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Energy level promotion in the correlation from the tunnelling-doubledharmonic oscillator to the bi-rotor: application to internal rotation inmolecules
Stephen C. Rossab and Koichi M. T. Yamadacd
Received 12th July 2007, Accepted 7th September 2007
First published as an Advance Article on the web 20th September 2007
DOI: 10.1039/b710691b
A surprisingly rich variety of phenomena are revealed in the energy level correlation between the
limits of a tunnelling doubled harmonic oscillator and a bi-rotor. Some levels are found to have
their vibrational quantum number promoted upon removal of the barrier to rotation, other
levels, which we dub invariant, are found to be completely independent of the barrier, while yet
other levels exhibit a smooth transition between these limits. The general nature of these features
can be understood in terms of the different degeneracies of the limiting cases. The elucidation of
these effects aids the understanding of the rotationalvibrational energy levels of molecules having
two internal rotor moieties.
I. Introduction
A variety of simple solutions of the Schro dinger equation are
valuable as textbook examples and also as the basis for
understanding more complicated systems. The hydrogen
atom, the particle-in-a-box, and the harmonic oscillator are
prime examples. Less commonly cited is the free rotor. The
free rotor problem is analogous to that of a particle-in-a-box,
but with periodic boundary conditions. These result in quali-
tative differences in the solutions of the Schro dinger equation.
Although the energy level spectrum in both systems is of the
form Ep n2, the lowest level for the free rotor is a unique level
of the form f constant. All higher levels have sinusoidalwavefunctions, as do all the levels of the particle-in-a-box
problem, but in the free rotor case these levels are doubly
degenerate, consisting of 901 phase-shifted pairs. As we will
show below, this difference in degeneracies leads to unusual
features in the energy level spectrum of molecules possessing
two internal rotor moieties.
Increasing the dimensionality of these systems enriches their
physical features. For example, angular momentum appears
naturally in a two-dimensional isotropic harmonic oscillator.
The case of a double free rotor is the analogous extension of
the free rotor to a higher dimensionality and forms the
unhindered limiting case for internal rotation seen in many
classes of molecules. Moieties exhibiting internal rotation canbe symmetric tops (as in ethane, C2H6) or consist of a single
atom as in the systems we consider here, exemplified by HSSH,
shown in Fig. 1. We concentrate on the case of equivalent
rotor moieties, briefly mentioning the non-equivalent case at
the end.
Internal rotation can be hindered to a greater or lesser
extent. The energy level patterns vary dramatically with the
potential energy barrier to internal rotation, H, correlating
with two limiting cases: a tunnelling-doubled harmonic oscil-
lator within the limit of a high barrier and a pair of free rotors
within the limit of no barrier. More physical intermediate
cases have presented a continuing challenge in molecular
physics. The assignment of molecular rotationalvibrational
energy levels, both in experiment and in theoretical calcula-
tions, is not trivial. For example, in the case of non-equivalentrotors, recourse has been made to the extended group
method of Hougen and DeKoven.1 Their method could be
Fig. 1 Schematic diagram of HSSH type of molecules. The signed
quantum numbers ni, i 1, 2, are used for the internal rotation of each
end of the molecule. Unlike what is shown here, the GSRB calcula-
tions have the x-axis fixed along one of the SH bonds.
a Department of Physics and Centre for Laser, Atomic, and MolecularSciences, University of New Brunswick, P. O. Box 4400,Fredericton, New Brunswick, Canada E3B 5A3
b Infrared Free Electron Laser Research Center, Tokyo University ofScience, 2641 Yamazaki, Noda, Chiba, 278-8510, Japan
c National Institute of Advanced Industrial Science and Technology(AIST), EMTech, AIST Tsukuba West, Onogawa 16-1, Tsukuba,305-8569, Japan
dA&A Laboratories, Sengataki Naka-ku 523-1, Karuizawa, Nagano,389-0111, Japan
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dubbed the rational approximation group theory technique
relying, as it does, on a rational approximation to the ratio of
the moments of inertia of the rotor moieties. Here we wish to
proceed more directly to an understanding of the energy
structure and wavefunction form for such molecules.
In 1928 Wigner and Witmer2 and Mulliken,3 building on
work of Hund,46 showed that consideration of the correlation
between the limit of atomic electronic states and the limit of
molecular electronic states is a powerful tool for understand-
ing the electronic structure of diatomic molecules. Here we
show that consideration of the correlation between the limit of
an unhindered bi-rotor (i.e. one with no barrier to internal
rotation) and that of a tunnelling-doubled harmonic oscillator
with a strong barrier to internal rotation is also very useful. In
the correlation we find invariant levels, i.e. levels unaffected
by the barrier height, along with levels that present a feature
analogous to the promotion of quantum orbitals that can
occur as two atoms, brought together to form a diatomic
molecule.3 Combined with other levels that behave more
normally, this correlation provides a coherent picture of the
otherwise complicated energy level structure in molecules
possessing two internal rotor moieties and an intermediate
degree of hindering.
II. Correlation diagram obtained from GSRB
We explore this correlation using the generalised semirigid
bender (GSRB) program. The GSRB was recently used in the
elucidation of the nature of quantum monodromy in quasi-
linear molecules7,8 and is an extension of the semirigid bender
model of Bunker and Landsberg9 which is itself an extension
of the HougenBunkerJohns (HBJ) Hamiltonian.10 The HBJ
approach treats the rotation and one large-amplitude vibra-
tion of the molecule by introducing a four-dimensional gen-
eralisation of the moment of inertia matrix which depends on
the coordinate for the large-amplitude motion r. We choose r
as the angle between the two HSS moieties. One molecular
reference axis is fixed in the plane of one of the HSS moieties
of the molecule, leading to strong coupling of the vibrational
rotational basis and requiring the use of a somewhat larger
basis set. However, this choice also means that increasing r
from 0 to 2p brings the system identically back to its initial
situation at r 0. (If the reference axis is fixed in the bi-rotor
bisector plane then r would have to increase to 4p to bring the
reference axes and the molecule back to the initial situation,
giving rise to unphysical levels that must later be discarded.11)
The figures below give GSRB results for a molecule with
geometry and masses akin to those of HSSH.
For a conceptual understanding of the GSRB results we
take a simplified view of the two limits of the bi-rotor system.
First we consider the free rotor (unhindered, or H 0) case.
As shown in Fig. 2 we use ni, i 1, 2, as signed quantum
numbers for the internal rotation of each end of the molecule,
with the sign indicating the sense of the rotation. Such internal
rotation gives rise to a component, k, of angular momentum
along the z-axis, k n1 n2. If n1 a n2 the molecule is
internally twisted and the resulting signed torsional quantum
number is n n1 n2. Molecules of the type HSSH typically
have potential energy minima at two skew angles with two
potential energy barriers to twisting (properly called inter-
nal rotation) when the SH moieties are parallel or antiparallel.
For simplicity of illustration, we consider the case of equal
potential energy barriers at 90 and 2701. In a molecule with
identical rotor moieties and H 0 the energy due to internal
rotation takes the particularly simple form of a sum of
squares,
E An2 Ak2 2An21 n22 2Aw: 2:1
Here A is the inertial constant about the z-axis of the full
molecule and w n12 n2
2. The squares in this expression
mean that we can use the unsigned quantum numbers Ni
|ni|
to label the bi-rotor energy states as [N1, N2]. For our
prototype HSSH molecule these bi-rotor energy state labels
Fig. 2 The correlation of energy levels for J 3 from the free rotor limit (H 0 cm1) to the deeply hindered case, plotted relative to the energy of
the lowest J 0 level. The broad, slightly curved, line is the potential energy barrier. Bi-rotor state labels [ N1, N2] are given at the left, tunnelling
doubled harmonic oscillator labels v are given at the right in (b). For each Klevel the approximate degeneracy (on the scale of this figure) is given
in parentheses. The quadruply degenerate K 3 level divides into two pairs of approximately degenerate levels. In (b) barrier-invariant levels are
labelled invariant while promoted levels are labelled promotion.
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8/3/2019 Stephen C. Ross and Koichi M. T. Yamada- Energy level promotion in the correlation from the tunnelling-doubled h
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are shown in the left of each panel of Fig. 2. We label the
individual substates of each bi-rotor energy state using signed
quantum numbers as (n1, n2), where the first entry represents
the signed angular momentum quantum number of one rotor
moiety, and the second that of the other rotor moiety. As for
the two-dimensional harmonic oscillator, bi-rotor states can
be degenerate. With identical rotor moieties a bi-rotor state
[N1, N2] of given energy can be formed in eight distinct ways:
(N1, N2), (N1, 8N2), and the same with N1 and N2switched. However, if N1 N2, or if one of them is 0, there
are only four distinct combinations, if both are 0 there is only
one combination. Thus, at a particular energy, we either have
eight-fold degeneracy, four-fold degeneracy, or no degeneracy.
For higher levels accidental degeneracies, such as 02 52
w 32 42 and 12 72 w 52 52, give rise to non-
equivalent polyads, further enriching the energy level struc-
ture.
The picture just presented is somewhat modified when we
account for the energy of the end-over-end rotation of the
molecule, not just the component along the z-axis. The total
angular momentum quantum number, J, must be at least K
|k | |n1 n2 | , leading to the relationship between the internal
rotation quantum state and the allowed rotation levels J Z
K |n1 n2 | . This is analogous to the link between the
possible values of the angular momentum and the vibrational
quantum numbers in the case of the two-dimensional harmo-
nic oscillator. End-over-end rotation increases the level energy
given in eqn (2.1) to give
E 2Aw BJJ 1 K2; 2:2
where B is the effective inertial constant for rotation perpen-
dicular to the z-axis. The K2 term breaks a given [N1, N2] state
into separate Klevels, thereby reducing the degeneracy, as seen
on the left part of Fig. 2a and b.
The other bi-rotor limit is the case of high barrier. The
rightmost panel of Fig. 3b shows that within this limit there
are two, nearly harmonic, equivalent potential energy wells
separated by a cyclic barrier. We therefore expect almost
degenerate symmetric and antisymmetric combinations of
harmonic oscillator wavefunctions for each well. These are
the tunnelling doublet states which we label with the vibra-
tional quantum number of the wavefunction in one well, v,
along with a or superscript for the symmetric and
antisymmetric combinations, respectively, that is as v. We
write the corresponding wavefunctions as |vi. With v defined
in this manner the energy levels within this limit are, approxi-
mately,E v 1=2hw: 2:3
Fig. 2 shows the GSRB calculated energies as a function of
barrier height for J 3. All quantities in this figure are plotted
relative to the energy of the J 0 level correlating with the
[0, 0] bi-rotor level. Thus the barrier energy, H, slopes upward
in a curve rather than in a straight line. The energy levels are
labelled on the left by the appropriate bi-rotor label, [N1, N2],
and in the right part of Fig. 2b by the appropriate tunnelling
doubled harmonic oscillator label v. Each level is also
labelled with K, the unsigned quantum number for the projec-
tion of the angular momentum along the molecular z-axis.
III. Degeneracy, invariance, and promotion
On the scale of Fig. 2 most of the levels shown are degenerate
with the degree of degeneracy indicated in parentheses. For the
K 3 level shown here we write the degeneracy as (2 2) since
splitting into two pairs of levels is just becoming visible. This
breaking of the degeneracy is briefly discussed further below.
For even-K in the free rotor limit the lowest bi-rotor state is
[K/2, K/2]. For a particular K this level only has 2-fold
degeneracy: (K/2, K/2) and (K/2, K/2), (so only 1-fold if
K 0). In the hindered limit the degeneracy must be 4-fold
(2-fold if K 0) due to the K-doubling (i.e. k K and K
both exist) combined with the tunnelling doubling. As the
potential energy barrier is lowered this means that there are
twice as many sub-levels for each even-K than allowed in the
free rotor limit. Therefore, as clearly visible in Fig. 2b, half of
the sub-levels for each even-K are promoted as the barrier
drops to zero, analogous to the promotion of orbitals that can
occur as two atoms are brought together to form a diatomic
molecule.3 The behaviour of the other half of the sub-levels,
those that correlate to the free rotor state [K/2, K/2], is
completely different and initially surprising: in Fig. 2b these
sub-levels are seen to be completely barrier-invariant. These
sub-levels are (K/2, K/2) and (K/2, K/2) for which the
torsional quantum number is n n1 n2 0. This means that
the two moieties are co-rotating and the entire molecule is
simply rotating around the z-axis, which is not affected by the
torsional potential energy function.
For odd-K the lowest state in the free rotor limit is
[(K 1)/2, (K 1)/2], which for each K has the same 4-fold
degeneracy as the correlating hindered limit. Thus, as visible in
Fig. 2b, the odd-K levels more-or-less stick together as the
potential energy barrier is swept from one limit to the other.
Fig. 3, plotted relative to the potential energy minimum,
illustrates how this quantum level re-ordering plays out in
terms of wavefunctions. Fig. 3a shows the energy correlation
for the lowest K 0 levels. The quantum labels for the free and
strongly hindered limits are given at the left and right of the
graph, respectively. Fig. 3b and c show the GSRB torsional
energies and their wavefunctions for K 0 for selected
potential energy barriers. Fig. 3b includes the potential energy
function, whereas for clarity of labelling Fig. 3c does not. In
both panels the level energies are shown by horizontal lines,
connected by diagonal lines for different values of H. Note
that the horizontal line labelled | [0, 0]i for H 0 is the
wavefunction of the lowest level which has an approximately
constant value ofE1/O
2p. Levels and wavefunctions corre-
lating in the high barrier limit to symmetric harmonic oscilla-
tor wavefunction combinations are show with solid lines, while
those correlating to antisymmetric wavefunctions are shown
by dashed lines.
Following the progression from left to right in Fig. 3b and c,
we see that with increasing barrier (or, equivalently, increasing
well depth) the lowest level concentrates in the potential well
between the barriers, finally taking the form of a symmetric
combination of v 0 harmonic oscillator wavefunctions
within the limit of the high barrier. Within this limit the
antisymmetric combination must also appear. Since the lowest
free rotor level [0, 0] K 0 is single, the antisymmetric level 0
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8/3/2019 Stephen C. Ross and Koichi M. T. Yamada- Energy level promotion in the correlation from the tunnelling-doubled h
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must come from above, and it is one of the degenerate pairs
of first excited free rotor wavefunctions |[1,1]i that do this.
The other wavefunction of the excited pair begins to acquire a
dimple, eventually taking the form of a symmetric combina-
tion ofv 1 harmonic oscillator wavefunctions, correlating to
11 within the limit of the high barrier. This borrowing from
above continues up the energy level stack and is a direct
consequence of the uniqueness of the lowest free rotor level.
Viewed in the reverse direction, from high to low barrier, we
say that half of each degenerate hindered-rotor level is
promoted to a higher vibrational level as the barrier is
removed, exactly as expected from the discussion of degen-
eracy given above.
In this rich variety of behaviours: promotion or invariance
for even-K versus regular change for odd-K, we find an
explanation for the observed staggering of energy splittings
in HSSH type molecules. Experimentally even-K levels are
observed to have larger splittings than odd-Klevels.12 This we
now see is due to the wider separation between promoted and
invariant sub-levels compared to the odd-K sub-levels which
move as a group between the two limiting cases.
IV. Discussion
The discussion above has been somewhat simplified for clarity.
Thus we neglected the effects of vibrational averaging over
different geometries and of the dependence of the GSRB
moment of inertia tensor on the angle r. These lead to splitting
of the degeneracy, even in the case of zero barrier. Thus the
degeneracy labels in Fig. 2 are approximations. It is important
Fig. 3 (a) Correlation ofK 0 energy levels between the unhindered bi-rotor limit (left) and the tunnelling doubled harmonic limit (right). Unlike
Fig. 2, these are plotted relative to the potential minimum. Bi-rotor state labels [N1, N2] are given on the left, tunnelling doubled harmonic
oscillator labels v are given on the right (b) (which also includes the potential energy function), and (c) show the GSRB torsional energies
(horizontal lines) and their wavefunctions for K 0 for several potential energy barriers. The horizontal line labelled | [0, 0]i for H 0 in (c) is the
wavefunction of the lowest level which has an almost constant value ofE1/O2p. Level energies for different barrier values are connected bydiagonal lines. Levels correlating to symmetric harmonic oscillator wavefunction combinations in the high barrier limit are shown with solid lines,
as are their wavefunctions, those correlating to antisymmetric combinations are shown with dashed lines.
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8/3/2019 Stephen C. Ross and Koichi M. T. Yamada- Energy level promotion in the correlation from the tunnelling-doubled h
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to note that these effects are accounted for in the GSRB
calclations.
The calculated results presented here represent a relatively
simple situation as they are restricted to a molecule having
identical rotor moieties, potential energy minima at exactly 901
relative orientations of these moieties, and equal potential
energy barriers. However, the GSRB program handles the
general case. For example, changing the torsional angle of the
potential energy minima from 90 and 2701 leads to greater
effective asymmetry in the molecule with the consequence that
various Ksub-levels shown as approximately degenerate in the
scale of Fig. 2 separate further than they do there. In the case
of non-equivalent rotor moieties eqn (2.1) is no longer valid. In
this case the energy of the state [N1, N2] is not equal to that of
[N2, N1] and the staggering pattern of K-splittings is then
different than it is for HSSH, although still present and easily
understandable by examining the appropriate correlation dia-
gram. For example, in the case of DSSH, for J 6 it is the
levels with K 0, 3, and 6 that exhibit approximate invariance
and promotion and which have wider splittings. In general
cases invariance and promotion would show up at non-integer
values ofK. Since such levels do not exist the effects are spread
over the levels with an adjacent integer K, leading to a more
complicated but now still understandable energy level pattern.
We conclude that the features identified in the present work
appear in the more general case, albeit with some modification
in detail. Promotion and degeneracy continue to play a role,
and examination of a correlation diagram such as Fig. 2 allows
a unified and qualitative understanding of the energy level
pattern in molecules possessing two internal rotors. This is
particularly important for the otherwise difficult intermediate
cases of internal molecular rotation encountered experimen-
tally. In addition, the GSRB program can perform quantita-
tive calculations and should allow the determination of the
potential energy function and the torsional dynamics (how the
bond lengths and angles depend on r) by least-squares fitting
to experimental data.
AcknowledgementsWe acknowledge the support of the Natural Sciences and
Engineering Research Council of Canada and the practical
and friendly assistance of the staff of the AIC (AIST) in
Tsukuba.
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