Cite this Article
Luo Wang, Li Rui, Gai Zhiqiang, Ai RuiBo, Zhang Hongmin, Zhang Xiaomei, Yan Bing. Configuration interaction studies on the spectroscopic properties of PbO including spin–orbit coupling. Chinese Physics B, 2016, 25(7): 073101  
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Configuration interaction studies on the spectroscopic properties of PbO including spin–orbit coupling
Luo Wang1, 2, Li Rui1, †, , Gai Zhiqiang3, Ai RuiBo1, Zhang Hongmin1, Zhang Xiaomei2, 4, Yan Bing2, 4, ‡,
College of Science, Qiqihar University, Qiqihar 161006, China
Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China
Department of Electronic Information, Jiangsu University of Science and Technology, Zhenjiang 212003, China
Jilin Provincial Key Laboratory of Applied Atomic and Molecular Spectroscopy, Jilin University, Changchun 130012, China

 

† Corresponding author. E-mail: lirei01@163.com

‡ Corresponding author. E-mail: yanbing@jlu.edu.cn

Project supported by the National Natural Science Foundation of China (Grant Nos. 11404180 and 11574114), the Natural Science Foundation of Heilongjiang Province, China (Grant No. A2015010), the University Nursing Program for Young Scholars with Creative Talents in Heilongjiang Province, China (Grant No. UNPYSCT-2015095), and the Natural Science Foundation of Jilin Province, China (Grant No. 20150101003JC).

Abstract
Abstract

Lead oxide (PbO), which plays the key roles in a range of research fields, has received a great deal of attention. Owing to the large density of electronic states and heavy atom Pb including in PbO, the excited states of the molecule have not been well studied. In this work, high level multireference configuration interaction calculations on the low-lying states of PbO have been carried out by utilizing the relativistic effective core potential. The effects of the core-valence correlation correction, the Davidson modification, and the spin–orbital coupling on the electronic structure of the PbO molecule are estimated. The potential energy curves of 18 Λ-S states correlated to the lowest dissociation limit (Pb (3Pg) + O(3Pg)) are reported. The calculated spectroscopic parameters of the electronic states below 30000 cm−1, for instance, X1Σ+, 13Σ+, and 13Σ−, and their spin–orbit coupling interaction, are compared with the experimental results, and good agreements are derived. The dipole moments of the 18 Λ-S states are computed with the configuration interaction method, and the calculated dipole moments of X1Σ+ and 13Σ+ are consistent with the previous experimental results. The transition dipole moments from 11Π, 21Π, and 21Σ+ to X1Σ+ and other singlet excited states are estimated. The radiative lifetime of several low-lying vibrational levels of 11Π, 21Π, and 21Σ+ states are evaluated.

PACS: 31.50.Df;31.15.aj;31.15.ag
Keyword:lead oxide;MRCI+Q method;dipole moments;transitional properties
1. Introduction

Over the last several decades, the lead oxide molecule (PbO) has received considerable attention of experimental and theoretical studies due to the extensive use in making glass.[1,2] The percentage of PbO in glass can affect the refractive index and the viscosity index of the glass, which can also influence the ability of glass absorbing x-rays. On the other hand, the chemiluminescence spectra of PbO are located in the ultraviolet and visible range, which have been observed in the chemical reaction of atomic Pb with O3 under single-collision conditions.[3,4] Hence, the electronic structures and transitional properties of low-lying states of PbO can help to understand the detailed reaction mechanisms.

The early experimental investigations mainly focused on the spectroscopic properties of the ground state X1Σ+, 13Σ+, 13Σ−, and 13Π. Horiai et al.[5] observed the vibration–rotation spectrum in the infrared region using a tunable laser spectrometer, and obtained the Dunham coefficients of X1Σ+. The infrared spectrum of X1Σ+ was photographed by Chertihin and Andrews[6] via chemical reactions of laser ablated Pb and O2, and the spectroscopic constant of X1Σ+ was evaluated. Knöckel et al.[7] used molecular-beam laser spectroscopy to observe the 13Σ+−X1Σ+ transition of PbO, and obtained the precise spectroscopic parameters of the 13Σ+ state. Later, Dorko et al.[8] photographed the chemiluminescence of PbO by using the reaction between lead vapor and the 1Δ excited state of O2. From the observed vibronic spectrum, they estimated the vibronic molecular constants of X1Σ+, 13Σ+, 13Σ−, 13Π, and 21Σ+. DeMille et al.,[9] Kawall et al.,[10] Hunter et al.,[11] and Leanhardt et al.[12] adopted a series of new approaches to investigate the molecular dipole moments (DM) and hyperfine constants of X1Σ+ and 13Σ+ states.

Along with the experimental works, a larger number of theoretical studies have been carried out to investigate the electronic structures of the low-lying states of PbO. Chertihin et al.[6] and Barandiara et al.[13] computed the geometry and vibrational frequency of X1Σ+ employing density functional theory (DFT/B3LYP) and quasi-relativistic ab initio model potential method, respectively. Subsequently, Iliaš et al.[14] studied the electronic structure of X1Σ+ using a high-level coupled-cluster method with single, double, and noniterative triple excitations (CCSD(T)), and obtained the more accurate spectroscopic constant of X1Σ+. Kello et al.[15,16] and Petrov[17] calculated the DM of X1Σ+ using a relativistic high-level-correlated method and a semi-empirical approach, respectively. Navati[18] calculated the potential energy curves of X1Σ+ and 13Π1 using the Morse–Korwar–Navati (MKN) oscillator model. Using the configuration interaction method, Balasubramanian and Pitzer[4] systematically calculated the electronic structures of the 11 electronic states of PbO, applying the relativistic effective core potential. In their calculations, the spin–orbit coupling (SOC) interactions of the low-lying states were also taken into account.

Despite the great many experimental and theoretical works that have been performed over the last few decades, information about the low-lying states of PbO is still too limited. The previous experimental studies were mainly focused on the ground state X1Σ+, 13Σ+, 13Σ−, and 13Π, while the previous theoretical investigations were concentrated on X1Σ+ except Ref. [4]. Hence, our knowledge of the other excited states of PbO is still lacking. As illuminated in the literature of the isovalent molecules CS,[19] CSe,[20] and GeO,[21] the low-lying electronic states of PbO present a large density, which can lead to strong coupling interactions of these states. The strong coupling interactions cause more complicated electronic configurations of the excited states, making it a challenge to compute the electronic structure of these states.

In this work, we carry out high-level configuration interaction calculations on the low-lying states of PbO. On the basis of the calculated potential energy curves (PECs) of the low-lying states, the spectroscopic parameters of the bound states are fitted. The radiative lifetimes of several low-lying vibrational levels of 11Π, 21Π, and 21Σ+ are evaluated.

2. Methods and computational details

In the present study, the electronic structure calculations are performed with the Molpro2010[22] program designed by Werner et al. The symmetry group of PbO is C∞v; however, due to the limitation of the Molpro package, the calculations are done in the C2v subgroup. The relationships of the irreducible representations of C2v and C∞v are Σ+ = A1, Π = B1 + B2, Δ = A1 + A2, and Σ− = A2. The single-point energies of the low-lying states are calculated to construct the PECs, where the contracted aug-cc-pwCV5Z-PP[23] with relativistic effective core potential (RECP) ECP60MDF and aug-cc-pwCV5Z[24] basis sets are selected for atoms Pb and O, respectively. With the help of the Winger–Witmer principle, the relationship between 18 low-lying Λ-S states and the lowest dissociation limit is given by

In order to obtain the accurate structure of PbO, the energy eigenvalues of the electronic states at a set of internuclear distances are computed through the following three steps. First, the single-configuration wavefunction is calculated by the Hartree–Fock method. Second, the multi-configuration wavefunction is obtained by the state-averaged complete active space self-consistent field (CASSCF) method.[25] Finally, by utilizing the CASSCF energies as the reference, the energy eigenvalues of the Λ-S states are computed with the internally contracted multireference configuration interaction approach (MRCI)[26] including Davison size-extensivity correction (+Q).[27] In the CASSCF and MRCI calculations, the active space is selected as 4A1, 2B1, and 2B2, which are correlated to Pb 6s6p and O 2s2p atomic orbitals. The 5 MOs correlating to the 5d atomic orbital of Pb are placed in the closed shell. That is, there are a total of 20 electrons of PbO taking into account in the electronic correlation calculations.

The SOC effect is taken into consideration through the state interaction method utilizing the full Breit–Pauli Hamilton (HBP)[28] operator. The spin–orbit part of the HBP operator can be written as

where e is the electron charge, me is the electronic quality, c is the speed of light, ZK is the number of nuclear charges of nucleus K, rij is the distance between electrons i and j, and riK = ri − RK is the position vector of electron i with respect to nucleus K. In the SOC calculations, the spin–orbit eigenstates are obtained by diagonalizing the Ĥel + Ĥso matrix in the basis of the Λ-S wavefunctions. Additionally, the diagonal Ĥel matrix elements are evaluated by the MRCI+Q method, and the off-diagonal Ĥso matrix elements are generated from the MRCI wavefunctions.

Based on the PECs of the bound and quasibound Λ-S and Ω states, the spectroscopic parameters, including equilibrium bond length Re, harmonic vibrational frequencies ωe, anharmonic vibrational frequencies ωeχe, balanced rotation constant Be are determined by the numerical solution of the one-dimensional nuclear Schrödinger equations utilizing LeRoy’s LEVEL[29] program.

3. Results and discussion
3.1. PECs and spectroscopic properties of Λ-S states

The lowest 18 Λ-S electronic states of PbO, which are correlated to the dissociation limit (Pb(3Pg) + O(3Pg)), are calculated with the MRCI+Q method. The PECs of these Λ-S electronic states are plotted in Fig. 1. It can be seen that the typical bound states (13Σ+, 13Δ, 13Σ−, 11Σ−, 11Δ, 13Π, and 11Π) are dominantly located in the excitation energy range of 17500–29000 cm−1, which exhibit deep potential wells. In the excitation energy > 36000 cm−1, there are no typical bound states. The 21Σ+ and 23Π states are weakly bound, which have a shallow potential well ∼3000 cm−1. The 23Σ+ and 21Π states are also weakly bound at R = 2.2 Å, whereas they become repulsive at a large bond length. The quintet states (15Π, 15Σ+, 25Σ+, 15Δ, 15Σ−, and 15Π) are all typical repulsive states. The spectroscopic parameters of these states are determined by numerical solutions of the radical Schrödinger equations of nuclear motion, and the results are tabulated in Table 1. The previously available results in the literature and the electronic configurations of the bound states are also listed in Table 1.

Fig. 1. The calculated PECs of the low-lying Λ-S electronic states of PbO employing the MRCI+Q method.
Table 1.

Computed and experimental spectroscopic parameters of PbO.

.

As shown in Table 1, the ground state X1Σ+ is mainly described by the closed-shell configuration 6σ27σ28σ23π4. Our calculated spectroscopic parameters of X1Σ+ are in good accordance with the latest experimental data. The ωe, ωexe, and Be are computed to be 725.1593 cm−1, 2.9293 cm−1, and 0.3039 cm−1, which differ from the corresponding experimental values by only 5.1593 cm−1,[30] 0.5000 cm−1, and 0.0034 cm−1,[5] respectively; Re is merely overestimated by 0.0109 Å, as compared to the experimental figure of 1.9218 Å.[5]

The first excited state 13Σ+ originates from the open-shell configuration 6σ27σ28σ23π34π1 corresponding to the 3π → 4π one electron excitation. Our calculated Te is 18162 cm−1, as compared to the previous experimental result of Ω component average of 16118.5 cm−1. The overestimation of 2043.5 cm−1 most likely arises from the SOC effect missing in the calculation of the Λ-S state. With comparison to the averaged Ω-component of experimental results,[30,31] the deviations of the calculated ωe and ωexe are 60.27 cm−1 and 0.2986 cm−1, respectively. The equilibrium distance is calculated to be 2.1146 Å, which is reasonably consistent with the earlier experimental value of 2.16 Å[30] and the theoretical value of 2.23 Å.[4] For the other spectroscopic constants of the Λ-S 13Σ+ state, deviations from the experimental values are shown in Table 1, while better agreement can be expected by introducing the spin-orbit coupling.

The following four excited states 13Δ, 13Σ−, 11Δ, and 11Σ− are mainly described by the electronic configuration 6σ27σ28σ23π34π1. For the four excited states, only the spectroscopic data of 13Σ− have been reported in the previous investigations. The Te, ωe, Be, and Re of the 13Σ− state are calculated to be 21812 cm−1, 519.0418 cm−1, 0.2498 cm−1, and 2.1321 Å, respectively, which are in good agreement with the previous experimental values of the Ω component average of 24665.25 cm−1, 503.5 cm−1, 0.2495 cm−1, and 2.1325 Å.[31] However, the computed ωexe of 13Σ− is 2.5928 cm−1, which is 0.6322 cm−1 smaller than the experimental value of Ω component average of 3.225 cm−1.[31]

The 13Π and 11Π states both mainly originate from the electronic configuration 6σ27σ28σ13π44π1. The Be and Re of 13Π are computed to be 0.2807 cm−1 and 2.0113 Å, respectively, being a little larger than the experimental results of 0.2586 cm−1 and 2.094 Å. While the calculated Te = 24422 cm−1, ωe = 585.1766 cm−1, and ωexe = 2.3595 cm−1 are evidently a departure from the experimental Te = 19894, ωe = 420 cm−1, and ωexe = 0.54 cm−1. The relatively large deviation of 13Π is most probably caused by the SOC between 13Π and other higher Λ-S electronic states. As for the 11Π state, our computed ωe and Be are 589.2317 cm−1 and 0.2768 cm−1, respectively, which are only 58.7317 cm−1 and 0.0057 cm−1 larger than the reported experimental values;[31] the values of Te, ωexe, and Re are computed to be 26996 cm−1, 2.6505 cm−1, and 2.0253 Å, respectively, which are in reasonable agreement with the previous experimental results of 30198.7 cm−1, 2.92 cm−1, and 2.046 Å.[31]

The permanent dipole moments (DM) of the 18 Λ-S states are also obtained in the MCRI calculations. The variations of DMs with bond lengths are plotted in Fig. 2. At the equilibrium position Re = 1.9327 of the ground state X1Σ+, the DM of the state is computed to be −1.46 a.u., indicating Pbδ+Oδ− polarity of the PbO molecule. Compared with the reported experimental value of −1.83 a.u.,[16] our overestimation of 0.37 a.u. may be caused by the deviations in the RECP basis set applied for the Pb atom. At the equilibrium position Re = 2.1146 of the first excited state 13Σ+, the computed DM of the state is −1.03 a.u., which is in accordance with the latest experimental value of −1.29 a.u.[11] It can be seen from Fig. 2 that the DMs of all 18 Λ-S states tend to zero as internuclear distance R → ∞, illuminating that the dissociation products are neutral Pb and O atoms.

Fig. 2. Dipole moment curves of low-lying Λ-S states.
3.2. PECs and spectroscopic parameters of Ω states

As is well known, the SOC interactions lead to the energy level splitting of the multiplet state. Especially, for the diatomic molecule containing a heavy atom, the SOC splitting of one multiplet state can reach a size comparable with the energy gap between two different electronic states. In the present study, the SOC is taken into account by perturbation theory using the full Breit–Pauli Hamilton (HBP) operator. In the SOC calculation, the electronic states X1Σ+, 13Σ+, and 13Σ− are simultaneously included. The three Λ-S states can generate five Ω states , and the PECs of the five Ω states are plotted in Fig. 3. The calculated spectroscopic parameters of these Ω and the dominant Λ-S state compositions at their equilibrium bond lengths are listed in Table 2.

Fig. 3. The PECs of low-lying Ω states of PbO.
Table 2.

Calculated spectroscopic parameters of low-lying Ω states of PbO.

.

The ground state mainly arises from X1Σ+ (98.5%). Compared with the spectroscopic parameters obtained with the MRCI+Q method, the ωe, ωexe, Be, and Re of the state are corrected by −3.4936 cm−1, 0.4685 cm−1, 0.0022 cm−1, and −0.0067 Å, respectively, which are closer to the reported experimental results of 720 cm−1,[30] 3.42929 cm−1,[5] 0.30730347 cm−1,[5] and 1.9218 Å.[31]

The SOC effect gives rise to the splitting of the triplet state 13Σ+, and two Ω states and are obtained. Both the two Ω states and are generated from Λ-S states 13Σ+ (∼70%) and 13Σ− (∼30%). The lack of configuration mixing between 13Σ+ and 13Σ− is probably the main reason of the deviations for the spectroscopic constants of the 13Σ+ Λ-S state. The energy separation of is 510 cm−1, which is consistent with the previous experimental measurement of 429.1 cm−1.[31] However, the present calculated energy separation of has a large departure from the experimental value of 270 cm−1,[30] which has a large error bar (120 cm−1). For and , the calculated Te are only underestimated by 424 cm−1 (2.6%) and 184 cm−1 (1.1%), as compared with the experimental Te of 16051(60) cm−1 and 16321(60) cm−1.[30] The calculated Re of and are only 0.0155 Å (0.7%) and 0.0309 Å (1.4%) shorter than the most recent experimental results of 2.155 Å and 2.175 Å.[30] The ωe of is calculated to be 507.9586 cm−1, which is larger than the experimental result (480(20) cm−1) by 27.9586 cm−1(5.8%).[30] Moreover, our calculated ωe (507.9586 cm−1) of is well consistent with the upper limit of error bar of the experiment (500 cm−1).[30] Compared with the experimental results,[30] our calculated deviation of ωe of (27.9586 cm−1 (5.8%)) is slightly larger than that of the previous theoretical result (7 cm−1(1.5%)),[4] but the calculated deviations of Te and Re of are only 184 cm−1 (1.1%) and 0.0309 Å (1.4%), which are evidently smaller than those of the previous theoretical results of 1860 cm−1 (11.4%) and 0.055 Å (2.5%).[4]

Taking into account the SOC effect, the triplet state 13Σ− splits into two Ω states and The is composed of 13Σ− (97%) and X1Σ+ (3%), while is generated from 13Σ− (71%) and 13Σ+ (29%). The energy separation of is 1713 cm−1, as compared with the experimental figure of 1127 cm−1. The overestimation of 586 cm−1 is most likely caused by the relatively few electronic states X1Σ+, 13Σ+, and 13Σ− included in the SOC calculations. For , the calculated Te, ωe, and Re are 21697 cm−1, 507.1535 cm−1, and 2.1457 Å, respectively, which are closer to the experimental results of 23820 cm−1, 532 cm−1, and 2.11 Å[31] than the previous theoretical results of 20747 cm−1, 612 cm−1, and 2.23 Å.[4] For , the calculated Te is 1537 cm−1 (6.2%) smaller than the experimental value of 24947 cm−1;[31] the computed ωe and Re are 6 cm−1 (1.2%) and 0.0082 Å (0.4%) larger than the experimental results of 494 cm−1 and 2.14 Å.[31]

3.3. Transitional properties and radiative lifetimes

The transition dipole moments (TDMs) of some transitions to the ground state X1Σ+ and several low-lying excited states are computed as a function of bond length R. The TDM curves of 11Π−X1Σ+, 21Π−X1Σ+, 21Σ+−X1Σ+, 21Σ+−11Π, 21Σ+−21Π, and 21Π−11Π transitions are mapped in Fig. 4.

Fig. 4. The transition dipole moment between singlet states of PbO.

As depicted in Fig. 4, in the Franck−Condon region (1.75 Å−2.25 Å), both TDMs of 11Π−X1Σ+ and 21Π−X1Σ+ transitions have large values (> 0.23 a.u.), while the TDM of the 21Σ+−X1Σ+ transition decreases rapidly as the Pb–O distance increases and then gradually increases at large internuclear distance. Hence, the TDM curve of the 21Σ+−X1Σ+ transition exhibits a minimum around the equilibrium distance. In general, the calculated TDMs of the 11Π−X1Σ+ transition are obviously larger than those of the 21Σ+−X1Σ+ transition in the Franck–Condon region. On the basis of TDMs, the Franck–Condon factors (FCF), and the vibrational energy difference between higher and lower electronic states, the radiative lifetime of the vibrational level of the excited state is calculated with the following formula:

where TDM is the averaged electronic transition dipole moment in atomic unit, qv′v″ is the FCF between vibrational energy levels v′ and v″, and ΔEv′v″ is the energy difference of vibrational energy levels v′ and v″. The calculated lifetimes of the five lowest vibrational levels of 11Π, 21Π, and 21Σ+ states are listed in Table 3. The lifetimes of the low-lying states of PbO have not been reported in the previous experimental and theoretical works. While the lifetimes of isovalent molecule SnO[32] have been computed by Giri et al. using multireference singles and doubles configuration interaction (MRDCI) method, and the calculated lifetimes of A1Π and E1Σ+ of SnO were on the order of 10−7 s and 10−8 s. In this work, the MRCI computed radiative lifetimes of 11Π and 21Σ+ at the low-lying vibrational levels are on the order of 10−7 s and 10−6 s, respectively. The lifetimes of 11Π of PbO and those of A1Π of SnO[32] are on the same order, however, the lifetimes of 21Σ+ of PbO are two orders of magnitude larger than those of E1Σ+ of SnO.[32] As shown in Eq. (3), the radiative lifetime is directly proportional to the reciprocal of TDM, FCF, and ΔEv′v″. The equilibrium distance of 11Π is close to that of 21Σ+, so the FCFs of the 11Π−X1Σ+ and 21Σ+−X1Σ+ transitions are almost the same. As discussed in the properties of TDMs, the TDMs of the 11Π−X1Σ+ transition are obviously larger than those of the 21Σ+−X1Σ+ transition, and the ΔEv′v″ of the 11Π−X1Σ+ transition is slightly smaller than that of the 21Σ+−X1Σ+ transition, which make the radiative lifetimes of 11Π smaller than those of 21Σ+.

Table 3.

Radiative lifetimes of several electronic states at low-lying vibrational levels of PbO.

.
4. Conclusion

In summary, high-level MRCI+Q calculations on the 18 Λ-S states correlating to the lowest dissociation limitation of PbO have been carried out. The PECs of the low-lying Ω states are computed by employing the state interaction method with the aid of the Breit–Pauli operator. From the PECs of the bound states, the corresponding spectroscopic parameters are evaluated by solving the nuclear Schrödinger equations, and found to be in good accord with the reported experimental results. The permanent DMs of the 18 Λ-S states are determined by the MRCI method. Our calculated permanent DMs of the ground state X1Σ+ and 13Σ+ are −1.46 a.u and −1.03 a.u., respectively, which are consistent with the experimental measurements of −1.83 a.u. and −1.29 a.u. The TDMs of three singlet states (11Π, 21Π, and 21Σ+) to the ground state X1Σ+ and other low-lying exited states are investigated. Based on TDMs, FCFs, and energy differences between higher and lower vibrational levels, the radiative lifetimes of the five lowest vibrational levels of 11Π, 21Π, and 21Σ+ states are determined. The present theoretical investigations reveal more detailed information on the electronic structures and transitional properties of the PbO molecule.

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