1. BREAKING THE SYMMETRY IN JAHN-TELLER ACTIVE MOLECULES
BY ASYMMETRIC ISOTOPIC SUBSTITUTION:
SPLITTING THE ZERO-POINT VIBRONIC LEVEL
DMITRY G. MELNIK, JINJUN LIU AND TERRY A. MILLER
The Ohio State University, Dept. of Chemistry, Laser Spectroscopy Facility,
120 W. 18th Avenue, Columbus, Ohio 43210
ROBERT F. CURL,
Department of Chemistry and Rice Quantum Institute, Rice University, Houston, Texas 77005

2. Problem outline:
Objective:
analysis of the vibrational level structure of methoxy radicals
study isotopic effects on vibrational energy level structure
Motivation:
vibrational (vibronic) level structure samples molecular PES with
the isotopic substitution
helps to study dynamics of unimolecular decomposition
methoxy is a combustion intermediate, whose kinetics
and thermodynamics depend on its PES.
test case for systems exhibiting conical intersection.

3. Summary of the vibrational analysis
Standard approach:
formulate full Hamiltonian excluding rotational and translational parts in terms of the parameters characterizing molecular PES
formulate isotopic relationships
set up the Hamiltonian in the appropriate basis set
adjust the parameters of the Hamiltonian to adequately predict experimentally observed energies.
Challenges (methoxy):
complex interaction pattern due to competing spin-orbit and Jahn-Teller
interaction
9 vibrational modes makes the task computationally challenging
rapidly increasing density of states makes vibrational assignment difficult

4. Spin-vibronic level structure of JT active molecule
Sym Sym Sym Asym Asym
Harmonic JT JT1+JT JT1+JT JT1+JT2 +SO
v(E)=1 ? ?
v=0
DE=E(A’)-E(A”)
(a) T. Barckholtz and T. A. Miller, Intl. Rev. Phys. Chem., 17, 435 (1998)

5. Vibronic Hamiltonian R is a displacement vector in an arbitrary set of
internal coordinates spanning 3N-6 space. R=0
indicates a “reference” nuclear configuration.
Using the basis, the vibronic Hamiltonian writes in matrix form
Harmonic part
Jahn-Teller coupling
Where
At this point we disregard spin-orbit interaction, which can be trivially added later.

6. Hamiltonian reduction for the isotopic problem
The values of can be measured experimentally. Suppose there
is an isotopologue for which all three matrices are known. We will call
it a “reference” isotopologue, and the corresponding values are labeled
with “(0)”:
“Potential reduction”: Hev can be rewritten for any isotopologue using unique sets
of normal coordinates
“Kinetic reduction”: Potential energy is unchanged upon isotopic substitution,
but kinetic energy is:

7. Jahn-Teller interaction matrices
Bilinear JT coupling A
modes E
modes
Symmetric to sxz
Quadratic and cross-quadratic JT coupling
Antisymmetric to sxz
Linear JT coupling
Relationships to the experimentally observed values

8. Vibronic Hamiltonian for 3-mode (A+E) model (H3)
A model with a single pair of A and E modes.
2. Normal modes are mass-weighted symmetry
modes.
3. Transforming internal coordinates to symmetry
coordinates,
we write explicitly: y 2 R3 1 R1 x R2 3

9. Calculations of the vibronic level structure of isotopologues of H3
Parameters of Hev are chosen to approximately replicate fundamental
frequencies and strength of vibronic coupling in v3+v6 pair in CH3O
The Hamiltonian is set up in the vibrational basis vx,y,z = 0…10, resulting
in the vibronic matrix 2000x2000. Vibrational matrix elements:

10. Calculations of the zero point splitting, DE = E(A’) – E(A”)
DE(b,g;k=-1.75*104)
DE(b,k;g=-9*104)
H2D
H2D
HD2
HD2
Predicted by Scharf(a) and observed in CH3O(b) and C5H5(c)
(a) B. Scharf et al, J. Chem. Phys, 77, 2226 (1982)
(b) D. Melnik et al., J. Chem. Phys., submitted
(c) L. Yu et. al., J. Chem. Phys., 98, 2682, (1993)

11. Isotopic problem for CH3O
Use the “potential reduction” to obtain fundamental frequencies and
Jahn-Teller parameters.
Fundamental frequencies are square roots of the eigenvalues of GF matrix.
Write potential energy as the function of the internal coordinates,
RCO qj Ri
jij
3. Fit f-parameters of the potential to the experimentally measured
deperturbed fundamental frequencies of CH3O an CD3O.
4. Diagonalize G(k)F for all isotopologues, obtain U(k) and fundamental
frequencies

12. Isotopic relationships for JT parameters in CH3O
Transformation between the normal coordinates of different isotopologues:
where U(m) is the unitary transformation which diagonalizes
We can express parameters of one isotopologue (l), in terms those of another:

13. Vibrational parameters of CHD2O
Fit fundamental frequencies
PES parameters (dyn/cm)
CH3O
CD3O
Predicted values of fundamental frequencies and JT coupling for CHD2O
LJT QJT A’ A”
(a) E. Wilson, J.D.Decius P.C.Cross, “Molecular Vibrations”

14. Vibrational level structure of CHD2O
v5+v6+v8+v9
Experiment
v4+v6+v8+v9

15. Summary
Vibronic calculations for a simplified 3-mode molecule show semi-quantitative
agreement with the observed splitting of the ground vibronic level.
The ground vibronic level splitting is primarily cause by bilinear coupling,
hence the latter needs to be included in the vibronic analysis to correctly
analyze the vibrational level structure of JT-active moelcules. This necessitates
the multimode approach.
To correctly interpret spectra of the asymmetrically substituted molecules,
all isotopologues need to be analyzed globally
NEXT: set up multimode spin-vibronic problem and use nonlinear fit to obtain
parameters of the PES that correctly predict the observed spectra.

16. Acknowledgements
Colleagues:
Dr. Gabriel Just,
Dr. Phillip Thomas,
Ming-Wei Chen,
Terrance Codd,
Neal Kline
Rabi Chhantyal-Pun
OSU
NSF