Ab initio study of band strength distribution for the D2Σ+–A2Π transition of AlO and the effect of R dependence of the electronic transition moment on the distribution

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  • Accepted Manuscript

    Ab initio study of band strength distribution for the D2 +-A2 transition of

    AlO and the effect of R dependence of the electronic transition moment on the

    distribution

    Nobumitsu Honjou

    PII: S2210-271X(14)00522-2

    DOI: http://dx.doi.org/10.1016/j.comptc.2014.11.019

    Reference: COMPTC 1683

    To appear in: Computational & Theoretical Chemistry

    Received Date: 28 September 2014

    Revised Date: 24 November 2014

    Accepted Date: 27 November 2014

    Please cite this article as: N. Honjou, Ab initio study of band strength distribution for the D2 +-A2 transition of

    AlO and the effect of R dependence of the electronic transition moment on the distribution, Computational &

    Theoretical Chemistry (2014), doi: http://dx.doi.org/10.1016/j.comptc.2014.11.019

    This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers

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    http://dx.doi.org/10.1016/j.comptc.2014.11.019http://dx.doi.org/http://dx.doi.org/10.1016/j.comptc.2014.11.019

  • 1

    Ab initio study of band strength distribution for the D2+-A2

    transition of AlO and the effect of R dependence of the electronic

    transition moment on the distribution

    b y Nobumitsu Honjou

    Oita University, 700 Dannoharu, Oita 870-1192, Japan

    Abstract: The band strength distribution for the D2+ v -A2 v electronic transition of AlO,

    and the effect of R dependence of the electronic transition moment on the distribution, are

    studied based on the ab initio band strengths and the Franck-Condon factors (FCFs) for

    bands with vibrational quantum numbers v=0-18 and v=0-18. The band strength

    distribution exhibits a Condon parabola of the bands involving v=0-9 and 11 of all the

    inner-well levels of the D2+ double potential well. Five unobserved bands calculated

    among the ten largest band strength bands are found to involve the v levels near the D2+

    potential barrier top. The effect of the R dependence is examined for the FCF maximum

    bands in the v-progressions for v=0-9 and 11-18 (excluding v=10 of the D2+ outer-well

    level) by using the ratio of the band strength to the FCF, both relative to the 0-0 band.

    Large (Small) ratios of value 1.276-1.559 (0.619-0.841) are found for ratio>1 (ratio

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    1. Introduction

    The emission (absorption) intensity of a band in an electronic transition of a molecule,

    divided by the fourth (first) power of the transition energy, is referred to as the band

    strength in emission (absorption) [1]. Theoretical band strengths [2] are useful in predicting

    and interpreting experimental spectra for electronic transitions of molecules, as they

    provide intensity relations for bands independent of experimental conditions such as the

    populations of the initial vibrational levels and the sensitivity of the detection system.

    For the aluminum monoxide (AlO) molecule, theoretical band strengths have been

    reported for the X2+-A2 [2], X2+-B2+ [2] and F2+-A2 [3,4] transitions. Relative band

    strengths for the X2+-B2+ transition have been determined by theoretical calculations [5]

    and experiment [6]. For the D2+-A2 transition, in which emission bands have been

    observed [7,8], intensities are useful for a proper interpretation of emission spectra, but no

    experimental or theoretical band strengths are available.

    The D2+ state has a double potential well, whose barrier top at 4.229 a0 (7944 cm-1

    above the D2+ inner-well minimum) is a result of an avoided crossing with the F2+ state in

    the neighborhood of 4.0 a0 [9]. The R dependence of the electronic transition moment for

    the F2+-A2 transition is responsible for a major breakdown of the Condon approximation

    [4]. A theoretical electronic transition moment for the D-A transition between 2.8-4.2 a0

    shows considerable R dependence [10]. The effect of R dependence on the band strength

    distribution for this transition is unknown.

    We have carried out ab initio calculations of the band strengths and the Franck-Condon

    factors (FCFs) for the D2+ v -A2 v bands with vibrational quantum numbers v=0-18

    and v=0-18. We report below the theoretical band strength distribution for the D-A

    transition and the effect of the R dependence of the electronic transition moment on the

    band strengths. In Section 3.1, the theoretical band strength distribution is characterized.

    In Section 3.2, the effect of the R dependence on the band strengths is examined by

    focusing on the FCF maximum bands in the v-progressions for v values of the D2+

    inner-well and double-well levels, using the ratio of the band strength to the FCF, both

    relative to the 0-0 band. In Section 3.3, the effect of the R dependence is analyzed using a

    model of transition matrix element [4] to identify R regions important for band strengths,

    and is explained according to the difference in electronic transition moments between the

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    important R regions. A limitation of the model where some bands involve vibrational levels

    above or just below the D2+ potential barrier top is discussed. In Section 3.4, the Einstein

    A coefficients which have been evaluated for the D-A bands from the calculated band

    strengths and transition energies are presented.

    2. Theoretical approach

    We carried out ab initio calculations of the band strengths Svv and the Franck-Condon

    factors (FCFs) qvv from the upper D2+ state with v' to the lower A2 state with v" involving

    the v'=0-18 and v"=0-18 vibrational levels. The method of calculation of the band strength

    and the FCF is the same as for our previous calculations of the F2+-A2 bands [3,4]. The

    band strength is calculated from the transition matrix element v'v" of the electronic

    transition matrix element function (R) for the D2+-A2 transition between the vibrational

    wavefunction for the upper state v' and that for the lower state v": v'v" v'|(R)|v".

    Here, the electronic transition matrix elements of D|x|A (=D|y|A) in a.u.

    mean the nonvanishing components of the electronic transition dipole moment (the

    electronic transition moment, in short), where x , y and z respectively denote the x, y and z

    components of the position vector of an electron in the molecule fixed system whose

    internuclear axis is the z axis [2]. We computed the transition matrix element v'v" and the

    overlap integral v'|v" from the (R) function and the vibrational wavefunctions

    calculated using the same method as in our previous studies (Ref. [9] for D and Ref. [3]

    for A ).

    The present (R) function consists of the cubic natural spline (CNS) curves for 2.4-6.0

    a0 and the CNS linear extrapolation function for 2.2-2.4 a0. The CNS curves were fitted to

    the electronic transition matrix elements in 2.4-6.0 a0 at intervals of 0.2 a0, so as to cover

    the range of R in which the vibrational wavefunctions for the v=0-18 and v=0-18 states are

    clearly non-zero. The CNS linear extrapolation function has gradient equal to the CNS

    curve at 2.4 a0 and passes through the point at 2.4 a0.

    To calculate the electronic transition matrix elements, we used the multi-reference

    configuration interaction (MRCI) wavefunctions for the D [9] and A [3] states

    obtained in previous studies. The ALCHEMY computer program package [11] was used

  • 4

    for the electronic transition matrix element calculations [12]. A computer program

    constructed by the author [13] was used for computations of the vibrational wavefunctions,

    the FCFs and the transition matrix elements.

    The accuracy of the vibrational wavefunctions and vibrational energies, which were

    obtained from the MRCI potential energy curves (PECs) [3,9], was checked by a

    comparison with the available experimental values [14-16]. The present FCFs for the D-A

    transition were compared with the FCFs obtained from the PECs using

    Rydberg-Klein-Rees (RKR) approach for 33 bands with v=0-5 by Murty [14] and for 19

    bands with v=0-4 by Joshy et al. [15]. The present FCFs agree well with Murty [14] within

    0.025 and with Joshy et al. [15] within 0.079. The theoretical separation values of

    successive vibrational levels (vibrational spacing values) were compared with the

    experimental values derived from the vibrational term values for the D v=0-5 and A v=0-4

    levels, obtained from the RKR PECs by Ito [16]. The theoretical vibrational term values for

    v=0-18 of the A state, which were calculated in the previous study [3], are listed in Table 1,

    and those for the D state were taken from Ref. [9]. The theoretical values agree well with

    the experimental values for the D state within 4 cm-1 and those for the A state within 2 cm-1.

    The errors in the theoretical vibrational spacing values are estimated to be within 4 cm-1 for

    v=0-17 of the D state and within 2 cm-1 for v=0-17 of the A state.

    3. Results and discussion

    3.1. Band strength distribution

    Fig. 1 shows the theoretical electronic transition matrix element (R) for the D2+-A2

    transition as a function of in