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

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

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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

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

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

S₂vapor, 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

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

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

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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 Barrow¹¹for

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.¹²on 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 Barrow¹¹referred 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.¹⁴where

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 ⌺_ustate varied between

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_._G_r_e_e_n_a_n_d_C_._M_._W_e_s_t_e_r_n_:_A_n_a_l_y_s_i_s_o_f_B₃_⌺_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

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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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

3

4

5

6

7

8

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 ³S₂because 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-

4167^a

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 S₂signals.

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

393 ͑36͒

1661͑222͒

1729

b

two tests. First we measured lifetimes of the vϭ0 and 1 of

11

338 ͑21͒

358 ͑39͒

3

1

the A ⌸ state of SO; these are known to have very long

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_._G_r_e_e_n_a_n_d_C_._M_._W_e_s_t_e_r_n_:_A_n_a_l_y_s_i_s_o_f_B₃_⌺_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

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

Ϫ

͓

from the BЉ ⌸ –X ⌺_g͑9-0͒ 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_._G_r_e_e_n_a_n_d_C_._M_._W_e_s_t_e_r_n_:_A_n_a_l_y_s_i_s_o_f_B3⌺_u^Ϫ

853

ground state constants taken from Fourier transform infrared

spectra²¹for 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 2

ˆ ˆ

HϭB͑JϪLϪS͒ ϩAL•S

ˆ2 ˆ 2 ˆ 2

ˆ ˆ

ϭB͓J͑Jϩ1͒ϩS͑Sϩ1͒ϪJ ϪL ϪS ͔ϩAL S

z z

z

ˆ ˆ

1

2

ˆ ˆ

Ϫ

ϪB͓͑J L ͒ϩ͑J L ͔͒ϩ͑ AϩB͒

͕

L S ϩL S_ϩ͖

,

Ϫ

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.²²The minor terms are

TABLE II. Reﬁned constants for the X ³⌺ vЉϭ0–5, 7 levels. Figures in parentheses are three standard deviations.

Ϫ

g

T_v_,₀͑cm^Ϫ¹

͒

BЉ ͑cm^Ϫ¹

͒

DЉ ͑10^Ϫ⁷cm^Ϫ¹

͒

␥Љ ͑cm^Ϫ¹

͒

␭Љ ͑cm^Ϫ¹

͒

␭Љ ͑10 cm^Ϫ¹͒

5

XvЉ

D

a

1.96 ͑2͒^a

1.97 ͑2͒^a

1.96 ͑17͒

1.97 ͑12͒

2.046͑62͒

2.12^b

Ϫ0.007 157 ͑9͒^a

Ϫ0.007 148͑28͒^a

Ϫ0.005 93 ͑91͒

Ϫ0.006 6 ͑14͒

Ϫ0.007 22 ͑51͒

Ϫ0.007 283^b

11.7931 ͑4͒^a

11.8659͑10͒^a

11.937 ͑14͒

12.021 ͑20͒

12.091 ͑19͒

12.163^b

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_._G_r_e_e_n_a_n_d_C_._M_._W_e_s_t_e_r_n_:_A_n_a_l_y_s_i_s_o_f_B₃_⌺_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

͉

H

ˆ

͉

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

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 ⌸₂vЈϭ4

,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

͗

BЉ⌳ϭ1

͉

␣Ј L_ϩ͉B⌳ϭ0

͘

,

.

͑12͒

͑13͒

D

␤Ј L_ϩ͉B⌳ϭ0

D

ˆ

ˆ 4

2

3

ˆ 2 ˆ2

1

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 e^ϩ²ⁱ^␾͒,

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 L_Ϯin 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

͗

BЉ⌳ϭ1

͉

͑ AϩB͒L_ϩ͉B⌳ϭ0

͘

,

͑3͒

͑4͒

ˆ

BL_ϩ͉B⌳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⌺

͘

Ϯ͑Ϫ1͒^J^Ϫ^S^ϩ^s

͉

JϪ⍀␩Ϫ⌳SϪ⌺

,

͑5͒

͘

͖

J. Chem. Phys., Vol. 104, No. 3, 15 January 1996

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_M_._E_._G_r_e_e_n_a_n_d_C_._M_._W_e_s_t_e_r_n_:_A_n_a_l_y_s_i_s_o_f_B3⌺_u^Ϫ

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

Pati n˜ o and Barrow^c

Local ﬁt^a

Global ﬁt^b

e levels

f levels

T_v_,₀͑cm^Ϫ¹

͒

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 ͑cm^Ϫ¹

͒

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 ͑cm^Ϫ¹

͒

BvЈϭ1

BЉvЈϭ3

BЉvЈϭ4

4.922 ͑90͒

8.942 ͑90͒

7.077 ͑45͒

4.724͑75͒

5.88͑12͒

5.624͑36͒

AЈ ͑cm^Ϫ¹

͒

BЉvЈϭ3

BЉvЈϭ4

195.815 ͑69͒

192.561 ͑81͒

193.698͑94͒

190.986͑57͒

␣

␤

͑cm^Ϫ¹

͒

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 ͑cm^Ϫ¹

BЉvЈϭ3

BЉvЈϭ4

0

1.2

͑6͒

1.695 ͑18͒

q ͑cm^Ϫ¹

͒

BЉvЈϭ3

BЉvЈϭ4

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 T_v_,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_._G_r_e_e_n_a_n_d_C_._M_._W_e_s_t_e_r_n_:_A_n_a_l_y_s_i_s_o_f_B₃_⌺_uϪ

TABLE IV. Deperturbed constants from ﬁt 1 for the BvЈϭ0–6 states. Figures in parentheses are three standard deviations.

T_v_,₀͑cm^Ϫ¹

͒

BЈ ͑cm^Ϫ¹

͒

DЈ ͑10^Ϫ⁷cm^Ϫ¹

͒

␭Ј ͑cm^Ϫ¹

͒

␥Ј ͑cm^Ϫ¹

͒

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 Barrow¹¹in 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 ⌺_ustate 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

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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857

TABLE V. Deperturbed constants from ﬁt 1 for the BЉvЈϭ2–12 states. Figures in parentheses are three standard deviations.

T_v_,₀͑cm^Ϫ¹

͒

BЈ ͑cm^Ϫ¹

͒

DЈ ͑10^Ϫ⁷cm^Ϫ¹

͒

AЈ ͑cm^Ϫ¹

͒

␭Ј ͑cm^Ϫ¹

͒

␥Ј ͑cm^Ϫ¹

͒

oЈ ͑cm^Ϫ¹

͒

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

2.40͑20͒

2.15͑12͒

0

0.39͑23͒

0

1.50͑18͒

0

1

0

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

͉

͑ AϩB͒L_ϩ͉B⌳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

͉

͑ AϩB͒L_ϩ͉B⌳ϭ0

͘

.

͑15͒

BvЈ–BЉvЈ

␣ ͑cm^Ϫ¹

͒

␣_D͑10^Ϫ⁴cm^Ϫ¹

͒

␤ ͑cm^Ϫ¹

͒

␤_D͑10^Ϫ⁶cm^Ϫ¹

͒

0

1

2

3

4

5

–2

–3

–4

–5

–6

–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

–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_._G_r_e_e_n_a_n_d_C_._M_._W_e_s_t_e_r_n_:_A_n_a_l_y_s_i_s_o_f_B₃_⌺_uϪ

TABLE VII. Coefﬁcients from least-squares ﬁt of observed against calcu-

TABLE IX. Calculated ␤ parameters ͑in cm^Ϫ¹͒ 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_/_c_mϪ1

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.²⁵The 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

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

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 cm^Ϫ¹͒ 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_._G_r_e_e_n_a_n_d_C_._M_._W_e_s_t_e_r_n_:_A_n_a_l_y_s_i_s_o_f_B3⌺_u^Ϫ

859

mined from the Fourier transform infrared data.²¹The ﬁnal

values, given in Table XII, are consistent with previous

values.

included in the basis. Their omission leads to changes of 2

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.¹⁴suggested 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

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

͚_i͑c ͒,

͑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.¹⁴derived 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.

T_v_,₀͑cm^Ϫ¹

͒

BЈ ͑cm^Ϫ¹

͒

DЈ ͑10^Ϫ⁷cm^Ϫ¹

͒

AЈ

␭Ј

␥Ј

oЈ

vЈ

a

3.5^a

198.5^a

5.5^a

0^a

0

a

0

1

2

3

4

5

6

7

8

9

31 069

31 402

0.208

0.206

2

a

3.5^a

197.5^a

5.5^a

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

5.5^a

0.105͑66͒

0.191͑21͒

0.048͑11͒

0.190͑40͒

0.486͑68͒

0.059͑14͒

0.113͑67͒

0.27͑13͒

1.7^a

1.36͑20͒

1.681͑96͒

1.2

1.0

1.60͑14͒

0.6

0.4

0.21͑18͒

a

1

0

Ϫ0.022͑17͒

0.149͑89͒

0.0^a

a

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_._G_r_e_e_n_a_n_d_C_._M_._W_e_s_t_e_r_n_:_A_n_a_l_y_s_i_s_o_f_B₃_⌺_uϪ

TABLE XI. Deperturbed constants from ﬁt 2 for the BvЈϭ0–6 states. Figures in parentheses are three standard

deviations.

T_v_,₀͑cm^Ϫ¹

͒

BЈ ͑cm^Ϫ¹

͒

DЈ ͑10^Ϫ⁷cm^Ϫ¹

͒

␭Ј ͑cm^Ϫ¹

͒

␥Ј ͑cm^Ϫ¹

͒

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

␣ ͑cm^Ϫ¹

͒

␣_D͑10^Ϫ⁴cm^Ϫ¹

͒

␤ ͑cm^Ϫ¹

͒

␤_D͑10^Ϫ⁶cm^Ϫ¹

͒

0

1

2

3

4

5

–2

–3

–4

–5

–6

–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

–10

–11

–12

Ϫ6.3͑11͒

•••

Ϫ5.74͑96͒

•••

J. Chem. Phys., Vol. 104, No. 3, 15 January 1996

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_M_._E_._G_r_e_e_n_a_n_d_C_._M_._W_e_s_t_e_r_n_:_A_n_a_l_y_s_i_s_o_f_B3⌺_u^Ϫ

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⁾^w^h^e^r^e^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 ␶_fis inversely proportional to ⌺ (c ) as

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

J. Chem. Phys., Vol. 104, No. 3, 15 January 1996

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862

_M_._E_._G_r_e_e_n_a_n_d_C_._M_._W_e_s_t_e_r_n_:_A_n_a_l_y_s_i_s_o_f_B₃_⌺_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

ϭ

⌬

͉

⌬₀

͉

Ϯ͑B ϪB ͒J͑Jϩ1͒

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

Ϸϯ

Ϫ

͑B ϪB ͒J͑Jϩ1͒,

͑18͒

2

B

BЉ

⌬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

ϩ

1

Ϫ

27,28

BЉ states, the AЈ ⌬ , A ⌺ , and c ⌺ states,

all three

u

3

of which could perturb the BЉ ⌸ state via spin–orbit mix-

u

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_M_._E_._G_r_e_e_n_a_n_d_C_._M_._W_e_s_t_e_r_n_:_A_n_a_l_y_s_i_s_o_f_B3⌺_u^Ϫ

863

ing. The origins of these states lies some 10 000 cm^Ϫ¹below

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 O₂.³³Mixing 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

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

Field²²͑Sec. 2.4.2͒ by expressing the spin–orbit operator in

its many electron form,

ˆ

i

H

_S_Oϭ_͚_i

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

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.³⁴Using 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

B ⌺

•••5␴ 2␲ 2␲ ,

͑19͒

͑20͒

͑21͒

g

u

g

3

2

4

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-

_m_e_n_t_._A_s_d_i_s_c_u_s_s_e_d_b_y_H_e_r_z_b_e_r_g₃0 and Mulliken,

31,32

a paral-

lel transition from a bonding ␲ orbital to the corresponding

u

antibonding ␲_gorbital, 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

^B^Љ^⌸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 ⌺_u͑vЈϭ0–6͒ and

3

BЉ ⌸_u͑vЈϭ2–12͒ 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_._G_r_e_e_n_a_n_d_C_._M_._W_e_s_t_e_r_n_:_A_n_a_l_y_s_i_s_o_f_B₃_⌺_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

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-

ingham, H. J. Hurst, R. F. Barrow, and co-workers.

J. M. Ricks, D. Phil. thesis, Oxford University, 1970.

K. N. Rosser, Q. Wang, and C. M. Western, J. Chem. Soc. Faraday Trans.

16

17

89, 391 ͑1993͒.

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,

315 ͑1986͒.

͑

1

20

M. A. A. Clyne and J. P. Liddy, J. Chem. Soc. Faraday Trans. 2 78, 1127

͑1982͒.

E. H. Fink, H. Kruse, and D. A. Ramsay, J. Mol. Spectrosc. 119, 377

2

1

2

3

͑

1986͒.

ACKNOWLEDGMENTS

H. Lefebvre-Brion and R. W. Field, Perturbations in the Spectra of Di-

atomic Molecules ͑Academic, New York, 1986͒

B. G. Wicke, R. W. Field, and W. Klemperer, J. Chem. Phys. 56, 5758

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

E. Hirota, J. M. Brown, J. T. Hougen, T. Shida, and N. Hirota, Pure Appl.

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

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