Potential energy curves, transition dipole moments, and radiative lifetimes of KBe molecule

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Wan Ming-Jie, Jin Cheng-Guo, Yu You, Huang Duo-Hui, Shao Ju-Xiang. Potential energy curves, transition dipole moments, and radiative lifetimes of KBe molecule. Chinese Physics B, 2017, 26(3): 033101 Copy to clipboard

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Potential energy curves, transition dipole moments, and radiative lifetimes of KBe molecule

Wan Ming-Jie^{1, †}, Jin Cheng-Guo¹, Yu You², Huang Duo-Hui^{1, 3}, Shao Ju-Xiang^{1, 3}

1 Computational Physics Key Laboratory of Sichuan Province, Yibin University, Yibin 644007, China

2 College of Optoelectronic Technology, Chengdu University of Information Technology, Chengdu 610225, China

3 School of Physics and Electronic Engineering, Yibin University, Yibin 644007, China

† Corresponding author. E-mail: wanmingjie1983@sina.com

Abstract

An ab initio calculations on the ground and low-lying excited states (X²Σ⁺, 2²Σ⁺, 3²Σ⁺, 1⁴Π, 2⁴Π, 1⁴Σ⁺, 2⁴Σ⁺, and 3⁴Σ⁺) of KBe molecule have been performed using multireference configuration interaction (MRCI) plus Davidson corrections (MRCI+Q) approach with all electron basis set aug-cc-pCV5Z-DK for Be and def2-AQZVPP-JKFI for K. The 3²Σ⁺, 1⁴Π, 2⁴Π, 1⁴Σ⁺, 2⁴Σ⁺, and 3⁴Σ⁺ states are investigated for the first time. Inner shell electron correlations are computed on the potential energy curves (PECs) calculations. The spectroscopic and molecular parameters are also predicted. In addition, The transition properties including transition dipole moment, Franck–Condon factors q_υ′υ″, Einstein coefficients A_υ′υ″, and the radiative lifetimes τ_υ′ for the , , and transitions are predicted at the same time.

PACS: 31.15.A-;31.15.V-;33.15.-e;33.70.Ca

Keyword:potential energy curves;transition dipole moment;Franck-Condon factors;Einstein coefficients

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1. Introduction

In the last twenty years, a number of interesting concepts have been discussed in cold and ultracold molecules formed from alkali-metal atoms (Alk) of group I and alkaline-earth-metal atoms (AEM) of group II. Photoassociation (PA)^[1] is one of the most common approaches to obtain ultracold molecules from ultracold atoms. PA spectroscopy of KRb,^[2] Yb₂,^[3,4] YbRb,^[5] and RbCs^[6,7] has been observed.

Spectroscopic parameters, rovibrational levels, transition dipole moments, and Franck–Condon factors are also important data in the study of interactions between ultracold molecules.^[8,9] Recently, several theoretical investigations have focused on the electronic structures of ultracold molecules,^[8–22] such as MgAlk molecules,^[9] SrAlk molecules,^[10] AEMLi molecules,^[11] KRb,^[12] LiBe,^[13,14] NaBe,^[13,15] KLi,^[16] KBe,^[17,18] RbBe,^[19] CsBe,^[19] RbYb,^[20] LiYb,^[21,22] and our previous works for MgCl,^[23] MgBr,^[23] BeCl,^[24] BeBr,^[24] BeI,^[25] MgI.^[25] There are a few theoretical investigations on the potential energy curves (PECs) of KBe molecule. Bruna et al.^[17] have given the equilibrium distance and dipole moment for the ground state of KBe molecule for the first time. The PECs of four doublet states with different inner electron correlations were investigated by Xiao et al.^[18] Four sets of the frozen core orbitals (FCOs) for each electronic state were considered in their work. At first, they kept the 1s orbital of Be and 1s2s2p3s3p orbitals of K as frozen core orbital; then, 1s orbital of Be, 1s2s2p3s orbitals of K; 1s orbital of Be, 1s2s2p orbitals of K, and 1s2s2p orbitals of K were considered FCO. They believed that the spectroscopic parameters based on the last one were more reliable.

In this work, we have carried out the multi-reference configuration interaction (MRCI)^[26,27] computations on three , two ⁴Π, and three states of KBe molecule based on inner shell electron correlations. At the present time, Li,^[28] Zhu,^[29] Wang,^[30] Liu,^[31] and Xing^[32] have adopted the MRCI method to calculate the ground and low-lying states of CS⁺, AsO⁺, P₂, GeO, and CF⁺, and the obtained results conform to the experimental values. No experimental data have yet been found in the literature for these states, and our main goal is to provide accurate spectroscopic parameters for these states. In addition, the dipole moments, the transition dipole moments (TDMs), Einstein coefficients (A_υ′υ″), and Franck–Condon factors (q_υ′υ″) are also presented.

2. Computational details

The electronic structure calculations are performed with the MOLPRO 2010 program package.^[33] Working in C_2ν point group symmetry, which has four irreducible representations (A₁, B₁, B₂, and A₂). The A₁ irreducible representation yields Σ⁺ states and a component of Δ states; B₁ gives the Π states, and A₂ yields Σ⁻ states and the other component of Δ states.

Complete active space self-consistent-field (CASSCF) calculations are carried out. Using the CASSCF^[34,35] wave functions as a zero-order function, the energies of the Λ–S states are computed via the MRCI method plus Davidson corrections (MRCI+Q).^[36,37] Eight electronic states (X²Σ⁺, 2²Σ⁺, 3²Σ⁺, 1⁴Π, 2⁴Π, 1⁴Σ⁺, 2⁴Σ⁺, 3⁴Σ⁺) are investigated in the present work. The PECs for the , , and ⁴Π states are calculated separately. To improve the quality of PECs, core–valence correlations are computed in PECs calculations in MRCI+Q step. In CASSCF step, the 1s2s2p orbitals of K are doubly closed, which are frozen orbitals, 1s of Be and 3s3p of K are closed-shell orbitals, the 2s2p orbitals of Be and 4s4p3d orbitals of K are the active orbitals. Considering electron correlation effect in the MRCI+Q step, the 1s2s2p orbitals of K are core orbitals, 1s of Be and 3s3p of K are closed-shell orbitals, therefore, 13 electrons are correlation electrons.

In the present work, the all electron basis set aug-cc-pCV5Z-DK for Be^[38] and def2-AQZVPP-JKFI for K^[39] are chosen in all calculations. The spectroscopic and molecular parameters are determined with Le Roy’s LEVEL 8.0 program.^[40] The PECs calculations are performed with an interval of 0.1 Å over the distance from 2.3 Å to 30 Å for three and two ⁴Π states and 3.5 Å to 30 Å for three states, and the value decreases to 0.02 Å near R_e.

3. Results and discussion

3.1. PECs

The PECs for three , two ⁴Π, and three electronic states of KBe molecule are performed, which are illustrated in Figs. 1–3, respectively. We can see that the PECs for all states are smooth. For convenient of discussion, the eight bound electronic states are divided into two categories in the discussion, i.e., the PECs for the doublet () and quartet (⁴Π, ) electronic states, respectively.

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	Fig. 1. (color online) PECs for the X²Σ⁺, 2²Σ⁺, and 3²Σ⁺ states of KBe.

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	Fig. 2. (color online) PECs for the 1⁴Π and 2⁴Π states of KBe.

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	Fig. 3. (color online) PECs for the 1⁴Σ⁺, 2⁴Σ⁺, and 3⁴Σ⁺ states of KBe.

3.2. PECs for the

states

The dominant electronic configurations at R_e for these eight states are tabulated in Table 1. We can see that the dominant compositions for the ground state X²Σ⁺ at R_e is 4σ²5σ²6σ²7σ²8σ2π⁴; the dominant compositions for the 2²Σ⁺ state can be represented by three main configurations, 4σ²5σ²6σ²7σ²8σ2π⁴, 4σ²5σ²6σ²7σ²9σ2π⁴, and 4σ²5σ²6σ²7σ²10σ2π⁴; the 3²Σ⁺ state can be represented by 4σ²5σ²6σ²7σ²9σ2π⁴ and 4σ²5σ²6σ²7σ²10σ2π⁴. The electronic transition and both arise from and electron promotions.

Table 1.

Important electronic configurations of KBe at the equilibrium separations.

The PECs for the X²Σ⁺, 2²Σ⁺, and 3²Σ⁺ states of KBe are plotted in Fig. 1. To our best knowledge, there are not experimental data on spectroscopic parameters of KBe. In previous theoretical studies, Xiao et al.^[18] have investigated PECs for the X²Σ⁺ and 2²Σ⁺ states at the same level, they also given the results of spectroscopic parameters. Compared the results for the X²Σ⁺ and 2²Σ⁺ states between ours and Xiao’s,^[18] the percentage errors are: for the X²Σ⁺ state, (Ref. [18]), (Ref. [18]), (Ref. [18]), for 2²Σ⁺ state, (Ref. [18]), (Ref. [18]), (Ref. [18]). There are large errors of D_e and ω_e between ours and Xiao’s results, and the reason may be that the 3²Σ⁺ state is considered in the CASSCF and MRCI+Q steps. We also calculate only two states at the same level. R_e increased and D_e decreased for the ground state when the third 3sigma is considered. , , , and The 3²Σ⁺ state may play a very important role, and influence the spectroscopic properties for the X²Σ⁺ and 2²Σ⁺.

Augustovičvá et al.^[9] and Guérout et al.^[10] have investigated the ground state for AEM–Alk ( and Sr; , Na, K, Rb, and Cs) at RCCSD(T) and FCI level, respectively, for both MgAlk and SrAlk molecules. R_e increases and D_e decreases with the increase of the atomic weight of Alk, i.e., R_e are 3.100, 3.529, 4.087, 4.286, and 4.518 Å; D_e are 1414.6, 808.4, 637.5, 578.5, and 558.2 cm⁻¹ for LiMg, NaMg, KMg, RbMg, and CsMg,^[9] respectively. R_e is 3.476, 3.819, and 4.407 Å; D_e is 2587, 1597, and 1166 cm⁻¹ for LiSr, NaSr, and KSr,^[10] respectively. Be, Mg, and Sr in periodic table belongs to group II, the properties of AEM–Alk are similar. R_e and D_e for the ground state of LiBe, NaBe, RbBe, and CsBe molecules are 2.607,^[13] 3.052,^[13] 3.762,^[19] 4.049 Å,^[19] and 2340,^[13] 1129.2,^[13] 761.2,^[19] and 418.6 cm⁻¹^[19] by Bauschlicher et al. and Yang et al., respectively. The results for the ground state of KBe molecule are as follows: and , respectively. Same to MgAlk and SrAlk molecules, R_e(CsBe) > R_e(RbBe) > R_e(KBe) > R_e(NaBe) > R_e(LiBe), and D_e(CsBe) < D_e(RbBe) < D_e(KBe) < D_e(NaBe) < D_e(LiBe).

The PECs for the 3²Σ⁺ electronic state are studied for the first time. We have given the spectroscopic parameters based on inner electrons correlation. All spectroscopic parameters for the three states are listed in Table 2. Thought there are not experimental data of spectroscopic parameters, we have provided other reference values of spectroscopic parameters for the first three states.

Table 2.

Spectroscopic parameters for eight low–lying states of KBe.

3.3. The PECs for quartet states

The dominant electronic configurations at R_e for the 1⁴Π and 2⁴Π are 4σ²5σ²6σ²7σ8σ2π⁴3π and 4σ²5σ²6σ²7σ9σ2π⁴3π, which is electron promotion for the electronic transition. The 1⁴Π is the ground quartet state, which corresponds to the dissociation channel Be(³P) + K(²P), and has the lowest energy at R_e, because the lower dissociation channel Be(¹S) + K(²P) can only obtain one doublet Π state.

The dominant electronic configurations at R_e for the 1⁴Σ⁺ are 4σ²5σ²6σ²7σ8σ9σ2π⁴ and 4σ²5σ²6σ²7σ8σ10σ2π⁴; the dominant electronic configurations at R^e for the 2⁴Σ⁺ are 4σ²5σ²6σ²7σ8σ9σ2π⁴, 4σ²5σ²6σ²7σ8σ10σ2π⁴, and 4σ²5σ²6σ²7σ2π⁴3π4π; and the dominant electronic configurations at R^e for the 3⁴Σ⁺ is 4σ²5σ²6σ²7σ2π⁴3π4π. Therefore, the electronic transition from a state to another at R_e can be obtained.

The PECs and spectroscopic parameters for quartet states have been investigated for the first time, the 1⁴Π and 2⁴Π states in Fig. 2 and the 1⁴Σ⁺, 2⁴Σ⁺, and 3⁴Σ⁺ states in Fig. 3. From Fig. 3, it can be seen that the avoided crossing of 2⁴Σ⁺ and 3⁴Σ⁺ states occurred at about 6.0 Å. The energy difference between these two states is only 1.909 cm⁻¹ at . All spectroscopic parameters for two ⁴Π and three states are listed in Table 2. D_e for the 1⁴Σ⁺, 2⁴Σ⁺, and 3⁴Σ⁺ states are 24.927, 98.330, and 22.178 cm⁻¹, respectively. They are shallow potential wells.

3.4. Molecular parameters

Based on the reliable PECs for the bound states (X²Σ⁺, 2²Σ⁺, 3²Σ⁺, 1⁴Π, 2⁴Π, 1⁴Σ⁺, 2⁴Σ⁺, and 3⁴Σ⁺), we have determined their vibrational levels G_υand inertial rotation constants B_υ when . G_υ and B_υ for the ground state together with Xiao’s results^[18] are collected in Table 3, the other data are collected in Supplementary Material. Compared with the results of Xiao et al.,^[18] we overestimate the G_υ for the ground state. From Supplementary Material, the first 21 levels of G_υ for the 2²Σ⁺ state are close to the results of Xiao et al.^[18] The largest error is only about 2.6%, which is in agreement with the results of D_e.

Table 3.

The vibrational levels and inertial rotation constants B_v for the ground state X²Σ⁺ of KBe.

The highest vibrational level for the ground state X²Σ⁺ is in our work. This level lies 0.7035 cm⁻¹ below the dissociation limit, or the energy of 99.91% for D_e. The level is for the 2²Σ⁺ state, which lies 0.013 cm⁻¹ below the dissociation limit, the energy of almost 100% for D_e. The highest vibrational level for the ground state X²Σ⁺ obtained by Xiao et al.^[18] is , which lies 1.02 cm⁻¹ below the dissociation limit, the energy of 99.85% for D_e.

Inertial rotation constants B_υ for states (X²Σ⁺, 2²Σ⁺, 3²Σ⁺, 1⁴Π, 2⁴Π, 1⁴Σ⁺, 2⁴Σ⁺, and 3⁴Σ⁺) are also listed in Supplementary Material. Based on the accuracy of calculations, we believe that the values of G_υ and B_υ for these eight states are reliable.

3.5. Dipole moments, transition dipole moments, and radiative lifetimes

The dipole moment as a function of R is plotted in Fig. 4, for the X²Σ⁺, 2²Σ⁺, 3²Σ⁺, 1⁴Π, 2⁴Π, 1⁴Σ⁺, 2⁴Σ⁺, and 3⁴Σ⁺ states. It is clearly seen that the avoided crossing position between the 2⁴Σ⁺ and 3⁴Σ⁺ states occurs at about 6.0 Å. The dipole moments for eight states approach zero at large interatomic distance. We can say that the dissociation channels for these eight states are the sum of neutral K atom and Be atom. Dipole moments at R_e for eight states are listed in Table 4.

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	Fig. 4. (color online) Dipole moments for the X²Σ⁺, 2²Σ⁺, 3²Σ⁺, 1⁴Π, 2⁴Π, 1⁴Σ⁺, 2⁴Σ⁺, and 3⁴Σ⁺ states of KBe.

Table 4.

Dipole moments at the equilibrium distance for the eight states of KBe.

Notable, for one-electron and two-electron systems, the dipole moments for the ground state at R_e for KBe and LiYb^[22] are 1.748 and 0.03 Debye (D), respectively; and the electronegativities (EN) of Li, K, Be, and Yb atoms are 0.98, 0.82, 1.57, and 1.06 eV, respectively. It is clearly seen that the difference of EN between K and Be is 0.75 eV, but the value is only 0.08 eV between Li and Yb. Therefore, KBe has a sizable dipole moment, but LiYb does not. For Alk-Be series, the dipole moments are 2.415,^[14] 1.978,^[15] 1.748, 1.302,^[19] and 1.046 D^[19] for the ground state of LiBe, NaBe, KBe, RbBe, and CsBe molecules, respectively. It can be seen that the dipole moments for the ground state decrease with the increase of the atomic weight of Alk.

Figure 5 shows the transition dipole moment functions (TDMs) for the 2²Σ⁺–X²Σ⁺, 3²Σ⁺–X²Σ⁺, and 2⁴Π−1⁴Π systems of KBe molecule. Notable the transitions for the 2²Σ⁺–X²Σ⁺ and 2⁴Π−1⁴Π⁺ have the same asymptotic limit, and come from an allowed transition of K atom at large interatomic distance. Our calculation results at are 7.8185 and 7.8124 D, respectively. Xiao et al.^[18] obtains the value 8.17 D for the 2²Σ⁺–X²Σ⁺ bond system, which is about 0.25 D bigger than our results. The 3²Σ⁺––X²Σ⁺ system is transition of Be atom at large interatomic distance, which is a forbidden transition. The TDMs for the 3²Σ⁺–X²Σ⁺ system decreases and tends to be zero at large interatomic distance.

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	Fig. 5. (color online) Transition dipole moments for the 2²Σ⁺–X²Σ⁺, 3²Σ⁺–X²Σ⁺, and 2⁴Π−1⁴Π transitions of KBe.

Transition probabilities for emission including Einstein coefficients A_υ′υ″ and Franck–Condon factors q_υ′υ″ for the 2²Σ⁺–X²Σ⁺, 3²Σ⁺–X²Σ⁺, 3²Σ⁺−2²Σ⁺, and 2⁴Π−1⁴Π band systems have been evaluated, which are tabulated in Table 5. The values for transitions from the vibrational levels of a higher state to the vibrational levels of a lower state are listed in Table 5. For the first five vibrational levels () for the 2²Σ⁺–X²Σ⁺ system, the stronger transitions occur from to , 1, 2, 3; to , 2, 3, 4, 5, 6; to , 1, 2, 6, 7, 8; to , 1, 4; and to , 3. We can clearly see that both A_υ′υ″ and q_υ′υ″ are concordant on this prediction for the calculated transition systems.

Table 5.

Einstein spontaneous emission coefficients A_υ′υ″ (in s⁻¹) and Franck–Condon factors q_υ′υ″ (in italics) for the , , , , and band systems of KBe.

υ′/υ″	^a
υ′/υ″	0	1	2	3	4	5	6	7	8	9
0	5.889(6)	5.365(6)	2.932(6)	1.292(6)	5.143(5)	1.974(5)	7.601(4)	2.999(4)	1.229(4)	5.266(3)
	3.343(−1)	3.262(−1)	1.860(−1)	8.540(−2)	3.541(−2)	1.415(−2)	5.667(−3)	2.325(−3)	9.892(−4)	4.398(−4)
1	6.070(6)	1.511(5)	1.216(6)	2.639(6)	2.474(6)	1.708(6)	1.028(6)	5.810(5)	3.211(5)	1.771(5)
	3.261(−1)	8.171(−3)	7.171(−2)	1.604(−1)	1.553(−1)	1.106(−1)	6.858(−2)	3.987(−2)	2.260(−2)	1.276(−2)
2	3.819(6)	1.447(6)	2.522(6)	3.996(5)	1.219(5)	8.821(5)	1.371(6)	1.390(6)	1.158(6)	8.662(5)
	1.916(−1)	7.556(−2)	1.355(−1)	2.178(−2)	7.368(−3)	5.314(−2)	8.429(−2)	8.732(−2)	7.422(−2)	5.651(−2)
3	1.877(6)	3.736(6)	3.880(5)	8.029(5)	1.661(6)	8.742(5)	1.389(5)	1.265(4)	1.899(5)	3.669(5)
	8.846(−2)	1.824(−1)	1.942(−2)	4.187(−2)	8.858(−2)	4.744(−2)	7.499(−3)	8.372(−4)	1.153(−2)	2.236(−2)
4	7.749(5)	3.941(6)	4.130(5)	2.054(6)	3.724(5)	1.274(5)	7.623(5)	9.225(5)	6.537(5)	3.330(5)
	3.441(−2)	1.810(−1)	1.964(−2)	1.001(−1)	1.850(−2)	6.610(−3)	3.999(−2)	4.912(−2)	3.522(−2)	1.807(−2)
υ′/υ″
υ′/υ″	0	1	2	3	4	5	6	7	8	9
0	8.010(6)	2.639(6)	6.370(5)	1.507(5)	3.870(4)	1.128(4)	3.777(3)	1.441(3)	6.165(2)	2.902(2)
	6.999(−1)	2.282(−1)	5.465(−2)	1.274(−2)	3.182(−3)	8.890(−4)	2.825(−4)	1.018(−4)	4.117(−5)	1.840(−5)
1	2.857(6)	2.756(6)	3.356(6)	1.582(6)	5.878(5)	2.109(5)	7.942(4)	3.232(4)	1.431(4)	6.853(3)
	2.537(−1)	2.424(−1)	2.895(−1)	1.354(−1)	4.993(−2)	1.769(−2)	6.529(−3)	2.586(−3)	1.109(−3)	5.136(−4)
2	4.567(5)	4.087(6)	1.859(5)	2.193(6)	2.071(6)	1.205(6)	6.031(5)	2.923(5)	1.443(5)	7.395(4)
	4.139(−2)	3.689(−1)	1.701(−2)	1.901(−1)	1.770(−1)	1.023(−1)	5.089(−2)	2.447(−2)	1.195(−2)	6.045(−3)
3	4.598(4)	1.392(6)	3.245(6)	4.190(5)	4.592(5)	1.414(6)	1.421(6)	1.024(6)	6.482(5)	3.909(−2)
	4.581(−3)	1.298(−1)	3.006(−1)	3.636(−2)	4.066(−2)	1.217(−1)	1.209(−1)	8.650(−2)	5.451(−2)	3.273(−2)
4	3.588(3)	2.551(5)	2. 385(6)	1.164(6)	1.586(6)	9.571(4)	2.246(5)	7.169(5)	8.693(5)	7.625(5)
	4.137(−4)	2.619(−2)	2.297(−1)	1.130(−1)	1.436(−1)	8.093(−3)	1.989(−2)	6.174(−2)	7.396(−2)	6.442(−2)
υ′/υ″
υ′/υ″	0	1	2	3	4	5	6	7	8	9

0	9.420(5)	2.672(6)	3.820(6)	3.381(6)	2.063(6)	8.853(5)	2.601(5)	4.932(4)	5.365(3)	2.287(2)
	6.643(−2)	1.888(−1)	2.707(−1)	2.406(−1)	1.474(−1)	6.346(−2)	1.868(−2)	3.534(−3)	3.811(−4)	1.571(−5)
1	2.580(6)	2.735(6)	5.284(5)	2.971(5)	2.121(6)	2.896(6)	1.965(6)	7.651(5)	1.712(5)	1.845(4)
	1.820(−1)	1.932(−1)	3.759(−2)	2.087(−2)	1.505(−1)	2.067(−1)	1.407(−1)	5.482(−2)	1.223(−2)	1.301(−3)
2	3.504(6)	5.865(5)	6.444(5)	1.827(6)	3.650(5)	4.363(5)	2.360(6)	2.642(6)	1.351(6)	3.336(5)
	2.476(−1)	4.150(−2)	4.534(−2)	1.295(−1)	2.618(−2)	3.071(−2)	1.679(−1)	1.887(−1)	9.658(−2)	2.372(−2)
3	3.129(6)	1.201(5)	1.742(6)	1.329(5)	9.426(5)	1.294(6)	4.290(3)	1.416(6)	2.908(6)	1.878(6)
	2.215(−1)	8.458(−3)	1.230(−1)	9.519(−3)	6.653(−2)	9.224(−2)	3.515(−4)	1.003(−1)	2.072(−1)	1.338(−1)
4	2.092(6)	1.337(6)	6.300(5)	6.785(5)	9.983(5)	8.115(4)	1.401(6)	3.379(5)	6.838(5)	2.902(6)
	1.486(−1)	9.452(−2)	4.453(−2)	4.773(−2)	7.084(−2)	5.616(−3)	9.960(−2)	2.448(−2)	4.805(−2)	2.074(−1)

The values in parentheses are the exponents, e.g., 5.889(6) equals

Table 5.

Einstein spontaneous emission coefficients A_υ′υ″ (in s⁻¹) and Franck–Condon factors q_υ′υ″ (in italics) for the , , , , and band systems of KBe.

The radiative lifetimes can be calculated as the reciprocal of total Einstein spontaneous emission coefficient,

The lifetime of the first 11 vibrational levels for the 2²Σ⁺–X²Σ⁺, 3²Σ⁺–X²Σ⁺, and 2⁴Π−1⁴Π transitions of KBe are listed in Table 6. We can see that three transitions are strong. When , the lifetimes are 61.28, 87.01, and 71.03 ns for the 2²Σ⁺–X²Σ⁺, 3²Σ⁺–X²Σ⁺, and 2⁴Π−1⁴Π transitions, respectively.

Table 6.

Radiative lifetime (in ns) of the transitions 2²Σ⁺–X²Σ⁺, 3²Σ⁺–X²Σ⁺, and 2⁴Π−1⁴Π band systems of KBe molecule.

4. Conclusion and perspectives

In this paper, the potential energy curves (PECs) for electronic states (X²Σ⁺, 2²Σ⁺, 3²Σ⁺, 1⁴Π, 2⁴Π, 1⁴Σ⁺, 2⁴Σ⁺, and 3⁴Σ⁺) of KBe molecule are performed by using highly accurate multireference configuration interaction (MRCI) with aug-cc-pCV5Z-DK for Be atom and def2-AQZVPP-JKFI for K atom.

The spectroscopic parameters for these states are also predicted. The 1⁴Π is the ground quartet state, three quartet Σ⁺ electronic states are shallow potential wells. Based on the accurate results of PECs, molecular parameters for these eight states are also obtained by solving the radial Schrödinger equation. The dipole moments, transition dipole moments, Franck–Condon factors q_υ′υ″, Einstein coefficients A_υ′υ″ and the radiative lifetimes τ_υ′ for the 2²Σ⁺–X²Σ⁺, 3²Σ⁺–X²Σ⁺, and 2⁴Π−1⁴Π transitions are also predicted at the same time. The 3²Σ⁺, 1⁴Π, 2⁴Π, 1⁴Σ⁺, 2⁴Σ⁺, and 3⁴Σ⁺ states are investigated for the first time.

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