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Extensive ab initio study of the electronic states of BSe radical including spin–orbit coupling
Extensive ab initio study of the electronic states of BSe radical including spin–orbit coupling Siyuan Liu, Hongsheng Zhai, Yufang Liu PII: DOI: Reference:
S1386-1425(16)30110-X doi: 10.1016/j.saa.2016.03.008 SAA 14321
To appear in: Received date: Revised date: Accepted date:
21 December 2015 1 March 2016 6 March 2016
Please cite this article as: Siyuan Liu, Hongsheng Zhai, Yufang Liu, Extensive ab initio study of the electronic states of BSe radical including spin–orbit coupling, (2016), doi: 10.1016/j.saa.2016.03.008
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ACCEPTED MANUSCRIPT Extensive ab initio study of the electronic states of BSe radical
Siyuan Liu, Hongsheng Zhai, Yufang Liu
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including spin–orbit coupling
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College of Physics and Electronic Engineering, Henan Normal University, Xinxiang 453007, China
Abstract
The internally contracted multi-reference configuration interaction method (MRCI)
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with Davidson modification and the Douglas-Kroll scalar relativistic correction has been used to calculate the BSe molecule at the level of aug-cc-pV5Z basis set. The
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calculated electronic states, including 9 doublet and 6 quartet Λ-S states, are correlated to the dissociation limit of B(2Pu ) + Se(3Pg) and B(2Pu ) + Se(1Dg). The
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Spin-orbit coupling (SOC) interaction is taken into account via the state interaction
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approach with the full Breit-Pauli Hamiltonian operator, which causes the entire 15 Λ-S states to split into 32 Ω states. This is the first time that the spin-orbit coupling
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calculation has been carried out on BSe. The potential energy curves of the Λ-S and Ω electronic states are depicted with the aid of the avoided crossing rule between electronic states of the same symmetry. The spectroscopic constants of the bound Λ-S
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and Ω states were determined, which are in good agreement with the experimental data. The transition dipole moments (TDMs) and the Frank-Condon factors (FCs) of the transitions from the low-lying bound Ω states A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 to the ground state X2Σ+1/2 have also been presented. Based on the previous calculations, the radiative lifetimes of the A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 were evaluated.
Keywords: MRCI, Potential energy curve, Spin-orbit coupling effect (SOC), Spectroscopic constants
1. Introduction
Corresponding author. E-mail address: [email protected] . 1
ACCEPTED MANUSCRIPT Boron selenides are an increasing crucial area in the technology of high-temperature use，semiconductor materials[1-3] due to its status. Selenium is the rare earth elements[4] and plays an important part in the human healthy. Naturally, the
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theoretical studies of accurate electronic structure of BSe are important to the comprehension and improvement of this technology. Previous experimental
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investigations of the B-Se system have been restricted to the condensed state. The first experimental study of BSe was made by Uy and Drowart[3], they obtained BSe(g) by the gas-phase reaction of boron with yttrium selenide and the energy of dissociation
employed solid-state
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De (BSe, g, 0 K)=4.74±0.15 eV was determined. Subsequently, HÜRTER et al[1] B NMR techniques to study the structural nature of various
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phases obtained in the system of boron-selenium including BSe. Melucci and Wahlbeck[5] identified the boron selenides BSe(g) in studies of the vaporisation
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behaviour of solid B2Se3 with a time-of-flight mass spectrometer over the range 100
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to 700 ºC. Previous experimental studies about BSe mainly focused on vaporisation behavior at High-temperature. Previous experimental studies on the excited states of
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BSe are limited, thus an extensive theoretical study on the spectroscopic and the transition properties of BSe low-lying electronic states would be indispensable. As far as we know, the spectroscopic properties of Boron-Chalcogenide and
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Boron-Halogenide have been investigated in large quantities. Hanner[6] studied the ultrafast V–E transfer in boron oxide (BO) and found that the spin orbit interaction in BO facilitated a route for rapid intramolecular energy transfer. Then Karna et al[7] performed a configuration-interaction calculation on low-lying states of BO, and particular attention was given to the perturbations in the C2Π state. Yang and Boggs[8] have calculated the electronic structures and the transition properties of the low-lying excited states of the BS radical and determined the spectroscopic constants at the MR-CISD+Q level. More recently, Yang and co-worker[9] have investigated the spectroscopic parameters of BBr and BCl in order to prove that it would be a promising candidate for laser cooling. However, to the best of our knowledge, the systematic study on BSe molecule is rather sparse, where the spin-orbit interaction was not considered. It is well known that SOC plays an important role in the 2
ACCEPTED MANUSCRIPT spectroscopy and dynamics of the molecules, even in light molecules that containing only atoms of the first row of the periodic table. In addition, S and Se are oxygen family element and the B-S and B-Se system shows similar vaporization behavior[5],
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so compare the calculation results of BSe to the part of theoretical studies of BS[8] would be necessary.
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Our work will concentrate on the theoretical investigations of the electronic structures and the transition properties of the entire 32 Ω states generated from all of the 15 Λ-S states of BSe. We used internally contracted multi-reference configuration
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interaction method (MRCI) with Davidson modification and the Douglas-Kroll scalar relativistic correction for single point energy calculations. The potential energy curves
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(PECs) and spectroscopic constants were fitted after considering the avoided crossing rule between states of the same symmetry. The transition dipole moments (TDMs) and
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the Franck–Condon (FC) factors of the transitions from the low-lying bound Ω states
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A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 to the ground state X2Σ+1/2. were also calculated. Finally, the corresponding single-channel radiative lifetimes were derived from above
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results.
2. Computational details
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The ab initio calculation on the electronic structure of BSe is launched by using the ab initio quantum chemistry program package MOLPRO 2010.1[10]. The spectroscopic parameters with the transition dipole moments(TDM), Frank-Condon factors(FCs) and radiative lifetime are determined by using the Le Roy’s LEVEL 8.0 program[11]. In order to obtain the PECs of BSe and guarantee the accuracy, the uncontracted Gaussian type all-electron aug-cc-pV5Z basis set is selected for both atom B[15s,9p,5d,4f,3g,2h][12, 13];Se[27s,18p,14d,4f,3g,2h][14] in the calculation of the Λ-S and Ω electronic states. The bond length is circulated with the step of 0.05Å to scan a series of the single-point energy over the internuclear distance range from 1.3 to 6.25Å. The ground state molecule orbitals (MOs) are calculated firstly by adopting 3
ACCEPTED MANUSCRIPT restricted Hartree-Fock (RHF) method. Then the state-averaged complete active space self-consistent field (SA-CASSCF)[15, 16] method is carried out using previous RHF orbitals as starting guess for orbital optimization. Finally, the energies of Λ-S states
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are calculated by using the internally contracted multi-reference configuration interaction (MRCI)[17, 18] approach base on the previous SA-CASSCF energies as
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reference values. The Douglas-Kroll scalar relativistic one-electron integrals have been taken into account, and the Davidson modification[19-21] (MRCI+Q) is employed to correct the size-extensively error.
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Due to the limitation of MOLPRO program package, the subgroup C2v point group symmetry has been considered for the BSe molecule although it belongs to a
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higher symmetry group in the calculation. The C2v point group symmetry holds A1, B1, B2, A2 irreducible representations. For the BSe molecule, 5a1, 2b1 and 2b2
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symmetry molecule orbitals (MOs) are selected as the active space, corresponding to
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the B 2s2p3s and Se 4s4p shells. The outmost 2s2p1 electrons of B atom and 4s24p4 of Se atom are placed in the active space, and the remaining 30 electrons are frozen and
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not correlated. Namely, 9 electrons are used in the correlation energy calculation in total. The PECs of 15 Λ-S electron states are plotted by connecting the calculated points with the aid of the avoided crossing rule between electron states of same
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symmetry.
In addition, the spin-orbit matrix elements and eigenstates in the present work are calculated by using the state interaction method and full Breit-Pauli Hamiltonian operator (HBP) after the MRCI+Q calculations. The state interaction is employed in our SOC calculations, which means that the SOC eigenstates are obtained by diagonalizing the matrixes Hel + Hso on the basis of eigenfunction of Hel. In this process, the Hel and Hso are obtained from MRCI+Q calculations and CASSCF wave functions, respectively. The SOC potential energy curves are drawn with the aid of the avoided crossing rule of the same symmetry. The spectroscopic constants of the bound Λ-S and Ω states, including the equilibrium internuclear distance Re, excitation energy Te, the harmonic and anharmonic vibrational constants ωe and ωeχe, the rotational constants Be, and the dissociation energy De are determined by the numerical solutions 4
ACCEPTED MANUSCRIPT of the one-dimensional nuclear Schrödinger equation. The transition dipole moments (TDMs) and the Frank-Condon factors (FCs) of the transitions from the low-lying bound Ω states A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 to the ground state X2Σ+1/2 are
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X2Σ+1/2, 2Π(I)1/2- X2Σ+1/2 and C2Δ(I)3/2-X2Σ+1/2 are evaluated.
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presented. Based on the previous calculations, the radiative lifetimes of the A2Π(I)3/2-
3. Results and Discussion
3.1. Results and analysis of the PECs of the 15 Λ-S electron states
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The PECs of the 15 Λ-S states[B2Σ+(II), 2Δ(II), 2Π(III), X2Σ+, A2Π(I), C2Δ(I), 2 -
Σ (I), 2Π(II), 2Σ-(II), 4Σ+, 4Π(I), 4Π(II), 4Δ, 4Σ-(I), 4Σ-(II)] of the BSe molecule are
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shown in Fig. 1. The B2Σ+(II), 2Δ(II) and 2Π(III) states are correlated with the dissociation limit of the first excited state Se(1Dg) and the ground state B(2Pu), while
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the other six doublet and six quartet Λ-S states are correlated with the dissociation
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limit of the atom ground states Se(3Pg) and B(2Pu) . Table 1 shows the calculated electronic states of the BSe molecule and their dissociation relationships. The
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calculated energy gap of 9545.64 cm-1 between the atom state B(2Pu) and Se(1Dg) is in good agreement with the experimental result of 9576.149 cm-1. The spectroscopic constants of the 15 bound Λ-S states of the BSe molecule are
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fitted and summarized in Table 2, in which the main electronic configurations of these states are also presented. The amplified view of the energy degeneracy of B2Σ+(II), C2Δ(I) , 2Σ-(I), 2Δ(II) and 2Σ-(II) states are shown in Fig. 2. As we can see, the ground states X2Σ+ crossed the first exited state A2Π(I) when the bond length is about 2.3 Å. It’s mainly characterized by the electronic configuration 1σαβ2σαβ1παβαβ3σα2π04σ05σ0 with the configuration weight factor of 0.8 around equilibrium position. The depth of the well without incorporating spin-orbit coupling is 4.84 eV and the well holds about 66 vibrational levels, which result is in good agreement with the dissociation energy D0 of 4.74±0.15 eV obtained by UY and DROWART[3] in experiment. (De = D0 +1/2ωe, where ωe is about 0.12 eV). It should be noted that there is one important crossing region between the X2Σ+ state and the first exited state A2Π(I) at 2.26 Å on 5
ACCEPTED MANUSCRIPT the PECs of the Λ-S states, and corresponding predissociation pathway is possibility X2Σ+(ν′17)A2Π(I) after we calculated. The states B2Σ+(II), C2Δ(I) and 2Σ-(I) are energy degeneracy near bond length 2
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Å and their first well have the same electron configuration 1σαβ2σαβ1παββ3σα2πα4σ05σ0
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and 1σαβ2σαβ1παβα3σα2πβ4σ05σ0 around the corresponding equilibrium position,
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excitation energy Te of which are 32284.4376 cm-1, 32388.9211 cm-1 and 32407.8901 cm-1 respectively. The excitation energy interval betwwen the C2Δ(I) and 2Σ-(I) states is as small as 18.969 cm-1, while the value between B2Σ+(II) and C2Δ(I) states is
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104.4835 cm-1. Especially the C2Δ(I) and 2Σ-(I) states not only have nearly same electron configuration around equilibrium position but also have almost same
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spectroscopic constants. Moreover, the B2Σ+(II) state cross more other states, which is a Rydberg state and dissociates to the first excited atomic state B(2Pu) + Se(1Dg).
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In addition, Both the C2Δ(I) and 2Σ-(I) states have an unconspicuous potential
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barrier near bond position 2.8 Å which gives rise to the double well. But the second well of the C2Δ(I) and 2Σ-(I) states is too shallow to observe in PECs shape. The depth
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and equilibrium position of the second well of the C2Δ(I) and 2Σ-(I) states is only 0.01 eV at 3.9435 Å and 0.0082 eV at 4.245 Å, and the corresponding vibrational levels are merely 4 and 2, respectively.
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As for second root of 2Δ and 2Σ- symmetry, the 2Δ(II) and 2Σ-( II) states have the same properties as the C2Δ(I) and 2Σ-(I) states. The 2Δ(II) and 2Σ-(II) states appear energy degeneracy when the bond length is less than 2.8 Å. They have the same electron configuration and similar spectroscopic constants expect well depth De near their equilibrium position, with being their excitation energy interval is only 101.6055 cm-1. Based on the shape of PECs, the 2Π(II) and 2Π(III) states obviously avoided crossing at the bond length of about 2.3 Å. This result is accord well with Yang’s[8] prediction. The 2Π(II) and 2Π(III) states have potential barrier at bond length 2.36 Å and 2.8 Å, respectively, the former formation of a clear potential barrier because of the avoided crossing rule. The potential barrier leads to the double well for the 2Π(II) and 2Π(III) states. The bond position of the wells locate at 1.855 Å and 3.6385 Å on 6
ACCEPTED MANUSCRIPT the 2Π(II) state, and the corresponding well depth are 0.9216 eV and 0.0305 eV, respectively. While The bond position of the wells locate at 2.3515 Å and 3.494 Å on the 2Π(II) state, and the corresponding well depth are 2.4943 eV and 0.0927 eV,
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respectively.
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Quartet Λ-S states correlated with the dissociation limit of the ground state
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Se(3Pg) and B(2Pu) , which have more smooth PECs than doublet state, and they are bound states except the 4Σ-(II) state. The 4Σ+, 4Δ and 4Σ-(I) states have similar shape of PEC, their Te and Re both increasing in turn. It was found that 4Σ+,4Δ and 4Σ-(I) not
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only have the same main electron configuration 1σαβ2σαβ1παβα3σα2πα4σ05σ0, but also have very close configuration weight factor(0.86,0.87,0.87) near their equilibrium
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bond length.
As for the excited states 4Π(I),4Π(II) and 4Σ-(II) states, the 4Σ-(II) state is the only
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repulsive state of all electronic states. The 4Π(I) state has the lowest well which only
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holds 3 vibrational levels at equilibrium bond length 3.0685 Å. 4Π(I) crosses 4Π(II) at a bond length of about 1.9Å because of the avoided crossing rule. At the same time, Π(I) has crossing with PECs of the bound state 4Σ+,4Δ, 4Σ-(I) and ground state X2Σ+
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from bond length 3.5-3.8 Å. The 4Π(II) state has a particularly shallow potential well in which De is 0.27 eV. Meanwhile, the state 4Π(II) is an unstable well due to an
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avoided crossing with the 4Π(I) state, and the bottom of 4Π(II) is 14105.4 cm-1 higher than the dissociation limit Se(3Pg) + B(2Pu). The characteristic of 4Π(II) indicate that it is hardly be observed in experiment. As the second root of 2Π symmetry, bound state 2Π(II) arises from the electronic configuration 1σαβ2σαβ1παβαβ3σ02πα4σ05σ0 (68.40%),1σαβ2σα1παβαβ3σβ2πα4σ05σ0 (8.25%), 1σαβ2σαβ1παββ3σ02παα4σ05σ0 (3.08%) and 1σαβ2σα1παβα3σ02παβ4σ05σ0 (2.89%) for the first well at the equilibrium point 1.855 Å, which indicate that the 2Π(II) state is obviously multi-configurational in nature and that it is necessary to use the multi-reference configuration interaction in our calculation.
3.2. Results and analysis of the PECs of the 32 Ω states Introduction of the SOC effect into the calculation causes 15 Λ-S states split into 7
ACCEPTED MANUSCRIPT 32 Ω states, namely, 1/2, 3/2, 5/2, and 7/2 symmetries, which hold eight different dissociation limits. The possible Ω states and corresponding energy separations are given in Table 3. The two highest dissociation limits 2P1/2 + 1D2 and 2P3/2 + 1D2 are
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splitting from the second atomic state 2Pu + 1Dg, with the energy interval is 9545.64
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and 9574.52 cm-1, respectively, and the results are in good agreement with
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corresponding energy interval value 9576.149, 9591.436 cm-1 obtained from experiment[22]. Meanwhile, the computed PECs for the 15 states of Ω = 1/2, 11 states of Ω = 3/2, 5 states of Ω = 5/2 and one state of Ω = 7/2 are plotted in Fig. 3 separately.
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The spectroscopic constants limits of the bound Ω states obtained by solving the nuclear Schrödinger equations are shown in Table 4.
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The SOC ground state X2 Σ1/2 almost has the same spectroscopic constants as
Λ-S ground states X2Σ+ and the ground state X2Σ+ has no splitting, which indicates
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that SOC calculation has little influence on the ground Λ-S states X2Σ+, and the
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spectroscopic constants of A2Π(I), A2Π(I)1/2 and A2Π(I)3/2 show the same conclusion as well.
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The Ω states do not have smooth PEC shape compared with Λ-S states because the avoided crossing rule of the same symmetry. With increasing bond length, these Ω states cross and mix, and the compositions become complicated. Because of the
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splitting of the atomic ground states of B and Se, the theoretical De result for the ground states X2Σ+1/2 is about 0.12 eV lower than that for the X2Σ+ state. The X2Σ+1/2 and A2Π(I)1/2 are crossing at the bond length about 2.2 Å due to the avoided crossing rule.
4. Transitions properties analysis Transitions dipole moments (TDMs) of A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 to the ground state X2Σ+1/2 of BSe as functions of the internuclear distance are diagrammatically depicted in Fig. 4. TDMs of A2Π(I)3/2-X2Σ+1/2, 2Π(I)1/2-X2Σ+1/2 are larger than that of C2Δ(I)3/2-X2Σ+1/2around the equilibrium position.
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The radiative lifetime can be calculated by the following formula[23, 24] with a
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given vibrational level ν′：
3h τ ν' ν' ν'' ν'' 64 4 | a0 e TDM |2 Σ ν'' qν' ,ν'' ΔEν' ,ν'' 4.936 105
| TDM |2 Σ ν'' qν' ,ν'' ΔEν' ,ν''
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g' g ''
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g' g ''
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where the qν′,ν′′ is the Frank-Condon factor, TDM is the averaged nondegenerate transition dipole moments in atomic units, g′ and g′′ are the degeneracy of the upper
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and lower states, respectively. The energy difference ΔEν′,ν′′ is in cm-1, and τν′ is in second(s).
Table 5 lists the computed radiative lifetimes for the transitions from the
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A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 in their lower vibrational levels to the ground state
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X2Σ+1/2at single-channel. The radiative lifetimes of the B2Π(I)1/2 and C2Δ(I)3/2 states
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are much shorter than that of the A2Π(I)3/2 state. The intensity distribution in a band system can mainly be explained by the Frank-Condon principle. This can be illustrated by the Frank-Condon factors
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assuming that the electronic transition moment does not vary over the band system. The Frank-Condon factors of the A2Π(I)3/2-X2Σ+1/2, 2Π(I)1/2-X2Σ+1/2, C2Δ(I)3/2- X2Σ+1/2 transitions are evaluated by the LEVEL 8.0 program and are listed in Table 6.
5. Conclusions The 15 Λ-S electronic states of BSe have been investigated using a MRCI/ aug-cc-pV5Z method. The spin-orbit coupling effect is evaluated by introducing the full Breit-Pauli Hamiltonian operator in the calculation to split 15 Λ-S electronic states to 32 Ω states. With the help of the avoided crossing rule of the same symmetry, we draw the diagram of the PECs of all Λ-S and Ω states, then the spectroscopic constants of the bound states of Λ-S and Ω state are determined. The calculated spectroscopic constants of the X2Σ+ state are very close to the experiment values. The Λ-S electronic states expect ground state X2Σ+ are reported in our work for the first 9
ACCEPTED MANUSCRIPT time. Furthermore, the B2Σ+(II), C2Δ(I) and 2Σ-(I) states appear energy degeneracy near bond length 2 Å, the 2Π(II) and 2Π(III) states obviously avoided crossing at the bond length of about 2.3 Å and they all have double well. We hope that our findings
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will be useful for further spectroscopy property research. The spin-orbit coupling effect calculation suggests that the SOC effect has little influence on the low-lying
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states of BSe. In particular, the spectroscopic constants of X2Σ+ and X2Σ+1/2, A2Π(I),A2Π(I)1/2 and A2Π(I)3/2 are almost same. The predictive radiative lifetimes of A2Π(I)3/2 are longer than those of B2Π(I)1/2 and C2Δ(I)3/2. It is expected that the
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excited states of the BSe molecule.
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present theoretical study could stimulate further experimental interests on various
Acknowledgments
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This work was supported by the National Natural Science Foundation of China
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(Grant No. 11274096), Innovation Scientists and Technicians Troop Construction Projects of Henan Province of China (Grant No. 124200510013) and Science and
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Technology Research Key Project of Education Department of Henan Province of China (Grant No. 13A140690), the Foundation for Key Program of Education Department of Henan Province(Grant No. 13A140519). The calculation about this
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work was supported by the High Performance Computing Center of Henan Normal University.
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ACCEPTED MANUSCRIPT Figure caption: Figure 1. PECs of 15 Λ-S electron states of BSe (a) The doublet Λ-S states (b) The quartet Λ-S states of BSe.
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Figure 2.The amplified view of the the energy degeneracy of B2Σ+(II), C2Δ(I) , 2Σ-(I), 2Δ(II) and 2 -
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Σ (II) states.
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Figure 3. The PECs of 32 Ω states. (a) The Ω=1/2 states. (b) The Ω=3/2 states (The dot line is the ground 1/2 states). (c) The Ω=5/2 states (The dot line is the ground 1/2 states). (d) The Ω=7/2 states (The dot line is the ground 1/2 states). 2
2 +
Figure 4. Transitions dipole moments of A Π(I)3/2-X Σ
1/2,
2
Π(I)1/2-X2Σ+1/2,
2
Δ(I)3/2-
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X2Σ+1/2 of the BSe molecule as the functions of the internuclear distance.
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Figure 1
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T
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AC
CE P
TE
D
MA
Figure 2.
14
Figure 3.
AC
CE P
TE
D
MA
NU
SC R
IP
T
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15
MA
NU
SC R
IP
T
ACCEPTED MANUSCRIPT
AC
CE P
TE
D
Figure 4
16
ACCEPTED MANUSCRIPT
Table 1 The dissociation limits of the Λ-S sates Λ-S Sates
Pu + 3Pg
4 +4
Σ , Π(I),4Π(II),4Δ, 4Σ-(I),4Σ-(II) 2
Pu + 1Dg
0
9545.64
SC R
B2Σ+(II), 2Δ(II)，2Π(III)
T
2
IP
X2Σ+,A2Π(I), C2Δ(I),2Σ-(I), 2Π(II),2Σ-(II),
9576.149a
CE P
TE
D
MA
NU
Experimental value from Ref[22].
0
AC
a
Energy/cm-1
Atomic state (B + Se)
17
ACCEPTED MANUSCRIPT
Table 2 The spectroscopic constants of the bound Λ-S states of BSe. Λ-S states
Te (cm-1)
Re(Å)
ωe( cm-1)
ωeχe( cm-1
Be( cm-1)
De(eV)
0.6185
4.8395
Main Electron configuration(%)
0
1.753
997.3197
5.3021
1.6646
1σαβ2σαβ1παβα3σα2πα4σ05σ0(85.77)
0.4945
1.284
1σαβ2σαβ1παβα3σα2πα4σ05σ0(87.43)
0.4776
0.989
1σαβ2σαβ1παβα3σα2πα4σ05σ0(86.63)
0.4858
1.8815
1σαβ2σαβ1παββ3σα2πα4σ05σ0(64.03)
643.6443
3.613
0.4846
4 +
Σ
25652.5403
1.93
726.9045
6.8067
0.51
4
28734.4483
1.96
675.4452
7.8469
4 -
Σ (I)
31106.7277
1.9945
627.936
9.11
B2Σ+(II)
32284.4376
1.9855
745.5837
7.9141
39032.4579
3.9435
668.0405
25.4499
2 -
Π(II)
1st well
2nd well
35122.8826
38849.372
666.4063
4.245
1.855
1.2922
8.8135
40.276
6.9634
TE
1.9855
CE P
2
39037.71
AC
2nd well
32407.8901
9.0095
D
Σ (I)
1st well
NU
1.987
MA
2nd well
32388.9211
770.1451
SC R
1.9795
1st well
1σαβ2σαβ1παββ3σα2πα4σ05σ0(4.2) 1σαβ2σαβ1παβα3σαβ2π04σ05σ0(83.76)
12387.3031
C2Δ(I)
1σαβ2σαβ1παβαβ3σα2π04σ05σ0(80.09)
3.3037
A2Π(I) Δ
IP
XΣ
T
) 2 +
0.4816
0.1197
0.8758
1σαβ2σαβ1παβα3σα2πβ4σ05σ0(20.97) 1σαβ2σαβ1παββ3σα2πα4σ05σ0(74.59) 1σαβ2σαβ1παβα3σα2πβ4σ05σ0(10.45)
0.01
1σαβ2σαβ1παβα3σα2πβ4σ05σ0(76.06) 1σαβ2σαβ1παββ3σα2πα4σ05σ0(11.1)
0.4821
0.8912
1σαβ2σαβ1παββ3σα2πα4σ05σ0(70.51) 1σαβ2σαβ1παβα3σα2πβ4σ05σ0(14.60)
0.1086
0.0082
1σαβ2σαβ1παβα3σα2πβ4σ05σ0(65.03) 1σαβ2σαβ1παββ3σα2πα4σ05σ0(9.27) 1σαβ2σαβ1παβ3σαβ2π04σα5σ0(12.95)
8.7625
0.5519
0.9216
1σαβ2σαβ1παβαβ3σ02πα4σ05σ0(68.40) 1σαβ2σα1παβαβ3σβ2πα4σ05σ0 (8.25) 1σαβ2σαβ1παββ3σ02παα4σ05σ0(3.08) 1σαβ2σα1παβα3σ02παβ4σ05σ0(2.89)
3.6385
59.3446
3.3847
0.1438
0.0305
1σαβ2σαβ1παα3σαβ2πβ4σ05σ0(74.6) 1σαβ2σαβ1παβα3σα2π04σβ5σ0(9.45) 1σαβ2σαβ1παββ3σα2π04σα5σ0(2.27)
4
Π(I)
38356.5369
3.0685
91.1793
2.9171
0.2018
0.0865
1σαβ2σαβ1παα3σαβ2πα4σ05σ0(84.50)
2
Δ(II)
41671.1812
2.0485
364.1738
3.4713
0.453
0.7132
1σαβ2σαβ1παβα3σα2πβ4σ05σ0(60.72) 1σαβ2σαβ1παββ3σα2πα4σ05σ0(21.66)
2 -
Σ (II)
41772.7867
2.053
357.2092
3.0721
0.451
0.4778
1σαβ2σαβ1παβα3σα2πβ4σ05σ0(71.59) 1σαβ2σαβ1παββ3σα2πα4σ05σ0(10.79)
2
Π(III)
1st well
42903.7375
2.3515
856.4139
45.891
0.3444
2.4943
1σαβ2σαβ1παα3σαβ2πβ4σ05σ0(53.60) 1σαβ2σαβ1παβαβ3σ02πα4σ05σ0(17.53)
2nd well
46698.8802
3.494
115.2793
4.3219
0.1557
0.0927
1σαβ2σαβ1παββ3σα2π04σα5σ0(73.89) 1σαβ2σαβ1παβα3σαβ2π04σ05σ0(4.32) 1σαβ2σαβ1παβα3σα2π04σβ5σ0(3.12) 1σαβ2σαβ1παβ3σαβ2πα4σ05σ0(2.36) 18
ACCEPTED MANUSCRIPT 53138.5565
1.9255
620.6627
---
0.512
0.2695
1σαβ2σα1παβαβ3σα2πα4σ05σ0(69.09)
TE
D
MA
NU
SC R
IP
T
1σαβ2σαβ1παβα3σ02παα4σ05σ0(12.02)
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Π(II)
AC
4
19
ACCEPTED MANUSCRIPT Table 3 The dissociation limit relationships of Ω electronic states Energy(cm-1)
Ω states
Atomic state (B + Se)
Expta
Theory
1/2(4),3/2(4),5/2(2)
0
2
P1/2 + 3P2
1/2(8),3/2(6),5/2(4),7/2(2)
25.65
P3/2 + 3P2
2
P1/2 + 3P1
0 15.287
1887.24
1989.497
3
1/2(6),3/2(4),5/2(2)
1924.47
2004.784
2
3
1/2(2)
2756.5
2534.36
2
3
2772.68
2549.647
9545.64
9576.149
9574.52
9591.436
P1/2 + P0 P3/2 + P0
2
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1/2(4),3/2(2)
2
P3/2 + P1
1/2(2),3/2(2)
1
B( Pu) + Se( Dg) 2
P1/2 + 1D2
1/2(2),3/2(2)
1
P3/2 + D2
1/2(2),3/2(2),5/2(2)
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Experimental value from the Ref[22].
NU
2
a
IP
2
T
B(2Pu) + Se(3Pg)
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ACCEPTED MANUSCRIPT
Re(Å)
ωe( cm-1)
ωeχe( cm-1)
Be( cm-1)
De(eV)
X1/2
0
1.753
996.6327
5.4998
0.6183
4.7189
3/2
11687.9885
1.9795
653.4243
0.8814
0.4845
3.2714
1/2(2)
13174.2045
1.9765
641.5118
---
0.4859
3.0878
1/2(3)
25522.3111
1.9315
724.691
7.0398
0.5095
3/2(2)
25602.463
1.9315
725.4032
6.9946
0.5096
1.5483
7/2
28195.4525
1.963
670.2226
8.0016
IP
1.5579
0.4929
1.2271
5/2
28451.5297
1.963
672.3696
8.1061
0.4933
1.2119
1/2(4)
28810.0059
1.9615
674.8791
8.2087
0.494
1.1683
3/2(3)
29346.6685
1.957
682.6811
8.5371
0.4963
1.1025
1/2(5)
31017.4677
1.99
621.4168
8.7479
0.4795
0.8962
3/2(4)
31236.7331
1.9915
632.4383
9.5642
0.8695
32169.9256
1.987
633.5023
7.1401
0.4847
0.7965
3/2(5)
32258.5291
1.9615
NU
0.4788
1/2(6)
684.8187
10.7035
0.4917
0.8382
1/2(7)
32512.9845
1.978
722.3482
14.1017
0.4859
0.8617
5/2(2)
32839.7521
1.9855
437.1151
---
0.4814
0.8224
1/2(8)
35078.9584
1.858
840.5188
8.3812
0.5486
0.6809
3/2(6)
35575.3104
1.8595
836.0552
41.2053
0.5483
0.6312
5/2(3)
39265.1536
2.587
152.8676
9.6578
0.2981
0.1997
1/2(9)
39663.3313
3.3955
76.2714
4.8266
0.1581
0.3551
1/2(10)
39770.5297
3.5575
56.9437
3.7604
0.1486
0.3117
3/2(7)
39795.5462
3.4585
53.0412
3.9403
0.151
0.4001
3/2(8)
39809.4338
3.6865
61.2801
6.0866
0.1361
0.5315
1/2(11)
39960.4708
4.819
4625.9533
792.8899
---
0.5623
1/2(12)
39967.1983
4.7275
---
---
5/2(4)
40543.5389
3.7015
96.8595
9.5692
0.1417
0.5929
3/2(9)
40586.8787
3.664
---
---
---
1.581
40741.5119
4.4545
---
---
---
1.5706
43481.8911
3.0745
380.2026
11.5678
0.1981
1.4672
45640.9627
2.7925
321.2137
7.2645
0.2472
1.2191
1/2(15)
45813.7022
2.7625
303.2262
14.8033
0.2455
1.2175
5/2(5)
46146.2246
2.8
314.0246
17.4002
0.2501
1.1975
1/2(16)
46446.9233
2.7415
265.4431
26.7355
0.2872
2.0521
3/2(10) 1/2(14)
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1/2(13)
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Te (cm-1)
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Ω states
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Table 4 Spectroscopic constants of the Ω states
0.5664
21
ACCEPTED MANUSCRIPT Table 5 Radiative lifetimes for the transitions from the A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 states to the ground state X2Σ+1/2. Radiative lifetimes (μs) Transition
1
2
3
4
1/2
91.79
82.08
76.49
70.45
64.55
1/2
1.66
1.84
2.46
2.68
2.93
1/2
1.61
1.81
2.47
2.73
2.96
A Π(I)3/2-X Σ
2 +
B Π(I)1/2-X Σ 2
2 +
C Δ(I)3/2-X Σ
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ν′
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2
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ACCEPTED MANUSCRIPT
Table 6 Frank-Condon factors for the transitions from the A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 states to the ground state X2Σ+1/2. 0
1
2
3
4
5
6
2 +
A2Π(I)3/2 -X Σ
Transition
1/2
7
8
9
T
ν′
IP
ν′′
0.0042
0.0266
0.0783
0.1467
0.1960
0.1986
0.1588
0.1023
0.0537
0.0232
1
0.0190
0.0812
0.1429
0.1251
0.0425
0.0001
0.0466
0.1232
0.1555
0.1295
2
0.0450
0.1226
0.1055
0.0158
0.0177
0.0889
0.0804
0.0133
0.0119
0.0845
3
0.0762
0.1201
0.0293
0.0137
0.0818
0.0443
0.0010
0.0601
0.0817
0.0207
4
0.1086
0.0828
0.0001
0.0665
0.0458
0.0024
0.0651
0.0461
0.0004
0.0565
SC R
0
2 +
B Π(I)1/2 - X Σ 2
1/2
0.0058
0.0326
0.0873
0.1520
0.1934
0.1910
0.1518
0.0987
0.0527
0.0231
1
0.0274
0.1008
0.1541
0.1162
0.0309
0.0016
0.0535
0.1231
0.1473
0.1200
2
0.0612
0.1364
0.0902
0.0051
0.0320
0.0946
0.0679
0.0067
0.0180
0.0902
3
0.0986
0.1153
0.0122
0.0316
0.0863
0.0278
0.0065
0.0684
0.0721
0.0131
4
0.1265
0.0605
0.0067
0.0768
0.0278
0.0111
0.0691
0.0311
0.0033
0.0599
MA
0
NU
Transition
2 +
C Δ(I)3/2-X Σ 2
Transition 0.0056
0.0341
0.0971
1
0.0205
0.0836
0.1384
2
0.0473
0.1206
3
0.0797
0.1152
4
0.1117
0.0762
0.1730
1/2
0.2147
0.1970
0.1397
0.0794
0.0374
0.0150
0.1081
0.0253
0.0051
0.0744
0.1489
0.1611
0.1204
0.0946
0.0096
0.0258
0.0958
0.0718
0.0059
0.0229
0.1008
0.0238
0.0163
0.0783
0.0355
0.0034
0.0683
0.0783
0.0146
0.0005
0.0617
0.0377
0.0036
0.0634
0.0406
0.0010
0.0582
AC
CE P
TE
D
0
23
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Graphical abstract
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The amplified view of the the energy degeneracy of B2Σ+(II), C2Δ(I) , 2Σ-(I), 2Δ(II) and 2Σ-(II) states
24
ACCEPTED MANUSCRIPT Highlights
The PECs of the 15 Λ–S states have been calculated for the first time.
The PECs of the 32 Ω states arising from 15 Λ-S states have been calculated.
The energy degeneracy among B2Σ+(II), C2Δ(I) and 2Σ-(I) states have been analyzed.
The double well of C2Δ(I) ,2Σ-(I), 2Π(II) and 2Π(III) states have been studied.
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S1386-1425(16)30110-X doi: 10.1016/j.saa.2016.03.008 SAA 14321
To appear in: Received date: Revised date: Accepted date:
21 December 2015 1 March 2016 6 March 2016
Please cite this article as: Siyuan Liu, Hongsheng Zhai, Yufang Liu, Extensive ab initio study of the electronic states of BSe radical including spin–orbit coupling, (2016), doi: 10.1016/j.saa.2016.03.008
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ACCEPTED MANUSCRIPT Extensive ab initio study of the electronic states of BSe radical
Siyuan Liu, Hongsheng Zhai, Yufang Liu
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including spin–orbit coupling
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College of Physics and Electronic Engineering, Henan Normal University, Xinxiang 453007, China
Abstract
The internally contracted multi-reference configuration interaction method (MRCI)
NU
with Davidson modification and the Douglas-Kroll scalar relativistic correction has been used to calculate the BSe molecule at the level of aug-cc-pV5Z basis set. The
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calculated electronic states, including 9 doublet and 6 quartet Λ-S states, are correlated to the dissociation limit of B(2Pu ) + Se(3Pg) and B(2Pu ) + Se(1Dg). The
D
Spin-orbit coupling (SOC) interaction is taken into account via the state interaction
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approach with the full Breit-Pauli Hamiltonian operator, which causes the entire 15 Λ-S states to split into 32 Ω states. This is the first time that the spin-orbit coupling
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calculation has been carried out on BSe. The potential energy curves of the Λ-S and Ω electronic states are depicted with the aid of the avoided crossing rule between electronic states of the same symmetry. The spectroscopic constants of the bound Λ-S
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and Ω states were determined, which are in good agreement with the experimental data. The transition dipole moments (TDMs) and the Frank-Condon factors (FCs) of the transitions from the low-lying bound Ω states A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 to the ground state X2Σ+1/2 have also been presented. Based on the previous calculations, the radiative lifetimes of the A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 were evaluated.
Keywords: MRCI, Potential energy curve, Spin-orbit coupling effect (SOC), Spectroscopic constants
1. Introduction
Corresponding author. E-mail address: [email protected] . 1
ACCEPTED MANUSCRIPT Boron selenides are an increasing crucial area in the technology of high-temperature use，semiconductor materials[1-3] due to its status. Selenium is the rare earth elements[4] and plays an important part in the human healthy. Naturally, the
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theoretical studies of accurate electronic structure of BSe are important to the comprehension and improvement of this technology. Previous experimental
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investigations of the B-Se system have been restricted to the condensed state. The first experimental study of BSe was made by Uy and Drowart[3], they obtained BSe(g) by the gas-phase reaction of boron with yttrium selenide and the energy of dissociation
employed solid-state
11
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De (BSe, g, 0 K)=4.74±0.15 eV was determined. Subsequently, HÜRTER et al[1] B NMR techniques to study the structural nature of various
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phases obtained in the system of boron-selenium including BSe. Melucci and Wahlbeck[5] identified the boron selenides BSe(g) in studies of the vaporisation
D
behaviour of solid B2Se3 with a time-of-flight mass spectrometer over the range 100
TE
to 700 ºC. Previous experimental studies about BSe mainly focused on vaporisation behavior at High-temperature. Previous experimental studies on the excited states of
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BSe are limited, thus an extensive theoretical study on the spectroscopic and the transition properties of BSe low-lying electronic states would be indispensable. As far as we know, the spectroscopic properties of Boron-Chalcogenide and
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Boron-Halogenide have been investigated in large quantities. Hanner[6] studied the ultrafast V–E transfer in boron oxide (BO) and found that the spin orbit interaction in BO facilitated a route for rapid intramolecular energy transfer. Then Karna et al[7] performed a configuration-interaction calculation on low-lying states of BO, and particular attention was given to the perturbations in the C2Π state. Yang and Boggs[8] have calculated the electronic structures and the transition properties of the low-lying excited states of the BS radical and determined the spectroscopic constants at the MR-CISD+Q level. More recently, Yang and co-worker[9] have investigated the spectroscopic parameters of BBr and BCl in order to prove that it would be a promising candidate for laser cooling. However, to the best of our knowledge, the systematic study on BSe molecule is rather sparse, where the spin-orbit interaction was not considered. It is well known that SOC plays an important role in the 2
ACCEPTED MANUSCRIPT spectroscopy and dynamics of the molecules, even in light molecules that containing only atoms of the first row of the periodic table. In addition, S and Se are oxygen family element and the B-S and B-Se system shows similar vaporization behavior[5],
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so compare the calculation results of BSe to the part of theoretical studies of BS[8] would be necessary.
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Our work will concentrate on the theoretical investigations of the electronic structures and the transition properties of the entire 32 Ω states generated from all of the 15 Λ-S states of BSe. We used internally contracted multi-reference configuration
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interaction method (MRCI) with Davidson modification and the Douglas-Kroll scalar relativistic correction for single point energy calculations. The potential energy curves
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(PECs) and spectroscopic constants were fitted after considering the avoided crossing rule between states of the same symmetry. The transition dipole moments (TDMs) and
D
the Franck–Condon (FC) factors of the transitions from the low-lying bound Ω states
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A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 to the ground state X2Σ+1/2. were also calculated. Finally, the corresponding single-channel radiative lifetimes were derived from above
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results.
2. Computational details
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The ab initio calculation on the electronic structure of BSe is launched by using the ab initio quantum chemistry program package MOLPRO 2010.1[10]. The spectroscopic parameters with the transition dipole moments(TDM), Frank-Condon factors(FCs) and radiative lifetime are determined by using the Le Roy’s LEVEL 8.0 program[11]. In order to obtain the PECs of BSe and guarantee the accuracy, the uncontracted Gaussian type all-electron aug-cc-pV5Z basis set is selected for both atom B[15s,9p,5d,4f,3g,2h][12, 13];Se[27s,18p,14d,4f,3g,2h][14] in the calculation of the Λ-S and Ω electronic states. The bond length is circulated with the step of 0.05Å to scan a series of the single-point energy over the internuclear distance range from 1.3 to 6.25Å. The ground state molecule orbitals (MOs) are calculated firstly by adopting 3
ACCEPTED MANUSCRIPT restricted Hartree-Fock (RHF) method. Then the state-averaged complete active space self-consistent field (SA-CASSCF)[15, 16] method is carried out using previous RHF orbitals as starting guess for orbital optimization. Finally, the energies of Λ-S states
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are calculated by using the internally contracted multi-reference configuration interaction (MRCI)[17, 18] approach base on the previous SA-CASSCF energies as
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reference values. The Douglas-Kroll scalar relativistic one-electron integrals have been taken into account, and the Davidson modification[19-21] (MRCI+Q) is employed to correct the size-extensively error.
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Due to the limitation of MOLPRO program package, the subgroup C2v point group symmetry has been considered for the BSe molecule although it belongs to a
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higher symmetry group in the calculation. The C2v point group symmetry holds A1, B1, B2, A2 irreducible representations. For the BSe molecule, 5a1, 2b1 and 2b2
D
symmetry molecule orbitals (MOs) are selected as the active space, corresponding to
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the B 2s2p3s and Se 4s4p shells. The outmost 2s2p1 electrons of B atom and 4s24p4 of Se atom are placed in the active space, and the remaining 30 electrons are frozen and
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not correlated. Namely, 9 electrons are used in the correlation energy calculation in total. The PECs of 15 Λ-S electron states are plotted by connecting the calculated points with the aid of the avoided crossing rule between electron states of same
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symmetry.
In addition, the spin-orbit matrix elements and eigenstates in the present work are calculated by using the state interaction method and full Breit-Pauli Hamiltonian operator (HBP) after the MRCI+Q calculations. The state interaction is employed in our SOC calculations, which means that the SOC eigenstates are obtained by diagonalizing the matrixes Hel + Hso on the basis of eigenfunction of Hel. In this process, the Hel and Hso are obtained from MRCI+Q calculations and CASSCF wave functions, respectively. The SOC potential energy curves are drawn with the aid of the avoided crossing rule of the same symmetry. The spectroscopic constants of the bound Λ-S and Ω states, including the equilibrium internuclear distance Re, excitation energy Te, the harmonic and anharmonic vibrational constants ωe and ωeχe, the rotational constants Be, and the dissociation energy De are determined by the numerical solutions 4
ACCEPTED MANUSCRIPT of the one-dimensional nuclear Schrödinger equation. The transition dipole moments (TDMs) and the Frank-Condon factors (FCs) of the transitions from the low-lying bound Ω states A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 to the ground state X2Σ+1/2 are
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X2Σ+1/2, 2Π(I)1/2- X2Σ+1/2 and C2Δ(I)3/2-X2Σ+1/2 are evaluated.
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presented. Based on the previous calculations, the radiative lifetimes of the A2Π(I)3/2-
3. Results and Discussion
3.1. Results and analysis of the PECs of the 15 Λ-S electron states
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The PECs of the 15 Λ-S states[B2Σ+(II), 2Δ(II), 2Π(III), X2Σ+, A2Π(I), C2Δ(I), 2 -
Σ (I), 2Π(II), 2Σ-(II), 4Σ+, 4Π(I), 4Π(II), 4Δ, 4Σ-(I), 4Σ-(II)] of the BSe molecule are
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shown in Fig. 1. The B2Σ+(II), 2Δ(II) and 2Π(III) states are correlated with the dissociation limit of the first excited state Se(1Dg) and the ground state B(2Pu), while
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the other six doublet and six quartet Λ-S states are correlated with the dissociation
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limit of the atom ground states Se(3Pg) and B(2Pu) . Table 1 shows the calculated electronic states of the BSe molecule and their dissociation relationships. The
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calculated energy gap of 9545.64 cm-1 between the atom state B(2Pu) and Se(1Dg) is in good agreement with the experimental result of 9576.149 cm-1. The spectroscopic constants of the 15 bound Λ-S states of the BSe molecule are
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fitted and summarized in Table 2, in which the main electronic configurations of these states are also presented. The amplified view of the energy degeneracy of B2Σ+(II), C2Δ(I) , 2Σ-(I), 2Δ(II) and 2Σ-(II) states are shown in Fig. 2. As we can see, the ground states X2Σ+ crossed the first exited state A2Π(I) when the bond length is about 2.3 Å. It’s mainly characterized by the electronic configuration 1σαβ2σαβ1παβαβ3σα2π04σ05σ0 with the configuration weight factor of 0.8 around equilibrium position. The depth of the well without incorporating spin-orbit coupling is 4.84 eV and the well holds about 66 vibrational levels, which result is in good agreement with the dissociation energy D0 of 4.74±0.15 eV obtained by UY and DROWART[3] in experiment. (De = D0 +1/2ωe, where ωe is about 0.12 eV). It should be noted that there is one important crossing region between the X2Σ+ state and the first exited state A2Π(I) at 2.26 Å on 5
ACCEPTED MANUSCRIPT the PECs of the Λ-S states, and corresponding predissociation pathway is possibility X2Σ+(ν′17)A2Π(I) after we calculated. The states B2Σ+(II), C2Δ(I) and 2Σ-(I) are energy degeneracy near bond length 2
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Å and their first well have the same electron configuration 1σαβ2σαβ1παββ3σα2πα4σ05σ0
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and 1σαβ2σαβ1παβα3σα2πβ4σ05σ0 around the corresponding equilibrium position,
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excitation energy Te of which are 32284.4376 cm-1, 32388.9211 cm-1 and 32407.8901 cm-1 respectively. The excitation energy interval betwwen the C2Δ(I) and 2Σ-(I) states is as small as 18.969 cm-1, while the value between B2Σ+(II) and C2Δ(I) states is
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104.4835 cm-1. Especially the C2Δ(I) and 2Σ-(I) states not only have nearly same electron configuration around equilibrium position but also have almost same
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spectroscopic constants. Moreover, the B2Σ+(II) state cross more other states, which is a Rydberg state and dissociates to the first excited atomic state B(2Pu) + Se(1Dg).
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In addition, Both the C2Δ(I) and 2Σ-(I) states have an unconspicuous potential
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barrier near bond position 2.8 Å which gives rise to the double well. But the second well of the C2Δ(I) and 2Σ-(I) states is too shallow to observe in PECs shape. The depth
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and equilibrium position of the second well of the C2Δ(I) and 2Σ-(I) states is only 0.01 eV at 3.9435 Å and 0.0082 eV at 4.245 Å, and the corresponding vibrational levels are merely 4 and 2, respectively.
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As for second root of 2Δ and 2Σ- symmetry, the 2Δ(II) and 2Σ-( II) states have the same properties as the C2Δ(I) and 2Σ-(I) states. The 2Δ(II) and 2Σ-(II) states appear energy degeneracy when the bond length is less than 2.8 Å. They have the same electron configuration and similar spectroscopic constants expect well depth De near their equilibrium position, with being their excitation energy interval is only 101.6055 cm-1. Based on the shape of PECs, the 2Π(II) and 2Π(III) states obviously avoided crossing at the bond length of about 2.3 Å. This result is accord well with Yang’s[8] prediction. The 2Π(II) and 2Π(III) states have potential barrier at bond length 2.36 Å and 2.8 Å, respectively, the former formation of a clear potential barrier because of the avoided crossing rule. The potential barrier leads to the double well for the 2Π(II) and 2Π(III) states. The bond position of the wells locate at 1.855 Å and 3.6385 Å on 6
ACCEPTED MANUSCRIPT the 2Π(II) state, and the corresponding well depth are 0.9216 eV and 0.0305 eV, respectively. While The bond position of the wells locate at 2.3515 Å and 3.494 Å on the 2Π(II) state, and the corresponding well depth are 2.4943 eV and 0.0927 eV,
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respectively.
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Quartet Λ-S states correlated with the dissociation limit of the ground state
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Se(3Pg) and B(2Pu) , which have more smooth PECs than doublet state, and they are bound states except the 4Σ-(II) state. The 4Σ+, 4Δ and 4Σ-(I) states have similar shape of PEC, their Te and Re both increasing in turn. It was found that 4Σ+,4Δ and 4Σ-(I) not
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only have the same main electron configuration 1σαβ2σαβ1παβα3σα2πα4σ05σ0, but also have very close configuration weight factor(0.86,0.87,0.87) near their equilibrium
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bond length.
As for the excited states 4Π(I),4Π(II) and 4Σ-(II) states, the 4Σ-(II) state is the only
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repulsive state of all electronic states. The 4Π(I) state has the lowest well which only
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holds 3 vibrational levels at equilibrium bond length 3.0685 Å. 4Π(I) crosses 4Π(II) at a bond length of about 1.9Å because of the avoided crossing rule. At the same time, Π(I) has crossing with PECs of the bound state 4Σ+,4Δ, 4Σ-(I) and ground state X2Σ+
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4
from bond length 3.5-3.8 Å. The 4Π(II) state has a particularly shallow potential well in which De is 0.27 eV. Meanwhile, the state 4Π(II) is an unstable well due to an
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avoided crossing with the 4Π(I) state, and the bottom of 4Π(II) is 14105.4 cm-1 higher than the dissociation limit Se(3Pg) + B(2Pu). The characteristic of 4Π(II) indicate that it is hardly be observed in experiment. As the second root of 2Π symmetry, bound state 2Π(II) arises from the electronic configuration 1σαβ2σαβ1παβαβ3σ02πα4σ05σ0 (68.40%),1σαβ2σα1παβαβ3σβ2πα4σ05σ0 (8.25%), 1σαβ2σαβ1παββ3σ02παα4σ05σ0 (3.08%) and 1σαβ2σα1παβα3σ02παβ4σ05σ0 (2.89%) for the first well at the equilibrium point 1.855 Å, which indicate that the 2Π(II) state is obviously multi-configurational in nature and that it is necessary to use the multi-reference configuration interaction in our calculation.
3.2. Results and analysis of the PECs of the 32 Ω states Introduction of the SOC effect into the calculation causes 15 Λ-S states split into 7
ACCEPTED MANUSCRIPT 32 Ω states, namely, 1/2, 3/2, 5/2, and 7/2 symmetries, which hold eight different dissociation limits. The possible Ω states and corresponding energy separations are given in Table 3. The two highest dissociation limits 2P1/2 + 1D2 and 2P3/2 + 1D2 are
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splitting from the second atomic state 2Pu + 1Dg, with the energy interval is 9545.64
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and 9574.52 cm-1, respectively, and the results are in good agreement with
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corresponding energy interval value 9576.149, 9591.436 cm-1 obtained from experiment[22]. Meanwhile, the computed PECs for the 15 states of Ω = 1/2, 11 states of Ω = 3/2, 5 states of Ω = 5/2 and one state of Ω = 7/2 are plotted in Fig. 3 separately.
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The spectroscopic constants limits of the bound Ω states obtained by solving the nuclear Schrödinger equations are shown in Table 4.
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The SOC ground state X2 Σ1/2 almost has the same spectroscopic constants as
Λ-S ground states X2Σ+ and the ground state X2Σ+ has no splitting, which indicates
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that SOC calculation has little influence on the ground Λ-S states X2Σ+, and the
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spectroscopic constants of A2Π(I), A2Π(I)1/2 and A2Π(I)3/2 show the same conclusion as well.
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The Ω states do not have smooth PEC shape compared with Λ-S states because the avoided crossing rule of the same symmetry. With increasing bond length, these Ω states cross and mix, and the compositions become complicated. Because of the
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splitting of the atomic ground states of B and Se, the theoretical De result for the ground states X2Σ+1/2 is about 0.12 eV lower than that for the X2Σ+ state. The X2Σ+1/2 and A2Π(I)1/2 are crossing at the bond length about 2.2 Å due to the avoided crossing rule.
4. Transitions properties analysis Transitions dipole moments (TDMs) of A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 to the ground state X2Σ+1/2 of BSe as functions of the internuclear distance are diagrammatically depicted in Fig. 4. TDMs of A2Π(I)3/2-X2Σ+1/2, 2Π(I)1/2-X2Σ+1/2 are larger than that of C2Δ(I)3/2-X2Σ+1/2around the equilibrium position.
8
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The radiative lifetime can be calculated by the following formula[23, 24] with a
1
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given vibrational level ν′：
3h τ ν' ν' ν'' ν'' 64 4 | a0 e TDM |2 Σ ν'' qν' ,ν'' ΔEν' ,ν'' 4.936 105
| TDM |2 Σ ν'' qν' ,ν'' ΔEν' ,ν''
3
g' g ''
3
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g' g ''
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where the qν′,ν′′ is the Frank-Condon factor, TDM is the averaged nondegenerate transition dipole moments in atomic units, g′ and g′′ are the degeneracy of the upper
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and lower states, respectively. The energy difference ΔEν′,ν′′ is in cm-1, and τν′ is in second(s).
Table 5 lists the computed radiative lifetimes for the transitions from the
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A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 in their lower vibrational levels to the ground state
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X2Σ+1/2at single-channel. The radiative lifetimes of the B2Π(I)1/2 and C2Δ(I)3/2 states
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are much shorter than that of the A2Π(I)3/2 state. The intensity distribution in a band system can mainly be explained by the Frank-Condon principle. This can be illustrated by the Frank-Condon factors
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assuming that the electronic transition moment does not vary over the band system. The Frank-Condon factors of the A2Π(I)3/2-X2Σ+1/2, 2Π(I)1/2-X2Σ+1/2, C2Δ(I)3/2- X2Σ+1/2 transitions are evaluated by the LEVEL 8.0 program and are listed in Table 6.
5. Conclusions The 15 Λ-S electronic states of BSe have been investigated using a MRCI/ aug-cc-pV5Z method. The spin-orbit coupling effect is evaluated by introducing the full Breit-Pauli Hamiltonian operator in the calculation to split 15 Λ-S electronic states to 32 Ω states. With the help of the avoided crossing rule of the same symmetry, we draw the diagram of the PECs of all Λ-S and Ω states, then the spectroscopic constants of the bound states of Λ-S and Ω state are determined. The calculated spectroscopic constants of the X2Σ+ state are very close to the experiment values. The Λ-S electronic states expect ground state X2Σ+ are reported in our work for the first 9
ACCEPTED MANUSCRIPT time. Furthermore, the B2Σ+(II), C2Δ(I) and 2Σ-(I) states appear energy degeneracy near bond length 2 Å, the 2Π(II) and 2Π(III) states obviously avoided crossing at the bond length of about 2.3 Å and they all have double well. We hope that our findings
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will be useful for further spectroscopy property research. The spin-orbit coupling effect calculation suggests that the SOC effect has little influence on the low-lying
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states of BSe. In particular, the spectroscopic constants of X2Σ+ and X2Σ+1/2, A2Π(I),A2Π(I)1/2 and A2Π(I)3/2 are almost same. The predictive radiative lifetimes of A2Π(I)3/2 are longer than those of B2Π(I)1/2 and C2Δ(I)3/2. It is expected that the
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excited states of the BSe molecule.
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present theoretical study could stimulate further experimental interests on various
Acknowledgments
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This work was supported by the National Natural Science Foundation of China
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(Grant No. 11274096), Innovation Scientists and Technicians Troop Construction Projects of Henan Province of China (Grant No. 124200510013) and Science and
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Technology Research Key Project of Education Department of Henan Province of China (Grant No. 13A140690), the Foundation for Key Program of Education Department of Henan Province(Grant No. 13A140519). The calculation about this
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work was supported by the High Performance Computing Center of Henan Normal University.
References [1]. H.U. Huerter, B. Krebs, H. Eckert, W. Mueller-Warmuth, Solid-state boron-11 NMR studies on boron-chalcogenide systems, Inorg. Chem. 24 (1985) 1288-1292. [2]. V. KORYAZHKIN, On spectra and structure of BO 2 BS 2 and BSe 2 molecules and CO+ 2 and CS 2+ ions, Cand. Sci (Chem) Thesis. Moscow, (1967). [3]. O.M. Uy, J. Drowart, Mass spectrometric determination of the dissociation energies of the boron monochalcogenides, High. Temp. Sci. 2 (1970) 293-298. [4]. H. Bergmann, H. Hein, P. Kuhn, U. Vetter, Rare Earth Elements and Selenium, in:
Sc, Y, La-Lu
Rare Earth Elements C 9, Springer, 1985, pp. 1-526. [5]. R.C. Melucci, P.G. Wahlbeck, Mass spectrometric observations of gaseous boron selenides, Inorg. Chem. 9 (1970) 1065-1068. 10
ACCEPTED MANUSCRIPT [6]. A. Hanner, J. Gole, Evidence for ultrafast V–E transfer in boron oxide (BO), J. Chem. Phys. 73 (1980) 5025-5039. [7]. S. Karna, F. Grein, BO: low-lying electronic states obtained by configuration-interaction studies, J. Mol. Spectrosc. 122 (1987) 356-364.
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[8]. X. Yang, J.E. Boggs, Ab initio study of the electronic states of BS including spin–orbit coupling, Chem. Phys. Lett. 410 (2005) 269-274.
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[9]. R. Yang, Y. Gao, B. Tang, T. Gao, The ab initio study of laser cooling of BBr and BCl, Phys. Chem. Chem. Phys. 17 (2015) 1900-1906.
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[10]. P. Knowles, F. Manby, M. Schütz, P. Celani, G. Knizia, T. Korona, R. Lindh, A. Mitrushenkov, G. Rauhut, T. Adler, MOLPRO, version 2010.1, a package of ab initio programs, See http://www. molpro. net, (2010).
[11]. R. Le Roy, LEVEL 8.0: A computer program for solving the radial Schrödinger equation for
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bound and quasibound levels, University of Waterloo Chemical Physics Research Report CP-663, (2007).
[12]. T.H. Dunning Jr, Gaussian basis sets for use in correlated molecular calculations. I. The atoms
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boron through neon and hydrogen, J. Chem. Phys. 90 (1989) 1007-1023. [13]. R.A. Kendall, T.H. Dunning Jr, R.J. Harrison, Electron affinities of the first‐row atoms revisited. Systematic basis sets and wave functions, J. Chem. Phys. 96 (1992) 6796-6806. [14]. A.K. Wilson, D.E. Woon, K.A. Peterson, T.H. Dunning Jr, Gaussian basis sets for use in
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correlated molecular calculations. IX. The atoms gallium through krypton, J. Chem. Phys. 110
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(1999) 7667-7676.
[15]. P.J. Knowles, H.-J. Werner, An efficient second-order MC SCF method for long configuration expansions, Chem. Phys. Lett. 115 (1985) 259-267.
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[16]. H.J. Werner, P.J. Knowles, A second order multiconfiguration SCF procedure with optimum convergence, J. Chem. Phys. 82 (1985) 5053-5063. [17]. P.J. Knowles, H.-J. Werner, An efficient method for the evaluation of coupling coefficients in configuration interaction calculations, Chem. Phys. Lett. 145 (1988) 514-522.
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[18]. H.J. Werner, P.J. Knowles, An efficient internally contracted multiconfiguration–reference configuration interaction method, J. Chem. Phys. 89 (1988) 5803-5814. [19]. P.J. Bruna, S.D. Peyerimhoff, R.J. Buenker, The ground state of the CN+ ion: a multi-reference CI study, Chem. Phys. Lett. 72 (1980) 278-284. [20]. P.J. Knowles, H.-J. Werner, Internally contracted multiconfiguration-reference configuration interaction calculations for excited states, Theor. Chim. Acta. 84 (1992) 95-103. [21]. S.R. Langhoff, E. Davidson, Configuration Interaction Calculations on the Nitrogen Molecule, Int. J. Quantum. Chem. 8 (1974) 61-72. [22]. C.E. Moore, Atomic Energy Levels, National Bureau of Standards, Washington DC Vol. III. (1971) [23]. M.C. Heaven, Fluorescence decay dynamics of the halogens and interhalogens, Chem. Soc. Rev. 15 (1986) 405-448. [24]. H. Okabe, Photochemistry of Small Molecules, Willey, New York. (1978)
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ACCEPTED MANUSCRIPT Figure caption: Figure 1. PECs of 15 Λ-S electron states of BSe (a) The doublet Λ-S states (b) The quartet Λ-S states of BSe.
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Figure 2.The amplified view of the the energy degeneracy of B2Σ+(II), C2Δ(I) , 2Σ-(I), 2Δ(II) and 2 -
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Σ (II) states.
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Figure 3. The PECs of 32 Ω states. (a) The Ω=1/2 states. (b) The Ω=3/2 states (The dot line is the ground 1/2 states). (c) The Ω=5/2 states (The dot line is the ground 1/2 states). (d) The Ω=7/2 states (The dot line is the ground 1/2 states). 2
2 +
Figure 4. Transitions dipole moments of A Π(I)3/2-X Σ
1/2,
2
Π(I)1/2-X2Σ+1/2,
2
Δ(I)3/2-
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X2Σ+1/2 of the BSe molecule as the functions of the internuclear distance.
12
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D
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Figure 1
13
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Figure 2.
14
Figure 3.
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15
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Figure 4
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Table 1 The dissociation limits of the Λ-S sates Λ-S Sates
Pu + 3Pg
4 +4
Σ , Π(I),4Π(II),4Δ, 4Σ-(I),4Σ-(II) 2
Pu + 1Dg
0
9545.64
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B2Σ+(II), 2Δ(II)，2Π(III)
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2
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X2Σ+,A2Π(I), C2Δ(I),2Σ-(I), 2Π(II),2Σ-(II),
9576.149a
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Experimental value from Ref[22].
0
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a
Energy/cm-1
Atomic state (B + Se)
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Table 2 The spectroscopic constants of the bound Λ-S states of BSe. Λ-S states
Te (cm-1)
Re(Å)
ωe( cm-1)
ωeχe( cm-1
Be( cm-1)
De(eV)
0.6185
4.8395
Main Electron configuration(%)
0
1.753
997.3197
5.3021
1.6646
1σαβ2σαβ1παβα3σα2πα4σ05σ0(85.77)
0.4945
1.284
1σαβ2σαβ1παβα3σα2πα4σ05σ0(87.43)
0.4776
0.989
1σαβ2σαβ1παβα3σα2πα4σ05σ0(86.63)
0.4858
1.8815
1σαβ2σαβ1παββ3σα2πα4σ05σ0(64.03)
643.6443
3.613
0.4846
4 +
Σ
25652.5403
1.93
726.9045
6.8067
0.51
4
28734.4483
1.96
675.4452
7.8469
4 -
Σ (I)
31106.7277
1.9945
627.936
9.11
B2Σ+(II)
32284.4376
1.9855
745.5837
7.9141
39032.4579
3.9435
668.0405
25.4499
2 -
Π(II)
1st well
2nd well
35122.8826
38849.372
666.4063
4.245
1.855
1.2922
8.8135
40.276
6.9634
TE
1.9855
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2
39037.71
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2nd well
32407.8901
9.0095
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Σ (I)
1st well
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1.987
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2nd well
32388.9211
770.1451
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1.9795
1st well
1σαβ2σαβ1παββ3σα2πα4σ05σ0(4.2) 1σαβ2σαβ1παβα3σαβ2π04σ05σ0(83.76)
12387.3031
C2Δ(I)
1σαβ2σαβ1παβαβ3σα2π04σ05σ0(80.09)
3.3037
A2Π(I) Δ
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XΣ
T
) 2 +
0.4816
0.1197
0.8758
1σαβ2σαβ1παβα3σα2πβ4σ05σ0(20.97) 1σαβ2σαβ1παββ3σα2πα4σ05σ0(74.59) 1σαβ2σαβ1παβα3σα2πβ4σ05σ0(10.45)
0.01
1σαβ2σαβ1παβα3σα2πβ4σ05σ0(76.06) 1σαβ2σαβ1παββ3σα2πα4σ05σ0(11.1)
0.4821
0.8912
1σαβ2σαβ1παββ3σα2πα4σ05σ0(70.51) 1σαβ2σαβ1παβα3σα2πβ4σ05σ0(14.60)
0.1086
0.0082
1σαβ2σαβ1παβα3σα2πβ4σ05σ0(65.03) 1σαβ2σαβ1παββ3σα2πα4σ05σ0(9.27) 1σαβ2σαβ1παβ3σαβ2π04σα5σ0(12.95)
8.7625
0.5519
0.9216
1σαβ2σαβ1παβαβ3σ02πα4σ05σ0(68.40) 1σαβ2σα1παβαβ3σβ2πα4σ05σ0 (8.25) 1σαβ2σαβ1παββ3σ02παα4σ05σ0(3.08) 1σαβ2σα1παβα3σ02παβ4σ05σ0(2.89)
3.6385
59.3446
3.3847
0.1438
0.0305
1σαβ2σαβ1παα3σαβ2πβ4σ05σ0(74.6) 1σαβ2σαβ1παβα3σα2π04σβ5σ0(9.45) 1σαβ2σαβ1παββ3σα2π04σα5σ0(2.27)
4
Π(I)
38356.5369
3.0685
91.1793
2.9171
0.2018
0.0865
1σαβ2σαβ1παα3σαβ2πα4σ05σ0(84.50)
2
Δ(II)
41671.1812
2.0485
364.1738
3.4713
0.453
0.7132
1σαβ2σαβ1παβα3σα2πβ4σ05σ0(60.72) 1σαβ2σαβ1παββ3σα2πα4σ05σ0(21.66)
2 -
Σ (II)
41772.7867
2.053
357.2092
3.0721
0.451
0.4778
1σαβ2σαβ1παβα3σα2πβ4σ05σ0(71.59) 1σαβ2σαβ1παββ3σα2πα4σ05σ0(10.79)
2
Π(III)
1st well
42903.7375
2.3515
856.4139
45.891
0.3444
2.4943
1σαβ2σαβ1παα3σαβ2πβ4σ05σ0(53.60) 1σαβ2σαβ1παβαβ3σ02πα4σ05σ0(17.53)
2nd well
46698.8802
3.494
115.2793
4.3219
0.1557
0.0927
1σαβ2σαβ1παββ3σα2π04σα5σ0(73.89) 1σαβ2σαβ1παβα3σαβ2π04σ05σ0(4.32) 1σαβ2σαβ1παβα3σα2π04σβ5σ0(3.12) 1σαβ2σαβ1παβ3σαβ2πα4σ05σ0(2.36) 18
ACCEPTED MANUSCRIPT 53138.5565
1.9255
620.6627
---
0.512
0.2695
1σαβ2σα1παβαβ3σα2πα4σ05σ0(69.09)
TE
D
MA
NU
SC R
IP
T
1σαβ2σαβ1παβα3σ02παα4σ05σ0(12.02)
CE P
Π(II)
AC
4
19
ACCEPTED MANUSCRIPT Table 3 The dissociation limit relationships of Ω electronic states Energy(cm-1)
Ω states
Atomic state (B + Se)
Expta
Theory
1/2(4),3/2(4),5/2(2)
0
2
P1/2 + 3P2
1/2(8),3/2(6),5/2(4),7/2(2)
25.65
P3/2 + 3P2
2
P1/2 + 3P1
0 15.287
1887.24
1989.497
3
1/2(6),3/2(4),5/2(2)
1924.47
2004.784
2
3
1/2(2)
2756.5
2534.36
2
3
2772.68
2549.647
9545.64
9576.149
9574.52
9591.436
P1/2 + P0 P3/2 + P0
2
SC R
1/2(4),3/2(2)
2
P3/2 + P1
1/2(2),3/2(2)
1
B( Pu) + Se( Dg) 2
P1/2 + 1D2
1/2(2),3/2(2)
1
P3/2 + D2
1/2(2),3/2(2),5/2(2)
AC
CE P
TE
D
MA
Experimental value from the Ref[22].
NU
2
a
IP
2
T
B(2Pu) + Se(3Pg)
20
ACCEPTED MANUSCRIPT
Re(Å)
ωe( cm-1)
ωeχe( cm-1)
Be( cm-1)
De(eV)
X1/2
0
1.753
996.6327
5.4998
0.6183
4.7189
3/2
11687.9885
1.9795
653.4243
0.8814
0.4845
3.2714
1/2(2)
13174.2045
1.9765
641.5118
---
0.4859
3.0878
1/2(3)
25522.3111
1.9315
724.691
7.0398
0.5095
3/2(2)
25602.463
1.9315
725.4032
6.9946
0.5096
1.5483
7/2
28195.4525
1.963
670.2226
8.0016
IP
1.5579
0.4929
1.2271
5/2
28451.5297
1.963
672.3696
8.1061
0.4933
1.2119
1/2(4)
28810.0059
1.9615
674.8791
8.2087
0.494
1.1683
3/2(3)
29346.6685
1.957
682.6811
8.5371
0.4963
1.1025
1/2(5)
31017.4677
1.99
621.4168
8.7479
0.4795
0.8962
3/2(4)
31236.7331
1.9915
632.4383
9.5642
0.8695
32169.9256
1.987
633.5023
7.1401
0.4847
0.7965
3/2(5)
32258.5291
1.9615
NU
0.4788
1/2(6)
684.8187
10.7035
0.4917
0.8382
1/2(7)
32512.9845
1.978
722.3482
14.1017
0.4859
0.8617
5/2(2)
32839.7521
1.9855
437.1151
---
0.4814
0.8224
1/2(8)
35078.9584
1.858
840.5188
8.3812
0.5486
0.6809
3/2(6)
35575.3104
1.8595
836.0552
41.2053
0.5483
0.6312
5/2(3)
39265.1536
2.587
152.8676
9.6578
0.2981
0.1997
1/2(9)
39663.3313
3.3955
76.2714
4.8266
0.1581
0.3551
1/2(10)
39770.5297
3.5575
56.9437
3.7604
0.1486
0.3117
3/2(7)
39795.5462
3.4585
53.0412
3.9403
0.151
0.4001
3/2(8)
39809.4338
3.6865
61.2801
6.0866
0.1361
0.5315
1/2(11)
39960.4708
4.819
4625.9533
792.8899
---
0.5623
1/2(12)
39967.1983
4.7275
---
---
5/2(4)
40543.5389
3.7015
96.8595
9.5692
0.1417
0.5929
3/2(9)
40586.8787
3.664
---
---
---
1.581
40741.5119
4.4545
---
---
---
1.5706
43481.8911
3.0745
380.2026
11.5678
0.1981
1.4672
45640.9627
2.7925
321.2137
7.2645
0.2472
1.2191
1/2(15)
45813.7022
2.7625
303.2262
14.8033
0.2455
1.2175
5/2(5)
46146.2246
2.8
314.0246
17.4002
0.2501
1.1975
1/2(16)
46446.9233
2.7415
265.4431
26.7355
0.2872
2.0521
3/2(10) 1/2(14)
TE
CE P
AC
1/2(13)
SC R
T
Te (cm-1)
D
Ω states
MA
Table 4 Spectroscopic constants of the Ω states
0.5664
21
ACCEPTED MANUSCRIPT Table 5 Radiative lifetimes for the transitions from the A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 states to the ground state X2Σ+1/2. Radiative lifetimes (μs) Transition
1
2
3
4
1/2
91.79
82.08
76.49
70.45
64.55
1/2
1.66
1.84
2.46
2.68
2.93
1/2
1.61
1.81
2.47
2.73
2.96
A Π(I)3/2-X Σ
2 +
B Π(I)1/2-X Σ 2
2 +
C Δ(I)3/2-X Σ
AC
CE P
TE
D
MA
NU
SC R
2
IP
0
T
ν′
2 +
2
22
ACCEPTED MANUSCRIPT
Table 6 Frank-Condon factors for the transitions from the A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 states to the ground state X2Σ+1/2. 0
1
2
3
4
5
6
2 +
A2Π(I)3/2 -X Σ
Transition
1/2
7
8
9
T
ν′
IP
ν′′
0.0042
0.0266
0.0783
0.1467
0.1960
0.1986
0.1588
0.1023
0.0537
0.0232
1
0.0190
0.0812
0.1429
0.1251
0.0425
0.0001
0.0466
0.1232
0.1555
0.1295
2
0.0450
0.1226
0.1055
0.0158
0.0177
0.0889
0.0804
0.0133
0.0119
0.0845
3
0.0762
0.1201
0.0293
0.0137
0.0818
0.0443
0.0010
0.0601
0.0817
0.0207
4
0.1086
0.0828
0.0001
0.0665
0.0458
0.0024
0.0651
0.0461
0.0004
0.0565
SC R
0
2 +
B Π(I)1/2 - X Σ 2
1/2
0.0058
0.0326
0.0873
0.1520
0.1934
0.1910
0.1518
0.0987
0.0527
0.0231
1
0.0274
0.1008
0.1541
0.1162
0.0309
0.0016
0.0535
0.1231
0.1473
0.1200
2
0.0612
0.1364
0.0902
0.0051
0.0320
0.0946
0.0679
0.0067
0.0180
0.0902
3
0.0986
0.1153
0.0122
0.0316
0.0863
0.0278
0.0065
0.0684
0.0721
0.0131
4
0.1265
0.0605
0.0067
0.0768
0.0278
0.0111
0.0691
0.0311
0.0033
0.0599
MA
0
NU
Transition
2 +
C Δ(I)3/2-X Σ 2
Transition 0.0056
0.0341
0.0971
1
0.0205
0.0836
0.1384
2
0.0473
0.1206
3
0.0797
0.1152
4
0.1117
0.0762
0.1730
1/2
0.2147
0.1970
0.1397
0.0794
0.0374
0.0150
0.1081
0.0253
0.0051
0.0744
0.1489
0.1611
0.1204
0.0946
0.0096
0.0258
0.0958
0.0718
0.0059
0.0229
0.1008
0.0238
0.0163
0.0783
0.0355
0.0034
0.0683
0.0783
0.0146
0.0005
0.0617
0.0377
0.0036
0.0634
0.0406
0.0010
0.0582
AC
CE P
TE
D
0
23
ACCEPTED MANUSCRIPT
D
MA
NU
SC R
IP
T
Graphical abstract
AC
CE P
TE
The amplified view of the the energy degeneracy of B2Σ+(II), C2Δ(I) , 2Σ-(I), 2Δ(II) and 2Σ-(II) states
24
ACCEPTED MANUSCRIPT Highlights
The PECs of the 15 Λ–S states have been calculated for the first time.
The PECs of the 32 Ω states arising from 15 Λ-S states have been calculated.
The energy degeneracy among B2Σ+(II), C2Δ(I) and 2Σ-(I) states have been analyzed.
The double well of C2Δ(I) ,2Σ-(I), 2Π(II) and 2Π(III) states have been studied.
AC
CE P
TE
D
MA
NU
SC R
IP
T
25