Laser-rf double resonance spectroscopy of174YbF in the X2∑+ state: Spin-rotation, hyperfine interactions, and the electric dipole moment

Article abstract of DOI:10.1063/1.472569

We report the first laser-radio-frequency double resonance spectrum of a simple lanthanide compound, the paramagnetic radical ¹⁷⁴YbF. Measurements of the rf intervals as a function of the rotational quantum number N allow us to determine precise spin-rotation and fluorine hyperfine interaction coupling constants for the X²∑⁺(v =0 and v = 1) ground states of ¹⁷⁴YbF. The results for v=0 are γ₀=-13.424 00(16) MHz, γ₁=3.9823(11) kHz, γ₂=-25(1) mHz, b₀= 141.795 6(5) MHz, b₁, = -0.510(11) kHz, c = 85.4026(14) MHz, C=20.38(13) kHz. For v = 1 they are γ₀=-33.811 8(7) MHz, γ₁=4.323(6) kHz, γ₂=-28(9) mHz, b₀= 139.89(4) MHz, b₁=-0.7(4) kHz, c=86.75(5) MHz, C=18.3(1) kHz. A direct microwave measurement of the first rotational interval in X²∑⁺(v=0) gives the rotational constant B₀=0.241 292 7(7) cm^-1. Finally, the Stark shift of hyperfine transitions in the first two rotational states of X²∑⁺(v=0) are analyzed to determine the electric dipole moment μ_e=3.91(4) D. We find that although the gross structure of YbF in its ground state resembles that of an alkaline earth monofiuoride, the 4f shell is evidently important even in the low-lying excited states and leads to unusual behavior in the ground-state magnetic structure.

Full text of DOI:10.1063/1.472569

Laserrf double resonance spectroscopy of 174YbF in the X 2Σ+ state: Spinrotation,
hyperfine interactions, and the electric dipole moment
B. E. Sauer, Jun Wang, and E. A. Hinds
Citation: The Journal of Chemical Physics 105, 7412 (1996); doi: 10.1063/1.472569
View online: http://dx.doi.org/10.1063/1.472569
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2
Laser-rf double resonance spectroscopy of 174YbF in the X ⌺؉ state:
Spin-rotation, hyperfine interactions, and the electric dipole moment
B. E. Sauer, Jun Wang, and E. A. Hinds
Sussex Centre for Optical and Atomic Physics, University of Sussex, Brighton, BN1 9QH, United Kingdom
͑Received 2 May 1996; accepted 30 July 1996͒
We report the first laser-radio-frequency double resonance spectrum of a simple lanthanide
compound, the paramagnetic radical 174YbF. Measurements of the rf intervals as a function of the
rotational quantum number N allow us to determine precise spin-rotation and fluorine hyperfine
2
interaction coupling constants for the X ⌺ϩ͑ ϭ0 and ϭ1͒ ground states of 174YbF. The results for
v
v
ϭ0 are ␥ ϭϪ13.424 00͑16͒ MHz, ␥ ϭ3.982 3͑11͒ kHz, ␥ ϭϪ25͑1͒ mHz, b ϭ141.795 6͑5͒
v
0
1
2
0
MHz, b1ϭϪ0.510͑11͒ kHz, cϭ85.402 6͑14͒ MHz, Cϭ20.38͑13͒ kHz. For ϭ1 they are
v
␥0ϭϪ33.811 8͑7͒ MHz, ␥1ϭ4.323͑6͒ kHz, ␥2ϭϪ28͑9͒ mHz, b0ϭ139.89͑4͒ MHz, b1ϭϪ0.7͑4͒
kHz, cϭ86.75͑5͒ MHz, Cϭ18.3͑1͒ kHz. A direct microwave measurement of the first rotational
2
interval in X ⌺ϩ͑ ϭ0͒ gives the rotational constant B ϭ0.241 292 7͑7͒ cmϪ1. Finally, the Stark
v
0
shift of hyperfine transitions in the first two rotational states of X ⌺ϩ͑ ϭ0͒ are analyzed to
2
v
determine the electric dipole moment ␮eϭ3.91͑4͒ D. We find that although the gross structure of
YbF in its ground state resembles that of an alkaline earth monofluoride, the 4f shell is evidently
important even in the low-lying excited states and leads to unusual behavior in the ground-state
magnetic structure. © 1996 American Institute of Physics. ͓S0021-9606͑96͒02641-4͔
I. INTRODUCTION
nuclear spin is zero͒. We found a strong variation with rota-
tion N of the spin-rotation ‘‘constant’’ ␥, which was well
described by an expansion in powers of N(Nϩ1),
The spectroscopy of lanthanide diatomics is generally
complicated by the large number of low-lying configurations
resulting from the partially filled 4f shell of the lanthanide.
With its filled 4f shell ytterbium ͑ground-state configuration
͓Xe͔4f146s2͒ is an exception and YbF might reasonably be
expected to resemble the alkaline earth fluorides. In some
respects it does, but previous work in our group1 has shown
that the ground state spin-rotation structure is anomalous; its
magnitude is small and it has enormous centrifugal distor-
tion. That work, to which we will refer as ͑I͒ was based on
analysis of Doppler free optical spectra of the
␥ϭ␥ ϩ␥ N Nϩ1͒ϩ␥ N Nϩ1͒ 2ϩ••• .
͓ ͑
͑2͒
͑
͔
0
1
2
After fitting the spectra of 31 rotational transitions between
Nϭ6 and 75, we obtained the values shown in Table I for the
2
various parameters in the X ⌺ϩ( ϭ0) state. While b and c
v
are consistent with an earlier matrix isolation experiment,5 ␥0
is 10–20 times smaller than expected and the centrifugal
distortion term ␥1 is enormous; so large that ␥ changes sign
at Nϭ60. The constant C in Eq. ͑1͒ was too small to mea-
2
2
A ⌸1/2–X ⌺ϩ͑0-0͒ band. Here we present new results de-
rived from much higher precision laser-radio-frequency
double resonance ͑LRDR͒ spectra for both the 0-0 and 1-1
bands. We have also used the Stark effect of hyperfine tran-
sure. Our fit also included the splitting
⑀ of the
2
A ⌸1/2( ϭ0) upper state, shown in the last row of the
v
table.
The LIF spectrum is quite congested near the bandhead
because there are seven stable Yb isotopes, five of which
have more than 10% natural abundance. As a result, none of
the 174YbF lines below Nϭ6 was sufficiently well isolated to
be included in the analysis of ͑I͒ outlined above. By contrast,
the LRDR measurements reported here reveal the magnetic
substructure of the P and Q lines all the way down to Nϭ0
with a radio-frequency linewidth of approximately 35 kHz.
This allows us not only to determine ␥0, ␥1, b, and c with
much improved accuracy, but also to correct a sign error in ␥
and to measure the constant C, the centrifugal distortion of
b, and the quartic centrifugal correction ␥2. These measure-
sitions within the Nϭ0 and Nϭ1 rotational states of the ϭ0
v
ground state to determine the molecular dipole moment and
have directly measured the ground state rotational constant
by means of laser-microwave spectroscopy.
In ͑I͒ we reported the first molecular beam, laser induced
fluorescence ͑LIF͒ measurements on the paramagnetic radi-
cal 174YbF. The optical resonances observed in that work
2
were in the P and Q branches of the A ⌸1/2( ϭ0͒
v
–X 2⌺ϩ͑ ϭ0͒ transition at 552 nm.2 The residual Doppler
v
broadening was small enough to reveal the magnetic sub-
structure of each rotational line. The splittings of the 174YbF
lines were understood using the effective ground state
Hamiltonian3,4
ments were done on both the ϭ0 and ϭ1 states. Replacing
v
v
the rf oscillator by a microwave source, we were able to
2
observe the rotational transition Nϭ1→0 of X ⌺͑ ϭ0͒,
v
Hϭ␥S•NϩbI•SϩcIzSzϩCI–N,
͑1͒
from which we determined the rotational constant B. Finally,
where S is the electron spin, N is the X-state rotational an-
gular momentum, I is the 19F nuclear spin ͑Iϭ1/2͒ and z
components are along the internuclear axis ͑the 174Yb
an electric field was added to the rf interaction region in
order to study the Stark effect of the magnetic splittings of
2
X ⌺͑ ϭ0͒ in the Nϭ0 and Nϭ1 rotational states. Analysis
v
7412
J. Chem. Phys. 105 (17), 1 November 1996
0021-9606/96/105(17)/7412/9/$10.00
© 1996 American Institute of Physics
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Sauer, Wang, and Hinds: Spectroscopy of 174YbF
7413
2
2
TABLE I. Measured constants for the X ⌺ϩ and A ⌸1/2 states of YbF with standard deviations in parentheses.
The magnetic constants ␥i , bi , c, and C are defined by Eqs. ͑1͒ and ͑2͒. ␮ is the electric dipole moment, B is
e
2
the rotational constant, and ⑀ is the hyperfine splitting of the A ⌸1/2 state.
ϭ0a
ϭ0
ϭ1
v
v
v
␥0 ͑MHz͒
␥1 ͑kHz͒
␥2 ͑mHz͒
b0 ͑MHz͒
b1 ͑kHz͒
c ͑MHz͒
C ͑kHz͒
13.31͑8͒
Ϫ3.801͑15͒
Ϫ13.424 00͑16͒
3.982 3͑11͒
Ϫ25͑1͒
141.795 6͑5͒
Ϫ0.510͑11͒
85.402 6͑14͒
20.38͑13͒
Ϫ33.811 8͑7͒
4.323͑6͒
Ϫ28͑9͒
139.89͑4͒
Ϫ0.7͑4͒
142͑2͒
84͑8͒
2
X ⌺ϩ
86.75͑5͒
18.3͑1͒
␮ ͑D͒
3.91͑3͒
0.241 292 7͑7͒
e
B ͑cmϪ1
͒
2
A ⌸1/2
⑀ ͑MHz͒
2.8͑1.8͒
Ϫ1.0͑1.0͒
15.1͑8͒
aReference 1.
of the observed shifts yields the first experimental value for
mm ␾͒ with a 10 mm long and 1 mm internal diameter
molybdenum tube screwed into the center of the end cap
through which the beam effuses. This is placed inside a 10
cm long ceramic tube with 2 m of 0.25 mm tungsten heater
wire wrapped in a spiral groove on the outside. The whole
assembly is surrounded by spun alumina insulation ͑Carbo-
rundum Fiberfrax͒ and packed into a water-cooled jacket.
Typically, the heater requires 5 A at 50 VAC to keep the
oven at 1200 °C. The oven is loaded with ϳ3 g of YbF3 and
ϳ0.4 g of Al metal, with the idea that the Al will act as a
reducing agent to produce the radical YbF. Black-body ra-
diation from the oven, a source of stray background light in
the fluorescence detector, is carefully baffled to make sure
that there is no direct path to the photomultiplier.
the electric dipole moment ␮ of this molecule. Our new
e
values for these various constants are summarized in Table I,
where b0 and b1 are defined by analogy with Eq. ͑2͒ to
account for the centrifugal distortion of b.
II. APPARATUS
Figure 1 is a schematic view of our apparatus in the
standard LRDR arrangement.6 YbF molecules formed in an
effusive oven pass through a pump laser beam which de-
pletes one of the X-state sublevels. Next the beam passes
through a resonance region where rf transitions can be driven
to repopulate that sublevel. Finally the probe beam, tuned to
the same wavelength as the pump, induces fluorescence
which we detect with a photomultiplier. The intensity of the
fluorescence is a direct measure of the efficiency of the rf
transition in repopulating the sublevel.
B. Optical system
Light at 552 nm from a single mode cw laser ͑Spectra
Physics 380D using R110 dye͒ is set to the correct wave-
length using a wavemeter and transported through an optical
fiber to the molecular beam apparatus. There it is expanded,
collimated and apertured by a circular knife edge, to produce
a parallel beam of 1 cm diam and reasonably ͑ϳ10%͒ uni-
form intensity. This is split into a pump beam ͑ϳ43 mW͒ and
a probe beam ͑ϳ14 mW͒ which pass into the vacuum chamber
through Brewster-angle windows and intersect the molecular
beam 7 cm and 14 cm downstream from the nozzle. Consid-
erable care must be taken to prevent stray laser light from
scattering into the fluorescence detector. The light beams are
therefore allowed to emerge from the vacuum through two
more Brewster windows and are absorbed by beam stops 2 m
from the apparatus. Two stable low finesse confocal Fabry–
Perot cavities, one 10 cm long and one 50 cm long, are used
to monitor the change of the optical frequency when the laser
is scanned.
A. Source and vacuum chamber
The experiment takes place in an aluminum vacuum
chamber pumped by untrapped oil diffusion pumps to a pres-
sure of ϳ5ϫ10Ϫ7 Torr when the oven is hot. The oven
chamber is a cylindrical molybdenum7 crucible ͑18 mmϫ21
The fluorescence induced by the probe beam is collected
through optics inside the vacuum chamber. A spherical mir-
ror ͑28 mm radius of curvature, 40 mm ␾͒ is placed below
the center of the probe region, as shown in Fig. 1, to retrore-
flect the fluorescence that is emitted downwards. An
antireflection-coated aspheric lens ͑24.5 mm effective focal
FIG. 1. Schematic view of our laser-radio-frequency double resonance ap-
paratus.
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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7414
Sauer, Wang, and Hinds: Spectroscopy of 174YbF
2
2
FIG. 2. Laser-induced fluorescence spectrum of the A
⌸ ( ϭ0)–X ⌺
ϩ( ϭ0) transition at 552 nm in the vicinity of the P bandhead.
v v
1/2 1
length, 38.1 mm ␾͒ above the region collimates the fluores-
cence light, while a second identical lens, 7 cm above the
first forms an image of the probe region just outside the
vacuum chamber. A narrow slit in the image plane ͑1 mm
wide and 3 mm along the beam direction͒ defines the region
from which fluorescence is collected, thereby controlling
both the Doppler width of the LIF signal and the amount of
stray light. A pulse-counting photomultiplier tube ͑Ham-
mamatsu R878͒ just above the slit detects the fluorescence
signal. Although the black-body light from the oven is care-
fully baffled, enough of it reaches the detector that we benefit
from the use of a 55 nm bandpass interference filter ͑centered
at 548 nm͒, placed between the two lenses where the fluo-
rescence light is nominally collimated. The net efficiency of
this detection system is estimated to be 0.8%, taking into
account quantum efficiency of the photocathode, the trans-
mission of the filter, and the reflection losses from the optical
interfaces.
balls. These boards were sufficiently flat that the spacing was
uniform to better than 0.015 mm over the whole of the beam,
allowing the static electric field to be known to 1% once the
potential difference between the plates was determined. One
of the plates was split into three strips, of which the central
one ͑1 cm wide͒ was used as the shorted rf transmission line.
The outer two maintained a uniform static electric field
throughout the fringe region of the rf field.
2
III. LRDR MEASUREMENT OF THE X ⌺؉ MAGNETIC
CONSTANTS
A. 0-0 band
2
Figure 2 shows the LIF spectrum of the A ⌸1/2( ϭ0)
v
–X ⌺ϩ( ϭ0) transition at 552 nm in the vicinity of the P
2
v
1
bandhead, obtained using the probe beam without any pump
beam or rf field. The P12 branch forms a separate bandhead
approximately 1 THz to the red side of this region and the R1
branch begins 40 GHz to the blue. Even at this relatively
high resolution the spectrum is quite congested, with many
small overlapping peaks due to the various isotopes of Yb.
C. rf regions
P
The rf magnetic field required for these experiments has
a frequency in the range 30–230 MHz and an amplitude of
ϳ30 mG. We use a 0–60 MHz synthesizer ͑Hewlett–
Packard 3325B͒ as the oscillator with various frequency dou-
blers, filters, and amplifiers, to generate power at the neces-
sary frequencies. Semirigid coaxial cable feeds the rf current
through the vacuum wall and is terminated by a hairpin loop
5 cm high, 1 mm wide, and 1 cm along the beam direction,
as shown in Fig. 1. This shorted transmission line has a
magnetic antinode in the vicinity of the beam with the mag-
netic field polarized along the beam direction. The loop is
enclosed in a cylindrical magnetic field shield ͑25 mm diam,
100 mm long, 1 mm wall͒ of high permeability ͑Advance
Magnetics AD-MU80͒, which reduces the ambient magnetic
field to a few mG, thereby avoiding any significant broaden-
ing or shifting of the rf lines.
The most prominent features are the P1- and Q12-lines of
the most abundant isotope 174YbF, which are labeled more
simply as 174P(N) in Fig. 2. Each 174P(N) line exhibits
four peaks, although at this scale the two low-frequency
PQ12 components are unresolved. These are associated
with the four total angular momentum states of
2
X ⌺ϩ͑FЉϭNϩ1,N,N,NϪ1 formed by coupling rotation N,
electron spin S, and fluorine nuclear spin I͒ which differ in
energy because of the magnetic interactions in Eq. ͑1͒. The
2
splitting due to hyperfine structure in A ⌸1/2 is very small
and cannot be resolved in this spectrum. At the low values of
N shown here, the spin doubling and hyperfine interactions
are of comparable strength, making the coupling intermedi-
ate between b␤J and b␤S . The much smaller Q1͑0͒ and
[Q1(1),QR12(1)] lines are shown in Fig. 2 as 174Q(N). They
are somewhat obscured by P-lines from other even isotopes;
174Q͑0͒ overlaps with 172P͑8͒ and 176P͑9͒, while 174Q͑1͒ is
mixed with 172P͑9͒ and 176P͑10͒. Figure 3 shows a high-
resolution scan over the region of 174Q͑0͒ in which one can
also clearly see the four components of 172P͑8͒ and 176P͑9͒.
Q͑0͒ exhibits only two lines since there are only two total
angular momentum states ͑FЉϭ0 and FЉϭ1͒ when Nϭ0.
A different design of the rf region was needed in our
measurement of the electric dipole moment because we had
to apply an electric field. For those experiments, we built a
parallel plate structure from two rectangular printed circuited
boards ͑2 cmϫ7.5 cm͒ with their copper surfaces facing each
other and accurately spaced 1.549 mm apart by sapphire
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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Sauer, Wang, and Hinds: Spectroscopy of 174YbF
7415
2
2
FIG. 3. High-resolution LIF spectrum of the A
⌸
( ϭ0)–X
v
⌺
ϩ( ϭ0)
v
1/2
transition in the region of the 174YbF Q͑0͒ lines.
FIG. 4. Radio-frequency resonance line for the Nϭ0 ground state X͑ ϭ0͒,
v
measured by pumping and probing on the lower-frequency Q͑0͒ component
͑F ϭ1͒.
Љ
Figure 4 shows the rf resonance line for the Nϭ0 state,
measured by pumping and probing on the lower frequency
Q͑0͒ component ͑FЉϭ1͒. This was the weakest line we ob-
served because of the small number of molecules having
Nϭ0. Each point in the figure represents 1 s of integration
time while the solid line is a least-squares fit to a Lorentzian
profile with a sloping background. The center frequencies of
this line and 35 other lines in states up to Nϭ10 are listed in
Table II under the headings f␣ , f␤ , f␥ , and f␦ , according to
the pairs of levels coupled in the transition. Figure 5 defines
the intervals ␣, ␤, ␥, and ␦ that were measured. In order to
obtain good agreement between these measurements and the
intervals predicted by Eq. ͑1͒, it is necessary to expand the
spin-rotation constant ␥ in terms of ␥0, ␥1, ␥2, as in Eq. ͑2͒.
The constant b is similarly expanded with coefficients b0 and
b1. These constants, together with c and C were adjusted to
obtain a least-squares fit to the measured transition frequen-
cies ͓details of the eigenvalues and eigenfunctions of Eq. ͑1͒
are given in the appendix͔ and the ground-state structure de-
termined in this way is plotted in Fig. 5. Not surprisingly,
these high-resolution LRDR data allowed much more accu-
rate determination of the constants than our previous LIF
measurements described in ͑I͒. However, we were able to
use the LIF measurements to improve even further on the
value of ␥2 by fixing ␥0, ␥1, b0, b1, c, and C at their new
LRDR values and reanalyzing the LIF spectrum. This is
quite sensitive to ␥2, because it goes much higher in N ͑up to
Nϭ73͒ than the LRDR spectrum. In this fit, we allowed the
2
A ⌸1/2͑ ϭ0͒ upper level to have a fixed splitting ⑀, which
v
was a second fitting parameter. This parameter was consis-
tent with zero, both for the P lines and for the Q lines. The
results for all these constants are listed in the ϭ0 column of
v
Table I.
We note that the signs of the spin-rotation constants ␥0
and ␥1 were incorrect in ͑I͒. The old LIF data did not have
the combination of high resolution and low N needed to
determine the sign of ␥, but we overlooked this ambiguity
and unfortunately chose the wrong sign. With our LRDR
data, N goes all the way down to zero and there is no uncer-
tainty as to the sign of ␥.
B. 1-1 band
We recorded the LIF spectra for nineteen (Q1 ,QR12)
lines of the 1-1 band in the range Nϭ8 to Nϭ49. Lines of
2
TABLE II. Measured frequencies f␣ , f␤ , f␥ , f␦ of the rf intervals in X ⌺ϩ ͑ ϭ0͒ with standard deviations in
v
parentheses. The transitions ␣,␤,␥,␦, are defined in Fig. 5.
N
f␣ ͑MHz͒
f␤ ͑MHz͒
f␥ ͑MHz͒
f␦ ͑MHz͒
0
1
2
3
4
5
6
7
8
•••
•••
•••
170.254 0͑4͒
161.183 6͑8͒
154.212 4͑4͒
148.252 1͑4͒
142.989 5͑3͒
138.291 9͑5͒
134.075 1͑5͒
130.288 4͑4͒
126.881 1͑1͒
121.051 5͑4͒
155.660 9͑14͒
177.507 0͑16͒
188.363 6͑14͒
197.963 3͑5͒
207.341 7͑18͒
216.752͑5͒
226.247 9͑10͒
235.835 4͑10͒
255.143 4͑12͒
30.954͑4͒
39.773͑4͒
48.467͑3͒
57.286͑3͒
66.293͑3͒
75.496͑2͒
84.870 2͑14͒
94.374 6͑12͒
113.599 7͑3͒
192.137 9͑10͒
193.989 7͑5͒
196.719 9͑4͒
200.275 0͑2͒
204.585 0͑3͒
209.572 7͑8͒
215.157 4͑3͒
221.254 1͑4͒
234.650 4͑1͒
10
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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7416
Sauer, Wang, and Hinds: Spectroscopy of 174YbF
of the magnetic structure in the X͑ ϭ1͒ ground state ͑upside
v
down͒, except for the relatively minor point that we have so
far neglected any upper state hyperfine splitting in the analy-
sis. On comparing Figs. 5 and 6 it is evident that the ϭ0
v
and ϭ1 ground states have significantly different magnetic
v
splittings.
In order to determine the ϭ1 constants more accu-
v
rately, we also made LRDR measurements on this band. We
determined all the transition frequencies f␤ in the range
Nϭ8–13, stopping at this point because our oscillator fre-
quency would not go any higher. These results and the fre-
quencies of six other transitions are shown in Table III. Our
analysis of these LIF and LRDR data used the procedures
described for the 0-0 band and gave the ϭ1 constants re-
v
ported in Table I.
On comparing the ϭ0 and ϭ1 results in Table I it is
v
v
striking that ␥0 changes by a factor of 3, whereas the coeffi-
cients ␥1 and ␥2 are practically constant. In the last section of
this paper we propose a possible explanation for this unusual
behavior of ␥. It is also notable that the hyperfine splitting ⑀
in the upper states A ⌸1f /2 is no longer too small to see; it is
2
approximately 15 MHz and is essentially independent of J.
A fuller analysis of this structure will be published sepa-
rately.
FIG. 5. Energies of the four ground-state levels determined from our laser-
radio-frequency double resonance spectra.
IV. MICROWAVE MEASUREMENT OF B
The rf experiments above allowed us to determine the
intervals within one rotational manifold N but were not di-
rectly sensitive to the rotational structure. In order to mea-
sure the rotational constant B we coupled a 14 GHz micro-
wave generator ͑Gigatronics model 900͒ to the rf loop in the
hope of driving the Nϭ1→0 transition directly. With the
pump and probe beams tuned to the lower Q͑0͒ peak of the
0-0 band ͑NЉϭ0, FЉϭ1͒, electric dipole transitions are pos-
sible from all four hyperfine states in NЈϭ1 ͑FЈϭ1,2,0,1͒.
However, the transition from the lower-lying FЈϭ1 level is
relatively weak and we did not measure its center. For the
other three transitions we found the following line centers:
lower N are obscured by the P12͑42͒ lines of the 0-0 band,
while the higher-N lines overlap the P1 band of 0-0. Once
again there are four components to each (Q1 ,QR12) line,
corresponding now to the four magnetic levels in the ϭ1
v
ground state. For each of the 19 lines we determined the
centers of the four components, and the total data set was
fitted by a least-squares adjustment to the Hamiltonian of Eq.
͑1͒. The result of this fit, shown in Fig. 6, is essentially a map
N ϭ1, F ϭ0→N ϭ0, F ϭ1: 14 452 566.0 5͒ kHz,
͑
Ј
Ј
Љ
Љ
N ϭ1, F ϭ2→N ϭ0, F ϭ1: 14 458 087.8 3͒ kHz,
͑
Ј
Ј
Љ
Љ
N ϭ1, F ϭ1→N ϭ0, F ϭ1: 14 489 042.0 2͒ kHz.
͑
Ј
Ј
Љ
Љ
͑3͒
TABLE III. Measured frequencies f␣ , f␤ , f␥ , f␦ of the rf intervals in
2
X ⌺ϩ͑ ϭ1͒ with standard deviations in parentheses. The transitions
v
␣,␤,␥,␦, are defined in Fig. 5.
N
f␣ ͑MHz͒
f␤ ͑MHz͒
f␥ ͑MHz͒
f␦ ͑MHz͒
8
9
10
11
12
13
384.62͑2͒
414.81͑3͒
245.077͑1͒
275.217͑1͒
305.517͑1͒
335.870͑1͒
366.188͑1͒
396.400͑35͒
347.674͑6͒
374.983͑8͒
36.9401͑6͒
39.8215͑3͒
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
•••
2
2
FIG. 6. The four LIF peaks of the A
⌸ ( ϭ1)–X ⌺
ϩ( ϭ1) transition
v v
1/2
vs rotational quantum number N. Solid line is a fit to the ground-state
Hamiltonian given in Eq. ͑1͒.
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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Sauer, Wang, and Hinds: Spectroscopy of 174YbF
7417
Although our rf structure was not designed for this purpose,
very little power is required to drive these strongly allowed
electric dipole transition and we had to be careful to avoid
using excessive microwave power which broadens the lines
and shifts them. The line centers were determined by fitting
each of the measured spectra to a Lorentzian profile plus a
sloping background.
The level shifts due to hyperfine interactions are known
from our measurements of b, c, and C given in Table I,
together with the theory given in the appendix. The upper
level is also affected by the spin-rotation interaction and the
centrifugal stretching energy 4Dvϭ28.6 kHz.8 When our
measurements are corrected for all these contributions, we
find a value for the rotation constant.
FIG. 7. Frequency of the Nϭ0 hyperfine transition ͑F ϭ1,
Љ
B ϭ7 233.800 7 10͒ MHz
͑
v
MF ϭϮ1͒←͑F ϭ0͒ as a function of static electric field strength. Data ob-
Љ
Љ
2
2
tained using LRDR on the Q͑0͒ line of A
⌸ ( ϭ0)–X ⌺
ϩ( ϭ0).
v v
1/2
ϭ0.241 293 62 3͒ cmϪ1
.
͑4͒
͑
Solid line shows the calculated shift using our values for the magnetic
interaction constants. The only free parameter was the dipole moment ␮
e
which was chosen to give the best fit to our data.
This result is not entirely consistent with the value
Bvϭ0.241 416͑25͒ cmϪ1 given by Dunfield et al.,8 presum-
ably because they fitted an optical spectrum with unresolved
magnetic structure, whereas we measure the Nϭ1→0 transi-
tion directly with sufficient resolution to account for the
spin-rotation and hyperfine interactions.
states and the two fluorine nuclear spin states, is then used as
a basis ͉N ϭ0͘
͉
Sϭ 12,mS
͉
Iϭ 1,m for diagonalizing Eq. ͑1͒
͘
͘
2
I
to determine the energies of the four ͉N ϭ0͘ hyperfine lev-
els. This approximation neglects magnetic interactions be-
tween manifolds of different N . In zero electric field it
amounts to neglect of the ⌬Nϭ2 tensor interaction discussed
in the Appendix, which shifts the ͑Nϭ0, Fϭ1͒ states down
by 9.3 kHz according to Eq. ͑15͒. This additional shift varies
with the applied electric field and is unfortunately rather
complicated to calculate explicitly. Our qualitative estimate,
however, is that it does not change by more than a few tens
of kHz over the range of electric fields in our experiment and
V. STARK EFFECT AND DETERMINATION OF ␮
e
Next, we turned to a measurement of the electric dipole
moment ␮e , which required the addition of a static electric
field to the rf region, as described in Sec. II C. With this
interaction region we were able to study the shift of the Nϭ0
hyperfine transition ͑FЉϭ1, MF ϭϮ1͒←͑FЉϭ0͒ as a func-
Љ
contributes much less to the final uncertainty in ␮ than the
e
tion of the electric field strength. Using the LRDR on the
Q͑0͒ line, we measured this transition frequency at 45 field
settings up to 18 kV/cm, with the results shown in Fig. 7.
Although the Nϭ0 levels shift quite strongly ͑ϳ20 GHz at
20 kV/cm͒, the two hyperfine levels involve ͑essentially͒ the
same rotational state and shift almost equally. The small
difference—the Stark shift we observe—is due primarily to
the tensor part of the hyperfine interaction, which affects
only the triplet state ͑i.e., the state corresponding to FЉϭ1 in
zero electric field͒ and which changes as the molecular rota-
tion is polarized by the field. This differential shift is of order
c/6BϷ10Ϫ3 relative to the full Stark shift of the levels. Fig-
ure 7 also shows a fit to our data of the calculated differential
shift based on the Hamiltonian H in Eq. ͑1͒ and the interac-
tion Ϫ␮e•E. Since the magnetic interaction constants are al-
ready known from our experiments in zero electric field, the
only fitting parameter is the electric dipole moment ␮e . For
this calculation, we first diagonalize the rigid rotor states
with an electric field E along the z-axis, i.e., with the effec-
tive Hamiltonian BN(Nϩ1)ϩ␮E cos ␪. This produces
1% uncertainty in the electric field calibration. We are there-
fore able to ignore this effect when we fit the data without
compromising the accuracy of the experiment. Our result is
␮ ϭ1.97 2͒ MHz/͑V/cm͒
͑
e
ϭ3.91 4͒ D.
͑5͒
͑
As a check on this result, we have also studied the Stark
shift of three rf transitions near 160 MHz in the N ϭ1 mani-
fold. In the notation (FЈ,mF ←FЉ,mF ), they are
Ј
Љ
͑2,1←1Ϫ,0͒, ͑2,0←1Ϫ,1͒, and ͑2,1←1Ϫ,1͒, where Fϭ1Ϫ in-
dicates the lower-lying of the two Fϭ1 states. ͑The last of
these is a ⌬mϭ0 transition which requires higher power be-
cause the rf field is mainly perpendicular to the electric
field.͒ The Stark shift of these transition frequencies is not
suppressed, as it was in the rotational ground state, because
the states involved here are not of the same rotational char-
acter: they have different mixtures of ͉N ϭ1, mNϭ0͘ and
͉N ϭ1, mNϭϮ1͘ which have opposite Stark shifts. Conse-
quently we only needed to raise the field to 300 V/cm to
achieve shifts of several MHz. In order to deduce the value
a
Stark-mixed rotational ground state
͉N ϭ0
͘
ϱ
ϭ ͚Nϭ0aN͉N,m ϭ0 in which it was sufficiently accurate
͘
N
for our purpose to neglect states having NϾ6. This Stark-
of ␮ from these measurements, it was necessary to calculate
e
mixed rotational state, together with the two electron spin
the N ϭ1 Stark effect. Although this is essentially as de-
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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7418
Sauer, Wang, and Hinds: Spectroscopy of 174YbF
2
TABLE IV. Constants determined in this work for the X ⌺ϩ states of YbF
TABLE V. Molecular dipole moment ␮e , the quantity ␮ ͑electron charge
0
in comparison with those of the alkaline-earth monofluorides.
times bond length͒ and the difference between the two for YbF and the
alkaline-earth monofluorides.
YbF
MgFa
155
CaFb
SrFc
BaFd
64
␮ ͑D͒
␮ ͑D͒
␮0Ϫ␮ ͑D͒
e
0
e
b00 ͑MHz͒
b10 ͑MHz͒
b01 ͑kHz͒
143
Ϫ1.9
Ϫ0.5
109
98
MgF
CaF
SrF
BaF
YbF
3.2
3.1
3.5
3.2
3.9
8.4
9.4
10.0
10.3
9.7
5.2
6.3
6.5
7.1
5.8
Ϫ1.4
Ϫ0.5
Ϫ1.2
Ϫ0.3
c00 ͑MHz͒
c10 ͑MHz͒
85
1.3
178
40
1.1
30
0.8
8
C00 ͑kHz͒
20.4
29
2.3
00
␥
␥
␥
␥
͑MHz͒
͑MHz͒
͑kHz͒
͑Hz͒
Ϫ3.2
Ϫ20
3.8
51
Ϫ1
40
75
81
Ϫ0.058
0.11
10
01
11
Ϫ0.32
0.010
Ϫ2.1
Ϫ0.46
0.059
Ϫ3.7
341
To assist in the comparison, we have recast our constants for
YbF in the same way ͑even though this makes little sense for
␮ ͑D͒
3.9
3.2e
3.1f
3.5g
0.25
3.2
e
␥ because it changes so strongly with ͒.
v
B ͑cmϪ1
͒
0.24
0.52
0.34
0.22
First, let us look at the hyperfine interaction of the F
nucleus, characterized by the constants b and c. The main
point to notice is that these constants are all very small com-
pared with the hyperfine interaction of a free F atom, indi-
cating that the amount of unpaired electron spin density near
the F nucleus is very small because the bonds are all strongly
ionic. As one progresses down the IIa sequence from MgF to
BaF there is a monotonic increase in ionicity, and this is
clearly seen as a corresponding decrease in the hyperfine
constants b and c in Table IV. The same point can be found
in the quantity c/(3bϩc) which is the ratio of the dipolar to
the isotropic hyperfine interaction. This increases from 0.04
in BaF to 0.28 in MgF indicating in the lighter species a
more hybridized orbital that can participate in covalent bond-
ing. Turning now to Yb, the atom is about 10% smaller in
radius than barium ͓͑Xe͔6s2͒ because of the lanthanide con-
traction, and as a result its radius and electronegativity are
between those of Mg and Ca. When we look at the magni-
tudes of b and c and at the ratio c/(3bϩc) this is indeed
where YbF comes, so from this point of view it fits well into
the family of alkaline earth monofluorides.
aReference 18.
bReference 4.
cReference 11.
dReference 19.
eReference 20.
fReference 21.
gReference 22.
scribed above for N ϭ0, the details are considerably more
complicated because of the higher angular momenta in-
volved and we refrain from reproducing them here. The
point of this check was that in such weak fields, there is no
uncertainty due to the magnetic interactions between mani-
folds of different N . All three of the transitions we mea-
sured yielded the same result as Eq. ͑5͒ to better than 0.2%,
giving us some confidence that both the experimental and the
computational parts of the measurement are being done cor-
rectly. Differences at that level are, of course, insignificant in
comparison with the 1% uncertainty associated with the
magnitude of the electric field.
Our value in Eq. ͑5͒ for the dipole moment of YbF can
be compared with the ab initio pseudopotential calculations
of Dolg et al.9 who obtained values between 3.40 D and 3.75
D, with 3.55 D being the result of their most sophisticated
calculation. Using a density functional approach Wang and
Schwarz have calculated ␮eϭ3.64 D.10
The electric dipole moment ␮ in group IIA monofluo-
e
rides is reasonably well described by a very simple ionic
bonding model12,13 in which ␮eϭ␮0Ϫ(␮MϪ␮F). Here
␮0ϭRee is due to the charges of the ions ͑Re is the internu-
clear distance͒, while ␮ and ␮ are the dipole moments
M
F
VI. DISCUSSION
induced in each ion by the field of the other. Table V shows
The Yb atom in its ground state configuration
͓Xe͔4f146s2 is similar to an alkaline earth, having a pair of
ns outer electrons and an empty (nϪ1)d shell. The differ-
ence, of course, is in the 4f shell. We can gain some insight
into the structure of YbF by comparing the constants we
have measured with those of the alkaline earth fluorides
listed in Table IV. For CaF ͑Ref. 4͒ and SrF,11 the magnetic
constants are given as the coefficients of a Dunham expan-
that ␮ for YbF lies between CaF and SrF while the polar-
0
ization contribution ␮0Ϫ␮ has a value in between MgF and
e
CaF indicating a relatively small Ybϩ polarization in keeping
with its small radius.
Although YbF fits comfortably into the alkaline-earth
monofluoride series with regard to bond length, dipole mo-
ment, and hyperfine constants, it is obvious that ␥ is quite
different. It has the ‘‘wrong’’ sign, it almost triples between
sion, i.e., constant A for the vibrational state and rotational
v
ϭ1 and ϭ2 and its centrifugal correction is a hundred
v
v
state N is
times too large. The electron spin-rotation interaction is pre-
sumably due to the usual second order perturbation,14
1
2
␭␮
␭
␮
A
,N͒ϭ
A
ϩ ͒ N Nϩ1͒ .
͓ ͑ ͔
͑6͒
͑v
͑v
͚
␭,␮
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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Sauer, Wang, and Hinds: Spectroscopy of 174YbF
7419
2
2
2
2
X ⌺ϩ
͉
B r͒LϪ
͉
͑
n ⌸1/2
,
n ⌸
, ͉ ͉
A r͒Lϩ X ⌺ϩ
v ͑
͘
͑
v
͗
͗͘
1/2
2
␥Ϸ
ϩh.c.
͑7͒
͚
2
E n ⌸ , ͒ϪE n ⌺ϩ͒
v
͑
n,v
1/2
in which the L-uncoupling operator B(r)J–L and the spin–
orbit interaction A(r)L–S admix excited 2⌸1/2 configurations
ture which would indicate the violation of time reversal sym-
metry.
2
into the ⌺ ground state. In ͑I͒ we assumed that only the
(6p␲)2⌸1/2 part of the A-state is admixed and used Eq. ͑7͒
to make the rough estimate ␥Ϸ270 MHz. In the alkaline-
earth monofluorides this kind of estimate is reliable to better
than 40% but for YbF it does not even give the correct sign,
let alone the magnitude, which is much smaller in reality.
The inclusion of more (np␲)2⌸1/2 states will not improve
the agreement since all the factors involved are positive. A
reduction of ␥ could occur through interference between the
contributions of p and d orbitals but the amount of d␴ char-
acter in the X and A states is simply too small to make a
substantial contribution. We are therefore led to believe that
the small value of ␥ indicates the presence of a more com-
plicated 2⌸1/2 state, perhaps involving a hole in the f shell. It
is likely that the X state has one or two percent of 4f␴
ACKNOWLEDGMENTS
We thank H. J. Metcalf for lending us the microwave
synthesizer and I. B. Khriplovich and R. W. Field for useful
discussions concerning the spin-rotation interaction. We are
also indebted to C. Linton for conveying results of his YbF
experiments prior to publication.
APPENDIX
The magnetic structure of the YbF X ⌺ϩ ground state is
given by Eq. ͑1͒ for Yb isotopes of zero nuclear spin. In
order to write down the eigenstates and energies, it is conve-
nient to adopt a coupling scheme ͉(NS)J,I;F,M in which
͘
rotation N and electron spin S form an intermediate angular
momentum J, coupled in turn to fluorine nuclear spin I to
give the total angular momentum F with z-component M.
2
2
character.15 This would couple to such a ⌸1/2 state to pro-
duce a negative contribution to ␥ because the spin–orbit in-
teraction of the f hole is inverted. Moreover, the strength of
the 4f spin–orbit interaction in Ybϩ is large enough
͑␨4fϭ2949 cmϪ1͒ that a complete cancellation of ␥ is ex-
pected for 1%–2% of 4f␴ in the ground state wave
function.16 This idea seems particularly plausible since the
recent discovery of an ⍀ϭ1/2 state nearly degenerate with
We abbreviate these states to ͉(J)F,M . Since Sϭ1/2 and
͘
Iϭ1/2, there are four levels in the rotational manifold N,
characterized by Fϭ(Nϩ1),N,N,(NϪ1). For the FϭNϮ1
states, the eigenfunctions are simply
͉
͉
Nϩ1 ϭ
͉
͉
Nϩ1/2͒Nϩ1,M ,
͑
͑
͘
͘
2
A ⌸1/2, whose vibrational and rotational frequencies are
consistent with those of an f13s2 configuration.8 The new
NϪ1 ϭ
NϪ1/2͒NϪ1,M ,
͑A1͒
͘
͘
state also provides a plausible basis for the observed varia-
with eigenvalues
2
tion of ␥ with because the A ⌸1/2͑ ϭ1͒ state is strongly
v
v
␥N
b
c
CN
admixed with the ϭ0 state of the perturber, whereas
v
E
E
Nϩ1ϭ
ϩ
ϩ
ϩ
,
2
2
4
4 2Nϩ3͒
2
͑
A ⌸1/2͑ ϭ0͒ is relatively unperturbed.
v
The admixture of a perturbing state does not help to
explain the large size of the distortion corrections ␥1 which
are very similar for both vibrational states. Although some
ideas exist in the literature as to the origin of the distortion
correction to ␥,1 they do not account for the observed distor-
tion in even the simpler alkaline earth fluorides.
␥ Nϩ1͒
b
ϩ Ϫ
c
C Nϩ1͒
͑
͑
NϪ1ϭϪ
Ϫ
. ͑A2͒
2
4
4 2NϪ1͒
2
͑
For the remaining two states,
the explicit Hamiltonian for any value of M is
͉
(J)F,M ϭ
͉
(NϮ 1)N,M ,
͘
͘
2
N
0
␥
In this paper we have reported the first laser-radio-
frequency double resonance spectrum of a simple lanthanide
compound. Although the gross structure of YbF in its ground
state resembles that of an alkaline earth monofluoride, the 4f
shell is evidently important even in the low-lying excited
states and leads to unusual behavior in the magnetic struc-
ture. It would be interesting to go on to look at hyperfine
interactions in the odd isotopomers of YbF in order to shed
further light on the details of the ground state structure. One
might also consider similar experiments on other lanthanide
compounds in which the ground state involves an open f
shell. Finally we note that our original motivation for study-
ing this molecule was related to elementary particle physics;
YbF is an extremely sensitive system in which to search for
a permanent electric dipole moment of the electron,17 a fea-
Hϭ
ͫ
ͬ
4ͱN Nϩ1͒
2
0
Ϫ Nϩ1͒
͑
4ͱN Nϩ1͒
͑
Ϫ 2Nϩ3͒
͑
b
ϩ
ϩ
ϩ
ͫ
ͫ
ͫ
ͬ
ͬ
4 2Nϩ1͒
͑
Ϫ 2NϪ1͒
͑
͑
2ͱN Nϩ1͒
͑
Ϫ1
c
4 2Nϩ1͒
͑
2ͱN Nϩ1͒
͑
1
Ϫ2ͱN Nϩ1͒
ϪN 2Nϩ3͒
͑
͑
C
,
ͬ
2 2Nϩ1͒
͑
Ϫ2ͱN Nϩ1͒
͑
2NϪ1͒ Nϩ1͒
͑
͑
͑A3͒
with the diagonal element for JϭNϩ 12 at the top left. The
eigenvalues are
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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7420
Sauer, Wang, and Hinds: Spectroscopy of 174YbF
ENϮϭϪ 14 ␥ϩbϩC͒
With the above expressions ͑A2͒, ͑A4͒, and ͑A8͒ for the four
͑
hyperfine eigenenergies, we were able to calculate the tran-
sition frequencies to be compared with experiment.
Ϯ 41
2Nϩ1͒2ϩ 2bϩcϪ2C͒ 2bϩcϪ2␥͒.
͑ ͑
2
ͱ ␥ϪC͒
͑
͑
͑A4͒
1 B. E. Sauer, Jun Wang, and E. A. Hinds, Phys. Rev. Lett. 74, 1554 ͑1995͒.
2 R. F. Barrow and A. H. Chojnicki, J. Chem. Soc. Faraday Trans. II 71,
728 ͑1975͒; H. U. Lee and R. N. Zare, J. Mol. Spectrosc. 64, 233 ͑1977͒.
3 W. Weltner, Jr., Magnetic Atoms and Molecules ͑Dover, New York,
1983͒, pp. 42 ff.
The corresponding eigenfunctions can be written as
͉
NϮ,M ϭxϮ Nϩ 1͒N,M ϩyϮ NϪ 1͒N,M
͉ ͉
͑ ͑
͘ ͘
2 2
͑A5͒
͘
4 W. J. Childs, G. L. Goodman, and L. S. Goodman, J. Mol. Spectrosc. 86,
365 ͑1981͒.
in which the ratio of the coefficients, given by
Nϩ 1͒N,M NϪ 1͒N,M
͉H͉
͑
͘
2
5 R. J. Van Zee et al., J. Phys. Chem. 82, 1192 ͑1978͒.
6 W. J. Childs, Phys. Rep. 211, 113 ͑1992͒.
xϮ
yϮ
͑
͗
2
ϭϪ
,
͑A6͒
Nϩ 1͒N,M
͉
H
͉
Nϩ 1͒N,M ϪE
Ϯ
7 Zr also works well as an oven material but it is destroyed more quickly
than Mo by reactions with the chemicals.
͑
͑
͗
͘
2
2
N
8 K. L. Dunfield, C. Linton, T. E. Clarke, J. McBride, A. G. Adam, and J. R.
D. Peers, J. Mol. Spectrosc. 174, 433 ͑1995͒.
can be determined straightforwardly from Eq. ͑11͒ together
with the off-diagonal matrix elements of Eq. ͑10͒.
9 M. Dolg, H. Stoll, and H. Preuss, Chem. Phys. 165, 21 ͑1992͒.
10 S. G. Wang and W. H. E. Schwarz, J. Phys. Chem. 99, 11 687 ͑1995͒.
11 W. J. Childs, L. S. Goodman, and I. Renhorn, J. Mol. Spectrosc. 87, 522
͑1981͒.
In this treatment, we have diagonalized the magnetic in-
teractions H within the rotational manifold N. This is a good
approximation because the coupling of the manifold N with
other rotational states NЈ is small compared with the rota-
tional splitting. In fact this coupling is zero for the two states
͉NϮ͘ having FϭN. However, the state ͉Nϩ1͘ ͑FϭNϩ1 in
manifold N͒ is coupled by H to ͉NЈϪ1͘ ͑FЈϭNЈϪ1 in
manifold NЈ͒ when NЈϭNϩ2 ͑that is, when FЈϭF͒. Simi-
larly ͉NϪ1͘ is coupled to ͉NЈϩ1͘ when NЈϭNϪ2. The two
matrix elements
12 E. S. Rittner, J. Chem. Phys. 19, 1030 ͑1951͒.
13
¨
T. Torring, W. E. Ernst, and S. Kindt, J. Chem. Phys. 81, 4614 ͑1984͒.
14 H. Lefebvre-Brion and R. W. Field, Perturbations in the Spectra of Di-
atomic Molecules ͑Academic, New York, 1986͒.
15 M. Dolg ͑private communication͒.
16 We are indebted to I. B. Khriplovich for pointing this out to us.
17 E. A. Hinds, in Quantum Dynamics of Simple Systems, edited by G.-L.
Oppo, S. Barnett, E. Riis, and M. Wilkinson, SUSSP44 School ͑IOP,
Bristol, 1996͒.
18 M. A. Anderson, M. D. Allen, and L. M. Ziurys, J. Chem. Phys. 100, 824
c ͱ NϮ1͒ Nϩ1Ϯ1͒
͑
͑
͑1994͒.
N ϯ1
Ј
͉
H
͉
NϮ1 ϭ
͑A7͒
͗
͘
19
¨
For spin-rotation constants, see Ch. Ryzlewicz and T. Torring, Chem.
2
2Nϩ1Ϯ2
Phys. 51, 329 ͑1980͒. For hyperfine constants and dipole moment, see W.
¨
¨
E. Ernst, J. Kandler, and T. Torring, J. Chem. Phys. 84, 4769 ͑1986͒.
are due to the rank-2 tensor part of the interaction cIzSz .
Treating this as a second-order perturbation to the energies
of the two affected levels, we obtain the small corrections
20
´
Calculated value: A. D. Buckingham, and R. M. Olegario, Chem. Phys.
Lett. 212, 253 ͑1993͒.
21 W. J. Childs, L. S. Goodman, U. Nielsen, and V. Pfeuer, J. Chem. Phys.
80, 2283 ͑1984͒.
c2 NϮ1͒ Nϩ1Ϯ1͒
͑
͑
22
¨
¨
W. E. Ernst, J. Kandler, S. Kindt, and T. Torring, Chem. Phys. Lett. 113,
351 ͑1985͒.
⌬ENϮ1ϭϯ
.
͑A8͒
3
8B
2Nϩ1Ϯ2͒
͑
J. Chem. Phys., Vol. 105, No. 17, 1 November 1996
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