Improved molecular constants for the X1∑+ and a1∑+ states of NaH

Article abstract of DOI:10.1007/s100530050546

Improved molecular constants for the X¹∑⁺ and A¹∑⁺ states of the NaH molecule are presented. NaH molecules are produced by reactive scattering of H and Na₂ in a crossed beam experiment. High vibrational levels (6 ≤ υ″ ≤ 9) of the NaH molecules are predominantly populated. Their excitation spectrum in the range 630 nm ≤ λ ≤ 670 nm has been measured using a new variant of Doppler spectroscopy. The transition frequencies involving the vibrational levels 2 ≤ υ′ ≤ 8 in the A¹∑⁺ and 6 ≤ υ″ ≤ 9 in the X¹∑⁺ state have been determined with an accuracy of better than 0.01 cm^-1. Using also previously published data a new set of molecular constants for the X²∑⁺ and A¹∑⁺ state is derived. In particular, the vibrational dependence of the rotational constants B, D and H as well as some of υ″-υ′ band origins for 0 ≤ υ″ ≤ 9 and 0 ≤ υ′ ≤ 25 is determined. The transition frequencies measured here or published previously are reproduced by these new coefficients with an accuracy of ≤ 0.1 cm^-1 [rms value] with a maximum deviation of 0.4 cm^-1. New RKR potential energy curves have been calculated up to the turning points of the levels υ″ = 9 in the X^-1∑¹+ state and υ′ = 25 in the A¹∑⁺ state.

Full text of DOI:10.1007/s100530050546

Eur. Phys. J. D 10, 247–257 (2000)
THE EUROPEAN
PHYSICAL JOURNAL D
EDP Sciences
Societ a` Italiana di Fisica
Springer-Verlag 2000
ꢀ
c
Improved molecular constants
1
+
1
+
for the X Σ and A Σ states of NaH
a
F.P. Pesl, S. Lutz, and K. Bergmann
Fachbereich Physik, Universit ¨a t Kaiserslautern, 67663 Kaiserslautern, Germany
Received 21 August 1999
1
+
1
+
Abstract. Improved molecular constants for the X Σ and A Σ states of the NaH molecule are pre-
sented. NaH molecules are produced by reactive scattering of H and Na in a crossed beam experiment.
2
0
0
High vibrational levels (6 ≤ v ≤ 9) of the NaH molecules are predominantly populated. Their excitation
spectrum in the range 630 nm ≤ λ ≤ 670 nm has been measured using a new variant of Doppler spec-
0
1
+
00
troscopy. The transition frequencies involving the vibrational levels 2 ≤ v ≤ 8 in the A Σ and 6 ≤ v ≤ 9
1
+
−1
in the X Σ state have been determined with an accuracy of better than 0.01 cm . Using also previously
1
+
1
+
published data a new set of molecular constants for the X Σ and A Σ state is derived. In particular,
00
0
the vibrational dependence of the rotational constants B, D and H as well as some of v –v band origins
00
0
for 0 ≤ v ≤ 9 and 0 ≤ v ≤ 25 is determined. The transition frequencies measured here or published
−
1
previously are reproduced by these new coefficients with an accuracy of ≤ 0.1 cm
[rms value] with a
maximum deviation of 0.4 cm . New RKR potential energy curves have been calculated up to the turning
−
1
00
1
+
0
1
+
points of the levels v = 9 in the X Σ state and v = 25 in the A Σ state.
PACS. 33.15.Mt Rotation, vibration, and vibration-rotation constants – 33.20.Kf Visible spectra
1
Introduction
state was reported by Olsson [4], who analysed the isotopic
shift between levels of NaH and NaD. The data obtained
by Hori and Olsson from absorption spectroscopy in the
spectral region of 358 nm ≤ λ ≤ 445 nm cover vibrational
Spectroscopic investigations of the alkali hydrides in the
gas phase have played an important role in the develop-
ment of molecular spectroscopy and for the analysis of
molecular structure. In fact, the NaH molecule was re-
ferred to by Dunham [1] himself as a test for his approach
to the analysis of the spectra of diatomic molecules. One
reason for the particular interest in the alkali hydrides
0
0
0
levels in the range 0 ≤ v ≤ 3 and 1 ≤ v ≤ 20. Pankhurst
[
5] extended the data by adding results from dispersed
fluorescence measurements in the range of 460 nm ≤ λ ≤
00
6
1
45 nm including the vibrational levels 0 ≤ v ≤ 8 and
0
0
≤ v ≤ 7. Experimental data involving v = 0 in the
1
+
spectral region of 600 nm ≤ λ ≤ 700 nm was first re-
lies in the avoided crossing between the X Σ and the
1
+
ported by Orth et al. [6].
A Σ state due to a strong interaction between the ionic
and covalent branches of the potential curves. The A-
state is flat-bottomed and highly anharmonic. This re-
sults in a negative anharmonicity constant ωexe as well as
1
+
Accurate information about the X Σ state of NaH
resulted from the rovibrational spectra obtained by
Sastry et al. [7] and Maki et al. [8] based on conven-
tional spectroscopic methods like millimeter range and
Fourier spectroscopy, respectively. Precise data for the
electronic ground state acquired by infrared and far in-
frared laser spectroscopy was reported for the first time in
the late eighties in [9,10]. The publications on the spec-
troscopy of NaH in the gas phase before 1990 were re-
viewed by Stwalley et al. [11]. These authors recommended
in an anomalous variation of the vibrational level spacings
∆
quantum number v . For low lying vibrational levels the
rotational constants Bv0 and Dv0 initially increase with v
before they decrease in the usual manner. For high vibra-
tional levels also the shape of the X Σ state potential
curve is affected by the avoided crossing. Therefore, NaH
and other alkali hydrides have been studied by experimen-
talists and theorists for many decades [2–23].
0
G(v ) and the rotational constants with the vibrational
0
0
1
+
1
+
the molecular constants of Maki et al. for the X Σ state
1
+
and those of Orth et al. for the A Σ state.
Recently, Lochbrunner et al. [12] have used the De-
generate Four Wave Mixing method to observe spectra of
NaH in the near UV and in the blue region. These authors
noticed that the Dunham coefficients from Stwalley et al.
One example of important classical spectroscopy on
NaH is the work of Hori [2,3], who observed and anal-
1
+
1
+
ysed X Σ –A Σ bands in absorption and emission. An
1
+
adjustment of the vibrational assignment in the A Σ
[
11] did not fit their spectra. Discrepancies between cal-
a
−1
e-mail: bergmann@rhrk.uni-kl.de
culated and measured line positions up to 60 cm were

248
The European Physical Journal D
observed especially for high rotational levels. Such large
discrepancies where observed only for vibrational levels,
which were not accessible in the previous experiments.
They found that the recommended Dunham coefficient
Y12 correctly describes the vibrational dependence of the
0
constant Dv0 up to v = 7. Therefore, they determined
1
+
new molecular constants for the A Σ state based on the
0
vibrational levels 0 ≤ v ≤ 25 by combining their mea-
surements with data from the literature. New Dunham
1
+
coefficients and a RKR potential curve of the A Σ state
were derived by Al-Tuwirqi et al. [13], as well. These au-
thors combined the results of Rafi et al. [14] for the lev-
0
0
els v = 14–25 with the data of Orth et al. for v = 0,
0
0
Pankhurst for v = 1–7 and Olsson for v = 8–13. They
covered the same region of vibrational levels in the excited
state as Lochbrunner et al. In spite of these improvements
1
+
Fig. 1. Essential components of the experimental set-up. LIF:
high speed detector for laser induced NaH fluorescence; DL:
autoscan 699 dye-laser; OD: optical diode; PM: power meter;
I : iodine reference cell; BS: beam splitter (80:20); SM: single
2
mode fibre; FP fibre polarizer; HR: high reflection mirror.
of the molecular constants for the A Σ state accurate
prediction of transition frequencies is restricted to transi-
tions from the lowest vibrational levels of the electronic
ground state. Transitions from higher vibrational levels,
−1
such as those measured by Orth, deviate up to 40 cm
from the calculated ones. These facts suggested the need
for more accurate experimental data including higher vi-
brational levels of the electronic ground state.
The H-atoms are produced in a 2.45 GHz microwave
discharge. The water cooled microwave cavity is placed in-
side the vacuum apparatus. The pressure in the source is
typically 3 mbar and the discharge is normally driven with
a microwave power of 40 W. The gas expands through a
0.4 mm orifice into the vacuum and is collimated by a
skimmer of 1 mm diameter at a distance of 5 mm from
the nozzle, resulting in a collimation of the H-beam of
The present spectroscopic work is a first step in a de-
00
00
tailed study of the Na2(v , J ) + H → NaH(v, J) + H
reaction dynamics [24]. The exothermicity of the reaction
(
1.22 eV) is sufficient to populate vibrational levels up to
00
1 +
v = 10 in the X Σ state of NaH. We present a study of
the NaH spectroscopy in the gas phase with a resolution
−1
of better than 0.01 cm for transitions from vibrational
◦
00
8.0 (full width). The degree of dissociation is measured
levels 6 ≤ v ≤ 9 in the electronic ground state.
with a quadrupole mass spectrometer (QMS) by compar-
ing the signal of mass 2 with the discharge turned on and
off. Liquid nitrogen cooling of the quartz walls of the dis-
charge tube is essential in order to dissociate more than
2
Experiment
2
5% of the hydrogen molecules. The velocity distribution
which is characterized by a flow velocity of uH = 1891 m/s
and a width of the distribution αH = 1730 m/s (FWHM)
is determined by time-of flight analysis, using a chopper
wheel.
The essential components of the experimental set-up are
shown in (Fig. 1). The NaH molecules are produced in the
crossing region of a Na2 molecular beam and a H-atom
0
0
00
beam by the reaction Na2(v = 0, J ) + H → NaH(v, J)
+
Na. The supersonic sodium beam source with an ori-
The products are analysed in the reaction zone by
fice of 0.5 mm diameter is operated at a temperature of laser-induced fluorescence using a single mode dye laser.
◦
about 650 C which corresponds to a vapor pressure of A single mode fiber (5 µm core diameter) delivers
about 60 mbar. About 12% of the particles in the beam about 160 mW of power from DCM laser (Coherent 699
are dimers [25] with 99% of them found in the vibrational Autoscan system, typical linewidth 1 MHz) to the inter-
0
0
level v = 0 of the electronic ground state and about 1% action region inside the vacuum chamber. The detector
00
◦
in v = 1. The beam is collimated to 0.9 by two skimmers consists of a lens system, which collects 9% of the fluo-
with a diameter of 2 mm and 1 mm at a distance of 36 mm rescence light and images it onto a photomultiplier which
and 69 mm, respectively, from the nozzle. The vibrational- is integrated into the detector. Spatial filtering through
rotational population distribution as well as the velocity an aperture of 3 mm diameter serves to suppress back-
distribution are monitored by laser induced fluorescence ground radiation. After collimation the laser beam crosses
0
0
at a distance of 700 mm from the nozzle [26]. For v = 0 the reaction region in the scattering plane parallel to the
the rotational level population distribution corresponds to mean relative velocity of Na2 and H, which is at an an-
00
◦
a rotational temperature of 17 K for J ≤ 10 and 31 K for gle of β = (31 ꢀ 8) relative to the H-atom beam axis.
00
00
J
> 10. The maximum population is found in J = 7. The laser is subsequently reflected at normal incidence
00
00
The state (v = 0, J = 7) carries about 10% of the total by the mirror HR outside the vacuum. The mirror is ad-
Na2 population. The flow velocity is uNa = 1432 m/s and justed to maximize the fraction of light coupled back into
2
the width of the velocity distribution is αNa = 257 m/s the fiber and observed at the opposite end of the fiber.
2
(
FWHM).
An optical diode avoids optical feedback to the dye laser.

1
F.P. Pesl et al.: Improved molecular constants for the X Σ and A Σ states of NaH
+
1
+
249
Fig. 2. Part of the NaH-spectrum. The
NaH-lines are distinguished from the
Na
2
-lines by their linewidth and their
-lines
lineprofile. The width of the Na
2
is substantially smaller than that of
the NaH-lines. Chopping the H-beam
in combination with lock-in detection
also allows unambiguous identification
of lines related to NaH spectra.
This alignment procedure assures that the incoming and
the reflected beam overlap in the region of the reaction
center. The laser wavelength was calibrated with an accu-
−1
racy of 0.001 cm using an iodine cell [27].
The excitation spectrum of NaH is measured by laser
induced fluorescence (LIF) in the range 620 nm ≤ λ ≤
6
70 nm. Since the density of Na2 is about three orders of
magnitude larger than the density of NaH and the absorp-
tion spectra of these molecules overlap, the emission from
Na2 needs to be suppressed by filters. Almost all of the
fluorescence light of Na2 is shifted to the red with respect
to the excitation wavelength, while a large fraction of the
NaH fluorescence is shifted to the blue. Two subsequent
shortpass filters with substantial suppression of radiation
with λ > 600 nm were combined to strongly discriminate
against the Na2 fluorescence. Nevertheless, the Na2 lines
are still observable. Unambiguous identification of spec-
tral features as NaH-lines is based on the line shapes (see
Fig. 2) or by chopping the H-atom beam and detecting
only the modulated components of the spectrum. Bands
Fig. 3. Typical line shape recorded using the back reflection
method.
1
+
1
+
of the electronic transition X Σ –A Σ , including the
The center frequency ω of a spectral line is Doppler
0
0
0
0
levels 6 ≤ v ≤ 9 and 2 ≤ v ≤ 8, are observed with shifted to:
−1
an accuracy of better than 0.01 cm , using the iodine
spectra for calibration. Use of the new NaH ab initio po-
tentials of Dern et al. [28], which reproduced the data
of Orth et al., Maki et al. and Lochbrunner et al. with
a remarkable accuracy of 1 cm or better, allowed the
unambiguous identification of the observed transitions.
NaH
ω(k) = ω0 + k · Vs + k · vcm
(1)
where ω0 is the resonance frequency in the molecular
frame, k is the wave vector of the radiation, Vs is the
−
1
NaH
velocity of the center of mass and vcm is the velocity of
the product molecule in the center-of-mass system. Upon
reflection of the laser beam (k ⇒ −k) a mirror image
of the spectrum, induced by the incoming laser, is pro-
duced. A spectral component ω(k) is mapped into one at
3
Analysis of the line profiles
+
−
ω(−k). The two spectral profiles, I (ω) and I (ω), are
The NaH molecules are formed in a reactive process. The symmetric with respect to the resonance frequency ω0.
profiles of their absorption lines are therefore determined The centers of gravity of the two spectral lines are sepa-
by the velocity and angular distribution of the products rated by ∆ω = 2|k · Vs| (see Fig. 3), with typical values
−1
−1
with respect to the direction of propagation of the laser of 0.05 cm ≤ ∆ω ≤ 0.06 cm . The width of the line
−1
−1
beam. The analysis of these profiles to obtain differential profiles ranges from 0.03 cm
to 0.08 cm depending
cross-sections for the reactive events will be the subject of on the kinetic energy of the product molecule, which is
another publication [24]. Here, we need only to consider related to its internal energy. Details of the line profiles
the consequences of the Doppler shift and broadening for are determined by the reaction dynamics, in particular by
the determination of the transition frequencies.
the angular distribution of the products.

250
The European Physical Journal D
0
Now we will discuss the determination of the resonance rotational quantum number J are given by [30]:
ꢁ
frequency ω0 from the spectral lines. The profiles I (ω)
are scaled for equal area before the center of gravity of
the two profiles, expected at ω0, is determined. From the
observed spectral profiles of a given transition we calculate
1
2
00
0
0
∆F (J ) = 4B 00 (J +
)
v
0
2
0
2
0
2
02
− Dv00 [(J + 1) (J + 2) − (J − 1) J ]
R
3
0
3
0
3
0
3
0
+
00
−
Hv [(J + 1) (J + 2) − (J − 1) J ].
I (ω)dω
λ = R
(2)
(
8)
A least squares fit procedure, based on equation (8) and
−
I (ω)dω
using integration boundaries which do not overlap and
scale the intensities according to
00
0
using the measured values ∆F (J ) yields the rotational
constants Bv00 , Dv00 and Hv00 . A total of about 2800 tran-
(3) sition frequencies are known. Here, we include only those
+
+
−
−
s (ω) = I (ω) and s (ω) = λI (ω).
1
000 transitions (including 283 from our experiment) for
This is done to correct for any difference in the inten-
which the accuracy is known [5,6,8,12,14]. An iterative
sity of the incoming and reflected laser beam and thus
R
procedure is used to minimize the correlation between Bv00
ꢁ
ꢁ
to assure that the area A
=
s (ω)dω are identical. We
and Dv00 or Hv00
.
+
−
consider s (ω), s (ω) as different segments of s(ω), which
cover both components of the profile. We then choose a
frequency ωs close to the estimated frequency ω0 and min-
imize the quantity
In a first step, the rotational constants Bv00 , Dv00 and
Hv00 are determined, using equation (8). The Dunham co-
efficients Yk1 are then obtained by a fit procedure using:
ꢀ
ꢁ
k
X
1
2
00
N
Bv00
=
Yk1 v +
.
(9)
X
2
S(ωs) =
|s(ωs + i∆ω) − s(ωs − i∆ω)|
(4)
k=0
i=1
The next step starts with the insertion of the constants
B 00 using the Yk1 and equation (9) followed by another
cal
v
by variation of ωs, where ∆ω is a small interval, typically
−4
−1
1
.7×10 cm wide. The numbers N must be sufficiently fit procedure to determine a modified value of Dv00
.
large, so that the interval [ωs − N∆ω, ωs + N∆ω] covers
Next, the Dunham coefficients Yk2 are determined
both components of the line profile. The desired frequency from a fit according to
ω0 is found, if
ꢀ
ꢁ
k
X
1
2
00
Dv00 = −
Yk2 v +
.
(10)
S(ω0) = min S(ωs).
(5)
ωs
k=0
cal
v
The numerical error of the procedure is estimated as fol- Finally, the constants D (using the Yk2 from Eq. (10))
00
cal
lows. The error of the sum S(ω0) is calculated using the
law of propagation of errors and uncertainties [29]. After-
wards, all frequencies ωs with a sum S(ωs) in the interval
and Bv00 from equation (9) are inserted in equation (8)
and the fit procedure is repeated to yield improved con-
stants Hv , based on which the Dunham coefficients Yk3
are determined by a fit according to
00
Ierror = [S(ω0) − ∆S(ω0), S(ω0) + ∆S(ω0)]
are determined. The ωs with the largest deviation from ω0
yields the error of the resonance frequency:
(6)
ꢀ
ꢁ
k
X
1
2
00
Hv
00
=
Yk3 v +
.
(11)
k=0
∆
ω0 = max |ω0 − ωs| ∀ ωs with S(ωs) ∈ Ierror. (7) The same procedure is applied for the determination of the
ωs
Dunham coefficients for the A-state. Here, the differences
0
00
of the termvalues ∆F (J ) of R- and P-lines of a vibra-
The algorithm is insensitive to the choice of the starting
value ωs as well as to the scaling parameter λ for typical
experimental conditions. Typical values of ∆ω0 are in the
00
tional band with the same rotational quantum number J
30], obtained from the experiment, are used.
[
00
0
−1
Finally, the band origins ν¯ 0(v , v ) are obtained from
the differences of the transition frequencies between levels
range of 0.005 cm . In measurements with low counting
rates, resulting in a poor signal to noise ratio, the error
0
0
00
00
00
−1
(
v , J ) and (v , J ) and extrapolation to J = 0 [30]. The
rarely exceeds values of 0.01 cm . Therefore, the line
Dunham coefficients Yk0 for the electronic ground state are
positions are determined with an accuracy of better than
−1
obtained by a fit procedure based on
0
.01 cm
.
"
#
ꢀ
ꢁ
ꢀ
ꢁ
k
k
X
3
2
1
2
0
0
00
00
∆
G(v ) =
Yk0
v +
−
v +
,
(12)
4
Determination of the molecular constants
k=1
0
0
using the difference of transition frequencies ∆G(v ) =
4
.1 Fit procedure
0
0
0
00
0
ν¯ 0(v + 1, v ) − ν¯ 0(v , v ) for the band origins of the vi-
1
+
1
+
00
00
The A Σ → X Σ LIF-spectra of NaH show only R- brational levels v and v + 1. An analogous procedure is
00
0
and P-lines. The combination differences ∆F (J ) be- used to derive the equivalent Dunham coefficients for the
tween R- and P-lines of a vibrational band with the same electronically excited state.

1
F.P. Pesl et al.: Improved molecular constants for the X Σ and A Σ states of NaH
+
1
+
251
−1
1
+
Table 1. Rotational constants in cm for the X Σ -state.
00
−4
−8
v00 / 10
v
B
v00
D
v00 / 10
H
9
8
7
6
5
4
3
2
1
0
3.66340 ꢀ0.00006 3.112 ꢀ0.002 1.462 ꢀ0.013
3.79275 ꢀ0.00006 3.112 ꢀ0.001 1.554 ꢀ0.006
3.9214 ꢀ0.0006
4.0486 ꢀ0.0004
4.1777 ꢀ0.0016
4.3050 ꢀ0.0027
4.4365 ꢀ0.0009
4.5670 ꢀ0.0006
3.135 ꢀ0.003 1.592 ꢀ0.010
3.151 ꢀ0.002 1.668 ꢀ0.014
3.189 ꢀ0.031 1.700 ꢀ0.176
3.222 ꢀ0.052 1.790 ꢀ0.306
3.271 ꢀ0.018 1.850 ꢀ0.102
3.314 ꢀ0.016 1.922 ꢀ0.190
4.70002 ꢀ0.00004 3.356 ꢀ0.001 1.950 ꢀ0.009
4.83502 ꢀ0.00002 3.404 ꢀ0.007 1.896 ꢀ0.007
4
.2 Rotational constants
The data of Lochbrunner et al. [12] show the least devia-
tion between experimental data and calculated transition
frequencies based on Dunham coefficients reported in the
literature. Therefore, we compare our results to the data
reported in reference [12] only.
1
+
4
.2.1 Rotational constants of the X Σ state
Our experimental data cover vibrational levels in the
0
0
range 6 ≤ v ≤ 9. The data base is extended to low vibra-
tional levels by including the experimental results of Maki
0
0
et al. [8] which cover the range (0 ≤ v ≤ 2). Further-
more, Pankhurst [5] reports on combination differences
00
for vibrational levels in the range 3 ≤ v ≤ 5. The latter
author do not offer an error analysis. Therefore, his data is
not considered for the fit. However, his results are used to
assess the quality of our new set of Dunham coefficients.
1
+
We estimate the error of the transition frequencies calcu- Fig. 4. Rotational constants of the X Σ -state of the NaH
lated with their molecular constants to be of the order of molecule, with the range of vibrational levels, experimentally
−
1
0
.05 cm . The new set of rotational constants is shown covered by various authors indicated. (1) Maki et al. [8], (2)
in Figure 4 and Table 1. Obviously, the data of Pankhurst Pankhurst [5], (3) this work; full dots: experimental data; solid
line: results from the analysis in this work; dashed line: results
by Lochbrunner et al. [12].
are helpful to verify the quality of the new results.
1
+
The coefficients reported by Maki et al. for the X Σ
state, recommended by Stwalley et al. and used by
Lochbrunner et al. are based on data restricted to the
00
range v ≤ 3. Therefore, it is not surprising, that these co-
efficients are not adequate for transitions involving higher
vibrational levels. The deviation of their rotational con-
stants Bv00 from our results is substantial for vibrational
0
[
14] (16 ≤ v ≤ 25). The new rotational constants as well
as the data of Rafi et al. are indicated in Table 2 and
00
levels v > 4. Similarly, the new coefficients Dv00 and Hv00
Figure 5. The unusual shape of the potential energy curve
1
+
00
of the A Σ -state is apparent from the vibrational level
are consistent with previously reported results for v ≤ 5,
0
0
0
0
dependence of rotational constants Bv and Dv , which
but show substantial improvement for v ≥ 6.
0
0
show a pronounced maximum near v = 6 and v = 3,
respectively.
1
+
4
.2.2 Rotational constants of the A Σ state
The data of Lochbrunner et al. for Bv0 are not dis-
The fit procedure for the determination of the molecular tinguishable from our results (and therefore not shown in
constants for the first electronically excited state is based Fig. 5). However, their results fail to adequately character-
0
0
on our data which cover the range 2 ≤ v ≤ 8 and the ex- ize the variation of Dv0 with v . Our results confirm that
0
perimental data by Orth et al. [6] (v = 0, 1), Lochbrunner Lochbrunner et al. justifiably neglected the constants Hv0
12] (9 ≤ v ≤ 15) and the molecule constants of Rafi et al. for the analysis of their data.
0
[

252
The European Physical Journal D
−1
1
+
Table 2. Rotational constants in cm for the A Σ -state.
0
−4
−8
v0 /10
v
B
v0
D
v0 /10
H
1
1
1
1
1
1
5
4
3
2
1
0
1.78338 ꢀ0.00530 1.5222 ꢀ0.0100 –0.4000 ꢀ0.0850
1.80424 ꢀ0.00265 1.5139 ꢀ0.0288 –0.4000 ꢀ0.0980
1.84068 ꢀ0.00013 1.5253 ꢀ0.0770 –0.3500 ꢀ0.0700
1.85680 ꢀ0.00046 1.6100 ꢀ0.0673 –0.3300 ꢀ0.0900
1.87792 ꢀ0.00204 1.6205 ꢀ0.0345 –0.3260 ꢀ0.0870
1.89332 ꢀ0.00520 1.6509 ꢀ0.0317
1.90560 ꢀ0.00688 1.7070 ꢀ0.0500
1.92374 ꢀ0.00146 1.8257 ꢀ0.0380
1.93847 ꢀ0.00154 1.9200 ꢀ0.0261
1.94166 ꢀ0.00045 2.0340 ꢀ0.0012
1.93987 ꢀ0.00012 2.1308 ꢀ0.0012
1.92901 ꢀ0.00030 2.1982 ꢀ0.0031
1.90981 ꢀ0.00017 2.2696 ꢀ0.0034
1.87321 ꢀ0.00041 2.2598 ꢀ0.0034
1.82186 ꢀ0.00059 2.2014 ꢀ0.0020
1.75080 ꢀ0.00017 1.9720 ꢀ0.0020
0.1512 ꢀ0.1100
0.4083 ꢀ0.2300
0.9480 ꢀ0.2200
1.4610 ꢀ0.1030
1.8405 ꢀ0.0050
2.5148 ꢀ0.0030
3.1539 ꢀ0.0120
3.7330 ꢀ0.0090
4.1390 ꢀ0.0070
4.3285 ꢀ0.0040
4.5135 ꢀ0.0040
9
8
7
6
5
4
3
2
1
0
0
00
Table 3. Band origins (v −v ).
−1
band origin ν
this work
0
/ [cm
others
]
band
9–8
9–7
9–6
9–5
8–6
8–5
8–4
8–3
7–5
7–4
7–3
7–2
6–4
6–2
16176.7 ꢀ0.2
15817.2 ꢀ0.1
15459.88 ꢀ0.05
15105.86 ꢀ0.05
16300.5 ꢀ0.1
15946.4 ꢀ0.1
15596.9 ꢀ0.3
15253.1 ꢀ0.3
16823.2 ꢀ0.2
16473.8 ꢀ0.1
16130.0 ꢀ0.3
15792.8 ꢀ0.4
17386.4 ꢀ0.2
16705.5 ꢀ0.4
-
-
-
-
16300.5 [5]
15946.4 [5]
-
-
16823.4 [5]
16473.7 [5]
16130.1 [5]
-
1
+
Fig. 5. Rotational constants of the A Σ -state of the NaH-
molecule, with the range of vibrational levels, experimentally
covered by various authors indicated. (1) Orth et al. [6], (2)
this work, (3) Lochbrunner et al. [12], (4) Rafi et al. [14]; full
dots: experimental data; solid line: results from the analysis in
this work; dashed line: results by Lochbrunner et al. [12].
17386.6 [5]
16705.6 [5]
00
Table 4. Variation of the vibrational level spacing ∆G(v )
+
00
1
with v for the X Σ -state.
4.3 Vibrational constants
0
0
−1
∆
G(v )/[cm ]
0
0
The determination of the vibrational constants is based on
our data for the band origins (Tab. 3) and data of Maki
et al. [8], Orth et al. [6] and Lochbrunner et al. [12]. Ex-
cellent agreement with the previously published data for
the band origins and vibrational level spacings (Tabs. 3,
v
this work
840.4 ꢀ0.2
876.5 ꢀ0.3
912.8 ꢀ0.3
948.2 ꢀ0.6
984.7 ꢀ0.4
others
-
8
7
6
5
4
3
2
1
0
876.9 ꢀ0.1 [6]
912.8 ꢀ0.1 [6]
948.8 [5]
00
4 and 5 as well as Fig. 6) is found up to level v = 8 and
0
984.4 [5]
v ≈ 25, respectively.
1021.0 ꢀ0.2 1021.0 [8]
1057.8 ꢀ0.2 1057.8 [8]
1095.1 ꢀ0.2 1095.1 [8]
1133.2 ꢀ0.2 1133.2 [8]
4
.4 Dunham coefficients
1
+
The new set of Dunham coefficients of the X Σ -state
1
+
and the A Σ -state is listed in Table 6. The electronic

1
F.P. Pesl et al.: Improved molecular constants for the X Σ and A Σ states of NaH
+
1
+
253
0
Table 5. Variation of the vibrational level spacing ∆G(v )
0
1
+
with v for the A Σ -state.
0
−1
]
∆
G(v )/[cm
0
v
this work
others
1
1
1
1
1
4
348.7 ꢀ0.5
352.0 ꢀ0.7
354.9 ꢀ0.7
357.8 ꢀ0.7
359.3 ꢀ0.6
360.2 ꢀ0.6
360.7 ꢀ0.8
359.0 ꢀ0.3
357.2 ꢀ0.2
354.0 ꢀ0.1
349.7 ꢀ0.4
343.6 ꢀ0.4
337.2 ꢀ0.7
329.9 ꢀ0.2
348.5 ꢀ0.8 [12]
351.8 ꢀ0.7 [12]
354.9 ꢀ0.7 [12]
357.8 ꢀ0.7 [12]
359.3 ꢀ0.6 [12]
360.2 ꢀ0.6 [12]
360.7 ꢀ0.6 [12]
359.2 [5]
3
2
1
0
9
8
7
6
5
4
3
2
1
0
357.5 ꢀ0.7 [12]
353.9 ꢀ0.6 [12]
349.7 ꢀ0.7 [12]
343.6 ꢀ0.7 [12]
337.2 ꢀ0.7 [12]
329.9 [5]
323.76 ꢀ0.2 323.84 [6]
Fig. 7. Deviation of calculated term values from the measured
ones, using the Dunham coefficients from this work and from
Lochbrunner et al. [12].
excitation energy Tel is required for the calculation of the
transition frequencies. According to Stwalley et al. [11]
−
1
this energy difference is Tel = 22 713ꢀ5 cm . With Tel =
−1
−1
2
2 713.0 cm we find a constant deviation of 0.2 cm of
the measured and calculated transition frequencies. Ex-
−1
cellent agreement is found for Tel = 22 713.2 cm
.
The quality of the new Dunham coefficients is demon-
strated by inspection of the deviation of a representative
set of measured term value differences from those calcu-
Fig. 6. The variation of vibrational level spacing ∆G for the lated. Figure 7a shows the deviation of calculated and
1
+
1
+
X Σ - and A Σ -state of NaH. (1) Maki et al. [8], (2) this measured values for the 0–14 band, based on experimental
work, (3) Orth et al. [6], (4) Pankhurst et al. [5], (5) Lochbrun- data from reference [12]. The latter are determined with
−1
ner et al. [12], (6) Rafi et al. [14]. Full dots: experimental data; an accuracy of the line positions of 0.2 cm . Based on
solid line: results from the analysis in this work.
our new coefficients, the rms deviation of measured and
−1
calculated term value is ≤ 0.1 cm for the data shown
−1
in Figure 7a, with a maximum deviation of 0.4 cm . The
experimental data reported by Orth et al. [6], see Fig. 7b),

254
The European Physical Journal D
−1
Table 6. Dunham coefficients in cm for NaH.
coefficient
Lochbrunner et al. [8]
1
X Σ
this work Lochbrunner et al. [12]
1
A Σ
this work
+
+
T
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
el
22713 ꢀ 5
22713.2
0.45
00
0.59
1171.75909
−19.52352
0.12131
0.55
1171.968ꢀ0.012
−19.703 ꢀ 0.010
0.175ꢀ0.002
0.86
317.56
2.701
10
319.96ꢀ0.08
0.7783ꢀ0.083
0.908ꢀ0.031
−147.2 ꢀ 5.5
114ꢀ5.5
20
30
0.02731
−43.29
21.59
3
4
5
7
7
40 × 10
50 × 10
60 × 10
70 × 10
80 × 10
−0.59
−6.58 ꢀ 0.15
−2.235
−5.25
5.3
−50.7 ꢀ 3.0
121.7ꢀ8.5
−1.207 ꢀ 0.97
01
4.90336382
4.90327ꢀ0.00013
1.7121 1.70553ꢀ0.00056
11
−0.13709097 −0.137 ꢀ 0.00036
0.09152
−12.09
6.50
0.0971ꢀ0.0008
−13.7 ꢀ 0.3
8.21ꢀ0.4
3
4
5
7
4
6
8
8
8
9
1
8
1
1
1
1
1
1
1
1
21 × 10
31 × 10
41 × 10
51 × 10
1.111351
1.01ꢀ0.23
−0.007443 ꢀ 0.4
−0.355 ꢀ 0.2
−0.3492
−1.80
1.9
−2.55 ꢀ 0.21
3.06ꢀ0.39
02 × 10
12 × 10
22 × 10
32 × 10
42 × 10
52 × 10
62 × 10
03 × 10
13 × 10
23 × 10
33 × 10
43 × 10
53 × 10
63 × 10
73 × 10
83 × 10
−3.434285
4.9065
−3.425 ꢀ 0.013
4.4ꢀ1.2
−2.30
4.9
−1.808 ꢀ 0.016
−38.6 ꢀ 2.6
1020ꢀ120
−9.025
15.77ꢀ2.4
−2.9 ꢀ 1.4
−100 ꢀ 22
4.68ꢀ1.9
−1.013 ꢀ 0.75
7.78ꢀ11
2
2.1246
1.641
1.83ꢀ0.026
171ꢀ48
4.727ꢀ0.019
−593 ꢀ 50
403ꢀ41
1
1
0
2
1
3
4
6
−2.476
−75 ꢀ 22
0.96ꢀ0.4
−15.6 ꢀ 1.5
252ꢀ27
−4.23 ꢀ 1.88
−2.1 ꢀ 0.3
9.56ꢀ1.4
−2.26 ꢀ 0.4
2.1715ꢀ0.461
are reproduced with a rms deviation for the data shown is the Dunham correction [1]:
−
1
−1
of ≤ 0.1 cm and a maximum deviation of 0.2 cm . Fi-
2
2
nally, the experimental data from our work are reproduced
Y01 Y Y
11
Y Y
Y20
10
11 10
3
−1
Y00
=
−
+
+
·
(13)
with a rms deviation of ≤ 0.01 cm and a maximum dif-
4
12Y01
144Y
01
4
−1
ference of 0.02 cm . The deviation of the measured and
calculated term value difference, based on the coefficients
reported in [12] (see Figs. 7a and the inset in Figs. 7b and
The results based on the new rotational and vibrational
1
+
1
+
constants for the X Σ and A Σ states are given in
7
c), is significantly larger.
Tables 7 and 8, respectively.
5
Potential energy curves
6
Summary
The turning points of the RKR potential [31–33] for the We have demonstrated a novel laser spectroscopic
1
+
1
+
X Σ and A Σ states of NaH were calculated using sub-Doppler method, based on crossed beams reactive
a computer program reported by Tellinghuisen [34]. The scattering and capable of sub-Doppler resolution. Accu-
input parameters are the rotational constants Bv and the rate determination of the transition frequencies is possible,
vibrational part of the termvalues, G(v) + Y00, where Y00 when the Doppler shift |k · Vs| relative to the resonance

1
F.P. Pesl et al.: Improved molecular constants for the X Σ and A Σ states of NaH
+
1
+
255
Table 7. Classical turning points of the RKR-potential of the derived based on our data and on previously published
1
+
X Σ -state.
ones. They predict the known line positions to better
−1
00
00
−1
than 0.1 cm
(rms value) with a maximum deviation
˚
/A
˚
+
R /A
v
G(v )/ cm
R
−
−
1
00
of 0.4 cm for the vibrational levels 0 ≤ v ≤ 9 of the
0
1
2
3
4
5
6
7
8
9
581.63 ꢀ0.1 1.7284 ꢀ0.0001 2.0794 ꢀ0.0001
1714.73 ꢀ0.1 1.6336 ꢀ0.0001 2.2417 ꢀ0.0001
2809.80 ꢀ0.1 1.5741 ꢀ0.0005 2.3708 ꢀ0.0005
3867.59 ꢀ0.1 1.5298 ꢀ0.0011 2.4866 ꢀ0.0011
4888.67 ꢀ0.1 1.4943 ꢀ0.0008 2.5956 ꢀ0.0008
5873.45 ꢀ0.1 1.4646 ꢀ0.0005 2.7009 ꢀ0.0005
6822.20 ꢀ0.1 1.4392 ꢀ0.0005 2.8041 ꢀ0.0005
7735.02 ꢀ0.1 1.4170 ꢀ0.0006 2.9067 ꢀ0.0006
8611.86 ꢀ0.1 1.3975 ꢀ0.0005 3.0097 ꢀ0.0005
9452.50 ꢀ0.1 1.3799 ꢀ0.0004 3.1142 ꢀ0.0004
1
+
0
1
+
X Σ state and 0 ≤ v ≤ 25 of the A Σ state.
We have identified the cause for the sometimes large
deviation between the measured and calculated line po-
sitions (known from previous work): the previously avail-
00
able accuracy of the rotational constants Bv00 (for v > 2)
and Dv (in the range where it exhibit a maximum) was
0
inadequate. This deficiency is overcome with our work.
We thank W. Meyer, P. Dern, S. Lochbrunner, H. Theuer, S.
Scherer and P. Dittmann for valuable discussions. This work
was supported by the Deutsche Forschungsgemeinschaft. Par-
tial support was also received through the European Union
under HCM ERB-CHR-XCT-94-0603.
Table 8. Classical turning points of the RKR-potential of the
1
+
A Σ -state.
0
0
−1
˚
/A
˚
+
R /A
v
G(v )/cm
R
−
0
160.73 ꢀ0.3 2.8573 ꢀ0.0001 3.5141 ꢀ0.0002
Appendix A: Wavenumbers of the measured
line positions
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
484.54 ꢀ0.3
814.65 ꢀ0.3
1151.82 ꢀ0.3
1495.67 ꢀ0.3
1845.23 ꢀ0.3
2199.26 ꢀ0.3
2556.50 ꢀ0.3
2915.76 ꢀ0.3
3275.96 ꢀ0.3
3636.14 ꢀ0.3
3995.44 ꢀ0.3
4353.04 ꢀ0.3
4708.19 ꢀ0.3
5060.16 ꢀ0.3
5408.32 ꢀ0.3
5752 ꢀ1.1
2.608 ꢀ0.001
2.448 ꢀ0.002
2.330 ꢀ0.002
2.237 ꢀ0.002
2.161 ꢀ0.001
2.098 ꢀ0.001
2.044 ꢀ0.002
1.997 ꢀ0.002
1.956 ꢀ0.006
1.919 ꢀ0.009
1.886 ꢀ0.010
1.857 ꢀ0.010
1.829 ꢀ0.010
1.804 ꢀ0.010
1.781 ꢀ0.010
1.760 ꢀ0.009
1.740 ꢀ0.008
1.722 ꢀ0.006
1.706 ꢀ0.006
1.691 ꢀ0.006
1.679 ꢀ0.007
1.668 ꢀ0.006
1.660 ꢀ0.006
1.653 ꢀ0.006
1.646 ꢀ0.006
3.746 ꢀ0.001
3.905 ꢀ0.002
4.038 ꢀ0.002
4.157 ꢀ0.002
4.268 ꢀ0.001
4.375 ꢀ0.001
4.479 ꢀ0.002
4.581 ꢀ0.002
4.682 ꢀ0.006
4.782 ꢀ0.009
4.882 ꢀ0.010
4.983 ꢀ0.010
5.084 ꢀ0.010
5.185 ꢀ0.010
5.288 ꢀ0.010
5.392 ꢀ0.009
5.498 ꢀ0.008
5.606 ꢀ0.006
5.714 ꢀ0.006
5.823 ꢀ0.006
5.933 ꢀ0.007
6.042 ꢀ0.006
6.153 ꢀ0.006
6.265 ꢀ0.006
6.384 ꢀ0.006
0
0
00
0
00 0
band: v =6–v =4
J
band: v =6–v =2
R-branch P-branch
R-branch
−1
P-branch
−1
−1
−1
[cm
]
[cm
]
[cm
]
[cm ]
21
22
23
24
25
26
27
28
29
30
31
15796.213 15650.829
15707.757 15556.905
15615.599 15459.430
15519.840 15358.489
15420.572 15254.163
15317.870 15146.545
15211.864 15035.743
15102.642
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
15863.279
15755.036
15829.490 15643.719
15529.437
14756.834
6091 ꢀ1.8
6425 ꢀ2.6
0
0
00
0
00 0
band: v =7–v =3
6754 ꢀ3.1
J
band: v =7–v =2
R-branch P-branch
7079 ꢀ3.5
R-branch
−1
P-branch
−1
−1
−1
7399 ꢀ4.8
[cm
]
[cm
]
[cm
]
[cm ]
7715 ꢀ6.2
11
12
13
14
15567.762 15484.221
15522.632 15432.323
15473.506 15376.574
8027 ꢀ7.5
8335 ꢀ8.9
15719.076
8636 ꢀ10.3
15420.447 15316.990 15765.857 15660.396
15363.474 15253.625 15709.980 15597.998
15302.631 15186.522 15650.290 15531.936
15
16
17
18
19
20
21
22
23
24
15237.966 15115.726
15462.231
frequency is sufficiently large to allow the spectral reso-
lution of the profiles induced by counterpropagating laser
beams.
15169.519 15041.287 15519.668 15388.952
15097.362 14963.267 15448.817 15312.146
15021.539 14881.727 15374.358 15231.871
15296.343 15148.190
We have presented new spectroscopic data for the A–X
transition with an experimental accuracy of the individual
15214.840 15061.168
−1
transition frequencies of better than 0.01 cm , involv-
15129.922 14970.876
00
ing relatively high vibrational levels 6 ≤ v ≤ 9 in the
15041.287
electronic ground state. New Dunham coefficients were

256
The European Physical Journal D
0
0
00
0
00
0
00
00
0
00 0
band: v =9–v =6
J
band: v =7–v =4
R-branch P-branch
band: v =7–v =5
J
band: v =9–v =5
R-branch
−1
P-branch
−1
R-branch
−1
P-branch
−1
R-branch
−1
P-branch
−1
−1
−1
[cm
]
[cm
]
[cm
]
[cm
]
[cm
]
[cm
]
[cm
]
[cm ]
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
16066.234
1
2
3
4
5
6
7
8
9
15110.233 15098.599 15464.265
15107.204
16114.901 16008.134
16059.756 15946.366
16000.840 15880.972
15938.211 15812.004
15871.896 15739.495
15663.511
15100.712 15073.640 15454.783 15427.673
15090.784 15056.026 15444.868 15410.076
15077.400 15035.013 15431.518 15389.077
15060.574 15010.612 15414.724 15364.699
15040.319 14982.845 15394.518 15336.961
15016.643 14951.733 15370.899 15305.886
14989.568 14917.311 15343.888 15271.498
14959.102 14879.574 15313.503 15233.831
14925.275 14838.584 15279.770 15192.905
14888.107 14794.360 15242.713 15148.763
14847.621 14746.940 15202.354 15101.434
15158.732 15050.965
15728.439 15584.100 16083.750 15938.277
15651.401 15501.324 16007.318 15856.055
15570.905 15415.248 15927.465 15770.559
15487.031 15325.939 15844.243 15681.855
15399.837 15233.469 15757.723 15590.015
15309.404 15137.915 15667.987 15495.123
15215.822 15039.363 15575.108 15397.251
15479.175 15296.476
10
11
12
13
14
15
15111.878 14997.385
1
6
15061.827 14940.748
0
0
00
0
00 0
band: v =8–v =4
J
band: v =8–v =3
R-branch P-branch
0
0
00
0
00 0
band: v =9–v =8
R-branch
−1
P-branch
−1
J
band: v =9–v =7
R-branch P-branch
−1
−1
[cm
]
[cm
]
[cm
]
[cm ]
R-branch
−1
P-branch
−1
−1
−1
2
3
4
5
6
7
8
9
15253.293 15234.225
15245.785 15219.133
15234.509 15200.307
15219.457 15177.754
15200.634 15151.491
[cm
]
[cm
]
[cm
]
[cm ]
4
5
6
7
8
9
15767.266
15788.603 15746.158
15771.764 15721.820
15751.501 15694.048
15727.822 15662.921
15700.751 15628.480
15670.307 15590.753
15178.055 15121.538 15523.386 15466.233
15151.725 15497.439 15432.901
15121.662 15050.612 15467.805 15395.970
15009.704 15434.498 15355.457
15050.382 14965.244 15050.382 14965.244
15356.949 15263.808
10
11
12
13
14
15
16
1
1
1
1
1
1
1
1
1
1
2
2
0
1
2
3
4
5
6
7
8
9
0
1
15636.515 15549.771 15994.520 15908.127
15599.402 15505.566 15957.212 15863.751
15558.993 15458.179 15916.608 15816.185
15515.320 15407.650 15872.734 15765.464
15468.426 15354.022 15825.630 15711.640
15418.344 15297.337 15775.340
15312.751 15212.736
15264.973 15158.212
15213.656 15100.274
15158.829 15038.964
15100.531 14974.327
References
. J.L. Dunham, Phys. Rev. 41, 713 (1932).
15038.815 14906.413
1
2
3
4
5
6
14973.717 14835.272
14905.299 14760.958
. T. Hori, Z. Phys. 62, 353 (1930).
. T. Hori, Z. Phys. 71, 478 (1931).
. E. Olsson, Z. Phys. 93, 206 (1935).
. R.C. Pankhurst, Proc. Phys. Soc. Lond. A 62, 191 (1949).
. F.B. Orth, W.C. Stwalley, S.C. Yang, Y.K. Hsieh, J. Mol.
Spec. 79, 314 (1980).
14833.612 14683.535
0
0
00
0
00
0
J
band: v =8–v =5
R-branch P-branch
band: v =8–v =6
R-branch P-branch
−1
−1
−1
−1
[cm
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