Prof. Per Jensen, Ph.D.
Theoretical Chemistry
Research

Molecular Symmetry

Research on molecular symmetry focusses on the application of the molecular
symmetry group whose elements consist of nuclear permutations with and without the inversion.
A recent application has been to
spherical top
molecules such as CH_{4}, SF_{6} and C_{60}.
The second edition of the book `Molecular Symmetry and
Spectroscopy' has been written in collaboration with P. R. Bunker, and this was published by
NRC RESEARCH PRESS
in mid August 1998. The first edition, written by P. R. Bunker, was published by Academic Press in 1979.
The
preface and table of contents
are available from the NRC Research Press Web Site.

The molecular symmetry group of the
C_{60} molecule is the icosahedral group I_{h}(M);
it has 120 elements.

The ammonia dimer (NH_{3})_{2}, however, has the molecular
symmetry group G_{144} with 144 elements. Thus we can argue
that (NH_{3})_{2} has higher symmetry than C_{60}.
The symmetries of C_{60}, (NH_{3})_{2} and of many
other molecules are discussed in
`Molecular Symmetry and Spectroscopy'.

Recent publications on molecular symmetry

(111) Per Jensen and P. R. Bunker:
The Symmetry of Molecules,
prepared by invitation for
"Encyclopedia of Chemical Physics and Physical Chemistry",
(J. H. Moore and N. D. Spencer, Eds.),
IOP Publishing, Bristol, in press.

(107) Per Jensen and P. R. Bunker: Nuclear Spin Statistical Weights
Revisited, Mol. Phys., 97, 821-824 (1999).

(105) P. R. Bunker and Per Jensen: Spherical top molecules and the
molecular symmetry group, Mol. Phys., 97, 255-264 (1999).

(71) Per Jensen and P.R. Bunker: The Molecular Symmetry Group for Molecules
in High Angular Momentum States, J. Mol. Spectrosc. 164, 315 (1994).

The Formation of Fourfold Rovibrational Energy Level Clusters
in Triatomic Molecules

It is now an established, experimentally verified fact
that in the vibrational states of the H_{2}Se molecule, at high
J and K_{a} values
the rotational energies form four-member groups
of nearly degenerate levels, so-called energy clusters. Realistic
quantum mechanical calculations have shown that the H_{2}S and
H_{2}Te molecules exhibit similar effects. In recent years
we have been concerned with
the theoretical description of the energy
clusters, mostly by variational calculations, i.e., calculations
of the rotation-vibration energies by diagonalization of a matrix
representation of the rotation-vibration Hamiltonian.
The four-fold clusters were initially predicted by classical
and semi-classical theory, and we have shown how these predictions are
borne out by experiment and by
quantum mechanical calculations. Analysis of
rotation-vibration wavefunctions obtained from variational
calculations provides a simple picture of the rotational motion
in the cluster states: The molecule rotates around one
of its two bonds in a clockwise or an
anticlockwise manner. The two choices for the bond, and the two choices for
the sense of the rotation provide a total of four equivalent situations
corresponding to a four-fold energy cluster.

The energy level structure of a rigidly rotating
H_{2}^{130}Te molecule.
The term
values are
plotted relative to the highest term value for each
J multiplet.

The rotational energy level structure in the vibrational ground state of the
H_{2}^{130}Te molecule, calculated directly from the potential
energy function of the molecule.
The term
values are
plotted relative to the highest term value for each
J multiplet. The calculated spacings are in good agreement with
values derived from experiment.

Comparison of the two figures shows that when we allow the molecule to vibrate,
its rotational energy structure changes drastically at high J:
four-fold energy clusters are formed.

Recent Publications on Four-fold Energy Clusters

(114) Per Jensen: An Introduction to the Theory of Local Mode Vibrations,
Mol. Phys., in press. Article prepared by invitation.

Per Jensen, G. Osmann, and I. N. Kozin:
The Formation of Four-fold Rovibrational
Energy Clusters in H_{2}S, H_{2}Se, and H_{2}Te,
in: "Advanced Series in Physical Chemistry", vol. 9,
"Vibration-Rotational Spectroscopy and Molecular Dynamics"
(D. Papousek, Ed., ISBN 981-02-1635-1),
pp. 298-351, World Scientific
Publishing Company, Singapore, 1997.

(99) P. C. Gomez, L. F. Pacios, and Per Jensen:
Fourfold Clusters of Rovibrational Energies in H_{2}Po Studied
with an ab initio Potential Energy Function,
J. Mol. Spectrosc. 186, 99 (1997).

(98) P. C. Gomez and Per Jensen:
A Potential Energy Surface for the Electronic
Ground State of H_{2}Te
Derived from Experiment,
J. Mol. Spectrosc. 185, 282 (1997).

The Renner Effect

The effect on the spectrum of electronic orbital and spin angular momentum
in triatomic molecules is being investigated in collaboration with P. R. Bunker,
W. P. Kraemer (Max Planck Institute of Astrophysics, Garching, Germany),
R. J. Buenker (University of Wuppertal) and others.
This is generally
termed the Renner effect. We have developed a computer program with which we
can calculate both the positions and intensities of the lines
in a spectrum that arise from transitions between the two halves of a
Renner state. Applications to free radicals and molecular ions are being
undertaken using potential energy surfaces calculated by ab initio methods.
We have predicted the electronic
spectra of the NH_{2}^{+} and CH_{2}^{+} ions, and these predictions will be of
assistance in their search.

The diagram shows how the
X^{2}A_{1} and
A^{2}B_{1} electronic states of
CH_{2}^{+} become degenerate at linear configurations;
the abscissa is the supplement of the bond angle. These two electronic states
are subject to the Renner effect.

For CH_{2}^{+}
it has been conjectured, on the basis of the interpretation of data obtained using
the Coulomb explosion imaging (CEI)
method, that there is a large nonadiabatic contribution to the low-lying
wavefunctions beyond that coming from the Renner effect.
Very recently,
we have calculated the energies of the lowest excited electronic
states and find, in agreement with results already in the literature, that the
excited electronic states of
CH_{2}^{+}
are at much too high an energy (greater than 6 eV)
for such nonadiabatic interaction to be significant. To compare with the CEI results
we calculate the Boltzmann averaged bending angle
distribution using our previously calculated ab initio potential
energy curves of the X,A pair of Renner interacting
potentials, and make full allowance for the Renner effect in the calculation of the wavefunctions.
This ab initio calculation leads to a distribution that is significant
over a narrower range of bending angles than that obtained experimentally by the
CEI method.
Depending on the accuracy of the CEI distribution
this could indicate an error in the ab initio potential energy surfaces.
We have modified the shape of the X-state surface in order to approximately
reproduce the CEI result, and the change we have to make is rather large. An experimental
determination of some of the bending energy level separations for
CH_{2}^{+}
would
be a more definitive way of testing the shape of the potential surface.

The diagram shows the principle of a CEI experiment. Molecular ions are
accelerated and "shot" through a foil, where they lose several
electrons. The remaining, highly unstable system "explodes" due
to repulsive Coulomb forces, and by detection of the fragments the molecular
geometry in the instant of the explosion can be determined.

The HO_{2} molecule in the
X^{2}A'' and
A^{2}A' electronic states is the
subject of further calculations.

The diagram shows the bending potential energy curves in these
two electronic states.

The diagram shows reduced energies for the K_{a} = 0 states
in the A(0,0,0) (filled triangles and diamonds)
and X(1,1,2) (empty triangles and diamonds) vibronic states
of HO_{2}.
A diamond represents a state with positive parity
[symmetry A' in C_{s}(M)] and
a triangle represents a state with negative parity
[symmetry A'' in C_{s}(M)]. The calculations
predict a local perturbation of the A(0,0,0) levels around
J = 51/2, in good agreement with experimental findings of
E. H. Fink and D. A. Ramsay
[J. Mol. Spectrosc.185, 304-324 (1997)]. Based on our
theoretical calculations, the perturbing state can be
identified as X(1,1,2).

Fink and Ramsay
[J. Mol. Spectrosc.185, 304-324 (1997)]
also found a number of lines with anomalous selection rules;
apparently these transitions take place between states of the same
parity indicating that they are magnetic dipole lines.
Our theoretical calculations confirmed that in the
X^{2}A'' <-
A^{2}A' electronic transition of HOO,
the magnetic dipole transitions have intensities typically around 1/10
of that
of the electric dipole lines. The diagram shows the simulated electric
dipole spectrum in black and the magnetic dipole spectrum in green.

Recent Publications on the Renner Effect

(115) Per Jensen, R. J. Buenker, J.-P, Gu,
G. Osmann and P. R. Bunker:
Refined Potential Energy Surfaces for the
X^{2}A'' and
A^{2}A' Electronic States of
the HO_{2} Molecule,
Can. J. Phys., submitted for publication.

(111) G. Osmann, P. R. Bunker,
W. P. Kraemer, and Per Jensen:
Coulomb Explosion Imaging: The
CH_{2}^{+},
H_{2}O^{+} and
NH_{2}^{+} Ions as Benchmarks,
Chem. Phys. Lett. 318, 597-606 (2000).

(108) G. Osmann, P. R. Bunker, W. P. Kraemer, and Per Jensen:
Coulomb Explosion Imaging and the
CH_{2}^{+} Molecule,
Chem. Phys. Lett., 309, 299-306 (1999).

(105) G. Osmann, P. R. Bunker, Per Jensen, R. J. Buenker, J.-P. Gu, and G. Hirsch:
A Theoretical Investigation of the Renner Interactions and
Magnetic Dipole Transitions in the A - X
Electronic Band System of HO_{2},
J. Mol. Spectrosc., 197, 262-274 (1999).

(103) J.-P. Gu, G. Hirsch, R. J. Buenker, M. Brumm, G. Osmann, P. R. Bunker and P. Jensen:
A theoretical study of the absorption spectrum of singlet CH_{2},
J. Mol. Struct., 517, 247-264 (2000).

(101) G. Osmann, P. R. Bunker, P. Jensen and W. P. Kraemer: An Ab Initio
Study of the NH_{2}^{+} Absorption Spectrum. J. Mol. Spectrosc. 186, 319 (1997)

(100) G. Osmann, P. R. Bunker, P. Jensen and W. P. Kraemer: A Theoretical
Calculation of the Absorption Spectrum of CH_{2}^{+}. Chem. Phys. 225, 33 (1997).

(85) J.-P. Gu, R. J. Buenker, G. Hirsch, P. Jensen and P. R. Bunker:
An ab initio calculation of the BH_{2}^{-} rovibronic energies: a very small
singlet-triplet splitting. J. Mol. Spectrosc. 178, 172 (1996).

(79) M. Kolbuszewski, P. R. Bunker, W. P. Kraemer, G. Osmann and P.
Jensen: An ab initio calculation of the rovibronic energies of the BH_{2}
molecule. Mol. Phys. 88, 105 (1996).

Quasibound States

We have implemented the stabilization method of Mandelshtam, Taylor and co-workers
to calculate the quasibound states of a triatomic molecule.
So far, the resulting computer program has been applied to
^{1}B_{2} ozone and to
H_{2}O^{++} in its electronic ground state.

We have also done calculations for the
C^{1}B_{2} electronic state of
SO_{2}; the diagram shows a cut through the
potential energy surface of this state.

Recent Publications on Quasibound States

(112) O. Bludský, P. Nachtigall, J. Hrusak, and Per Jensen:
The Calculation of the Vibrational States of SO_{2}
in the
C^{1}B_{2}
Electronic State up to the
SO(^{3}^{-})+O(^{3}
P)
Dissociation Limit,
Chem. Phys. Lett. 318, 607-613 (2000).

(106) P. R. Bunker, O. Bludský, Per Jensen,
S. S. Wesolowski, T. J. Van Huis, Y. Yamaguchi,
and H. F. Schaefer III:
The H_{2}O^{++} Ground State Potential Energy Surface,
J. Mol. Spectrosc., 198, 371-375 (1999).

(95) O. Bludský and Per Jensen:
The Calculation of the Bound and Quasibound Vibrational States
of Ozone in its ^{1}B_{2} Electronic State,
Mol. Phys. 91, 653 (1997).