The isotope shift is composed of the mass shift (MS) and the field shift (FS). The former is caused by the motion of nucleus with the finite mass and the latter by the nuclear charge distribution.

In the approximation of first-order perturbation theory, the MS Hamiltonian including the relativistic nuclear recoil corrections is given by^[27]

(2)

(4)

|Ψi(0)|2

(5)

(6)

Δ|Ψ(0)|2=|Ψ(0)|u2−|Ψ(0)|l2

In the multi-configuration Dirac–Hartree–Fock method, an atomic state wave function (ASF) is composed of configuration state wave functions (CSFs) with the same parity P, the total angular momentum J and its component along the z direction M_J,

(7)

(8)

The active space (AS) approach is applied to capture the electron correlation effects. According to the perturbation theory,^[32] the CSFs obtained by single (S), double (D), triple (T), and quadruple (Q) substitutions from the occupied in the reference configuration to the active set can be regarded as the first-order and the higher-order set, respectively. Furthermore, the first-order correlation wave function can be classified into several particular pair correlations, i.e., valence–valence (VV), core–valence (CV), and core–core (CC) correlations. In this work, the reference configurations are {1s²2s²} and {1s²2s2p} for the ground state and the excited state, respectively. The 1s orbital is treated as the core, and the others are valence orbitals. At the first step, the configuration space is expanded by the single and restricted double (SrD) substitutions from the reference configuration for taking into account the CV and VV correlations. The restricted double excitation means that only one electron in the 1s core can be excited to the active set. The active set is enlarged layer by layer up to n=10 to monitor the convergence of the calculated energies and mass shift factors. Each correlation layer is label as nSrD. The calculation starts from the Dirac–Hartree–Fock (DHF) approximation, and the orbitals in the reference configuration are treated as spectroscopic. The virtual orbitals in the active set are optimized as correlation orbitals. Subsequently, CC and higher-order correlations, labeled as “CInSD” and “CInTQ”, respectively, and the Breit interaction, are considered in the RCI computation. All calculations are performed by using the GRASP2K package.^{[33, 34]} The isotope shift factors are calculated by the RIS module.^[35]

The isotope shift factors of the 2s2p 3,1P1−2s2 1S0 transitions in B II are shown in Fig. 1 as the functions of the computational model. It can be seen from this figure that the mass shift factors are sensitive to core–valence and valence–valence correlations. The core–core and higher-order correlations are also important, especially for the specific mass shift factors.^[23] For the field shift factors it is more stable. We also made a comparison for the mass and the field shift factors of the ground state. The present results, displayed in Table 1, are in excellent agreement with other theoretical values.

Fig. 1.

	Figure Option K^NMS (in unit GHz), and FS factor F (in units GHz·fm−2) of the 2s2p 3,1P1−2s2 1S0 transitions for B II as functions of the computational models. (a) NMS and SMS factors; (b) FS factors. The unit 1 fm=10−15 m. '>ViewDownloadNew Window
K^NMS (in unit GHz), and FS factor F (in units GHz·fm−2) of the 2s2p 3,1P1−2s2 1S0 transitions for B II as functions of the computational models. (a) NMS and SMS factors; (b) FS factors. The unit 1 fm=10−15 m. '>	Fig. 1. (color online) NMS factor K^NMS (in unit GHz), and FS factor F (in units GHz·fm−2) of the 2s2p 3,1P1−2s2 1S0 transitions for B II as functions of the computational models. (a) NMS and SMS factors; (b) FS factors. The unit 1 fm=10−15 m.

Table 1.

Table 1.

Table 1. .   KNMS KSMS ρe(0) This work 24.3467 0.5968 73.22 Other methods 24.3488[16] 0.59554(40),[16] 0.59514[17] 72.517(30),[16] 71.9(2)[18]

Table 1. .

The mass shift factor and the field shift factor for the 2s2p 3,1P1−2s2 1S0 and  1S0 states are given in Table 2. We found that the contribution of the relativistic correction of the mass shift factor is less than 1%. The effect of the Breit interaction is less than 0.5%. The normal mass shift factors can be estimated from the scaling law

2s2p 1P1−2s2 1S0

 10B

Table 2.

Table 2.

Table 2. .    1S0  3P1  1P1  3P1−1S0  1P1−1S0 E/a.u. ΔK^NMS ΔK^SMS ΔF ΔK^NMS ΔK^SMS ΔF DF −24.24516 −24.12704 −23.91541 −61.69 −1998.5 −44.03 −61.70 −1586.6 −44.03 This work −24.35567 −24.18563 −24.02099 −615.53 −1842.56 −39.453 −1206.10 −1174.90 −44.479 +Breit −24.35450 −24.18445 −24.01983 −615.65 −1842.63 −39.442 −1205.97 −1175.20 −44.467 Ref. [21] −24.35345 −24.18360 −24.01858 −613.49 −1840.16 −39.490 −1183.04 −1174.99 −44.427 Scaling law       −614.08     −1207.01

Table 2. .

Table 3.

Table 3.

Table 3. .   δσIS/cm−1 in experiment δσMS/cm−1 in this work δσMS/cm−1 in other theories 1P1−1S0 0.70(3)[14] 0.7179a 0.711,[21] 0.7145[14] 0.74(19)[15] 0.718(2)b 0.7146,[19] 0.7165,[20] 0.7106(22)[22] a Breit results in this work b for uncertainty estimation, see text.

Table 3. .

Furthremore, we investigated the relation between the energies and the MS factors as expansion of the configuration space. The results are illustrated in Fig. 2. For convenience, the results obtained from the RCI computation are shown in the large figure. δK^MSk is defined as δK^MSk=|K^MS(α)−K^MS(ζ)| and δEk=|E(α)−E(ζ)| for the state, respectively. α and ζ represents the configuration space used in each computational step and the largest configuration space “CI7TQ″ for two states, respectively. One can see from Fig. 2 that δK^MS and δE for these two states approximate to a linear correlation. A similar relation can also be found for the 2s2p1P1-2s2 1S0 transition as shown in Fig. 3. It is worth noting that the linear correlation only occurs in the large-scale calculation. In addition, the linear correlations between the MS factors and the energies for the transition are not as good as those for the level since the transition MS factors are too small and thus sensitive to the electron correlations.

Fig. 2

	Figure Option 2s2 1S0 SMS; (c) 2s2p 1P1 NMS; (d) 2s2p 1P1 SMS. '>ViewDownloadNew Window
2s2 1S0 SMS; (c) 2s2p 1P1 NMS; (d) 2s2p 1P1 SMS. '>	Fig. 2 (color online) The convergence of the NMS and SMS factor (in unit GHz) versus the corresponding energy eigenvalues (in unit Hartree). (a) 2s2 1S0 SMS; (c) 2s2p 1P1 NMS; (d) 2s2p 1P1 SMS.

Fig. 3

	Figure Option 2s2p 1P1−2s2 1S0 transition versus the corresponding transition energy (in unit Hartree). (a) NMS; (b) SMS. '>ViewDownloadNew Window
2s2p 1P1−2s2 1S0 transition versus the corresponding transition energy (in unit Hartree). (a) NMS; (b) SMS. '>	Fig. 3 (color online) The convergence of the NMS and SMS factors (in unit GHz) in 2s2p 1P1−2s2 1S0 transition versus the corresponding transition energy (in unit Hartree). (a) NMS; (b) SMS.

Based on the discussion above, we evaluated the uncertainties in the MS factors. Two methods were used for this purpose. From the linear correlation between the transition energies and the MS factors, the uncertainty of the MS factors with the experiment transition energies of the NIST database can be estimated. The advantage of this method is to evaluate the uncertainty of the unknown and sensitive MS factors based on the known and exact experimental transition energy with the linear correlation. In our work, the results of MS factors are 8 GHz and 36 GHz for the  3P1−1S0 transition, respectively. On the other hand, we can also estimate the uncertainty through the convergence trend of the MS factors by the difference of the result obtained with the “CI7TQ″ and the “CI6TQ″ model. Because the all-order electron correlation effects were considered in the model, the uncertainties mainly come from the truncation error reflecting on the convergence trend as the expansion of the computational model. The corresponding uncertainties of the MS factors are 3 GHz and 6 GHz, which are quite a bit smaller than the ones obtained with the former method. It can be seen that the uncertainties in the former method estimated by these two methods have a positive relation. Therefore, we can say that the uncertainty estimation of the MS factor based on the linear correlation is possible. However, due to the relation being worse between the MS factors and the energies for the transition, we use the latter method to present the final results. The reason for this relation being worse is that the transition MS factors are so tiny as to be sensitive to the electron correlations.