A deperturbation analysis of the B3∑u- =(v′ = 0-6) and the B″ 3∏u(v′ = 2-12) states of S2

Article abstract of DOI:10.1063/1.470810

Laser-induced fluorescence spectra of ³²S₂ have been obtained, covering v′=0-6 of the B ³-∑_u^- X ³∑_g^- transition and v′=2-12 of the B″ ³∏_u-X ³∑_g^- transition, using static cell and supersonic free jet techniques. The spectra include transitions to the previously unseen B″ ³∏_2,u components. Analysis of the many perturbations between the two upper electronic states has been carried out using a Hamiltonian matrix including all the B and B″ states simultaneously rather than deperturbing individual pairs of vibronic states. This takes into account longer range interactions and gives deperturbed molecular constants that vary smoothly with vibrational state. Our model for the B″ v′ = 2-12 and B v′=0-6 levels covering J up to 100 can fit all 3320 observed transitions with an average error of 0.064 cm^-1. The widely ranging fluorescence lifetimes of the B″ state vibronic levels provide independent information about the state mixing and confirm the model, since the observed B″-X fluorescence arises almost entirely via this mixing. However, lifetime measurements of the newly observed Ω=2 components of the B″ state showed little variation in lifetimes, about an average of 4.2±0.4 μs. This indicates a small intrinsic B″-X transition strength as the Ω=2 components are essentially not mixed with the B state. A model for the perturbation parameters is developed, based on Franck-Condon factors between the two states. The magnitude of the perturbations and transition moments are discussed in relation to the electronic configurations of the B and B″ states. 1996 American Institute of Physics.

Full text of DOI:10.1063/1.470810

AdeperturbationanalysisoftheB3Σu−(v′=0–6)andth B″3Πu(v′=2–12) states
of S2
M. E. Green and C. M. Western
Citation: J. Chem. Phys. 104, 848 (1996); doi: 10.1063/1.470810
View online: http://dx.doi.org/10.1063/1.470810
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3
؊
u
A deperturbation analysis of the B ⌺ (vЈ؍0–6) and the
3
BЉ ⌸ (vЈ؍2–12) states of S
u
2
M. E. Green and C. M. Western
School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS, United Kingdom
͑
Received 1 August 1995; accepted 10 October 1995͒
Laser-induced fluorescence spectra of 32S have been obtained, covering vЈϭ0–6 of the
2
3
Ϫ
u
3
Ϫ
g
3
3
Ϫ
g
B ⌺ –X ⌺ transition and vЈϭ2–12 of the BЉ ⌸ –X ⌺ transition, using static cell and
supersonic free jet techniques. The spectra include transitions to the previously unseen BЉ ⌸2
u
3
,u
components. Analysis of the many perturbations between the two upper electronic states has been
carried out using a Hamiltonian matrix including all the B and BЉ states simultaneously rather than
deperturbing individual pairs of vibronic states. This takes into account longer range interactions
and gives deperturbed molecular constants that vary smoothly with vibrational state. Our model for
the BЉ vЈϭ2–12 and B vЈϭ0–6 levels covering J up to 100 can fit all 3320 observed transitions
Ϫ1
with an average error of 0.064 cm . The widely ranging fluorescence lifetimes of the BЉ state
vibronic levels provide independent information about the state mixing and confirm the model, since
the observed BЉ–X fluorescence arises almost entirely via this mixing. However, lifetime
measurements of the newly observed ⍀ϭ2 components of the BЉ state showed little variation in
lifetimes, about an average of 4.2Ϯ0.4 ␮s. This indicates a small intrinsic BЉ–X transition strength
as the ⍀ϭ2 components are essentially not mixed with the B state. A model for the perturbation
parameters is developed, based on Franck–Condon factors between the two states. The magnitude
of the perturbations and transition moments are discussed in relation to the electronic configurations
of the B and BЉ states. © 1996 American Institute of Physics. ͓S0021-9606͑96͒01303-5͔
I. INTRODUCTION
The B–X absorption bands in S are analogous to the
2
extensively studied Schumann–Runge bands in O . The
3
Ϫ
3
Ϫ
2
The B ⌺ –X ⌺ transition of S in the ultraviolet is a
u
g
2
transition may be readily observed as a strong absorption in
S2 vapor, and appears as a long progression of bands. The
coarse vibrational structure of these bands is fairly straight-
forward with the bands becoming diffuse above vЈϭ9, due to
their being above the dissociation limit as shown in Fig. 1,
strong transition which should be ideal for laser-induced
fluorescence ͑LIF͒ probing of S . However, despite the fact
that this transition has been studied for over 80 years, a full
rotational analysis has never been carried out. This is hinder-
ing work on dynamical studies involving the measurement of
internal state distributions of S as a reaction product, as
mentioned in investigations into the photodissociation of CS
Ref. 2͒ and the reaction of OCS with S͑ D͒ atoms. The
recent observation of this transition by the Hubble Space
Telescope in the perturbed stratosphere of Jupiter after the
2
1
5
and therefore predissociated. The rotational analysis of these
2
bands, however, has proved to be far from trivial. Early
2
6
1
3
work, as summarized by Barrow and du Parcq mainly con-
͑
sisted of partial rotational analyses of relatively unperturbed
regions of the system. Investigations into the perturbed na-
ture of the system were first carried out by Meakin and
4
collision of the comet Shoemaker–Levy 9, has also illus-
trated the need for an accurate model of the S absorption
bands. The problem with the analysis lies with the extensive
perturbations in the system which are now known to be due
7
Barrow, who found many rotational perturbations in the vi-
2
brational levels vЈϭ7 and 8, and suggested that they were
3
due to a perturbing state of ⌸ symmetry, now labeled BЉ.
u
8
3
Further investigations by Meyer and Crosley illustrated the
to a single state, the BЉ ⌸ . The potential energy curves of
u
erratic behavior of some of the B state molecular constants
due to the extensive perturbations in the system. For example
they found that the spin–spin splitting constant ␭ of the B
state, which would normally vary smoothly with vibrational
the B and BЉ states are given in Fig. 1, where the two states
can be seen to overlap over a wide range of vibrational lev-
els. The interaction between the B and BЉ states is analyzed
in detail in this paper and provides a model for understand-
ing highly perturbed systems. It also provides some interest-
ing insights into the electronic configurations of the states
involved, which turn out to be not entirely straightforward.
The traditional molecular configurations of the three states
involved are given below
Ϫ1
state, behaved anomalously having values of ␭ϷϪ4.7 cm
Ϫ1
for vЈϭ0,2,4 and ␭Ϸϩ9.5 cm for vЈϭ1,3,5. This alterna-
tion gives a quite different appearance to vibrational bands
T
with odd and even v as the satellite branches such as R31
having significant intensity where ␭ has the opposite sign to
the ground state.
3
Ϫ
u
2
3
3
B ⌺
•••5␴ 2␲ 2␲ ,
g u g
Although spin and orbitally allowed, the BЉ–X transi-
tions are much weaker than the B–X transitions and were
not seen directly for many years. Bondybey and English9
were the first to directly observe BЉ–X transitions in their
3
2
4
1
1
BЉ ⌸
•••5␴ 2␲ 2␲ 5␴ ,
g u g u
u
3
Ϫ
2
4
2
X ⌺
•••5␴ 2␲ 2␲ .
g u g
g
848
J. Chem. Phys. 104 (3), 15 January 1996
0021-9606/96/104(3)/848/17/$6.00
© 1996 American Institute of Physics
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M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
849
interaction strengths between the two states were evaluated.
It was assumed throughout these calculations that the intrin-
sic transition moment for the BЉ–X transition is negligibly
small and therefore that the BЉ–X transitions are only ob-
served due to the borrowing of transition moment from the
B–X transition.
In this paper we describe rotationally cold LIF spectra of
BЉ–X and B–X bands of S produced in a supersonic free
2
jet and rotationally hot B–X spectra from S produced by
2
photolysis. In analyzing the spectra we also use the results of
earlier work by Barrow and co-workers1
5,16
who obtained
FIG. 1. RKR potential energy curves of the B and BЉ states of S . Vibra-
absorption ͑plate͒ spectra of sulfur vapor at 1025 K using a
Jarrell–Ash 3.4 m Ebert spectrograph. The range of J cov-
ered in this work is therefore extensive with low J term
values for both BЉ–X and B–X transitions being obtained
from the molecular beam spectra, and high J information on
the B state being provided by both the room temperature
photolysis work and the higher temperature work of Barrow
and co-workers. A particular feature of this work is the ob-
servation of the formerly unseen ⍀ϭ2 bands which has led
to a more complete picture of the system, including the de-
termination of the BЉ spin–spin constants and hence the
modification of the band origins and spin–orbit constants.
The measurement of the fluorescence lifetimes of these ⍀ϭ2
states also indicates some intrinsic transition moment for the
BЉ–X transition which was previously assumed to be negli-
gible.
2
tional levels in the B state above vЈϭ9 are above the dissociation limit and
hence predissociated. The solid lines represent observed levels whereas the
dashed lines represent levels extrapolated from the measured data.
study of S isolated in rare gas matrices. By studying fluo-
2
rescence emission and excitation spectra, they reassigned
fluorescence that had previously been attributed to transitions
from the B state as being from the BЉ state and tabulated a
vibrational progression of this state for vЈϭ0–6. Quick and
1
0
Weston measured fluorescence lifetimes for S vapor ex-
2
cited to the B state and observed additional low intensity
peaks with increased lifetimes and attributed these to pertur-
bations by the BЉ state. The first deperturbation analysis of
the B–X system was carried out by Pati n˜ o and Barrow11 for
the vЈϭ1 levels of the B state. They derived constants for the
3
The above data are deperturbed to provide a model for
the position and intensity of all the observed transitions. Al-
perturbing state confirming that it was ⌸ and, by measur-
u
ing isotopic shifts of the deperturbed levels, they estimated
Ϫ1
though in previous work individual states have been
T for the BЉ state to be 31 070 cm , which was found to be
e
deperturbed,1
1,12
the states involved were treated as though
9
consistent with the matrix work. A deperturbation analysis
was also carried out by Heaven et al.12 on the B vЈϭ2 state
for Jр30, again obtaining constants for the perturbing state.
They used supersonic free jet methods to produce molecules
with low rotational temperatures, leading to less congested
spectra.
they existed in isolation, so that interactions with more dis-
tant states where there are no avoided crossings were not
included in the calculations. As there are large homogeneous
͑J independent͒ interactions which can mix distant states,
this type of deperturbation is incomplete and yields only ef-
fective molecular constants which often vary erratically with
vibrational state and/or parity. An example of this is found in
the work of Pati n˜ o and Barrow11 referred to above; they had
As a consequence of the BЉ–X transitions being much
weaker than the B–X transitions, the radiative lifetimes of
the BЉ states are longer than for the B states. By combining
supersonic free jet methods with lifetime separation methods
to fit e and f parity components separately to give a good fit.
It is clear from the analysis performed here that many BЉ
states interact with more than one B state, and so the deper-
turbation must be extended to include these long-ranging in-
teractions. The ideal deperturbation procedure is to fit the
experimental data for as many states as possible in a simul-
taneous variation of all the parameters in an appropriate
combined Hamiltonian matrix. Here we describe such a de-
perturbation of the vibrational levels of the B state from
vЈϭ0 to 6 and of the BЉ state from vЈϭ2 to 12. The density
of states increases rapidly above this level, and so the higher
levels vЈϭ7–9 of the B state and vЈϭ13–18 of the BЉ state
will be analyzed in a separate fit and form the basis of a
separate publication.
1
3
Matsumi et al. produced greatly simplified LIF spectra,
which enabled the weak BЉ–X transitions to be seen. By
setting a time gate across the fluorescence decay curve to
measure the fluorescence produced a certain time after the
laser had fired, fluorescence from a BЉ state could be selec-
tively measured, while discriminating against B state fluores-
cence. From the spectra of the BЉ–X bands, rotational con-
stants and term values for the BЉ⍀ϭ0 and 1 states were
obtained for vЈϭ3–18. A time-resolved study of the B and
BЉ states was also carried out by Matsumi et al.14 where
fluorescence decay rates were measured for individual rovi-
bronic levels of the two electronic states. They found that the
fluorescence lifetimes depended strongly on v and ⍀ but not
3
Ϫ
J. The measured values for the B ⌺u state varied between
3
3
4.2 and 49.3 ns whereas those of the BЉ ⌸ states varied
u
II. EXPERIMENT
to a much greater extent, between 106 and 3888 ns. This
variation of lifetimes was interpreted in terms of the homo-
geneous interactions between the B and the BЉ states, and
Information on the B–X transitions came from three
sources. LIF spectra at ϳ300 K were obtained from S gen-
2
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850
M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
erated by a photoinitiated reaction of H S in a static cell. A
were set across the fluorescence decay curve, allowing the
fluorescence produced at different times to be sampled. In
this way discrimination could be obtained between the short
lived ͑ϳ30 ns͒ fluorescence from the B state and the longer
lived ͑Ͼ100 ns͒ fluorescence from the BЉ states.
2
2
5% mixture of H S in argon at a pressure of ϳ10 Torr was
2
photodissociated by 266 nm light from a Nd:YAG ͑Spectron
Laser Systems SL802͒ to produce S , most probably via the
2
radical–radical reaction mechanism described by Heaven
1
2
et al. The S produced was then probed by the frequency
Wavelength calibration for both supersonic free jet and
the static cell experiments was obtained by simultaneously
recording the laser-induced fluorescence spectrum of iodine
vapor ͑using undoubled light from the dye laser͒ and inter-
2
doubled output of a dye laser ͑Spectra Physics PDL3͒
pumped by a Nd:YAG ͑Quanta-Ray GCR 18S͒. The spectral
Ϫ1
width of the UV light was approximately 0.1 cm and its
Ϫ1
power was adjusted to avoid gross saturation. A delay of
ϳ10 ␮s between the photolysis beam and the pump beam
was needed to allow time for secondary photochemical reac-
ference fringes from a 1 cm free spectral range etalon. The
line positions were taken directly from the spectra by using a
center of mass algorithm on the digitally recorded data. This
allowed line positions to be determined to better than the
tions to occur and produce S . By varying this delay, the
2
Ϫ1
degree of collisional cooling and hence the rotational tem-
perature could be controlled to some extent. The fluores-
cence produced was collected at 90° to the laser beams and
detected with a photomultiplier ͑Thorn EMI 9125QB͒. The
signal from the photomultiplier was sent directly to a Lecroy
experimental linewidth of ϳ0.1 cm . The iodine calibration
Ϫ1
spectra were typically fitted to 0.01 cm in the visible, lead-
Ϫ1
ing to a measurement accuracy of 0.02 cm in the ultravio-
let for strong, unblended lines.
In order to measure the fluorescence lifetimes, the laser
was tuned to the frequency of a single low J transition and
the fluorescence collected as before. The output of the pho-
tomultiplier was sent to the digital oscilloscope where the
fluorescence decay profile was averaged over 100–2000
shots. For each lifetime measurement, a recording of the
background light was obtained by recording the same num-
ber of shots of the output from the photomultiplier with the
laser slightly detuned from the transition of interest. This was
subtracted from the fluorescence decay profile, leaving a re-
sultant decay curve from which a lifetime value was obtained
by least-squares fitting to a single exponential.
͑9400A͒ digital oscilloscope where a gate was set across the
fluorescence decay curve. The time gate was set to start col-
lecting signal ϳ10 ns after the laser had fired to minimize
scattered light and had a width of ϳ50 ns. The spectra ob-
tained in this way gave information on the rotational levels
of the B state up to JϷ50. The congested nature of the spec-
tra often resulted in the overlap of vibrational bands making
assignment difficult. In these cases a monochromator was
placed before the photomultiplier which allowed discrimina-
tion between transitions to different lower vibrational states,
thereby allowing individual vibrational bands to be observed.
Information on the B–X transition also came from static cell
absorption spectra of S at temperatures over 1000 K, taken
2
15,16
by Barrow and co-workers
some years before. These ro-
III. RESULTS
tationally hot spectra provided term values for levels with J
up to 100, allowing many high J avoided crossings to be
observed. The lowest J levels, which could not be obtained
from the static cell spectra due to the high level of conges-
tion around the bandheads, were taken instead from the less
congested molecular beam spectra.
LIF spectra of B–X bands vЈϭ0–6 and BЉ–X bands
vЈϭ2–12 were taken. Transitions to the ⍀ϭ0 and 1 levels of
the BЉ state were observed as well as transitions to the very
weak and hence previously unobserved ⍀ϭ2 levels. The su-
personically cooled spectra of the BЉ bands had rotational
temperatures typically in the range of 5–30 K and hence
gave information up to JϷ20. The vibrational cooling in the
supersonic jet was not as efficient as the rotational cooling,
with the result that many of the observed bands were hot
bands. This proved advantageous as many of the Franck–
Condon factors between low vibrational levels of the ground
Rotationally cold laser-induced fluorescence spectra of
both the B–X and BЉ–X transitions were obtained using
supersonic free jet methods. A mixture of H S in argon
2
͑ϳ1%͒ was passed through a pulsed electric discharge nozzle
and then through a 1 mm pinhole to a region of low pressure
Ϫ4
͑
10 Torr͒. The pulsed electric discharge nozzle has been
17
18
described in detail by Western et al., but basically consists
of a plastic ͑Delrin͒ piece with a ϳ1-mm-diam, 1-cm-long
channel for the gas, mounted on a pulsed molecular beam
valve. The electric discharge is provided by a pair of tung-
sten electrodes mounted halfway down the channel, across
which a voltage ͑Ͻ1 kV͒ is applied as the gas is passed
through. The pressure behind the nozzle was variable in the
range of 0–40 psi and this gave some control over the rota-
tional temperature in the supersonic free jet. A probe beam,
produced as before, crossed the molecular beam ϳ10 mm
downstream from the nozzle and the fluorescence was col-
lected at 90° to both the molecular beam and laser beam. The
signal was again collected by a photomultiplier, and sent
directly to the digital oscilloscope, where various time gates
states and B vibrational states are very low. A typical spec-
trum showing the BЉ–X ͑10–0͒ and the B–X ͑7–1͒ bands
with a rotational temperature of 15 K is given in Fig. 2. The
upper trace was taken using a time gate set to collect fluo-
rescence produced 200–2000 ns after the laser was fired, and
hence shows only the BЉ state with a long fluorescence life-
time. The time gate for the lower trace was set to collect
fluorescence produced immediately after the scattered light
from the laser, and hence shows both the BЉ–X band and the
B state with a much shorter fluorescence lifetime. Although
the two bands in the lower trace have similar intensity, this is
misleading as the B–X transitions are many times stronger
than the BЉ–X transitions. The apparent similarity in inten-
sity is due both to saturation of the B–X transition, and the
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M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
851
FIG. 2. LIF molecular beam spectra of the BЉ–X͑10–0͒ and the B–X͑7–1͒ bands; a time gate was set to collect fluorescence produced ͑a͒ 200–2000 ns after
the laser was fired and ͑b͒ immediately after the laser was fired.
TABLE I. Fluorescence lifetimes of the BЉ 3⌸u state of S2 . The figures in
parentheses for the data of Matsumi et al. are three standard deviations. The
estimated errors for the lifetimes measured here are Ϯ10%.
fact that the time gate was set to discriminate against scat-
tered light, which therefore cut out much of the B state fluo-
rescence.
Spectra of the mixed isotope 34S 32S species ͑8% natural
abundance͒ were not observed in this work. This is perhaps
surprising, given the high signal to noise obtained in most
␶f/ns
͑this work͒
␶f/ns
Matsumi et al.
c
vЈ
⍀Ј
2
2
2
3
3
4
4
4
5
5
5
6
6
6
7
7
7
8
8
8
9
9
9
0
1
2
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
623
182
3
4
32
spectra ͑see Figs. 2 and 6͒. However the 2
S S species has
twice the number of rotational lines as 3 S2 because of the
nuclear spin statistics. ͑The symmetric isotope has alternate
rotational levels missing.͒ With our resolution this means
that vibronic bands from the mixed isotope will have poorly
resolved rotational structure, and will appear as a weak con-
4167a
322
296 ͑24͒
4375
125
1025
b
247
738
4296
773
118
4042
495
1039
4243
167
887
4123
907
154
3982
393
155 ͑42͒
1126 ͑99͒
32
tinuous background under the stronger S2 signals.
204 ͑21͒
717 ͑93͒
Fluorescence lifetimes of BЉ bands were measured in the
molecular beam and are given in Table I. We found the same
wide range of lifetimes ͑100–1600 ns͒ as Matsumi et al.14
for the ⍀ϭ0 and 1 levels. Measurement of the lifetimes of
the ⍀ϭ2 levels required particular care; not only were the
signals weak because of the weak excitation transitions, but
also the fluorescence lifetimes are sufficiently long that mol-
ecules can fly out of the viewing region before fluorescing,
artificially depressing the apparent fluorescence lifetime. The
819 ͑57͒
108 ͑9͒
556 ͑72͒
1057 ͑99͒
164 ͑9͒
896 ͑75͒
⍀ϭ2 state transients were therefore recorded with all the
947 ͑45͒
139 ͑12͒
normal fluorescence collection optics removed, so there was
only a flat window between the photomultiplier and the mo-
lecular beam. To check that this arrangement allowed accu-
rate measurement of the fluorescence decay we performed
10
10
10
393 ͑36͒
1661͑222͒
1729
b
two tests. First we measured lifetimes of the vϭ0 and 1 of
11
338 ͑21͒
358 ͑39͒
3
1
1
the A ⌸ state of SO; these are known to have very long
1
1
9,20
11
4076
lifetimes
of 39.2 and 17.1 ␮s, respectively. Measure-
ments on our apparatus gave lifetimes of 29 and 13.5 ␮s,
though the picture was slightly confused by the observation
of strong quantum beats. These figures suggest the require-
a
Corrected value—see the text.
States with strongly J-dependent lifetimes.
Reference 14.
b
c
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852
M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
FIG. 5. LIF spectrum of the B–X͑5–0͒ band, produced by the photolysis of
H S in a static cell.
2
measurements to be 0.4 ␮s. Given this uncertainty the small
variation shown in lifetimes for the ⍀ϭ2 states in Table I is
probably not significant and the most useful number is the
average over all the ⍀ϭ2 measurements, 4.16 ␮s.
For BЉ vϭ2, ⍀ϭ2 the signal was so weak that the life-
time could not be measured without the fluorescence collec-
tion optics. The value quoted in Table I was therefore mea-
sured with the optics and corrected by comparing
measurements for BЉ vϭ3 with the same optics. We do not
quote lifetimes for BЉ vϭ4 and 10 as these showed a very
strong variation with J. This is readily understood in terms of
the model developed below as both these states undergo a
strong, J-dependent mixing at low J with B state levels that
are very close in energy.
FIG. 3. Fluorescence intensity ͑a͒ and the log of the fluorescence intensity
3
3
Ϫ
͑
b͒ plotted against time for a BЉ ⌸2 –X ⌺ ͑9–0͒ transition recorded
,u g
using both helium and argon mixtures.
ment for some corrections to lifetimes much longer than
those we observed for any of the BЉ states.
Second some BЉ ⍀ϭ2 lifetimes were recorded using two
different carrier gases for the molecular beam, helium, and
argon. As the gas mixtures were very dilute this implies a
velocity difference of ϳ3 between the two mixtures, but nev-
ertheless very similar fluorescence decay curves were ob-
served, as shown in Fig. 3. This shows the same transition
3
3
Ϫ
͓
from the BЉ ⌸ –X ⌺g ͑9-0͒ band͔ recorded using he-
2,u
lium and argon mixtures. Note that the helium curve shows
nonlinearity in the log plot after 20 ␮s; this corresponds to a
distance of around 3 cm. The lifetimes obtained from these
two curves are close, being 3.93 ␮s ͑Ar͒ and 4.16 ␮s ͑He͒.
This sort of variation was also typical of repeated measure-
ments taken with the same gas mixture, and probably arises
from incomplete subtraction of background light, and prob-
lems produced in the photomultiplier by the very strong sig-
nals at tϭ0 from strong B state fluorescence. Taking these
factors into account we estimate the error on these lifetime
A spectrum of a ⍀ϭ2 band ͓BЉ–X ͑6–3͔͒ is given in
Fig. 4͑a͒, taken using a time gate set to collect fluorescence
produced 1630–4130 ns after the laser had fired. A simula-
tion of this band is shown in Fig. 4͑b͒ using the model de-
scribed below.
A typical spectrum of a B–X band of S produced by the
2
photolysis of H S in a static cell is shown in Fig. 5. The
2
spectrum was taken using a monochromator in order to dis-
criminate against overlapping bands, and shows the B–X
͑5–0͒ band with a rotational temperature of ϳ300 K. A por-
FIG. 6. ͑a͒ A simulation and ͑b͒ the observed spectrum of part of the
B–X͑5–0͒ band shown in Fig. 5. Assignments are shown for some of the
branches present in the spectrum; note that some weak transitions from the
BЉ–X͑10–0͒ band also show in this region.
FIG. 4. LIF molecular beam spectrum ͑a͒ and simulation ͑b͒ of the
BЉ 3⌸2 –X ⌺ ͑6–3͒ band; the spectrum was taken using a time gate set
3
Ϫ
,u
g
to collect fluorescence produced 1630–4130 ns after the laser had fired.
J. Chem. Phys., Vol. 104, No. 3, 15 January 1996
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M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
853
ground state constants taken from Fourier transform infrared
spectra21 for vЉϭ0 and 1 and from emission studies for the
other vibrational states. Rotational combination differences
were found within each vibrational upper state, and to con-
firm vibrational assignments, vibrational combination differ-
ences were found for each B and BЉ vibrational state. The
reduced term values, the observed upper state term values
16
Ϫ1
minus 0.2J(Jϩ1) cm , are plotted in Fig. 7 as a function of
J for vЈϭ0–6 of the B state and vЈϭ2–12 of the BЉ states.
The solid lines in Fig. 7 are term values calculated from the
model described below. The ground state term values re-
quired to convert the observed transitions to upper state term
values were calculated from the refined ground state con-
stants given in Table II, described below.
It is clear from this energy level diagram why analysis of
the B state is so difficult; each vibrational level of the B state
is perturbed by at least two different vibrational levels of the
BЉ state. This gives several avoided crossings in the rota-
tional levels of the B state, to a degree that varies dramati-
cally with vibrational state depending on which ⍀ compo-
nent crosses the B state and the J value of the crossing. An
expanded view of the rotational term values of one B state
vibronic level is shown in Fig. 8; note the contrast between
the small avoided crossing from the BЉ ⍀ϭ2 state ͑shown in
the inset͒, and the much larger crossing with the ⍀ϭ1 state.
FIG. 7. Observed upper state reduced term ͓term values Ϫ0.2J(Jϩ1)͔ plot-
ted against J for vЈϭ0–6 for the B state and vЈϭ2–12 for the BЉ. The छ
and ϫ are observed term values of e and f parity, respectively, and the solid
lines are calculated term values, using the model described in the text. The
large numbers on the right-hand side of the diagram indicate the position of
the B state vibrational levels. The smaller numbers on the diagram indicate
v and ⍀ for the BЉ states; the numbers on the left-hand side refer to the
curve immediately below whereas the numbers in the middle refer to the
curve immediately above.
B. Hamiltonian
The system has been modeled by a computer program
PGOPHER, which uses a fairly conventional effective Hamil-
tonian for a linear molecule. The major terms include the
ˆ ˆ
spin–orbit coupling ͑AL•S͒ and the rotational kinetic energy
ˆ 2
͑
BR ͒;
tion of this spectrum is shown in Fig. 6 together with a
simulation using the model developed below. This shows
some weak transitions from the BЉ vЈϭ10 state, as well as
stronger transitions from the B vЈϭ5 state.
ˆ
ˆ
ˆ
ˆ 2
ˆ ˆ
HϭB͑JϪLϪS͒ ϩAL•S
ˆ2 ˆ 2 ˆ 2
ˆ ˆ
ϭB͓J͑Jϩ1͒ϩS͑Sϩ1͒ϪJ ϪL ϪS ͔ϩAL S
z z
z
z
z
ˆ ˆ
ˆ ˆ
1
2
ˆ ˆ
ˆ ˆ
Ϫ
ϪB͓͑J L ͒ϩ͑J L ͔͒ϩ͑ AϩB͒
͕
L S ϩL Sϩ͖
,
Ϫ
ϩ
ϩ
Ϫ
ϩ
Ϫ
IV. ANALYSIS
A. Assignment
͑
1͒
Rotational analysis and assignment was carried out using
the method of combination differences based initially on
where J, L, S, and N have the conventional definitions as
given in Lefebvre-Brion and Field.22 The minor terms are
TABLE II. Refined constants for the X 3⌺ vЉϭ0–5, 7 levels. Figures in parentheses are three standard deviations.
Ϫ
g
Tv,0 ͑cmϪ1
͒
BЉ ͑cmϪ1
͒
DЉ ͑10Ϫ7 cmϪ1
͒
␥Љ ͑cmϪ1
͒
␭Љ ͑cmϪ1
͒
␭Љ ͑10 cmϪ1͒
5
XvЉ
D
a
a
1.96 ͑2͒a
1.97 ͑2͒a
1.96 ͑17͒
1.97 ͑12͒
2.046͑62͒
2.12b
Ϫ0.007 157 ͑9͒a
Ϫ0.007 148͑28͒a
Ϫ0.005 93 ͑91͒
Ϫ0.006 6 ͑14͒
Ϫ0.007 22 ͑51͒
Ϫ0.007 283b
11.7931 ͑4͒a
11.8659͑10͒a
11.937 ͑14͒
12.021 ͑20͒
12.091 ͑19͒
12.163b
a
0
1
2
3
4
5
7
0
0.294 592 3 ͑4͒
0.292 997 5͑23͒
1.05͑5͒
1.05
a
719.995͑90͒
1434.25͑10͒
2142.85͑11͒
2845.61͑14͒
3542.94͑52͒
4920.03͑22͒
0.291 423 ͑55͒
0.289 646 ͑72͒
0.288 376 ͑44͒
0.284 4
͑27͒
0.283 241 ͑49͒
1.987͑63͒
Ϫ0.009 62 ͑59͒
12.247 ͑25͒
a
Reference 21.
Reference 16.
b
J. Chem. Phys., Vol. 104, No. 3, 15 January 1996
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854
M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
Ϫ
where s is 1 for a ⌺ state and 0 otherwise. This is consistent
with the phase convention of Lefebvre-Brion and Field.22
Note that this is different from the phase convention used in
earlier work on S ͑Refs. 11 and 14͒, which used matrix
2
23
elements taken from Wicke et al. In our symmetrized basis
the matrix elements off diagonal in the electronic state are
3⌸2
3⌸1
3⌸0
3⌸0
͉
͉
͉
͉
͉
H
ˆ
͉
͉
͉
͉
͉
3⌺
3⌺
3⌺
3⌺
Ϫ
1
ϭϪ͑xϪ2͒1/2␤,
ϭ21/2͑␣ϩ␤͒,
ϭϯx1/2␤,
͑6͒
͑7͒
͑8͒
͑9͒
͗
͗
͗
͗
͗
͘
͘
͘
͘
͘
ˆ
Ϫ
1
H
ˆ
Ϫ
1
H
ˆ
Ϫ
0
H
ϭ͑1Ϯ1͒͑␣ϩ␤͒,
3⌸1
H
ˆ
3⌺
Ϫ
0
ϭϪ͑1Ϯ1͒͑
1x͒1/2␤,
͑10͒
2
where xϭJ(Jϩ1) and the upper sign corresponds to e parity
and the lower sign to f parity.
It was found necessary to include centrifugal distortion
of the perturbation parameters to model the high J crossings
accurately. The required operators in the Hamiltonian were
defined by analogy to the centrifugal distortion of other fine
structure constants as defined by Hirota et al.:24
1
2
ˆ 2
Ј ͓R ,L S ϩL S ͔ ϩ ␤Ј ͓R ,L J ϩL J ͔ .
ϩ Ϫ Ϫ ϩ ϩ ϩ Ϫ Ϫ ϩ ϩ
2
1
ˆ 2
␣
D
D
FIG. 8. Expanded view of the observed reduced term values for the B state
͑
11͒
3
vЈϭ2 and its neighboring BЉ states. The inset shows the BЉ ⌸2 vЈϭ4
,u
3
Ϫ
,u
state crossing the B ⌺1 vЈϭ2 state. The छ and ϫ are observed term
The parameters for these terms are defined by the following
matrix elements:
values of e and f parity respectively and the solid lines are calculated term
values.
␣D
␤D
ϭ
ϭ
͗
͗
BЉ⌳ϭ1
BЉ⌳ϭ1
͉
͉
␣Ј Lϩ͉B⌳ϭ0
͘
͘
,
.
͑12͒
͑13͒
D
␤Ј Lϩ͉B⌳ϭ0
D
ˆ
ˆ 4
2
3
ˆ 2 ˆ2
1
2
2
3
ˆ 2 ˆ2 ˆ 2
z
ϩ
HЈϭϪDR ϩ ␭͑3S ϪS ͒ϩ ␭ ͓ ͑3S ϪS ͒,N ͔
z
D
1
2
ˆ 2 Ϫ2i␾ ˆ 2
ϩS eϩ2i␾͒,
C. Fitting procedure
ˆ ˆ
ϩ␥N•SϪ o͑S e
͑2͒
ϩ ϩ
We present two fits here; the first, is a straightforward fit
to the parameters determined directly by the data. The sec-
ond fit, described in Sec. IV D, includes several estimated
parameters to take account of interactions that are not di-
rectly observed. These other interactions are large enough to
have a significant effect on the determined parameters, so the
end result is a more physically reasonable set of values.
All the parameters which could be directly determined
by the data were simultaneously varied in least-squares fits
of calculated against observed data, until the best fit possible
was obtained. The excited energy levels were calculated us-
ing large basis sets containing all the excited state levels of a
given parity and J including the B states vЈϭ0–6 as well as
the BЉ states vЈϭ2–12. The ground state levels were calcu-
lated similarly using matrices containing all states of a given
J and parity for vЉϭ0–5 and 7. This simultaneous calcula-
tion is not required for the ground state levels, but it removes
the need for extra programming. Despite the large numbers
of parameters floated few problems with convergence of the
fits were encountered. The only problems were with B vЈϭ6
and BЉ vЈϭ12 where one or two parameters had to be varied
manually to find the best value for the parameter and its
error. This probably reflects the relatively small number of
levels measured for B vЈϭ6.
ˆ ˆ
ˆ ˆ ˆ ˆ
where [O,Q] ϭOQϩQO. This includes centrifugal distor-
ϩ
tion (D), spin–spin coupling ͑␭͒, and its centrifugal distor-
tion ͑␭ ͒, spin–rotation coupling ͑␥͒, and the ⍀ϭ0 lambda
D
doubling, o. We do not include the terms p and q conven-
tionally used to describe ⌳ doubling in the ⌸ state; this is
discussed further below.
The terms containing LϮ in Eq. ͑2͒ give matrix elements
off-diagonal in electronic state and are responsible for the
perturbations between the B and BЉ states. These matrix el-
ements cannot be evaluated directly and so are taken as pa-
rameters in the fit. The spin–orbit parameter ͑␣͒ and the
L-uncoupling parameter ͑␤͒ are defined as
1
2
ˆ
␣
␤
ϭ
ϭ
͗
͗
BЉ⌳ϭ1
BЉ⌳ϭ1
͉
͉
͑ AϩB͒Lϩ͉B⌳ϭ0
͘
,
͑3͒
͑4͒
ˆ
BLϩ͉B⌳ϭ0
͘
.
Note that these are defined in an unsymmetrized basis. The
Hamiltonian matrix is made up in a symmetrized Hund’s
case ͑a͒ basis with the phase of the electronic wave functions
defined such that states of Ϯ parity are given by
Ϯ
1
&
͉
J⍀␩⌳S⌺
͘
ϭ
͕
͉
J⍀␩⌳S⌺
͘
Ϯ͑Ϫ1͒JϪSϩs
͉
JϪ⍀␩Ϫ⌳SϪ⌺
,
͑5͒
͘
͖
J. Chem. Phys., Vol. 104, No. 3, 15 January 1996
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M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
855
TABLE III. Comparison of fits to the B 3⌺ vЈϭ1 state and the BЉ ⌸ vЈϭ3 and 4 states. Figures in parentheses are three standard deviations.
Ϫ
3
u
u
Pati n˜ o and Barrowc
Local fita
Global fitb
e levels
f levels
Tv,0 ͑cmϪ1
͒
BvЈϭ1
BЉvЈϭ3
BЉvЈϭ4
32 114.12 ͑20͒
32 035.55 ͑10͒
32 342.149 ͑63͒
32 102.71͑23͒
32 039.70͑13͒
32 344.732͑53͒
BЈ ͑cmϪ1
͒
BvЈϭ1
BЉvЈϭ3
BЉvЈϭ4
0.221 145 ͑48͒
0.201 72 ͑10͒
0.200 405 ͑63͒
0.223 429͑80͒
0.201 702͑79͒
0.199 143͑84͒
0.221 90͑30͒
0.206 6 ͑12͒
0.202 28͑12͒
0.221 54 ͑9͒
0.197 9͑192͒
0.201 1 ͑36͒
␭
Ј ͑cmϪ1
͒
BvЈϭ1
BЉvЈϭ3
BЉvЈϭ4
4.922 ͑90͒
8.942 ͑90͒
7.077 ͑45͒
4.724͑75͒
5.88͑12͒
5.624͑36͒
AЈ ͑cmϪ1
͒
BЉvЈϭ3
BЉvЈϭ4
195.815 ͑69͒
192.561 ͑81͒
193.698͑94͒
190.986͑57͒
␣
␤
͑cmϪ1
͑cmϪ1
͒
͒
͒
B–BЉ͑1–3͒
B–BЉ͑1–4͒
Ϫ7.51 ͑54͒
10.810 ͑96͒
18.43͑29͒
Ϫ11.291͑60͒
Ϫ12.15
͑10͒
12.26
͑6͒
B–BЉ͑1–3͒
B – BЉ͑1–4͒
Ϫ0.125 0 ͑60͒
Ϫ0.018 4 ͑18͒
Ϫ0.080 8͑18͒
0.029 55͑97͒
Ϫ0.0598 ͑6͒
Ϫ0.034
Ϫ0.0852 ͑11͒
Ϫ0.0272 ͑2͒
͑6͒
p ͑cmϪ1
BЉvЈϭ3
BЉvЈϭ4
0
0
0
0
1.2
͑6͒
1.695 ͑18͒
q ͑cmϪ1
͒
BЉvЈϭ3
BЉvЈϭ4
0
0
0
0
Ϫ0.034 9 ͑18͒
Ϫ0.003 7 ͑4͒
a
Fit to these three states only.
From the global fit to all B and BЉ states ͑Tables VIII–XII͒.
Reference 11. This fit calculated individual BЉ components separately so the Tv,0, ␭, and A values are not directly comparable.
b
c
As the Hamiltonian matrix becomes rather large when
all the states are included the computational time becomes
significant. For the 3320 transitions included in the final fit,
approximately 200 independent diagonalizations are required
tion time. This is overcome in the distribution logic by work-
ing through the states in order of J ͑varying fastest͒ then e/f
then upper/lower state.
The significant computational effort invested in calculat-
ing all the states simultaneously is justified by obtaining
more physically reasonable parameters which vary smoothly
with vibrational state, and by requiring fewer parameters.
This is due to two related effects: first, any given BЉ state is
typically close to two B state vibrational levels, so that the
interaction with one B state level is changed by virtue of its
interaction with the other. Second, some of the B–BЉ pertur-
bations, particularly the J-independent ones, are large
enough to give measurable effects for levels more than one
vibrational quantum away. These latter interactions with
more distant states typically vary smoothly with J and could
simply be modeled by an adjustment to the effective con-
stants, though the effective constants would then be less easy
to interpret.
The necessity for fitting all the states simultaneously
may be illustrated by comparing constants obtained for a
state deperturbed in a global fit with those obtained from an
individual state fit. For example, if the B vЈϭ1 state is dep-
erturbed in isolation from the surrounding B states, only in-
cluding interactions with the BЉ vЈϭ3 and 4 states in the
model, the constants obtained are different from those for the
same state deperturbed in a global fit. The constants from
both these fits are shown in Table III. Also given in Table III
͑
two parities for J up to 100 in the upper and lower states͒
and each calculation must be repeated for each parameter
171 times͒ to calculate the numerical derivatives. For this
͑
reason, use was made of a parallel computer, a Meiko Com-
puting Surface 2. This computer is a distributed memory
system, essentially several independent computers ͑6–8
were used for this work͒, so the calculation must be split into
several independent parts with minimal communication be-
tween them. Fortunately this is easy to do for this calcula-
tion; work on each Hamiltonian matrix in one cycle of the
least-squares fitting process can take place independently of
the calculation on all the other matrices. Thus, given n ma-
trices to diagonalize, the matrices are distributed equally
over the p processors so that each processor calculates n/p
levels. The results are then combined on one processor to do
the least-squares correction. Given that this latter step is a
small part of the total computational time, a speed up relative
to a single processor close to p is obtained. In principle the
repeated calculations to find the numerical derivatives could
also be split over different processors, though this is not done
here as the calculation is already split into enough parts. The
only potential problem is in the distribution of the matrices
over the processors, as there is a considerable variation in
size ͑from 8 for f parity ground states to 53 for e parity
excited states͒, implying a considerable variation in calcula-
J. Chem. Phys., Vol. 104, No. 3, 15 January 1996
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856
M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
TABLE IV. Deperturbed constants from fit 1 for the BvЈϭ0–6 states. Figures in parentheses are three standard deviations.
Tv,0 ͑cmϪ1
͒
BЈ ͑cmϪ1
͒
DЈ ͑10Ϫ7 cmϪ1
͒
␭Ј ͑cmϪ1
͒
␥Ј ͑cmϪ1
͒
vЈ
0
1
2
3
4
5
6
31 682.852͑75͒
32 108.69͑23͒
32 530.23͑23͒
32 947.82͑17͒
33 354.96͑34͒
33 768.55͑22͒
34 173.550͑86͒
0.224 201͑53͒
0.223 227͑77͒
0.222 105͑84͒
0.219 588͑82͒
0.219 83͑15͒
0.215 967͑95͒
0.213 37͑11͒
2.442͑67͒
2.72͑10͒
4.03͑14͒
3.21͑13͒
6.08͑34͒
3.35͑22͒
2.56͑34͒
2.877͑96͒
3.530͑77͒
2.42͑13͒
3.15͑14͒
3.45͑19͒
2.58͑16͒
1.42͑19͒
Ϫ0.0280͑19͒
Ϫ0.0185͑12͒
Ϫ0.0072͑28͒
Ϫ0.0028͑29͒
Ϫ0.0089͑51͒
Ϫ0.0009͑39͒
0.0029͑25͒
are the constants obtained by Pati n˜ o and Barrow11 in their
deperturbation of the B vЈϭ1 state. In order to obtain a
reasonable fit for this single state, they found it necessary to
float the e and f parities separately and to include very large
values of the ⌳ doubling parameters p and q for the BЉ
states.
L-uncoupling mixing term ͑␤͒ would need to be included in
the fit in order to model the avoided crossing. However,
whenever a ⍀ϭ2 component of a BЉ state crosses a B state,
the selection rules would allow mixing with the ⍀ϭ1 com-
ponent of the B state via the L-uncoupling mixing term, but
the spin–orbit mixing would not be involved. Hence only ␤
could be determined in this case.
The inclusion of the centrifugal distortion of the pertur-
bation parameters in the fit also required similar consider-
ations. The centrifugal distortion of the spin–orbit mixing
The model described here does not require these ⌳ dou-
bling parameters. These terms arise from the interaction of a
⌸
state with distant ⌺ states to give a splitting between states
of different parity. In the current case, we consider the inter-
3
Ϫ
action with the B ⌺u state directly, and the perturbations of
the ⌸ state could be viewed as an example of giant ⌳ dou-
bling. In principle interactions with more distant ⌺ states
would give rise to some additional shifts, but we estimate
that these effects would be small, and in any case, could not
be distinguished from the interactions with the several
͑␣ ͒ mixes states according to the selection rule ⌬⍀ϭ0,Ϯ1,
D
whereas the centrifugal distortion of the L-uncoupling ͑␤D͒
obeys the selection rules ⌬⍀ϭ0,Ϯ1,Ϯ2. These selection
rules restrict the parameters that have an effect in various
cases. Furthermore, in order for these parameters to be de-
termined for a particular interaction, there must be an
avoided crossing between the two states at a high value of J.
The number of cases for which ␣ and ␤ were required was
3
Ϫ
u
B ⌺ vibrational levels that are included as part of our
global fit.
D
D
The degree to which each constant and perturbation
could be determined by the data needed to be considered in
some detail in order to establish whether or not it could be
floated as a parameter in the fit. For example the parameter
therefore limited.
The sign of the matrix elements off-diagonal in elec-
tronic state are dependent on the phase convention chosen
for the electronic wave functions ͑given above͒ and the vi-
brational wave functions. The absolute sign of the perturba-
tion parameters is in any case not determined in our fit;
changing the signs of all the perturbation parameters simul-
taneously makes no difference to the calculated energy levels
or intensities. However several of the relative signs are de-
termined by our fit; obviously if a pair of states is connected
by more than one type of perturbation then the relative signs
of the different perturbations are determined. If both these
states are connected to a third state, then the relative signs of
all the parameters interconnecting all three states are deter-
mined. In our model we do not have any sets of three states
all interconnected in this way; however such connections do
exist via fourth or fifth states. These indirect connections
only have a very small effect on the model, so if, for ex-
ample, the signs of the perturbations between one particular
pair of B and BЉ states are all changed then all the eigenval-
ues change, but by an amount less than our experimental
error. We therefore chose to set the absolute signs of the ␣
perturbation parameters so that they were consistent with the
signs of the calculated vibrational overlap integrals between
the B and the BЉ states ͑with the vibrational wave function
chosen such that the outermost lobe is positive͒. A more
detailed model for ␣ is developed below, which confirms this
choice of signs.
ϩ
Ϫ
o, which gives the splitting between the 0 and 0 compo-
3
nents of the ⌸ state, had to be included to give a good fit
0
for those vibrational levels where both components are ob-
served. However, for most BЉ ⍀ϭ0 levels only the e parity
component was observed and so o could not be determined
3
in these cases. The ⌸ f component is rarely observed be-
0
cause of the ⌬⍀ϭ0 selection rule for the matrix elements of
the spin–orbit perturbation, Eqs. ͑7͒ and ͑9͒. This means that
the BЉ ⍀ϭ0 levels can only gain intensity via mixing with
the ⍀ϭ0 component of the B state, of which only an e
component exists. The f parity for the ⍀ϭ0 BЉ state was
only observed in those cases where a BЉ ⍀ϭ0 level crossed
a B ⍀ϭ1 level ͑for which both e and f components exist͒. In
the region of a crossing, the weak L-uncoupling perturbation
͓Eq. ͑8͔͒ becomes significant and causes mixing of the two
levels according to its selection rule ⌬⍀ϭϮ1. The f parity
component of the BЉ ⍀ϭ0 state can therefore gain intensity
in these cases by mixing with the f component of the B ⍀ϭ1
state. Hence the value of o is only determined for the vibra-
tional levels vЈϭ3, 4, 7, and 10 where such crossings occur.
The same considerations are important in deciding
which perturbations are determined. Whenever a ⍀ϭ0 or 1
component of a BЉ state crosses the ⍀ϭ0 or 1 components of
a B state, both the spin–orbit mixing term ͑␣͒, and the
J. Chem. Phys., Vol. 104, No. 3, 15 January 1996
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M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
857
TABLE V. Deperturbed constants from fit 1 for the BЉvЈϭ2–12 states. Figures in parentheses are three standard deviations.
Tv,0 ͑cmϪ1
͒
BЈ ͑cmϪ1
͒
DЈ ͑10Ϫ7 cmϪ1
͒
AЈ ͑cmϪ1
͒
␭Ј ͑cmϪ1
͒
␥Ј ͑cmϪ1
͒
oЈ ͑cmϪ1
͒
vЈ
2
3
4
5
6
7
8
9
31 725.55͑10͒
32 038.62͑13͒
32 344.02͑13͒
32 640.158͑73͒
32 931.19͑13͒
33 216.35͑18͒
33 485.03͑10͒
33 749.36͑24͒
33 998.813͑91͒
34 239.75͑10͒
34 453.183͑58͒
0.203 690͑87͒
0.201 698͑78͒
0.199 359͑90͒
0.197 475͑80͒
0.194 50͑17͒
0.190 82͑10͒
0.189 64͑23͒
0.186 51͑24͒
0.184 33͑15͒
0.179 20͑30͒
0.177 47͑10͒
3.82͑19͒
3.99͑11͒
3.31͑15͒
3.82͑15͒
3.35͑23͒
2.11͑31͒
5.36͑47͒
7.80͑53͒
4.53͑45͒
7.6͑5͒
198.976͑68͒
194.464͑93͒
191.409͑88͒
190.290͑90͒
186.094͑72͒
179.04͑12͒
177.53͑15͒
170.66͑15͒
165.60͑11͒
156.94͑13͒
136.69͑14͒
5.343͑94͒
6.34͑12͒
6.001͑82͒
5.342͑85͒
4.98͑11͒
4.52͑13͒
5.77͑13͒
5.54͑14͒
6.600͑37͒
5.30͑11͒
0
0.122͑68͒
0.199͑21͒
0.063͑12͒
0.148͑49͒
0.385͑63͒
0.037͑14͒
0.074͑67͒
0.28͑13͒
Ϫ0.042͑17͒
0.343͑70͒
0
0
2.40͑20͒
2.15͑12͒
0
0
0.39͑23͒
0
0
1.50͑18͒
0
0
1
0
1
1
1
2
0
Using considerations such as these a set of 172 param-
eters that are directly determined from our data was con-
structed, and a fit to these parameters ͑fit 1͒ produced con-
variation with v and it should also be possible to estimate the
matrix elements connecting the different vibronic states from
the vibrational wave functions. This is part of the way to-
ward directly fitting the underlying B and BЉ potential en-
ergy curves and r-dependent electronic interaction param-
eters rather than the individual fine-structure constants for
each state.
stants and perturbation values given in Tables IV–VI. This
Ϫ1
gave a good fit, with a standard deviation of 0.064 cm
.
D. Parameter model
The parameters that need to be estimated are o, ␣, and ␤.
Estimating the parameter o is straightforward as it should
show a smooth variation with vibrational state. We assumed
The drawback with fit 1 is that there are several param-
eters that we expect to be significant, but which cannot be
determined from the data. For example, the ␣ perturbation
parameter will act between all B and BЉ states with the same
a linear relationship with v, which reproduced the observed
Ϫ1
values to 0.4 cm
.
⍀
component. However it can only be well determined be-
The modeling of the perturbation parameters is not so
straightforward but, as suggested above, is actually rather
important as interactions with states more than one vibra-
tional quantum away can be significant. The parameters to be
determined are the matrix elements between vibronic states,
such as
tween those states with avoided crossings. For those cases
where states are close but do not cross the calculated energy
levels will be changed by omission of this parameter. The
changes can be dealt with by adjusting the values of other
constants, but this will give difficulties in interpretation of
these parameters. We therefore also provide an alternative fit,
fit 2, in which as many as possible of the significant param-
eters are either floated or set to estimated values. The basis of
the estimation is that the parameters in the Hamiltonian
should show a smooth variation with r. The constants for the
individual vibrational states should then show a smooth
1
2
ˆ
␣
ϭ
͗
BЉ⌳ϭ1,vЉ
͉
͑ AϩB͒Lϩ͉B⌳ϭ0,v
͘
.
͑14͒
If we initially assume that the electronic operator mixing the
two states is independent of r then the parameter will be
proportional to the overlap of the vibrational wave functions,
1
2
ˆ
TABLE VI. Perturbations determined from fit 1. Figures in parentheses are
three standard deviations.
␣ϭ
͗
vЉ
͉
v
͗͘BЉ⌳ϭ1
͉
͑ AϩB͒Lϩ͉B⌳ϭ0
͘
.
͑15͒
BvЈ–BЉvЈ
␣ ͑cmϪ1
͒
␣D ͑10Ϫ4 cmϪ1
͒
␤ ͑cmϪ1
͒
␤D ͑10Ϫ6 cmϪ1
͒
0
0
1
1
2
2
2
3
3
4
4
5
–2
–3
–3
–4
–4
–5
–6
–6
–7
–7
–8
–9
14.699͑71͒
Ϫ4.65͑37͒
18.36͑29͒
Ϫ11.289͑60͒
18.56͑59͒
Ϫ15.35͑11͒
10.77͑23͒
Ϫ17.980͑81͒
11.16͑22͒
Ϫ25.27͑49͒
14.78͑14͒
16.29͑11͒
Ϫ11.59͑21͒
•••
2.69͑45͒
Ϫ6.10͑58͒
Ϫ7.0͑14͒
Ϫ4.82͑29͒
Ϫ28.0͑17͒
Ϫ3.71͑29͒
•••
Ϫ0.0414͑63͒
Ϫ0.0635͑87͒
Ϫ0.0808͑18͒
0.0241͑15͒
Ϫ0.0497͑30͒
0.0492͑62͒
0.0240͑43͒
0.0776͑68͒
0.0066͑55͒
0.0369͑24͒
Ϫ0.0514͑53͒
Ϫ0.069͑12͒
Ϫ0.0037͑48͒
Ϫ0.0627͑84͒
0.0693͑71͒
0.0237͑20͒
Ϫ1.27͑50͒
13.0͑13͒
•••
1.70͑39͒
Ϫ22.3͑31͒
0.86͑62͒
•••
•••
•••
7.11͑74͒
26.4͑26͒
1.47͑53͒
•••
Ϫ5.1͑11͒
•••
Ϫ11.8͑18͒
•••
•••
•••
13.3͑22͒
•••
•••
5
6
6
6
–10
–10
–11
–12
Ϫ13.45͑14͒
11.84͑25͒
Ϫ4.62͑97͒
•••
FIG. 9. Plot of observed ␣ values against those calculated from Eq. ͑15͒
using the parameters in Table IX. The experimental errors in ␣ are less than
the size of the points.
•••
J. Chem. Phys., Vol. 104, No. 3, 15 January 1996
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858
M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
TABLE VII. Coefficients from least-squares fit of observed against calcu-
TABLE IX. Calculated ␤ parameters ͑in cmϪ1͒ set as fixed values in fit 2.
BЉv
lated ␣ and ␤ values using Eq. ͑16͒ with r ϭ2 Å. Figures in parentheses are
e
one standard deviation.
Bv
5
8
9
12
Coefficients
␣
␤
a0/cmϪ1
1
3
4
5
6
Ϫ0.004
Ϫ0.051
29.9 ͑1.4͒
187 ͑24͒
Ϫ421͑110͒
Ϫ0.126 ͑52͒
Ϫ3.33 ͑56͒
Ϫ1 Ϫ1
0.003
0.039
a1/cm
a2/cm
Å
Ϫ1 Ϫ2
0.009
Å
Ϫ0.008
We therefore calculated the Franck–Condon factors between
the BЉ and B states from our data. This required deriving
RKR potentials for both the B and the BЉ states from our
experimentally determined rotational and vibrational con-
stants, using the program AAEE.25 The program LEVEL was
then used to generate vibrational wave functions from the
RKR potentials for both states and also the Franck–Condon
factors between the vibrational levels of the two potentials.
These calculated Franck–Condon factors were found to
be broadly proportional to the measured ␣ values. However
by including an r dependence in the electronic operator a
better fit could be obtained, i.e., by modeling the ␣ values as
sary as the errors in the ␤ values were significantly larger
than for the ␣ values. This fit was not as satisfactory ͑average
Ϫ1
Ϫ1
error 0.009 cm in parameters of ϳ0.01–0.06 cm ͒ though
the correct sign is predicted for all except one case
(B0–BЉ3). This and three other points that were modeled
poorly were excluded from the fit. For the ␤ values a reduced
precision in the fit compared to the fit to the ␣ values is
acceptable since the ␤ perturbations are so weak except at
high J. Their long-range effects are thus negligible and very
few need to be included as fixed values in the fit. The coef-
ficients for the least-squares fit of calculated against observed
26
␤
values are given in Table VII, and the calculated ␤ values
2
␣
ϭ
͗
vЉ
͉
a ϩa ͑rϪr ͒ϩa ͑rϪr ͒
͉
v
͘
.
͑16͒
0
1
e
2
e
that were included in the fit are given in Table IX.
A small least-squares program was written to find the best
The calculated values of ␣, ␤, and o were then held fixed
while allowing the same parameters as before to float. The
entire process of estimating parameters was iterated several
times as the parameters from the revised fit were used to
produce a new estimate of the undetermined parameters. The
values of the coefficients a , a , and a using the vibrational
0
1
2
wave functions from LEVEL and the values of ␣ that were
well determined by the line position fit. This gave an excel-
Ϫ1
lent fit to the ␣ values ͑average error 0.36 cm in param-
Ϫ1
Ϫ1
eters of 10–20 cm ͒, as shown by Fig. 9.
final fit, fit 2, has a standard deviation of 0.064 cm , essen-
The coefficients obtained from the least-squares fit to
Eq. ͑15͒ against observed values of ␣ are given in Table VII
below. Using this polynomial, the ␣ values for all the pos-
sible BЉ–B pairs could then be calculated and included in
the fit as fixed parameters. These calculated values of ␣ are
given Table VIII. It is clear from these calculations that the
interactions between BЉvϭ0 and 1 and Bvϭ0 and 1 are
significant as the matrix elements between them are large.
For this reason the BЉvϭ0 and 1 were included in fit 2,
although they were not observed experimentally and their
positions had to be estimated by extrapolating the origin, B,
A, and ␭ from the higher states.
tially the same as the previous fit. However, the molecular
constants for this fit, given in Tables IX–XI, show a
smoother variation with vibrational state than both the pre-
vious fit and the constants prior to deperturbation. An ex-
ample of this can be seen in Fig. 12͑a͒ where three sets of BЉ
state origins are plotted against vibrational state. The highest
state in our fit, vЈϭ6 for the B state and vЈϭ11 and 12 for
the BЉ state are necessarily only partially deperturbed in our
fit as we have excluded higher states, though the parameters
derived from these states seem on the whole to be in line
with those of other states.
To complete the model, some of the ground state con-
stants for vЉϭ2–4 and 7 were also floated as our data al-
lowed a better determination of these constants. The values
for vЉϭ0 and 1 were not floated as they are better deter-
A similar analysis was performed for the ␤ perturbation
parameters, in this case using a fit weighted by the uncer-
tainty in the experimental ␤ values. This was deemed neces-
TABLE VIII. Calculated ␣ parameters ͑in cmϪ1͒ set as fixed values in fit 2.
BЉv
Bv
0
1
2
3
4
5
6
7
8
9
10
11
12
0
1
2
3
4
5
6
36.27
28.54
Ϫ27.27
11.78
2.15
Ϫ0.49
4.90
Ϫ25.45
Ϫ6.50
17.87
Ϫ1.51
8.28
0.10
Ϫ3.26
Ϫ16.78
Ϫ16.66
21.33
Ϫ5.68
0.65
Ϫ5.72
18.17
4.86
1.78
Ϫ8.63
0.19
3.59
11.76
12.63
Ϫ17.47
Ϫ0.48
6.03
3.56
Ϫ14.36
Ϫ4.65
Ϫ1.79
8.80
Ϫ9.51
14.88
J. Chem. Phys., Vol. 104, No. 3, 15 January 1996
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M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
859
mined from the Fourier transform infrared data.21 The final
values, given in Table XII, are consistent with previous
values.
included in the basis. Their omission leads to changes of 2
Ϫ1
Ϫ1
cm in ␭ and 10 cm in the origin for vЈϭ0 of the B state.
This is an extreme case as our calculations ͑Table VIII͒ re-
veal that the 0–0 matrix element is the largest and the higher
v levels have smaller matrix elements. We nevertheless ex-
pect some truncation errors for vЈϭ6 for the B state and
vЈϭ12 for the BЉ state and possibly for lower states. Deter-
mination of these must await analysis of the higher vibronic
levels.
1
6
Full details of both fits, including a tabulation of both the
observed and calculated line positions and a correlation ma-
trix for the determined parameters is available from the au-
thors. As mentioned above, both fits reproduce all the 3320
observed transitions well, with a maximum error of 0.4
Ϫ1
cm . The greatest discrepancies are typically at the highest
observed J for a particular state, where we do not model in
detail crossings at still higher values of J.
It should be emphasized that both the basis truncation
errors and the parameter estimation errors do not affect the
calculated position of any of the observed levels but only the
prediction of the unobserved levels.
The error bars given in the tables are taken directly from
the least-squares fits. However, for fit 2 we must consider
possible systematic errors due to uncertainties in the esti-
mated parameters and truncation of the vibronic basis. For
example, our model for the ␣ perturbation parameters repro-
E. Lifetimes
The work of Matsumi et al.14 suggested that there was
no intrinsic intensity to the BЉ state, and so the intensity of
any BЉ transition was due to mixing in of B state character.
This mixing can be calculated from our model, and thus
intensity measurements can give an independent check on
our model. It is difficult to measure absolute intensities of
different vibrational bands using laser-induced fluorescence,
but equivalent information can be obtained from the fluores-
cence lifetimes of the states concerned. The excited state can
only fluoresce to the ground state so the lifetime should be
inversely proportional to the B character of each state ͑as-
suming the transition moment is independent of r͒:
Ϫ1
duced the measured values to Ϯ0.36 cm . If a change is
Ϫ1
made to any one of the estimated ␣ values of 0.36 cm and
fit 2 repeated, the largest change seen is in the origins of the
states connected by that particular parameter, and is less than
twice the quoted error bars for the origins. A similar proce-
dure for the ␤ perturbation parameters also found the largest
changes in the other parameters to be within the original
error bars. Our estimation of the position of the BЉvЈϭ0 and
1
levels also has an effect on the constants we determine, but
again the effect is small. A change in the origin of BЉvЈϭ1
Ϫ1
Ϫ1
by 10 cm produces changes of 0.01 cm in ␭ and 0.2
Ϫ1
cm in the origin for BvЈϭ0. The estimated parameter with
the greatest uncertainty is o, as an adjustment in this param-
eter feeds through directly to a correction to the origin, A,
and ␭ for the BЉ state involved. However, our estimates for o
are less precise than for the other estimated parameters, lead-
ing to a possible systematic error in the origin, A, and ␭ for
the BЉ states up to 0.4 cm . The effect of this uncertainty on
the other parameters will be less than their error bars as for ␣
and ␤.
We can apply a similar procedure to estimate the effect
of truncating the vibronic basis at vЈϭ6 for the B state and
vЈϭ12 for the BЉ state. We comment above that interactions
with BЉvЈϭ0 and 1 are large and these states are therefore
2
␶fϰ1
͚i ͑c ͒,
͑17͒
Ͳ
i
where c is the coefficient of the ith state in our basis in the
i
actual state and the sum extends over only the B states in the
basis. This sum will depend on J as well as ⍀ and v, but as
the mixing in the low J levels for which lifetimes were mea-
sured is dominated by the J-independent spin–orbit pertur-
bation the choice of J is not normally important.
Ϫ1
In fact Matsumi et al.14 derived homogeneous spin–orbit
matrix elements from their lifetime data and band positions
by using a model containing two or, in some cases, three
TABLE X. Deperturbed constants from fit 2 for the BЉvЈϭ0–12 states. Figures in parentheses are three standard deviations.
Tv,0 ͑cmϪ1
͒
BЈ ͑cmϪ1
͒
DЈ ͑10Ϫ7 cmϪ1
͒
AЈ
␭Ј
␥Ј
oЈ
vЈ
a
a
3.5a
198.5a
5.5a
0a
0
a
a
0
1
2
3
4
5
6
7
8
9
31 069
31 402
0.208
0.206
2
2
a
a
3.5a
197.5a
5.5a
a
31 729.27͑10͒
32 039.70͑13͒
32 344.732͑53͒
32 642.155͑68͒
32 933.03͑16͒
33 214.074͑67͒
33 486.19͑12͒
33 749.22͑24͒
34 000.272͑90͒
34 239.34͑12͒
34 460.292͑43͒
0.203 549͑88͒
0.201 702͑79͒
0.199 143͑84͒
0.197 315͑80͒
0.194 41͑17͒
0.190 66͑11͒
0.189 71͑24͒
0.186 31͑24͒
0.184 03͑15͒
0.178 89͑31͒
0.176 09͑48͒
3.74͑19͒
4.04͑12͒
3.27͑15͒
3.79͑15͒
3.25͑25͒
0.65͑38͒
5.53͑48͒
7.42͑55͒
4.33͑46͒
6.5͑15͒
195.315͑67͒
193.698͑94͒
190.986͑57͒
188.430͑90͒
184.203͑93͒
180.324͑68͒
176.54͑18͒
170.85͑15͒
164.76͑10͒
157.36͑12͒
148.484͑55͒
5.237͑91͒
5.88͑12͒
5.624͑36͒
5.165͑76͒
5.07͑11͒
6.021͑30͒
5.52͑15͒
5.49͑14͒
5.794͑38͒
5.32͑13͒
5.5a
0.105͑66͒
0.191͑21͒
0.048͑11͒
0.190͑40͒
0.486͑68͒
0.059͑14͒
0.113͑67͒
0.27͑13͒
1.7a
1.36͑20͒
1.681͑96͒
1.2
1.0
1.60͑14͒
0.6
0.4
0.21͑18͒
a
a
a
a
1
0
Ϫ0.022͑17͒
0.149͑89͒
0.0a
a
1
1
0.1
0.0
a
1
2
Ϫ10.1͑21͒
aEstimated values; see the text.
J. Chem. Phys., Vol. 104, No. 3, 15 January 1996
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860
M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
TABLE XI. Deperturbed constants from fit 2 for the BvЈϭ0–6 states. Figures in parentheses are three standard
deviations.
Tv,0 ͑cmϪ1
͒
BЈ ͑cmϪ1
͒
DЈ ͑10Ϫ7 cmϪ1
͒
␭Ј ͑cmϪ1
͒
␥Ј ͑cmϪ1
͒
vЈ
0
1
2
3
4
5
6
31 672.426͑77͒
32 102.71͑23͒
32 525.644͑34͒
32 943.80͑18͒
33 359.406͑85͒
33 766.65͑22͒
34 170.925͑54͒
0.224 647͑54͒
0.223 429͑80͒
0.222 471͑74͒
0.219 572͑83͒
0.219 52͑16͒
0.216 068͑96͒
0.214 02͑13͒
2.571͑68͒
2.75͑11͒
4.35͑13͒
3.03͑14͒
6.74͑53͒
3.16͑22͒
3.24͑39͒
4.591͑96͒
4.724͑75͒
3.42͑11͒
3.88͑14͒
2.99͑23͒
2.96͑16͒
1.12͑15͒
Ϫ0.0324͑19͒
Ϫ0.0226͑11͒
Ϫ0.0036͑22͒
0.0022͑32͒
Ϫ0.0098͑52͒
0.0035͑39͒
Ϫ0.0133͑38͒
interacting states. The matrix elements they derive from this
simple model are consistent with ours, though their error
bars are larger as the lifetime data are less precise. This is
shown by Fig. 10, where the matrix elements from the two
models are plotted against each other. In Fig. 11 we plot the
lifetimes measured in our work as well as those of Matsumi
f parity ⍀ϭ0 levels but these are always coincident with the
much stronger e parity ⍀ϭ0 levels and therefore difficult to
observe.
We can now use our model to calculate the intensity of
all the individual rovibronic transitions. The relative intensi-
ties of the B–X bands are available as Franck–Condon fac-
1
4
18
et al. against our calculated B state character for ⍀ϭ0 and
states. The linearity of these plots demonstrates that Eq.
17͒ holds for both the B and BЉ states and the assumption of
tors from the work of Anderson et al. The direct BЉ–X
1
͑
contributions are included by using Franck–Condon factors
calculated from the known wave functions for the BЉ and X
states. These must be scaled down by the ratio of electronic
transition moments which we estimate from the state life-
times; for the B state we use the 32 ns calculated above and
for the BЉ state we use 4160 ns, the average of our measure-
ments. The relative intensities within each band, and the in-
tensity arising from the BЉ–B mixing are calculated from the
wave functions given by our model. This process also gives
the lifetime of each rotational state. A table of calculated line
strengths, positions, and upper state lifetimes is available
from the authors.
an r-independent transition moment is reasonable. The inter-
cept ͑at 100% B character͒ of these plots at 32Ϯ1 ns gives
the lifetime of a pure B state.
This model would predict a very low intensity for the
⍀ϭ2 components of the BЉ state. This is because the ⍀ϭ2
components can only mix with the B state via the
L-uncoupling mixing term ͓Eq. ͑6͔͒, which obeys selection
3
rules ⌬⍀ϭ1 and is ϳ10 times weaker than the spin–orbit
term for low J. Our model predicts a B state character of
Ϫ5
10
in typical ⍀ϭ2 states, corresponding to a lifetime of
ϳ3 ms. Our observations indicated that transitions to the
ϭ2 states were indeed weak, requiring higher laser powers
These calculated line strengths and positions should be
adequate to describe experimental spectra of the B and BЉ
states up to our highest observed J for each vibrational state.
However several points should be noted when using the cal-
culated line strengths in conjunction with experimental meth-
ods. No corrections should be needed for simple absorption
measurements, but if laser-induced fluorescence is used then
⍀
than the ⍀ϭ0 or 1 components, but the lifetimes of ϳ4 ␮s
indicate that there is in fact a small transition moment direct
to the BЉ state. It is sufficiently weak that it will not alter the
model above for the stronger transitions. In principle this
means that it should be possible to observe transitions to the
TABLE XII. Perturbations determined from fit 2. Figures in parentheses are three standard deviations.
BvЈ–BЉvЈ
␣ ͑cmϪ1
͒
␣D ͑10Ϫ4 cmϪ1
͒
␤ ͑cmϪ1
͒
␤D ͑10Ϫ6 cmϪ1
͒
0
0
1
1
2
2
2
3
3
4
4
5
–2
–3
–3
–4
–4
–5
–6
–6
–7
–7
–8
–9
14.874͑70͒
Ϫ5.02͑38͒
18.43͑29͒
Ϫ11.291͑60͒
•••
Ϫ15.27͑10͒
•••
Ϫ18.088͑85͒
10.69͑21͒
•••
14.49͑16͒
16.24͑12͒
Ϫ11.36͑22͒
•••
Ϫ13.41͑16͒
13.35͑40͒
2.50͑45͒
Ϫ5.51͑60͒
Ϫ7.6͑14͒
Ϫ4.88͑29͒
Ϫ29.3͑13͒
Ϫ3.79͑29͒
3.61͑32͒
•••
8.30͑71͒
48.1͑42͒
2.43͑50͒
•••
Ϫ0.040 4͑62͒
Ϫ0.052 5͑89͒
Ϫ0.080 8͑18͒
0.029 55͑97͒
Ϫ0.041 1͑21͒
0.056 6͑39͒
0.017 1͑47͒
0.086 2͑80͒
Ϫ0.002 9͑58͒
0.050 9͑61͒
Ϫ0.027 9͑68͒
Ϫ0.067͑13͒
0.000 7͑51͒
Ϫ0.033 7͑77͒
0.055 0͑94͒
•••
Ϫ1.08͑50͒
11.5͑13͒
•••
•••
Ϫ26.9͑20͒
•••
•••
•••
Ϫ9.6͑19͒
6.2͑36͒
Ϫ5.4͑16͒
•••
12.1͑24͒
•••
5
6
6
6
–10
–10
–11
–12
Ϫ6.3͑11͒
•••
Ϫ5.74͑96͒
•••
•••
•••
J. Chem. Phys., Vol. 104, No. 3, 15 January 1996
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M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
861
change. A shift such as this would be caused by the presence
of a nearby ⍀ϭ0 or ⍀ϭ1 component of a BЉ state which
would interact with the B state via the homogeneous spin–
orbit interaction. The value of ␭ for the B state would there-
fore vary depending on which BЉ states were close to it and
whether they lay at higher or lower energy. However, after
the interactions were included in the model, the deperturbed
value of ␭ showed little variation with vibrational level. We
also give in Fig. 12 plots of band origins and B values for the
B state, and A and B values for the BЉ state. These plots
show that all of the deperturbed constants follow smooth
trends when compared to the values before deperturbation.
The plots in Fig. 12 show that the deperturbation proce-
dure, in addition to smoothing the vibrational state depen-
dence of the constants, also shifts some of them by signifi-
cant amounts. Note for example the rotational constants for
the B states which are consistently shifted down with respect
to the deperturbed values, while the band origins lie on either
side of the deperturbed values. The reasons for this behavior
can be seen by considering just the J-independent mixing
between the states which is normally dominant. Second-
order perturbation theory indicates that this will shift the
FIG. 10. Plot of matrix elements determined in this work ͑fit 2͒ against
matrix elements calculated by Matsumi et al. ͑Ref. 14͒ from lifetime data.
The line yϭx is plotted to show the expected relationship. A sample error
bar of one standard deviation is given for the matrix elements of Matsumi
et al. ͑Ref. 14͒, estimated from errors in their measured lifetimes. The errors
in fit 2 are negligible on the scale of the plot.
the varying lifetime of the upper states can affect the ob-
served intensity. If all the fluorescence is collected there is no
problem, but a typical experiment will sample the fluores-
cence over a finite time interval or there could be collisional
quenching of the fluorescence. Both of these factors will usu-
ally discriminate against the longer lived states so a correc-
tion must be applied. The lifetime of the upper state involved
in each transition can be calculated from our model, and is
included in our table, so if the other factors are known the
correction can be calculated. Alternatively the tables can be
used to select lines with a reasonably constant upper state
lifetime. Note that our room temperature spectra ͑Figs. 5 and
2
origins of the states by Ϯ␣ /⌬ where ⌬ is the separation
between states, depending on whether the interacting state is
above or below the B state. However this interaction will
6͒ were necessarily taken at pressures where there would be
significant fluorescence quenching, so we do not expect
quantitative agreement between the simulated and observed
intensities, though the match is reasonable.
V. DISCUSSION
Due to the strong perturbations in the system, many of
the molecular constants for the B and BЉ states before dep-
erturbation varied irregularly with vibrational quantum num-
ber. This is because the constants were evaluated assuming
no interactions between the two electronic states. However,
by including the interactions in the model, the deperturbed
constants obtained varied smoothly with vibrational quantum
number, following simple analytical expressions as expected.
As shown earlier ͓Fig. 12͑a͔͒ the origins for the BЉ state
vary more smoothly with vibrational level after deperturba-
tion than before, with fit 2 showing a smoother trend than fit
1. The difference in the behavior of the constants before and
after deperturbation is perhaps most dramatic for the spin–
spin splitting constant ␭ for the B state shown in Fig. 12͑f͒,
which behaved erratically before deperturbation, changing
sign a number of times between with increasing vibrational
FIG. 11. Fluorescence lifetimes ͑␶ ͒ of ͑a͒ the B states and ͑b͒ the BЉ states
f
plotted against percentage of B state character calculated according to
8
13
2
⌺i(ci) where ci are the coefficients of the B state basis functions. The
level, as shown by Meyer and Crosley and Matsumi et al.
Since the splitting between the ⍀ϭ0 and ⍀ϭ1 components
2
nonlinear scale arises because ␶f is inversely proportional to ⌺ (c ) as
i
i
given in Eq. ͑17͒. The two filled symbols are lifetimes measured in this
work and the remainder are lifetimes measured by Matsumi et al. ͑Ref. 14͒.
The BЉ ⍀ϭ2 components are not included in these plots.
3
Ϫ
u
of the B ⌺ state is approximately 2␭, a shift in energy of
either component would cause the effective value of ␭ to
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862
M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
FIG. 12. Graphs of molecular constants before and after deperturbation against vibrational state. The constants before deperturbation ͑dashed lines͒ are taken
from Matsumi et al. ͑Ref. 13͒ and the constants after deperturbation ͑solid lines͒ are taken from this work ͑fit 2͒. ͑a͒ Band origins for the BЉ state. The dotted
Ϫ1
line represents the values after deperturbation using fit 1. The Matsumi data have been shifted down by 342 cm to correct for a difference in definition of
the band origins. ͑b͒ Band origins for the B state. ͑c͒ Rotational constants for the B state. ͑d͒ Rotational constants for the BЉ state. For the Matsumi et al. data
͑Ref. 13͒ values for both the ⍀ϭ0 ͑dashed line͒ and the ⍀ϭ1 ͑dotted line͒ states are plotted. ͑e͒ Spin–orbit splitting constants (A) for BЉ state. ͑f͒ Spin–spin
splitting constants ͑␭͒ for the B state.
also cause a correction to B as the separation between the
states, ⌬, changes with J as the rotational constants in the B
and BЉ states differ. The correction to the energy of the B
state is of the order of
equation that although the sign of the correction to the band
origin changes, the sign of the correction to B is always
negative. This is because a BЉ state below a B state will raise
the lower J states, lowering the effective value of B, while a
BЉ state above will depress the higher J states, again lower-
ing the effective value of B.
ϯ␣2
ϯ␣2
ϭ
⌬
͉
⌬0
͉
Ϯ͑B ϪB ͒J͑Jϩ1͒
B
B
Љ
A slight discontinuity appears in the B state rotational
constant around the levels vЈϭ3 and vЈϭ4 as well as in the
BЉ constants of B and D which show irregularities at vЈϭ7.
These irregularities may be caused by another electronic
state in this region perturbing either the B or the BЉ states.
There are three known states of u symmetry below the B and
␣2
␣2
Ϸϯ
Ϫ
͑B ϪB ͒J͑Jϩ1͒,
͑18͒
2
B
BЉ
⌬0
⌬0
where ⌬ is the energy difference between the interacting B
0
and BЉ levels at Jϭ0. The upper signs correspond to a BЉ
level situated above a B level, and the lower signs corre-
spond to a BЉ level below a B level. It can be seen from this
3
3
ϩ
1
Ϫ
27,28
BЉ states, the AЈ ⌬ , A ⌺ , and c ⌺ states,
all three
u
u
u
3
of which could perturb the BЉ ⌸ state via spin–orbit mix-
u
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M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
863
ing. The origins of these states lies some 10 000 cmϪ1 below
the B and BЉ states, so only the high vibrational levels of the
A, AЈ, and c states can give any effect. The vibrational over-
lap between the states will therefore be small so perturba-
tions will only be significant where there is a close resonance
between the energy levels. This is consistent with our obser-
vation of a single irregularity.
Since this deperturbation analysis only involves the vi-
brational levels of the B state vЈϭ0–6 and the BЉ vЈϭ2–12,
interactions between the B vЈϭ6 and higher BЉ states as well
as those between BЉ vЈϭ12 and higher B states have not
been included. The highest B and BЉ vibrational levels in this
work are therefore only partially deperturbed and the con-
stants for the states should be treated as preliminary. This has
led to anomalous values for their constants. For example, the
deperturbed value of ␭ for the B state vЈϭ0–5 ranges be-
tween 3.0 and 4.6, whereas the value for vЈϭ6 is 1.1.
1
4
2
1
3
Ϫ
u
•
••5␴ 2␲ 2␲ 5␴
͑ ⌺ ͒.
͑22͒
g
u
g
u
This would give a nonzero interaction matrix element with
the BЉ configuration shown above. It is also possible to get
configuration interaction in the BЉ state and, while this is not
considered explicitly by Swope et al., configuration mixing
is found in both of the analogous states in O2.33 Mixing with
the configuration:
2
g
2
u
3
g
1
u
3
•
••5␴ 2␲ 2␲ 5␴
͑ ⌸ ͒
͑23͒
u
would give spin–orbit interaction with configuration ͑19͒
and would if anything reduce the BЉ–X transition dipole
moment from the ground state as the transition
2
2
3
1
2
4
2
5
␴ 2␲ 2␲ 5␴ –5␴ 2␲ 2␲ is orbitally forbidden.
g u g u g u g
The matrix elements between the various states can be esti-
mated using the methods discussed by Lefebvre-Brion and
Field22 ͑Sec. 2.4.2͒ by expressing the spin–orbit operator in
its many electron form,
ˆ
ˆ
i
H
SOϭ͚i
al • sˆ .
͑24͒
i
VI. THE MOLECULAR CONFIGURATION QUESTION
During this work, uncertainties about the molecular con-
figurations of the B and BЉ states became apparent. Of the
The single electron spin–orbit coupling constant a, may be
Ϫ1
estimated at around 340 cm , given spin–orbit coupling
Ϫ1
few ab initio calculations performed on S , those by Swope
constants in the BЉ state of around 170 cm ͑see Table X͒.
2
2
9
et al. remain the most relevant to this work. They carried
The spin–orbit matrix elements between the various configu-
rations work out at 0.15a–0.35a, depending on the exact
states involved, and so our observed values of ␣ can be ac-
counted for with configuration mixing of the order of 5%–
35%. As the observed ␣ values are r dependent, the mixing
is also presumably r dependent. This is born out by the sig-
nificant variation in A with v, suggesting the configuration of
the BЉ state changes with r.
To shed more light on the configuration we performed
some multireference configuration interaction calculations
using MOLPRO.34 Using a large basis set ͑AVQZ͒ we could
reproduce the form of the potential energy curves around the
equilibrium region quite accurately, though there was signifi-
cant deviation from the RKR curves above 2.6 Å. The cal-
culation gave transition dipole moments at 2.3 Å of 1.91 D
for B–X and 0.090 D for BЉ–X, corresponding to lifetimes
of 27 ns and 12 ␮s, in reasonable agreement with our obser-
vations of 32 ns and 4.16 ␮s. The same calculation gave the
following natural orbital occupancies:
out self-consistent field and configuration interaction type
calculations on several low lying bound states of S , includ-
2
ing the B and BЉ states, predicting that the BЉ state lay about
Ϫ1
4
00 cm below the B state. The accepted configurations of
29
the three states as given by Swope et al. are as follows:
3
Ϫ
u
2
3
3
B ⌺
•••5␴ 2␲ 2␲ ,
͑19͒
͑20͒
͑21͒
g
u
g
3
2
4
1
1
BЉ ⌸
•••5␴ 2␲ 2␲ 5␴ ,
g u g u
u
3
Ϫ
g
2
4
2
X ⌺
•••5␴ 2␲ 2␲ .
g u g
The relative intensities of the B–X and BЉ–X transitions
predicted from these configurations agree well with experi-
ment. As discussed by Herzberg30 and Mulliken,
31,32
a paral-
lel transition from a bonding ␲ orbital to the corresponding
u
antibonding ␲g orbital, such as for the B–X transition,
should have considerable intensity. However, a transition in-
volving a ␲ to ␴ promotion, such as for the BЉ–X transition,
where the transition moment is perpendicular to the internu-
clear axis and the electron goes from one antibonding orbital
to another antibonding orbital, is expected to have a much
lower transition dipole moment. This is consistent with the
relative transition dipole moments estimated from fluores-
cence lifetime measurements in this work.
5
1.73
1.92
␴g
2␲u
3.08
3.55
2␲g
2.80
1.39
5␴u
0.28
1.03
3
Ϫ
B ⌺
u
3
BЉ ⌸u
These are consistent with both states undergoing significant
configuration mixing, along the lines suggested above.
However, interpretation of the data presented in this
work has established that there is strong spin–orbit mixing
between the B and BЉ states, which would not be possible
with these configurations. For any mixing to occur via the
one electron spin–orbit operator, the B and BЉ molecular
configurations must differ by only one spin orbital. This
problem could be overcome by including an additional con-
figuration for either the B or the BЉ state and, in fact, Swope
VII. CONCLUSIONS
3
Ϫ
A model of the interacting B ⌺u ͑vЈϭ0–6͒ and
3
BЉ ⌸u ͑vЈϭ2–12͒ states is presented here which fits the
positions of the 3320 experimentally measured lines to 0.064
Ϫ1
cm . This deperturbation analysis was carried out by a si-
2
9
et al. point to significant configuration interaction between
the configuration for the B state given above with
multaneous variation of all the constants and perturbations
defined by the data in appropriate Hamiltonian matrices. The
J. Chem. Phys., Vol. 104, No. 3, 15 January 1996
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864
M. E. Green and C. M. Western: Analysis of B 3⌺uϪ
8
deperturbed constants obtained vary smoothly with vibra-
tional number except for discontinuities in the constants be-
tween B state levels vЈϭ3 and 4, which we attribute to
weaker interactions with other states.
K. A. Meyer and D. R. Crosley, Can. J. Phys. 51, 2119 ͑1973͒.
V. E. Bondybey and J. H. English, J. Chem. Phys. 69, 1865 ͑1978͒.
C. R. Quick and R. E. Weston, J. Chem. Phys. 74, 4951, ͑1981͒.
P. Pati n˜ o and R. F. Barrow, J. Chem. Soc. Faraday Trans. 2 78, 1271
9
0
1
11
͑
1982͒.
12
Fluorescence lifetimes were measured for the BЉ states
and were found to be consistent with those of Matsumi
M. Heaven, T. A. Miller, and V. E. Bondybey, J. Chem. Phys. 80, 51
͑1984͒.
Y. Matsumi, T. Munakata, and T. Kasuya, J. Chem. Phys. 81, 1108 ͑1984͒.
Y. Matsumi, T. Suzuki, T. Munakata, and T. Kasuya, J. Chem. Phys. 83,
1
1
3
4
1
4
et al. These lifetime measurements provide an independent
verification of the state mixing in the system thereby con-
firming our model. Measurements of the fluorescence life-
times of the previously unseen ⍀ϭ2 components of the BЉ
state indicate a small intrinsic BЉ–X transition strength. This
allows us to construct a comprehensive model for the inten-
sities as well as the positions of all the rovibronic transitions.
The molecular configurations of the states involved have
been investigated and indicate configuration mixing for both
the B and BЉ states. Preliminary calculations using the pro-
gram MOLPRO have been carried out which support this pro-
posal as well as predicting transition dipole moments for the
two states which are consistent with experiment.
3798 ͑1985͒.
15
S. M. Harris, D. Phil. thesis, Oxford University, 1979; P. Pati n˜ o, D. Phil.
thesis, Oxford University, 1983; Y. de Vita Villamizar, D. Phil. thesis,
Oxford University, 1970; and unpublished data provided by J. M. Ketter-
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16
17
89, 391 ͑1993͒.
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1
8
9
W. R. Anderson, D. R. Crosley, and J. E. Allen, Jr., J. Chem. Phys. 71, 821
1979͒.
M. A. A. Clyne and P. H. Tennyson, J. Chem. Soc. Faraday Trans. 2 82,
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͑
1
20
M. A. A. Clyne and J. P. Liddy, J. Chem. Soc. Faraday Trans. 2 78, 1127
͑1982͒.
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͑
1986͒.
ACKNOWLEDGMENTS
H. Lefebvre-Brion and R. W. Field, Perturbations in the Spectra of Di-
atomic Molecules ͑Academic, New York, 1986͒
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We would like to thank the EPSRC for funding a stu-
dentship for M.E.G., and providing the parallel computer and
laser equipment used in this work. We are particularly grate-
ful to Dr. R. F. Barrow ͑Oxford͒ and his group including Dr.
J. M. Ketteringham, Dr. H. J. Hurst, Dr. J. M. Ricks, and Dr.
P. Pati n˜ o for providing details of their data and analyses. We
would also like to thank Gabriel Balint-Kurti for helpful dis-
cussions and assistance with the ab initio calculations and
Keith Rosser for help with the equipment.
͑
1972͒.
24
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27
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9
0
1
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͑1983͒.
4
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5
6
¨
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͑
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7
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J. Chem. Phys., Vol. 104, No. 3, 15 January 1996
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