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Vibronic coupling in cyclopentadienyl radical: A method for calculation of vibronic coupling constant and vibronic coupling density analysis
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10.1063/1.2150816
/content/aip/journal/jcp/124/2/10.1063/1.2150816
http://aip.metastore.ingenta.com/content/aip/journal/jcp/124/2/10.1063/1.2150816

Figures

Image of FIG. 1.
FIG. 1.

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.

Image of FIG. 2.
FIG. 2.

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

Image of FIG. 3.
FIG. 3.

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 .

Image of FIG. 4.
FIG. 4.

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 .

Image of FIG. 5.
FIG. 5.

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

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

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.

Image of FIG. 8.
FIG. 8.

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

Image of FIG. 9.
FIG. 9.

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.

Image of FIG. 10.
FIG. 10.

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.

Image of FIG. 11.
FIG. 11.

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.

Image of FIG. 12.
FIG. 12.

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

Image of FIG. 13.
FIG. 13.

(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.).

Image of FIG. 14.
FIG. 14.

(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.).

Image of FIG. 15.
FIG. 15.

(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.).

Image of FIG. 16.
FIG. 16.

(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

Generic image for table
Table I.

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.

Generic image for table
Table II.

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

Generic image for table
Table III.

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

Generic image for table
Table IV.

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.

Generic image for table
Table V.

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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/content/aip/journal/jcp/124/2/10.1063/1.2150816
2006-01-11
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Vibronic coupling in cyclopentadienyl radical: A method for calculation of vibronic coupling constant and vibronic coupling density analysis
http://aip.metastore.ingenta.com/content/aip/journal/jcp/124/2/10.1063/1.2150816
10.1063/1.2150816
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