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Journal of Molecular Spectroscopy 240 (2006) 58–63
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Velocity modulation spectroscopy of molecular ions II:
The millimeter/submillimeter-wave spectrum of TiF+ (X3Ur)
D.T. Halfen
*,1,
L.M. Ziurys
Department of Chemistry, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA
Department of Astronomy, Arizona Radio Observatory, and Steward Observatory,
University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA
Received 24 July 2006; in revised form 16 August 2006
Abstract
The pure rotational spectrum of the molecular ion TiF+ in its 3Ur ground state has been measured in the range 327–542 GHz using
millimeter-wave direct absorption techniques combined with velocity modulation spectroscopy. TiF+ was made in an AC discharge from
a mixture of TiCl4, F2 in He, and argon. Ten transitions of this ion were recorded. In every transition, fluorine hyperfine interactions, as
well as the fine structure splittings, were resolved. The fine structure pattern was found to be regular with almost equal spacing in frequency between the three spin components, in contrast to TiCl+, which is perturbed in the ground state. The data were fit with a case (a)
Hamiltonian and rotational, fine structure, and hyperfine constants were determined. The bond length established for TiF+,
r0 = 1.7775 Å, was found to be shorter than that of TiF, r0 = 1.8342 Å—also established from mm-wave data. The hyperfine parameters
determined are consistent with a d1p1 electron configuration with the electrons primarily located on the titanium nucleus. The nuclear
spin–orbit constant a indicates that the unpaired electrons are closer to the fluorine nucleus in TiF+ relative to TiF, as expected with the
decrease in bond length for the ion. The shorter bond distance is thought to arise from increased charge on the titanium nucleus as a
result of a Ti2+F configuration. A similar decrease in bond length was found for TiCl+ relative to TiCl.
2006 Elsevier Inc. All rights reserved.
Keywords: Millimeter/submm spectroscopy; Molecular ions; Rotational; Velocity modulation
1. Introduction
Metal-containing molecular ions play a major role in
many chemical processes. For example, FeO+ acts as a catalyst in the conversion of harmful nitrogen oxides and CO
into N2 and CO2. Gas-phase transition-metal oxide cations
are also used catalytically to activate C–H and C–C bonds,
for example, converting methane into methanol [1]. Nickel
cluster cations can be used to activate methane, as well.
This process is important industrially in the conversion of
methane into more complex chemical compounds [2]. In
addition, metal-bearing molecular ions are thought to be
*
1
Corresponding author. Fax: +1 520 621 5554.
E-mail address: [email protected] (D.T. Halfen).
NSF Astronomy and Astrophysics Postdoctoral Fellow.
intermediates leading to neutral species in the molecularrich gas surrounding old giant stars [3].
Previous studies of metal-containing molecular ions have
shown that these species may have different properties, as
compared to their neutral counterparts. For example, a significant change in bond length occurs for the TiCl and TiCl+
neutral-ion pair. From a rotational study of TiCl+ by Halfen and Ziurys [4], the bond length of this ion in its X3Ur
state was determined to be r0 = 2.1888 Å—much shorter
than the TiCl (X4Ur) bond length of r0 = 2.2675 Å, established from the pure rotational spectrum recorded by
Maeda et al. [5]. TiCl+ is created from TiCl by the removal
of an electron in the 11r orbital. This orbital is mostly
nonbonding 4s in character [6], and therefore, to a first
approximation, loss of this electron should not influence
the bond length. The reduction in bond distance occurs
nonetheless because of increased electrostatic interaction.
0022-2852/$ - see front matter 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.jms.2006.08.008
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Spectroscopy (2006), doi:10.1016/j.jms.2006.08.008
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Another ion of interest in this regard is TiF+. The first
spectrum of this ion was measured by Focsa et al., who
used velocity-modulation laser spectroscopy to record the
(17.6)3D–X3U transition with 0.005 cm1 resolution [7].
Three DR = 0 subbands were resolved for this transition,
and identified as 3D1–3U2, 3D2–3U3, D3–3U4. No intercombination (DR „ 0) subbands were detected; therefore, the
spectroscopic constants were determined separately for
the individual subbands. Focsa and Pinchemel then conducted dispersed laser-induced fluorescence experiments
on TiF+, measuring the (17.6)3D–B3D and (17.6)3D–C3P
transitions [8]. Ligand field calculations for this ion were
also carried out, which yielded predictions of the energies
of several low-lying electronic states. These energies were
found to be in reasonable agreement with the experimental
data [8]. Other theoretical investigations of this ion have
been performed, which suggested a 3U ground state, as
found experimentally [9,10].
Here, we present the first measurements of the pure
rotational spectrum of TiF+ in its X3Ur (v = 0) state. This
spectrum was obtained using a new velocity modulation
millimeter-wave spectrometer [11]. All three fine structure
components have been observed, from which spectroscopic
constants have been determined. In this paper, we report
our results and analysis.
2. Experimental
The pure rotational spectrum of TiF+ (X3Ur) was measured using the velocity modulation spectrometer of the
Ziurys group. This instrument, described in detail elsewhere [11], consists of a radiation source, gas cell, and
detector. The frequency source is a Gunn oscillator/Schottky diode multiplier combination, which generates radiation in the range of 65–650 GHz. The single-pass, glass
reaction chamber contains two ring discharge electrodes
and is cooled to 65 C with a methanol chiller. The radiation is launched from a scalar feedhorn, is directed
through the reaction cell using two Teflon lenses, and then
is focused into the detector, a helium-cooled InSb hot-electron bolometer. The system can be run in both source modulation and velocity modulation mode. For source
modulation, the radiation frequency is varied at a rate of
25 kHz with detection at 2f; in velocity modulation (VM)
mode, the r.f. discharge is changed at a rate of 20 kHz
and detection is at 1f. Source modulation is usually used
for initial searches, while velocity modulation is used to distinguish between ion and neutral signals (see Halfen and
Ziurys [4]).
TiF+ cation was produced in an AC discharge of gasphase TiCl4, argon, and a mixture of 10% F2 in helium.
Maximum signals were produced using 40 mTorr of argon,
less than 1 mTorr of TiCl4, and 1 mTorr of the F2/He
mixture. A power level of 250 W at 600 X and a modulation rate of 20 kHz was employed for the AC discharge.
The discharge plasma itself exhibited a bluish-purple color
when creating TiF+.
59
In Focsa et al. [7,8], TiF4 was used to produce TiF+. An
attempt was made to generate signals of TiF+ using this
precursor; however, it was not successful. This compound
is a solid, and thus needs to be heated to produce appreciable vapor. Focsa et al. warmed the solid to 400 C using an
oven. The current VM spectrometer is not easily adapted to
such a high-temperature oven, and hence an alternate
method was used.
Discharging TiCl4 is thought to first remove an electron to
form TiCl4 þ , followed by sequential elimination of the chlorine atoms. Eventually, a small amount of bare Ti+ is generated, along with other species [12]. Emission from Ti+ could
be responsible for the blue–white glow seen in the discharge
plasma that supports TiCl+ [4]. A possible formation mechanism for TiF+ is the reaction of F2 with Ti+ ions:
Tiþ þ F2 ! TiFþ þ F:
ð1Þ
+
The intensity of the TiF lines were found to be an order of
magnitude lower than those of TiCl+ from our previous
experiments [4]. This decrease could be explained if TiCl+
is created directly from the discharge products of TiCl4,
and TiF+ is formed from the secondary reaction of Ti+
and F2.
Initially a range of 2 GHz was searched in source modulation mode to locate TiF+ signals, based on predictions
using the constants of Focsa et al. [7]. The lowest energy
spin component, X = 2, was found and verified by velocity
modulation. Afterward, more selective searches were conducted for additional X = 2 components, and then finally
for the weaker X = 3 and 4 spin components. After locating the TiF+ features, precise rest frequencies were measured using 5 MHz wide scans taken in pairs so there was
always an equal number in increasing and decreasing frequency. Obtaining an adequate signal-to-noise ratio typically required using 2–16 scan pairs. Gaussian profiles
were fit to the line shapes to record the center frequency
and line width. Line widths were typically 1.1–1.6 MHz
in the range 327–542 GHz, and the experimental accuracy
is estimated to be ±100 kHz.
3. Results and analysis
The rotational spectrum of TiF+ was located based on
predicted frequencies using the constants of Focsa et al.
[7], as mentioned. The predictions were found to be accurate to within 120 MHz for all X ladders, indicating the
high quality of the optical data. As shown in Table 1, 10
transitions were measured for TiF+ in the range 327–
542 GHz. Because the ground state of this ion is 3Ur, each
transition consists of three fine structure components arising from spin–orbit and spin–spin coupling, labeled by
quantum number X. Fluorine hyperfine interactions further split every X component into doublets, each indicted
by quantum number F. All six hyperfine lines were measured for almost every transition recorded.
Unlike TiCl+, TiF+ exhibits a very regular case (a) pattern with the spin-components almost evenly separated.
Please cite this article as: D.T. Halfen, L.M. Ziurys, Velocity modulation spectroscopy of molecular ions II: ..., Journal of Molecular
Spectroscopy (2006), doi:10.1016/j.jms.2006.08.008
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Table 1
Observed rotational transitions of TiF+ (X3Ur)a
J00 fi J 0
X
F00 fi F 0
mobs
mobs mcalc
13 fi 14
2
2
3
3
4
4
12.5 fi 13.5
13.5 fi 14.5
12.5 fi 13.5
13.5 fi 14.5
12.5 fi 13.5
13.5 fi 14.5
327 710.973
327 707.350
328 985.124
328 980.122
330 355.520
330 349.580
0.071
0.052
0.065
0.040
0.034
0.132
14 fi 15
2
2
3
3
4
4
13.5 fi 14.5
14.5 fi 15.5
13.5 fi 14.5
14.5 fi 15.5
13.5 fi 14.5
14.5 fi 15.5
351 097.342
351 094.144
352 461.745
352 457.378
353 929.055
353 923.693
0.017
0.034
0.013
0.092
0.016
0.078
15 fi 16
2
2
3
3
4
4
14.5 fi 15.5
15.5 fi 16.5
14.5 fi 15.5
15.5 fi 16.5
14.5 fi 15.5
15.5 fi 16.5
374 479.280
374 476.522
375 933.836
375 929.925
377 497.933
377 493.444
0.059
0.026
0.012
0.006
0.062
0.088
16 fi 17
2
2
3
3
4
4
15.5 fi 16.5
16.5 fi 17.5
15.5 fi 16.5
16.5 fi 17.5
15.5 fi 16.5
16.5 fi 17.5
397 856.656
397 854.197
399 401.094
399 397.672
401 061.918
401 057.900
0.023
0.025
0.055
0.020
0.104
0.004
17 fi 18
2
2
3
3
4
4
16.5 fi 17.5
17.5 fi 18.5
16.5 fi 17.5
17.5 fi 18.5
16.5 fi 17.5
17.5 fi 18.5
421 229.119
421 226.873
422 863.294
422 860.398
—
—
0.041
0.020
0.100
0.087
18 fi 19
2
2
3
3
4
4
17.5 fi 18.5
18.5 fi 19.5
17.5 fi 18.5
18.5 fi 19.5
17.5 fi 18.5
18.5 fi 19.5
444 596.310
444 594.082
446 320.255
446 317.491
448 173.919
448 171.075b
0.074
0.167
0.007
0.008
0.076
19 fi 20
2
2
3
3
4
4
18.5 fi 19.5
19.5 fi 20.5
18.5 fi 19.5
19.5 fi 20.5
18.5 fi 19.5
19.5 fi 20.5
467 957.983
467 955.931
469 771.369
469 768.889
471 721.303
471 718.337
0.127
0.141
0.036
0.021
0.003
0.035
20 fi 21
2
2
3
3
4
4
19.5 fi 20.5
20.5 fi 21.5
19.5 fi 20.5
20.5 fi 21.5
19.5 fi 20.5
20.5 fi 21.5
491 313.740
491 312.059
493 216.631
493 214.250
495 262.484
495 259.583
0.098
0.026
0.074
0.045
0.090
0.085
21 fi 22
2
2
3
3
4
4
20.5 fi 21.5
21.5 fi 22.5
20.5 fi 21.5
21.5 fi 22.5
20.5 fi 21.5
21.5 fi 22.5
514 663.298
514 661.867
516 655.551
516 653.157
518 797.009
518 794.392
0.000
0.027
0.151
0.182
0.047
0.078
22 fi 23
2
2
3
3
4
4
21.5 fi 22.5
22.5 fi 23.5
21.5 fi 22.5
22.5 fi 23.5
21.5 fi 22.5
22.5 fi 23.5
538 006.435
538 005.219
540 087.839
540 085.575
542 324.875
542 322.270
0.093
0.017
0.209
0.169
0.187
0.131
a
b
In MHz.
Blended line, not included in fit.
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This pattern can be seen in Fig. 1, which shows the
J = 16 fi 17 transition near 399 GHz, measured using both
source and velocity modulation (upper and lower spectra).
The spin components are separated by 1.6 GHz due to
the fine structure interactions; hence, there are two frequency breaks in the spectra. Each feature appears as a
doublet, labeled by F, separated by 2–4 MHz as a result
of interactions of the 19F spin of I = 1/2. The ratio of relative intensities of the spin–orbit components is
X = 2:3:4 @ 4:2:1, indicating a discharge temperature near
415 K. No vibrationally excited lines were found in these
data. This result is expected because the vibrational separation is xe = 781 cm1 = 1125 K [7]—much higher than the
apparent discharge temperature.
TiF+ (X3Φr): J = 16
Ω=2
F = 16.5
17
Source Modulation
17.5
F = 15.5
16.5
Ω=3
F = 16.5
Ω=4
17.5
F = 15.5
16.5
F = 16.5
17.5
F = 15.5
16.5
Velocity Modulation
Ω=2
Ω=3
Ω=4
397.85
397.87 399.39
399.41
401.05
401.07
61
Detection of these features by velocity modulation indicates that they arise from ions. The profiles look somewhat
different in the source vs. velocity modulated spectra. This
difference arises because 2f detection is employed in source
modulation mode, producing second-derivative signals; for
the VM data, 1f detection is used, generating first-derivative spectra.
The spectrum of TiF+ was analyzed with an effective
Hund’s case (a) Hamiltonian that included rotational,
spin–orbit, spin–spin, and magnetic hyperfine interactions:
^ eff ¼ H
^ rot þ H
^ so þ H
^ ss þ H
^ mhf :
H
ð2Þ
The ordering of the fine structure components was based
on the data of Focsa et al. [7] and the respective signal
intensities. The F quantum number assignments of the
hyperfine doublets followed that for TiF [13]. Reversing
the doublet assignment, another possibility, made the fit
worse. All constants (rotational, fine structure, and hyperfine) were initially allowed to float in the fit. The analysis
produced a reasonable result, except the 3r uncertainties
for k and b were comparable to the values themselves.
Thus, k was held fixed to the fitted value (17 243 MHz).
The value found for b was small (30 MHz), so in this case
b was fixed to a value such that bF was equal to zero (Also
justified to a first approximation because no unpaired sr
electrons are present in this radical.). The constants established from this analysis are displayed in Table 2. The rms
of the fit was 85 kHz.
Also listed in Table 2 are the constants from the optical
work of Focsa et al. [7]. The B and D values in this work
are averages of the effective constants for each spin component, while A was calculated using the Mulliken formula:
Beff,X = B(1 + 2BR/AK). The optical values are in excellent
agreement with the millimeter-wave data. The effective constants for each spin component for each study are also listed in Table 2; again, the values are in good agreement
between the two studies, with the millimeter-wave constants slightly larger in magnitude by a few MHz. The effective hyperfine parameters hX for each spin component were
also determined, as shown in Table 2. From these values,
effective a and (b + c) constants were calculated—in reasonable agreement with the values established in the global
fit.
Frequency (GHz)
Fig. 1. Spectra of the J = 16 fi 17 transition of TiF+ (X3Ur) near
399 GHz taken in source modulation (upper panel: 2f detection) and in
velocity modulation (lower panel: 1f detection) modes. There are two
frequency breaks in the spectra such that all three spin–orbit components
can be displayed and plotted on the scale intensity scale. The spin–orbit
splitting is quite regular, with the fine structure components separated
uniformly in frequency. Each spin component is split into two lines due to
fluorine hyperfine interactions, labeled by quantum number F. The signalto-noise ratio is about a factor of four lower in the VM data, but is still
sufficient to establish these signals as arising from ions. The top spectrum
was created from two, four, and twelve scan averages, respectively. Each
scan was 110 MHz wide and acquired in 70 s, and then cropped to display
a 30 MHz frequency range. The lower spectrum was created from multiple
scan averages; each scan was 55 MHz wide and acquired in 35 s, and then
also cropped to be 30 MHz wide.
4. Discussion
The bond length of TiF+ was determined from the rotational constant to be r0 = 1.7775 Å, as listed in Table 2. In
comparison, the bond length of neutral TiF (X4Ur) measured in the mm-wave study is r0 = 1.8342 Å [13], an
increase of 0.057 Å. The electron configuration of TiF is
thought to be (core)8r23p49r11d14p1, while that of TiF+
is (core)8r23p41d14p1 [7,13]. The electron lost in TiF+
comes from the 9r orbital, which is considered to be primarily nonbonding in character, formed from the 4s atomic orbital [6,13]. Loss of a nonbonding electron should not
affect the bond distance to first order. The change in bond
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Table 2
Spectroscopic constants for TiF+ (X3Ur: v = 0)a
Parameter
MMW
Opticalb
B
D
A
AD
k
kD
a
b
bF
b+c
rms
r0 (Å)
Beff (X = 2)
Deff (X = 2)
h (X = 2)
Beff (X = 3)
Deff (X = 3)
h (X = 3)
Beff (X = 4)
Deff (X = 4)
h (X = 4)
aeff
(b + c)eff
11 755.5010(36)
0.0125765(45)
1 905 100(26 800)
0.59(23)
17243c
0.081(17)
111.0(2.6)
22c
0c
43(10)
0.085
1.7775
11 708.6709(51)
0.0121851(64)
362(15)
11 754.2936(69)
0.0125515(86)
333(14)
11 803.3878(76)
0.0129931(94)
297(12)
110.9(4.5)d
33(27)d
11 753.0(4.3)
0.01172(54)
1 894 700(72 000)
a
b
c
d
11 704.9(1.8)
0.01094(45)
11 751.9(1.8)
0.01157(63)
11 802.1(2.0)
0.01268(63)
In MHz; errors are 3r in the last quoted decimal places.
From Focsa et al. [7]. Values originally quoted in cm1.
Held fixed (see text).
Effective parameters derived from hX (see text).
length likely arises from the ionic nature of these two species, which can be characterized as Ti+F and Ti2+F
[7,13]. The higher charge on the titanium atom in the ion
increases the electrostatic attraction between the two
atoms, hence shortening the bond distance. This result is
also seen in TiCl+ relative to TiCl, which have bond lengths
of 2.1888 Å and 2.2675 Å, respectively, as mentioned, or a
difference of 0.079 Å [4].
The spin–orbit coupling constant A for TiF+ can be estimated if ionic bonding dominates. In this case, the 3d orbitals are not changed significantly on bond formation. The
A constant can then be calculated using the following relationship [14]:
A½X 3 Uð3d 2 Þ; pd ffi f3d ð3d 2 Þ=2:
2+
2
2
ð3Þ
1
For Ti (3d ), f3d (3d ) = 121 cm
[13], and therefore
A(X3U) @ 60.5 cm1—in good agreement with the millimeter-wave value of A(X3U) = 63.5(9) cm1. It should also be
noted that the spin–orbit splitting in TiF+ is very regular,
in contrast to TiCl+ [4]. Hence, the perturbing 3Dr state
of the chloride must be considerably higher in energy in
the fluoride, as predicted [8].
The proposed valence electron configuration of TiF+
suggest that the two unpaired electrons are in d and p orbitals. This configuration is consistent with the derived
^ and cð^I z S^z Þ
hyperfine constants, where parameters að^I LÞ
are non-negligible. This configuration also suggests that
the unpaired electrons are primarily on the titanium nucleus; the hf parameters, which are generated by interactions
with the fluorine spin, should be small, as found. Further
insight can be obtained by examining the nuclearPspin–orbital constant a, which is directly proportional to i h1=r3i i,
where ri is the distance between the interacting electron and
nucleus [15]. The value of Æ1/r3æ for the fluorine atom is
5.93 · 1031 m3 or 8.77 in units of a3
[16], which corre0
sponds to a distance between the 2p electron and the nucleus of r 0.5 a0. Using a fluorine magnetic moment of
lI = 2.628868 [17], Æ1/r3æ for TiF+ is 1.50(3) · 1030 m3 or
0.221(5) a3
0 , yielding an average electron–nucleus distance
of 1.65 a0. Therefore, the electrons on TiF+ are farther
away from the fluorine nucleus than in the atomic F case,
consistent with the notion that the d and p orbitals are centered on the titanium nucleus (This must be the case for the
d orbital.). In contrast, the value of a for TiF is
67.4(1.1) MHz, resulting in Æ1/r3æ = 0.134(3) a3
0 , or
r 1.95 a0 [13]. Hence, the electrons on TiF+ are closer
to the fluorine nucleus than those of TiF. This result agrees
with the decrease in bond length from TiF to TiF+.
The assumption that the 4p orbital is mainly Ti(3d) in
character can be tested by comparing the hyperfine parameters a and c. These constants can be related to the atomic
fluorine hyperfine term P(F) = 4400 MHz [16], which arises
exclusively from the 2pp orbital on fluorine, where
P ¼ gs lB gN lN hr13 i. The parameters a and c can be expressed
as [15],
1
1
1
a ¼ 2lB gN lN 2 3
þ 3
ð4Þ
3
r 1d
r 4p
and
3
c ¼ gs lB gN lN
2
1 3 cos2 h 1
3 cos2 h 1
þ
:
2
r3
r3
1d
4p
ð5Þ
Here, lB is the Bohr magneton, lN is the nuclear magneton, and gN is the nuclear g-factor for fluorine (gN = lI/
I). Because the nuclear spin is on the F atom, the major
contributor to a and c must be the orbitals created from
atomic fluorine orbitals. Hence, the 1d orbital can be
neglected in both cases because this orbital can only arise
from titanium. Furthermore, the fluorine contribution to
the 4p molecular orbital is the 2pp atomic orbital. Consequently, the expressions for a and c can be simplified to:
"
#
cF4p
1
a¼
2lB gN lN 3
r 2ppðFÞ
3
ffi
cF4p
P ðFÞ;
3
ð6Þ
and
"
#
3cF4p
3 cos2 h 1
c¼
gs lB gN lN
r3
4
2ppðFÞ
ffi
3cF4p
P ðFÞ;
10
ð7Þ
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Spectroscopy (2006), doi:10.1016/j.jms.2006.08.008
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where cF4p indicates the amount of 2p(F) character in the 4p
orbital, and the angular factor Æ3 cos2h 1æ2pp is 2/5 [16].
Using the values of a and c established here, cF4p ¼ 0:049
and 0.076 for TiF+, for an average of 0.062(8)% or
6.2(8)%. Therefore, to a good approximation, the assumption that the 4p orbital is chiefly 3d in character is probably
valid. This analysis can be performed for TiF as well, and
yields an average value of cF4p ¼ 3:3ð1:1Þ%. If these assumptions are valid, this result suggests that TiF+ is slightly
more covalent than TiF—perhaps due to more orbital
overlap from the shorter bond length.
5. Conclusion
The pure rotational spectrum of TiF+ in its 3Ur ground
state has been measured for the first time using millimeterwave techniques aided by velocity modulation. TiF+ was
created by the reaction of gas-phase TiCl4 and F2 in the
AC discharge. This study suggests that other titanium-containing molecular ions could be generated using TiCl4 as
the organometallic precursor. The bond length of TiF+ is
shorter than that of TiF, probably due to increased electrostatic attraction between the two atoms. This effect is
apparent in the fluorine hyperfine constants. The hyperfine
constant a for TiF+ is larger than in TiF, indicating that
the electrons are closer to the fluorine nucleus in the ion
vs. the neutral. The contribution of the fluorine 2p orbital
to the 4p orbital appears slightly higher in TiF+ as opposed
to TiF, as well, also a likely result of the decreased bond
length. This investigation, along with that of TiCl+ [4], suggests that the properties of metal-containing molecular ions
differ somewhat from those of the neutral species.
63
Acknowledgments
This research is supported by NSF Grant CHE-0411551
and NASA Grant NNX06AB64G. D.T.H. is supported by
an NSF Astronomy and Astrophysics Postdoctoral Fellowship under Award AST-0602282.
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Please cite this article as: D.T. Halfen, L.M. Ziurys, Velocity modulation spectroscopy of molecular ions II: ..., Journal of Molecular
Spectroscopy (2006), doi:10.1016/j.jms.2006.08.008