Extensive spectroscopic calculations of the 21 Λ-S and 74 Ω states of the AsN molecule including the spin–orbit coupling effect

Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 155–168

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Extensive spectroscopic calculations of the 21 Λ-S and 74 Ω states of the AsN molecule including the spin–orbit coupling effect Hui Liu a,b, Deheng Shi a,n, Jinfeng Sun a, Zunlue Zhu a a b

College of Physics and Electronic Engineering, Henan Normal University, Xinxiang 453007, China College of Physics and Electronic Engineering, Xinyang Normal University, Xinyang 464000, China

a r t i c l e i n f o

abstract

Article history: Received 7 June 2014 Received in revised form 18 September 2014 Accepted 19 September 2014 Available online 30 September 2014

The potential energy curves (PECs) of 74 Ω states generated from the 21 Λ-S states of AsN molecule are studied for the first time for internuclear separations from 0.1 to 1.0 nm. Of these 21 Λ-S states, the X1Σ þ , a0 3Σ þ , 15Σ þ , 13Δ, 13Σ  , a3Π, 15Π, 25Σ þ , 35Σ þ , 23Δ, 23Π, 33Π, 35Π, and A1Π states are found to be bound, and the 23Σ þ , 33Σ þ , 15Σ  , 15Δ, 25Δ, 25Π, and 17Σ þ states are found to be repulsive ones. The 33Π state possesses the double well. The 25Σ þ , 35Σ þ , 35Π, and 33Π states possess the shallow well. The a0 3Σ þ , 13Σ  , 23Π, 13Δ, 15Π, 25Π, 35Π, and 17Σ þ states are found to be the inverted ones with the spin–orbit coupling effect taken into account. The PECs are calculated using the CASSCF method, which is followed by the internally contracted MRCI approach with Davidson correction. Core–valence correlation and scalar relativistic corrections are included. The vibrational properties are evaluated for the 25Σ þ , 35Σ þ , and 35Π states and the second well of the 33Π state. The spin–orbit coupling effect is accounted for by the state interaction method with the Breit–Pauli Hamiltonian. The PECs are extrapolated to the complete basis set limit. The spectroscopic parameters are evaluated, and compared with available measurements and other theoretical results. The Franck–Condon factors and radiative lifetimes of the transitions from the a0 3Σ1þ , a3Π1, A1Π1, 13Δ1 and a3Π0  states to the X1Σ0þþ state are calculated for several low vibrational levels, and some necessary discussion is performed. Analyses show that the spectroscopic parameters reported in this paper can be expected to be reliably predicted ones. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Potential energy curve Spectroscopic parameter Franck–Condon factor Spin–orbit coupling effect Radiative lifetime

1. Introduction The AsN is one of the most important semiconductor materials, which can be used for the light-emitting and photovoltaic devices. As a result, the AsN molecule has attracted extensive attentions of physicists, chemists and material scientists. Much experimental [1–7] and theoretical works [8–13] have been done to obtain its various

n

Corresponding author. Tel./fax: þ86 373 3328876. E-mail addresses: [email protected], [email protected] (D. Shi).

http://dx.doi.org/10.1016/j.jqsrt.2014.09.020 0022-4073/& 2014 Elsevier Ltd. All rights reserved.

physical properties. However, little spectroscopic knowledge of the AsN has been known, and spectroscopic information is available only for few Λ-S states and Ω states up to now. As we know, all the applications as the semiconductor material need the accurate spectroscopic properties. For this reason, this paper in detail investigates the potential energy curves (PECs) so that some spectroscopic knowledge of the molecule can be extended. Early in 1934, for the first time, Spinks [1] observed nearly 30 red degraded bands of the AsN molecule in emission. In 1967 and 1968, D’incan and Fémelat made a partial analysis of the A1Π–X1Σ þ transitions, and observed

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a new transition involving the same lower state [2]. In 1970, Jones [2] photographed six bands of the strongly perturbed A1Π–X1Σ þ system of the molecule with a dispersion of approximately 0.4 Å/mm, and Dixit et al. [3] also photographed the 0–0 and 0–1 bands of the A1Π–X1Σ þ system at a dispersion of 0.38 Å/mm. In 1974, Fémelat and Jones [4] studied several transition bands originating from the A1Π–X1Σ þ system. Some spectroscopic parameters and molecular constants were evaluated from these measurements [1–4]. In 1979, Huber and Herzberg [14] summarized some accurate spectroscopic parameters and molecular constants of involved states as of that time. In 1982, Perdigon and Fémelat [5] made a detailed rotational analysis of the A1Π–X1Σ þ transition of As14N and As15N molecules so as to explain the perturbations observed in the A1Π state, and evaluated several spectroscopic parameters of a3Π and A1Π states. In addition, they also obtained the spin–orbit (SO) constant of a3Π state. In 1988, Henshaw et al. [6] identified two bands of AsN molecule in the chemiluminescence from the reaction, and evaluated several spectroscopic parameters of X1Σ þ and a0 3Σ þ states. In 1988, Saraswathy and Krishnamurty [7] photographed the ultraviolet bands of A1Π–X1Σ þ transition of the As14N and As15N under high dispersion. In combination with previous measurements, they reanalyzed the rotational structures of several states involved there. In 1999, Kerr and Stocker [15] thought that the accurate ground-state dissociation energy of AsN molecule should be 5.0370.02 eV. Summarizing these measurements [1–7,15], we find that the spectroscopic parameters of AsN molecule were evaluated only for the X1Σ þ , a0 3Σ þ , a3Π, and A1Π states, and the SO coupling constant was determined only for the a3Π state. In 1985, Ohanessian et al. [8] made the first ab initio calculations of this molecule. They calculated the PECs of eight states by the configuration interaction (CI) method with the basis set of double-zeta quality. In 1992, Toscano and Russo [9] obtained the PECs of six states by the linear combination of Gaussian type orbitals-model potentiallocal spin density (LCGTO-MP-LSD) approach. In 1995, Katsuki [10] determined the ground-state PEC using the model potential. They [8–10] evaluated some spectroscopic parameters of the involved states. In 2001, Martin and Sundermann [11] proposed a group of correlationconsistent valence basis sets with the Stuttgart–Dresden– Bonn (SDB) relativistic effective core potentials. In 2003, Peterson [12] developed a group of convergent basis sets with the relativistic pseudopotentials. With the basis sets, they [11,12] evaluated the ground-state spectroscopic

parameters. In 2010, Wang and Sun [13] computed the ground-state PEC, and evaluated the spectroscopic parameters. Summarizing the theoretical spectroscopic results available in the literature [9–13], we find the following. Firstly, most calculations are focused on the ground state, and few results achieve high quality. Secondly, no core– valence correlation correction has been included into the PEC calculations, though it can bring about the important effect on the accurate prediction of spectroscopic parameters. And thirdly, very few transition properties (such as Franck–Condon factors and radiative lifetimes of transitions) are calculated, though the transition properties are very useful in observing the excited states. Therefore, to improve the quality of spectroscopic parameters of the AsN molecule, more accurate calculations should be done. The aim of this work is to extend the spectroscopic knowledge of AsN molecule. Firstly, extensive ab initio calculations of the PECs will be made, in which both the core–valence correlation and scalar relativistic corrections are included so that the spectroscopic properties will be evaluated as accurately as possible. Secondly, the effect of SO coupling on the PECs will be introduced into the calculations since no PECs have been determined for any Ω states up to now. And thirdly, the Franck–Condon factors and radiative lifetimes of the transitions from several low-lying states to the ground state are calculated for several low vibrational levels since the theoretical transition properties are very few in the literature, though they are very useful in observing the AsN molecule in experiment. In the next section, we will briefly describe the theory and method used in this paper. In Section 3, the PECs of 21 states, X1Σ þ , a0 3Σ þ (10 3Σ þ ), 15Σ þ , 13Δ, 13Σ  , a3Π (13Π), 15Π, 25Σ þ , 35Σ þ , 23Δ, 23Π, 33Π, 35Π, A1Π, 15Σ  , 23Σ þ , 33Σ þ , 17Σ þ , 15Δ, 25Δ, and 25Π, of the AsN molecule are calculated using the complete active space self-consistent field (CASSCF) method, which is followed by the internally contracted multireference CI (icMRCI) approach [16,17] with Davidson correction (icMRCI þQ) [18,19]. The effect of core–valence correlation and scalar relativistic corrections on the PECs is taken into account. All the PECs are extrapolated to the complete basis set (CBS) limit. The spectroscopic parameters are evaluated, and compared with those available in the literature. The Franck–Condon factors and radiative lifetimes of transitions from the a0 3Σ1þ , a3Π1, A1Π1, 13Δ1 and a3Π0  states to the X1Σ0þþ ground state are calculated, and some necessary discussion is performed. And finally, some concluding remarks are given in Section 4.

Table 1 Dissociation relationships of a few states of the AsN molecule. Dissociation channel

Electronic state

Relative energy/cm  1

As(4Su) þN(4Su) As(2Du) þN(4Su) As(2Pu) þN(4Su) As(4Su) þN(2Du) As(2Du) þN(2Du)

X1Σ þ , a0 3Σ þ , 15Σ þ , 17Σ þ 13Δ, 15Δ, a3Π, 15Π, 23Σ þ , 25Σ þ 13Σ  , 15Σ  , 23Π, 25Π 23Δ, 25Δ, 33Π, 35Π, 33Σ þ , 35Σ þ A1Π, (21Σ þ , 31Σ þ , 41Σ þ , 43Σ þ , 53Σ þ , 63Σ þ , 11Σ  , 21Σ  , 13Σ  , 23Σ  , 11Π, 21Π, 31Π, 41Π, 33Π, 43Π, 53Π, 63Π, 11Δ, 21Δ, 31Δ, 33Δ, 43Δ, 53Δ, 11Φ, 21Φ, 13Φ, 23Φ, 11Γ, 13Γ)

0.0 10,590.7 7223.3 17,899.57 224.7 19,181.5 7 143.9 29,953.9

H. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 155–168

2. Theory and method To find out the dissociation channels of the electronic states involved here, we first deduce all the electronic states resulting from the lowest five dissociation channels of AsN molecule. These results are shown in Table 1. Among the 21 states involved here, four states (X1Σ þ , 03 þ a Σ , 15Σ þ , and 17Σ þ ) are attributed to the As(4Su) þN (4Su) dissociation channel. Six states (13Δ, 15Δ, a3Π, 15Π, 23Σ þ , and 25Σ þ ) belong to the As(2Du)þN(4Su) dissociation channel. Four states (13Σ  , 15Σ  , 23Π, and 25Π) dissociate into the As(2Pu) þN(4Su) dissociation channel. Six states (23Δ, 25Δ, 33Π, 35Π, 33Σ þ , and 35Σ þ ) correlate to the As(4Su)þ N(2Du) dissociation channel. And only the A1Π state is attributed to the As(2Du)þN(2Du) dissociation channel. All the PECs are calculated by the CASSCF method, which is followed by the icMRCI approach. Here, the CASSCF is employed as the reference wavefunction for the icMRCI calculations. The basis set used to calculate all the PECs is the aug-cc-pV5Z (AV5Z) [20,21]. All the PECs are calculated with the MOLPRO 2010.1 program package [22] for internuclear separations from about 0.1 to 1.0 nm. To determine accurately the PECs, the point spacing interval used here is 0.02 nm for each state, except around the internuclear equilibrium separation where the point spacing is 0.002 nm. Here, the smaller step is adopted around the equilibrium position of each state so that the properties of each PEC can be displayed more clearly. It should be pointed out that the point spacing stated here is suitable for all the PEC calculations, including the core–valence correlation and scalar relativistic correction calculations and the SO coupling calculations. We substitute the C1v symmetry with the C2v point group to perform the present calculations by orienting the molecule along the Z axis. The orbitals are optimized by the CASSCF method. The state-averaged technique is used in the CASSCF calculations. Eight outermost molecular orbitals (MOs), 4a1, 2b1, and 2b2, are put into the active space for the CASSCF and the subsequent icMRCI calculations. Five valence electrons in the 4s4p orbitals of As atom and five valence electrons in the 2s2p orbitals of N atom are put into the active space, which consists of full valence space. That is, 10 valence electrons in the molecule are distributed into the eight valence MOs. The energy ordering of the valence MOs is 8σ9σ10σ4π5π11σ. As a result, this active space is referred to as CAS (10, 8). The inner 30 electrons in the AsN molecule are put into the 15 closedshell MOs (8a1, 3b1, 3b2, and 1a2), which correspond to the 1–7σ, 1–3π, and 1δ MOs in the AsN molecule. In the calculations, the total number of external orbitals is 244, including 86a1, 60b1, 60b2, and 38a2 symmetry MOs. The 18 electrons in the 3s3p3d closed shell of As atom and the two electrons in the 1s closed shell of N atom are used as core electrons for the core–valence correlation calculations, while the 10 electrons in the 1s2s2p inner shell of the As atom are frozen. That is, when we make the core– valence correlation calculations [23,24], the 10 electrons in the 1s2s2p inner shell of As atom are frozen. When we make the frozen-core calculations, the 28 electrons in the 1s2s2p3s3p3d orbitals of As atom and two electrons in the

157

1s orbital of N atom are frozen. In summary, the 23 MOs (12a1, 5b1, 5b2, and 1a2) are used for the present PEC calculations of all the states, including the calculations of core–valence correlation and scalar relativistic corrections. At this time, we find that all the PECs are convergent. In general, the convergence of each PEC clarifies that the dissociation energy can be obtained by the difference between the total energy of AsN molecule at the equilibrium position (which is determined by fitting) and the total energy of the molecule at 1.0 nm. It should be pointed out that the core–valence correlation correction is calculated at the level of icMRCI theory with a cc-pCVTZ basis set [23,24], and is applied across the entire PEC of each state. Here, the contribution of core–valence correlation correction is denoted as CV. The preferred way to include the scalar relativistic correction is to employ the third-order Douglas–Kroll Hamiltonian (DKH3) approximation [25,26], since the total energy at the DKH3 approximation can best reproduce the full 4-component relativistic results. Therefore, we calculate the scalar relativistic correction in this way. The ccpVTZ-DK [26] basis set with the DKH3 approximation and the cc-pVTZ basis set with no DKH3 approximation are used to calculate the present scalar relativistic correction contribution, which is denoted as DK. It should be pointed out that the scalar relativistic correction is calculated at the level of icMRCI theory, and is applied across the entire PEC of each state. The total energies of a state are equal to the reference energy plus the correlation energy in the MRCI calculations. To obtain more accurate PECs, here we extrapolate the reference energy and the correlation energy, respectively, since the reference energy and the correlation energy have different convergent behaviors. Two correlationconsistent basis sets, aug-cc-pVQZ (AVQZ) and AV5Z [20,21], are used for the present two-point basis set extrapolation (which is denoted as Q5). The extrapolation formula is written as [27] ref  α ref ΔEref ; X ¼ E1 þ A X

ð1Þ

corr  β ΔEcorr ¼ Ecorr X : X 1 þA

ð2Þ

corr are the Hartree–Fock and correlation Here ΔEref X and ΔEX energies, respectively, obtained directly by the aug-cc-pVXZ corr (AVXZ) set in the present calculations. ΔEref are 1 and ΔE 1 the Hartree–Fock and correlation energies, respectively, obtained by extrapolating the basis set to the aug-cc-pV1Z (AV1Z). The extrapolation parameters α and β are taken as 3.4 and 2.4 for the present Hartree–Fock and correlation energies [27]. The SO coupling effect is included into the PEC calculations by the state interaction method with the Breit–Pauli Hamiltonian [28], which has been implemented in MOLPRO 2010.1 program package [22]. The calculations are made at the level of icMRCI theory with the all-electron aug-cc-pCVTZ (ACVTZ) basis set, and are applied across the entire PEC of each Ω state. The all-electron ACVTZ basis set with the Breit–Pauli Hamiltonian and the same basis set with no Breit–Pauli Hamiltonian are employed to calculate the SO coupling contribution. The difference between the

H. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 155–168

two energies generates the SO coupling splitting energy. Adding the SO coupling splitting energies to the icMRCIþ Q/Q5 þCV þDK results, we can acquire the final PEC of each Ω state. With these PECs, the spectroscopic parameters, including the excitation energy term Te referred to the ground state, equilibrium internuclear separation Re, harmonic frequency ωe, first- and second-order anharmonic constants ωexe and ωeуe, rotation–vibration coupling constant αe, rotational constant Be, and dissociation energy De, are evaluated. To evaluate accurately the spectroscopic parameters, all the PECs are fitted to an analytical form by cubic splines so that the rovibrational Schrödinger equation can be conveniently solved with Numerov's method [29]. That is, the rovibrational constants are first determined in a direct forward manner from the analytic potential by solving the rovibrational Schrödinger equation, and then the spectroscopic parameters are calculated by fitting the first ten vibrational levels whenever available. 3. Results and discussion Using the approach described in Section 2, we have calculated the PECs of 21 Λ-S states for internuclear separations from about 0.1 to 1.0 nm. At the same time, we have included the core–valence correlation and scalar relativistic corrections and extrapolation scheme into the PEC calculations. To show clearly the relationships of all the PECs, we depict them in Figs. 1 and 2. To make Figs. 1 and 2 more informative, we depict the PECs only over a small internuclear separation range from about 0.12 to 0.63 nm. As shown in Figs. 1 and 2, seven states (15Σ  , 23Σ þ , 33Σ þ , 17Σ þ , 15Δ, 25Δ, and 25Π) are repulsive. We discuss the effect of core–valence correlation and scalar relativistic corrections on the spectroscopic parameters of AsN molecule. To simplify the discussion, we only take the X1Σ þ , a0 3Σ þ , A1П, and a3Π states as examples at the level of icMRCI þQ theory. The Te, Re, ωe, and De of these four states are provided in Table 2. It should be pointed out that these four states selected for the present discussion are optional. As tabulated in Table 2, (1) Te is raised by 388.25– 970.15 cm  1 with only the core–valence correlation 10

Potential energy /Hartree

-2314.2

9 -2314.3

As(4Su)+N(2Du) As(2Pu)+N(4Su) As(2Du)+N(4Su)

8 7 5 6

-2314.4 3 4 2

-2314.5

-2314.6 0.1

As(4Su)+N(4Su)

1 0.2

0.3

0.4

0.5

0.6

-2314.25

Potential energy /Hartree

158

11

As(2Du)+N(2Du) As(4Su)+N(2Du) As(2Pu)+N(4Su)

-2314.30 8 7

-2314.35 -2314.40

0.1

10 5

As(2Du)+N(4Su)

6

4 3

-2314.45

9

2 1

0.2

0.3

0.4

0.5

0.6

Internuclear separation /nm Fig. 2. PECs of 11 states of AsN molecule. 1—13Δ; 2—a3Π; 3—A1Π; 4—15Π; 5—23Δ; 6—15Δ; 7—23Π; 8—33Π; 9—25Π; 10—35Π; 11—25Δ.

correction added, and by 164.52–224.52 cm  1 with only the scalar relativistic correction taken into account for the a0 3Σ þ , A1П, and a3Π states. From that, we can see that the effect of the two corrections on the Te is obvious. (2) ωe is raised by 0.13–14.11 cm  1 with only the core–valence correlation correction included, and is lowered by 3.66– 6.13 cm  1 with only the scalar relativistic correction added for the four states. On the whole, the contribution to the ωe by the core–valence correlation correction is larger than that by the scalar relativistic correction. (3) Core–valence correlation correction shortens the Re by 0.00095–0.00117 nm, but scalar relativistic correction shortens Re by at most 0.00037 nm. Therefore, the effect of core–valence correlation correction on the Re is obvious, but the effect of scalar relativistic correction on the Re is small. With the two corrections included into the calculations at the same time, the smallest shifts of Te, Re, and ωe reach 624.63 cm  1, 0.00106 nm, and 2.27 cm  1 for these four states, respectively. From that, great errors will be introduced if we exclude the effect of core–valence correlation and scalar relativistic corrections on the spectroscopic parameters. When we analyze the Te, Re, and ωe of other 10 Λ-S states involved in this paper, similar conclusion can be gained. To our knowledge, no previous spectroscopic calculations have taken into account the core–valence correlation correction. Then we study the convergence of this work since highquality ab initio calculations must be convergent with respect to the basis set. Otherwise, the results calculated may be insignificant because of low accuracy and low reliability. Due to length limitation, we only discuss such convergence in Supplementary material of this paper. From the discussion, we clearly see that the present calculations are convergent with respect to the basis set at the present level of theory. According to the discussion made above, we will use the PECs determined by the icMRCI þQ/Q5 þCVþDK calculations to make the following spectroscopic calculations. 3.1. Spectroscopic parameters of the 14 bound states

Internuclear separation /nm Fig. 1. PECs of 10 states of AsN molecule. 1—X1Σ þ ; 2—a0 3Σ þ ; 3—13Σ  ; 4—15Σ þ ; 5—17Σ þ ; 6—25Σ þ ; 7—23Σ þ ; 8—33Σ þ ; 9—15Σ  ; 10—35Σ þ .

The present calculations are involved with the first five dissociation channels of AsN molecule. We have determined

H. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 155–168

159

Table 2 Effect of core–valence correlation and scalar relativistic corrections on the spectroscopic parameters of X1Σ þ , a0 3Σ þ , A1П, and a3Π states at the icMRCI þQ level of theory. Te/cm  1 X1Σ þ icMRCI þ Q CV DK CV þDK A1 П icMRCI þ Q CV DK CV þDK

0.0 0.0 0.0 0.0 35,399.94 388.25 224.52 624.63

Re/nm

ωe/cm  1

De/cm  1

Te/cm  1

Re/nm

ωe/cm  1

De/cm  1

0.16343  0.00097  0.00037  0.00137

1053.12 14.11  6.13 7.86

39,775.22 257.29 310.52 582.33

a0 3Σ þ 19,957.93 970.15 164.52 1147.41

0.18036  0.00095  0.00009  0.00106

737.14 7.61  4.07 3.36

19,790.42  567.01  70.17  635.56

0.17018  0.00116  0.00009  0.00126

856.41 7.26  5.19 2.27

34,297.91 302.46  276.65 28.23

a3Π 29,057.12 562.96 213.11 786.16

0.16990  0.00117  0.00014  0.00132

873.70 0.13  3.66  3.43

21,326.90 37.91  316.98  274.23

the energy separations between each higher dissociation channel and the lowest one by the icMRCI þQ/Q5 þCV þDK calculations, and provided them in Table 1. As shown in Table 1, we can clearly see that the 21 states involved here can properly dissociate into the atomic fragments. It should be pointed out that the energy separations relative to the lowest one for the first four dissociation channels are obtained by averaging all the electronic states which dissociate into the same channel. The present energy separation between the fifth dissociation channel and the lowest one is obtained only by the A1Π state since only the A1Π state is involved here for the fifth channel. For convenience of discussion, the dominant valence configurations of 14 bound states as obtained from the icMRCI wavefunctions near the equilibrium positions are given in Table 3. Now we summarize the multiconfiguration characterizations of these states. According to Table 3, we think that the X1Σ þ , A1Π, 15Σ þ , 23Π, and 15Π states and the second well of 33Π state have poor multireference characterizations near the equilibrium positions. The a0 3Σ þ , a3Π, 13Δ, 23Δ, 25Σ þ , and 13Σ  states and the first well of 33Π state have the obvious multireference characterizations near the equilibrium positions when the spin orientation of electrons is included, whereas their multireference characterizations become poor when we ignore the spin orientation of electrons. As seen in Table 3, only the spin orientation of electrons in the 4π orbital is different for each dominant valence configuration of a0 3Σ þ , a3Π, 13Δ, 25Σ þ , and 13Σ  states. And only the spin orientation of electrons in the 4π and 5π orbitals is different for each main valence configuration of 23Δ state and the first well of 33Π state. Of these 14 states, only the 35Σ þ and 35Π states possess the multireference characterizations near the equilibrium positions whether the spin orientation of electrons is taken into account or not. From these configurations, we can instantly find out how the electronic transition occurs from one state to another. Using the PECs determined by the icMRCIþ Q/ Q5þCV þDK calculations, we have evaluated the spectroscopic parameters of all the states involved here by the approach outlined in Section 2. For convenience of discussion, we divide the 14 bound states into two categories. One is the X1Σ þ , a0 3Σ þ , a3Π, and A1Π, which are the states possessing the experimental spectroscopic results

[2,4–6,14,15]. The other is the 13Δ, 13Σ  , 15Σ þ , 15Π, 23Δ, 25Σ þ , 23Π, 33Π, 35Π, and 35Σ þ , which are the states without available spectroscopic measurements so far. 3.1.1. X1Σ þ , a0 3Σ þ , a3Π, and A1Π states For convenience of comparison, we tabulate the available theoretical results [8–13] in Table 4. Due to length limitation, we only collect some selected measurements [5,6,14,15] in Table 4 for discussion. As seen in Table 3, the dominant valence configuration of ground state is 8σ

αβ

9σ

αβ

10σ

αβ

4π

αβαβ

5π011σ0,

1

and the dominant valence configuration of A αβ

αβ

α

αβαβ

β

Π state

5π 11σ0 near the equilibrium is 8σ 9σ 10σ 4π separations. Thus, the electronic transition between the X1Σ þ and A1Π states arises from the 10σ-5π electron promotion. The present prediction of energy separation between the X1Σ þ and A1Π states is 36,098.53 cm  1, which deviates from the corresponding measurements [5] of 35,999 cm  1 by only 99.53 cm  1. The dominant valence configuration of a0 3Σ þ state can be regarded as 8σ29σ210σ24π35π111σ0 when the spin orientation of electrons is neglected. Thus, the transition between the X1Σ þ and a0 3Σ þ states can be viewed as arising from the 4π-5π electron promotion. As shown in Table 4, the present energy separation between the X1Σ þ and the a0 3Σ þ state is 21,269.51 cm  1, which deviates from the only measurements of 21,66478 cm  1 [6] by 394.49 cm  1. Only one group of theoretical work [9] determined the Te of a0 3Σ þ state to be 24,992 cm  1, which is larger than the measurements [6] by 3328 cm  1. Similar to the a0 3Σ þ , the main valence configuration of a3Π state is 8σ29σ210σ24π35π111σ0 when we ignore the spin orientation of electrons in the 4π orbital. Thus, the transition between the X1Σ þ and the a3Π state can also be viewed as arising from 4π-5π electron promotion. The present Te of a3Π state is 29,943.80 cm  1, which deviates from the measurements of 29,632 cm  1 [5] by 311.8 cm  1. In addition, the detailed PEC of a3Π state shows a potential barrier at about 0.29403 nm, which barrier height is about 134.98 cm  1. We cannot clearly see the barrier in Fig. 2, since it is too low. At least six groups of calculations [9–13] have been made to obtain the spectroscopic properties of these states, in particular for the ground state. By careful comparison, we

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Table 3 Dominant valence configurations of the 14 states at the energy minimum. State

Dominant valence configuration near the equilibrium position

X Σ a0 3Σ þ a3Π A1Π 15Σ þ 23Π 15Π 13Δ 23Δ 25Σ þ 13Σ 

8σαβ9σαβ10σαβ4παβαβ5π011σ0 (0.7827)a 8σαβ9σαβ10σαβ4πααβ5πα11σ0 (0.4104); 8σαβ10σαβ10σαβ4παβα5πα11σ0 (0.3982). 8σαβ9σαβ10σαβ4πααβ5πα11σ0 (0.4220); 8σαβ9σαβ10σαβ4παβα5πα11σ0 (0.4074). 8σαβ9σαβ10σα4παβαβ5πβ11σ0 (0.7622) 8σαβ9σαβ10σαβ4παα5παα11σ0 (0.8716) 8σαβ9σαβ10σα4παβαβ5πα11σ0 (0.7607) 8σαβ9σαβ10σαβ4παα5παα11σ0 (0.8837) 8σαβ9σαβ10σαβ4παβα5πα11σ0 (0.4140); 8σαβ10σαβ10σαβ4πααβ5πα11σ0 (0.4117) 8σαβ9σαβ10σαβ4παβ5παα11σ0 (0.5793); 8σαβ9σαβ10σαβ4παα5παβ11σ0 (0.3043) 8σαβ9σαβ10σα4παβα5πα11σα (0.3829); 8σαβ9σαβ10σα4πααβ5πα11σα (0.3827) 8σαβ9σαβ10σαβ4παβα5πα11σ0 (0.4032) ; 8σαβ9σαβ10σαβ4πααβ5πα11σ0 (0.4260)

33Π 1st well 2nd well

8σαβ9σαβ10σα4παβα5πβα11σ0 (0.4626); 8σαβ9σαβ10σα4παββ5παα11σ0 (0.2524). 8σαβ9σαβ10σα4παβαβ5πα11σ0 (0.2306)

1

þ

35Σ þ 35Π

8σαβ9σαβ10σα4πβα5παα11σα (0.2464); 8σαβ9σαβ10σαβ4παα5παα11σ0 (0.1676); 8σαβ9σαβ10σα4παα5πβα11σα (0.1649) 8σαβ9σαβ10σα4παβα5παα11σ0 (0.3468); 8σαβ9σαβ10σαβ4πα5παα11σα (0.1366); 8σαβ9σαβ10σα4παα5παβα11σ0 (0.1303); 8σαβ9σαβ10σα4πα5παα11σαβ (0.1241); 8σαβ9σαβ10σ04παβα5παα11σα (0.1054)

Table 4 Comparison of the spectroscopic parameters obtained by the icMRCI þ Q/Q5 þCV þDK calculations with available measurements and other theoretical results for the X1Σ þ , a0 3Σ þ , a3Π, and A1Π. Te/cm  1 X1Σ þ Exp. [5] Exp. [6] Exp. [14] Exp. [15] Cal. [8] Cal. [9] Cal. [10] Cal. [11] Cal. [12]c Cal. [13] 03

þ

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

De/eV

Re/nm

ωe/cm  1

ωexe/cm  1

102ωeуe/cm  1

Be/cm  1

103αe/cm  1

5.0267 – – – 5.03 70.02 4.22 6.17 1.66 5.09a 4.8902 4.97

0.16188 0.1618 – 0.16184

1065.60 1068.71 1075 74 1068.54

5.3470 5.45 5.4 70.6 5.41

10.79 –

0.54559 0.54566

3.867 3.659

–

0.54551

3.366

0.16606 0.16119 0.16230 0.16188b 0.16252 0.16259

1056 1064 1191 1079.1b 1073.32 1061.14

5.07 5.4715

–

0.53919

3.409

a Σ Exp. [6] Cal. [9]

21,269.51 21,664 7 8 24,992

2.4181 – 3.08

0.17904 – 0.17320

745.02 738 76 843

5.5549 4.5 71

1.560

0.4457

4.008

a3Π Exp. [5] Cal. [8] Cal. [9]

29,943.80 29,632 29,170 31,661

2.6494 – – 3.90

0.16834 0.1683 0.17129 0.16849

872.54 897 923 905

2.0289 6.9

36.20 –

0.5040 0.5045

4.358 6.1

A1Π Exp. [5] Exp. [14] Cal. [8] Cal. [9]

36,098.53 35,999 35,999.7 36,090 37,343

4.2800 – – – 6.06

0.16868 0.1689 0.1687 0.17087 0.16860

861.01 869 [853.3]d 856 870

0.9536 5.1 8.24

39.44 – –

0.5019 0.5006 0.501

4.049 6.8 9

a

SDB pseudopotential to the CBS limit. SDB pseudopotential. c cc-pV5Z-PP. d ΔG (1/2). b

find that few prior theoretical spectroscopic parameters are closer to the measurements than the present ones. For example, for the X1Σ þ state, no other theoretical De and ωe are closer to the measurements [15] than the present ones. Only the Re obtained by Martin and Sundermann [11] can be comparative with the present one in quality. For the a0 3Σ þ state, only one group of spectroscopic calculations [9]

can be found. As seen in Table 4, their results are inferior to the present ones when compared with the only measurements [6]. For the a3Π state, only the theoretical Te and Re obtained by Toscano and Russo [9] are slightly closer to the measurements [5] than the present ones. And for the A1Π state, no other Te and Re are closer to the measurements [5,14] than the present ones. As for the ωe, only the result

H. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 155–168

obtained by Toscano and Russo [9] is closer to the measurements [5] than the present one. For the X1Σ þ state, the present De of 5.0267 eV agrees favorably with the recent measurements of 5.0370.02 eV [15], and the difference between them is only 0.0033 eV (0.07%). The present Re of 0.16188 nm is in fair agreement with the measurements of 0.1618 nm [5], and the difference between them is only 0.00008 nm (0.05%). As for the ωe, the present result is 1065.60 cm  1, which deviates from the measurements of 1068.71 cm  1 [5] by 3.11 cm  1 (0.29%). For the A1Π state, the present Te and Re deviate from their respective measurements [5] by 99.53 cm  1 (0.28%) and 0.00022 nm (0.13%). For the a0 3Σ þ state, the present Te and ωe deviate from their respective measurements [6] by 394.49 and 7.02 cm  1. These deviations are slightly large. The similar situation exists for the a3Π state by comparison of Te and ωe with the measurements [5]. On the whole, the comparisons indicate that the present spectroscopic parameters should be high quality. We think that the reasons for achieving such high-quality spectroscopic parameters may be several aspects. Two main reasons are as follows. One is that the core–valence correlation and scalar relativistic corrections are included into the present calculations. The other is that the residual errors behind the basis set are eliminated by the totalenergy extrapolation to the CBS limit. 3.1.2. 13Δ, 13Σ  , 15Σ þ , 15Π, 23Δ, 25Σ þ , 23Π, 33Π, 35Π, and 35Σ þ states Only two groups of spectroscopic calculations [8,9] can be found for the 13Δ, 13Σ  , 15Σ þ , 15Π, 23Δ, 25Σ þ , 23Π, 33Π, 35Π, and 35Σ þ states. For convenience of comparison, we provide the spectroscopic parameters together with the theoretical results [8,9] in Table 5. Of these states, only the 33Π possesses the double well. One is at 0.19235 nm, which depth is about 2710.02 cm  1. The other is at 0.27277 nm, which depth is about

161

529.10 cm  1. The detailed PEC shows that the energies of the two wells at their respective equilibrium positions are smaller than that at the dissociation limit. Calculations have determined that the second well possesses only two vibrational states, which vibrational levels are 139.51 and 316.18 cm  1 for υ ¼0 and 1, respectively. To some extent, the second well is not easy to be observed in experiment, though it is stable. From Fig. 2, we can see that an avoided crossing appears between the PEC of 23Π and the PEC 33Π state. The second well of 33Π state also arises from the avoided crossing of the two states. In addition, the barrier on the PEC of 23Π state comes from the same avoided crossing. Calculations have predicted that the barrier of 23Π state is at about 0.26152 nm, and its barrier height is only 923.03 cm  1. Among the 14 bound states involved here, the 35Σ þ state possesses the shallowest well, which depth is only 74.20 cm  1. Calculations have determined that the state has only one vibrational state, which vibrational level is 31.60 cm  1. As a result, the 35Σ þ state should be very hard to be observed in experiment, though the energy of this state at the equilibrium position is smaller than that at the dissociation limit of this state. Each of the two states, 25Σ þ and 35Π, possesses one shallow well. Similar to the 35Σ þ state, we cannot clearly see these wells in Figs. 1 and 2 since they are too shallow. Detailed PECs tell us that the energies of their respective wells at the equilibrium positions are smaller than those at their respective dissociation limits. For the 25Σ þ state, the well depth is only 266.16 cm  1. However, this state has five vibrational states, which vibrational levels are 46.53, 121.54, 176.94, 218.19, and 255.58 cm  1 for υ ¼0–4, respectively. For the 35Π state, the well depth is only 229.87 cm  1. To our surprise, this state also possesses five vibrational states, which vibrational levels are 25.89, 78.43, 133.03, 179.05, and 221.66 cm  1 for υ ¼0–4, respectively. To some extent, the 25Σ þ and 35Π states can be observed in

Table 5 Comparison of the spectroscopic parameters obtained by the icMRCI þ Q/Q5 þ CVþ DK calculations with other theoretical results for the 13Δ, 13Σ  , 15Σ þ , 15Π, 23Δ, 25Σ þ , 23Π, 33Π, 35Π, and 35Σ þ . Te/cm  1

De/ eV

Re/nm

ωe/cm  1

ωexe/cm  1

102ωeуe/cm  1

Be/cm  1

103αe/cm  1

13 Δ Cal. [8] Cal. [9]

28,449.18 26,390 30,248

2.8780 – 3.73

0.17752 0.18034 0.17389

778.51 833 851

6.9529

30.48

0.4533

3.784

13 Σ  Cal. [8]

33,705.38 31,270

3.1128 –

0.17679 0.17981

788.35 849

5.4106

14.44

0.4571

3.487

15 Σ þ

34,714.96

0.7122

0.20482

480.10

9.6672

476.3

0.3408

1.005

15 Π Cal. [8]

45,428.18 44,150

0.7683 –

0.19101 0.18966

576.16 721

8.6394

55.96

0.3915

5.677

23 Δ 25 Σ þ 23 Π

45,709.32 51,395.47 52,846.64

1.7537 0.0330 0.7755

0.20249 0.29730 0.19135

526.85 103.21 583.41

4.9298 14.899 8.5804

139.7 105.1 46.55

0.3478 0.1621 0.3904

1.087 14.70 5.367

33 Π 1st well 2nd well

57,256.98 59,249.14

0.3360 0.0656

0.19235 0.27277

556.32 320.84

3.2012 87.761

17.81 972.5

0.3862 0.0645

3.436 591.4

35 Π 35 Σ þ

59,733.53 59,643.33

0.0285 0.0092

0.32426 0.34669

51.89 73.50

0.2087 –

30.45 –

0.1335 0.1239

0.363

162

H. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 155–168

Table 6 Dissociation relationships of possible Ω states of the AsN molecule obtained by the icMRCI þ Q/Q5 þCV þDK þSO calculations.

As( S3/2) þ N( S3/2) As(2D3/2) þ N(4S3/2) As(2D5/2) þ N(4S3/2) As(2P1/2) þ N(4S3/2) As(2P3/2) þ N(4S3/2) As(4S3/2) þ N(2D5/2) As(4S3/2) þ N(2D3/2) As(2D3/2) þ N(2D3/2)

þ

þ





3, 2, 2, 1, 1, 1, 0 , 0 , 0 , 0 3, 2, 2, 1, 1, 1, 0 þ , 0 þ , 0  , 0  4, 3, 3, 2, 2, 2, 1, 1, 1, 0 þ , 0 þ , 0  , 0  ,  1 2, 1, 1, 0 þ , 0  3, 2, 2, 1, 1, 0 þ , 0 þ , 0  , 0  ,  1 4, 3, 3, 2, 2, 2, 1, 1, 1, 0 þ , 0 þ , 0  , 0  ,  1 3, 2, 2, 1, 1, 1, 0 þ , 0 þ , 0  , 0  3, 2, 2, 1, 1, 1, 0 þ , 0 þ , 0  , 0  , …

experiment, though they are both weakly-bound states. We hope that some experiments will be made in the near future so that some results proposed here can be validated. 3.2. Spectroscopic parameters of the 45

Ω bound states

With the SO coupling effect added, the first dissociation channel of AsN molecule does not split. Each of the second, third and the fourth dissociation channels splits into two dissociation asymptotes, respectively. The fifth dissociation channel splits into four dissociation asymptotes, of which only the As(2D3/2) þN(2D3/2) is involved here. Table 6 shows the dissociation relationships of all the possible Ω states generated from the eight dissociation asymptotes. Using the method described in Section 2, we have calculated the energy separations relative to the lowest dissociation channel for each higher dissociation asymptote at the level of icMRCI þQ/Q5 þCVþ DK theory, which are tabulated in Table 6. For convenience of discussion, Table 6 also shows the experimental energy separations [20] between each of the higher dissociation asymptote and the lowest one. As seen in Table 6, the energy separation between each higher dissociation asymptote and the lowest one is in excellent agreement with the measurements [20]. The present 74 Ω states correlate to the eight dissociation asymptotes. In detail, the 10 Ω states (X1Σ0þþ , a0 3Σ1þ , a0 3Σ0þ , 15Σ2þ , 15Σ1þ , 15Σ0þþ , 17Σ3þ , 17Σ2þ , 17Σ1þ , and 17Σ0þ ) are attributed to the lowest dissociation asymptote. The 10 Ω states (25Σ0þþ , 25Σ1þ , 25Σ2þ , 13Δ3, 13Δ2, 13Δ1, 23Σ0þ , 23Σ1þ , a3Π0  , and a3Π0 þ ) correlate to the As (2D3/2)þN(4S3/2) dissociation asymptote. The 14 Ω states (15Π3, 15Π2, 15Π1, 15Π0 þ , 15Π0  , 15Π  1, a3Π2, a3Π1, 15Δ4, 15Δ3, 15Δ2, 15Δ1, 15Δ0 þ , and 15Δ0  ) are attributed to the As(2D5/2) þN(4S3/2) dissociation asymptote. The 5 Ω states (23Π2, 23Π1, 23Π0 þ , 23Π0  , and 13Σ1 ) correlate to the As(2P1/2) þN(4S3/2) dissociation asymptote. The 10 Ω states (13Σ0þ , 15Σ0 , 15Σ1 , 15Σ2 , 25Π3, 25Π2, 25Π1, 25Π0 þ , 25Π0  , and 25Π  1) correlate to the As(2P3/2)þN (4S3/2) dissociation asymptote. The 14 Ω states (35Π3, 35Π2, 35Π1, 35Π0 þ , 35Π0  , 35Π  1, 33Π2, 33Π1, 25Δ4, 25Δ3, 25Δ2, 25Δ1, 25Δ0 þ , and 25Δ0  ) correlate to the As (4S3/2)þN(2D5/2) dissociation asymptote. The 10 Ω states (35Σ2þ , 35Σ1þ , 35Σ0þþ , 23Δ3, 23Δ2, 23Δ1, 33Σ1þ , 33Σ0þ , 33Π0 þ , and 33Π0  ) belong to the As(4S3/2)þN(4D3/2) dissociation asymptote. Only the A1Π1 state is involved with the As(4D3/2)þ N(4D3/2) dissociation asymptote in this

This work

Exp. [30]

0.0 10,670.04 11,051.99 18,221.27 18,655.12 19,250.08 19,283.66 29,910.26

0.0 10,592.5 10,914.6 18,186.1 18,647.5 19,224.5 19,233.2 29,825.7

12

13

-2314.30

Potential energy /Hartree

4

-2314.35

10

8 5

-2314.40

As(4S3/2) + N(2D3/2) As(4S3/2) + N(2D5/2) As(2P3/2) + N(4S3/2) As(2P1/2) + N(4S3/2)

11

9

6

As(2D5/2) + N(4S3/2)

7

As(2D3/2) + N(4S3/2) 4

3

As(4S3/2) + N(4S3/2)

-2314.45 2 -2314.50

1 0.2

0.3

0.4

0.5

0.6

Internuclear separation /nm Fig. 3. PECs of Ω ¼0  states of AsN molecule. 1—a0 3Σ0þ ; 2—a3Π0  ; 3— 15Π0  ; 4—17Σ0þ ; 5—23Π0  ; 6—23Σ0þ ; 7—15Δ0  ; 8—33Π0  ; 9—25Π0  ;10 —35Π0  ; 11—33Σ0þ ;12—15Σ0 ; 13—25Δ0  .

-2314.3

Potential energy /Hartree

4

Relative energy/cm  1

Possible Ω states

Atomic state

10

11

As(4S3/2) + N(2D3/2) As(4S3/2) + N(2D5/2) As(2P3/2) + N(4S3/2) As(2P1/2) + N(4S3/2)

12 13

8 -2314.4

5 3

-2314.5

-2314.6

9

7 6

As(2D5/2) + N(4S3/2) As(2D3/2) + N(4S3/2) As(4S3/2) + N(4S3/2)

4

2

1 0.2

0.3

0.4

0.5

0.6

Internuclear separation /nm Fig. 4. PECs of Ω ¼0 þ states of AsN molecule. 1—X1Σ0þþ ; 2—a3Π0 þ ; 3— 13Σ0þ ; 4—15Σ0þþ ; 5—15Π0 þ ; 6—25Σ0þþ ; 7—23Π0 þ ; 8—33Π0 þ ; 9—15Δ0 þ ; 10—25Π0 þ ; 11—35Π0 þ ; 12—35Σ0þþ ;13—25Δ0 þ .

paper. Of these 74 Ω states, the 45 Ω states are the bound, and the 29 Ω states are the repulsive ones. The PECs of 69 Ω states are depicted in Figs. 3–8 and their respective dissociation asymptotes are labeled in the same figure. To avoid overlapping between the different PECs, we do not demonstrate the PECs of 15Δ4, 25Δ4, 15Π  1, 25Π  1, and 35Π  1 states in these figures. In addition, we do not either depict each of these PECs in a separate figure for the sake of length limitation. Similar to

H. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 155–168

9 10

-2314.30 -2314.35

As(2D3/2) + N(2D3/2) As(4S3/2) + N(2D3/2) As(2P3/2) + N(4S3/2)

8 7

As(2P1/2) + N(4S3/2)

-2314.40

As(2D3/2) + N(4S3/2) As(4S3/2) + N(4S3/2)

6 5

4 3

-2314.45 2 -2314.50

0.2

0.3

0.4

0.5

0.6

Internuclear separation /nm

Potential energy /Hartree

-2314.30

10 8

As(4S3/2) + N(2D3/2) As(4S3/2) + N(2D3/2)

9

7 6

5

-2314.45

2

As(2P3/2) + N(4S3/2) As(2P1/2) + N(4S3/2) As(2D5/2) + N(4S3/2) As(2D3/2) + N(4S3/2)

4 3

1

0.2

0.3

0.4

0.5

0.6

Internuclear separation /nm Fig. 6. PECs of Ω ¼ 1 states of AsN molecule. 1—13Δ1; 2—a3Π1; 3—15Π1; 4 —23Δ1; 5—23Π1; 6—15Δ1; 7—33Π1; 8—25Π1; 9—35Π1; 10—25Δ1.

14

Potential energy /Hartree

-2314.30 12 -2314.35

11

15 13

7

8

10 9

As(4S3/2) + N(2D3/2) As(4S3/2) + N(2D5/2) As(2P3/2) + N(4S3/2) As(2P1/2) + N(4S3/2) As(2D5/2) + N(4S3/2) As(2D3/2)+ N(4S3/2)

6 -2314.40

5

4

As(4S3/2) + N(4S3/2)

0.3

0.4

3 -2314.45

As(4S3/2) + N(2D3/2) As(4S3/2) + N(2D5/2)

-2314.35

6

4 5

-2314.40

As(2P3/2) + N(4S3/2) As(2D5/2) + N(4S3/2) As(2D3/2) + N(4S3/2)

3 2

As(4S3/2) + N(4S3/2)

-2314.45

2 1 0.2

0.5

0.2

0.3

0.4

0.5

0.6

Internuclear separation /nm

Fig. 5. PECs of Ω ¼ 1 states of AsN molecule. 1—a0 3Σ1þ ; 2—13Σ1 ; 3—15Σ1þ ; 4—A1Π1; 5—17Σ1þ ; 6—25Σ1þ ; 7—23Σ1þ ; 8—33Σ1þ ; 9—15Σ1 ; 10—35Σ1þ .

-2314.40

8 7

1

1

-2314.35

-2314.30

Potential energy /Hartree

Potential energy /Hartree

-2314.25

163

0.6

Internuclear separation /nm Fig. 7. PECs of Ω¼ 2 states of AsN molecule. 1—13Δ2; 2—a3Π2; 3—15Σ2þ ; 4 —17Σ2þ ; 5—15Π2; 6—23Δ2; 7—25Σ2þ ; 8—15Δ2; 9—23Π2; 10—33Π2; 11— 25Π2; 12—35Π2; 13—35Σ2þ ; 14—15Σ2 ; 15—25Δ2.

Figs. 1 and 2, Figs. 3–8 show the PECs of the present 69 Ω states only over a small internuclear separation range from about 0.12 to 0.63 nm so that some details of the PECs can be clearly demonstrated. Using the PECs determined by the icMRCIþ Q/ Q5þCV þDK þSO calculations, we have evaluated the spectroscopic parameters (Te, Re, ωe, and De) of these Ω bound states by the method given in Section 2. It should be pointed out that the 23Σ þ , 33Σ þ , 15Σ  , 15Δ, 25Δ, 25Π,

Fig. 8. PECs of Ω ¼3 states of AsN molecule. 1—13Δ3; 2—17Σ3þ ; 3—15Π3; 4—23Δ3; 5—15Δ3; 6—25Π3; 7—35Π3; 8—25Δ3.

and 17Σ þ are still the repulsive states with the SO coupling effect included. Of the seven repulsive states, the 17Σ þ and 25Π are found to be the inverted ones. We divide the present 45 Ω bound states into three categories according to their symmetries for convenience of discussion. The first group is the 14 Ω states, which are generated from the X1Σ þ , a0 3Σ þ , 13Σ  , 15Σ þ , 25Σ þ , and 35Σ þ states. The second group is the 25 Ω states, which are yielded by the A1Π, a3Π, 23Π, 33Π, 15Π, and 35Π states. And the last group is the 6 Ω states, which are produced from the 13Δ and 23Δ states. 3.2.1. 14 Ω States generated from the X1Σ þ , a0 3Σ þ , 13Σ  , 15Σ þ , 25Σ þ , and 35Σ þ states Table 7 tabulates the spectroscopic results of 14 Ω states, which are generated from the X1Σ þ , a0 3Σ þ , 13Σ  , 15Σ þ , 25Σ þ , and 35Σ þ states. For convenience of discussion, the table also provides the dominant Λ-S state compositions of each Ω state near the equilibrium positions. As seen in Table 7, the Λ-S state compositions of the X1Σ0þþ state is almost pure. As a result, the effect of SO coupling on its spectroscopic parameters is very small. Te of the a0 3Σ1þ state is smaller than, whereas Te of the 03 þ a Σ0  state is larger than that of the a0 3Σ þ state. Therefore, the a0 3Σ þ is an inverted state when the SO coupling effect is included. As seen in Table 7, the Λ-S state compositions of a0 3Σ0þ and a0 3Σ1þ Ω states are almost pure near the equilibrium positions. Therefore, the effect of SO coupling on the spectroscopic parameters should be small. In detail, Re of the two Ω states is the same as that of the a0 3Σ þ state. The largest deviation of ωe from that of the a0 3Σ þ state is only 0.29 cm  1. According to Table 7, the SO coupling constant of a0 3Σ þ state is only 31.61 cm  1. For the 13Σ  state, Te of the 13Σ1 Ω component is smaller than, and Te of the 13Σ0þ Ω component is larger than that of the 13Σ  state. Similar to the a0 3Σ þ , the 13Σ  is also an inverted state when the SO coupling effect is included. As tabulated in Table 7, the state compositions of its two Ω components slightly mix with other several Λ-S states near the equilibrium positions. For this reason, the 13Σ  state only demonstrates slight effect of SO coupling on its spectroscopic parameters. It can be clearly seen by comparison between the spectroscopic parameters given in Tables 5 and 7. In addition, the SO coupling constant of

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Table 7 Spectroscopic parameters determined by the icMRCI þQ/Q5 þ CVþ DKþ SO calculations for the 14 Ω states generated from the X1Σ þ , a0 3Σ þ , 13Σ  , 15Σ þ , 25Σ þ , and 35Σ þ states.

X Σ0þþ a0 3Σ1þ a0 3Σ0þ 13Σ1 13Σ0þ 15Σ0þþ 15Σ1þ 15Σ2þ 25Σ0þþ 25Σ1þ 25Σ2þ 35Σ0þþ 35Σ1þ 35Σ2þ 1

De/eV

Te/nm

Re/nm

ωe/cm  1

Dominant Λ-S state composition near the Re (%)

5.0276 2.4126 2.4088 3.1263 3.1402 0.7111 0.7085 0.6937 0.0388 0.0341 0.0315 0.0092 0.0087 0.0080

0.0 21,242.51 21,274.12 33,686.94 33,710.65 34,602.59 34,630.02 34,725.72 51,388.45 51,416.98 51,442.66 59,636.96 59,659.13 59,673.40

0.16189 0.17904 0.17904 0.17714 0.17700 0.20374 0.20349 0.20476 0.29730 0.29730 0.29730 0.34669 0.34673 0.34671

1063.17 744.82 744.73 788.79 789.38 478.99 480.54 479.39 105.15 102.87 101.16 69.61 67.78 68.21

X1Σ þ (99.96) a0 3Σ þ (99.66), 13Σ  (0.28) a0 3Σ þ (99.88), a3Π (0.12) 13Σ  (98.40), 15Σ þ (0.62), A1Π (0.29), a0 3Σ þ (0.28), a3Π (0.28) 13Σ  (98.50), 15Σ þ (0.84), 13Π (0.52) 15Σ þ (97.83), 13Σ  (1.81), 25Π (0.28) 15Σ þ (98.30), 13Σ  (1.36), 25Π (0.24) 15Σ þ (99.82) 25Σ þ (54.68) , a3Π (31.90), 15Π (8.68), 25Π (1.76), 15Σ  (0.94), 15Δ (0.90), 23Π (0.60), 13Σ  (0.47) 25Σ þ (63.86), 15Π (18.32), a3Π (8.61), 25Π (7.05), 13Δ(1.72), 23Π (0.44) 25Σ þ (64.98), 15Π (18.12), a3Π (10.72), 25Π (2.92), 13Δ (1.40), 15Σ  (1.08), 23Π (0.40), 15Δ (0.34) 35Σ þ (99.89) 35Σ þ (99.56), 15Δ (0.32) 35Σ þ (98.60), 15Δ (1.30)

the 13Σ  state is 23.71 cm  1, which is comparative with that of the a0 3Σ þ state. As provided in Table 7, near the equilibrium position, the Λ-S state compositions of each Ω state generated from the 15Σ þ state are almost pure; and those of each Ω state generated from the 25Σ þ state strongly mix with the a3Π and 15Π states. For the three Ω states generated from the 15Σ þ state, the largest deviation of ωe from that of the 15Σ þ state is 1.11 cm  1, which is tiny. For the three Ω states genereted from the 25Σ þ state, their Re is equal to that of the 25Σ þ state. And the largest deviation of ωe from that of the 25Σ þ state is only 2.05 cm  1, which is also very small. However, for the three Ω states genereted from the 15Σ þ state, the the largest deviation of Re from that of the 15Σ þ state amounts to 0.00133 nm, and the energy separations between two neighboring Ω states are 27.43 and 95.70 cm  1, which are relatively large. The potential well of the 35Σ þ state is very shallow. Therefore, the SO coupling can easily bring about the obvious effect on the spectroscopic parameters. For the 35Σ0þþ , 35Σ1þ , and 35Σ2þ states, the differences of ωe from that of the 35Σ þ state reach 3.89, 5.72, and 5.29 cm  1, though the differences of Re from that of the 35Σ þ state are only 0.00000, 0.00004, and 0.00002 nm, respectively. The energy separation between the 35Σ0þþ and 35Σ1þ states is 22.17 cm  1, and the energy separation between the 35Σ1þ and 35Σ2þ states is 14.27 cm  1. In conclusion, (1) only a0 3Σ þ and 13Σ  are the inverted states; (2) on the whole, the effect of SO coupling on the spectroscopic parameters of the X1Σ þ , a0 3Σ þ , 13Σ  , 25Σ þ , and 35Σ þ states is small except for the Re of the 35Σ þ state. The effect of SO coupling on the spectroscopic parameters of 15Σ þ state is slightly large. 3.2.2. 25 Ω States generated from the A1Π, a3Π, 23Π, 33Π, 15Π, and 35Π states Table 8 shows the spectroscopic parameters of 25 Ω states generated from these six Π states, and also tabulates the dominant Λ-S state compositions of each Ω state near the equilibrium positions. As seen in Table 8, the Λ-S state composition of A1Π1 component is almost pure near the

internuclear equilibrium position, and the effect of SO coupling on its spectroscopic parameters is small. The Λ-S state compositions of each Ω component generated from the a3Π state are almost pure near the equilibrium positions. The present energy separations between the two neighboring Ω states from the a3Π0  to the a3Π2 state are 14.48, 382.33, and 398.36 cm  1. Perdigon and Fémelat [5] obtained the effective SO coupling constant between the a3Π0 and a3Π1 states of As14N to be about 420 cm  1 in their experiment, where they did not distinguish the two Ω states, a3Π0  and a3Π0 þ . At this work, the energy separation between the a3Π0  and the a3Π1 state is 396.81 cm  1. To some extent, the present result compares well with the measurements [5]. In addition, Te of a3Π0  and a3Π0 þ states are 29,564.99 and 29,579.47 cm  1 in this work, respectively, which also compares well with the experimental T0 of a3Π0 of As14N, 29,659 cm  1 [5]. As seen in Tables 4 and 8, the largest differences of Re, ωe, and De is only 0.00001 nm, 1.86 cm  1, and 0.0318 eV, respectively. According to this, we think that the effect of SO coupling on the Re, ωe, and De of a3Π state is small except for the energy splitting. Te of the 23Π0  state is larger than, whereas Te of the 23Π2 state is smaller than that of the 23Π state. According to this, the 23Π is an inverted state. Similar to the a3Π, the Λ-S state compositions of each Ω state generated from the 23Π state are also almost pure around the equilibrium positions. The energy separations between the two neighboring Ω states from the 23Π0  to the 23Π2 are 0.88, 101.84, and 120.93 cm  1, respectively, which are obviously smaller than those of the a3Π state. The ωe of each Ω state is almost same as that of the 23Π state. And the deviation of Re of each Ω state from that of the 23Π state is also very small. On the whole, the effect of SO coupling on the spectroscopic parameters of 23Π state is not obvious. Only each Ω state yielded from the 33Π state possesses the double well. The energy separations between the two neighboring Ω states from the 33Π0  to the 33Π2 are 12.87, 124.30, and 150.56 cm  1 for the first well, and those are 3.95, 54.65, and 33.36 cm  1 for the second well. Near the equilibrium separations, the Λ-S state compositions are almost pure for the first well of each Ω state, but those

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165

Table 8 Spectroscopic parameters determined by the icMRCIþ Q/Q5 þ CVþ DKþ SO calculations for the 25 Ω states from the A1Π, a3Π, 23Π, 33Π, 15Π, and 35Π states. De/eV

Te/nm

Re/nm

ωe/cm  1

Dominant Λ-S state composition near the Re (%)

A Π1 a3Π0  a3Π0 þ a3Π1 a3Π2 23Π2 23Π1 23Π0 þ 23Π0 

4.2724 2.6388 2.6370 2.6176 2.6198 0.7803 0.7712 0.7659 0.7658

36,113.24 29,564.99 29,579.47 29,961.80 30,360.36 52,760.82 52,881.75 52,983.59 52,984.47

0.16884 0.16835 0.16835 0.16835 0.16833 0.19156 0.19138 0.19120 0.19120

860.96 874.27 874.40 873.43 873.25 583.69 583.80 583.77 583.49

A1Π (99.36), a3Π (0.49), 13Σ  (0.13) a3Π (99.16), 13Δ (0.81) a3Π (99.16), 13Δ (0.81) a3Π (98.18), 13Δ (1.20),13Σ  (0.32), A1Π (0.23) a3Π (99.84), a0 3Σ þ (0.10) 23Π (99.78), 15Π (0.16) 23Π (99.74), 15Π (0.16) 23Π (99.62), 15Π (0.19) 23Π (99.66), 15Π (0.12), 25Σ þ (0.14)

33Π0  1st well 2nd well

0.3297 0.1088

57,143.51 59,199.33

0.19267 0.26368

551.78 357.36

33Π (99.86), 23Π (0.14) 33Π (78.26) , 25Π (11.44), 23Π (10.12),13Π (0.12)

33Π0 þ 1st well 2nd well

0.3303 0.1083

57,156.38 59,203.28

0.19266 0.26377

551.53 357.34

33Π (99.88), 23Π (0.12) 33Π (72.08), 25Π (25.56), 23Π (2.28)

33Π1 1st well 2nd well

0.3488 0.0975

57,280.68 59,257.93

0.19235 0.26340

556.38 313.00

33Π (99.99) 33Π (73.55), 25Π (20.18), 23Π (6.13)

33Π2 1st well 2nd well

0.3511 0.0853

57,431.24 59,291.29

0.19204 0.26352

560.54 283.29

33Π (99.85) 33Π (78.26), 25Π (11.44), 23Π (10.12), 13Π (0.12)

15Π3 15Π2 15Π1 15Π0 þ 15Π0  15 Π  1 35Π3 35Π2 35Π1 35Π0 þ 35Π0  35 Π  1

0.9947 0.8120 0.8112 0.8103 0.8104 0.8096 0.0298 0.0297 0.0293 0.0287 0.0787 0.0783

45,222.31 45,338.19 45,451.66 45,564.47 45,573.51 45,677.50 59,750.87 59,754.16 59,757.89 59,762.06 59,763.01 59,766.45

0.19112 0.19103 0.19097 0.19093 0.19092 0.19091 0.32313 0.32375 0.32450 0.32509 0.32524 0.32619

571.82 575.47 577.89 579.34 579.87 580.01 51.45 51.63 52.13 51.69 51.80 51.58

15Π 15Π 15Π 15Π 15Π 15Π 35Π 35Π 35Π 35Π 35Π 35Π

1

strongly mix with the 25Π and 23Π states for the second well of each Ω state. At the same time, the depth of the second well is much shallower than that of the first well. As a result, the effect of SO coupling on the Te, Re, and ωe of the second well is more obvious than that of the first well. This can be clearly seen by comparison between the spectroscopic parameters tabulated in Tables 5 and 8. Te of the 15Π3 is smaller than, but Te of the 15Π0  is larger than that of the 15Π state. Similar to the a0 3Σ þ , 13Σ  , and 23Π state, the 15Π is also an inverted state. For the 15Π state, as seen in Table 8, the Λ-S state compositions of each Ω state more or less mix with other several Λ-S states near the equilibrium positions. The energy separations between the two neighboring Ω states from the 15Π  1 to the 15Π3 are 103.99, 9.04, 112.81, 113.47, and 115.88 cm  1, respectively. The largest deviations of ωe and Re of these Ω states from those of the 15Π state are 0.00011 nm and 3.85 cm  1, respectively. On the whole, the effect of SO coupling on the spectroscopic parameters of 15Π state are not obvious. Similar to the 15Π, the 35Π is also an inverted state with the SO coupling effect taken into account. Different from the 15Π, the Λ-S state compositions of each Ω state generated from the 35Π state are almost pure near the equilibrium positions. The energy separations between the two neighboring Ω states from the 35Π  1 to the

(88.84 ), 23Δ (6.18), 25Π (4.70), 13Δ (0.20) (88.48), 25Π (5.88), 23Δ (5.02), a3Π (0.18), 23Π (0.12), 13Δ (0.10) (79.76), 25Π (19.75), a3Π (0.20), 23Π (0.13) (98.26), 25Π (0.92), 13Δ (0.29), a3Π (0.20 ), 23Π (0.16), 15Σ þ (0.12) (98.06), 25Π (1.52), a3Π (0.16), 23Π (0.16) (98.40), 25Π (1.17), 13Δ (0.28) (99.92) (99.84), 33Π (0.10) (99.82), 33Π (0.15) (99.80), 33Π (0.12) (99.84), 33Π (0.10) (99.92)

35Π3 are only 3.44, 0.95, 4.17, 3.73, and 3.29 cm  1, and the deviations of ωe from those of the 35Π state are also only 0.31, 0.09, 0.20, 0.24, 0.26, and 0.44 cm  1 for the states from the 35Π  1 to the 35Π3. Therefore, we think that the effect of SO coupling on the Te and ωe of 35Π state is very small. However, for the sake of its shallow well, some effect on the Re can be clearly seen by comparison between the results tabulated in Tables 5 and 8. In conclusion, (1) the 23Π, 15Π, and 35Π are the inverted states with the SO coupling effect taken into account; (2) on the whole, the effect of SO coupling on the spectroscopic parameters of these six Π states is not obvious except for the energy splitting in the a3Π state. In addition, the effect of SO coupling on the Re of 35Π state and the second well of 33Π state can aslo be clearly seen. 3.2.3. 6 Ω States generated from the 13Δ and 23Δ states Table 9 shows the spectroscopic parameters of 6 Ω states generated from the 13Δ and 23Δ states. For convenience of discussion, similar to Tables 7 and 8, in Table 9, we also tabulate the Λ-S state compositions of each Ω state near the equilibrium positions. As clearly shown in Table 9, Te of the 13Δ3 is smaller than, whereas Te of the 13Δ1 is larger than that of the 13Δ state. Therefore, the 13Δ is an inverted state with the SO coupling effect included. The Λ-S state compositions of the

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Table 9 Spectroscopic parameters determined by the icMRCI þQ/Q5 þ CVþ DKþ SO calculations for the 6 Ω states generated from the 13Δ and 23Δ states.

Transition dipole moments /a.u.

1 Δ3 13Δ2 13Δ1 23Δ1 23Δ2 23Δ3

Te/cm  1

Re/nm

ωe/cm  1

Dominant Λ-S state compositions near the Re (%)

2.7963 2.8042 2.7955 1.7834 1.7499 1.7488

28,276.89 28,442.81 28,508.00 45,664.77 45,760.90 45,821.04

0.17750 0.17754 0.17764 0.20587 0.20720 0.20789

776.70 779.60 778.83 525.00 527.51 526.21

13Δ 13Δ 13Δ 23Δ 23Δ 23Δ

0.20

Transition dipole moments /a.u.

3

De/eV

0.18

0.16

0.14

0.12 1.5

1.6

1.7

1.8

1.9

2.0

2.1

(98.78), a3Π (1.10), A1Π (0.10) (99.13), a3Π (0.80) (99.82) (93.76), 15Π (3.95), 25Π (1.38), A1Π (0.71), a3Π (0.11) (92.76), 15Π (4.70), 25Π (1.66), a3Π (0.72) (93.76), 25Π (3.56), 15Π (1.98), 13Δ (0.34), 15Σ þ (0.34)

0.008

a'3Σ+1-X1Σ+0 0.006

0.002

13Δ1-X1Σ+0+ 0.000 1.5

3.3. Transition properties Here we only study the spectroscopic transition properties of the five transitions, a0 3Σ1þ –X1Σ0þþ , a3Π1–X1Σ0þþ ,

1.6

1.7

1.8

1.9

2.0

2.1

2.2

Internuclear separation /nm

Internuclear separation /nm

13Δ3, 13Δ2, and 13Δ1 states are almost pure. Accordingly, the effect of SO coupling on the spectroscopic parameters should be small. For example, the largest devitions of the Re and ωe from the 13Δ state are 0.00002 nm and 1.81 cm  1, and the energy separations between the 13Δ3 and the 13Δ2 state and between the 13Δ2 and the 13Δ1 state are 165.92 and 65.19 cm  1, respectively. Different from the 13Δ, the 23Δ is a regular state. The Λ-S state compositions of the 23Δ3, 23Δ2, and 23Δ1 components slightly mix with several other Λ-S states. Some effect of SO coupling on the spectroscopic parameters is displayed for the 23Δ state, in particular for the Re. In detail, the deviations of Re from that of the 23Δ state are 0.00338, 0.00471, and 0.0054 nm for the 23Δ1, 23Δ2, and 23Δ3 states. Such deviations are very large. However, the deviations of ωe from that of the 23Δ state are only 1.85, 0.66, and 0.64 cm  1 for the 23Δ1, 23Δ2, and 23Δ3 states, respectively. Such deviations are very small. The energy separations between the 23Δ1 and the 23Δ2 state and between the 23Δ2 and the 23Δ3 state are 96.13 and 60.14 cm  1, which are smaller than those of the 13Δ state. As a conclusion, the 13Δ is an inverted state with the SO coupling effect included. The effect of SO coupling on the spectroscopic parameters of 13Δ state is small, whereas the effect of SO coupling on the spectroscopic parameters of the 23Δ state is relatively obvious, in particular for the Re.

a3Π1-X1Σ+0+

0.004

2.2

Fig. 9. TDM versus R for the A1Π1–X1Σ0þþ transitions of AsN molecule.

a3Π0--X1Σ+0+

Fig. 10. TDM versus R for the a3Π0  –X1Σ0þþ , a0 3Σ1þ –X1Σ0þþ , a3Π1–X1Σ0þþ and 13Δ1–X1Σ0þþ transitions of AsN molecule.

A1Π1–X1Σ0þþ , 13Δ1–X1Σ0þþ and a3Π0  –X1Σ0þþ due to the length limitation. For this reason, we calculate the transition dipole moments (TDMs) versus internuclear separation for the five transitions, and depicted the results in Figs. 9 and 10. As demonstrated in Figs. 9 and 10, the TDMs of A1Π1–X1Σ0þþ transitions are large near the equilibrium position, which is consistent with the fact that the present singlet-singlet transition is allowed. The TDMs of a0 3Σ1þ – X1Σ0þþ , a3Π1–X1Σ0þþ , 13Δ1–X1Σ0þþ and a3Π0  –X1Σ0þþ transitions are very small, which agrees well with the fact that the triplet-singlet transitions are forbidden. It should be pointed out that the TDMs depicted in Figs. 9 and 10 are calculated by the Breit–Pauli Hamiltonian in combination with the all-electron ACVTZ basis set at the level of icMRCI theory. We employ the following formula [31] to calculate the radiative lifetimes for a given vibrational υ0 :

τυ0 ¼ ðAυ0 Þ  1 ¼ ¼

3h

g0

64π 4 ja0 eTDMj2 ∑υ0 qυ0 ; υ″ ðΔEυ0 ; υ″ Þ3 g″

g0 ; jTDMj2 ∑υ0 qυ0 ; υ″ ðΔEυ0 ; υ″ Þ3 g″ 4:936  105

ð3Þ

where qυ0 , υ″ denotes the Franck–Condon factor. TDM is the averaged TDM in atomic unit. g0 and g″ are the degeneracy of upper and lower states, respectively. The energy separation ΔEv0 , v″ is in cm  1 and the radiative lifetime τv0 is in s. Using the PECs determined by the icMRCI þQ/ Q5þ CVþDK þSO calculations, with the help of LEVEL Program [32], we have evaluated the Frank–Condon factors at the different vibrational levels for the a0 3Σ1þ –X1Σ0þþ ,

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167

Table 10 Frank–Condon factors for the a0 3Σ1þ –X1Σ0þþ , a3Π1–X1Σ0þþ , A1Π1–X1Σ0þþ , 13Δ1–X1Σ0þþ , and a3Π0  –X1Σ0þþ transitions. υ″ ¼0

υ″ ¼1

υ″ ¼2

υ″¼ 3

υ″ ¼4

υ″ ¼5

υ″ ¼6

υ″ ¼ 7

υ″ ¼ 8

υ″ ¼9

0.0097 0.0378 0.0793 0.1183 0.1405

0.0514 0.1243 0.1459 0.1021 0.0393

0.1293 0.1582 0.0608 0.0005 0.0275

0.2018 0.0791 0.0017 0.0656 0.0771

0.2207 0.0027 0.0762 0.0707 0.0022

0.1806 0.0359 0.0976 0.0009 0.0510

0.1146 0.1257 0.0226 0.0512 0.0604

0.0578 0.1672 0.0092 0.0893 0.0003

0.0236 0.1372 0.0089 0.0244 0.0525

0.0079 0.0801 0.1495 0.0076 0.0800

a3Π1–X1Σ0þþ υ0 ¼ 0 0.4779 υ0 ¼ 1 0.3358 υ0 ¼ 2 0.1336 υ0 ¼ 3 0.0399 υ0 ¼ 4 0.0100

0.3632 0.0298 0.2451 0.2127 0.1009

0.1276 0.3028 0.0257 0.0902 0.1999

0.0271 0.2339 0.1423 0.1156 0.0084

0.0039 0.0792 0.2666 0.0303 0.1161

0.0004 0.0160 0.1407 0.2306 0.0003

0.0000 0.0021 0.0385 0.1927 0.1559

0.0000 0.0003 0.0065 0.0707 0.2207

0.0000 0.0000 0.0007 0.0150 0.1087

0.0000 0.0000 0.0000 0.0020 0.0287

A1Π1–X1Σ0þþ υ0 ¼ 0 0.4635 υ0 ¼ 1 0.3535 υ0 ¼ 2 0.1320 υ0 ¼ 3 0.0370 υ0 ¼ 4 0.0099

0.3574 0.0235 0.2441 0.2148 0.1045

0.1377 0.2827 0.0339 0.0784 0.1942

0.0350 0.2284 0.1265 0.1324 0.0028

0.0058 0.0879 0.2583 0.0193 0.1662

0.0006 0.0209 0.1496 0.2148 0.0033

0.0000 0.0028 0.0465 0.1998 0.1338

0.0000 0.0003 0.0082 0.0823 0.2213

0.0000 0.0000 0.0008 0.0185 0.1229

0.0000 0.0000 0.0000 0.0024 0.0349

13Δ1–X1Σ0þþ υ0 ¼ 0 0.0195 υ0 ¼ 1 0.0615 υ0 ¼ 2 0.1086 υ0 ¼ 3 0.1467 υ0 ¼ 4 0.1640

0.0892 0.1603 0.1371 0.0656 0.0099

0.1890 0.1413 0.0183 0.0115 0.0667

0.2422 0.0272 0.0348 0.0947 0.0443

0.2129 0.0141 0.1162 0.0266 0.0130

0.1385 0.1146 0.0538 0.0213 0.0804

0.0697 0.1807 0.0013 0.0997 0.0133

0.0278 0.1553 0.0797 0.0465 0.0330

0.0087 0.0905 0.1602 0.0023 0.0916

0.0022 0.0385 0.1500 0.0823 0.0248

a3Π0  –X1Σ0þþ υ0 ¼ 0 0.4770 υ0 ¼ 1 0.0294 υ0 ¼ 2 0.1340 υ0 ¼ 3 0.0401 υ0 ¼ 4 0.0101

0.3636 0.3360 0.2443 0.2127 0.1013

0.1279 0.3026 0.0261 0.0894 0.1994

0.0272 0.2344 0.1418 0.1161 0.0081

0.0039 0.0794 0.2669 0.0300 0.1612

0.0004 0.0160 0.1410 0.2308 0.0003

0.0000 0.0021 0.0385 0.1929 0.1560

0.0000 0.0002 0.0065 0.0707 0.2210

0.0000 0.0000 0.0007 0.0150 0.1087

0.0000 0.0000 0.0001 0.0020 0.0286

Σ1þ –X1Σ0þþ

03

a υ0 ¼ 0 υ0 ¼ 1 υ0 ¼ 2 υ0 ¼ 3 υ0 ¼ 4

Table 11 Radiative lifetimes of the transitions from the a0 3Σ1þ , a3Π1, A1Π1, a3Π0  , and 13Δ1 states to the X1Σ0þþ state for several low vibrational states. Transitions

a0 3Σ1þ –X1Σ0þþ a3Π1–X1Σ0þþ A1Π1–X1Σ0þþ a3Π0  –X1Σ0þþ 13Δ1–X1Σ0þþ

Radiative lifetimes/μs υ0 ¼0

υ0 ¼ 1

υ0 ¼2

υ0 ¼3

υ0 ¼ 4

4810 2210 0.626 379 652

4750 2240 0.634 385 660

4680 2280 0.646 391 667

4630 2310 0.656 397 672

4640 2340 0.664 402 685

a3Π1–X1Σ0þþ , A1Π1–X1Σ0þþ , 13Δ1–X1Σ0þþ , and a3Π0 –X1Σ0þþ transitions. For the lowest five vibrational levels of selected transitions, we have also evaluated their radiative lifetimes. For convenience of discussion, here we collect these results in Tables 10 and 11, respectively. From Tables 10 and 11, for the A1Π1–X1Σ0þþ , we can see that a number of transitions, such as 0–0, 0–1, 0–2, 2–1, 3–1, 3–2, 3–3, 4–2, 4–4, 5–2, 5–3, 6–3, 6–4, 7–4, and 8–4, are ramarkably strong with the short lifetimes. And there are still some transitions, such as 3–0, 4–0, 5–0, 6–0, 7–0, 8–0, 4–1, 5–1, 6–1, 7–1, and 8–1, which are very weak since their Frank–Condon factors are small. For the a0 3Σ1þ – X1Σ0þþ , only few transitions, such as 2–0, 3–0, 4–0. 5–0, 6–0, 1–1, 2–1, 7–1, 8–1, 3–0, 3–0, and 4–0, are strong. On the whole, the strength of the A1Π1–X1Σ0þþ transitions is much stronger than that of the a0 3Σ1þ –X1Σ0þþ since the

former Frank–Condon factors are much greater than the latter, which can be clearly seen in Table 10. Accordingly, the lifetimes of A1Π1–X1Σ0þþ transitions are much shorter than those of the a0 3Σ1þ –X1Σ0þþ . Similar analyses are suitable for the other three groups of forbidden transitions as shown in Table 10. 4. Conclusions In this paper, the PECs of 21 Λ-S states and 74 Ω states have been investigated for internuclear separations from about 0.1 to 1.0 nm using the CASSCF method, which is followed by the icMRCI approach with Davidson correction. The X1Σ þ , a0 3Σ þ , 15Σ þ , 13Δ, 13Σ  , a3Π, 15Π, 25Σ þ , 35Σ þ , 23Δ, 23Π, 33Π, 35Π, and A1Π are found to be bound states, whereas the 23Σ þ , 33Σ þ , 15Σ  , 15Δ, 25Δ, 25Π, and

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17Σ þ are found to be repulsive ones. The 33Π state possesses the double well. The 25Σ þ , 35Σ þ , 35Π, and 33Π states possess the shallow well. And the a0 3Σ þ , 13Σ  , 13Δ, 15Π, 23Π, 25Π, 35Π, and 17Σ þ are found to be the inverted states with the SO coupling effect taken into account. The SO coupling effect is accounted for by the Breit–Pauli Hamiltonian with the all-electron ACVTZ basis set. To determine more reliable spectroscopic results, the effect of core–valence correlation and scalar relativistic corrections on the PECs is taken into account. In detail, scalar relativistic correction is calculated using the DKH3 approximation at the level of a cc-pVTZ basis set. Core– valence correlation correction is included with the ccpCVTZ basis set. The spectroscopic parameters have been evaluated for the 14 Λ-S bound states and for the 45 Ω bound states, and compared in detail with those available in the literature. Excellent agreement has been found between the present results and available measurements. The Franck–Condon factors and radiative lifetimes of transitions from the a0 3Σ1þ , a3Π1, A1Π1, 13Δ1 and a3Π0  states to the X1Σ0þþ ground state are calculated, and some necessary discussion is done. The analyses demonstrate that the spectroscopic parameters of 14 Λ-S states and 45 Ω states reported here can be expected to be reliably predicted ones, and can be good references for the future laboratory research and theoretical work.

Acknowledgments This work was sponsored by the National Natural Science Foundation of China under Grant no. 11274097 and the Program for Science and Technology of Henan Province in China under Grant no. 142300410201. Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j. jqsrt.2014.09.020.

References [1] Spinks JWT. Über ein ultraviolettes bandensystem yon AsN. Z Phys 1934;88:511–4. [2] Jones WE. Rotational analysis of the 1Π–X1Σ þ system of AsN. J Mol Spectrosc 1970;34:320–6 (and references therein). [3] Dixit MN, Krishnamurty G, Narasimham NA. Rotational analysis of the 1Π–X1Σ þ system of AsN. Proc Ind Acad Sci A 1970;71:23–31. [4] Fémelat B, Jones WE. Extension of the rotational analysis of the 1Π– X1Σ þ system of AsN. J Mol Spectrosc 1974;49:388–400. [5] Perdigon P, Fémelat B. The a3Π and A1Π states of the (σπn) configuration in the AsN molecule. J Phys B 1982;15:2165–85. [6] Henshaw TL, McElwee D, Stedman DH, Coombe RD. Chemiluminescent reaction of As (4Su) atoms with azide radicals. J Phys Chem 1988;92:4606–10. [7] Saraswathy P, Krishnamurty G. Rotational analysis of A1Π–X1Σ þ bands of As14N and As15N: perturbation studies in the A1Π state. Pramana—J Phys 1988;31:493–504. [8] Ohanessian G, Durand G, Volatron F, Halwick P, Malrieu JP. Theoretical study of the AsN spectrum. Chem Phys Lett 1985;115:545–8.

[9] Toscano M, Russo N. Geometric and electronic structure of ground and excited states of group VA diatomics. A theoretical LCGTO-MPLSD study. Z Phys D 1992;22:683–92. [10] Katsuki S. Model potential in spectral representation form. Its fidelity to the frozen-core operator and comparison with other model potentials. Can J Chem 1995;73:696–702. [11] Martin JML, Sundermann A. Correlation consistent valence basis sets for use with the Stuttgart–Dresden–Bonn relativistic effective core potentials: the atoms Ga–Kr and In–Xe. J Chem Phys 2001;114: 3408–20. [12] Peterson KA. Systematically convergent basis sets with relativistic pseudopotentials. I. Correlation consistent basis sets for the post-d group 13–15 elements. J Chem Phys 2003;119:11099–112. [13] Wang JM, Sun JF. Multireference configuration interaction study on spectroscopic parameters and molecular constants of AsN (X1Σ þ ) radical. Acta Phys Sin 2011;60:123103. [14] Huber KP, Herzberg G. Molecular spectra and molecular structure IV, constants of diatomic molecules. Princeton (NJ): Van NostrandReinhold; 1979 (and references therein). [15] Kerr JA, Stocker DW. Handbook of Chemistry and Physics. Boca Raton: Chemical Rubber Crop; 1999 (and references therein). [16] Werner H-J, Knowles PJ. An efficient internally contracted multiconfiguration-reference configuration interaction method. J Chem Phys 1988;89:5803–14. [17] Knowles PJ, Werner H-J. An efficient method for the evaluation of coupling coefficients in configuration interaction calculations. Chem Phys Lett 1988;145:514–22. [18] Langhoff SR, Davidson ER. Configuration interaction calculations on the nitrogen molecule. Int J Quantum Chem 1974;8:61–72. [19] Richartz A, Buenker RJ, Peyerimhoff SD. Ab initio MRD-CI study of ethane: The 14–25 eV PES region and Rydberg states of positive ions. Chem Phys 1978;28:305–12. [20] Kendall RA, Dunning TH, Harrison RJ. Electron affinities of the firstrow atoms revisited. Systematic basis sets and wave functions. J Chem Phys 1992;96:6796–806. [21] Wilson AK, Woon DE, Peterson KA, Dunning TH. Gaussian basis sets for use in correlated molecular calculations. IX. The atoms gallium through krypton. J Chem Phys 1999;110:7667–76. [22] Werner HJ, Knowles PJ, Lindh R, Manby FR, Schütz M, Celani P, Korona T, Mitrushenkov A, Rauhut G, Adler TB, Amos RD, Bernhardsson A, Berning A, Cooper DL, Deegan MJO, Dobbyn AJ, Eckert F, Goll E, Hampel C, Hetzer G, Hrenar T, Knizia G, Köppl C, Liu Y, Lloyd A W, Mata R A, May A J, McNicholas SJ, Meyer W, Mura ME, Nicklass A, Palmieri P, Pflüger K, Pitzer R, Reiher M, Schumann U, Stoll H, Stone AJ, Tarroni R, Thorsteinsson T, Wang M, Wolf A. MOLPRO 2010.1 is a package of ab initio programs, http://www.molpro.net/. [23] Woon DE, Dunning TH. Gaussian basis sets for use in correlated molecular calculations. V. Core–valence basis sets for boron through neon. J Chem Phys 1995;103:4572–85. [24] DeYonker NJ, Peterson KA, Wilson AK. Systematically convergent correlation consistent basis sets for molecular core–valence correlation effects: the third-row atoms gallium through krypton. J Phys Chem A 2007;111:11383–93. [25] Wolf A, Reiher M, Hess BA. The generalized Douglas–Kroll transformation. J Chem Phys 2002;117:9215–26. [26] De Jong WA, Harrison RJ, Dixon DA. Parallel Douglas–Kroll energy and gradients in NWChem: estimating scalar relativistic effects using Douglas–Kroll contracted basis sets. J Chem Phys 2001;114: 48–53. [27] Oyeyemi VB, Krisiloff DB, Keith JA, Libisch F, Pavone M, Carter EA. Size-extensivity-corrected multireference configuration interaction schemes to accurately predict bond dissociation energies of oxygenated hydrocarbons. J Chem Phys 2014;140:044317. [28] Berning A, Schweizer M, Werner H-J, Knowles PJ, Palmieri P. Spin– orbit matrix elements for internally contracted multireference configuration interaction wave functions. Mol Phys 2000;98: 1823–33. [29] González JLMQ, Thompson D. Getting started with Numerov's method. Comput Phys 1997;11:514–5. [30] Moore CE. Atomic energy levels, vol. 1. Washington: National Bureau Standards; 1971 (National Standard Reference Data Series). [31] Okabe H. Photochemistry of small molecules. New York: Wiley Interscience; 1978. [32] Le Roy, RJ. LEVEL 7.5: A computer program for solving the radial Schrödinger equation for bound and quasi-bound levels. University of Waterloo Chemical Physics Research Report No. CP-642R3. Waterloo (ON, Canada): University of Waterloo; 2001.