4974
M. Li and J. A. Coxon: The A 2⌸(100)–(020) Fermi resonance in CaOD
˜
Evaluation of equilibrium bond lengths of an X–Y –Z
type molecule requires values of the equilibrium moments of
inertia for two isotopic molecules, which are expressed by
thors. The equations employed in this work are referred to
Eqs. ͑22͒–͑24͒ in Ref. 6. However, Eq. ͑24͒ of Ref. 6, which
was obtained from the work of Penny and Sutherland,34 is in
error by a factor of 4, as pointed out previously by
2
12
2
13
2
I ϭ͑m m r ϩm m r ϩm m r ͒/M.
͑19͒
25
e
1
2
1
3
2
3
23
Herzberg; the correct equation is
Here, Mϭm ϩm ϩm , and r denote the equilibrium in-
1
2
3
ij
ternuclear separations with the labels 1, 2, and 3 correspond-
ing to Ca, O, and H/D atoms, respectively. The moments of
2
2
2
2
Ϫ1
r
r
r
3
13
12
2
2
2
2
2
f ϭ4 c r r
2 12 23
ͭ
ϩ
ϩ
ͮ
.
͑22͒
22
2
m1 m2 m3
inertia, I ϭh/(8 cB ), can be calculated directly from the
e
e
equilibrium rotational constants B , defined by
e
1
2
1
2
Once f , f , and f were found, the unknown frequency
for CaOD was calculated from Eqs. ͑22͒ and ͑23͒ in Ref. 6,
11
13
33
3
B ϭB ͑000͒ϩ ␣ ϩ ␣ ϩ␣ Ϫ␥ .
22
͑20͒
e
v
1
3
2
There is a ␥22 term in Eq. ͑20͒; this is different from the
normal expressions25 which only include ␣ terms. Like ␣ ,
and used in further calculations.
˜ ˜
For each electronic state ͑A or X͒, two f values have
22
i
2
the parameter ␥22 describes the v dependence of the rota-
2
been obtained from the two isotopomers and are in excellent
agreement, as expected. These values are as follows:
f ͑CaOH͒ϭ0.0633 and f ͑CaOD͒ϭ0.0647 mdyn Å for the
tional constant B , as shown by Eq. ͑10͒. The contribution of
v
the ␥22 term to B is important in CaOH and is dominant,
v
2
A
2
22
compared with the ␣ term, in CaOD. The values of ␥ are
˜
2
22
state; f ͑CaOH͒ϭ0.0605 and f22͑CaOD͒ϭ0.0617
22
listed in Table IV.
˜
mdyn Å for the X state. The f22 value averaged over the two
isotopomers for each state will be used.
2
Hilborn et al. obtained the first estimates of the equilib-
rium bond lengths based on rotational analysis of the CaOH
The harmonic force constants, f , f , f , and f ,
11
13
33
22
˜
˜
˜
˜
˜
A(000)–X(000) band and partial analysis of the CaOD
evaluated above are listed in Tables VII and VIII for the A
A(000)–X(000) band. Now, the values of B (000), ␣ and
˜
v
1
and X states, respectively. As shown in the two tables, the
␣2
have been determined from a more precise and much
˜
constants f , f , and f in the A state have values that are
1
1
33
22
larger data base, including several excited vibrational levels.
˜
very close to those in the X state. This is expected since the
two states have similar potentials. However, f13 has quite
different values between the two states. Owing to the large
˜
˜
The results of the present calculations for the A and X states
are listed in Tables VII and VIII, respectively.
The results show that while the O–H bond length is
essentially the same in the two states, the Ca–O bond length
in the A ⌸ state is slightly shorter than that in the X ⌺
difference in frequency, the coupling between the and 3
1
modes is expected to be very weak, and f , then, has a very
13
˜
2
˜ 2
ϩ
small magnitude. It was noticed that f13 is sensitive to the
uncertainties of the values. The uncertainties of f in the
state, which indicates that the ‘‘nonbonding’’ valence elec-
1
13
˜
tron is actually slightly antibonding in the X state.
˜
˜
A and X states are probably comparable with the magnitudes
of the constants. Thus the f13 values are to be considered
with caution, while f , f , and f have been determined
The potential energy function of the CaOH molecule can
be expressed in curvilinear internal coordinates in the form,
1
1
22
33
1
2
2
1
2
2
with confidence.
Vϭ f ͑⌬r ͒ ϩf ͑⌬r ͒͑⌬r ͒ϩ f ͑⌬r ͒
11
12
13
12
23
33
23
The Coriolis coupling constants contain valuable infor-
mation on the harmonic force field. Alternatively, as in the
present work, the Coriolis coupling constants can be calcu-
lated from the harmonic force field. The values of and 23
in the A and X states for both isotopomers, calculated using
Eqs. ͑25͒ and ͑26͒ in Ref. 6, are listed in Tables VII and VIII,
respectively.
ϩ f ͑⌬␣͒ ϩf ͑⌬r ͒͑⌬␣͒ ϩf ͑⌬r ͒͑⌬␣͒2
1
2
2
2
22
122
12
322
23
ϩ••• ,
͑21͒
21
˜
˜
where ⌬r represent the displacements from the equilibrium
ij
nuclear separations and ⌬␣ is the angular displacement as-
sociated with the bending vibration. Since the molecular
force field expressed in the curvilinear internal expansion is
isotopically invariant, it is possible to determine the force
field using spectroscopic information from more than one
isotopic species. The second property of the force field in
these true bond-stretching and angle-bending ͑curvilinear͒
coordinates is that the quadratic force field alone gives a
good representation of the anharmonicity and, in conse-
quence, the cubic and quartic interaction terms are mini-
mized and the simplest expression for the force field is
The l-type doubling constants are dependent on the Co-
riolis coupling constants and some other constants, , ,
1
2
, and B , as expressed by Eq. ͑15͒ in Ref. 6. These con-
3
e
stants are now available and summarized in Tables VII and
VIII. An interesting comparison is possible, therefore, be-
tween calculated l-type doubling constants and those deter-
mined experimentally. As shown in Table IX͑a͒, it turns out
v
˜
˜
that the calculated q values for the A and X states of CaOH
and CaOD are all in excellent agreement with the values
determined from experimental data. This gives perhaps the
most eloquent proof of the quality of the data sets and the
deperturbation models in the present work.
2
8
obtained.
The three harmonic force constants, f , f , and f ,
may be determined from the harmonic frequencies and 3
11
13
33
1
for CaOH, combined with for CaOD. The harmonic force
constant f22 can be calculated from the equilibrium bond
Of further interest is a comparison between the experi-
mental centrifugal distortion constants D and those calcu-
lated from the derived Coriolis coupling constants and other
1
lengths and of either isotopomer. The methods and equa-
2
tions for such evaluations have been discussed by many au-
molecular constants. The expression35 for D is
e
J. Chem. Phys., Vol. 104, No. 13, 1 April 1996
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Li, Mingguang
