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Curl Condition for a Four-State Born−Oppenheimer System Employing the Mathieu Equation

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    Curl Condition for a Four-State Born−Oppenheimer System Employing the Mathieu Equation
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    Department of Chemistry, Indian Institute of Technology, Guwahati, Guwahati 781 039, India; Department of Chemistry, North-Eastern Hill University, Shillong 793 022, India; and Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032, India
    * To whom correspondence should be addressed. Fax: +91-33-24732805. E-mail: [email protected].
    †Indian Institute of Technology.
    ‡North-Eastern Hill University.
    §Indian Association for the Cultivation of Science.
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    The Journal of Physical Chemistry A

    Cite this: J. Phys. Chem. A 2008, 112, 40, 9868–9885
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    https://doi.org/10.1021/jp8029709
    Published September 11, 2008
    Copyright © 2008 American Chemical Society
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    Abstract

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    When a group of four states forms a subspace of the Hilbert space, i.e., appears to be strongly coupled with each other but very weakly interacts with all other states of the entire space, it is possible to express the nonadiabatic coupling (NAC) elements either in terms of s or in terms of electronic basis function angles, namely, mixing angles presumably representing the same sub-Hilbert space. We demonstrate that those explicit forms of the NAC terms satisfy the curl conditions—the necessary requirements to ensure the adiabatic−diabatic transformation in order to remove the NAC terms (could be often singular also at specific point(s) or along a seam in the configuration space) in the adiabatic representation of nuclear SE and to obtain the diabatic one with smooth functional form of coupling terms among the electronic states. In order to formulate extended Born−Oppenheimer (EBO) equations [ J. Chem. Phys. 2006, 124, 074101] for a group of four states, we show that there should exist a coordinate independent ratio of the gradients for each pair of ADT/mixing angles leading to zero curls and, thereafter, provide a brief discussion on its analytical validity. As a numerical justification, we consider the first four eigenfunctions of the Mathieu equation to demonstrate the interesting features of nonadiabatic coupling (NAC) elements, namely, the validity of curl conditions and the nature of curl equations around CIs.

    Copyright © 2008 American Chemical Society

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    This article is cited by 47 publications.

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    Cite this: J. Phys. Chem. A 2008, 112, 40, 9868–9885
    Click to copy citationCitation copied!
    https://doi.org/10.1021/jp8029709
    Published September 11, 2008
    Copyright © 2008 American Chemical Society
    Request reuse permissions

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