^{1}and A. J. C. Varandas

^{1,a)}

### Abstract

The Jahn-Teller(JT) coupling effects in the triply degenerate ground electronic state of methane radical cation are investigated theoretically within a quadratic vibronic coupling approach. The underlying potential energy surfaces over the two-dimensional space of nuclear coordinates, subject to the *T* _{2} ⊗ (*e* + *t* _{2} + *t* _{2}) Jahn-Teller effect, are established from extensive *ab initio* calculations using the multi-reference configuration interaction method and then employed to determine the various parameters of a diabatic Hamiltonian of this system. Our previous investigation [T. Mondal and A. J. C. Varandas, J. Chem. Phys.135, 174304 (2011)10.1063/1.3658641], relying on the linear vibronic coupling approach augmented by only a diagonal second-order term of the totally symmetric mode, are extended here by including all possible quadratic coupling constants of JT active *e* and *t* _{2} modes. Inclusion of these quadratic couplings is found to be important to reproduce correctly the broad vibrational structure and for a better description of dynamical JT effect in the first vibronic band of this radical cation. The impact of large amplitude motions (which are responsible for floppiness of the molecule) on the vibronic structure and dynamics of the first photoelectron band have been examined via readjustment of their linear coupling parameters up to ±10%.

This work is financed by FEDER through “Programa Operacional Factores de Competitividade - COMPETE” and national funds under the auspices of Fundação para a Ciência e a Tecnologia, Portugal (Project Nos. PTDC/QUI-QUI/099744/2008 and PTDC/AAC-AMB/099737/2008).

I. INTRODUCTION

II. THE VIBRONIC HAMILTONIAN AND DYNAMICAL OBSERVABLES

III. DETERMINATION OF COUPLING PARAMETERS

IV. ADIABATIC POTENTIAL ENERGY SURFACES

V. PHOTOELECTRON SPECTRUM

VI. SUMMARY AND OUTLOOK

### Key Topics

- Photoelectron spectra
- 17.0
- Manifolds
- 15.0
- Jahn Teller effect
- 14.0
- Ab initio calculations
- 11.0
- Eigenvalues
- 8.0

## Figures

Adiabatic potential energies of the electronic state of along the dimensionless normal coordinate for the totally symmetric vibrational mode ν_{1}. Energy of the ground electronic state of CH_{4} () at equilibrium configuration (**Q = 0**) is set to zero. The present vibronic model is shown by the solid line and the computed *ab initio* data by the solid dots.

Adiabatic potential energies of the electronic state of along the dimensionless normal coordinate for the totally symmetric vibrational mode ν_{1}. Energy of the ground electronic state of CH_{4} () at equilibrium configuration (**Q = 0**) is set to zero. The present vibronic model is shown by the solid line and the computed *ab initio* data by the solid dots.

Same as in Fig. 1 along both the components (θ and ε) of *e* and ζ component of *t* _{2} vibrational modes, respectively. See panels a–d.

Same as in Fig. 1 along both the components (θ and ε) of *e* and ζ component of *t* _{2} vibrational modes, respectively. See panels a–d.

Same as in Fig. 1 along the simultaneous displacement of two coordinates of *e* and *t* _{2} vibrational modes, respectively. See panels a–c.

Same as in Fig. 1 along the simultaneous displacement of two coordinates of *e* and *t* _{2} vibrational modes, respectively. See panels a–c.

Same as in Fig. 1 along the simultaneous displacement of two coordinates of one component of *e* mode and one component of *t* _{2} vibrational modes. See panels a–d.

Same as in Fig. 1 along the simultaneous displacement of two coordinates of one component of *e* mode and one component of *t* _{2} vibrational modes. See panels a–d.

Vibronic band of the electronic state of . The intensity (in arbitrary units) is plotted along the energy (relative to minimum of the state of CH_{4}) of the final vibronic states.

Vibronic band of the electronic state of . The intensity (in arbitrary units) is plotted along the energy (relative to minimum of the state of CH_{4}) of the final vibronic states.

Vibronic band of the electronic state of : (a) Spectrum reproduced from Ref. 31, (b) spectrum calculated by employing LVC Hamiltonian augmented by diagonal quadratic coupling parameter of totally symmetric ν_{1} vibration only, and (c) spectrum obtained from the present QVC Hamiltonian. The intensity is same as in Fig. 5.

Vibronic band of the electronic state of : (a) Spectrum reproduced from Ref. 31, (b) spectrum calculated by employing LVC Hamiltonian augmented by diagonal quadratic coupling parameter of totally symmetric ν_{1} vibration only, and (c) spectrum obtained from the present QVC Hamiltonian. The intensity is same as in Fig. 5.

Vibrational energy levels of the electronic manifold of : (a) partial spectrum computed with the totally symmetric *a* _{1} vibrational mode ν_{1}, (b) partial spectrum computed with the JT active doubly degenerate *e* vibrational mode ν_{2}, and (c) partial spectrum computed with two JT active triply degenerate *t* _{2} vibrational modes ν_{3}–ν_{4}. The intensity (in arbitrary units) is plotted as a function of the energy of the final vibronic state. The zero of energy corresponds to the equilibrium of the electronic ground state CH_{4}. The theoretical stick spectrum in each panel in convoluted with a Lorentzian function of 40 FWHM to generate the spectral envelope.

Vibrational energy levels of the electronic manifold of : (a) partial spectrum computed with the totally symmetric *a* _{1} vibrational mode ν_{1}, (b) partial spectrum computed with the JT active doubly degenerate *e* vibrational mode ν_{2}, and (c) partial spectrum computed with two JT active triply degenerate *t* _{2} vibrational modes ν_{3}–ν_{4}. The intensity (in arbitrary units) is plotted as a function of the energy of the final vibronic state. The zero of energy corresponds to the equilibrium of the electronic ground state CH_{4}. The theoretical stick spectrum in each panel in convoluted with a Lorentzian function of 40 FWHM to generate the spectral envelope.

Vibronic band of the electronic state of . The intensity is same as in Fig. 5. (a) Spectrum computed from the present QVC model Hamiltonian, (b) spectrum as in (a) but with readjustment of linear coupling parameters by −5% for ν_{2} mode and +5% for ν_{4} mode, (c) spectrum as in (a) but with readjustment of linear coupling parameters by −10% for ν_{2} mode and +10% for ν_{4} mode, and (d) spectrum as in (a) but with readjustment of linear coupling parameters by −10% for ν_{1} mode, −5% for ν_{2} mode, +10% for ν_{3} mode and +5% for ν_{4} mode, respectively.

Vibronic band of the electronic state of . The intensity is same as in Fig. 5. (a) Spectrum computed from the present QVC model Hamiltonian, (b) spectrum as in (a) but with readjustment of linear coupling parameters by −5% for ν_{2} mode and +5% for ν_{4} mode, (c) spectrum as in (a) but with readjustment of linear coupling parameters by −10% for ν_{2} mode and +10% for ν_{4} mode, and (d) spectrum as in (a) but with readjustment of linear coupling parameters by −10% for ν_{1} mode, −5% for ν_{2} mode, +10% for ν_{3} mode and +5% for ν_{4} mode, respectively.

## Tables

Description of the vibrational modes of the electronic ground state of CH_{4} computed at the MP2/cc-pVTZ level of theory. While the theoretical frequencies are harmonic in nature the experimental ones are fundamentals. All values are in eV.

Description of the vibrational modes of the electronic ground state of CH_{4} computed at the MP2/cc-pVTZ level of theory. While the theoretical frequencies are harmonic in nature the experimental ones are fundamentals. All values are in eV.

Coupling parameters in Eqs. (1)–(4) and (5a)–(5f) as obtained from CASSCF/MRCI calculations. The vertical ionization energy of the electronic state is also given. All values are in eV.

Coupling parameters in Eqs. (1)–(4) and (5a)–(5f) as obtained from CASSCF/MRCI calculations. The vertical ionization energy of the electronic state is also given. All values are in eV.

Normal mode combinations, sizes of both primitive and single-particle basis used in the WP propagation within the MCTDH framework in the coupled electronic manifold using the complete vibronic Hamiltonian of Eqs. (1)–(4) and (5a)–(5f). First column denotes the vibrational degrees of freedom (DOF) which are combined to particles. Second column gives the number of primitive basis functions for each DOF. Third column gives the number of single particle functions (SPFs) for each JT splitted electronic state.

Normal mode combinations, sizes of both primitive and single-particle basis used in the WP propagation within the MCTDH framework in the coupled electronic manifold using the complete vibronic Hamiltonian of Eqs. (1)–(4) and (5a)–(5f). First column denotes the vibrational degrees of freedom (DOF) which are combined to particles. Second column gives the number of primitive basis functions for each DOF. Third column gives the number of single particle functions (SPFs) for each JT splitted electronic state.

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