THE JOURNAL OF CHEMICAL PHYSICS 127, 194308 共2007兲
The rotational spectrum of CoF in all three spin-orbit components
of the X 3⌽i state
Jeremy J. Harrison and John M. Brown
The Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford OX1 3QZ, United Kingdom
M. A. Flory, P. M. Sheridan,a兲 S. K. McLamarrah, and L. M. Ziurysb兲
Department of Chemistry and Steward Observatory, Department of Astronomy, University of Arizona,
933 North Cherry Avenue, Tucson, Arizona 85721, USA
共Received 9 July 2007; accepted 30 August 2007; published online 21 November 2007兲
The pure rotational spectrum of cobalt monofluoride in its X 3⌽i electronic state has been measured
in the frequency range of 256– 651 GHz using direct absorption techniques. CoF was created by
reacting cobalt vapor with F2 in helium at low pressure 共25– 30 mTorr兲. All three spin components
were identified in the spectrum of this species, two of which exhibited lambda doubling. Each spin
component showed hyperfine splittings from both nuclei: an octet pattern arising from the 59Co spin
of I = 7 / 2, which is further split into doublets due to the 19F nucleus 共I = 1 / 2兲. The data were fitted
close to experimental precision using an effective Hamiltonian expressed in Hund’s case 共a兲 form,
and rotational, fine structure, hyperfine, and lambda-doubling parameters were determined. There is
evidence that the rotational levels of the highest spin component 3⌽2 are perturbed. The r0 bond
length of CoF was estimated from the rotational constant to be 1.738 014共1兲 Å. This value is in
good agreement with previous studies but much more accurate. The matrix elements necessary for
the complete treatment of ⌳ doubling in a ⌽ state have been derived and are presented for the first
time. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2789427兴
I. INTRODUCTION
Diatomic molecules that contain cobalt are of great interest because their ground states possess large spin and orbital electronic angular momenta: CoH共X 3⌽i兲, CoF共X 3⌽i兲,
CoCl共X 3⌽i兲, CoO共X 4⌬i兲, and CoS共X 4⌬i兲. Since relatively
few ⌬ and ⌽ states have been investigated using rotational
spectroscopy, studies of such molecules are of particular interest to molecular physicists. In order to test an effective
Hamiltonian, the experimental observation of the molecule
in all available spin components is usually required; yet, for
many of the above molecules, the higher spin components
were not accessed until recently. The cobalt magnetic hyperfine interaction 共mhf兲 is also of interest. In particular, the
large magnetic moment 共+4.63␮N兲 and high nuclear spin
共I = 7 / 2兲 of this nucleus result in a broad and often complicated hyperfine pattern. In order to determine a complete set
of hyperfine parameters 共a , b , b + c兲, resolved spectra of at
least three spin components are desirable. While CoH, CoCl,
CoO, and CoS spectra have been analyzed in detail,1–4 a full
rotational analysis of CoF has not yet been achieved.
Cobalt monofluoride has been the subject of several previous studies. In 1994, Adam et al.5 performed the first high
resolution study of its electronic spectrum using laser induced fluorescence 共LIF兲 techniques; they obtained three rotationally resolved bands assigned to the three main components of a 3⌽i – X 3⌽i transition with the selection rule ⌬⍀
= 0. This was followed by two Fourier transform infrared
a兲
Present address: Department of Chemistry and Biochemistry, Canisius
College, Buffalo, NY, USA.
b兲
Electronic mail: [email protected]
0021-9606/2007/127共19兲/194308/11/$23.00
emission spectroscopy studies, in which the C 3⌬i – X 3⌽i,
G 3⌽i – X 3⌽i, and D 3⌬i – X 3⌽i transitions were detected
and analyzed.6,7 The first Hund’s case 共a兲 analysis was reported in 2001 with the observation of the 关18.8兴 3⌽i – X 3⌽i
electronic transition.8 Two years later, Zhang et al. detected
the 关20.6兴 3⌫5 – X 3⌽4 transition using LIF techniques.9 Okabayashi and Tanimoto recorded the first millimeter-wave
spectrum of CoF in the lowest ⍀ = 4 spin component, with
well-resolved cobalt and fluorine hyperfine structures.10
Quite recently, Steimle et al. have observed the hyperfine
splittings within the ⍀ = 3 spin component arising from the
59
Co nucleus11 at molecular beam resolution in an optical
experiment.
In a continuation of the study of the 3d transition metal
diatomic molecules, we have recorded the pure rotational
spectrum of CoF in all three spin components 共⍀ = 4 , 3 , 2兲 of
its ground 3⌽ state. Spectroscopic parameters, including
⌳-doubling and hyperfine effects, have been extracted from
the data using an effective Hamiltonian approach, which required the extension of existing ⌳-doubling theory to ⌽
states. These parameters have provided further insight into
the nature of the bonding and electronic structure of the molecule.
II. EXPERIMENTAL
The pure rotational spectrum of CoF was recorded using
the high-temperature, millimeter/submillimeter direct absorption system of Ziurys et al. at the University of Arizona.
This instrument has been described in detail elsewhere.12
The CoF radical was produced by reacting cobalt vapor
127, 194308-1
© 2007 American Institute of Physics
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194308-2
J. Chem. Phys. 127, 194308 共2007兲
Harrison et al.
with a 10% mixture of F2 in He. Metal vapor was obtained
by melting chips of cobalt 共Aldrich, 99.5%兲 in a Broida-type
oven through resistive heating. Because of the metal’s high
melting point 共⬃1495 ° C兲, the crucible was wrapped in zirconia to provide thermal insulation. A F2 : He gas mixture at
about 25– 30 mTorr pressure was added over the top of the
crucible; the use of an auxiliary dc discharge was not
necessary.
The spectroscopic constants obtained by Ram et al.7 for
individual spin components of the X 3⌽ state aided the initial
search for rotational lines. Transition frequencies were measured by averaging an equal number of scans taken in increasing and decreasing frequency directions. These scans
were 5 – 7 MHz in coverage, and typically two or four scans
were necessary to obtain an adequate signal to noise ratio.
The frequencies of the line centers were determined by fitting the observed lines to Gaussian profiles. The instrumental
accuracy is approximately ±100 kHz with typical linewidths
of 0.6– 2.0 MHz over the range of 256– 651 GHz.
Sample spectra are given in Figs. 2 and 3. Figure 2 displays the J = 27← 26 transition of CoF near 627 GHz for all
three spin-orbit components. The cobalt hyperfine splitting is
clearly exhibited, but the fluorine hyperfine splitting is not
resolved for these high-J transitions. Although no ⌳ doubling is observed for the ⍀ = 4 component, it is clearly evident for both the ⍀ = 3 and 2 substates, being substantially
larger for the latter.
Figure 3 shows the J = 16← 15 transition for CoF in the
⍀ = 4 spin-orbit component near 372 GHz. In this figure,
both the cobalt and fluorine hyperfine splittings are clearly
resolved and are larger than the splittings observed in the
higher-J transition shown in Fig. 2. The relative intensities of
the hyperfine lines point to the F quantum number assignments, shown in the figure. The features marked by a dagger
at high frequency on the figure are rotational transitions arising from the ⍀ = 3 spin component.
IV. ANALYSIS
III. RESULTS
Selected transition frequencies for CoF共X 3⌽i兲 in its
three spin components can be found in Table I. The complete
list is available electronically on EPAPS.13 A total of 32 rotational transitions were measured 共at least ten for each spin
component兲, spanning the frequency range of 256– 651 GHz.
The spectra show hyperfine structure arising from both nuclei: an octet of lines from 59Co共I = 7 / 2兲, each of which is
further split into doublets from the 19F nucleus 共I = 1 / 2兲.
Both the ⍀ = 2 and 3 spin components exhibit ⌳ doubling,
although in the latter case this only becomes apparent at
higher rotational levels 共J ⬇ 20兲 and is accompanied by a
collapsing of the fluorine hyperfine structure, somewhat simplifying the spectrum. Rotational lines that could not be resolved due to spectral congestion, particularly in the region
where rotational lines of the ⍀ = 3 and 4 systems overlap,
were not included in the fit. In total, 545 line frequencies
were recorded. Of this set, 89 lines in the high-J region were
given zero weight in the analysis because they are significantly perturbed, as will be discussed later; most of these are
in the ⍀ = 2 component. The transition frequencies measured
for the 3⌽4 component agree well with those measured earlier by Okabayashi and Tanimoto;10 the present observations
extend the data set to higher frequencies.
Figure 1 shows a stick diagram of the spectral pattern of
the J = 28← 27 transition for all spin components near
650 GHz. Lambda doubling is shown, but not hyperfine
structure. The figure illustrates the uneven splitting of the
fine structure and also gives the relative size of the ⌳ doubling. The sign of the q̃⌽共⍀ = 3兲 parameter and hence the e
and f assignments for the ⍀ = 3 spin component in the figure
can be predicted using perturbation theory and by recognizing the nature of the interacting sigma electronic state 共 3⌺−兲.
Such assignments cannot be made reliably for the
⍀ = 2 spin component due to significant interactions with
other unknown electronic states, so the labels a and b,
corresponding to the lower and upper lambda doublets
respectively, are used instead.
The energy levels of the X 3⌽ state 共v = 0兲 were modeled
using an effective N2 Hamiltonian,
Heff共X 3⌽兲 = Hso + Hrot + Hss + Hld + Hmhf + HQ ,
共1兲
where Hso represents the spin-orbit interaction, Hrot the rotational kinetic energy, Hss the electron spin-spin interaction,
Hld the lambda doubling, Hmhf the magnetic hyperfine interaction, and HQ the nuclear electric quadrupole interaction.
The detailed forms are
1
1
Hso = ALzSz + AD关LzSz,N2兴+ + AH关关LzSz,N2兴+,N2兴+
2
4
1
+ AL关关关LzSz,N2兴+,N2兴+,N2兴+ ,
8
Hrot = BN2 − DN4 + HN6 ,
共2兲
共3兲
2
1
Hss = ␭兵3Sz2 − S2其 + ␭D关兵3Sz2 − S2其,N2兴+
3
3
1
+ ␭H关关兵3Sz2 − S2其,N2兴+,N2兴+
6
+
1
␭L关关关兵3Sz2 − S2其,N2兴+,N2兴+,N2兴+
12
共4兲
1
1
HId = q̃⌽共J−6 + J+6兲 + q̃⌽D关共J−6 + J+6兲,N2兴+
2
4
1
1
− p̃⌽共J−5S− + J+5S+兲 − p̃⌽D关共J−5S− + J+5S+兲,N2兴+
2
4
1
1
+ õ⌽共J−4S−2 + J+4S+2兲 + õ⌽D关共J−4S−2 + J+4S+2兲,N2兴+ ,
2
4
共5兲
HQ =
eQq0
共3I2 − I2兲,
4I1共2I1 − 1兲 1z 1
共6兲
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194308-3
J. Chem. Phys. 127, 194308 共2007兲
Rotational spectrum of CoF
TABLE I. Sample rotational transition frequencies 共in MHz兲 for the X 3⌽ state of CoF.
⍀=4
F1 + 1 ← F1
⍀=3
F+1←F
␯
␯obs-calc
16← 15
15← 14
15← 14
14← 13
14← 13
13← 12
13← 12
12← 11
12← 11
11← 10
11← 10
10← 9
10← 9
9←8
9←8
8←7
16← 15
15← 14
15← 14
14← 13
14← 13
13← 12
13← 12
12← 11
12← 11
11← 10
11← 10
10← 9
10← 9
9←8
9←8
8←7
279 049.861
279 056.185
279 090.896
279 096.816
279 127.591
279 133.097
279 159.932
279 165.182
279 187.940
279 192.992
279 211.541
279 216.569
279 230.687
279 235.976
279 245.322
279 251.344
−0.119
0.052
−0.034
0.096
0.011
0.046
0.020
0.056
0.039
0.040
0.030
0.026
−0.003
0.031
−0.032
0.064
Par
⍀=2
␯
a
␯obs-calc
␯
␯obs-calc
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
b
b
b
b
b
b
279 384.937
279 387.479
279 433.044
279 435.414
279 476.247
279 478.515
279 514.435
279 516.656
279 547.444
279 549.623
279 575.183
279 577.386
279 597.508
279 599.831
0.205
0.218
0.146
0.089
0.085
0.016
0.053
0.005
0.017
−0.040
0.009
−0.042
0.005
−0.030
b
b
b
b
b
b
b
b
b
279 522.427
279 553.259
279 555.418
279 581.071
279 583.302
279 603.436
279 605.821
279 620.306
279 622.858
−0.094
−0.042
−0.120
0.018
−0.008
0.048
0.073
0.117
0.067
Par
a
J = 12← 11
15.5← 14.5
15.5← 14.5
14.5← 13.5
14.5← 13.5
13.5← 12.5
13.5← 12.5
12.5← 11.5
12.5← 11.5
11.5← 10.5
11.5← 10.5
10.5← 9.5
10.5← 9.5
9.5← 8.5
9.5← 8.5
8.5← 7.5
8.5← 7.5
15.5← 14.5
15.5← 14.5
14.5← 13.5
14.5← 13.5
13.5← 12.5
13.5← 12.5
12.5← 11.5
12.5← 11.5
11.5← 10.5
11.5← 10.5
10.5← 9.5
10.5← 9.5
9.5← 8.5
9.5← 8.5
8.5← 7.5
8.5← 7.5
31.5← 30.5
30.5← 29.5
29.5← 28.5
28.5← 27.5
27.5← 26.5
26.5← 25.5
25.5← 24.5
24.5← 23.5
31.5← 30.5
30.5← 29.5
29.5← 28.5
28.5← 27.5
32← 31
31← 30
31← 30
30← 29
30← 29
29← 28
29← 28
28← 27
28← 27
27← 26
27← 26
26← 25
26← 25
25← 24
25← 24
24← 23
32← 31
31← 30
31← 30
30← 29
30← 29
29← 28
29← 28
28← 27
b
b
279 151.398
279 155.526
279 197.691
279 201.460
279 238.197
279 241.834
279 273.005
279 276.684
279 302.563
279 306.172
279 326.249
279 329.962
279 344.133
279 348.342
−0.141
−0.007
0.064
0.017
0.012
−0.029
−0.163
−0.084
0.036
0.032
0.042
−0.012
−0.003
0.047
b
279 616.886
279 390.579
279 393.105
279 438.792
279 441.131
279 482.009
279 484.284
−0.013
−0.006
−0.011
0.036
−0.054
−0.016
−0.080
b
650 211.309
0.010
J = 28← 27c
e
650 357.947
650 217.954
0.012
e
650 365.576
0.274
a
650 996.587
−12.962
650 224.239
0.011
e
650 372.608
0.230
a
651 002.390
−13.234
650 230.183
0.018
e
650 379.335
0.323
a
651 007.815
−13.461
650 235.799
0.035
e
650 385.601
0.391
a
651 012.784
−13.728
650 241.053
0.021
e
650 391.455
0.474
a
651 017.438
−13.899
650 245.979
0.001
e
650 396.661
0.330
a
651 021.655
−14.099
650 250.506
−0.102
e
650 400.889
−0.378
a
651 025.368
−14.402
f
650 361.695
−0.104
b
651 211.866
11.198
f
650 369.069
−0.259
b
651 219.487
12.246
f
650 376.033
−0.372
b
651 226.465
13.077
f
650 382.586
−0.452
b
651 233.061
13.943
0.175
a
650 990.355
−12.691
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194308-4
J. Chem. Phys. 127, 194308 共2007兲
Harrison et al.
TABLE I. 共Continued.兲
⍀=4
F1 + 1 ← F1
F+1←F
27.5← 26.5
␯
⍀=3
␯obs-calc
28← 27
27← 26
27← 26
26← 25
26← 25
25← 24
25← 24
24← 23
26.5← 25.5
25.5← 24.5
24.5← 23.5
⍀=2
␯
␯obs-calc
Par
f
650 388.718
−0.519
f
650 394.936
f
f
Par
a
␯
␯obs-calc
b
651 239.034
14.600
−0.071
b
651 244.554
15.213
650 400.853
0.495
b
651 249.456
15.613
650 405.076
−0.218
b
651 253.884
15.939
a
a
The a and b parity labels correspond to the lower and upper lambda doublets, respectively, for the ⍀ = 2 spin component. The e and f parity labels for the
⍀ = 3 spin component were predicted from perturbation theory 共see text for further details兲.
b
Blended lines.
c
Data from the J = 28← 27 lines of the ⍀ = 2 and 3 spin components were not included in the fit. The ␯obs-calc. values from the global fit indicate the extent of
the perturbations of these levels.
兺
Hmhf =
⍀⬘=2,3,4
1
2
␦⍀⍀⬘h⍀⬘共1兲I1z + b共1兲共I1+S− + I1−S+兲
1
− d⌽共1兲共J+3I1+S+2 + J−3I1−S−2兲
2
+
兺
⍀⬘=2,3,4
1
2
␦⍀⍀⬘h⍀⬘共2兲I2z + b共2兲共I2+S− + I2−S+兲.
共7兲
Note that spin-rotation terms are not used because of the
correlations with those describing centrifugal distortion corrections to the spin-orbit coupling parameter.14 Also note that
square brackets around two operators, 关A , B兴+, indicate that
the anticommutator should be taken to ensure that the Hamiltonian is Hermitian. Definitions of most of these parameters
can be found in Ref. 14. Labels 共1兲 and 共2兲 in the hyperfine
Hamiltonian refer to nucleus 1 共cobalt兲 and nucleus 2 共fluo-
FIG. 1. A stick diagram showing the fine structure pattern of the
J = 28← 27 transition of CoF in the X 3⌽ state near 650 GHz. Consistent
with other 3⌽ Co-containing molecules, the ⍀ = 3 component is shifted to
lower frequency relative to the mean of the ⍀ = 4 and 2 features. The
lambda-doubling splittings are shown, labeled a and b for the ⍀ = 2 component and e and f for ⍀ = 3.
FIG. 2. A sample spectrum of the J = 27← 26 transition of CoF in the X 3⌽
state near 627 GHz, showing data for all three spin components. The 59Co
hyperfine interactions are visible in all components, but the fluorine hyperfine is not resolved for these high-J transitions. As is seen in the lower two
panels, ⌳ doubling is present for both ⍀ = 3 and 2 components, but the
doublets in the former case are intermixed. In ⍀ = 2, the doublets are separated in frequency by 180 MHz, and the data displayed have a frequency
break. The ⍀ = 4 and 3 spectra were acquired in single, 60 s scans. The
⍀ = 2 component required 12 scans to obtain an adequate signal to noise
ratio.
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194308-5
J. Chem. Phys. 127, 194308 共2007兲
Rotational spectrum of CoF
FIG. 3. The J = 16← 15 transition of CoF in the ⍀ = 4 component of the
X 3⌽ state near 372 GHz. This figure shows the high resolution achieved in
the experiment: both the 59Co and 19F hyperfine structures are resolved.
Individual transitions are labeled by quantum number F. There are two
unidentified lines at the left of the spectrum, marked with asterisks, as well
as features arising from the ⍀ = 3 component, indicated by a dagger. These
data were acquired in a single scan, 60 s in duration.
rine兲, respectively. The extension of the effective Hamiltonian approach to the analysis of the X 3⌽ state of CoF
required the addition of several new terms, which will be
outlined below.
In Eq. 共8兲, the angular momenta have their usual meanings and the ladder operators N± = Nx ± iNy, etc., are defined
in a molecule-fixed coordinate system. The factors of 21 are
chosen for consistency with the p and q terms in Mulliken
and Christy’s expressions for 2⌸ states.18,19 The effects of
the operators in Eq. 共8兲 are to be evaluated on the implicit
understanding that they link the ⌳ = 3 and ⌳ = −3 components
of the 2S+1⌽ state only. For example, matrix elements of N+6
and N−6 must be taken to mean 具⌳ = −3兩N+6兩⌳ = 3典 and
具⌳ = 3兩N−6兩⌳ = −3典, respectively. Angular momentum constraints show that only the first three terms of Eq. 共8兲 are
relevant for a 3⌽ state.
It is much more likely that any experimental example of
a molecule in a 2S+1⌽ state will conform more closely to a
Hund’s case 共a兲 coupling scheme. There is also a marked
preference for performing diatomic molecule energy level
calculations in a case 共a兲 basis set 兩⌳S⌺J⍀典. We therefore
recast the Hamiltonian in Eq. 共8兲 in a form appropriate for
Hund’s case 共a兲 coupling by replacing N with 共J − S兲 to give
Hld = 21 q̃⌽共J+6 + J−6兲 − 21 p̃⌽共J+5S+ + J−5S−兲 + 21 õ⌽共J+4S+2
+ J−4S−2兲 − 21 ñ⌽共J+3S+3 + J−3S−3兲 + 21 m̃⌽共J+2S+4 + J−2S−4兲
− 21 l̃⌽共J+S+5 + J−S−5兲 + 21 k̃⌽共S+6 + S−6兲,
where
q̃⌽ = q⌽ ,
A. Lambda-type doubling
p̃⌽ = p⌽ + 6q⌽ ,
Lambda-doubling effects in ⌽ states are expected to be
much smaller in magnitude than those in ⌸ or ⌬ states because they occur through sixth-order mixing of electronic
states. Nevertheless, increasing experimental resolution and
the study of molecules with higher densities of electronic
states have revealed a few examples of lambda-type doubling in ⌽ states.1,15,16 In their study of the rotational spectrum of CoH in its 3⌽ ground state, Beaton et al.15 were able
to resolve the lambda-type doubling in the 3⌽3 spin component but not in the 3⌽4 component; they did not observe any
transitions in the highest 3⌽2 component. They therefore
modeled their observations with one additional parameter in
the effective Hamiltonian. In the present study of CoF in its
ground 3⌽ state, observations have been made on all three
spin components, and lambda-type doubling is observed in
two of them 共3⌽3 and 3⌽2兲. In this work, we have developed
a full description of lambda-type doubling in 2S+1⌽ states for
implementation in the effective Hamiltonian.
The effective Hamiltonian representing lambda-type
doubling for a molecule in a 2S+1⌽ state can be derived along
the same lines as for a 2S+1⌬ state,17 but in this case using
sixth-order perturbation theory to connect the degenerate
兩⌳ = 3典 and 兩⌳ = −3典 components via one or more 2S+1⌺ states.
The operator is assumed to act only within the ⌽ state of
interest and can be written in Hund’s case 共b兲 form as
Hld =
1
6
2 q⌽共N+
+
N−6兲
−
1
5
2 p⌽共N+S+
+
N−5S−兲
+
1
4 2
2 o⌽共N+S+
−
+
N−S−5兲
+
1
6
2 k⌽共S+
+
S−6兲.
õ⌽ = o⌽ + 5p⌽ + 15q⌽ ,
ñ⌽ = n⌽ + 4o⌽ + 10p⌽ + 20q⌽ ,
m̃⌽ = m⌽ + 3n⌽ + 6o⌽ + 10o⌽ + 15q⌽ ,
l̃⌽ = l⌽ + 2m⌽ + 3n⌽ + 4o⌽ + 5p⌽ + 6q⌽ ,
k̃⌽ = k⌽ + l⌽ + m⌽ + n⌽ + o⌽ + p⌽ + q⌽ .
3
共10兲
The case 共a兲 matrix elements of Hld for a molecule in a
⌽ state can be evaluated to give
具⌳ ⫿ 6;S⌺⬘ ;J,⍀ ⫿ 6兩Hld兩⌳;S⌺;J,⍀典
= ␦⌺⬘⌺ 2 q̃⌽关兵J共J + 1兲 − 共⍀ ⫿ 6兲共⍀ ⫿ 5兲其兵J共J + 1兲
1
− 共⍀ ⫿ 5兲共⍀ ⫿ 4兲其兵J共J + 1兲 − 共⍀ ⫿ 4兲共⍀ ⫿ 3兲其
⫻兵J共J + 1兲 − 共⍀ ⫿ 3兲共⍀ ⫿ 2兲其兵J共J + 1兲
− 共⍀ ⫿ 2兲共⍀ ⫿ 1兲其兵J共J + 1兲 − 共⍀ ⫿ 1兲⍀其兴1/2 ,
共11兲
具⌳ ⫿ 6;S⌺ ± 1;J,⍀ ⫿ 5兩Hld兩⌳;S⌺;J,⍀典
= − 21 p̃⌽关兵J共J + 1兲 − 共⍀ ⫿ 5兲共⍀ ⫿ 4兲其兵J共J + 1兲
− 共⍀ ⫿ 4兲共⍀ ⫿ 3兲其兵J共J + 1兲 − 共⍀ ⫿ 3兲共⍀ ⫿ 2兲其
+ N−4S−2兲 − 21 n⌽共N+3S+3 + N−3S−3兲 + 21 m⌽共N+2S+4 + N−2S−4兲
1
5
2 l⌽共N+S+
共9兲
共8兲
⫻兵J共J + 1兲 − 共⍀ ⫿ 2兲共⍀ ⫿ 1兲其兵J共J + 1兲
− 共⍀ ⫿ 1兲⍀其兵S共S + 1兲 − 共⌺ ± 1兲⌺其兴1/2 ,
共12兲
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194308-6
J. Chem. Phys. 127, 194308 共2007兲
Harrison et al.
TABLE II. Lambda-doubling matrix elements for 3⌽ states. 关x = J共J + 1兲.
Upper and lower signs refer to the ⫹ and ⫺ parity levels, respectively.兴
具⌳ ⫿ 6;S⌺ ± 2;J,⍀ ⫿ 4兩Hld兩⌳;S⌺;J,⍀典
= 21 õ⌽关兵J共J + 1兲 − 共⍀ ⫿ 4兲共⍀ ⫿ 3兲其兵J共J + 1兲
− 共⍀ ⫿ 3兲共⍀ ⫿ 2兲其兵J共J + 1兲 − 共⍀ ⫿ 2兲共⍀ ⫿ 1兲其
⫻兵J共J + 1兲 − 共⍀ ⫿ 1兲⍀其兵S共S + 1兲 − 共⌺ ± 1兲⌺其兵S
具 3⌽ 4兩
兩 3⌽ 3典
兩 3⌽ 2典
¯
¯
共−1兲J
⫿ 2 q̃⌽x共x − 2兲共x − 6兲1/2共x − 12兲1/2
共−1兲J
⫿ 2 q̃⌽x共x − 2兲共x − 6兲
共−1兲J
± 冑 p̃⌽x共x − 2兲共x − 6兲1/2
2
共13兲
⫻共S + 1兲 − 共⌺ ± 1兲共⌺ ± 2兲其兴1/2 .
具 3⌽ 3兩
The matrix elements of the four remaining operator terms in
Eq. 共9兲 have a very similar form and can easily be derived
for higher spin multiplicities.
In practice, we take linear combinations of Hund’s case
共a兲 basis functions that preserve parity,14
具 3⌽ 2兩
− ⌺;J,− ⍀典其.
The effective Hamiltonian also includes a paritydependent hyperfine d⌽ term for the cobalt nucleus, analogous to the d and d⌬ terms in previous studies of diatomic
molecules.14 Since the inclusion of this term is only required
for the X 3⌽2 spin component, we have chosen the form of
this operator 关Eq. 共7兲兴 such that it makes a diagonal, paritydependent contribution to 3⌽2 only. The selection rules for
this term are ⌬⌳ = ± 6, ⌬⌺ = ⫿ 2, ⌬⍀ = ± 4, ⌬J = 0, ±1,
⌬F = 0, and the matrix elements can be evaluated as
共14兲
Here the upper and lower sign choices refer to ⫹ and ⫺
parity states, respectively. When the matrix elements are expressed in this basis set, the lambda-type doubling terms add
and subtract by equal amounts. In the case of a 3⌽ molecule,
the matrix elements evaluate to those given in Table II.
冎
再
⫿共−1兲Jõ⌽x共x − 2兲
B. Nuclear hyperfine effects
兩⌳;S⌺;J,⍀; ± 典 = 2−1/2兵兩⌳;S⌺;J,⍀典 ± 共− 1兲J−S兩− ⌳;S,
再
兩 3⌽ 4典
冉
冊
2 S
I J⬘ F
S
3
关I共I + 1兲共2I + 1兲兴1/2关共2J⬘ + 1兲共2J + 1兲兴1/2 兺 ␦⌳⬘,⌳⫿6共− 1兲S−⌺⬘
关共2S − 1兲
共10兲−1/2d⌽共− 1兲J+I+F
16
J I 1
− ⌺⬘ 2q ⌺
q=±1
冉
冊
⫻共2S兲共2S + 1兲共2S + 2兲共2S + 3兲兴1/2 共− 1兲J−⍀⬘
⫻共2J + 4兲兴1/2 + 共− 1兲J⬘−⍀⬘
冉
J⬘
3
J⬘
− ⍀⬘ − 3q ⍀⵮
J⬘
1
J
− ⍀⬘ − q ⍀⬙
共− 1兲J⬘−⍀⵮
冉
J⬘
冊
共− 1兲J−⍀⬙
1
J
冉
冊
− ⍀⵮ − q ⍀
J
3
J
− ⍀⬙ − 3q ⍀
冊
关共2J − 2兲共2J − 1兲 ¯ 共2J + 3兲
冎
关共2J⬘ − 2兲共2J⬘ − 1兲 ¯ 共2J⬘ + 3兲共2J⬘ + 4兲兴1/2 .
共15兲
C. Least-squares fit
Matrix elements of the effective Hamiltonian in Eq. 共1兲
were determined using a Hund’s case 共a␤J兲 coupling scheme,
which can be expressed in vector form as
J + I1共 59Co兲 = F1 ,
F1 + I2共 19F兲 = F.
共16兲
Several minor modifications have been made to the effective Hamiltonian used in this work because of the presence of perturbations in the lambda doubling and hyperfine
structure, notably within the X 3⌽2 spin-orbit component. Instead of using the diagonal hyperfine parameters a and b + c,
we have chosen to use h2, h3, and h4. In a nonperturbed
system, these parameters are related by
h⍀ = a⌳ + 共b + c兲⌺.
共17兲
This choice has the advantage of introducing an additional
degree of freedom to model the hyperfine splittings. In order
to obtain the best possible fit, it was necessary to use two
different sets of lambda-doubling parameters, one for the
⍀ = 2 and another for the ⍀ = 3 spin component.
The measured transition frequencies were fitted directly
using a modified version of the least-squares fitting program
20
HUNDA2SPIN. There is no direct information on the spinorbit splittings of CoF in the X 3⌽ state in the rotational
spectrum. Only the X 3⌽3 – X 3⌽4 interval is known from the
optical spectrum.8 In order to use this interval in our fit, we
chose to use previous data5,8 to create a “dummy point” corresponding to the X 3⌽3 – X 3⌽4 interval. The Q共4兲 line was
given a value of 703.12 cm−1 with an uncertainty of
0.1 cm−1. The values for the various determined parameters
are given in Table III. The complete data set contains 32 pure
rotational transitions and 545 total line frequencies, of which
89 in the high-J region have been given zero weight since
they are significantly perturbed, as mentioned previously. En
route to the final fit, least-squares fits were also performed
for each individual spin-orbit component, which greatly
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194308-7
J. Chem. Phys. 127, 194308 共2007兲
Rotational spectrum of CoF
TABLE III. Spectroscopic parameters 共in MHz兲 for the X 3⌽i state of CoF.
B
D
H
A
AD
AH
AL
␭
␭D
␭H
␭L
õ⌽ 共⍀ = 2兲
p̃⌽ 共⍀ = 2兲
q̃⌽ 共⍀ = 2兲
q̃⌽ 共⍀ = 3兲
p̃⌽D 共⍀ = 2兲
h2共Co兲
h2D共Co兲
d⌽ 共⍀ = 2兲
b共Co兲
h3共Co兲
h3D共Co兲
h4共Co兲
h4D共Co兲
eq0Q共Co兲
h2共F兲
h2D共F兲
b共F兲
h3共F兲
h3D共F兲
h4共F兲
h4D共F兲
Std. dev.
X 3⌽ 4a
X 3⌽ 4b
X 3⌽ 3a
X 3⌽3,4c
X 3⌽ 2a
X 3⌽2,3,4a
11 648.040 4共12兲d
0.015 412 1共27兲
−6.8共1.8兲 ⫻ 10−9
11 635.320 99共51兲
0.015 343 7共10兲
−6.038⫻ 10−9e
11 638.498 64共59兲
0.015 565 62共50兲
11 647.757 5共16兲
0.015 382 7共33兲
−6.038⫻ 10−8e
−6 981 354e
11 637.267共11兲
0.015 351共39兲
2.14共49兲 ⫻ 10−7
11 641.258 6共39兲
0.015 439共14兲
6.6共1.8兲 ⫻ 10−8
−8 130 100共1100兲
1.269 8共20兲
−8.3共7.1兲 ⫻ 10−6
−3.53共91兲 ⫻ 10−8
1 701 500e
4.437 8共31兲
8.8共1.1兲 ⫻ 10−5
4.9共1.4兲 ⫻ 10−8
−4.29共10兲 ⫻ 10−4
1.36共10兲 ⫻ 10−3
3.32共17兲 ⫻ 10−4
−3.932共51兲 ⫻ 10−8
−9.92共46兲 ⫻ 10−7
2 489.4共4.3兲
0.434共37兲
−9.5共2.2兲 ⫻ 10−5
−606.8共8.3兲
1 686.10共37兲
0.0
972.59共59兲
0.0
−82.4共3.3兲
190共19兲
−0.106共88兲
−215共50兲
200.3共5.4兲
−0.082共26兲
218.9共4.1兲
0.0
0.137
−20 890e
−4.189共90兲 ⫻ 10−4
7.78共59兲 ⫻ 10−4
3.08共14兲 ⫻ 10−4
−3.934共39兲 ⫻ 10−8
−7.62共32兲 ⫻ 10−7
2 477.5共5.3兲
0.622共39兲
−5.1共2.2兲 ⫻ 10−5
1 684.21共48兲
0.056 3共23兲
973.70共46兲
−0.167 9共18兲
−81.0共3.1兲
974.9共1.8兲
−0.167 5共11兲
−77.50共91兲
1 571共12兲
973.069共66兲
−85.4共4.4兲
−78共3兲
−79共23兲
171共17兲
−0.124共77兲
199.6共4.2兲
−0.063共20兲
219.3共3.0兲
−5.9共1.0兲 ⫻ 10−2
0.085
233.52共32兲
0.029
233共1兲
0.092
0.226
a
This work.
Reference 10.
c
Reference 11.
d
Values in parentheses are 1␴ errors to the places reported.
e
Held fixed in fit.
b
aided in identifying the location of the perturbations. The
results of these fits are also given in Table III. The assigned
line frequencies, with their residuals, are provided in the
supplementary material.13
The standard deviations for the fits of the individual spin
components are 85, 92, and 226 kHz for the ⍀ = 4, 3, and 2
levels, respectively; the experimental uncertainty is 100 kHz.
In the global fit, the ⍀ = 2 data were given a lower weighting
in line with the poorer quality of the individual fit 共uncertainty of 226 kHz兲. The standard deviation of the final fit was
137 kHz.
V. DISCUSSION
This work contains a number of notable firsts. For the
first time, a full description of lambda-type doubling in 2S+1⌽
states has been implemented in the effective Hamiltonian.
Also, the pure hyperfine-resolved rotational lines for all three
spin components of CoF in the X 3⌽ state have been reported, and a fit has been attempted for these three spin com-
ponents. Previous analyses of both the fine and hyperfine
structures for this system have concentrated largely on the
⍀ = 4 spin component,10 with some recent work on ⍀ = 3 by
Steimle et al.11 The data presented in this paper are fitted
close to experimental uncertainty over a wide range of J
values.
The data set that Steimle et al. used in their analysis11
consists of 87 pure rotational data points measured by Okabayashi and Tanimoto 共X 3⌽4兲, along with 738 optical transitions from the X 3⌽3 and X 3⌽4 levels to an excited 3⌽
state. Of these 738 optical transitions, only 138 provided any
information on the X 3⌽3 component, with a comparatively
large experimental uncertainty of 0.05 cm−1 共⬇1.5 GHz兲.
The X 3⌽3 parameters reported in this work are therefore
considerably more accurate than those determined by
Steimle et al.11
A. Rotational constant
In their analysis, Okabayashi and Tanimoto 共X 3⌽4兲 neglected all off-diagonal matrix elements, making it difficult
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194308-8
J. Chem. Phys. 127, 194308 共2007兲
Harrison et al.
to compare their parameters directly with those determined
in this work. Because these authors did not use the spinuncoupling term in the rotational Hamiltonian, only effective
B values for each component of a multiplet degenerate state
could be determined. Using perturbation theory, the following approximate relationship between Beff and B0 can be
derived:
冉
eff
⬇ B0 1 +
B⍀,0
冊
2B0⌺
.
A⌳
共18兲
eff
= 11 635.320 99 MHz,10 one obUsing Eq. 共18兲 and B⍀=4.0
tains B0 = 11 648.277 63 MHz, which is to be compared with
the value of 11 648.0404 MHz obtained from the individual
fit of the ⍀ = 4 spin-orbit component performed in this study.
Since all three spin-orbit components have been included in the least-squares fit, we now have a more reliable
value for the rotational parameter, B0 = 11 641.2586 MHz.
Assuming that B0 contains no contributions from nearby
electronic states, this value corresponds to a zero-point Co–F
bond length r0 of 1.738 014共1兲 Å, slightly different from the
value of 1.737 82 Å quoted by Steimle et al.11
h4D ⬇ −
bB
,
2共E4 − E3兲
共19兲
and the following identity also holds:
2h2D + 3h3D + 4h4D ⬇ 0.
共20兲
It is easily verified from the h⍀D parameters for the three
spin-orbit components in Table III that Eq. 共20兲 does not
hold. This result is due to the significant perturbations in the
hyperfine energy levels for the ⍀ = 2 spin component, in particular, the inadequacy of the bI · S term to correctly describe
the off-diagonal interactions with ⍀ = 2, so that the perturbation expression for h2D no longer holds. We can still assume
that the perturbation expressions for h3D and h4D are approximately correct. Since E3 − E4 is known experimentally, it is
possible to obtain a value for E3 − E2 using the above equation for h3D. By equating these values with expressions determined from the diagonal Hund’s case 共a兲 matrix elements,
a value for ␭ can be obtained,
E3 − E2 ⬇ 3A − 2␭ − 4B,
B. Fine structure parameters
Adam and Hamilton8 have previously reported values
for both spin-orbit A and spin-spin ␭ parameters for CoF in
the X 3⌽ state. However, since only the 3⌽3 – 3⌽4 interval
was known experimentally, the determination of the parameter ␭ from the optical spectrum was not justified. It was
observed during the analysis in the present work that both A
and ␭ were very highly correlated when both were included
in the least-squares fit. However, we have been able to determine an approximate value for ␭ from the Co hyperfine
structure using perturbation theory, as outlined below.
As noted by previous workers, the approach of fitting
rotational data for each spin-orbit component individually
leads to large centrifugal distortion corrections to the hyperfine parameters because of the neglect of the off-diagonal
⌬⌺ = ± 1 matrix elements of the bI · S operator. Using
second-order perturbation theory,10,11 approximate relationships can be derived for the centrifugal distortion corrections, h⍀D, for all three spin-orbit components. In a global fit
of all spin-orbit components for a well-behaved system, one
would expect the centrifugal distortion corrections to fall to a
negligibly small value if the off-diagonal part of the bI · S
term was included in the Hamiltonian. The justification for
this assumption is that the hyperfine parameters a, b, b + c for
the 59Co nucleus are essentially independent of the bond
length and so will not show any genuine centrifugal distortion effects. For a well-behaved system, the perturbation expressions are
bB
,
h2D ⬇ −
E2 − E3
h3D ⬇ −
冋
册
2bB
1
1
+
,
3 E3 − E2 E3 − E4
E4 − E3 ⬇ 3A + 2␭ − 8B.
共21兲
Using this method, the value of ␭ was found to be approximately 1700 GHz. The large magnitude of this value is explicable as the effect of second-order spin-orbit mixing with
low-lying electronic states of CoF. This value was held fixed
in the global least-squares fit. The resultant values for E3
− E4 and E2 − E3 are 21.0834 THz 共compared with the value
of 21.0790 THz determined from the parameters in Ref. 8兲
and 27.8369 THz, respectively.
The experimental data encompass a wide range of J values. Although the effective Hamiltonian fits the ⍀ = 3 and 4
data to within experimental uncertainty, the ⍀ = 2 data only
fit to three times this value, greater than that from the fit for
the ⍀ = 2 data alone 共standard deviation= 226 kHz兲. This
points to significant perturbations of the levels of the ⍀ = 2
spin-orbit component and contributes amongst other things
to the large nonzero value for h2D. Small, systematic residuals persist in the Co hyperfine structure of all three spin
components even after the least-squares fit was completed
共refer to the supplementary material13兲. This suggests that the
effective Hamiltonian for a molecule in a 3⌽ state is not fully
adequate to model the energy levels of CoF.
C. Lambda-doubling effects
Although one would expect lambda-doubling interactions in ⌽ states to be very small, the X 3⌽ state of the CoF
molecule exhibits sizable splittings. Figure 4 gives a plot of
the lambda-doubling splitting versus J for the ⍀ = 2 spinorbit component. The splitting within a lambda doublet
equates to the difference in splittings between the upper and
lower rotational levels of the corresponding transition.
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194308-9
J. Chem. Phys. 127, 194308 共2007兲
Rotational spectrum of CoF
= 4 and 3 components of CoF has been determined
experimentally,8 whereas that between ⍀ = 3 and 2 has not.
We have therefore used the values for the hyperfine parameters for the ⍀ = 4 and 3 components to determine the three
fundamental hyperfine parameters, a, bF, and c,
h4 = 3a + 共b + c兲,
h3 = 3a,
FIG. 4. A plot of the variation in lambda-doubling splitting per J level for
the ⍀ = 2 spin component. The solid curve depicts a lambda-doubling energy
increase proportional to J4, as expected for the ⍀ = 2 level. The points show
the actual splittings, which deviate from the ideal curve, indicating the presence of perturbations in these levels.
Therefore, it is possible to determine experimental splittings
for each rotational level, although this requires the lowest
splitting to be calculated using the best-fit parameters. Normally, one would expect the lambda doubling to increase as
J2兩⍀兩; however, it is clear that the experimental splitting increases more rapidly than described by this relationship.
There are also small but significant deviations from the
model behavior, showing that there are perturbations of the
individual levels of the ⍀ = 2 spin component. Because of
such perturbations, it was necessary to use different values of
the lambda-doubling parameters for the ⍀ = 2 and ⍀ = 3 transitions. The irregular behavior of the splittings in the ⍀ = 2
spin component requires four lambda-doubling parameters to
model them. Consequently, the values of these parameters
are not physically meaningful. The lambda doubling in the
⍀ = 3 component is modeled better by a single parameter, q␾
共see Table II兲. This parameter probably does carry reliable
structural information. At the very least, we can determine its
sign from the perturbation theory17 and the nature of the
electronic states involved 共3⌬, 3⌸, and 3⌺−兲 and hence the
absolute parities of each lambda doublet of the ⍀ = 3 spin
component. Although this is the first time that the hyperfine
lambda-doubling parameter, d⌽, has been determined for a
molecular system, it is unlikely to be particularly meaningful
because of the perturbations in the lambda-doubling structure
of the ⍀ = 2 component.
bF = b + 31 c.
共23兲
Values for all three parameters have been determined for the
first time for both 59Co and 19F nuclei and are given in Table
V. The previous attempt to determine the 59Co parameters by
Steimle et al.11 was not complete because they had to assume
that the dipolar parameter c was zero. Though the parameter
is small in magnitude, it is not zero 共−106.7 MHz兲. In addition, their analysis was limited by the quality of the optical
measurements of the Co hyperfine splittings in the 3⌽3 component 共⬃100 MHz兲, whereas the present measurements are
accurate to 100 kHz. The inadequacy of the standard magnetic nuclear hyperfine Hamiltonian to model the splittings
of CoF in the X 3⌽2 state can be gauged from the experimental value of h2 for Co, 2489.4共43兲 MHz, compared with that
calculated from the values for a, bF, and c, which is
2399.61共79兲 MHz.
Steimle et al.11 have discussed the 59Co hyperfine parameters in terms of the two electronic configurations that
are thought to contribute to the ground 3⌽ state. These are
configuration A, 共3d␴兲2共3d␦兲3共3d␲兲3 with two open shells,
and configuration B, 共3d␴兲1共3d␦兲3共3d␲兲3共4s␴兲1 with four
open shells. Treating the electron holes, configuration A
gives rise to one 3⌽ state represented by the Slater determinant,
⌿共A兲 = 兩␦+2␲+1兩,
共24兲
while configuration B gives rise to three 3⌽ states represented by the spin-adapted linear combinations of four Slater
determinants,
D. Nuclear hyperfine structure
It became clear during the course of this work that the
nuclear hyperfine splittings in the rotational spectrum of CoF
could not be fitted to experimental accuracy using the standard magnetic hyperfine effective Hamiltonian,
Hmhf = 共aLz + 兵b + c其Sz兲Iz + 21 b共I+S− + I−S+兲.
⌿共B1兲 = 共2兲−1/2关兩␴¯␦+2␲+1␴兩 − 兩¯␴␦+2␲+1␴兩兴,
¯ +1␴兩 − 兩␴¯␦+2␲+1␴兩 − 兩¯␴␦+2␲+1␴兩兴,
⌿共B2兲 = 共6兲−1/2关2兩␴␦+2␲
共22兲
As a result, we used a modified Hamiltonian 关Eq. 共7兲兴 containing the parameters h2, h3, h4, and b for each nucleus, as
well as various centrifugal corrections. The advantage of this
approach is the introduction of an additional degree of freedom to model the hyperfine splittings.
It has already been established that the ⍀ = 2 spin component is significantly more perturbed than the ⍀ = 4 and 3
ladders. In addition, the spin-orbit splitting between the ⍀
¯ +1␴兩 − 兩␴¯␦+2␲+1␴兩
⌿共B3兲 = 共12兲−1/2关3兩␴␦+2␲+1¯␴兩 − 兩␴␦+2␲
− 兩¯␴␦+2␲+1␴兩兴.
共25兲
In this work, we have extended the approach of Steimle
et al. to include the fluorine hyperfine parameters and the Co
electric quadrupole interaction. In this simple approach to the
prediction of the CoF parameters from those for atomic Co
and F, the molecular orbitals are approximated as follows:
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194308-10
J. Chem. Phys. 127, 194308 共2007兲
Harrison et al.
TABLE IV. Molecular hyperfine parameters in terms of the Co and F atomic hyperfine parameters 共see text for
further information兲.
⌿共A兲
⌿共B1兲
⌿共B2兲
⌿共B3兲
2
cF2p
␲ 01
3 a2p共F兲
2
cF2p␲ 10
2 a2p共F兲
2
cF2p
␲ 01
3 2a2p共F兲
2
cF2p␲ + cF2p␴ 10
a2p共F兲
2
2
cF2p
␲ 01
3 2a2p共F兲
2
cF2p␲ + cF2p␴ 10
a2p共F兲
2
2
cF2p
␲ 01
3 a2 2p共F兲
2
c
+c
− F2p␲ 4 F2p␴ a10
2p共F兲
2
3cF2p
␲ 12
a2p共F兲
− 10
2
2
3共2cF2p
␴ − cF2p␲兲 12
a2p共F兲
10
a共Co兲
2
2 + cCo3d
␲ 01
a3d共Co兲
3
2
2 + cCo3d
␲ 01
a3d共Co兲
3
2
2 + cCo3d
␲ 01
a3d共Co兲
3
2
2 + cCo3d
␲ 01
a3d共Co兲
3
bF共Co兲
2
1 + cCo3d
␲ 10
a3d共Co兲
2
2
cCo4s
␴ 10
2 a4s 共Co兲
2
c
␲ 10
+ Co3d
2 a3d共Co兲
2
cCo4s
␴ 10
2 a4s 共Co兲
2
4 − cCo3d␲ 10
+
a3d共Co兲
6
c2 ␴ 10
− Co4s
4 a4s 共Co兲
2
5共2 + cCo3d
␲兲 10
+
a3d共Co兲
12
2
3共cCo3d
␲ − 2兲 12
a3d共Co兲
14
2
3cCo3d
␲ 12
14 a3d共Co兲
c2 ␲ 12
− Co3d
14 a3d共Co兲
2
5cCo3d
␲ 12
28 a3d共Co兲
2
2共3cCo3d
␲ − 2兲 02
b3d共Co兲
7
2
2共3cCo3d
␲ − 4兲 02
b3d共Co兲
7
2
2共3cCo3d
␲ − 4兲 02
b3d共Co兲
7
2
2共3cCo3d
␲ − 4兲 02
b3d共Co兲
7
a共F兲
bF共F兲
c共F兲
c共Co兲
eQq0共Co兲
兩9␴典 = 兩Co 3d␴典,
兩1␦典 = 兩Co 3d␦典,
兩4␲典 = cCo3d␲兩Co 3d␲典 + cF2p␲兩F 2p␲典,
兩10␴典 = cCo4s␴兩Co 4s␴典 + cF2p␲兩F 2p␴典.
共26兲
Note that due to the simplicity of this approach, we have
neglected contributions from those atomic orbitals for which
no hyperfine data exist. Molecular hyperfine parameters are
determined from Eq. 共27兲. Note that, because of the involvement of the rotational angular momentum, this approach is
not useful for the d⌽ parameter,
a=
bF =
c=
冓 冏兺 冏 冔 冒
冓 冏兺 冏 冔 冒
␮0
2gN␮B␮N ⌳
4␲
lzir−3
⌳
i
i
␮0 8␲
g eg N␮ B␮ N ⌳
4␲ 3
␮0 3
g eg N␮ B␮ N
4␲ 2
冓 冏兺
⫻ ⌳
szi␦共r兲 ⌳
i
S,
s
冏 冔冒
⌳
szi共3 cos2 ␪i − 1兲r−3
i
i
eQq0 = −
⌳,
l
S,
s
冓 冏兺
e 2Q
⌳
4␲␧0
i
冏冔
共3 cos2 ␪i − 1兲r−3
⌳ .
i
共27兲
l
The atomic hyperfine parameters used for 59Co and 19F
belong to the 3d84s1 and 2p5 configurations, respectively.
These are “effective” constants, in which the radial expectation values are treated as free parameters.21 In this approach,
the bF parameter can also be used for non-s electrons, where
−
2
2
2cF2p
␴ + 3cF2p␲ 12
a2p共F兲
10
2
2
10cF2p
␴ + 3cF2p␲ 12
a2p共F兲
10
it represents an induced contact interaction, due to polarization of closed s shells. Because of the electronegativity difference between Co and F, the bonding is largely ionic in
nature. It would therefore be preferable to have hyperfine
data for Co+ 共3d74s1兲, but such data are not available.
The 59Co atomic hyperfine parameters are explained in
detail in Ref. 21. The important parameters used in this work
10
10
01
are a3d
= −210.2 MHz, a4s
= 4410.6 MHz, a3d
= 617.9 MHz,
12
02
a3d = 857.1 MHz, and b3d = 409.2 MHz. The 19F parameters
are found in Ref. 22, and after redefining these in a
similar manner to those of Co above, we obtain
10
01
a2p
= 1887.966 MHz,
a2p
= 3085.027 MHz,
and
12
a2p = 5070.122 MHz.
Table IV gives calculated expressions for the molecular
hyperfine parameters in terms of the coefficients above.
Table V gives numerical values for these expressions using
the appropriate atomic hyperfine parameters. Note that only
the diagonal magnetic hyperfine matrix elements are calculated here because the true eigenvectors, involving electron
exchange and configuration interactions, are not known. The
values of cCo3d␲共=0.854兲 and cF2p␲共=0.255兲 in Eq. 共26兲 were
chosen so as to reproduce the experimental a共F兲 and a共Co兲
values. The coefficients determined for the 兩4␲典 orbital are
not normalized to unity. This simplification is not too significant when it is realized that other atomic contributions to this
orbital have been neglected and that this method is approximate only. In particular, we have calculated the expectation
values of the hyperfine operators in a chosen basis set 关Eqs.
共24兲 and 共25兲兴. Only if this choice is close to the true eigenvector will the results be meaningful.
Our calculated c共Co兲 parameters do not agree with those
of Steimle et al., listed in Table II of Ref. 11. Using our
values, their proposed wavefunction 关Eq. 共14兲兴, 兩X 3⌽典
= 0.391⌿共B2兲 + 0.920⌿共B3兲, gives a value of ⬃100 MHz for
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194308-11
J. Chem. Phys. 127, 194308 共2007兲
Rotational spectrum of CoF
TABLE V. Predicted and experimental nuclear hyperfine parameters 共in MHz兲 for CoF in the X 3⌽state,
determined from a simple MO approach 共see text兲.
a共F兲
bF共F兲
c共F兲
a共Co兲
bF共Co兲
c共Co兲
eQq0共Co兲
Expt.
⌿共A兲
⌿共B1兲
⌿共B2兲
⌿共B3兲
共2兲−1/2 兵⌿共A兲 + ⌿共B3兲其
66.7共18兲
−137共53兲
234共50兲
562.03共12兲
−642.4共87兲
−106.7共83兲
−82.4共33兲
66.7
61.2
−98.7
562.03
−181.7
−233.5
21.8
66.7
265.4
559.4
562.03
1651.6
133.8
−212.1
66.7
265.4
−318.0
562.03
1613.6
−44.6
−212.1
66.7
−132.7
597.7
562.03
−1103.1
111.5
−212.1
66.7
−35.7
249.5
562.03
−642.4
−61.0
−95.1
c共Co兲, whereas the experimental value is −106.7共83兲 MHz.
Their wavefunction also predicts values for c共F兲 and
eQq0共Co兲 that are in poor agreement with the measured
ones. An inspection of Table V reveals that it is more likely
that the X 3⌽ state of CoF arises predominantly from an
admixture of both the A and B configurations. If we choose
the X 3⌽ state to be represented by
兩X 3⌽典 = 共2兲−1/2兵⌿共A兲 + ⌿共B3兲其,
共28兲
we obtain the parameters in the final column of Table V.
Note that cCo3d␴共=0.885兲 has been chosen to reproduce the
experimental bF共Co兲 value and cF2p␴共=0.465兲 to satisfy the
normalization condition. The remaining parameters show
quite good agreement for such a simple method, although the
predictions for c共Co兲 and bF共F兲 are the least successful.
However, it is reassuring that we have managed to reproduce
the correct signs.
A comparison of the CoF hyperfine parameters determined in this work with those of similar systems, e.g., CoCl
or CoH, would be very informative. However, at present, the
hyperfine analyses of such systems lag behind that of CoF.
Once this situation improves, comparison with the present
results will provide invaluable insight into the bonding of Co
with simple ligands.
ACKNOWLEDGMENTS
This research was supported by NSF Grant Nos. CHE
04-11551 and CHE 07-18699. The authors would like to
thank Tim Steimle for helpful correspondence in the treatment of the Co hyperfine structure. One of the authors
共J.J.H.兲 thanks the Leverhulme Trust for financial support
and Christ Church College, Oxford for a nonstipendiary
fellowship.
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