First stretching overtone of BiH3: An extreme local-mode case for XH3-type molecule?

Article abstract of DOI:10.1063/1.1647055

The first stretching overtone region of BiH₃ was investigated using Fourier transform infrared spectroscopy. It was observed that ratios of vibration-rotation interaction parameters and the appropriate rotational constant differences were in go

Full text of DOI:10.1063/1.1647055

First stretching overtone of BiH 3 : An extreme local-mode case for XH 3 -type
molecule?
W. Jerzembeck, H. Bürger, V. Hänninen, and L. Halonen
Citation: The Journal of Chemical Physics 120, 5650 (2004); doi: 10.1063/1.1647055
View online: http://dx.doi.org/10.1063/1.1647055
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 120, NUMBER 12
22 MARCH 2004
First stretching overtone of BiH3 : An extreme local-mode case
for XH3-type molecule?
a)
¨
W. Jerzembeck and H. Burger
¨
Anorganische Chemie, FB 10, Bergische Universitat, D-42097 Wuppertal, Germany
¨
V. Hanninen and L. Halonen
Laboratory of Physical Chemistry, P.O. Box 55 (A.I. Virtasen aukio 1), FIN-00014 University of Helsinki,
Finland
͑Received 17 November 2003; accepted 17 December 2003͒
The first stretching overtone region of short-lived, formerly inaccessible BiH3 near 3405 cmϪ1 has
been measured by Fourier-transform infrared spectroscopy with a resolution of 0.0066 cmϪ1. Only
the 2␯1(A1)/␯1ϩ␯3(E) band system has been observed. Rotational analysis, with transitions
reaching JЈmaxϭ14, has revealed almost perfect local-mode behavior for the upper states denoted as
(200A1 /E) in the local-mode notation. Ratios of vibration–rotation interaction parameters qeff /␣e(BffB)
¯
and reff /␣(eBffC) , and the appropriate rotational constant differences, are in good agreement with
theoretical local-mode limit values. A simple stretching vibrational model reproduces the observed
vibrational term values well, and the potential parameters obtained are close to true values.
© 2004 American Institute of Physics. ͓DOI: 10.1063/1.1647055͔
I. INTRODUCTION
mode limit we label the states (200A1 /E), and (110A1 /E),
where the digits 200 and 110 denote the distribution of
quanta in the three bonds.
The concept of vibrational energy localization upon
single and, in particular, multiple excitation of the stretching
motion has been successfully applied to hydrides XH2 ,
XH3 , and XH4 .1–3 The X–H stretching motions are anhar-
monic and therefore the ⌬ ϭ1 vibrational selection rule is
no longer strictly valid and ⌬ Ͼ1 transitions are sufficiently
intense for observation by infrared spectroscopy. On the
other hand, stretch–bend combination bands are almost ab-
sent at least at high vibrational excitation indicating that the
local-mode type stretching states are largely decoupled from
motions involving bending states. At the same time, coupling
between different X–H oscillators should also be small due
to a heavy central atom X and right angle HXH bond angles
in XH2 and XH3 systems.1–3 This favors the local mode
behavior within the stretching vibrational manifold.
Localized vibrational states exert patterns of the rota-
tional energy that are unique for molecules of the respective
symmetry.4–8 There is dynamic symmetry lowering in the
excited local-mode state, e.g., from C3v in the ground state
Of the series of the XH3 species, where X is a group 15
element, stretching overtones of AsH3 and SbH3 have been
studied in some detail while stretching overtones of PH3 still
await a thorough investigation. The NH3 spectra do not re-
veal rovibrational structure as described above for localized
vibrations. Although the first overtone states (200A1 /E) of
SbH3 fulfill the above-mentioned local-mode requirements
quite well, the corresponding states of AsH3 are less local
and particularly the (110A1 /E) states of AsH3 differ dis-
tinctly from local-mode behavior.9–11 Closer agreement of
the (n00A1 /E) stretching states with local-mode expecta-
tions becomes evident for higher excitation, nу4 and nу3
in AsH3 ͑Refs. 12 and 13͒ and SbH3 ,14,15 respectively.
The hitherto unavailable BiH3 molecule might show
local-mode behavior at lower stretching excitation than other
XH3 species. The HBiH bond angle should be very close to
90°, indicating a small kinetic energy coupling between the
bond oscillators. Furthermore, the heavy Bi atom ͑atomic
mass 209 u͒ most likely damps the potential energy
HBi/BiHЈ coupling. Thus, the net coupling between bond
oscillators will be small when compared with the BiH bond
anharmonicity (Y20ϭϪ31.92 cmϪ1 in diatomic BiH͒.16 This
should lead to near local-mode behavior.3,17,18 It would be a
particularly interesting task to investigate stretching vibra-
tional localization in this conceivably extreme case in the
series of XH3 molecules, if BiH3 were available.
v
v
of an XH species to C in a local-mode state of XH H with
Ј
3
s
2
the excitation localized in the X–HЈ stretching motion.
Therefore, effects of vibrational angular momentum are
quenched, and both z and x, y axis Coriolis couplings are
negligible. Rotational constants of the almost degenerate
stretching vibrational local-mode states of XH3 , e.g.,
2␯1 (A1) and ␯1ϩ␯3 (E) corresponding to the (200A1 /E)
states, become equal. Simple relations between vibration–
rotation interaction parameters are then obeyed.5,8 Here and
throughout this paper we use the following labels for the
vibrational states: In the normal mode limit we label the
states 2␯1 (A1), ␯1ϩ␯3 (E), and 2␯3 (A1ϩE). In the local
The synthesis of the BiH3 molecule was first reported in
196119 but could never be repeated in spite of several at-
tempts by different researchers including ourselves. The fail-
ure to prove or disprove the first observation lies in synthetic
difficulties due to the most demanding nature of the prepa-
ration and the thermal and photolytic instability of BiH3 . In
a͒
Author to whom all correspondence should be addressed. Electronic mail:
buerger1@uni-wuppertal.de
0021-9606/2004/120(12)/5650/7/$22.00
5650
© 2004 American Institute of Physics
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J. Chem. Phys., Vol. 120, No. 12, 22 March 2004
BiH3 : An extreme local-mode case
5651
the gas phase, this molecule decomposes rapidly slightly
above the temperature ϳϪ40 °C, which is also necessary for
the BiH3 formation.19 Accordingly, neither any spectroscopic
data nor its structure could be determined to date.
Recently, we have been able to repeat the reported syn-
thesis of BiH3 so that high-resolution infrared and
millimeter-wave spectra could be measured.20 Ground state
rotational and centrifugal distortion constants as well as the
r0 and re structures have been obtained thereof,21 and the
vibrational fundamentals ␯1 , ␯2 , ␯3 , and ␯ have been mea-
4
sured in the infrared region with an optical resolution of
22
5.5–6.6ϫ10Ϫ3 cmϪ1
.
Expectation that BiH3 may be close to being vibra-
tionally localized already in the (BiH)ϭ2 overtone stretch-
v
ing state is supported by the finding that the HBiH angle,
FIG. 1. Survey spectrum of the ⌬ (BiH)ϭ2 band of BiH .
v
␪eϭ90.32°,21 is close to a right angle and the stretching
3
fundamentals
are
almost
degenerate,
(␯1Ϫ␯3)
z
ϭϪ1.21 cmϪ1
, and the Coriolis parameters, C␨
3
III. DESCRIPTION AND ASSIGNMENT
OF THE SPECTRUM AND DATA FIT
x
ϭϪ0.0231 and B⍀␨1,3ϭϮ0.0656 cmϪ1 are small in magni-
tude. We have therefore searched for this first stretching
overtone band near 3400 cmϪ1 and eventually observed it in
good quality in spite of the only ϳ1% intensity relative to
The first stretching overtone band of BiH3 is shown in
Fig. 1. It is centered at ϳ3405 cmϪ1 and reveals well pro-
nounced Q branch transitions (⌬JϭJ ϪJ ϭ0) over a spec-
Ј
Љ
the ␯1 /␯ band system and the experimental difficulties as-
3
tral range of 4–5 cmϪ1 degraded to smaller wave number
from the sharp edge at 3406.8 cmϪ1. The label J is the total
angular momentum quantum number, and the prime and
double prime refer to the upper and lower states, respec-
tively. Toward the large wave numbers ⌬Jϭϩ1, R branch
clusters are well pronounced and can be followed up to about
sociated with the measurements at high resolution. In this
contribution, we report on our observations and the analysis
of the spectrum.
Section II will give some experimental details, in Sec. III
the observed spectrum, the assignment, and the data fit are
described. Section IV will report on rotational local-mode
effects, Sec. V presents a simple vibrational calculation, and
a discussion in Sec. VI concludes the study.
J ϭ8. The structure of these J clusters becomes more com-
Љ
plex as J increases. Some prominent lines stand out near the
low wave number edges of these R branch clusters; at low J
these can be associated with RRK(JϭK) lines of the (200E)
band.
The ⌬JϭϪ1, P branch region contains wider-spread
less compact J clusters than the R branch. Moreover, the
spectrum available to us is noisier in the P branch than in the
R and ͑very dense͒ Q branch regions. At first inspection, the
Q branch looks chaotic, and assignments could only be made
when reliable predictions became available. We estimate that
the intensity of the overtone band is 1%–2% of that of the
II. EXPERIMENTAL METHOD
Bismuthine BiH3 , which is a highly reactive, short-lived
species, rapidly decomposing above Ϫ35 °C to give Bi ͑de-
posited on all surfaces͒ and H2 , was prepared as described in
Refs. 19 and 20. Since no multipass cell could be employed
owing to the immediate deposition of Bi on metal mirrors,
the region of the first stretching overtones was studied using
a double-jacketed glass cell of 1.2 m length with CaF2 win-
dows and external cooling. The wall temperature was ad-
justed to Ϫ36 °C. The cell had an inner diameter of 70 mm
and was placed in the external parallel beam of a Bruker 120
HR interferometer in a setup similar to that described in Ref.
23. The interferometer was equipped with a halogen source,
a CaF2 beam splitter, an InSb detector, and a 2.5–3.5 ␮m
optical bandpass filter. The resolution was adjusted to 0.0066
cmϪ1 ͑1/maximum optical path difference͒. Altogether 120
scans were collected. The total pressure in the cell ͑contain-
ing some H2) was adjusted to 60 Pa, and the absorption cell
was refilled three times with a fresh BiH3 sample.
␯1 /␯ fundamental band system.
3
The first assignments of the spectrum were made in the
region of the compact R(0) to R(3) clusters revealing a
simple structure. Use was made of ground state combination
differences employing the spectroscopic parameters reported
in Ref. 21. Comparison with the spectrum of the closely
related ⌬ ϭ2 band of SbH also helped in finding rotational
v
3
assignments.11 Moreover, it was useful during the first steps
of the assignment to make use of an approximate prediction
of the BiH3 spectrum obtained from scaled parameters for
the (200A1 /E) state of SbH3 .11 An empirical scaling factor
sϭ0.85 was used for the appropriate most important
vibration–rotation parameters ͑see the following͒. This value
was obtained by a comparison of the ␯ and ␯ parameters of
3
BiH3 ͑Ref. 22͒ with those of SbH3 .251 A transition moment
ratio ␮(A1):␮(E)ϭ1.0 was chosen, in analogy to that of
SbH3 , 1:1.042.11 After several loops of assignments and
simulations based on the least-squares optimization of spec-
Calibration was carried out using H2O lines in the 3600
cmϪ1 region.24 Wave number precision of medium to strong
unblended lines is better than 5ϫ10Ϫ4 cmϪ1
.
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5652
J. Chem. Phys., Vol. 120, No. 12, 22 March 2004
Jerzembeck et al.
TABLE I. Ground state parameters ͑all in cmϪ1͒ of BiH3 , B reduction.a
operator H04 both for the ground state and for the upper
states. Its matrix elements are (iϭ0, 200A1 , or 200E)
B
C
2.641 601 718
2.601 402 9
53.803 18
Ϫ64.037 2
67.713 2
3.177 3
Ϫ3.162
Ϫ3.278
͑i͒
i,Jkϩ3͉H04 /hc͉i,Jk
͘
͗
DJϫ106
D
JKϫ106
ϭ⑀ 2kϩ3͒F J,k͒F J,kϩ1͒F J,kϩ2͒,
͑3͒
͑
͑
͑
͑
DKϫ106
HJϫ109
i
1/2
where F(J,k)ϭ J(Jϩ1)Ϫk(kϩ1)
.
͔
͓
H
H
JKϫ109
KJϫ109
For the upper state off-diagonal elements of the Hamil-
tonian, we just give the formulas and refer to Ref. 11 for a
more detailed discussion.
HKϫ109
⑀ϫ106
5.896
12.883 3
The Coriolis interaction matrix elements between the E
and A1 bands take the form ͑with one higher-order term in
aFrom Ref. 21.
(K)
eff
␨
)
200E ᐉϭϩ1͒,Jkϩ1͉ H ϩH ͒/hc͉200A ,Jk
͑
͘
21 23 1
͑
͗
troscopic model parameters altogether 634 transitions in-
volving also Q-branch lines were assigned. The maximum J
Ј
ϭϪ 200A ,Jkϩ1
͉
H ϩH ͒/hc
͉
200E ᐉϭϪ1͒,Jk
͑
͑
͗
͘
1
21
23
and K values, which could be reached, were 14 and 13,
Ј
͑K͒
͑A͒
respectively, and no medium to strong features of the spec-
trum remained unassigned. We did not observe any features
at larger wave numbers which could be associated with the
ϭ 21/2 B␨
⍀
effϩ␨ G k F J,k ,
͑4͒
͓
͑
͒
͑ ͔͒ ͑
͒
eff
where G(k)ϭk2ϩ(kϩ1)2.
All H22 operators are included in our model. Of these,
the ᐉ- and r-resonance terms couple different vibrational an-
gular momentum states in the E vibrational band. The
⌬ ϭ2 band reaching the (110A /E) state.
v
1
The fit was performed using the same nonlinear least-
squares computer program that had been employed for the
(200A1 /E) overtone spectrum of SbH3 .11 Ground state pa-
rameters were taken from Ref. 21. For convenience, these are
gathered in Table I.
For the model used in assigning the spectrum, see Refs.
4 and 11. The following ground and upper state matrix ele-
ments were employed.
The diagonal matrix elements for the ground and for the
nondegenerate excited A1 state take the customary form (s
ϭ0 or 200A1)
ᐉ-resonance matrix elements ͑with one high-order term in
(J)
¯
qeff) are
200E ᐉϭϩ1͒,Jkϩ2͉ H ϩH ͒/hc͉200E ᐉϭϪ1͒,Jk
͑ ͑
22 24
͑
͗
͘
ϭϪ 12 q ϩq J Jϩ1͒ F J,k͒F J,kϩ1͒
͑5͒
͑
eff
J͒
¯ ¯
eff
͓
͑
͔ ͑
͑
and the r-resonance elements are
200E ᐉϭϪ1͒,Jkϩ1
͉
H ϩH ͒/hc
͉
200E ᐉϭϩ1͒,Jk
͑
͑
͑
͗
͘
22
24
͑J͒
ϭ2 2kϩ1͒ r ϩr J Jϩ1͒͒
͔
͓͑
͑
͑
͕
eff
eff
H ϭ s,Jk
͉
H/hc
͉s,Jk
͘
͗
s
͑K͒
3
ϩreff kϩ1͒3ϩ kϩ2͒ F J,k͒.
͑6͒
͓͑
͑
͔
͑
͖
2
ϭ␯ ϩB J Jϩ1͒ϩ C ϪB ͒K2ϪD J2 Jϩ1͒
͑
͑
͑
s
s
s
s
Js
3
ϪDJKsJ Jϩ1͒K2ϪDKsK4ϩHJsJ3 Jϩ1͒
There are two other H22 operators, the ␣-resonance terms,
which couple the rotational states of the 200A1 and 200E
vibrational states. The corresponding matrix elements are
͑
͑
ϩHJKsJ2 Jϩ1͒2K2
͑
ϩHKJsJ Jϩ1͒K4ϩHKsK6.
͑1͒
͑
200E ᐉϭϩ1͒,Jkϩ1͉ H ϩH ͒/hc͉200A ,Jk
͑
͘
22 24 1
͑
͗
The corresponding expression for the doubly degenerate E
state is (tϭ200E)
ϭ 200A ,Jkϩ2
͉
H ϩH ͒/hc
͉
200E ᐉϭϪ1͒,Jk
͑
͑
͗
͘
1
22
24
ϭ 122Ϫ1/2
␣
ϩ␣eff J Jϩ1͒ϩ␣
G k͒
͑ ͔
͑
eff
BC͒
͑
BCJ͒
͑
BCK͒
͓
͑
eff
H ϭ t,Jk
͉
H/hc
͉t,Jk
͘
͗
t
ϫ 2kϩ1͒F J,k͒
͑7͒
͑
͑
ϭ␯ ϩB J Jϩ1͒ϩ C ϪB ͒K2Ϫ2 C␨͒ kᐉ
͑
͑
͑
t
t
t
t
t
and
ϪDJtJ2 Jϩ1͒2ϪD J Jϩ1͒K2ϪD K4
͑
͑
JKt
Kt
200E ᐉϭϪ1͒,Jkϩ2
͉
H22 /hc
H22 /hc
F J,k͒F J,kϩ1͒.
͉
200A ,Jk
͑
͗
͘
ϩHJtJ3 Jϩ1͒3ϩHJKtJ2 Jϩ1͒2K2ϩHKJtJ
1
͑
͑
ϭ 200A ,Jkϩ2
͉
͉200E ᐉϭϩ1͒,Jk
͑
͘
͗
ϫ Jϩ1͒K4ϩH K6ϩ␩ J Jϩ1͒kᐉϩ␩ k3ᐉ.
1
͑
͑
Kt
Jt
Kt
ϭ 122Ϫ1/2
␣
eff
͑8͒
͑
BB͒
͑
͑
͑2͒
In these equations,
is a product of vibrational (
eigenfunctions (͉Jk ). The quantum label ᐉ is the vibrational
angular momentum quantum number, the k quantum number
corresponds to the component of the total angular momen-
tum in the molecular fixed axis system, and Kϭ
also included one rotationally off-diagonal quartic distortion
͉
i,Jk ϭ
͉
i
͉
Jk (iϭ0, 200A , or 200E)
Only parameters, which have been determined with sta-
͘
͘
͘
1
͉
i ) and symmetric top rigid rotor
͘
tistical significance, have been optimized with the nonlinear
least-squares method. The selection of fitted parameters is
similar to that employed for SbH3 although some higher or-
der terms are different. The achieved rms deviation ␴͑Fit͒ of
ϳ0.8ϫ10Ϫ3 cmϪ1 is better than for SbH3 ͑2.4 and 2.2
ϫ10Ϫ3 cmϪ1 for 121SbH3 and 123SbH3 , respectively͒.11
͘
͉k͉. We have
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J. Chem. Phys., Vol. 120, No. 12, 22 March 2004
BiH3 : An extreme local-mode case
5653
TABLE II. Excited state parameters ͑in cmϪ1͒ of BiH3 .a,b
(200A1)
(200E)
␯
B
C
3406.700 391͑222͒
2.595 129 1͑120͒
2.558 983 5͑274͒
3406.361 162͑150͒
2.595 587 1͑187͒
2.559 593 0͑232͒
Ϫ6.562 3͑257͒
51.125͑107͒
0
C␨ϫ103
DJϫ106
53.965͑121͒
Ϫ58.163͑480͒
66.902͑726͒
D
JKϫ106
Ϫ56.480͑996͒
66.65͑130͒
DKϫ106
␩Kϫ106
9.063͑688͒
⑀ϫ106
13.447͑119͒
13.788 4͑753͒
Ϫ22.587 9͑111͒
2.458͑123͒
7.660 67͑748͒
0.569 7͑693͒
Ϫ3.377͑163͒
3
¯
qϫ10
q
(J)ϫ106
¯
rϫ103
r(J)ϫ106
r(K)ϫ106
FIG. 2. The R(6) cluster of the ⌬ (BiH)ϭ2 band of BiH . Lower trace:
v
3
experimental spectrum. Upper trace: simulated spectrum with the transition
moment ratio ␮(A1):␮(E)ϭ1.0.
(A)ϫ103
ͱ2B⍀␨
␨
␣
␣
␣
␣
Ϫ14.158 8͑518͒
(K)ϫ106
Ϫ4.419 ͑335͒
43.646 1͑305͒
2.345͑231͒
Ϫ16.053͑605͒
33.401 7͑282͒
634
(BC)ϫ103
(BCJ)ϫ106
(BCK)ϫ106
(BB)ϫ103
͉
␮ 200A ͒:␮ 200E͒
͉
ϭ21/2
where ␤ϭ␲Ϫarc sin (2/31/2)sin(␣/2) and ␣ϭ90.32° is the
͉cos ␤/sin ␤͉ϭ1:1.008, ͑9͒
͑
͑
1
No. of data
͓
͔
␴(Fit)ϫ103
0.79
observed HBiH bond angle.21 The transition moments given
are defined as in the paper by diLauro and Mills for Coriolis
interacting A1 and E fundamentals.27 It is clear from Fig. 2
that the band contour simulations confirm the usefulness of
this dipole moment model and prove that the spectrum has
been correctly assigned.
J
max /Kmax
14/13
aStandard deviations ͑1␴͒ in parentheses refer to the least significant digits.
bAll sextic centrifugal distortion coefficients are contrained to their ground
state values given in Table I.
The results of the data fit are gathered in Table II. The
rms deviation of ϳ0.8ϫ10Ϫ3 cmϪ1 is only 1.5 times larger
than the quoted precision of unblended medium-to-strong
lines. This is fully satisfactory in view of the many weak and
blended lines included as data in the calculation.
We have performed spectral simulations. Experimental
and calculated spectra of the R(6) cluster near 3440 cmϪ1
are illustrated in Fig. 2. Details of the assignment of the
respective transitions are available as supplementary
material.26 The transition moment ratio ␮(A1):␮(E) was set
to 1.0, and any significant modification of this value made
the agreement between observed and simulated spectra
worse. This agrees well with a bond dipole moment model3
that gives a simple interpretation for the ratio of the A1 and E
vibrational state transition moments as follows:11
IV. ROTATIONAL LOCAL-MODE EFFECTS
The vibrational dependence of the leading rovibrational
parameters can be modeled with a simple approach based on
the treatment of local-mode vibrations in the normal-
coordinate representation. The key element in this treatment
is the inclusion of quartic anharmonic resonance terms,
Darling–Dennison interaction operators, to couple stretching
vibrational overtone and combination states that are in strong
resonance. Thus, in the harmonic normal-mode basis, one
ends up in a block-diagonal Hamiltonian matrix model
where only states with the same symmetry and the same total
stretching vibrational quantum number are coupled. Details
TABLE III. Local-mode analysis of vibration–rotation parameters of BiH3 .a
(100A1 /E)
(200A1 /E)
Model I
Calc.
Model I
Calc.
Model II
Calc.
Obs.
Obs.
␯(A1) (cmϪ1
␯(E) (cmϪ1
B0ϪBv(A1) (cmϪ1
C0ϪCv(A1) (cmϪ1
)
1733.25
1734.47
1733.25
1734.46
3406.70
3406.36
3406.54
3406.55
)
)
)
0.022 92
0.027 14
0.023 17
0.046 47
0.042 42
0.046 01
0.041 81
Ϫ0.022 59
0.007 66
0.043 65
0.033 40
Ϫ0.010 01
Ϫ0.006 56
0.046 17
0.040 57
0.046 17
0.040 28
Ϫ0.022 01
0.007 81
0.044 18
0.031 46
Ϫ0.001 24
Ϫ0.000 60
0.046 17
0.040 19
0.046 17
0.040 19
Ϫ0.022 09
0.007 81
0.044 18
0.031 24
0.0
B0ϪBv(E) (cmϪ1
)
)
C0ϪCv(E) (cmϪ1
0.016 57
Ϫ1
¯
q (cm
)
)
Ϫ0.005 005
0.003 852
0.022 24
0.019 89
Ϫ0.046 39
Ϫ0.023 11
r (cmϪ1
␣
␣
(BC) (cmϪ1
)
)
(BB) (cmϪ1
B␨⍀ (cmϪ1
)
C␨ (cmϪ1
)
0.0
aThe quantities B0 ͑or C0) and Bv ͑or Cv) are the rotational constants of the ground and excited states,
respectively.
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5654
J. Chem. Phys., Vol. 120, No. 12, 22 March 2004
Jerzembeck et al.
of the vibrational model are given in Refs. 1, 3, and 28. The
eigenvectors of these matrices can be used to calculate effec-
tive spectroscopic parameters of vibrational overtone and
combination states employing spectroscopic constants of the
stretching fundamentals. Rotational and Coriolis constants,
and the coefficients of vibrationally off-diagonal H22 Hamil-
tonian resonance coefficients, have been calculated in this
way. This approach has been described before for XH3-type
molecules in Ref. 11.
where B0 and C0 are rotational constants of the ground vi-
brational state and the parameters on the very right-hand side
are from the upper state analysis of stretching fundamentals,
see for example Ref. 4, and B0ϪB (A1)ϭ␣(B) and B0
␯
1
1
ϪB (E)ϭ␣(B) , etc.31 We have used Eq. ͑11͒ to calculate
␯
3
3
effective vibration–rotation parameters of the (200A1 /E)
stretching vibrational overtone states (nϭ2). The results are
given in Table III ͑model II͒.
It is interesting to note that both models I and II give
essentially identical results which on the whole agree well
with experimental spectroscopic parameters. Thus, it is un-
necessary to use the more sophisticated Darling–Dennison
resonance model ͑model I͒ in the case of a near local-mode
molecule like BiH3 . The simple local-mode limit expres-
sions in Eq. ͑11͒ are adequate. The only disagreement with
experiment concerns the two Coriolis coupling coefficients.
Experimental values are about one order of magnitude larger
than results from model I. However, this is not serious, be-
cause it would be unlikely that the present simple theoretical
approach would reproduce accurately these small experimen-
tal parameters. The magnitudes of Coriolis coefficients are
normally about 0.1–1.0.
The harmonic wave number ␻ϭ1795.62 cmϪ1, anhar-
monicity constant ␻xϭ30.78 cmϪ1, and the bond stretch os-
cillator coupling parameter ␭ϭϪ0.404 cmϪ1 of BiH3 have
been used to compute standard normal-mode theory vibra-
tional parameters including Darling–Dennison resonance co-
efficients as explained in Refs. 3 and 28. Table III contains
both calculated vibrational term values and effective rovibra-
tional parameters of the 200A1 and 200E states ͑model I͒.
Spectroscopic parameters for the stretching fundamentals21
used in the calculation are also given in Table III.
The BiH3 molecule is close both to the vibrational local-
mode limit, where the bond oscillators are vibrationally un-
coupled (log10͉␻x/␭͉ϭ1.88, see Ref. 29͒, and to the rovibra-
tional local-mode limit, where neither vibration nor rotation
couples the bond oscillators.8 The rovibrational local-mode
limit is formally defined in a way that vibration–rotation
parameters fulfill the following special relations:
V. VIBRATIONAL CALCULATIONS
We have adopted a simple vibrational model to obtain
information of the stretching vibrational potential energy sur-
face of bismuthine. Following earlier work,1,29 our vibra-
tional Hamiltonian is
␯
A ͒ϭ␯ E͒,
͑
1
v
͑
v
Bv A ͒ϭB E͒,
͑
͑
1
v
3
1
2
2
͑rr͒
Cv A ͒ϭC E͒,
͑
͑
1
v
H/hcϭ
g
pi ϩDeyi
ͩ
ͪ
͚
2hc
iϭ1
͑BB͒
¯
ͱ
␣
␣
ϭϪ 2qeff
,
͑10͒
eff
1
pipjϩfrr aϪr 2y y
,
͑12͒
͑rr
͒
Ј
͑BC͒
eff
ϩ
g
ͩ
ͪ
͚
i
j
ͱ
Ј
ϭ4 2reff
,
hc
iϽj
Ј
B␨⍀͒ ϭ0,
͑
͑
eff
where
g(rr)ϭ1/mHϩ1/mBi
,
g(rr )ϭcos ␪e /mBi ,
pj
ϭϪiបץ/ץ⌬rj is the momentum operator conjugate to the
bond stretching displacement coordinate ⌬rj for the jth
C␨͒ ϭ0,
eff
bond, and the Morse variable yjϭ1ϪeϪa ,⌬r . The masses
r
j
where the subindex ϭn00 and n is the local-mode vibra-
v
tional quantum number. This means that the vibrational term
values (␯v) and the corresponding rotational constants of the
excited A1 and E states (Bv or Cv) are equal, and there is no
Coriolis coupling between rotational states. In addition, the
special relations between the q, r, and ␣ parameters ensure
that the rotational energy level structure is that of a single
vibrational state of an asymmetric rotor instead of coupled
rotational states of nondegenerate A1 and E vibrational states
of a symmetric rotor.1,7 In the case, the local-mode limit
vibrational wave functions are used and matrix elements are
approximated by harmonic oscillator formulas, the following
simple expressions for the effective overtone parameters are
obtained:30
mH and mBi refer to hydrogen and bismuth, respectively, ␪
e
ϭ90.3215° is the equilibrium bond angle,21 De is the Morse
dissociation energy, ar is the Morse steepness parameter, and
frr is a harmonic force constant, which describes the
Ј
strength of the potential energy coupling between the bond
oscillators. The weight function for the volume element of
integration is one.
We have also considered the effects of constraints in the
Hamiltonian unlike in the earlier work on XH3-type systems.
The deep role of constraints is controversial in quantum
mechanics32 but in the present types of vibrational problems
the solution is known at least in classical mechanics. The
vibrational tensor elements g(cij) in the presence of rigid con-
straints can be obtained from the unconstrained elements
g(ij) as33
͑B͒
1
͑B͒
B0ϪBv A ͒ϭB ϪB E͒ϭ n/3͒ ␣ ϩ2␣3 ͒,
͑
͑
͑
͑
1
0
v
3
3
͑C͒
1
͑C͒
C0ϪCv A ͒ϭC ϪC E͒ϭ n/3͒ ␣ ϩ2␣3 ͒,
͑
͑
v
͑
͑
1
0
͑rr͒
͑rr͒
͑r␪ ͒
͑r␪ ͒
j
i
gc ϭg
Ϫ
g
Mijg
,
͚ ͚
͑11͒
iϭ1 jϭ1
͑BB͒
͑BB͒
13
¯
q ϭϪ 1/ͱ2͒␣ ϭ n/3͒ q Ϫͱ2␣
͑
͑
͑
͒,
eff
3
eff
3
3
͑13͒
͑rr
͒
͑BC͒
͑BC͒
13
͑
rr
͒
͑r␪ ͒
͑r ␪ ͒
Ј
j
Ј
Ј
i
r ϭ 1/4ͱ2͒␣ ϭ n/6͒ 2r ϩ 1/ͱ2͒␣
͑
gc ϭg
Ϫ
g
Mijg
,
,
͔
͑
͓
͑
͚ ͚
eff
3
eff
iϭ1 jϭ1
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J. Chem. Phys., Vol. 120, No. 12, 22 March 2004
BiH3 : An extreme local-mode case
5655
TABLE IV. Constrained and unconstrained stretching vibrational metric ten-
sor elements.
TABLE V. Vibrational energy level fit of BiH3 . All observations are in
cmϪ1
.
Molecule g(rr) (uϪ1
)
g(crr) (uϪ1
)
g(crr (uϪ1 ÅϪ1
)
)
g(rr ) (uϪ1 ÅϪ1
)
Ј
Ј
State
Obs.
Obs.Ϫcalc.
100A1
100E
200A1
200E
1733.25
1734.47
3406.70
3406.36
Ϫ0.01
0.01
0.20
1.063 63
1.024 52
1.005 58
1.000 51
0.997 02
1.058 06
1.023 49
1.005 40
1.000 44
0.997 00
Ϫ0.020 516
Ϫ0.001 943
Ϫ0.000 482
Ϫ0.000 222
Ϫ0.000 040
Ϫ0.020 103
Ϫ0.002 343
Ϫ0.000 560
Ϫ0.000 253
Ϫ0.000 051
NH3
PH3
AsH3
SbH3
BiH3
Ϫ0.20
six-dimensional variational vibrational calculation with an
exact kinetic energy operator and an accurate state-of-the-art
ab initio potential energy surface on this system ͑see Refs.
39 and 40 for this kind of approach applied to NH3 and
H3Oϩ). A comparison with SbH3 shows that, although the
separation of the stretching fundamentals in stibine is larger,
the (200A1) and (200E) states of SbH3 are more degenerate.
But interestingly, also in stibine the order of the A1 and E
states is reversed when compared with stretching fundamen-
tals. A stretching vibrational model similar to the one
adopted in the present contribution does not account for this
energy level order change.41
where the indices i and j are over the three constrained bend-
ing degrees of freedom (⌬␪1 , ⌬␪2 , and ⌬␪ being the ap-
3
propriate bond angle displacement coordinates͒ and Mij is an
element of the symmetrical matrix
Ϫ1
͑␪ ␪
͒
͒
͒
͑␪ ␪
͒
͒
͒
͑␪ ␪
͒
͒
͒
1
1
2
3
1
2
2
3
1
3
3
3
g
g
g
g
g
g
g
g
g
͑␪ ␪
͑␪ ␪
͑␪ ␪
1
2
2
Mϭ
.
͑14͒
ͩ
ͪ
͑␪ ␪
͑␪ ␪
͑␪ ␪
1
2
3
All unconstrained metric tensor elements can be obtained
from Ref. 33.
We have calculated the constrained and the uncon-
strained stretching vibrational metric tensor elements for
NH3 , PH3 , AsH3 , SbH3 , and BiH3 at the respective equi-
librium geometry which have been taken from Refs. 21, 34–
37. The results are gathered in Table IV. Although we have
computed the constrained tensor elements only at the equi-
librium geometry, it is clear that in BiH3 there would be no
real difference whether constrained or unconstrained metric
tensor elements are used in Eq. ͑12͒. Therefore, all calcula-
tions have been performed with unconstrained elements.
The eigenvalues of the vibrational Hamiltonian in Eq.
͑12͒ with the unconstrained metric tensor elements have
been computed variationally with a symmetrized Morse os-
cillator product basis as described in Ref. 29. All matrix
elements have been obtained with analytical formulas.38 All
basis functions with the maximum stretching quantum num-
ber 6 have been included. The nonlinear least-squares
method has been employed to optimize the parameters ␻,
VI. CONCLUSION
We have been able to obtain a high-resolution infrared
spectrum of hitherto inaccessible BiH3 in spite of its reactiv-
ity and fast decomposition. In addition to the fundamental
bands,21 the first stretching overtone spectrum could also be
recorded. This has been assigned as the (200A1 /E) local-
mode band. The 2␯3 (A1 /E) stretching overtone band reach-
ing the (110A1 /E) state in the local-mode notation is appar-
ently too weak to be observed with the experimental set-up
available to us.
The nonlinear least-squares optimization of vibrational
parameters of a simple model leads to a rms deviation of
ϳ0.8ϫ10Ϫ3 cmϪ1. This is about 1/10 of the resolution
(6.6ϫ10Ϫ3 cmϪ1) used, and the body of the data does not
reveal any nonrandom residuals. Therefore, we consider the
quality of the fit as fully satisfactory.
The quartic centrifugal distortion parameters of the
(200A1) and (200E) states agree within 5.3% at worst.
These are also similar to the ground state values with differ-
ences in the absolute value being between 1% and 15%. We
noticed that constraining the J and K dependencies of the r
and ␣(BC) parameters to zero changes the DJ , DJK , and DK
constants of the (200E) state to physically unsatisfactory
values without altering the rms deviation of the fit. We have
rejected this alternative solution when fitting the data. Sup-
posedly, these two fit approaches respond in a slightly differ-
ent manner to perturbational effects exerted by unidentified
dark states.
The rotational analysis of the (200A1 /E) state clearly
disclosed the expected local-mode behavior. Simple local
mode limit formulas explain well the vibrational dependence
of the effective vibration–rotation parameters. It is of interest
to know whether BiH3 is closer to the local-mode limit than
␻x, and frr instead of the true potential energy constants
Ј
De , ar , and frr in order to reduce correlation between the
Ј
parameters. However, this is not problematic because simple
relations between ␻ and ␻x, and potential energy constants
De and ar allow one to readily convert one parameter set to
the other.1 Experimental vibrational term values have been
employed as data. We obtain the parameter values
␻ϭ1795.65͑66͒ cmϪ1
,
␻xϭ30.78(23) cmϪ1
,
and frr
Ј
ϭϪ42.0(149) cmϪ1 ÅϪ2. The uncertainties in parentheses
are one-standard deviation in the least significant digit. The
fit to the experimental vibrational term values is given in
Table V.
An inspection of the vibrational results reveals that the
overall vibrational term value fit is good and the stretching
potential energy parameters obtained are close to true values.
However, the small splitting of the (200A1) and (200E)
states is calculated to be almost absent and the order of the
states is also incorrectly predicted. This indicates a slightly
different coupling between different stretching states and the
bending manifold. It would be of interest to perform a full
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5656
J. Chem. Phys., Vol. 120, No. 12, 22 March 2004
Jerzembeck et al.
10 S. Yang, H. Lin, D. Wang, and Q.-S. Zhu, J. Chem. Soc., Faraday Trans.
TABLE VI. Local-mode parameters and relations of the (200A1 /E) states
of some XH3 molecules.
94, 1397 ͑1998͒.
11
¨
M. Halonen, L. Halonen, H. Burger, and P. Moritz, J. Chem. Phys. 95,
a
b
c
121SbH3
209BiH3
Parameter
75AsH3
Theoryd
7099 ͑1991͒.
12 J.-X. Cheng, X.-G. Wang, H. Lin, J.-L. Liao, and Q.-S. Zhu, Spectrochim.
Acta, Part A 54, 1947 ͑1998͒.
␯(A1)Ϫ␯(E) (cmϪ1
(C␨)/C0ϫ103
)
Ϫ1.164
Ϫ3.94
Ϯ6.50
Ϫ0.47
0.073
Ϫ2.43
Ϯ3.44
1.38
0.339
Ϫ2.52
Ϯ3.79
Ϫ0.46
0.0
0.0
0.0
0.0
13 J.-X. Han, O. N. Ulenikov, S. Yurchinko, L.-Y. Hao, X.-G. Wang, and
(B⍀␨)/B0ϫ103
Q.-S. Zhu, Spectrochim. Acta, Part A 53, 1705 ͑1997͒.
(B(A )ϪB(E))ϫ103
1
14
¨
M. Halonen, L. Halonen, H. Burger, and P. Moritz, Chem. Phys. Lett. 203,
(cmϪ1
)
157 ͑1993͒.
(C(A )ϪC(E))ϫ103
Ϫ4.47
Ϫ1.13
Ϫ0.61
0.0
1
15
¨
J. Lummila, T. Lukka, L. Halonen, H. Burger, and O. Polanz, J. Chem.
(cmϪ1
)
Phys. 104, 488 ͑1996͒.
(BB)
¯
/q
Ϫ1.5163
5.461
Ϫ1.3932
5.569
Ϫ1.4787 Ϫ1.4142
5.697 5.657
␣
␣
eff
16 R. D. Urban, P. Polomsky, and H. Jones, Chem. Phys. Lett. 181, 485
(BC)/reff
eff
͑1991͒.
17 M. S. Child and R. T. Lawton, Faraday Discuss. Chem. Soc. 71, 273
aFrom Ref. 9 with the ground state parameters from Ref. 35.
bFrom Ref. 11.
cThis work.
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¨
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W. Jerzembeck, H. Burger, L. Constantin, L. Margules, J. Demaison, J.
Breidung, and W. Thiel, Angew. Chem., Int. Ed. 44, 2550 ͑2002͒.
the lighter XH3 species AsH3 ͑Refs. 9 and 10͒ and SbH3 .11
This comparison can be made on the basis of vibrational,
rotational, and rovibrational resonance parameters as given
in Table VI. It is evident that BiH3 is closer to the theoretical
values for perfect local-mode behavior than AsH3 but the
difference between SbH3 and BiH3 is small.
21
22
¨
`
W. Jerzembeck, H. Burger, F. L. Constantin, L. Margules, and J. Demai-
son, J. Mol. Spectrosc. ͑to be published͒.
¨
W. Jerzembeck, H. Burger, J. Breidung, and W. Thiel, J. Mol. Spectrosc.
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may be found in the online article’s HTML reference section. The docu-
ment may also be reached via the EPAPS homepage ͑http://www.aip.org/
pubservs/epaps.html͒ or from ftp.aip.org in the directory /epaps/. See the
EPAPS homepage for more information.
Vibrational analysis with a simple stretching vibrational
model produces good results. The effects of the neglected
bending vibrations are shown to be small and the stretching
vibrational potential energy parameters obtained are close to
true values. This is pleasing because, if the very highest pre-
cision is not needed, the bending vibrations can safely be
decoupled from the stretches.
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ACKNOWLEDGMENTS
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We thank the Deutsche Forschungsgemeinschaft and the
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also grateful to the Finnish Cultural and Magnus Ehrnrooth
Foundations for Fellowships and L.H. thanks the Academy
of Finland for financial support.
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