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Prof. Per Jensen, Ph.D.
Research

Document current to: June 2000
Molecular Symmetry


BUNKER AND JENSEN   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 CH4, SF6 and C60. 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 C60 molecule is the icosahedral group Ih(M); it has 120 elements.

C60

 

The ammonia dimer (NH3)2, however, has the molecular symmetry group G144 with 144 elements. Thus we can argue that (NH3)2 has higher symmetry than C60.  

NH3 dimer

 

 

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).


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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 H2Se molecule, at high J and Ka values the rotational energies form four-member groups of nearly degenerate levels, so-called energy clusters. Realistic quantum mechanical calculations have shown that the H2S and H2Te 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 H2130Te 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 H2130Te 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 H2S, H2Se, and H2Te, 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 H2Po 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 H2Te Derived from Experiment, J. Mol. Spectrosc. 185, 282 (1997).


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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 NH2+ and CH2+ ions, and these predictions will be of assistance in their search.

CH2+ potentials

The diagram shows how the X2A1 and A2B1 electronic states of CH2+ 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 CH2+ 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 CH2+ 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 CH2+ would be a more definitive way of testing the shape of the potential surface.

CEI experiment

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 HO2 molecule in the X2A'' and A2A' electronic states is the subject of further calculations.

CEI experiment

The diagram shows reduced energies for the Ka = 0 states in the A(0,0,0) (filled triangles and diamonds) and X(1,1,2) (empty triangles and diamonds) vibronic states of HO2. A diamond represents a state with positive parity [symmetry A' in Cs(M)] and a triangle represents a state with negative parity [symmetry A'' in Cs(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).

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 X2A'' and A2A' Electronic States of the HO2 Molecule, Can. J. Phys., submitted for publication.

(111) G. Osmann, P. R. Bunker, W. P. Kraemer, and Per Jensen: Coulomb Explosion Imaging: The CH2+, H2O+ and NH2+ 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 CH2+ 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 HO2, 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 CH2, 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 NH2+ 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 CH2+. 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 BH2- 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 BH2 molecule. Mol. Phys. 88, 105 (1996).


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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 1B2 ozone and to H2O++ in its electronic ground state.

SO2 PES

We have also done calculations for the C1B2 electronic state of SO2; 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 SO2 in the C1B2 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 H2O++ 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 1B2 Electronic State, Mol. Phys. 91, 653 (1997).

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