^{1,a)}, Carlo Callegari

^{1}, Pavel Soldán

^{2}and Wolfgang E. Ernst

^{1,a)}

### Abstract

The potassium trimer is investigated in its lowest electronic doublet states, employing several high-level *ab initio* methods (coupled cluster with single, double, and noniterative triple excitations, multiconfiguration self-consistent field, and multireference Rayleigh–Schrödinger perturbation theory of second order). One-dimensional cuts through the lowest 12 electronic states at symmetry give insight in the complex electronic structure of the trimer, showing several (pseudo-)Jahn–Teller distortions that involve two or three excited states. Contour plots of the involved molecular orbitals are shown to prove the validity of the shell model frequently used for a qualitative description of metallic clusters.

The authors wish to thank Horst Köppel and Bernd Engels for helpful discussions. A.W.H. gratefully acknowledges support from the Graz Advanced School of Science, a cooperation project between Graz University of Technology and the University of Graz. P.S. appreciates the support of the European Science Foundation, the Czech Science Foundation (EUROCORES program EuroQUAM, project QuDipMol, Grant No. QUA/07/E007), and the Ministry of Education, Youth and Sports of the Czech Republic (Research Project No. 0021620835).

I. INTRODUCTION

II. BASIS SET CONSIDERATIONS

III. EXTREMAL POINTS

A. Normal coordinates

IV. ELECTRONICALLY EXCITED STATES

V. SHELL MODEL

VI. CONCLUSION

### Key Topics

- Excited states
- 15.0
- Shell model
- 13.0
- Sodium
- 13.0
- Ground states
- 12.0
- Basis sets
- 9.0

## Figures

Potential energy curves scanned along the coordinate at and . The curves are labeled according to the point group except for , where the nomenclature is applied. They are sorted into three pictures with respect to their Jahn-Teller distortion pattern. Left: Nondegenerate states, center: two-state distortion, right: Three-state distortion.

Potential energy curves scanned along the coordinate at and . The curves are labeled according to the point group except for , where the nomenclature is applied. They are sorted into three pictures with respect to their Jahn-Teller distortion pattern. Left: Nondegenerate states, center: two-state distortion, right: Three-state distortion.

Comparison of the electronic structure of potassium and sodium trimers. The colors red, green, blue, and yellow correspond to the electronic state symmetries , , , and , respectively. On the left hand side the CASSCF(3,23) results for are plotted, in the center the curves obtained with are shown. The values in the last picture were taken from Ref. 25. Its energy scale is shrunk down by a factor of 1.302, corresponding to the ratio of the lowest excitations of atomic K and Na. This confirms the applicability of approximative scaling procedures for alkali trimers suggested by Reho *et al.* ^{63}

Comparison of the electronic structure of potassium and sodium trimers. The colors red, green, blue, and yellow correspond to the electronic state symmetries , , , and , respectively. On the left hand side the CASSCF(3,23) results for are plotted, in the center the curves obtained with are shown. The values in the last picture were taken from Ref. 25. Its energy scale is shrunk down by a factor of 1.302, corresponding to the ratio of the lowest excitations of atomic K and Na. This confirms the applicability of approximative scaling procedures for alkali trimers suggested by Reho *et al.* ^{63}

Contour plots of the SOMOs obtained in the state-averaged CASSCF calculation at the global-minimum geometry. Each SOMO plot is labeled with its corresponding electronically excited state. Red lines show positive, blue lines negative amplitudes. The potassium atoms are plotted with an atomic diameter of . Note the different cut surfaces for and states. The strongly delocalized and atomic-shaped orbitals exhibit the provisory applicability of a shell model.

Contour plots of the SOMOs obtained in the state-averaged CASSCF calculation at the global-minimum geometry. Each SOMO plot is labeled with its corresponding electronically excited state. Red lines show positive, blue lines negative amplitudes. The potassium atoms are plotted with an atomic diameter of . Note the different cut surfaces for and states. The strongly delocalized and atomic-shaped orbitals exhibit the provisory applicability of a shell model.

Schematic energy level diagram showing the progressive lifting of electronic state degeneracies as the applied MO theory becomes more sophisticated. From left to right, the spherical symmetric shell model (a) that undergoes an oblate distortion (b) is compared with the calculated state order at equilateral geometry (c), and finally with the order obtained at the equilibrium geometry for the ground state (d). The SOMOs, which we assign to the corresponding state in the shell-model interpretation, are given in brackets. The color code in (d) corresponds to the one used in Figs. 1 and 2.

Schematic energy level diagram showing the progressive lifting of electronic state degeneracies as the applied MO theory becomes more sophisticated. From left to right, the spherical symmetric shell model (a) that undergoes an oblate distortion (b) is compared with the calculated state order at equilateral geometry (c), and finally with the order obtained at the equilibrium geometry for the ground state (d). The SOMOs, which we assign to the corresponding state in the shell-model interpretation, are given in brackets. The color code in (d) corresponds to the one used in Figs. 1 and 2.

## Tables

Potassium basis set comparison: The CASPT2 atomic excitation energies (in ) and CCSD(T) electric dipole polarizability (in a.u.). The results obtained with the augmented ECP10MDF basis set are closest to the experimental values.

Potassium basis set comparison: The CASPT2 atomic excitation energies (in ) and CCSD(T) electric dipole polarizability (in a.u.). The results obtained with the augmented ECP10MDF basis set are closest to the experimental values.

Potassium basis set comparison: The CCSD(T) optimization of the dimer singlet ground state. The equilibrium distances and the binding energies as obtained with the basis set candidates are listed. All but the Park basis set are reasonably close to the experiment.

Potassium basis set comparison: The CCSD(T) optimization of the dimer singlet ground state. The equilibrium distances and the binding energies as obtained with the basis set candidates are listed. All but the Park basis set are reasonably close to the experiment.

Potassium basis set comparison: The CCSD(T) optimization results for the trimer global minimum . The obtained absolute energies are printed together with their corresponding geometries defined by the isoseles bond length and the apical angle .

Potassium basis set comparison: The CCSD(T) optimization results for the trimer global minimum . The obtained absolute energies are printed together with their corresponding geometries defined by the isoseles bond length and the apical angle .

Potassium basis set comparison: Same as Table III, but for the local minimum of . is the energetic difference between global and local minimum.

Potassium basis set comparison: Same as Table III, but for the local minimum of . is the energetic difference between global and local minimum.

Potassium trimer: Comparison of extremal points on the ground-state potential energy surface with DFT results in Ref. 23. The geometries are defined by the isosceles bond length and the apical angle . Note the large discrepancy for the saddle point geometry. However, the energetic difference between minimum and saddlepoint is nearly the same in both calculations.

Potassium trimer: Comparison of extremal points on the ground-state potential energy surface with DFT results in Ref. 23. The geometries are defined by the isosceles bond length and the apical angle . Note the large discrepancy for the saddle point geometry. However, the energetic difference between minimum and saddlepoint is nearly the same in both calculations.

Potassium trimer in doublet states: Energy order of the pseudocanonical MOs at the global-minimum geometry including the shell-model labeling.

Potassium trimer in doublet states: Energy order of the pseudocanonical MOs at the global-minimum geometry including the shell-model labeling.

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