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
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
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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
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) . Theoretical band strengths  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 , X2+-B2+  and F2+-A2 [3,4] transitions. Relative band
strengths for the X2+-B2+ transition have been determined by theoretical calculations 
and experiment . 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 . The R dependence of the electronic transition moment for
the F2+-A2 transition is responsible for a major breakdown of the Condon approximation
. A theoretical electronic transition moment for the D-A transition between 2.8-4.2 a0
shows considerable R dependence . 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  to identify R regions important for band strengths,
and is explained according to the difference in electronic transition moments between the
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 . 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.  for D and Ref. 
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  and A  states
obtained in previous studies. The ALCHEMY computer program package  was used
for the electronic transition matrix element calculations . A computer program
constructed by the author  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  and for 19
bands with v=0-4 by Joshy et al. . The present FCFs agree well with Murty  within
0.025 and with Joshy et al.  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 . The theoretical vibrational term values for
v=0-18 of the A state, which were calculated in the previous study , are listed in Table 1,
and those for the D state were taken from Ref. . 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