3 Energy levels

Since the accuracy of the computed level energies gives a first indication of the quality of the wavefunctions, we make a detailed comparison with experiment and other calculations. Our aim is to obtain level energies to an accuracy of 5% or better and to be able to reproduce adequately the characteristic spectral features of the sequence. Present results are compared with the spectroscopic values, MVGK and CKD in Table 2 and in the histograms shown in Fig. 1. Apart from the $\sp2{\rm P}\sp0_{1/2}$ground levels, the present dataset contains 322 levels for Z=6-28 of which 97% agree with experiment to within 5% (see Fig. 1a). The larger differences are concentrated in C II ($\sim 8$%) and N III ($\sim 6$%) for the 2s2p$\sp2\ \sp4$P$_{\rm J}$ and $\sp2$P$_{\rm J'}$ levels. Similarly, as shown in Fig. 1b, MVGK give 196 levels for Z=8-26 where all agree with experiment to the desirable accuracy. The larger differences are found in O IV for the 2p$\sp3\ \sp2$D$\sp0_{\rm J}$ and 2s2p$\sp2\ \sp2$D$_{\rm J'}$ levels. Finally, CKD list 308 levels for Z=7-28 of which 90% shows a 5% agreement with experiment (see Fig. 1c). The larger discrepancies with experiment are again encountered for the $\sp4$P$_{\rm J}$ levels ($\sim 9$%) for $Z\leq 10$ and, more seriously, for the 2s$\sp2$2p$\sp2$P$\sp0_{3/2}$(Z< 11) where they reach a factor of 2. Moreover, as depicted in Fig. 1a, in spite of the high accuracy in MVGK all the levels in this dataset are smaller than experiment whereas they should perhaps show the statistical scatter usually found in the ab initio approaches (see Figs. 1a-c).

  

Table 2: Experimental and calculated level energies (cm^-1) for the boron sequence. Expt: spectroscopic data by Moore (1970) for C II, Moore (1975) for N III and Edlén (1983) for the rest. Pres: present results. MVGK: Merkelis et al. (1995). CKD: Cheng et al. (1979)

 

Table 2: continued

 

Table 2: continued

 

Table 2: continued

Among the more interesting features of the B-sequence spectra are a distinctive avoided crossing between the 2s2p$\sp2$ $\sp2$S_1/2 and $\sp2$P_1/2 levels at $Z\sim 22$ and the level switchings that take place within both the 2s2p$\sp2\ \sp2$D$_{\rm J}$ and 2p$\sp3\ \sp2$D$\sp0_{\rm J}$ terms at $Z\sim 14$. In order to depict the avoided crossing, we follow Edlén (1983) by plotting as a function of Z the energy of the $\sp2$S_1/2 level for each ion relative to the center of gravity of the n=2 complex and scaled by its total width (Fig. 2). Taking the change of sign as reference, it can be seen that the experimental crossing takes place just at Z=22. MVGK reproduce this behaviour accurately while the slightly larger energies in the CKD and the present datasets place the crossing at Z=23. It will become clear in the discussions in Sect. 4 that the radiative properties of some transitions near the crossing are severely perturbed; consequently, the intervening interactions should be represented as accurately as possible, but the reliability of such radiative data will generally be very sensitive to small changes. Similarly, in Fig. 3 we show that small discrepancies between the theoretical and experimental level splittings can lead to different predictions of the position of the switching, and in the case of CKD, to an unobserved double switching of the $\sp2$D$_{\rm J}$ levels.

  

Figure 1: Histograms showing the percentage difference of the theoretical level energies relative to experiment. a) Present dataset, 322 levels for Z=6-28. b) MVGK, 196 levels for Z=8-26. c) CKD, 308 levels for Z=7-28 excluding 4 levels with assignment 2s$\sp2$2p$\sp2$P$\sp0_{3/2}$ for Z<11 which show differences greater than 10%

  

Figure 2: Energy of the 2s2p$\sp2\ \sp2$S1/2 level plotted as a function of Z showing the avoided crossing. For each ion, energies are referred to the center of gravity of the n=2 complex and scaled by its total width. Open circles: experiment. Filled squares: present results. Open squares: MVGK. Open triangles: CKD

  

Figure 3: Scaled energy splitting (Ryd) of the a) 2s2p$\sp2\ \sp2$D$_{3/2}-\sp2$D5/2 and b) 2p$\sp3\ \sp2$D$\sp0_{3/2}-\sp2$D$\sp0_{5/2}$ levels showing the level switching that takes place at $Z\sim 14$. Open circles: experiment. Filled squares: present results. Open squares: MVGK. Open triangles: CKD. It can be seen that for $\sp2$D$_{\rm J}$ CKD predict an unobserved double switching