^{1}, Dmitri Babikov

^{2}and Anna I. Krylov

^{3,a)}

### Abstract

The ground and electronically excited states of cyclic are characterized at the equilibrium geometry and along the Jahn-Teller distortions. Lowest excited states are derived from single excitations from the doubly degenerate highest occupied molecular orbitals (HOMOs) to the doubly degenerate lowest unoccupied molecular orbitals (LUMOs), which give rise to two exactly and two nearly degenerate states. The interaction of two degenerate states with two other states eliminates linear terms and results in a glancing rather than conical Jahn-Teller intersection. excitations give rise to two regular Jahn-Teller states. Optimized structures, vertical and adiabatic excitation energies, frequencies, and ionization potential (IP) are presented. IP is estimated to be , in agreement with recent experiments.

We are very grateful to Dr. Vladimir I. Pupyshev (Moscow State) for critical and insightful comments. We would also like to thank Alec Wodtke (UCSB) for helpful discussions and providing the experimental data. University of Southern California Center for High Performance Computing and Communications for making their computational resources available. A.I.K. acknowledges the support from the Department of Energy (DE-FG02-05ER15685).

I. INTRODUCTION

II. COMPUTATIONAL DETAILS

III. MOLECULAR ORBITAL PICTURE

IV. LOWEST EXCITED STATES

V. THE ANALYSIS OF THE PROBLEM IN CYCLIC

VI. IONIZATION POTENTIAL AND PHOTOELECTRON SPECTRUM

VII. CONCLUSIONS

### Key Topics

- Excited states
- 34.0
- Jahn Teller effect
- 15.0
- Wave functions
- 14.0
- Geometric phases
- 9.0
- Photoelectron spectra
- 9.0

## Figures

Molecular orbitals and the ground state electronic configuration of cyclic (equilateral triangle, ). Both HOMO and LUMO are doubly degenerate. labels are given in parentheses.

Molecular orbitals and the ground state electronic configuration of cyclic (equilateral triangle, ). Both HOMO and LUMO are doubly degenerate. labels are given in parentheses.

Leading electronic configurations of the ground (top panel) and the lowest excited states (lower panel). excitations give rise to four singlet and four triplet CSFs labeled , , , and . excitations yield two additional CSFs of each multiplicity: and .

Leading electronic configurations of the ground (top panel) and the lowest excited states (lower panel). excitations give rise to four singlet and four triplet CSFs labeled , , , and . excitations yield two additional CSFs of each multiplicity: and .

Changes in the excited states’ characters and potential energy surfaces upon distortions from an equilateral geometry to an obtuse (left) and an acute (right) isosceles triangles.

Changes in the excited states’ characters and potential energy surfaces upon distortions from an equilateral geometry to an obtuse (left) and an acute (right) isosceles triangles.

The EOM-CCSD/ potential energy surface scans along the bending normal coordinate for the ground (, shown on each plot) and the excited , , , states of cyclic (upper left, upper right, lower left, and lower right, respectively). Data points (squares, triangles, and filled circles) correspond to the calculated adiabatic surfaces; dashed lines represent approximate diabats and connect points with the same leading character of the wave function (see Fig. 2).

The EOM-CCSD/ potential energy surface scans along the bending normal coordinate for the ground (, shown on each plot) and the excited , , , states of cyclic (upper left, upper right, lower left, and lower right, respectively). Data points (squares, triangles, and filled circles) correspond to the calculated adiabatic surfaces; dashed lines represent approximate diabats and connect points with the same leading character of the wave function (see Fig. 2).

Adiabatic potential energy surfaces and contour plots of the and singlet (left) and triplet (right) states. Polar radius and angle are hyperspherical coordinates and , which are similar to the bending and asymmetric stretch normal mode, respectively. Stereographic projection is taken with fixed hyperradius (overall molecular size or symmetric stretch) corresponding to the cyclic ground state equilibrium geometry. Both surfaces feature conical intersection at . Stationary points on the surfaces are located along two distortions to acute and obtuse isosceles triangles. Energies of the transition state and the conical intersection relative to the minima are , and , on the singlet and tiplet PESs, respectively.

Adiabatic potential energy surfaces and contour plots of the and singlet (left) and triplet (right) states. Polar radius and angle are hyperspherical coordinates and , which are similar to the bending and asymmetric stretch normal mode, respectively. Stereographic projection is taken with fixed hyperradius (overall molecular size or symmetric stretch) corresponding to the cyclic ground state equilibrium geometry. Both surfaces feature conical intersection at . Stationary points on the surfaces are located along two distortions to acute and obtuse isosceles triangles. Energies of the transition state and the conical intersection relative to the minima are , and , on the singlet and tiplet PESs, respectively.

(Color) PESs of the ground and the first eight excited states of cyclic . Coordinates are as in Fig. 5. Three out of four states in each multiplicity are almost degenerate at geometry, two being exactly degenerate.

(Color) PESs of the ground and the first eight excited states of cyclic . Coordinates are as in Fig. 5. Three out of four states in each multiplicity are almost degenerate at geometry, two being exactly degenerate.

EOM-CCSD/ potential energy surface scans along symmetric stretch normal coordiante for the lowest and excited states. Singlets are shown on the left plot, triplets—on the right. Solid line shows two exactly degenerate states the and , i.e., the seam of the intersection. Circles and squares correspond to the nondegenerate and states, respectively. Big circles on the right plot show two tree-state PES intersections. is a bond length of equilateral triangle, vertical dashed line points at the cyclic ground state equilibrium geometry.

EOM-CCSD/ potential energy surface scans along symmetric stretch normal coordiante for the lowest and excited states. Singlets are shown on the left plot, triplets—on the right. Solid line shows two exactly degenerate states the and , i.e., the seam of the intersection. Circles and squares correspond to the nondegenerate and states, respectively. Big circles on the right plot show two tree-state PES intersections. is a bond length of equilateral triangle, vertical dashed line points at the cyclic ground state equilibrium geometry.

The Hamiltonian in the diabatic (left) and adiabatic (right) representations along the bending normal mode . Since the Hamiltonian is block diagonal, the pairs of states of the same symmetry do not interact with each other and form two noncrossing pairs (see text).

The Hamiltonian in the diabatic (left) and adiabatic (right) representations along the bending normal mode . Since the Hamiltonian is block diagonal, the pairs of states of the same symmetry do not interact with each other and form two noncrossing pairs (see text).

(not ZPE corrected) calculated as a difference between neutral’s and cation’s CCSD(T) total energies in the basis set limit. ZPE corrected IP, , is .

(not ZPE corrected) calculated as a difference between neutral’s and cation’s CCSD(T) total energies in the basis set limit. ZPE corrected IP, , is .

## Tables

Vertical excitation energies (eV) of the 12 lowest excited states of cyclic calculated at the EOM-CCSD/cc-pVTZ level of theory. Two out of four (, ) and two (, ) excited states are exactly degenerate pairs at . The ground state geometry and total energy are given in Table II.

Vertical excitation energies (eV) of the 12 lowest excited states of cyclic calculated at the EOM-CCSD/cc-pVTZ level of theory. Two out of four (, ) and two (, ) excited states are exactly degenerate pairs at . The ground state geometry and total energy are given in Table II.

constrained optimized geometries, harmonic vibrational frequencies, and total and adiabatic excitation energies of the ground and the lowest excited states calculated at the EOM-CCSD/cc-pVTZ level of theory. , , and are the frequencies of the symmetric stretch, bending, and asymmetric stretch, respectively.

constrained optimized geometries, harmonic vibrational frequencies, and total and adiabatic excitation energies of the ground and the lowest excited states calculated at the EOM-CCSD/cc-pVTZ level of theory. , , and are the frequencies of the symmetric stretch, bending, and asymmetric stretch, respectively.

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