The interaction
of symmetrical
of rotation and local mode tunneling in the overtone spectra
hydrides
Kevin K. Lehmann
Department
of Chemistry,
Princeton
University,
Princeton,
New Jersey 08544
(Received 1 March 1991; accepted 3 May 1991)
In the “local mode limit” where the tunneling time for vibrational energy exchange is long
compared to the classical rotational period, one expects that the effective rotational
Hamiltonian will reflect the reduced symmetry of the local mode state. Hamiltonians in the
local mode basis are given for interaction of rotation and local mode tunneling for molecules of
the XH, , XH, , and XH, type. Transformation of these Hamiltonians to a symmetrized basis
(which diagonalizes the vibrational problem), produces rotational couplings between the
vibrational states. Relations between the spectroscopic constants are derived that are less
restrictive than those given earlier by Halonen and Robiette, but reduce to them when the
assumptions of their model are met. The present algebraic procedure can be easily extended to
include higher order terms. The effect of these couplings is to reduce the size of the pure
vibrational splittings. This is due to the fact that in the rovibrational problem, in general, one
must reorient the angular momentum vector in the body frame as well as transfer the
vibrational action between bonds. This increases the length of the tunneling path and thus
decreases the rate of vibrational energy transfer. Model calculations show that a simple
semiclassical picture can rationalize the observed trends.
INTRODUCTION
The characterization of highly excited vibrational states
has become one of the central goals in chemical physics.
Such studies tie spectroscopy to intramolecular dynamics (a
connection that is always there but is often only implicit)
and reaction dynamics. Often, anharmonic interactions,
which in traditional spectroscopic theory are “perturbations,” change even the topological character of the motion.
Perhaps the best understood of these changes is the transition from normal to local modes of vibration that characterize the stretching overtone bands of almost all symmetric
hydrides.’
Despite the extensive literature on local mode vibrations, little is known of the implications of local mode vibrational motion on the rovibrational energy levels, and thus the
rotational dynamics. The first systematic study on this topic
was the work of Halonen and Robiette on XY,, XY,, and
XY, molecules.2 By assuming a vibrational potential equal
to a sum of uncoupled Morse oscillators, and then making
several very restrictive assumptions, Halonen and Robiette
derived a set of relations between the resonant terms of the
H,, operator.3 This operator contains terms of the form
qiqj J, Jb and dominates the vibrational dependence of the
rotational structure. An interesting prediction of this model
is that the rotational structure of a XH, molecule should
resemble a parallel transition of a symmetric top, including
the factor of 2 higher statistical weight for the K = 3n levels.
This effect has since been observed in the overtone spectra of
GeH, ,4 SiH, ,’ and SnH, .6 An independent derivation of
this effect has been given by Michelot er al.’ using an algebraic approach, and this model has been shown to fit the
observed first overtone band of SiH, spectrum with an rms
0.003 cm - ’with only two adjustable parameters in the effec-
tive Hamiltonian.
This symmetric top rotational structure can be understood in a simple physical way. In the traditional treatment
of the rotational motion, one assumes that the vibrational
motion is much faster than the rotation. As a result, the
rotational energy level structure is determined by a vibrationally averaged effective moment of inertia (the diagonal
terms of H,, ), But in the local mode limit, a molecule has
nearly degenerate vibrational energy levels.’ These can be
viewed as symmetrized combinations of “local-mode” states
wherein each X-H stretch has a constant vibrational action
unlike normal mode states which have constant vibrational
action in each normal mode. The splitting between the different symmetry representations of the local-mode states is
related to the time scale for tunneling to a wave function for
which the excitation has swapped bonds. If this tunneling
time is much longer than the rotational period, one would
expect that the rotational motion and thus energy level
structure will reflect the reduced symmetry of a single localmode state. The adiabatic separation works in reverse, since
rotational motion is now fast compared with tunneling. The
local-mode state of XH, with all the excitation in a single
bond will look like a slightly prolate symmetric top since, on
average, the excited bond will be longer. It should be remembered that changes in the vibrationally averaged structure
are only part of the vibrational dependence of the rotational
constants, but such a simplified model correctly predicts the
symmetry of the effective rotational Hamiltonian.
It is clear that the above argument depends only on a
separation of time scales, and thus does not depend upon the
restrictive assumptions about mass, structure, and bending
force constants that Halonen and Robiette made in their
work. Halonen and Robiette had given numerical evidence
that in fact the effective constants for overtone levels would
obey certain of the constraints of their local-mode limit
0021-9606/91 /I 62361-l 0$03.00
J. Chem. Phys. 95 (4), 15 August 1991
@ 1991 American Institute of Physics
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2362
Kevin K. Lehmann: Overtone spectra of hydrides
much more than the fundamentals; in particular, that the
effective rotational constants for each of the local-mode
states tends to become equal for each of the nearly degenerate local-mode vibrational states and that the effective Coriolis coupling of the states vanish. In this paper, we shall
examine the spectroscopic consequences of a model that invokes an effective rotational Hamiltonian for each localmode vibrational state plus vibrational tunneling between
local-mode states. The relationship of the present approach
to that of Halonen and Robiette is similar to the two independent derivations of the ‘X-K ” relations that allow a local-mode vibrational Hamiltonian to be expressed in a normal mode basis set. Mills and Robiette’ derived the x-K
relationships by starting with a simplified model for the potential energy function and then using perturbation theory.
My independent derivation” started with the Child and
Lawton harmonically coupled, anharmonic oscillator Hamiltonian, ” expressed as a polynomial in the bond mode raising and lowering operators, and by a symmetry transformation of these operators demonstrated the samex-Krelations.
Each method has its own advantages. The perturbation
method allows one to use a more general force field and
estimate the effect of the neglected terms. General perturbation expressions for all possible Darling-Dennison
(quartic) coupling terms between nondegenerate modes were recently published. l2 The algebraic method is much easier to
implement, however, and proves an exact equivalence for all
eigenvalues, which the perturbation method did not. Della
Valle has provided general formulas for the x-K relations for
any set of equivalent bonds using the algebraic approach. I3
The present study of vibration-rotation
interactions
will again start in a local-mode basis with a rotation-inversion Hamiltonian operator based upon general physical considerations, much as in Child and Lawton’s” treatment of
the vibrational coupling. Transformation of this Hamiltonian to a normal mode basis generates a set of relations
between the coefficients of the H22 terms. As with the x-K
relationships derived earlier, one does not expect the relationships to hold exactly in real spectra since the starting
Hamiltonian is only approximate. But the relationships will
be useful in attempts to predict and assign spectra, and as
possible constraints to be used in spectral fitting when parameter correlation or limited data sets prevents a free fit of
all the possible terms. More importantly, it provides a physical model for interpreting the rotational constants that are
derived from a fit of a local-mode spectrum.
An interesting result of the present treatment is that
rotation of a molecule can greatly reduce the tunneling rate,
stabilizing the molecule for a much longer time with vibrational motion localized along a single bond. It will be shown
that the qualitative features of this effect can be understood
by semiclassical analysis of motion on the rotational energy
surface, as used by Harter and Patterson.14 The physical
explanation for this effect is that to reach an isoenergetic
state, one must not only exchange all of the vibrational action from one bond to another, but in general, one must also
reorient the angular momentum in the body fixed frame.
Such a rotational suppression of local-mode tunneling appears to have been observed in the first overtone band of
stannane,‘j but it escaped explicit notice though it was correctly predicted by the fitted constants. The present study
provides a physical picture for this effect.
XH, MOLECULES
Consider an XH2 molecule in a local-mode state In, )
which means there are n quanta in bond 1. By symmetry the
state In,) will be degenerate with energy Go. These two
states will be coupled by tunneling, leading to a term in the
Hamiltonian R (n, ) (n, (. Here R is the rate at which the IZ
quanta of vibrational excitation tunnels from bond 1 to bond
2. It is known from earlier work on local modes that this
tunneling rate will decrease exponentially with increasing
nk.’ This tunneling term will lead to the vibrational eigenstates being $,,, = (In,) f In,))/+‘% with Es = Go +A
and E, = Go - /2. In each local-mode state, the rotational
motion is described by an asymmetric top Hamiltonian. But
due to the reduced dynamical symmetry ofthe (n, ) state, the
principle axes for this state will be rotated in the molecular
plane by an angle 19away from those of the ground state
(which are by symmetry the C,, axis and perpendicular to
it). We label the axes (X,JJ,Z) as theA,B,Caxes of the equilibrium structure and thus the C,, axis is along y, and the perpendicular to the molecular plane is z. To lowest order, the
inertial axis rotation comes about because the H22 operator
has a term q,q, (J, Jy + J,, J, ). The operator qsq, will have a
nonzero expectation value, despite being antisymmetric, because the local-mode state does not have C,, symmetry. Because of the longer average bond length of the excited bond,
one expects that the A axis will rotate towards this excited
bond. The state In*) by symmetry should have the same
rotational constants, but its principal axes will be rotated by
an equal but opposite amount compared to (n, ). Thus we
write an effective Hamiltonian to describe the rotationaltunneling dynamics of the In, ) and In, ) states as
H
=
[Go
+
+AJ:,
[Go
+BJ;,
+AJll
+CJ:,]IMn,I
+CJ:>]ln,)(n,I
b,>hl]7
+BJ;2
+A [IMn,l+
(1)
where the principal axes are given by
Jx, = cos t9J, + sin OJ,,,
J,, = - sin OJ, + cos BJ,,,
Jx, = cos OJ, - sin OJ,, (2)
(2)
Jy2 = sin OJ, + cos BJ,,
Jz, =J+=J*.
If we rewrite this Hamiltonian in terms of projections
onto the vibrational eigenstates and rotational operators on
axes determined by symmetry we find
H = [Go+~++A,J~+B,J~+C,Jt]Is>(sl
+ [Go -A+A,J:
++-L{J,,J,~[ld(~l
+B,J;
+
+C,J:]I4(al
Idbll,
where
CJx,J,)
= JxJ,
+ J,Jx,
J. Chem. Phys., Vol. 95, No. 4,15 August 1991
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(3)
Kevin K. Lehmann: Overtone spectra of hydrides
~,=A,=A~os~0+Bsin~t4
B, =
B, = A sin’ 0 + B COS’0,
c, = c, = c,
d, = (A - B) sin 20.
(4)
This is the same form as the Hamiltonian determined by
Halonen and Robiette, but the relationship between the parameters is not as restrictive. In absorption from the ground
state, one will have x polarized transition amplitude to the
I a) rovibrational basis states, and y polarized transition amplitude to the Is> rovibrational basis states. It is worth pointing out that, while one is tempted to predict the ratio ofx and
y matrix elements from the projections of the bonds on the x
and y axes (assuming the transition dipole is along the
bond), this is generally a poor approximation.
Several interactions are not present in this Hamiltonian,
and their absence needs to be justified. The first is an absence
of Coriolis interaction between states 1n’ ) and 1n, ) . By symmetry, there exists a Caxis Coriolis interaction between the s
and a stretching fundamentals. However, the Coriolis coupling constant is small, since without bend-stretch mixing, it
arises only from motion of the central atom,” and thus c T3
is on the order of (mn /m, ). The Coriolis interaction matrix
between the symmetrized local-mode basis functions will be
2<f, (nlplO)(Olqln).
The product of matrix elements
will
be approximately
given by
~~~lplwoJql~)
n!( w/lox] ) ’- “/n, where w is the bond mode harmonic frequency and wx the anharmonicity [see Eqs. (3.17)-( 3.19)
in Ref. 11. The ratio of o to wx is typically - 50 for a hydride.
Thus the Coriolis interaction will decrease exponentially
with increasing n. Halonen16 observed numerically that the
Coriolis interaction decreased rapidly as the local-mode limit was reached. The lack of Coriolis interaction, even in the
fundamentals, was part of the Halonen and Robiette model.
We can also consider rotational corrections to 1. But il is
already a term of order H2n,0 and rotational corrections
would have to be of even higher order and thus smaller still.
If we add rotational corrections toil proportional to J’, they
would split the rotational constants of the s and a states, but
they would not introduce any additional coupling between
the symmetrized local-mode states.
When ;I is much less than the rotational separations,
only local-mode rovibrational states with the same rotational energy will mix significantly. These states will be shifted by *;1 times the overlap of their rotational functions.
Because both local-mode states share the same C axis, only
rotational states that have the same symmetry of Kc, but
different symmetry of K, can have nonzero rovibrational
overlap. As a result, the Hamiltonian can be factored into
four symmetry blocks. If the principle axes rotation is small,
then the rotational function overlap will be near unity. But if
the XY, molecule is near an accidental symmetric top, as
many of the molecules with heavy X are, then this rotation
can be large. In the extreme, the A and B axes will switch
between the two local-mode states. In this case, the rotational overlap will decrease rapidly with increasing K, . This
represents a rotational quenching of the local-mode tunneling because in addition to having to transfer the vibrational
2363
action, one must move the direction of the body fixed angular momentum to find a state of the same rovibrational energyIn order to quantify this discussion, the rotation-tunneling energy levels have been calculated using parameters
that should give a reasonable estimate for the n = 3 overtone
band of SeH, . In particular, from the analysis of the interaction of the fundamentals by Gillis and Edwards,” it is estimated that for the n = 3 levels the spectroscopic constants
are A(s) = A(a) = A, - 3~4’= 7.83, B(s) = B(a) = B,
-3af=7.39,
C=C, -3&=3.74,
d,, =3d,, =0.6,
and /2 = 3A T3/( 8wx2) = - 0.06 cm-‘. These parameters
rotation
of
axis
inertial
imply
& 0.5 tanha;;d_/(As
-B,)]
= f 27” in then = 3 states.
Using the Hamiltonian in Eq. (2) and the parameters just
given, the 82 J = 20 levels have been calculated and are given
in Table I. In Fig. 1 is plotted both the tunneling splitting and
percent mixing of the vibrational character of the eigenstates
as a function of K, - Kc. Rotational energy increases monotonically from left to right in Fig. 1 This high J was chosen to
illustrate the rotational phase space structure of the levels.
The lowest energy levels correspond to K, z 0 and are localized near the Caxis. The s-u splitting is slightly smaller than
2;1, the value predicted without rotation. We can understand
this because these rotational wave functions are almost single oblate symmetric top functions and thus are the same for
both bond modes. With increasing energy, there is an increase in the mixing of the symmetric top functions. When
the rotational is viewed as motion of the angular momentum
vector on the rotational energy surface, the rotational trajectories increasingly dip down toward the B axis as K, is increased.13 This leads to a slow decrease in the effective tunneling splitting, down by a factor of two by Kc = 9. This
results in the a,s vibrational character being mixed by 25%75%. When Kc is decreased further, an interesting effect
occurs. The asymmetry splitting, which reflects tunneling
between the rotation about the + C and - C axes, grows
larger than the tunneling doubling. States for which the
asymmetry splitting is larger than the tunneling splitting are
marked by solid circles in Fig. 1, where it is obvious that
these levels have qualitatively different behavior. Since by
symmetry, the s and a vibrational states can only couple
across the asymmetry doublet, we see a rapid decrease in the
s-u splitting, reaching a value of only 0.0 1 cm - ’ at the level
20’,,6 which is closest to the separatrix separating C- and Atype rotation. The s and a vibrational states are here mixed
by 46%-54%. Not until K, reaches 19 does the asymmetry
splitting fall below the local mode splitting. The states with
K, = 19 and 20 have tunneling splittings of 0.05 1 and 0.02 1
cm-‘, respectively, showing that A-type motion strongly
suppresses the tunneling motion. The greater the difference
between A and B rotational constants, the greater the volume
of rotational phase space that is localized with A axis rotation. As a result, a greater fraction of the levels will show
suppressed inversion splittings. Figure 1 demonstrates an
almost exact inverse relationship between the inversion splitting and the degree of vibrational mixing. Last, it is noted
that when the tunneling splitting is not resolved, the nuclear
spin weights of the observed levels will be equal, i.e., the spin
J. Chem. Phys., Vol. 95, No. 4,15 August 1991
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2364
Kevin K.Lehmann:Overtonespectraofhydrides
TABLE I. Rotation-tunneling
Kp
KO
energy levels in cm - ’ for XH, J = 20.
AE
% mixing
1647.89
0.120
0
?&rib-s
4&b -a
1647.77
0
1
205
1
2
19
19 t
1797.96
1798.08
0.119
0
2
3
18
18 1
1940.41
1940.52
0.118
0
3
4
17
17 I
2075.09
2075.20
0.117
1
4
5
16
16 t
2202.00
2202.11
0.115
2
5
6
15
15 t
2321.12
2321.24
0.112
3
6
7
14
14 I
2432.44
2432.55
0.108
5
7
8
13
13 t
2535.92
2536.02
0.103
7
8
9
12
12 1
2631.50
2631.60
0.096
10
9
10
11
11 I
2719.13
2719.22
0.087
14
10
11
10
10 t
2798.70
2798.77
0.076
18
11
12
9
9I
2870.00
2870.06
0.063
25
12
13
8
8
2932.59
2932.85
2932.61
2932.87
0.023
0.023
42
38
13
14
7
7
2984.82
2987.23
2984.83
2987.25
0.014
0.019
44
42
14
15
15
6
6
5
3022.88
3035.10
3050.94
3022.89
3035.11
3050.96
0.012
0.010
0.022
45
46
40
16
16
5
4
3081.40
3084.89
3081.41
3084.95
0.007
0.058
47
26
17
17
4
3
3131.44
3131.81
3131.45
3181.85
0.010
0.042
47
32
18
18
3
2
3187.53
3187.56
3187.54
3187.56
0.005
0.001
48
49
19
19
2
11
3249.63
3249.68
0.051
28
20
20
1
0I
3317.29
3317.31
0.021
41
0.12
3
F
c
z
0.08
0.04
or
o-,,,.‘,‘,‘,,““,“-‘,
-10
-20
IO
0
20
Ka-Kc
alternation is lost because the hydrogens are now dynamically inequivalent.
XH, MOLECULES
In constructing an effective rotation inversion Hamiltonian for an XII, molecule the same physical assumptions
Ka-Kc
FIG. 1. Figure showing the (a) tunneling splitting and (b) the percent mixing of the vibrational character of the J = 20 eigenstates of the Hamiltonian
given in Eq. (3), using spectroscopic parameters estimated for the second
overtone of SeH,. States are plotted as a function ofK, - Kc, where Ka and
K, are the prolate and oblate labels of each asymmetric top level. States
represented by open circles have an asymmetry splitting less than the localmode tunneling splitting while the opposite is true of those plotted with
filled circles.
will be used. The differences are that we now have three
degenerate local-mode states that are connected by tunneling motion and that the effect of the vibrational motion is to
rotate the principle axes of the effective moment inertia tensor by an angle 8 in the plane defined by the C,, axis and the
bond that is excited. The axes for this molecule are laid down
with z along the C,, axis, and bond 1 in the y,z plane. The
axes are labeled as if the molecule is an oblate symmetric top
in the ground state. Thus for the 1n , ) state, the B axis will be
along x, the A axis rotated from y, and the C axis rotated
from z. The inertial axes for the states In, ) and In, ) are
related to those of 1n , ) by a rotation of f 120”around the z
axis. As a result, the a rotational-tunneling
Hamiltonian for
the three states Ink) can be written,
k#l
Jx, = cos BJ, + sin t9J,,
Jy, = Jy ,
Jz, = - sin 6J, + cos 0J,,
-$Jx
--iJy
1
+sinOJ,,
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Kevin K. Lehmann: Overtone spectra of hydrides
J Y2=aJx
2
Jz, = -sine
--$Jx
Jx, = cos e $f-Jx
[
J,, = -$Jx
A,B, and Care the effective rotational constants in the localmode state, and 13is the angle that the C inertial axis is rotated relative to the symmetry axis. Defining symmetrized local
mode states by
-+Jy,
[
1
-+Jy
-$Jy
2365
+coseJ,,
1+sinBJ,,
-+Jy,
-+Jy
1
H = [G,,+U+B,(J:+J;)
+coseJ,.
+C,J:]la)(ul+
(6)
I
we can transform the Hamiltonian
tional basis
[G, -A+B,(Jz
+J:)+C,J~][le+)(e+
+~[-~rCJ,~J~~+~J2,][l~~~~+I+l~+~~~l+l~+~~~-ll
++[ +idJ,,J+)+qJ’][IdbI + le- )@I +k- )(e+ II,
I
where
& =B, =~[B+AcosZe+Csin2e],
to a symmetrized vibra-
I+k->(e-
II
(8)
-r;l,
J- =J, +I;r,.
(9)
Once again, the form of the terms is the same as Halonen
and Robiette, but the present Hamiltonian is not as restrictive as to the relationship of the parameters. The relations of
Halonen and Robiette are recovered by setting B = C and
0 = 125.26” as required by their model. One obvious feature
is that the spectrum should, in the limit of small R, be that of
an asymmetric top with A,C hybrid band. This was predicted
by Ovchinnikova” who derived a model for the high overtone bands of NH, that is similar to the present one. Halonen and Robiette found that in their model, the local-mode
rotational structure of XH, should be that of a symmetric
top. This is due to Halonen and Robiette’s assumed geometry, which resulted in the molecule being an accidental
spherical top in the ground state. As a result, the vibrating
bond ensured that one axis would be unique, but the other
two retained the same moments of inertia and thus a symmetric top spectrum resulted. Notice, however, even in this
limit the quantization axis is one of the bonds, not the symmetry axis and thus the rotational levels will not have the 2: 1
nuclear spin weight alternation of the ground vibrational
state.
Like before, in the limit of small il, the tunneling
between rovibrational levels should be suppressed by a rotational overlap factor. Table II contains the J = 20 rotational-tunneling energy levels of a XH, molecule calculated
with A = 3.72, B = 3.61, C= 3.40, il = - 0.01 cm-‘, and
8 = 0 and 12”. These rotational constants, and the latter angle, are approximately those expected for ASH, at n = 3
based upon the analysis of the fundamentals by Olson et aLI9
When 8 = 0, the different local-mode states have their A
axes differing by 60” but have the same C axis. As for the
XH, case, the states with lowest energy (rotation around the
and the vibrational states are essentially unmixed. The degenerate E vibrational state is, however, split by a small
amount. As the energy rises, there is a decreasing tunneling
rate which drops to one-third the rotationless value when
K, = 13. When the asymmetry splitting grows to be larger
than the tunneling rate, the tunneling rate is strongly suppressed further, reaching a rate only 2% as large as the rotationless value for the state (2O,,,, ) localized on the separatix
between A and C axis motion. When asymmetry splitting
once again becomes negligible, at KP = 16, the tunneling
rate is back to 25% of the nonrotating value. With increasing
KP, the tunneling splitting again decreases, forming a pattern of A-E-E-A
levels, with a splitting 1:2:1. For
K, = J = 20, the total splitting of the pattern is only 6% of
the bare vibrational value. The rate of decrease of the tunneling rate for K, -J levels will be larger the more asymmetric
the top. This is because, while the distance that the A axes
must be rotated to overlap is constant at 60”, the more asymmetric the top, the more localized the highest rotational
states will be near the A axis. Again, if the tunneling splitting
is unresolved, the observed levels will have identical nuclear
spin weights unlike the 2-l alternation expected for a C,,
symmetric top. This is expected since the local-mode state
has no rotational axis of symmetry.
When we look at the energy levels for 8 = 12”, the most
striking change is that the lowest energy levels (K, w J) now
have a suppressed tunneling splitting. This is because the
rotation of the Caxis means the rotational wave functions of
different local-mode states no longer overlap strongly. By
the time K, decreases to a value of 13, the tunneling splitting
is close to that of the 8 = 0 case. For such values of K,, the
rotational functions are sufficiently spread out on the sphere
that a 12” rotation has little effect on the overlap. For the
state 201,,, , which is localized on the separatix, the C axis
rotation increases the overlap, and the tunneling splitting is a
factor of 10 larger than was calculated with no rotation of the
Caxis (0 = 0). The state with K,, = J has a further reduced
splitting. This can be understood since the A axes of the different local-mode states are now 64”instead of 60”apart. It is
the nature of tunneling that such a small change in rotation
C axis) have essentiallyan unperturbedtunneling splitting
angleleadsto a factor of 4 reduction in the tunneling rate.
C, =C,=Asin26+Ccos26,
r=(A-C)sin28,
q=Ac0s~8+Csin28-B,
J,
=J,
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Vol. 95,toNo.
4,15
Augustor1991
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Kevin K. Lehmann: Overtone spectra of hydrides
2366
TABLE II. Rotation-tunneling
energy levels for XH, J = 20.
R = 0.0
Energy
A = - 0.01 (cm-‘)
l9= 12’
A = -0.01 (cm-‘)
8=(r
rm
Energy
0.161 73
0.191 38
0.191 55
1433.18155
% *iz
Energy
l-N
% A
E
+A,
E
99
0
0
0.173 15
0.181 58
0.189 90
E
E
K. = 20
A, +A,
33
61
6
E
E
0.275 77
0.276 68
0.283 53
A, +A,
E
E
43
40
17
K. = 19
& = 18
I-N
4
95
0.259 22
0.288 10
1 0.288 65
A, +A,
98
2
0
0.823 84
0.851 32
1 0.852 27
A, +A,
E
E
95
4
1
0.837 54
0.838 41
0.851 49
A, +A,
50
47
3
1461.870 04
0.852 67
0.878 03
I 0.879 41
E
A, +A,
E
91
7
2
0.865 25
0.866 52
0.878 34
E
A, +A,
E
49
45
6
K,, = 17
1470.357 34
0.341 75
0.364 24
i 0.366 02
0.354 30
0.354 59
0.363 11
E
-4 i-4
E
43
43
14
K. = 16
A, +A,
85
10
4
0.285 65
0.30444
I 0.306 56
A, +A,
E
E
77
15
8
0.295 31
0.299 13
0.302 21
K, = 15
A, -i-A,
45
33
22
0.676 59
0.690 89
I 0.693 21
A, +A,
E
67
20
12
0.681 80
0.688 77
0.690 15
A, +A,
E
E
50
27
23
K. = 14
0.502
0.511
0.515
0.515
90
63
47
50
E
E
4
A,
57
28
15
15
E
1443.278 66
1452.842 48
1478.298 88
1485.686 90
1492.509 99
E
E
E
0.503
0.512
0.514
i 0.514
17
29
53
56
4
A,
56
25
18
18
E
E
E
E
E
E
K. = 13
1498.750 26
0.750 92
A, +E
A, +E
0.747
0.747
0.751
t 0.753
32
98
12
00
4
A,
E
E
43
43
31
26
0.741
0.752
0.753
0.757
73
61
27
11
A2
A,
E
63
25
25
12
1504.374 62
0.383 99
A, +E
A, +E
0.373
0.374
0.383
1 0.384
80
99
17
45
4
E
A,
E
36
32
36
32
0.365
0.374
0.379
0.389
CO
34
24
01
4
A,
E
E
65
65
19
16
A, +E
A, -I-E
0.290
0.294
0.384
1 0.388
12
41
24
69
A*
E
A,
E
43
29
43
10
0.290
0.297
0.385
0.391
36
50
23
86
E
1509.293 04
0.387 14
4
E
A,
44
18
38
18
1513.227 16
0.821 06
A, i-E
A, +E
0.224
0.232
0.817
i 0.827
54
37
83
53
E
A2
E
0.225
0.230
0.819
0.825
60
21
01
05
E
4
E
A,
42
16
44
12
A,
39
23
40
20
0.026 07
1 0.027 34
E
A2
35
30
0.025 94
0.026 78
4
E
35
33
(123)
0.969 01
1 0.969 73
A,
E
35
33
0.965 25
0.971 60
A,
E
47
26
(13,a)
0.703 16
0.704 57
‘6
E
36
32
0.703 58
0.705 14
E
4
35
30
(13,7)
1516.026 50
1517.969 48
1518.704 10
A, i-E
A, -I-E
A, A-E
K, = 12
K. = 11
K. = 10
K. =9
J. Chem. Phys., vol. 95, No. 4.15 August 1991
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Kevin K. Lehmann: Overtone spectra of hydrides
TABLE
II. (Continued.)
A = 0.0
A = - 0.01 (cm-‘)
O=o”
Energy
rN
Energy
A,
57
21
43
28
0.278
0.280
0.396
0.406
52
56
48
39
48
4
34
33
0.062
0.063
0.074
0.080
44
88
a4
67
52
37
24
26
0.393 60
0.396 47
0.400 41
0.40103
A,
37
36
27
32
0.248
0.251
0.256
0.259
64
13
61
40
4
57
45
21
10
0.592
0.595
0.601
0.604
42
36
24
17
4
49
41
25
17
0.404
0.405
0.407
0.408
69
65
56
52
‘42
36
35
32
30
0.661
0.662
0.662
0.662
94
04
25
36
4
0.272
0.282
0.399
0.404
05
a7
94
57
1527.063 68
0.076 02
A, +E
A, +E
0.059
0.072
0.075
0.076
35
35
a7
33
E
Al
E
0.392
0.397
0.400
0.400
03
36
46
68
A,
0.252
0.253
0.254
0.255
87
06
42
70
Al
1538.253 90
0.253 94
A, +E
A, +E
A, -I-E
A, +E
f 0.594
0.605
0.601 77
0.591
32
a5
17
E
4
E
A2
E
E
4
E
E
4
4
E
E
A,
1544.598 29
1551.406 60
1558.662 14
0.401
0.404
0.409
0.411
79
20
01
42
4
0.661
0.661
0.662
0.663
29
72
58
01
4
E
E
A,
E
E
A,
XH, MOLECULES
We now consider the general XH, molecule with tetrahedral symmetry. We now have four degenerate local-mode
states In,) with k = 1,4. Without vibrational excitation, the
molecule
is a spherical top. In each local-mode state, the
molecule will have its symmetry dynamically lowered to
C,,. It will thus have a prolate symmetric top rotational
Hamiltonian with the A aligned along the excited bond, We
set up the tetrahedral molecule such that the four X-H
bonds l-4 point in directions (l,l,l),
(1, - 1, - l),
( - l,l, - l), and ( - 1, - 1,l 1, respectively. Thus we
write an effective tunneling-rotation
Hamiltonian as
=
i
k=l
[Go
+BJ*+
Energy
% *o
A, +E
A, i-E
1532.397 65
0.398 47
A = - 0.01 (cm-‘)
e= 120
r,
1522.279 21
0.403 02
H
2367
r,
% *a
E
36
29
55
22
A,
A2
E
E
A,
E
*,
Kp = 14
41
33
33
la
A2
K, = 15
47
39
25
25
E
E
A*
K, = 16
51
43
24
15
E
E
A,
K, = 17
53
43
23
14
E
E
A,
K,=la
40
36
30
27
E
E
A,
K, = 19
34
34
33
33
E
E
A,
K, = 20
ek are unit vectors pointing along the four bonds. We introduce symmetrized vibrational states by
Ia) =i(ln,>
+ In*) + In,> + In,)),
I&) =:(b,)
+ In*) - 14) - ln4)),
IQ =I(br> - 1%) + I%> - In‘s)),
14) =:(I%) - In*>- I%> + In,)).
(12)
Substituting in for Ink) (nk I we find
f~ = [Go + 3A + B,J’]
la>(al
(A-B)(Bk.J)*]Ink)(nkI
+idcZt
(10)
B, =B,
(&
{Ji9Ji)[la>(rkl
=f(2B+A)
+
Itk)(al
+
Iti>(tjl],
da, ={(A--),
(13)
where
U,j,k)
is a cyclic
permutation
of (x,y,z). Introducing
J. Chem. Phys., Vol. 95, No. 4,15 August 1991
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Kevin K. Lehmann: Overtone spectra of hydrides
2366
TABLE III. Rotation-tunelliug
4
K=O
K=l
cl
E( rot)
% *”
AEX 10’
K,
A,
/ F,
4
4
1E
839.995 74
40.00123
36
22
549
K=
40.095 72
0.101 29
0.104 38
36
22
14
557
309
40.394 99
0.399 18
0.404 15
37
27
15
419
496
E
F2
14
K=2
energy levels in cm - ’ XH, J = 20.
r”r
E(rot)
% 4”
AE x 10’
E
6
F,
52.092 02
0.100 42
0.104 93
45
24
13
841
451
54.388
0.399
0.401
0.403
80
43
al
75
53
26
20
16
63
238
194
4
56.898 38
0.898 68
0.902 43
28
28
19
30
375
14
E
4
59.596 30
0.599 66
0.603 97
34
26
15
336
431
4
A2
62.494
0.498
0.501
0.505
89
33
71
02
38
29
21
12
344
340
331
E
Fl
65.594 10
0.599 99
0.605 94
40
25
10
589
595
68.896 86
0.900 03
0.903 la
33
25
17
317
316
72.398 76
0.899 60
0.400 43
0.40127
28
26
24
22
a4
a3
a4
11
K=12
1
4E
K=
13
4
Fz
AZ
t A,
40.897
0.899
0.902
0.908
20
11
51
52
32
27
18
4
191
339
601
E
4
I4
41.594 86
0.600 50
0.602 91
38
24
17
564
241
4
F2
1 E
42.499 03
0.499 16
0.592 71
28
27
la
13
355
F2
A2
8
f A,
43.598 66
0.599 72
OACO 12
0.603 98
28
26
25
16
lo6
40
386
K=7
E
F2
I 4
44.894 15
0.899 04
0.904 a7
40
27
13
489
583
K=
la
A,
4
4
i 4
K=8
4
F2
1 E
46.394 85
0.402 54
0.493 a9
38
19
15
769
135
K=
19
F2
E
I F*
76.099 90
0.10001
0.100 12
25
25
25
11
11
K=9
4
A,
4
i A2
48.096
0.099
0.100
0.109
78
31
24
67
33
27
24
1
253
93
843
F2
E
1 F1
ao.ooo oo
ao.ooo 01
ao.oco 01
25
25
25
1
0
E
4
i F2
49.994 93
50.000 46
0.002 a9
38
24
18
553
243
K=3
K=4
K=5
K=6
K=
10
the tensor operators of Robiette, Gray, and Birss,” one finds
the same relationship for their coefficients as found by Halonen and Robiette. This relationship has also been found to
be approximately correct for the XH, overtone spectra that
show symmetric top rotational structure.4*6*8
When A ( 2K(A - B) , the energy levels will cluster into
a groups of eight quasidegenerate states made up of basis
functions Ink,& f K >. These basis functions will be connected by tunneling integrals of two different magnitudes
A, =Ad;,,(lW)
and
A, = Ad ;, _ K ( 1090)
= Ad & (71”). For K = 3N, the phase factors of the inte-
K=
14
K=15
I
K=16
K=
4
17
i
K=20
4
E
4
grals will be equal, and these eight functions will produce a
cluster of four energy levels with shifts 3 (A, f A2 > (A, and
A,)and - (A, f/2,) (Fr andF,).ForK=3Nf
1,weget
three states F, , F2, and E. The order and splitting of these
levels depends upon the size of A, and A,. If we look at the
J= K levels, then the tunneling rate will fall off rapidly,
since d&(109”) = (1/3jJ and d&(71”) = (2/3)J. This
will produce the largest rotational quenching of the localmode tunneling, since the angular momentum vector needs
to move the furthest to reach a configuration of equal rotational energy.
J. Chem. Phys., Vol. 95, No. 4,15 August 1991
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Kevin K. Lehmann: Overtone spectra of hydrides
The rotation-inversion level structure will depend (except for J dependent scaling and shift) only on the ratio of
(A - B) and R. In Table III is given the rotation-inversion
energy levels calculated for a XH, top with A = 2.1,
B = 2.0, andil = - 0.01 cm - ‘. These constants where chosen to emphasize the local-mode limit. For these parameters,
the tunneling splitting of the nonrotating molecule is 0.04
cm-’ . Unlike the previous examples, all of the J = 20 levels
have a substantial reduction in the tunneling rate, the largest
splitting being only about 20% of the rotationless value. It is
also shown that the K = J levels have the smallest splitting,
predicted
above.
But
when
?i(J
+ l/2) = cos( 35”) = 0.82 (which implies K = 16.7
for J = 20), the classical trajectories for K with one bond
excited and - K with another bond excited overlap at a
tangent. The calculations reveal that the K = 16 is a local
maximum in the tunneling splitting. If one looks at
K/(J + l/2) = cos(54”) = 0.58(K=
11.8forJ= 20),the
classical trajectories for K with one bond excited and K with
another bond excited now overlap at a tangent. Again, the
splitting of the level K = 11 is a local maximum in the calculated splitting. The slow decrease in the splitting as one goes
to K = 0 can be understood by the fact that while the classical trajectories continue to cross, they cross at a steeper angle, and the width of the quantum states gets narrower, thus
leading to reduced overlap.
It is interesting that in the published figure of the stannane overtone,6 the R (6) transition clearly shows a reduction of the tunneling splitting as K increases. This spectrum
is only partially in the local-mode limit in that R and A - B
are comparable. Since there will be Stark transitions across
the tunneling levels (with matrix elements that scale with
the size of the vibrationally induced dipole of the local-mode
2369
state), some type of double resonance experiment would be
ideal for determining the splitting pattern.
EXTENSIONS
This initial work has presented a general procedure for
producing rotational-tunneling
Hamiltonians for a set of
nearly degenerate local mode states. One can clearly extend
the work to include quartic and sextic centrifugal distortions
terms in the local-mode Hamiltonians. For a quantitative
treatment of hydrides, both terms are usually required. The
most straightforward way to do a calculation is to use a basis
set of unsymmetrized local mode states and use rotational
wave functions for each vibrational state aligned with it’s
own inertial ellipsoid. Then vibrationally diagonal couplings
can be treated by a standard program. The tunneling matrix
elements will be izD JK,,K,(#,19,x), where #,19,x are the Euler
angles to convert one local-mode principle axis system into
another Alternatively, one can use a common set of rotational functions, aligned with the symmetry axes, for all the
local-mode vibrational functions, and use the rotation matrices to transform each of the rotational basis function. This is
likely the preferred method, since it allows a convenient calculation of transition intensities from the ground state,
though the basis functions are not of definite symmetry, and
symmetry assignments need to be made by examination of
the eigenvectors coefficients of symmetry related basis functions.
In order to work in a symmetrized local-mode basis, one
needs to convert higher powers of the rotational operators,
as give above for the quadratic terms. For the tetrahedral
molecule, the terms produced by including DJJ, DJK, and
DKK in the local mode, symmetric top, rotational Hamiltonian are given below:
I
HD = HD = -
D,,J4+DJKJ2(~k.J)2+D~~(~k.J)4](nk)(nk(,
D,J” -+dJ:
+ J; + J:)][ lh.4
+ 7 \td(b(,
-3
D.xJ* 2 CJi,JiI[ la)(f/cl +
Ifk)(al
+
Iti)(rjl]
(iik)
-- 1 DKK
DLl = D.,.,
9
(ijv )
-I- :D.,K
-I- Pm.
SCJf,CJi,JkII
+ 2CJj,JkI +2CJj,J:I
-7CJi,JkI] [ IQ>(fiI + Itt)(al A- Ifj)(fkl]9
(14)
I
Only some of these terms appear in the Hamiltonian used by
Halonen et al. to fit the spectrum of the first overtone of
stannane.6 Such selective inclusion of the terms will destroy
the high J,K local-mode energy patterns of the spherical top.
In fact, the rovibrational levels predicted by the final fitted
constants of Halonen etal. deviate from the expected pattern
of falling inversion splitting for the high J = K lines when J
become larger than those used in their fit.*l I believe this is
due to selective inclusion of terms. Including all of the terms,
but imposing the above constraints on the spectroscopic constants, has a firmer theoretical basis and may well prove a
nian is much easier to write (andprogram) in the unsymmetrized basis. As one goes to the sextic distortion terms and
beyond, which are often needed for a quantitative fit to the
rotational structure of a hyride, the advantage of the unsymmetrized basis only increases.
In fitting the stannane overtone spectrum, Halonen et
al. determined a diagonal (in the symmetrized basis) Coriolis term, but 2Bc3 was only - 0.0042 cm - ‘. Such terms are
easily included in the symmetrized local-mode Hamiltonian
since they take the same form as the interaction between the
fundamentals. In the unsymmetrized basis, one will have a
practical advantageas well. Clearly, the distortion Hamilto-
vibrational angular momentumbetweeneachpair on local-
J. Chem. Phys., Vol. 95, No. 4,15 August 1991
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2370
Kevin K. Lehmann: Overtone spectra of hydrides
mode vibrations, pointing in the direction of the cross product of the two bonds. One can use Morse oscillator matrix
elements to estimate the size of the expected Coriolis term.
Halonen ef aL6 also found that a Hz3 higher order Coriolistype interaction between the A and F, symmetrized localmode states could also be determined from the spectrum.
This term does not show up in the present treatment. It could
be generated by adding rotational corrections to the Coriolis
coupling between the bond modes, but it is desirable to have
a physical model for this interaction so that its magnitude
can be estimated from a model force field.
CONCLUSIONS
The present paper extends the earlier work of Halonen
and Robiette to provide a systematic treatment of the interaction of rotation and local-mode tunneling. Effective Hamiltonians are developed, both in an unsymmetrized and a
symmetrized basis. The unsymmetrized basis provides a natural framework for elucidating how rotational motion can
reduce the local-mode tunneling rate. At present, it is only
applicable when the interaction of the local-mode polyad
with other vibrational states can be treated perturbatively,
i.e., by an effective rotational Hamiltonian. Cases for Fermi
resonance, for example, will require extension but can likely
be handled in the same framework. Also, the treatment can
be generalized to cases of the local-mode combination bands,
with excitation in more than one bond mode, or with a bending mode excited.
Note added inproo$ Independently, M. S. Child and Q. Zhu
have shown that for an XH, molecule, the normal mode
Hamiltonian [ Eq. ( 131 and the Halonen and Robietie constraints can be transformed to the symmetric top, local mode
Hamiltonian [Eq. (lo] thus confirming one aspect of the
present work. [ Chem. Phys. Lett. (to be published) I.
ACKNOWLEDGMENTS
I want to thank the Department of Physical Chemistry,
University of Helsinki, for their hospitality during the initial
stage of this project; Lauri and Marjo Halonen and Quingshi
Zhu for helpful discussions on rotational structure in the
local-mode limit. This work was supported by the National
Science Foundation and the Donors of the Petroleum Research Fund, administered by the American Chemical Society.
’M. S. Child and L. Halonen, Adv. Chem. Phys. 57, 1 ( 1984).
2 L. Halonen and A. G. Robiette, J. Chem. Phys. 84,866l ( 1984).
’ F. W. Birss, Mol. Phys. 31,49 1 ( 1976).
4Q. Zhu, B. A. Thrush, and A. G. Robiette, Chem. Phys. Lett. 150, 181
(1988).
“Q. Zhu, B. Zhang, Y. Ma, and H. Qian, Chem. Phys. Lett. 564, 596
(1989).
bM. Halonen, L. Halonen, H. Burger, and S. Sommer, J. Chem. Phys. 93,
1607 (1990).
‘F. Michelot, J. Moret-Bailly, and A. DeMarino, Chem. Phys. Lett. 148,
52 (1988).
*M. Chevalier, A. DeMartino, and F. Michelot, J. Mol. Spectrosc. 131,382
(1988).
9 I. M. Mills and A. G. Robiette, Mol. Phys. 56, 743 ( 1985).
‘OK. K. Lehman, J. Chem. Phys. 79, 1098 (1983).
“M S. Child and R. T. Lawton, Faraday Discuss. Chem. Sot. 71, 273
(lb81).
“K. K. Lehmann, Mol. Phys. 66,1129 (1989).
“R. G. D. Della Valle, Mol. Phys. 63, 611 ( 1988).
I4 W. G. Hatter and C. W. Patterson, J. Chem. Phys. 80,424l (1984).
I5 See Eq. ( IO) of Ref. 2.
16L. Halonen, J. Chem. Phys. 86, 588 (1987).
“J. R. Gillis and T. H. Edwards, J. Mol. Spectrosc. 85, 74 ( 1981).
‘*M. Ya, Ovchinnikova, Chem. Phys. 120,249 ( 1988).
19W. B. Olson, A. G. Maki, and R. L. Sams, J. Mol. Spectrosc. 55, 252
(1975).
“A. G. Robiette, D. L. Gray, andF. W. Birss, Mol. Phys, 32,1591, ( 1976).
” M. Halonen and L. Halonen (private communication).
J. Chem. Phys., Vol. 95, No. 4,15 August 1991
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