^{1,a)}, Ken Tokunaga

^{2}and Kazuyoshi Tanaka

^{3}

### Abstract

A method of calculation of vibronic or electron-phonon coupling constant is presented for a Jahn-Teller molecule, cyclopentadienyl radical. It is pointed out that symmetry breaking at degenerate point and violation of Hellmann-Feynman theorem occur in the calculations based on a single Slater determinant. In order to overcome these difficulties, the electronic wave functions are calculated using generalized restricted Hartree-Fock and complete active space self-consistent-field method and the couplings are computed as matrix elements of the electronic operator of the vibronic coupling. Our result agrees well with the experimental and theoretical values. A concept of vibronic coupling density is proposed in order to explain the order of magnitude of the coupling constant from view of the electronic and vibrational structures. It illustrates the local properties of the coupling and enables us to control the interaction. It could open a way to the engineering of vibronic interactions.

Numerical calculation was partly performed in the Supercomputer Laboratory of Kyoto University.

I. INTRODUCTION

II. VIBRONIC HAMILTONIAN

III. METHOD OF CALCULATION

IV. SYMMETRY BREAKING AND WIGNER-ECKART THEOREM

V. HELLMANN-FEYNMAN THEOREM AND ENERGY GRADIENT

VI. RESULTS AND DISCUSSION

A. Geometrical and vibrational structures

B. Vibronic interaction matrix and Wigner-Eckart theorem

C. Vibronic coupling constant

D. Vibronic coupling density analysis

VII. CONCLUSION

### Key Topics

- Wave functions
- 25.0
- Jahn Teller effect
- 10.0
- Chemical bonds
- 5.0
- Vibration analysis
- 5.0
- Vibronic interactions
- 5.0

## Figures

Cross section of the Jahn-Teller potential. Jahn-Teller crossing is the nuclear configuration of the molecule without Jahn-Teller distortion, and energy minimum is the molecular structure with the lowest energy. The symmetry at is lowered into the at because of the Jahn-Teller effect. For the structure which is obtained after the optimization for the conical intersection, the energy difference is called Jahn-Teller stabilization energy , where is dimensionless vibronic coupling constant.

Cross section of the Jahn-Teller potential. Jahn-Teller crossing is the nuclear configuration of the molecule without Jahn-Teller distortion, and energy minimum is the molecular structure with the lowest energy. The symmetry at is lowered into the at because of the Jahn-Teller effect. For the structure which is obtained after the optimization for the conical intersection, the energy difference is called Jahn-Teller stabilization energy , where is dimensionless vibronic coupling constant.

Structure of cyclopentadienyl radical. Because of the Jahn-Teller effect, symmetry of the structure is lowered from to .

Structure of cyclopentadienyl radical. Because of the Jahn-Teller effect, symmetry of the structure is lowered from to .

orbitals of cyclopentadienyl radical. Because of the fivefold symmetry, the orbital level of HOMO is doubly degenerate , one of them is denoted as , and the other as . is transformed as , and is . Irreducible representations in the parentheses are those lowered into the subgroup : . is and is .

orbitals of cyclopentadienyl radical. Because of the fivefold symmetry, the orbital level of HOMO is doubly degenerate , one of them is denoted as , and the other as . is transformed as , and is . Irreducible representations in the parentheses are those lowered into the subgroup : . is and is .

Jahn-Teller active vibrational modes. The largest displacement locates on the carbon atoms for and , and on hydrogen atoms for and . The displacements of the bold arrows greatly contribute to the VCC. Inserted values are the magnitude of bold arrows. . is and is .

Jahn-Teller active vibrational modes. The largest displacement locates on the carbon atoms for and , and on hydrogen atoms for and . The displacements of the bold arrows greatly contribute to the VCC. Inserted values are the magnitude of bold arrows. . is and is .

Totally symmetric vibrational modes. The largest displacement locates on the carbon atoms for , and on hydrogen atoms for .

Totally symmetric vibrational modes. The largest displacement locates on the carbon atoms for , and on hydrogen atoms for .

Incorrect symmetry breaking of the frontier orbitals in cyclopentadienyl radical with symmetry. The left two diagrams are those of cyclopentadienyl anion, and the remaining ones are those of cyclopentadienyl radical. The HOMO should be degenerate because of the high symmetry. However, the HOMO calculated by the methods based on a single determinant, ROHF, ROB3LYP, UHF, and UB3LYP exhibit symmetry breaking.

Incorrect symmetry breaking of the frontier orbitals in cyclopentadienyl radical with symmetry. The left two diagrams are those of cyclopentadienyl anion, and the remaining ones are those of cyclopentadienyl radical. The HOMO should be degenerate because of the high symmetry. However, the HOMO calculated by the methods based on a single determinant, ROHF, ROB3LYP, UHF, and UB3LYP exhibit symmetry breaking.

Calculated electronic state of cyclopentadienyl radical state. Though the results using GRHF and state-averaged CASSCF give correct degeneracy, the calculations using ROHF, ROB3LYP, UHF, and UB3LYP exhibit an incorrect energy splitting.

Calculated electronic state of cyclopentadienyl radical state. Though the results using GRHF and state-averaged CASSCF give correct degeneracy, the calculations using ROHF, ROB3LYP, UHF, and UB3LYP exhibit an incorrect energy splitting.

Energy curves of the and states along the mode . The Jahn-Teller crossing is disappeared at where the two energy curves should cross.

Energy curves of the and states along the mode . The Jahn-Teller crossing is disappeared at where the two energy curves should cross.

Symmetry breaking in the lowest orbital of cyclopentadienyl radical and the anion (framed). The calculations were performed for the structure. The symmetric orbital becomes asymmetric when a single-determinant-based calculation is applied. This gives rise to the vibronic coupling matrix with a wrong symmetry (see text). The calculations were performed using STO-3G basis set.

Symmetry breaking in the lowest orbital of cyclopentadienyl radical and the anion (framed). The calculations were performed for the structure. The symmetric orbital becomes asymmetric when a single-determinant-based calculation is applied. This gives rise to the vibronic coupling matrix with a wrong symmetry (see text). The calculations were performed using STO-3G basis set.

Energy gradient and (VCI in the figure) using ROHF towards the minimum of the potential on the manifold. The unity of the displacement corresponds to that between the minimum of the potential and the origin, the Jahn-Teller crossing. The absolute value of at the Jahn-Teller crossing is larger than that of the energy gradient.

Energy gradient and (VCI in the figure) using ROHF towards the minimum of the potential on the manifold. The unity of the displacement corresponds to that between the minimum of the potential and the origin, the Jahn-Teller crossing. The absolute value of at the Jahn-Teller crossing is larger than that of the energy gradient.

Energy gradient and (VCI in the figure) using ROB3LYP towards the minimum of the potential on the manifold. The unity of the displacement corresponds to that between the minimum of the potential and the origin, the Jahn-Teller crossing. The absolute value of at the Jahn-Teller crossing is larger than that of the energy gradient.

Energy gradient and (VCI in the figure) using ROB3LYP towards the minimum of the potential on the manifold. The unity of the displacement corresponds to that between the minimum of the potential and the origin, the Jahn-Teller crossing. The absolute value of at the Jahn-Teller crossing is larger than that of the energy gradient.

Contour plot on the plane of the electron density of the frontier orbital calculated by the method.

Contour plot on the plane of the electron density of the frontier orbital calculated by the method.

(a) Contour map on the plane of the one-electron vibronic coupling operator with respect to the mode (a.u.). (b) Contour map on the plane of the vibronic coupling density (a.u.).

(a) Contour map on the plane of the one-electron vibronic coupling operator with respect to the mode (a.u.). (b) Contour map on the plane of the vibronic coupling density (a.u.).

(a) Contour map on the plane of the one-electron vibronic coupling operator with respect to the mode (a.u.). (b) Contour map on the plane of the vibronic coupling density (a.u.).

## Tables

Bond length (Å) of cyclopentadienyl anion calculated using and the neutral radical using . Note that the geometrical structure employed throughout this work is that of the anion.

Bond length (Å) of cyclopentadienyl anion calculated using and the neutral radical using . Note that the geometrical structure employed throughout this work is that of the anion.

Wave number of cyclopentadienyl anion calculated using and the neutral radical calculated using . Experimental values are taken from Ref. 8.

Wave number of cyclopentadienyl anion calculated using and the neutral radical calculated using . Experimental values are taken from Ref. 8.

Total energy and unscaled vibronic coupling constant of . Basis set employed is . The vibrational vectors are obtained by for the anion.

Total energy and unscaled vibronic coupling constant of . Basis set employed is . The vibrational vectors are obtained by for the anion.

Unscaled dimensionless vibronic coupling constant of calculated using basis set. The vibrational vectors employed in these calculations were obtained with for the anion. The values outside the parentheses were calculated using the vibrational frequencies of the anion obtained by , and the values in the parentheses using the vibrational frequencies of the radical evaluated by . Negative signs are neglected.

Unscaled dimensionless vibronic coupling constant of calculated using basis set. The vibrational vectors employed in these calculations were obtained with for the anion. The values outside the parentheses were calculated using the vibrational frequencies of the anion obtained by , and the values in the parentheses using the vibrational frequencies of the radical evaluated by . Negative signs are neglected.

Dimensionless scaled vibronic coupling constants for the degenerate modes calculated by . Negative signs are neglected. The calculations 1–3 and experimental values are taken from Ref. 8.

Dimensionless scaled vibronic coupling constants for the degenerate modes calculated by . Negative signs are neglected. The calculations 1–3 and experimental values are taken from Ref. 8.

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