5/2
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
ϩ
Ϫ
ϩ
ϩ
ϩ
ϩ
Ϫ
Ϫ
ϩ
Ϫ
ϩ
Ϫ
ϩ
ϩ
ϩ
ϩ
797.13
840.46
981.96
Ϫ1.9
0.2
Ϫ0.8
0.3
0.1
0.5
0.9
Ϫ0.7
1.2
0.4
1.7
Ϫ8.5
Ϫ8.6
Ϫ6.8
Ϫ6.7
Ϫ6.9
Ϫ8.6
Ϫ6.8
Ϫ8.6
Ϫ8.5
Ϫ7.0
Ϫ6.8
Ϫ5.0
Ϫ4.9
Ϫ6.8
Ϫ5.1
Ϫ6.7
Ϫ4.9
Ϫ5.2
Ϫ5.0
Ϫ5.1
Ϫ4.9
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0.1
1015.14
1034.24
1066.23
1070.53
1108.19
1142.79
1313.29
1356.19
1354.08
1396.86
1411.98
1424.55
1454.28
1473.89
1798.48
1859.66
1936.67
1997.68
Ϫ1/2
1/2
Ϫ1/2
1/2
3.1
0.3
Ϫ0.0
Ϫ0.9
1.7
Ϫ1.2
Ϫ1.2
Ϫ0.9
2.1
1/2
Ϫ1/2
Ϫ1/2
1/2
Ϫ1/2
1/2
Ϫ1/2
6.9
1
2
9
8
9
8
9
9
8
9
9
8
8
8
9
8
9
9
8
9
9
8
8
7
7
6
6
6
5
6
8
5
9
7
8
4
6
7
1/2 15/2
Ϫ1/2 15/2
1/2 13/2
Ϫ1/2 13/2
1/2 11/2
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
Ϫ
ϩ
Ϫ
Ϫ
Ϫ
ϩ
Ϫ
Ϫ
6 493 911.5
695.29
780.13
800.33
896.64
942.44
Ϫ0.3
0.4
Ϫ1.7
0.5
0.3
Ϫ0.7
0.3
Ϫ0.2
1.4
1.3
1.4
2.1
1.7
0.7
1.3
0.6
Ϫ0.1
0.5
Ϫ7.0
Ϫ7.0
Ϫ6.1
Ϫ6.1
Ϫ5.2
Ϫ5.1
Ϫ5.2
Ϫ4.3
Ϫ4.2
Ϫ4.4
Ϫ7.0
Ϫ4.2
Ϫ7.0
Ϫ6.1
Ϫ6.1
Ϫ3.3
Ϫ5.2
Ϫ5.2
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
R(7 )
Ϫ1/2 11/2
Ϫ1/2 11/2
976.75
1054.80
1143.76
1186.25
1234.53
1242.78
1282.22
1298.19
1428.20
1492.57
1509.75
1679.09
1754.51
1/2
Ϫ1/2
1/2
9/2
9/2
9/2
Ϫ1/2 15/2
Ϫ1/2 9/2
1/2 15/2
Ϫ1/2 13/2
1/2 13/2
Ϫ1/2
7/2
Ϫ1/2 11/2
1/2 11/2
2
stants B and D , which are better determined in the Fourier
determinable for atomic F in the P state and h3/2 is equal
0
0
3/2
4
18
2
transform investigation, possibly because more high-J lines
were included. The determination of the full set of magnetic
hyperfine parameters arises from the detection of resolved
hyperfine structures involving both spin states and both
lambda doublets. All these parameters have been fitted in the
present analysis and are shown in Table III. The zero-field
transitions predicted using the final parameters of Table III
are given in Table IV.
to aϩ(1/2)(bϩc). If the overlap integral is included,
and c turn out to be 0.226 and 0.844, respectively, and ͉A͉ is
188 cm , which is close to the experimental value. If the
calculation is repeated with different methods ͑such as esti-
mating c from atomic and molecular values for the Jϭ 2
and ⍀ϭ 2 components͒, ͉A͉ varies between 176 and 206
c
F
2
O
Ϫ1
1
F
1
Ϫ1
cm
.
The lambda-doubling parameters have been reliably de-
The magnitude ͉A͉ of the spin–orbit interaction term can
be estimated from the atomic spin–orbit constants ␰ and ␰
according to the equation
termined; however, the set of data is not extended enough to
allow the determination of the centrifugal distortion correc-
tion, which has been constrained to the literature value.3
The magnetic hyperfine constants are related to the elec-
F
O
2
F
2
O
͉
A
͉
ϭc
͉
␰F
͉
ϩc
͉
␰O
͉
,
͑13͒
20
tronic wave functions and provide useful information on
once the expansion coefficients of the atomic orbitals are
the distribution of the unpaired electron͑s͒ in the molecule.
known. Where possible, the coefficient c is evaluated from
The hyperfine parameter a is related to the orbital distribu-
F
Ϫ3
the hyperfine structure parameters. Although several meth-
tion
͗
r
͘l , where r is the distance between the unpaired
ods have been used in previous work,1
8,19
we think that the
electron and the F nucleus; the Fermi contact parameter b is
F
2
2
most reliable procedure is based on the relationship c
ϭ h /A3/2 , where A3/2 is the magnetic hyperfine parameter
a measure of the spin density at the nucleus
͉⌿(0)͉ ; c and
F
2
2
d are associated to the angular spin distribution, (3 cos ␪
͗
3
3/2
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J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
TABLE III. Molecular parameters obtained for FO in the ⌸ state.
Tamassia, Brown, and Evenson
7283
2
TABLE IV. Predicted fine-structure transitions of FO in zero magnetic field.
Parametera
This work
Previous works
␯
␯
Transition
͑MHz͒
p
͑MHz͒
p
B0
31 539.819͑94͒
0.129 79͑44͒
31 539.450͑33͒b
0.128 935͑33͒b
D0
1
5 317 782.5
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
5 313 496.8
5 374 103.4
5 427 654.6
5 488 721.5
5 544 447.0
5 605 972.3
5 663 869.2
5 725 848.7
5 785 911.0
5 848 337.9
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
P(10 )
07H
e
2
1
Ϫ0.39
0
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
5
370 246.3
P(9 )
Ϫ1
Ϫ193.28͑97͒b
16.8͑16͒b
A /cm
Ϫ196.108 686͑50͒
12.052͑10͒
5 431 083.1
5 485 721.6
5 547 018.2
5 603 829.9
0
P(8 )
AD
P(7 )
pϩ2q
428.104͑61͒
Ϫ0.45
Ϫ1.5255͑94͒
438.7͑18͒b
P(6 )
2
c
1
0 (p ϩ2q )
Ϫ0.45͑120͒
D
D
P(5 )
q
Ϫ2.1b
5
665 583.0
P(4 )
a
b
bϩc
d
779.4͑16͒
406.3͑20͒
Ϫ196.1͑22͒
977.79͑46͒
696͑18͒d
5 724 563.5
P(3 )
5
5
786 767.6
847 909.6
P(2 )
1
2
P(1 )
1
2
1
5
943 652.0
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
5 942 796.2
5 945 475.9
5 946 233.5
5 951 049.5
5 952 240.4
5 959 180.1
5 960 804.8
5 969 850.0
5 971 909.3
5 983 036.3
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
gL
gS
0.999 34͑14͒
2.001 11͑22͒
0.112 7͑28͒
Ϫ0.93͑17͒
0.477͑68͒
Q(1 )
5 944 193.1
947 942.2
Q(2 )
1
2
1
1
1
1
0 g
0 g
0 gЈ
0 (gЈϪgЈ)
l
1
5
4
Q(3 )
2
r
1
4
5 948 916.4
5 954 796.2
5 956 203.9
5 964 198.9
5 966 040.8
Q(4 2)
r
2
1
0.688 6͑37͒
5.257 73
l
r
Q(5 )
2
gN
1
Q(6 )
2
␴f
1
1.9
290
Q(7 )
2
No. of trans.
1
Q(8 )
2
1
5
5
976 130.2
978 407.1
aValues in MHz where appropriate. The numbers in parentheses represent
one standard deviation of the least-squares fit, in units of the last quoted
decimal place. Parameters with no standard deviation are held fixed in the
present analysis.
Reference 4.
Reference 3.
Reference 1. This is a value for h3/2ϭaϩ1/2(bϩc). Our value for this
parameter is 681.4͑19͒ MHz.
Calculated value.
Q(9 )
2
1
2
Q(10 )
1
6 101 077.5
167 571.8
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
ϩ
6 102 361.2
6 165 860.7
6 233 413.8
6 297 317.5
6 366 973.5
6 431 263.4
6 503 000.2
6 567 655.8
6 641 448.8
6 706 447.3
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
Ϫ
R(1 )
2
b
1
6
c
R(2 2)
1
d
6 231 275.6
6 299 882.6
R(3 2)
1
R(4 )
2
e
1
6
6
6
6
363 981.8
434 681.1
499 156.6
571 924.8
f
R(5 2)
␴
is the overall standard deviation of the fit.
1
R(6 )
2
1
R(7 )
2
1
R(8 )
2
Ϫ1)/r3͘s and
(sin ␪)/r3͘s , respectively, where ␪ is the angle
2
1
͗
6 636 754.9
711 565.7
R(9 )
2
between the vector r and the internuclear axis. The above-
mentioned expectation values for FO, CF,21 and the F atom
are given in Table V.
1
2
6
R(10 )
22
The spin average
͗
rϪ3͘s can be calculated from dϩc/3
a value of 4/5 and
2
2
by giving
͗
␲
͉
sin ␪
͉␲
͘
͗
␲
͉
3 cos ␪Ϫ1
͉␲
͘
a
The calculated values for gЈ (0.484ϫ10Ϫ4) and glЈ
ϪgЈ (0.6787ϫ10 ) are in excellent agreement with the
observed values 0.477ϫ10 and 0.6886ϫ10
The orbital g factor g can be expressed as a sum of a
main term ͑1.0͒ plus relativistic (
r
value of Ϫ2/5, which are correct for atomic p␲ orbitals and
good approximations for molecular ␲ orbitals. The ratios
Ϫ2
r
Ϫ4
Ϫ2
Ϫ3
2
3
2
3
.
͗
r
͘s
͗ ͗
: (3 cos ␪Ϫ1)/r ͘s : (sin ␪)/r ͘s , expected to be 1:
L
Ϫ0.4: 0.8 for a p atomic orbital are 1: Ϫ0.52: 0.84 for FO
␲
14
͗ ͘
␦g ), orbit–orbit
rel av
and 1: Ϫ0.35: 0.78 for CF. These values, greater than the
corresponding atomic ratios, suggest that the unpaired elec-
tron in FO is in an orbital which is slightly polarized from a
pure p orbital.
A full set of g factors, shown in Table III, has been
determined in the present work ͑see Refs. 16 and 23 for a
detailed description of the Zeeman parameters͒. Estimates of
some of the g factors can be obtained from the relationships
10,24
(
͗
␦g ), and nonadiabatic (⌬g ) contributions.
͘
The
electron spin g factor corrected for quantum electrodynamic
and relativistic effects is g ϭ2.002 32(1ϩ ␦g ). The
orb av
L
͗
͘
rel av
S
relativistic correction can therefore be estimated from the
experimental value for g and is Ϫ6.0ϫ10 for FO. As-
Ϫ4
S
suming that the orbit–orbit and nonadiabatic contributions
are negligible compared with the relativistic term,10 gL can
be calculated using the relationship g ϭ1.0ϩ ␦g
ϭ1.0–6.0ϫ10 ϭ0.9994, which is in excellent agreement
with the experimental value 0.999 34͑14͒. The nonadiabatic
͗ ͘
L
rel av
Ϫ4
glϭϪ␥/2B,
͑14͒
͑15͒
͑16͒
͑17͒
glЈӍp/2B,
contribution can also be estimated from the relationship
Ϫ5
grЈӍϪq/B,
⌬gLϭq/2B. The calculated value of 2.4ϫ10
that the contribution ␦g
contribution can be calculated from the corresponding
confirms
͗
͘
is dominant. This relativistic
rel av
gЈϪgЈӍ͑pϩ2q͒/2B.
l
r
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7284
J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Tamassia, Brown, and Evenson
TABLE V. Hyperfine structure parameters ͑in MHz͒ and expectation values
The constraint of the parameter A or ␥ to zero in the fit
D
Ϫ3
19
of distribution functions over the electronic wave function ͑in m ͒ of
F
has the effect of modifying several parameters in the Hamil-
tonian. If we apply Brown and Watson’s unitary transforma-
tion to the Zeeman Hamiltonian, we can show that the pa-
for FO, CF, and atomic fluorine.
Parameter
FOa
CFb
Fc
rameter g is one which is affected and so becomes an
l
a
bF
c
779.4͑16͒
205.5͑15͒
Ϫ602.4͑30͒
977.79͑46͒
10.485
3.2997
Ϫ5.4027
8.7691
10.452
705.94͑14͒
151.19͑49͒
Ϫ351.6͑14͒
792.195͑98͒
9.502
2.429
Ϫ3.151
7.100
9.075
d
d
effective parameter
˜
g where
l
d
␥
Ϫ30
Ϫ29
Ϫ30
Ϫ30
rϪ3
⌿(0)
(3 cos ␪Ϫ1)/r ͘s
(sin ␪)/r ͘s
˜
g ϭg Ϫ
͑g ϩg ͒,
͑21͒
͑22͒
1
1
1
1
1
0
0
0
0
0
͗
͘
4.96
4.84
l
l
L
S
l
͑
AϪ2B͒
2
2
3
͗
͗
2
3
if ␥ is constrained to zero or
Ϫ30 Ϫ3
͗
r
͘s
5.49
AD
a
˜g ϭg Ϫ
͑g ϩg ͒,
L S
This work.
Reference 21.
Reference 22.
Derived parameter.
l
l
2
B
b
c
d
if A is constrained to zero. Thus the value determined in the
D
fit, ˜g , will differ from the value g according to whichever
l
l
of these two two equations applies. If we accept the correct-
ness of Curl’s relationship, Eq. ͑14͒, and substitute for g in
Eq. ͑21͒, we can rearrange the result to give an expression
for ␥ in terms of experimentally determinable parameters:
l
atomic values for F10 and O,24 which are, respectively, Ϫ1.8
and Ϫ1.3ϫ10 . It is given by
Ϫ4
2
F
2
O
͗
␦g
͘
ϭc
͗
␦g
͘
ϩc
͗
␦g
͘
,
͑18͒
rel av,FO
rel av,F
rel av,O
2
B͑AϪ2B͒˜g
l
2
F
2
O
␥
ϭϪ
.
͑23͒
using values for the coefficients c and c determined from
͕
AϪ2B͓1Ϫ͑g ϩg ͔͒
͖
L S
the magnetic hyperfine parameters. The result is Ϫ1.5
Ϫ4
ϫ10 , which is remarkably different from the experimental
Substitution of values from Table III in Eq. ͑23͒ gives a
value for ␥ ͑the true spin–rotation constant͒ of Ϫ734.3 MHz,
Ϫ4
value of Ϫ6.0ϫ10 but consistent with the values calcu-
2
5
lated by Veseth for other molecules, such as OH, SH, NO,
NS, and ClO.
from which we determine that the true value for g ϭ
l
Ϫ␥/2B is 0.011 64. This latter value is now much closer to
If values for B and ␥ are known, the parameter g should
l
the effective value for
because the correction term on the right-hand side of Eq.
21͒ is comparatively small. If this determination of the
value for ␥ is valid, we can use it to derive a value for AD
˜g l of 0.011 27 than our earlier estimate
be predictable from Curl’s relationship, Eq. ͑14͒. Although
we know the value for B very precisely, we have not deter-
mined a value for the spin–rotation parameter ␥ in our fit
͑
because it is completely correlated with the parameter A .
D
˜
from that of the experimentally determined parameter A by
26
D
Brown and Watson have shown that, in a fit of the param-
eters of a ⌸ Hamiltonian to a set of data, one can either
use of Eq. ͑19͒. In this case, the value for A is 19.847 MHz,
D
2
˜
compared with A ϭ12.052 MHz. A detailed description of
D
chose to constrain ␥ to zero and determine a value for the
effective centrifugal distortion correction to the spin–orbit
the contact transformation of the Zeeman Hamiltonian is
given in the Appendix.
˜
coupling A , where
D
In this paper we have described the detection of the fine-
2
2
structure transition ⌸ ← ⌸ in the FO radical by far-
2
B
1/2
3/2
˜
A ϭA Ϫ␥
͑19͒
D
D
infrared laser magnetic resonance. The spin–orbit constant
͑
AϪ2B͒
A , all four magnetic hyperfine parameters a, b , c, d, and a
0
F
or one can constrain A to zero and determine a value for an
D
full set of g factors have been determined for the first time.
The recorded data form the most accurate measurements on
FO to date. Given sufficient sensitivity, it should be possible
to detect magnetic dipole, pure rotational transitions in the
microwave region. The predicted frequencies are given in
Table VI. We intend to make these observations in the near
future.
effective spin–rotation parameter
˜␥ , where
͑
AϪ2B͒
˜␥ ϭ␥ϪAD
.
͑20͒
2
B
We can estimate the value for ␥ by using Eq. ͑20͒ and sub-
˜
stituting the values of AD ͑strictly A ͒, A and B from Table
D
III. This is actually a value for ˜␥ but, if AD happened to be
very small, it would be close to the true value for ␥. Substi-
tution of this value in Eq. ͑14͒ gives a value for g of
Ϫ0.018, which is very different from the experimental value
l
ACKNOWLEDGMENTS
of 0.011 27. Similar behavior is shown for the energy levels
This work supported in part by NASA Contract No.
W-19, 167. We thank J. B. Burkholder for providing the
ozone used for the experiment and L. R. Zink for helping
with the measurements of the FIR laser lines. F.T. thanks the
European Community, TMR Programme, for his fellowship
͑Contract No. ERBFMBICT950167͒.
2
1
of CF; the authors in that case suggested that the discrep-
ancy was attributable to an inexact interpretation of the ef-
fective parameters in the Hamiltonian ͑in other words, the
assumption which we have made that A is very small is not
D
valid͒.
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J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Tamassia, Brown, and Evenson
7285
TABLE VI. Predicted magnetic dipole rotational transitions of FO in the
͑
ii͒ Ϫi␮ B ␪
͕
g ͓⌽ ͑J ϪS ͒ϩ⌽ ͑J ϪS ͔͒
2
B
0
S
Zx
x
x
Zy
y
y
⌸3/2
spin state.
ϩ͑g ϩg ͒͑⌽ J ϩ⌽ J ͒
͖
L S ,
͑A4͒
͑A5͒
͑A6͒
͑A7͒
l
r
Zx
x
Zy
y
z z
␯
JЈ
FЈ
←
J
F
p
͑MHz͒
͑
͑
͑
iii͒ i␮ B ␪͑g ϩg ͒⌽ L ͑J–SϪJ S ͒,
B
0
S
r
Zz
z
z z
5
5
5
5
/2
/2
/2
/2
3
3/2
3/2
3/2
3/2
2
2
1
1
ϩ
Ϫ
ϩ
Ϫ
156 827.1
156 828.1
157 019.5
157 020.1
2
z
2
iv͒ Ϫi␮ B ␪͑g ϩg ϩg ͒⌽ L ͑S ϪS ͒,
B
0
S
l
r
Zz
z
3
2
2
v͒ i␮ B ␪g ͑⌽ S ϩ⌽ S ͒L J ,
B
0
r
Zx
x
Zy
y
z z
where ⌽Ff is a direction cosine. Terms ͑i͒ and ͑iii͒ are con-
7
7
7
7
/2
/2
/2
/2
4
4
3
3
5/2
5/2
5/2
5/2
3
3
2
2
ϩ
Ϫ
ϩ
Ϫ
219 598.4
219 595.8
219 686.7
219 684.7
tributions to g and give rise only to matrix elements off-
l
diagonal in ⍀ and ⌺. The expectation value of term ͑ii͒ is
zero when calculated over the Hermitian average. Term ͑iv͒
is a contribution to g and term ͑v͒ is not considered because
it is negligible. For a ⌳ state, the matrix elements of term ͑i͒
9
9
9
9
/2
/2
/2
/2
5
5
4
4
7/2
7/2
7/2
7/2
4
4
3
3
ϩ
Ϫ
ϩ
Ϫ
282 339.4
282 345.0
282 390.9
282 395.3
L
2
diagonal in J and M and off-diagonal in ⍀ are of the form
J
͑JϪ⍀͒͑Jϩ⍀ϩ1͔͒1/2
͓
2
1
1
1
1
1/2
1/2
1/2
1/2
6
6
5
5
9/2
9/2
9/2
9/2
5
5
4
4
ϩ
Ϫ
ϩ
Ϫ
345 073.6
345 063.6
345 105.6
345 097.2
i␮ B ␪g ⌳ M
J
,
͑A8͒
B
0
L
2
J͑Jϩ1͒
where the corresponding matrix elements of term ͑iii͒ are
͑JϪ⍀͒͑Jϩ⍀ϩ1͔͒1/2
2J͑Jϩ1͒
͓
1
1
1
1
3/2
3/2
3/2
3/2
7
7
6
6
11/2
11/2
11/2
11/2
6
6
5
5
ϩ
Ϫ
ϩ
Ϫ
407 768.1
407 784.5
407 791.8
407 806.0
2
i␮ B ␪g ⌳ M
J
.
͑A9͒
B
0
S
Brown and Watson showed26 that, if AD is fitted and ␥ is
constrained to zero, the transformation parameter ␪ should be
1
1
1
1
5/2
5/2
5/2
5/2
8
8
7
7
13/2
13/2
13/2
13/2
7
7
6
6
ϩ
Ϫ
ϩ
Ϫ
470 476.0
470 450.9
470 490.8
470 468.8
2
set equal to ␥/͓(AϪ2B)⌳ ͔. Substituting these matrix ele-
ments in Eq. ͑A1͒, we obtain the effective values for
g ,
˜
g and
l
˜
L
␥
˜
g ϭg Ϫ
͑g ϩg ͒,
͑A10͒
͑A11͒
l
l
L
S
͑
AϪ2B͒
APPENDIX: TRANSFORMATION OF THE ZEEMAN
AND MAGNETIC HYPERFINE STRUCTURE
HAMILTONIANS
␥
˜g ϭg ϩ
͑g ϩg ϩg ͒.
S l r
L
L
͑AϪ2B͒⌳2
2
The same transformation also modifies the nuclear hy-
perfine Hamiltonian. The untransformed hyperfine Hamil-
tonian is given by
In a previous paper,26 Brown and Watson transformed
the spin–orbit Hamiltonian and demonstrated that ␥ and A
D
2
are totally correlated for a molecule in a ⌸ state. In this
HhfsϭaI L ϩb͑I S ϩI S ͒ϩ͑bϩc͒I S
z
z
x
x
y
y
z
z
section, we will apply the same contact transformation to the
Zeeman and magnetic hyperfine structure Hamiltonians.
Generally speaking, the procedure consists of calculating
1
2
Ϫ2i␾
2i␾
ϩ d͑e
I S ϩe I S ͒.
͑A12͒
ϩ
ϩ
Ϫ Ϫ
The first term does not contribute to the transformed Hamil-
tonian because it commutes with U. The second and fourth
terms have zero matrix elements when the Hermitian average
is taken. The third term is the only one which needs to be
calculated. The evaluation of the commutator gives
˜
an effective Hamiltonian H from an original operator H
through a unitary transformation of the form
iU
ϪiU
˜
Hϭe He
ϭHϩi͓U,H͔ϩ¯ ,
͑A1͒
where U is a suitable operator. The appropriate choice for U
is ␪ L (J S ϪJ S ) where ␪ is a parameter which will be
defined later.
For the moment let us consider just the Zeeman Hamil-
tonian, which has the form
͓
U,͑bϩc͒I S ͔ϭi␪͑bϩc͒͑J–SϪJ S ͒L I
͑A13͒
z
z
z
z
z z
͕
͖
y x
z
x
y
and the matrix elements diagonal in J, M , and M , and
J
I
off-diagonal in ⍀ are
͑JϪ⍀͒͑Jϩ⍀ϩ1͔͒1/2
͓
i␪͑bϩc͒⌳2
M M .
͑A14͒
J
I
2
J͑Jϩ1͒
H ϭg ␮ B L ϩg ␮ B S Ϫg ␮ B N
Z
Z
L
B
0
Z
S
B
0
Z
r
B
0
2
Substituting ␪ϭ␥/͓(AϪ2B)⌳ ͔ we obtain
ϩg ␮ ͑S B ϩS B ͒,
͑A2͒
l
B
x
x
y
y
␥
͓͑JϪ⍀͒͑Jϩ⍀ϩ1͔͒1/2
2J͑Jϩ1͒
where the smaller terms, depending on g , gЈ and gЈ have
not been included ͓see Eq. ͑10͔͒. The calculation of the com-
N
r
l
Ϫ
͑bϩc͒
M M .
J
I
͑
AϪ2B͒
͑
A15͒
mutator ͓U,H ͔ gives five terms:
Z
The magnetic hyperfine matrix elements off-diagonal in ⍀
and diagonal in J, M , and M are
2
z
͑
i͒ i␮ B ␪g ͑⌽ S ϩ⌽ S ͒L ,
͑A3͒
B
0
L
Zx
x
Zy
y
J
I
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7286
J. Chem. Phys., Vol. 110, No. 15, 15 April 1999
Tamassia, Brown, and Evenson
c ͓͑JϪ⍀͒͑Jϩ⍀ϩ1͔͒1/2
10
R. J. Saykally, K. G. Lubic, A. Scalabrin, and K. M. Evenson, J. Chem.
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ͩ
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͑A16͒
3
2J͑Jϩ1͒
1
1
1
1
1
1
1
2
3
4
5
6
͑
1998͒.
Since b Ϫc/3ϭb we can finally write the effective hyper-
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F
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͑A17͒
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˜
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21
1
22
23
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28
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