Stephen C. Ross and Koichi M. T. Yamada- Energy level promotion in the correlation from the...

8/3/2019 Stephen C. Ross and Koichi M. T. Yamada- Energy level promotion in the correlation from the tunnelling-doubled h

1/5

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

This journal is c the Owner Societies 2007 Phys. Chem. Chem. Phys., 2007, 9, 58095813 | 5809

PAPER www.rsc.org/pccp | Physical Chemistry Chemical Physics


2/5

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.

5810 | Phys. Chem. Chem. Phys., 2007, 9, 58095813 This journal is c the Owner Societies 2007


3/5

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



4/5

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.

5812 | Phys. Chem. Chem. Phys., 2007, 9, 58095813 This journal is c the Owner Societies 2007


5/5

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.

References

1 J. T. Hougen and B. M. DeKoven, J. Mol. Spectrosc., 1983, 98,375.

2 E. Wigner and E. E. Witmer, Z. Phys., 1928, 51, 859.3 R. S. Mulliken, Phys. Rev., 1928, 32, 186.4 F. Hund, Z. Phys., 1926, 36, 657.5 F. Hund, Z. Phys., 1927, 40, 742.6 F. Hund, Z. Phys., 1927, 42, 93.7 B. P. Winnewisser, M. Winnewisser, I. R. Medvedev, M. Behnke,

F. C. D. Lucia, S. C. Ross and J. Koput, Phys. Rev. Lett., 2005, 95,243002.

8 M. Winnewisser, B. P. Winnewisser, I. R. Medvedev, F. C. D.Lucia, S. C. Ross and L. M. Bates, J. Mol. Spectrosc., 2006, 798, 1.

9 P. R. Bunker and B. M. Landsberg, J. Mol. Spectrosc., 1977, 67,374.

10 J. T. Hougen, P. R. Bunker and J. W. C. Johns, J. Mol. Spectrosc.,1970, 34, 136.

11 K. M. T. Yamada, G. Winnewisser and P. Jensen, J. Mol. Struct.,2004, 695, 323.

12 G. Winnewisser and K. M. T. Yamada, Vib. Spectrosc., 2004, 1,263.


Stephen C. Ross and Koichi M. T. Yamada- Energy level promotion in the correlation from the...

Documents

Transcript of Stephen C. Ross and Koichi M. T. Yamada- Energy level promotion in the correlation from the...