Extensive theoretical study on the excited states of the PCl+ molecule including spin-orbit coupling

Journal of Quantitative Spectroscopy & Radiative Transfer 196 (2017) 142–148

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Extensive theoretical study on the excited states of the PCl þ molecule including spin-orbit coupling Xiaomei Zhang, Hongsheng Zhai, Siyuan Liu, Yufang Liu n College of Physics and Materials Science, Henan Normal University, Xinxiang 453007, China

art ic l e i nf o Article history: Received 20 January 2017 Received in revised form 3 April 2017 Accepted 7 April 2017 Available online 8 April 2017 Keywords: PCl þ MRCI þQ Spectroscopic constant Spin-orbit coupling effect Transition property

a b s t r a c t The entire 23 Λ-S states of the PCl þ molecule have been studied by using the high-level relativistic MRCI þQ method with full-electron aug-cc-pCVQZ-DK basis set. The potential energy curves(PECs) and wavefunctions of the states have been calculated. From the PECs, the spectroscopic constants of the bound states are also determined, and the good agreements could be found with the experiments. The high density region of states exhibits many PECs' crossings, which lead to complicated interaction of the states. Here, the interactions arising from the dipolar interaction and spin-orbit coupling (SOC) effect have been discussed in detail. Under the influence of the SOC effect, the A2Π state is perturbed by the 14Σ- state. Considering the SOC effect, total 45 Ω states are generated from the original 23 Λ-S states. The transition properties are also predicted, including the transition dipole moments, Franck-Condon factors, and radiative lifetimes. The lifetimes of the transitions A2Π1/2-X2Π1/2 and A2Π3/2-X2Π3/2 are determined to be 478.9 ns and 487.0 ns(v'¼0), respectively. & 2017 Elsevier Ltd. All rights reserved.

1. Introduction Phosphinidene is the important transient species [1–3], which can form stable complexes with transition metal fragments. The chloro-phosphinidene(PCl) plays an important role in many chemical reactions involved in the phosphinidene complex [4,5]. The electronic structures and transition properties of the molecule therefore deserve great attention. Until now, the PCl molecule has been studied systematically in previous work [6–8]. However, only a few experimental and theoretical studies have focused on its cation PCl þ . The experiment investigations on the PCl þ molecule were carried out by Coxon et al. [9,10]. They recorded the v' ¼0 and 1 progressions with 10 r v'' r 20 for the transitions A2Π1/2X2Π1/2 and A2Π3/2-X2Π3/2, however, the perturbation that the excited A2Π state suffered from was unidentified in their work. On the other hand, the pioneering theoretical work was performed by Nguyen [2]. They performed the Møller-Plesset perturbation theory calculation to study the low-lying bound states of the PCl þ cation. Then, the MRCI method was used by Kim and Hirst [11] to study the ground and low-lying excited states of the PCl þ molecule. In their work, the spectroscopic constants and transition properties of the states have been derived. Recently, Niu et al. [12] also used the MRCI method to calculate the potential energy n

Corresponding author. E-mail address: [email protected] (Y. Liu).

http://dx.doi.org/10.1016/j.jqsrt.2017.04.011 0022-4073/& 2017 Elsevier Ltd. All rights reserved.

curves (PECs) of the 12 Λ-S and 27 Ω states. However, the interactions of excited states have not been studied. Here, we extend the previous theoretical work [12] and calculate the PECs of the 23 Λ-S states for the PCl þ molecule with more accurate MRCIþQ/ACVQZ-DK method. These Λ-S states are totally correlated to three dissociation limits. From PECs, the spectroscopic constants of the bound Λ-S states are computed. The interactions among excited Λ-S states are quantitatively analyzed with the aid of the transition dipole moment and spin-orbit matrix elements. The SOC effect also leads to the splitting of the Λ-S states, and the original 23 Λ-S states split into the 45 Ω states. Finally, the transition properties of PCl þ for are analyzed as well.

2. Calculation method All the ab initio calculations of the electronic structures for PCl þ were performed with the MOLPRO 2010 program [13]. The calculations were performed to determine the potential energy curves (PECs), and the aug-cc-pCVQZ-DK [14] basis set is selected for P and Cl. The detailed calculation process is as the following: Firstly, the restricted Hartree-Fock method is used to generate the single-configuration wavefunction for ground state; Then, multiconfiguration wavefunction is derived with the state-averaged complete active space self-consistent field (SA-CASSCF) method [15,16]; Finally, the internally contracted multi-reference configuration interaction (MRCI) approach [17,18] is employed to calculate dynamics correlation energy. The calculation also includes

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scalar relativistic effect by using the second-order Douglas-Kroll integrals [19]. The size-consistency error of the MRCI method is treated with Davidson correction ( þQ) [20]. Due to the limitation of program, the C2ν symmetry has been employed in the calculation. For PCl þ , the 4a1, 2b1, and 2b2, symmetry molecular orbitals (MOs) are determined as the active space, corresponding to 3s3p atomic shells for P and Cl. The outermost 3s23p2 electrons of P and 3s23p5 electrons of Cl are placed in the active space. The inner 2s22p6 electrons for P and Cl are placed in the closed shell. These orbitals are doubly occupied in all reference configuration state functions, and correlated via single and double excitations. Therefore, total 27 electrons are included in the correlation energy calculation. The spin-orbit coupling (SOC) effect is considered with the state interaction approach [21]. The off-diagonal SO matrix elements are calculated with the CASSCF wavefunctions, while the diagonal SO matrix elements are composed of MRCI þQ energies. Thus, the 45 Ω states are generated from the original 23 Λ-S states. The PECs of the Λ-S and Ω states are plotted with the help of the avoided crossing rule. The spectroscopic constants of the bound states were determined by solving nuclear Schrödinger equations using the LEVEL program [22], including adiabatic excitation energy (Te), equilibrium internuclear distance (Re), vibrational constants (ωe and ωeχe), balanced rotation constant (Be), and ro-vibrational coupling constant (αe). The dissociation energy (De) is obtained by subtracting energy at Re from energy at a large separation. The transition dipole moments (TDMs) of the transitions are calculated with MRCI wavefunction, and the FranckCondon factors are evaluated with the LEVEL program. Based on the calculated TDMs and Franck-Condon factors, the radiative lifetimes are predicted.

3. Results and discussion 3.1. The PECs and spectroscopic constants of the

Λ-S states

Total 23 Λ-S states of PCl þ have been studied by the icMRCI þQ/ACVQZ-DK method. They are correlated to three dissociation limits including P þ (3Pg)þCl(2Pu), P þ (1Dg) þCl(2Pu), and P þ (1Sg)þCl(2Pu), respectively. The detailed dissociation relationships have been tabulated in Table 1. The energy intervals of P þ (3Pg)-P þ (1Dg) and P þ (3Pg)-P þ (1Sg) are calculated to be 8268.21 cm  1 and 21,176.08 cm  1, which are in good agreement with the observed values 8882.31 cm  1 and 21,575.63 cm  1 [23]. Fig. 1 shows the PECs of the Λ-S states. The spectroscopic constants of the bound states have been listed in Table 2. The ground state X2Π mainly consists of the electron configuration 9s210s°3π44π1. The calculated ωe ¼688.6 cm  1 and ωeχe ¼2.619 cm  1 reproduce the experiment values 689.8 cm  1 and 2.6 cm  1 [9] well. The determined Re is also much close to the experiment result [10], of which the deviation is only 0.003 Å. At the same time, the resulting results are also in good agreement with the recent theoretical results reported by Niu et al. [12]. To the best of our knowledge, the dissociation energy De, has not been reported in Table 1 The dissociation relationships of the Λ-S states. Λ-S State

Atomic State

Energy(cm  1) Calc.

Expt.[23]

Σ þ , 2Σ-, 2Σ-, 2Π, 2Π, 2Δ, 4Σ þ , 4Σ-, 4Σ-, P þ (3Pg) þ Cl(2Pu) 0 0 4 Π, 4Π, 4Δ 2 þ 2 þ 2 - 2 2 2 2 2 2 þ 1 2 Σ , Σ , Σ , Π, Π, Π, Δ, Ф, Δ P ( Dg) þ Cl( Pu) 8268.21 8882.31 2 Π, 2Σ þ P þ (1Sg)þ Cl(2Pu) 21,176.08 21,575.63 2

Fig. 1. The PECs of the Λ-S states (a) The doublet Λ-S states (b) The quartet Λ-S states.

experiment. The predicted De of 4.54 eV in the absence of the SOC effect is quite close to the theoretical value 4.60 eV [12]. The first excited 14Π state is mainly characterized by the electron configuration 9s210s°3π34π2. This state also has a deeper potential well with De ¼1.535 eV. The excitation energy Te is determined to be 24814 cm  1. Unfortunately, no corresponding experimental value has been found in the literature. The discrepancy with the theoretical result [12] is only 536.74 cm  1. The A2Π state has the same dissociation limit with the X2Π state. This state is mainly dominated by the same electron configuration 9s210s°3π34π2 with 14Π. The wavefunction indicates this state is generated from one-electron 3π-4π excitation of the X2Π state. The bonding 3π and anti-bonding 4π orbitals determine that the A2Π state has the much shallower well than X2Π. The evaluated Te of 28511 cm  1 is in accordance with the measurement of 28753.3 cm  1 [9]. For Re and ωe, the deviations with the experiments [9,10] are only 0.0121 Å and 1.296 cm  1. The 12Σ þ state shows the obvious multi-configuration character, of which the wavefunction contains two important electron configurations: 9s210s13π44π°(55.5%) and 9s210s13π34π1(29.4%). The state has a rather shallow potential well with De being only 0.605 eV, which could be attributed to the anti-bonding 10s orbital. In addition, only one 2Ф state has been obtained in the calculation, which is correlated to the second dissociation limit P þ (1Dg)þCl (2Pu). It is mainly consist of the electronic configuration 9s210s°3π34π2. This state has an obvious potential well and the depth is determined to be 1.529 eV. The 32Π state has the same electron configuration with the 2Ф state. The excitation energy Te is

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Table 2 The spectroscopic constants of the bound Λ-S states. Λ-S state

Te/cm  1

Re/Å

ωe/cm  1

ωeχe/cm  1

Be/cm-1

103αe/cm-1

De/eV

Main CSFs at Re(%)

X2 Π

0

1.903

688.6

2.619

0.283

1.561

4.540

7s28s29s210s°3π44π1(83.8) 7s28s29s210s°3π34π2(1.5)

Expt. [9] Expt. [10] Calc. [2] Calc. [11] Calc. [12] 14 Π Calc. [2] A2Π

0 0 0 0 0 24,814 24,277.28 28,511

689.8

2.6

1.900 1.912 1.915 1.9017 2.314 2.325 2.346

0.284

Expt. [9] Expt. [10] Calc. [11] 14ΣCalc. [11] Calc. [12] 12 Σ þ

28,753.3

681.3 690.29 344.3 337.7 319.3

2.6 2.5321 2.497 2.2 1.419

0.28

1.471 2 2

0.192 0.19 0.186

1.522 2 1.296

4.60 1.535 0.990

7s28s29s210s°3π34π2(78.2) 7s28s29s210s°3π44π1(5.2) 7s28s29s110s13π44π1(4.2)

0.947

7s28s29s110s°3π44π2(87.1)

320.6

27,987.43 29,175 27,987 29,461 32,057

2.334 2.365 2.038 2.04 2.034 2.266

312.8 409.0 433.7 418.75 308.5

1.5 7.249 7.3 6.7100 3.511

0.18 0.247 0.25 0.24797 0.199

1 3.790 4 3.4466 1.825

0.954 0.6045

Calc.[12] Ф 32 Π

31,602.24 33,549 35,812

2.225 2.316 2.468

342.31 339.0 289.0

7.8256 1.998 1.810

0.207 0.191 0.169

3.0504 1.476 1.286

0.693 1.529 1.272

42 Π

40,717

2.365

304.3

3.227

0.183

1.889

0.602

62 Π

54,906

3.172

176.4

3.633

0.102

1.577

0.440

42 Σ þ

56,398

2.797

189.3

6.810

0.131

3.123

0.150

2

1/2, 3/2 1/2,1/2,1/2,3/2,3/2,5/2 1/2,1/2,1/2,1/2,3/2,3/2,3/2,5/2,5/2,7/2 1/2 1/2,1/2,3/2 1/2,1/2,3/2,3/2,5/2 1/2,1/2,1/2,1/2,3/2,3/2,3/2,5/2,5/2,7/2 1/2,1/2,3/2,3/2,5/2 1/2,3/2 1/2

Atomic State

P þ (3P0)þCl(2P3/2) P þ (3P1)þCl(2P3/2) P þ (3P2)þCl(2P3/2) P þ (3P0)þCl(2P1/2) P þ (3P1)þCl(2P1/2) P þ (3P2)þCl(2P1/2) P þ (1D2)þCl(2P3/2) P þ (1D2)þCl(2P1/2) P þ (1S0) þCl(2P3/2) P þ (1S0) þCl(2P1/2)

7s28s29s210s13π44π°(55.5) 7s28s29s210s13π34π1(29.4) 7s28s29s110s23π44π°(1.6) 7s28s29s210s°3π34π2(89.6) 7s28s29s210s°3π34π2(75.2) 7s28s29s110s13π44π1(10.8) 7s28s29s210s°3π44π1(2.4) 7s28s29s210s°3π34π2(82.3) 7s28s29s110s13π44π1(3.3) 7s28s°9s210s°3π34π4(1.7) 7s28s29s210s23π34π°(1.6) 7s28s29s210s23π34π°(55.4) 7s28s29s210s°3π34π2(32.6) 7s28s°9s210s°3π34π2(3.3) 7s28s29s110s°3π44π2(42.1) 7s28s29s110s23π44π°(28.7) 7s28s29s210s13π44π°(10.7) 7s28s29s210s13π34π1(5.9) 7s28s°9s110s23π44π2(2.2)

3.2. The PECs' crossing and coupling

Table 3 The dissociation relationships of the Ω states.

Ω State

7s28s29s210s°3π34π2(88.7)

Energy(cm-1) Calc.

Expt. [23]

0 158.77 313.73 899.15 1057.92 1212.88 8638.04 9537.19 21343.64 22242.79

0 164.90 469.12 882.35 1047.25 1351.47 8882.31 9764.66 21575.63 22457.98

a little larger than that of 2Ф. The dissociation energy De is calculated to be 1.272 eV. The high-density region of states is distributed in the energy interval 30,000 cm  1  50,000cm  1. In this region, the PECs of the many states are rather near and even cross between each other. Among them, the PECs of the 12Δ and 22Δ states avoid crossing at R¼ 2.449 Å. And the 22Σ þ and 32Σ þ also exhibit the PECs' avoided intersection at R ¼2.351 Å. The states 62Π and 42Σ þ with transition energies larger than 50000 cm  1 are both weakly bound states. The computed De values are only 0.440 eV and 0.150 eV. It should be mentioned that the multi-configuration characters for the two excited states are especially obvious.

Fig. 2 shows the enlarged PECs' crossings in the energy region of 30,000–40,000 cm  1. It well known that these PECs' crossings could lead to complicated interaction of states [6,24–29]. Here, we focus on interactions resulting from the dipolar interaction and the SOC effect. The calculated R-dependent TDM(in Fig. 3(a)) and SO (in Fig. 3(b)) matrix elements are applicable for quantitative description of the interaction. As shown in Fig. 2, the A2Π and 14Σ- states have similar Te but rather different Re, which lead to the unavoidable PECs' crossing. The crossing point is located between the vibrational states v' ¼2 and 3 for A2Π, and the corresponding SO matrix element has the value 148 cm  1 at the intersection(R ¼2.167 Å). The lowest two vibrational states v' ¼0 and 1 are therefore unperturbed by the interaction with the 14Σ- state. This could account for the fact that only the vibrational states v' ¼0 and 1 have been observed in previous experiments [9,10]. The 2Ф and 32Π states are both associated with the second dissociation limit P þ (1Dg) þCl(2Pu). In Fig. 2, they exhibit the complicated PECs' crossings with repulsive states 12Σ-, 4Σ þ , 24Π, 22Σ-, 4Δ, 14Σ- and 2Δ, which are all correlated to the lowest dissociation limit P þ (3Pg)þ Cl(2Pu). According to the corresponding selection rules of the dipole transition and SOC effect [30], the SOC effect or the dipolar interaction cannot induce interaction of the lowest two states 4Σ þ and 12Σ- with the 2Ф state. The 4Δ state has the common Ω components of 5/2 and 7/2 with the 2Ф state, and the SO matrix element has the value of 122 cm  1. Thus, it is

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145

interaction. In addition, as illustrated in Fig. 2, no PECs' crossing has been found in the Franck-Condon region of 12Σ þ , and the transition to X2Π has large TDM around Re. We therefore speculate this excited state is likely to be observed in future experiments. 3.3. The PECs and spectroscopic constants of the

Fig. 2. The enlarged PECs' crossing in the high density region of electronic states.

Fig. 3. The TDMs(a) and SO matrix elements(b) of the interacting systems.

expected that the predissociation of the 2Ф state could be induced by the SOC effect via the 4Δ channel. At the same time, under the influence of the dipolar interaction, the 2Ф state is also easier to be predissociated by the adjacent 12Δ state. It should be pointed out that the lowest 12Σ- state exhibits the obvious PECs' overlap at the bottom of the potential well for the 32Π state, and the corresponding TDMs in this region are calculated to be about 0.0280.045a.u. As a result, the 12Σ- state could provide the predissociation channel for the 32Π state as influenced by the dipolar

Ω states

The SOC effect makes the original 23 Λ-S states split into 45 Ω states, including 21 Ω ¼1/2 states, 15 Ω ¼3/2 states, 7 Ω ¼5/2 states, and 2 Ω ¼7/2 states. At the same time, the dissociation limits split into ten asymptotes. In detail, the dissociation limit P þ (3Pg) þCl(2Pu) splits into six limits P þ (3P0)þCl(2P3/2), P þ (3P1) þ Cl(2P3/2), P þ (3P2)þCl(2P3/2), P þ (3P0)þ Cl(2P1/2), P þ (3P1) þCl(2P1/2), and P þ (3P2)þ Cl(2P1/2); P þ (1Dg)þ Cl(2Pu) splits into P þ (1D2)þ Cl (2P3/2) and P þ (1D2)þCl(2P1/2); P þ (1Sg) þCl(2Pu) splits into P þ (1S0)þCl(2P3/2) and P þ (1S0) þCl(2P1/2). The detailed dissociation relationships of the Ω states have been listed in Table 3 . The calculated atomic energy levels of P þ (3P0)-P þ (3P1), P þ (3P0)P þ (3P2), P þ (3P0)-P þ (1D2), P þ (3P0)-P þ (1S0), and Cl(2P3/2)-Cl(2P1/2) are 158.77 cm  1, 313.73 cm  1, 8638.04 cm  1, 21,343.64 cm  1, and 899.15 cm  1, in good agreement with the observed values of 164.90 cm  1, 469.12 cm  1, 8882.31 cm  1, 21,575.63 cm  1, and 882.35 cm  1 [23]. The PECs of the Ω states have been plotted in Fig. 4. The spectroscopic constants of the bound Ω states have been listed in Table 4, and the dominating Λ-S components of each Ω state are also included. The X2Π state splits into two Ω states including X11/2 and X23/2. The energy spacing at Re is calculated to be 353 cm  1. The value well confirms to the experimental prediction of  370 cm  1 [10], as well as the theoretical result of 346.11 cm  1 [12]. As shown in Table 4, the two Ω states are consist of X2Π(100%) around Re. Therefore, the spectroscopic constants almost keep unchanged. And the dissociation energy De also changes a little after considering the SOC effect. Similarly, the 4Π state splits into four Ω states including 5/2, 3/ 2, 1/2, and 1/2 states. Around Re, the energies of the Ω states increase in the order of 5/2, 3/2, 1/2, and 1/2. The calculated energy intervals of 5/2–3/2, 3/2-1/2, and 1/2-1/2 are 164.8 cm  1, 181.74 cm  1, and 201.0 cm  1, respectively. The 2Ф state splits into Ω ¼7/2 and 5/2 states. The determined SOC splitting of 550 cm  1 is much larger than that of X2Π. The PEC of the Ω ¼ 7/2(1) state has the potential peak at R ¼2.72 Å, which is generated from the avoided crossing with the Ω ¼7/2(2). This could be attributed to the PECs' crossing between the 4Δ and 2Ф states. The avoided crossing point is also the minimum energy(36293.3975 cm  1) of the 7/2(2) state, of which the wavefunction is composed of the Λ-S components 4Δ(88.1%) and 2Ф(11.8%). For PCl þ , under the influence of the SOC effect, many excited Ω states are generated, especially in the energy region of 30,000– 50,000 cm  1. The high-density Ω states and the non-crossing rule of electronic states lead to complex structures of the PECs. Some of them have two or more potential wells. These wells are rather shallow, and even cannot support one vibrational level. Therefore, it is expected that the spectra of the PCl þ cation in this energy region will be very diffuse and hard to be observed in experiment. Regarding the energy range of 50,000–60,000 cm  1, the 3/2(15) state mainly consists of the 62Π state over R. And the shape of the PEC is therefore similar to that of the 62Π state. 3.4. Transition properties of the excited states For the transitions 1/2(2)-X11/2, 1/2(3)-X11/2, 1/2(4)-X11/2, 3/2 (2)-X23/2, 3/2(3)-X23/2, and 5/2(1)-X23/2, the R-dependent transition dipole moments (TDMs) have been calculated. The TDM curves are plotted in Fig. 5. As shown in the figure, the transitions 1/2(4)-X11/2 and 3/2(3)-X23/2 have very large TDMs in the FranckCondon region, because they are mainly from dipole-allowed

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Table 4 The spectroscopic constants of the bound Ω states. Ω state

Te/cm-1

Re/Å

ωe/cm-1

ωeχe/cm-1

Be/cm-1

103αe/cm-1

De/eV

Main Λ-S components at Re

X11/2 Expt. [9] Expt. [10] X23/2 Expt. [10] 5/2(1) 3/2(2) 1/2(2) 1/2(3) 1/2(4) 3/2(3) 1/2(5) 3/2(4) 1/2(6) 7/2(1) 5/2(2) 1/2(5) 1/2(14) 7/2(2) 5/2(5) 1/2(15) 3/2(12) 5/2(6) 1/2(16) 5/2(7) 3/2(13) 3/2(15) 1/2(20) 1/2(21)

0 0 0 353  370 cm-1 24714 24879 25060 25261 28793 28938 30002 30097 32227 33445 33995 35914 38700 36293 39133 42233 42779 42813 43019 44363 44685 54815 55302 56836

1.903

688.8 689.8

2.615 2.6

0.283

1.556

4.551

X2Π(100)

1.900 1.903

688.1

2.610

0.28437 0.283

1.471 1.561

4.473

X2Π(100)

2.314 2.314 2.314 2.314 2.346 2.343 2.171 2.183 2.267 2.317 2.317 3.020 2.997 2.723 2.960 2.697 2.607 2.622 2.808 2.854 2.907 3.172 3.170 2.921

348.4 345.0 344.6 348.2 348.3 349.5 576.0 565.7 319.7 338.8 339.9 177.7 470.6 569.4 472.7 228.4 294.6 185.2 291.2 240.3 187.0 176.1 177.5 248.6

3.923 2.503 0.275 3.951 1.434 1.775 18.564 17.990 7.299 2.058 2.587 7.816 16.182 18.931 15.368 8.904 14.376 7.631 15.044 14.891 6.540 3.359 4.569 19.688

0.192 0.192 0.192 0.192 0.187 0.188 0.218 0.216 0.200 0.191 0.191 0.113 0.114 0.139 0.117 0.150 0.151 0.160 0.131 0.126 0.121 0.102 0.102 0.119

1.580 1.805 0.800 1.577 4.438 4.423 6.186 5.382 2.6518 1.487 1.541 1.817 0.579 0.930 0.614 – 8.590 – 2.471 2.757 2.630 1.610 1.517 4.788

1.506 1.354 1.440 1.456 0.901 0.877 0.837 0.740 0.483 –

14Π(100.0) 14Π(99.5) 14Π(99.4) 14Π(99.8) A2Π(99.36) A2Π(99.04) A2Π(91.6), 14Σ-(7.8) 14Σ-(91.4), A2Π(8.3) 32Π(99.7) 2 Ф(99.9) 2 Ф(99.8) 24Σ-(75.2), 12Σ-(20.1), 14Σ þ (2.6), 12Σ þ (1.3) 32Π(52.5), 24Π(46.1) 4 Δ(88.1), 2Ф(11.8) 24Π(97.84), 12Δ(1.4) 42Π(96.8), 22Σ þ (2.6) 22Δ(69.9), 24Π(29.4) 24Π(41.2), 22Δ(58.2) 42Π(83.3), 22Σ þ (14.8), 32Σ þ (0.7) 32Δ(98.1), 22Δ(1.7) 32Δ(98.0), 52Π(1.1) 62Π(99.9) 62Π(96.0), 42Σ þ (4.0) 42Σ þ (75.9), 62Π(24.0)

0.128 0.725 1.218 0.830 0.302 0.278 0.235 0.212 0.344 0.440 0.242 0.199

Fig. 4. The PECs of the Ω states (a) Ω¼ 1/2; (b) Ω ¼ 3/2; (c) Ω ¼5/2; (d) Ω ¼ 7/2.

(%)

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4. Conclusions The ab initio relativistic MRCI þQ/ACVQZ calculation has been carried out to study the ground and 22 excited Λ-S states of the PCl þ molecule. From PECs, the spectroscopic constants of the bound states have been determined. The calculated results are in good agreement with the available experiments. The SOC calculation for PCl þ has been performed utilizing the state interaction method. The calculated SO matrix element confirms that the A2Π state is perturbed by the 14Σ- state as influenced by the SOC effect. In addition, the 32Π state could be predissociated by the 12Σ- state via the dipolar interaction. The SOC effect makes the 23 Λ-S states split into the 45 Ω states. The SOC effect makes the significant impact on PCl þ , which leads to the complex structures of the states. The transition properties of the excited Ω states have been obtained. In the Franck-Condon region, the radiative lifetimes of the transitions A2Π1/2-X2Π1/2 and A2Π3/2-X2Π3/2 are at the level of nanosecond(ns).

Fig. 5. The TDMs of the selected transitions.

Table 5 The radiative lifetimes of the selected transitions. Transition

Acknowledgements

Radiative lifetimes

1/2(2)-X11/2 1/2(3)-X11/2 3/2(2)-X23/2 5/2(1)-X23/2 1/2(4)-X11/2 3/2(3)-X23/2

(μs) (ms) (μs) (ms) (ns) (ns)

v' ¼0

v'¼ 1

v' ¼ 2

136.2 32.1 158.0 114.9 478.9 487.0

130.4 30.8 151.0 109.2 463.4 460.8

129.6 30.6 149.9 106.8

transition A2Π-X2Π. Particularly, around Re of the upper states, the two transitions correspond to the A2Π1/2-X2Π1/2 and A2Π3/2X2Π3/2, respectively. Separately, the remaining transitions 1/2(2)X11/2, 1/2(3)-X11/2, 3/2(2)-X23/2, and 5/2(1)-X23/2 primarily arise from the spin-forbidden transition 14Π-X2Π and therefore have rather smaller TDMs. The Franck-Condon factors (see Table S1, S2, and S3 in Supplementary materials) of these transitions have been determined with the aid of LEVEL 8.0 program [22]. For the transitions A2Π1/2X2Π1/2 and A2Π3/2-X2Π3/2, the calculated Franck-Condon factors indicate that the 0–16, 0–17, 0–18, and 0–19(v'-v'') bands are the strongest bands, which are in good agreement with the experimental findings [9,10]. The radiative lifetime of the selected vibrational level v' for these transitions has been calculated by the following formula [31–33]:

τν′ = (A ν′)−1 =

=

3h 2

64π 4 a 0⋅e⋅TDM ∑ν ′′ q ν′, ν ′′ (ΔEν′, ν ′′)3 4.936 × 105 TDM ∑ν ′′ q ν′, ν ′′ (ΔEν′, ν ′′)3 2

where qν′,ν′′ is the Franck-Condon factor, TDM is the transition dipole moment in atomic unit(a.u.), the energy separation ΔEν′,ν′′ is in cm  1, and τν′ is in second. The computed radiative lifetimes have been tabulated in Table 5. Due to the large TDMs, the transitions A2Π1/2-X2Π1/2 and A2Π3/2X2Π3/2 have the short lifetimes of 478.9 and 487.0 ns. In contrast, the remaining transitions have much longer lifetimes. Among them, the transition 5/2(1)-X23/2 has the largest lifetime of 114.9 ms.

We very much appreciate the enthusiastic help of Prof. Wenli Zou (Northwestern University) for many years. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11274096 and 11604083), University Science and Technology Innovation Team Support Project of Henan Province (Grant No. 13IRTSTHN016), Postdoctoral Research Sponsorship of Henan province (Grant No. 2015072), University key science research projected of the Henan Province (Grant No. 16A140043), and Doctoral Scientific Research Foundation of Henan Normal University (Grant No. 5101029170297). The calculation of this work was supported by the High Performance Computing Center of Henan Normal University.

Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jqsrt.2017.04.011.

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